MATHS MATE. Skill Builder. J. B. Wright E D N TA

Transcription

1 MATHS MATE Skill Builder AL VA G E D ON UCATI ED A J. B. Wright THE 8 N TA

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3 THE EDUCATIONAL MATHS MATE /8 Skill Builder J. B. Wright Published by The Educational Advantage Pty Ltd P. O. Box 08 Echuca, VIC Australia Phone: Fax: info@mathsmate.net Website: Copyright J. B. Wright 000 All rights reserved. The publisher of these worksheets gives permission for schools to photocopy the following worksheets for use by any student who has purchased a Maths Mate Student Pad. Copying for use with the Maths Mate Program The Maths Mate Skill Builder series are sets of photocopiable masters designed to help individual students gain particular skills that the Maths Mate Program may have identified as being poorly grasped. Maths Mate users can download and duplicate copies, print and file copies of Skill Builders for easy access in class or at home. Material available for use in the Maths Mate Program NAME DESCRIPTION ISBN Maths Mate Student Pad - rd Ed Maths Mate Student Pad - rd Ed Maths Mate Student Pad - th Ed Maths Mate 8 Student Pad - th Ed Maths Mate 9 Student Pad - th Ed Maths Mate 9 Gold Student Pad - nd Ed Maths Mate 0 Student Pad - th Ed Maths Mate 0 Gold Student Pad - nd Ed Maths Mate Teacher Resource CD - Version For use with all student pads Maths Mate Teacher Resource Book - rd Ed Maths Mate Teacher Resource Book - rd Ed Maths Mate Teacher Resource Book - th Ed Maths Mate 8 Teacher Resource Book - th Ed Maths Mate 9 Teacher Resource Book - th Ed Maths Mate 9 Gold Teacher Resource Book - nd Ed Maths Mate 0 Teacher Resource Book - th Ed Maths Mate 0 Gold Teacher Resource Book - nd Ed Maths Mate Skill Builder / For use with Maths Mate & Maths Mate Skill Builder /8 For use with Maths Mate & 8 Maths Mate Skill Builder 9/0 For use with Maths Mate 9 & 0 A D VA N TA G E page i Maths Mate /8 Skill Builder

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5 FORWARD Why use Skill Builders? Too often, through the teaching, learning and assessment process, teachers identify weaknesses and gaps in student learning but the constraints of the classroom severely limit remediation opportunities. The Maths Mate Skill Builder series was prepared in response to requests from teachers and parents who want an easy but effective way to help students who identify skill deficiencies using the Maths Mate Program, and are motivated to do something about them. The Maths Mate record keeping sheets found at the start of each term in each Student Pad (and on each CD ~ Record Keeping Sheets, pages to ) enable students to find out what they know and what they still need to learn and practise. The Skill Builders extensively target through instruction and practice, all skills within the related Maths Mate Program except the problem solving questions. The Problem Solving Hints & Solutions (see CD ~ Problem Solving Hints & Solutions) can be used by teachers to develop students problem solving skills. The Skill Builders also contain a Glossary of important facts and reference material that will provide instant help when students present with difficulties. Background to the design of Maths Mate and Skill Builders Any question on the Maths Mate sheets is part of a set of similar questions in the term. For example, consider sheets,, and in year term. Question 0 on each sheet is similar in design, content and degree of difficulty. This grouping of question style is also true of the next set of four sheets and so on. Thus the Maths Mate tests made available in the Teacher Resource Book and CD (see CD ~ Test Masters, pages to and Test Answers, pages to ) also reflect this grouping of question style and substance. Generally too, the Skill Builders can be linked to each set of similar questions. These links are identified in the grid at the title of each skill. The grid shown here for example, would relate a skill to questions in the first sheets of term, the last sheets of term and the first sheets of term. Once understood, these links will be helpful to students in their selection of Skill Builders and to you in your allocation of Skill Builders to students. On each Maths Mate worksheet, questions through to get progressively harder. (Refer - How to use the Skill Builders, page iv) Suggestions for the preparation and organisation of Skill Builders Skill Builders can be downloaded from the internet. Teachers can either direct students to the internet to download and print their own copies or save the entire Skill Builder to disc and photocopy at will. Rather than photocopying Skill Builders one at a time, you may find it helpful to set up a file in a central area that contains perhaps five copies of each Skill Builder. In this way you will save time and be prepared in advance. The Glossary too can be downloaded or photocopied for students as a resource. How you can help We are confident that your students will be rewarded for the effort you have made in making these worksheets available to them either via the internet or through hardcopy. As with any program, however, there is always room for improvement and we place great value in feedback from people like yourself. Please, if you have any suggestions at all, contact us. page iii Maths Mate /8 Skill Builder

6 How to use Maths Mate Skill Builder. Determine which Maths Mate questions pose a difficulty If a student gets one or more incorrect answers, represented by one or more successive unshaded boxes on their worksheet results sheet, then that question is posing difficulty. For example, question in Sheets,, and is not shaded, so Skill. from Skill Builder needs to be handed to the student. For skill builder help go to MATHS MATE Worksheet Results Term Paul Wright Name:... B Class:... Miss Bourke Teacher:... Skill Builder links Sheet 8 Sheet Sheet Sheet Skill Builder links Sheet Sheet Sheet Sheet. [+ Whole Numbers to 0]... [ Whole Numbers to 0]... [ Whole Numbers to ]... [ Whole Numbers to ]... [Large Number +, ]... [Large Number, ]... [Decimal +, ].. 8. [Decimal, ] [Fraction +, ] , NUMBER & ALGEBRA 0. [Fraction, ]. [Percentages]. [Decimals / Fractions / Percents]. [Integers]. [Rates / Ratios] [Indices / Square Roots]... [Order of Operations]... [Exploring Number].. 8. [Multiples / Factors / Primes] , [Number Patterns] [Expressions]. [Substitution]. [Equations] [Coordinates].. MEASUREMENT & GEOMETRY. [Units of Measurement / Time]. [Perimeter]. [Area / Volume]. [Shapes] 8. [Exploring Geometry] STATISTICS & PROBABILITY 9. [Statistics] 0. [Probability] PROBLEM SOLVING. [Problem Solving ]. [Problem Solving ]. [Problem Solving ] Hints & Solutions Hints & Solutions Hints & Solutions Hints & Solutions Hints & Solutions Hints & Solutions Total Correct 0 page Maths Mate ~ Record Keeping Sheets. Find the relevant Skill Builder on the Maths Mate worksheet results sheet Check across the question that is posing difficulties on the worksheet results sheet to find the list of skills within the Skill Builder that are most relevant to that question. Obtain a copy of one or all of the skills listed for that question (pages to ). You can also double check with the grid at the right of each skill title, that the chosen skill is appropriate. Remember, students should work through the skills in order. The skills where possible are arranged in increasing degree of difficulty. Be aware that some skills may require the knowledge of previous skills, so when a student has several areas of weakness, they should work on the lowest numbered skill builders first. For example, a student struggling with Q0 and Q will need to build skills required for Q0 before they can improve Q.. [Substitution] Skill. Substituting one value into expressions involving + and Substitute the letters with numbers. Use the order of operations rule: Add ( + ) and/or subtract ( ) from left to right. Q. If a, find the value of a a) If p, find the value of + p + d) If m, find the value of m + g) If x, find the value of x + x j) If t, find the value of t + t + t A. a 8 substitute a b) If f, find the value of + f e) If g, find the value of g + h) If v, find the value of v + v k) If e, find the value of e + e + e c) If c, find the value of + c f) If z, find the value of z + i) If q, find the value of q + q l) If p 8, find the value of p + p + p m) If j 9, find the value of j + j 8 n) If k, find the value of k + k + o) If h 8, find the value of + h + h p) If m 8, find the value of m + m 9 q) If s, find the value of 9 + s + s r) If n, find the value of 8 + n + n page Maths Mate /8 Skill Builder page iv Maths Mate /8 Skill Builder

7 . Look up any unknown terms in the Skill Builder glossary The glossary (pages to ) is more than just a list of definitions. It contains a wealth of relevant information that may help the students to better understand the question at hand. Weaker students may find that referring to a copy of the glossary, and even building on it, is a helpful strategy for improving their overall mathematical competency. For example, a student might need to look up the word substitute before attempting to complete Skill. sq - su square units stem-and-leaf plot A unit of area equal to the area of a square with side lengths of unit. A diagram displaying data by place value. The data is in order from lowest to highest. A lw units units Area square units squared Multiplied by itself. squared is written as A number raised to the second power. statistics Numerical facts systematically collected, organised and analysed. Data is collected from a sample of the population, organised into a graph and interpreted to summarize some characteristic. Data set of elements: {, 8, 8, 9, 0,,,,,, 9, 0, } mode median (th element) range straight angle stem leaves lowest value median mode highest value An angle measuring 80. range high low 8 mean substitute To replace a number or function with another. Often used in algebra when a variable (letter) is replaced by a number. If x, the value of x + x is found by replacing the letter x with : + 8 subtract sum To take away or minus. The result when two or more numbers are added. If you subtract 0 from you are left with : 0 The sum of 0 and is : supplement of an angle An angle that, when added to an adjacent is the supplement of 0, angle, makes a straight angle (or 80 in total). because page Maths Mate /8 Glossary. Complete the relevant Skill Builder Work through the examples given for that skill, and complete the exercises. There are many techniques or methods that can be used to teach the same basic skills, even something as simple as adding and 9. It is good for a student to be given a range of alternatives appropriate for each skill but space restrictions make this impossible. These sheets often suggest an approach that may be different to a student s past experience. If a student feels more comfortable with his current technique, that is fine. In most cases it is the end result that counts. It is possible to take a very weak student back to a Skill Builder from a lower level if this is necessary. It is also possible to use a higher level book for students to have further practice if required.. Correct the relevant Skill Builders from the Skill Builder answer sheets (from page ). Circle the completed skill numbers on the Maths Mate worksheet results sheet. [Exploring Number].. 8. [Multiples / Factors / Primes] , [Number Patterns] [Expressions]. [Substitution]. [Equations] [Coordinates]... [Units of Measurement / Time]... [Perimeter]... Go back and repeat previous Maths Mate questions After completing a Skill Builder, students should be encouraged to go back and attempt again those particular questions on the recently completed Maths Mate worksheets. page v Maths Mate /8 Skill Builder

8 Dear Parents As part of their Mathematics program this year, all students have been given a weekly Maths Mate sheet. The program is now under way. The diagnostic nature of the worksheets helps students monitor their own progress. After they correct their worksheet and complete the record keeping sheet, over time, your child will be able to identify areas of strength and weakness in their mathematical learning. If your child is having difficulty with a question for consecutive weeks or believes that their understanding is not at the level they would like, then Skill Builder sheets will be made available to develop each of the skills in the Maths Mate program. Each Skill Builder focuses on and explores, one question from the Maths Mate sheets. Your child is encouraged to make full use of these resources by taking home any sheet that will help consolidate their understanding of a particular skill. Or, for your convenience, all worksheets are available on our website. Simply go to and follow the prompts to download the appropriate Skill Builder. As each question in the Maths Mate is generally more difficult than the last, finishing with the problem solving questions, then it would be advised that, if students are concerned with more than one question, they tackle lower numbered questions first. The Skill Builders may also help to motivate students to make another attempt at mastering skills that they have found too difficult in the past, given that it will become clear to them that they will be confronted by the same type of question on a regular basis. While we will be monitoring your child s progress and supporting their skill development in the school environment, it would be appreciated if you would complete the tear off slip at the bottom of this page so that we can be sure that you are aware of our expectations regarding both the Maths Mate worksheets and the availability of Skill Builder worksheets. We ask also that you continue to sign the completed worksheets each week so that we can ensure each student is working independently and regularly but with your support. We thank you in anticipation of your involvement and remind you that you are encouraged to call and discuss your child s progress at any time. Yours sincerely Class Teacher Principal Maths Mate Homework Program - Skill Builder Return Slip Student s Name: Class: As a parent / guardian I have signed this form to indicate that I am aware of the support Maths Mate Skill Builders can give my child in their mathematical development. Parent s Signature: Date:

9 CONTENTS Forward... iii How to use Maths Mate Skill Builders... iv Letter to Parents (sample)... vi Skill Builders... Glossary... Maths Facts... Symbols Number Facts Algebra Facts Measurement Facts Geometry Facts Answers... MM SB [Maths Mate - Mathematical strand] Question Skill No. Skill Builder - Skill description. [+ Whole Numbers to 0].... Adding whole numbers from to 0.. Adding whole numbers from to 0 to negative numbers.. [ Whole Numbers to 0].... Subtracting whole numbers from to 0.. Subtracting whole numbers from to 0 from negative numbers.. [ Whole Numbers to ].... Multiplying whole numbers from to.. Multiplying whole numbers from to by negative numbers.. [ Whole Numbers to ].... Dividing whole numbers from to.. Dividing whole numbers from to into negative numbers.. [Large Number +, ] Adding large numbers without carry over.. Subtracting large numbers without carry over.. Adding two large numbers with carry over.. Subtracting large numbers with carry over.. Adding and/or subtracting multiple large numbers with carry over.. [Large Number, ].... Multiplying a large number by a power of 0.. Dividing a large number by a power of 0.. Multiplying a large number by a single digit.. Dividing a large number by a single digit.. Multiplying a large number by a multiple of 0.. Dividing a large number by a multiple of 0.. Multiplying a large number by a two-digit number..8 Dividing a large number by a two-digit number..9 Multiplying a whole number by a large multiple of 0..0 Dividing a whole number - answer as a terminating decimal.. [Decimal +, ]... Adding decimal numbers.. Subtracting decimal numbers.. Subtracting a decimal number from a whole number. page vii Maths Mate /8 Skill Builder

10 MM SB [Maths Mate - Mathematical strand] Question Skill No. Skill Builder - Skill description 8. [Decimal, ].. 8. Multiplying a whole number by a decimal number. 8. Dividing a decimal number by a whole number. 8. Multiplying a decimal number by a decimal number. 8. Dividing a decimal number by a decimal number. 8. Dividing a whole number by a decimal number. 9. [Fraction +, ] Adding fractions with the same denominator. 9. Subtracting fractions with the same denominator. 9. Adding mixed numbers with the same denominator. 9. Subtracting mixed numbers with the same denominator. 9. Subtracting a fraction or a mixed number from a whole number. 9. Adding fractions with different denominators - one denominator divides evenly into the other denominator. 9. Adding fractions with different denominators - the denominators have no common factors other than (e.g. and ). 9.8 Subtracting fractions with different denominators - one denominator divides evenly into the other denominator. 9.9 Subtracting fractions with different denominators - the denominators have no common factors other than (e.g. and ). 0. [Fraction, ].. 0. Multiplying a fraction by a whole number. 0. Finding a fraction of a quantity. 0. Dividing a whole number by a fraction. 0. Multiplying two fractions. 0. Dividing a fraction by a whole number. 0. Dividing two fractions.. [Percentages]. Writing a number out of 00 as a percentage.. Finding the remaining percentage.. Finding a percentage of multiples of 00.. Finding a percentage of any number.. Working with percentages greater than 00%.. Working with percentages to find discounts and sale prices.. Writing one number as a percentage of another number..8 Calculating profit or loss as a percentage of the cost price.. [Decimals / Fractions / Percentages].... Illustrating fractions and percentages.. Simplifying fractions.. Finding equivalent fractions.. Writing a decimal number as a percentage.. Writing a percentage as a decimal number.. Writing a decimal number as a fraction in simplest form.. Writing a fraction as a terminating decimal..8 Writing a percentage as a fraction in simplest form..9 Writing a fraction as a percentage..0 Ordering decimal numbers.. Comparing and ordering fractions.. Converting between decimals, fractions and percentages.. Comparing decimals, fractions and percentages.. [Integers] Comparing and ordering integers.. Comparing integers using less than and greater than.. Modelling integer subtraction on a number line.. Finding the difference between a positive and a negative integer.. Modelling integer addition on a number line.. Solving word problems involving two or more integers.. Adding integers..8 Subtracting integers..9 Multiplying integers..0 Dividing integers. page viii Maths Mate /8 Skill Builder

11 MM SB [Maths Mate - Mathematical strand] Question Skill No. Skill Builder - Skill description. [Rates / Ratios] Simplifying ratios by comparing two numbers.. Simplifying ratios by comparing two quantities.. Solving questions involving distance, time and speed.. Simplifying ratios by comparing three numbers.. Deciding if two ratios are equivalent.. Completing equivalent ratios.. Solving word problems involving equivalent ratios..8 Finding the ratio of two quantities..9 Finding other rates.. [Indices / Square Roots].... Expressing powers as products and products as powers.. Squaring whole numbers.. Calculating powers of 0.. Finding square roots of whole numbers.. Evaluating powers of whole numbers.. Finding powers of negative whole numbers.. [Order of Operations].... Using order of operations mixing only and/or, or + and/or. Using order of operations mixing,, + and/or. Using order of operations mixing ( ) with + and/or. Using order of operations mixing ( ),,, + and/or. Using order of operations mixing powers, ( ),,, + and/or. Using order of operations involving negative numbers and mixing powers, ( ),,, + and/or. Using order of operations mixing square roots, powers,,, + and/or. [Exploring Number].... Comparing whole numbers.. Understanding and finding the place value of a digit in a number.. Writing word numbers as numerals.. Writing whole numbers in words.. Rounding whole numbers to a given place.. Rounding decimal numbers to a given place.. Recognising whole numbers and integers..8 Using inequality and equality signs to compare decimal numbers. 8. [Multiples / Factors / Primes] Finding the multiples of a number. 8. Finding the common multiples of two numbers. 8. Finding the lowest common multiple (LCM) of two numbers. 8. Finding the factors of a number. 8. Finding the common factors of two numbers. 8. Finding the highest common factor (HCF) of two numbers. 8. Recognising prime and composite numbers. 8.8 Expressing a number as a product of its prime factors using a factor tree. 8.9 Expressing a number as a product of its prime factors using consecutive divisions. 8.0 Expressing a number as a product of its prime factors using index notation. 9. [Number Patterns] Completing number patterns by adding the same number. 9. Completing number patterns by subtracting the same number. 9. Completing number patterns by adding or subtracting decimal numbers. 9. Completing number patterns in table format by adding the same number. 9. Completing number patterns by multiplying by the same number. 9. Completing number patterns by dividing by the same number. 9. Completing number patterns by using changing values in the rule. 9.8 Completing number patterns involving negative integers by adding or subtracting the same integer. 9.9 Finding a term in a number pattern. 9.0 Finding a particular term of a sequence given its general rule. page ix Maths Mate /8 Skill Builder

12 MM SB [Maths Mate - Mathematical strand] Question Skill No. Skill Builder - Skill description 0. [Expressions] Simplifying expressions by adding and subtracting like terms (coefficient ). 0. Simplifying expressions by adding and subtracting like terms (coefficient ). 0. Finding like terms. 0. Simplifying expressions by first grouping like terms. 0. Writing expressions to represent word problems.. [Substitution].... Substituting one value into expressions involving + and. Substituting one value into expressions involving and. Substituting one value into expressions involving +,, and. Substituting negative values into expressions.. Substituting two values into expressions involving + and. Substituting two values into expressions involving and. Substituting two values into expressions involving +,, and.8 Substituting into expressions involving powers..9 Substituting into expressions with brackets..0 Substituting into formulae.. [Equations].. 8. Finding the missing number in equations involving + and. Finding the missing number in equations involving. Finding the missing number in equations involving fractions.. Finding the missing number in equations involving +,, and/or brackets.. Finding the missing number in equations involving decimals.. Solving one-step equations by using the inverse operations of + and. Solving one-step equations by using the inverse operations of and.8 Solving two-step equations by using the inverse operations of +,, and. [Coordinates] Describing the position of ordered pairs on a Cartesian plane.. Using grid references to describe location on a map.. Using coordinates to describe location on a map.. Finding the coordinates of a point on a Cartesian plane.. Plotting ordered pairs on a Cartesian plane.. Writing linear expressions to represent real-life situations.. Completing a table of values for a linear rule..8 Graphing linear functions on a Cartesian plane..9 Using coordinates to visualise and draw transformations of two-dimensional shapes on a Cartesian plane..0 Plotting points from a table of values on a Cartesian plane.. [Units of Measurement / Time].... Converting units of time.. Converting units of length.. Converting units of mass.. Converting units of capacity.. Converting units of time, length, mass and capacity by using real-life facts.. Finding the elapsed time between two events.. Using time zones to calculate durations.. [Perimeter].... Finding the perimeter of polygons by measuring their side lengths.. Calculating the perimeter of polygons when all side lengths are given.. Calculating the perimeter of polygons by recognising congruent sides.. Calculating the perimeter of polygons using real life examples.. Calculating the perimeter of polygons using unit conversions.. Calculating an unknown side length when the perimeter of a polygon is given.. Calculating the circumference of circles..8 Calculating the perimeter of composite shapes. page x Maths Mate /8 Skill Builder

13 MM SB [Maths Mate - Mathematical strand] Question Skill No. Skill Builder - Skill description. [Area / Volume].... Calculating the area of polygons by counting squares and triangles on a square grid.. Comparing the area of polygons on a square grid.. Estimating the area of irregular shapes on a square grid.. Calculating the area of squares, rectangles and parallelograms.. Calculating the area of triangles.. Calculating the volume of rectangular prisms by counting cubes.. Calculating the volume of square and rectangular prisms..8 Calculating the area of composite shapes..9 Calculating the area of trapeziums and rhombii..0 Calculating the area of circles and composite circular shapes.. Calculating the volume of any prism.. [Shapes].... Measuring angles using a protractor.. Estimating the size of angles.. Recognising polygons and quadrilaterals.. Classifying and describing the properties of quadrilaterals.. Drawing lines and polygons.. Classifying and describing the properties of D shapes.. Classifying angles..8 Classifying and describing the properties of triangles..9 Working with vertically opposite angles and complementary angles..0 Working with supplementary angles.. Finding the size of angles inside a triangle.. Finding the size of angles inside a quadrilateral.. Describing the properties of circles. 8. [Exploring Geometry] Following directions and using compass bearings to describe location on a map. 8. Identifying and classifying symmetry in two-dimensional shapes. 8. Using a scale to calculate distance on a map. 8. Recognising basic transformations of two-dimensional shapes. 8. Drawing translations, reflections and rotations of objects on a grid. 8. Recognising nets of three-dimensional shapes. 8. Drawing the top, side and front views of three-dimensional shapes. 8.8 Recognising the shapes of cross sections through three-dimensional shapes. 8.9 Recognising congruence in two-dimensional shapes. 8.0 Recognising rotational symmetry in two-dimensional shapes. 9. [Statistics] Interpreting dot plots. 9. Interpreting pictograms. 9. Interpreting tables. 9. Interpreting bar graphs. 9. Interpreting stack graphs. 9. Calculating the mean and median of sets of data. 9. Calculating the mode and range of sets of data. 9.8 Interpreting line graphs. 9.9 Interpreting pie charts. 9.0 Interpreting stem-and-leaf plots. 9. Interpreting step graphs, histograms and scatter plots. 0. [Probability] 9 0. Describing the degree of likelihood of an event. 0. Recognising the likelihood of an event. 0. Finding the possible outcomes (sample spaces) of an event by completing tables. 0. Finding the possible outcomes (sample spaces) of an event by completing tree diagrams. 0. Calculating the probability of a simple event. 0. Calculating the probability of a simple event using probability scales. 0. Interpreting Venn diagrams. 0.8 Calculating the probability of complementary events. 0.9 Calculating the probability of mutually exclusive events. 0.0 Finding the possible outcomes of an event by applying the counting principle. page xi Maths Mate /8 Skill Builder

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15 . [+ Whole Numbers to 0] Skill. Adding whole numbers from to 0. EITHER Regroup into easier numbers Break a number up so that you can work with groups of 0 Example: OR Count on Start with the largest number and count on the smaller amount. Example: 8 + 9, 0,,, OR Use an addition table Move down the column and across the row to find the intersection Example: Hint: Addition tables are symmetrical. Q Add 8 to each of the numbers in the top row. A a) b) c) d) e) page Maths Mate /8 Skill Builder

16 Skill. Adding whole numbers from to 0 to negative numbers. Use a number line. Example: + is read as: negative plus Start at O Move units to the left (negative direction). Move units to the right (positive direction). You stop at + Start here negative integers positive integers (You can write positive integers with or without a + sign.) Q Add to each of the numbers on the top row. A a) b) c) d) e) f) page Maths Mate /8 Skill Builder

17 . [ Whole Numbers to 0] Skill. Subtracting whole numbers from to 0. EITHER Break down to easier numbers Break a number down so that you can work with groups of 0 Example: Make into 0 by taking from both and 9. OR Build up to easier numbers Build a number up so that you can work with groups of 0. Example: 9 Add to 9 to make 0 and another to get to. + OR Use an addition table Move down the column and across the row to find the intersection Example: 9? Reword the subtraction by turning it into an addition. What number when added to 9 will give? From the addition table, 9 + So 9 Q Subtract 9 from each of the numbers in the top row. A a) b) c) d) page Maths Mate /8 Skill Builder

18 Skill. Subtracting whole numbers from to 0 from negative numbers. Use a number line. Example: is read as: negative subtract OR negative minus Start at O Move units to the left (negative direction). Move units to the left again (negative direction). You stop at negative 9 Begin here negative integers positive integers (You can write positive integers with or without a + sign.) Q Subtract from each of the numbers on the top row. A a) b) c) d) e) f) page Maths Mate /8 Skill Builder

19 . [ Whole Numbers to ] Skill. Multiplying whole numbers from to. Use a multiplication table Find one of the numbers to be multiplied across the top row. Find the other number to be multiplied down the left hand side column. Follow the line of each number until they intersect at their product. Example: The product of and 9 is 9 Since 9 9 multiplication tables are symmetrical. Hint: This means you only need to learn half of your times tables MULTIPLICATION TABLE Q Multiply each of the numbers in the top row by 9. A a) b) c) d) e) page Maths Mate /8 Skill Builder

20 Skill. Multiplying whole numbers from to by negative numbers. Use a multiplication table Find one of the numbers to be multiplied across the top row. Find the other number to be multiplied down the left hand side column. Follow the line of each number until they intersect at their product. Then apply the rule: When multiplying a positive number by a negative number, the result is always negative. + + Example: The product of negative and is negative 8 8 Since 8 multiplication tables are symmetrical. MULTIPLICATION TABLE Q Multiply each of the numbers in the top row by. A Use the rule: A negative number multiplied by a positive number results in a negative number. a) b) c) d) e) f) page Maths Mate /8 Skill Builder

21 . [ Whole Numbers to ] Skill. Dividing whole numbers from to. Reword the division by turning it into a multiplication. Use a multiplication table. Convert the multiplication back to a division. Example: How many s go into?? Reworded: What number multiplied by equals?? From the multiplication table, So MULTIPLICATION TABLE Q Divide each of the numbers in the top row by. A a) b) c) d) e) page Maths Mate /8 Skill Builder

22 Skill. Dividing whole numbers from to into negative numbers. Reword the division by turning it into a multiplication. Use a multiplication table. Convert the multiplication back to a division. Then apply the rule: When dividing a negative number by a positive number, the result is always negative. + + Example: How many s go into negative 0? 0? Reworded: What number multiplied by gives negative 0?? 0 From the multiplication table, 0 So MULTIPLICATION TABLE Q Divide each of the numbers in the top row by. A Use the rule: A negative number divided by a positive number results in a negative number. a) 0 8 b) c) d) e) f) page 8 Maths Mate /8 Skill Builder

23 . [Large Number +, ] Skill. Adding large numbers without carry over. Always keep your working columns in line, aligning units with units, tens with tens, etc. Add from right to left. Q. + 0 A thousands hundreds tens units 9 Units: + 0 units Tens: tens Hundreds: + hundreds Thousands: + thousands a) b) + c) d) e) f) g) h) + + i) j) k) + + l) page 9 Maths Mate /8 Skill Builder

24 Skill. Subtracting large numbers without carry over. Always keep your working columns in line, aligning units with units, tens with tens, etc. Subtract from right to left. Q. 8 0 A thousands hundreds tens units Units: 8 units Tens: 0 ten Hundreds: hundreds Thousands: thousand a) 899 b) 0 c) d) e) 0 0 f) g) h) 9 9 i) j) k) l) 9 9 page 0 Maths Mate /8 Skill Builder

25 Skill. Adding two large numbers with carry over. Always keep your working columns in line, aligning units with units, tens with tens, etc. Add from right to left. Q A thousands hundreds tens units 0 Units: + 8 units ten and units units Carry the ten to the tens column. Tens: carry tens hundred and tens tens Carry the hundred to the next column. Hundreds: + + carry 0 0 hundreds thousand and 0 hundred 0 hundred Carry the thousand to the next column. Thousands: + + carry thousands a) b) + + c) d) e) f) g) + 98 h) i) j) + 8 k) l) page Maths Mate /8 Skill Builder

26 Skill. Subtracting large numbers with carry over (). continues on page Always keep your working columns in line, aligning units with units, tens with tens, etc. Subtract from right to left. Whenever a number cannot be subtracted from another number to give a positive result then use: EITHER Decomposition method - Borrow from a higher place value and give to a lower place value. Example: 9 Units:? Borrow ten from the tens column (reduce the tens to tens) and give it as 0 units to the units column to make units. units 9 Tens: 9? Borrow hundred from the hundreds column (reduce the hundreds to hundreds) and give it as 0 tens to the tens column to make tens. 9 tens Hundreds: hundreds Thousands: 0 thousands thousands hundreds units tens OR Equal addition method - Each time a number is added to the top it must also be added to the bottom but in different columns. Example: 9 Units:? Add 0 units to the and add 0 units ( ten) to the 9 (bottom number in the tens column) units Tens: (9 + )? Add 0 tens to the and add 0 tens ( hundred) to the (bottom number in the hundreds column) 0 tens Hundreds: ( + ) hundreds Thousands: 0 thousands 9 thousands hundreds tens units Q A Decomposition thousands hundreds tens units 0 OR Equal addition thousands hundreds units tens a) b) 08 c) Use decomposition d) e) f) page Maths Mate /8 Skill Builder

27 Skill. Subtracting large numbers with carry over (). continued from page g) 9 9 h) i) 8 8 j) 90 k) 08 9 l) m) 08 n) 009 o) p) 8 89 q) r) 9 s) 8 8 t) 9 u) v) w) 0 x) page Maths Mate /8 Skill Builder

28 Skill. Work from left to right. Adding and/or subtracting multiple large numbers with carry over. Q A Complete the addition first Then subtract 9 from thousands hundreds tens units 8 a) b) c) d) e) f) g) h) page Maths Mate /8 Skill Builder

29 . [Large Number, ] Skill. Multiplying a large number by a power of 0. When the multiplication is displayed in a horizontal line: Add the same number of zeros at the end of the given number as there are zeros in the power of 0. When the multiplication is displayed in a vertical algorithm: Move each digit of the given number as many places to the left as there are zeros in the power of 0. Add zeros as place holders in the vacated places. Q. 000 A Add zeros Move,, three places left Add zeros at the end of a) b) 00 0 c) Add zero d) 8 00 e) 0 00 f) Add zeros g) h) 000 i) j) k) l) page Maths Mate /8 Skill Builder

30 Skill. Dividing a large number by a power of 0. Remove as many zeros from the end of the given number as there are zeros in the power of 0. Hint: If the division is written as a fraction, simply cross off respective zeros from the top and bottom of the fraction. Q A OR Any division can be written as a fraction. Simplify by dividing both the numerator and denominator by 000. Cross off the respective zeros. a) b) c) d) e) f) g) h) i) j) k) l) page Maths Mate /8 Skill Builder

31 Skill. Multiplying a large number by a single digit. Multiply the number by the single digit working from right to left. If there is a carry over : First multiply. Then add on the carry over. Q. 09 A tens of thousands thousands hundreds tens units Units: 9 units tens and units units Carry the tens to the next column. Tens:, + tens hundred and tens tens Carry the hundred to the next column. Hundreds: hundred Thousands: 8 8 thousand a) 90 b) 9 c) Units first! 9 9 d) e) f) 0 8 Units first! 0 8 g) h) 0 i) j) k) l) 8 8 page Maths Mate /8 Skill Builder

32 Skill. Dividing a large number by a single digit. Divide from left to right across the digits one at a time. If any result is less than : Cross off the number being divided into. Carry over this amount to the next column. Add on the carry. Then try dividing again. Q. 8 8 A Read as: 8 divided by 8 OR How many times can 8 be taken from 8? OR How many 8 s go into 8? Divide 8 into. 8 doesn t divide into, so carry over the groups of 000 and make groups of divides into eight times with remainder. Write an 8 above the and carry the remaining groups of 00 to the tens column to make tens. Divide 8 into. 8 divides into nine times and remainder. Write a 9 above the and carry the remaining groups of tens to the units column to make 8 units. Divide 8 into 8. 8 divides into 8 six times and 0 remainder. Write a above the 8. a) 8 b) 8 8 c) d) e) 9 8 f) g) h) 0 i) j) k) l) m) 08 n) o) page 8 Maths Mate /8 Skill Builder

33 Skill. Multiplying a large number by a multiple of 0. Consider the zeros as making groups of 0 s or 00 s and place them at the end. Then multiply by the remaining digit as though it was a unit. Q. 00 A Consider 00 as groups of 00. Multiply by : To show we want groups of 00, place two zeros after. a) b) 0 c) d) 0 e) 0 f) g) 00 h) 8 00 i) j) 0 00 k) 00 l) m) 00 0 n) 00 0 o) page 9 Maths Mate /8 Skill Builder

34 Skill. Dividing a large number by a multiple of 0. Remove as many zeros from the end of the given number as there are zeros in the multiple of 0. Divide by the remaining digit working from left to right. Q A Divide both numbers by 0, by crossing off the zeros. Complete the division 8 divides into three times and remainder. Write a above the and carry the remaining groups of tens to the units column to make 8 units. divides into 8 eight times and 0 remainder. Write an 8 above the 8. a) 00 0 b) c) d) e) f) g) h) i) j) k) l) page 0 Maths Mate /8 Skill Builder

35 Skill. Multiplying a large number by a two-digit number (). Multiply by the unit digit first, working from right to left. Reminder: Put a zero in the units place before you start multiplying by the tens. Then multiply by the ten digit, working from right to left. Add the results last. continues on page Q. A Multiply by. Then multiply by 0. Remember: Put a 0 in the units place. Add these results. The question can be thought of as: plus a) as place holder b) c) d) e) 809 f) g) h) i) 0 0 page Maths Mate /8 Skill Builder

36 Skill. Multiplying a large number by a two-digit number (). continued from page j) k) 8 8 l) 9 9 m) n) o) p) 0 q) 08 8 r) s) 009 t) 0 u) page Maths Mate /8 Skill Builder

37 Skill.8 Dividing a large number by a two-digit number (). continues on page Work from left to right. Break down the division into smaller divisions by dividing into only as many digits as you need to get an answer greater than. It may be difficult, so guess the number of divisions, and multiply your guess to check. Subtract your answer from the original number to get the remainder, which must be less than the number you are dividing by. Continue in this way by bringing down the next digit to make the next number to divide into. Repeat until the result of the subtraction is zero. Q. 990 A Start at the left. 9 is too small to divide into, so consider 9. Divide 9? is a good guess. Check by multiplying 90 Subtract 9 90 Write above the. Bring down the 9. Divide 9? (Guess ) Check by multiplying 0 Subtract Write above the 9. Bring down the 0. Divide 90 (No remainder) Write above the 0. OR Work as a short division. a) b) 9 c) d) 0 e) 8 f) page Maths Mate /8 Skill Builder

38 Skill.8 Dividing a large number by a two-digit number (). continued from page g) 0 h) 808 i) j) k) l) 0 0 m) 0 n) 900 o) p) 8 q) 9 r) page Maths Mate /8 Skill Builder

39 Skill.9 Multiplying a whole number by a large multiple of 0. Consider the zeros as making groups of 0 s or 00 s and place them at the end. Multiply by the unit digit first, working from right to left. Then multiply by the ten digit, working from right to left. Add the results last. Q A Consider 00 as groups of 00. Work with the first. Multiply 0 by. Then multiply 0 by 0. Add these results. To show we want groups of 00, place two zeros after. a) b) 0 c) d) 0 0 e) f) g) h) i) page Maths Mate /8 Skill Builder

40 Skill.0 Dividing a whole number - answer as a terminating decimal. Line up the decimal point in your answer. Place a decimal point and more zeros at the end of the whole number to be divided. Divide into the whole number and continue until you get an exact division with no remainder. Hint: When no decimal point is shown it is always placed on the far right of the number. Q. 8 8 A Start at the left. Divide 8 into 8.00 Continue until you get an exact number with no remainder. a) 8. b) 0 c) d) 9 e) 8 f) g) 80 h) 8 i) j) 0 k) 8 l) m) 90 n) 89 o) 98 page Maths Mate /8 Skill Builder

41 . [Decimal +, ] Skill. Adding decimal numbers (). continues on page 8 Always keep your working columns in line, aligning the decimal points, the decimal places, units with units, tens with tens, etc. Add from right to left. Q A Hundredths: + 0 hundredths tens units tenths decimal point hundredths Tenths: 8 + tenths Carry over 0 tenths as unit Units: carry units Carry over 0 units as ten Tens: carry tens a) b) c) d) e). +. f) g) h) i) j) k).9 +. l) page Maths Mate /8 Skill Builder

42 Skill. Adding decimal numbers (). continued from page m) n). +. o) p) q) r) s) t). +. u) v) w) x) y) z) zz) page 8 Maths Mate /8 Skill Builder

43 Skill. Subtracting decimal numbers (). continues on page 0 Always keep your working columns in line, aligning the decimal points, the decimal places, units with units, tens with tens, etc. Subtract from right to left. Whenever a number cannot be subtracted from another number to give a positive result then use either the decomposition or equal addition method. (see skill., page ) Q...8 A Using the Equal Addition method Hundredths:? Add 0 hundredths to the and 0 hundredths ( tenth) to the 8 (bottom. number in the tenths column). 8 9 hundredths. 9 tens units tenths decimal point hundredths Tenths: (8 + )? Add 0 tenths to the and 0 tenths ( unit) to the (bottom number in the units column) 9 tenths Units: ( + )? Add 0 units to the and 0 units ( ten) to the (bottom number in the tens column) 8 units Tens: ( + ) ten a).. b).8 0. c) Use decomposition d) 8.. e)..9 f) g) 8..9 h)..8 i) page 9 Maths Mate /8 Skill Builder

44 Skill. Subtracting decimal numbers (). continued from page 9 j). 9. k) l) m).. n) o) p)..8 q)..9 r). 8.. s)..8 t). 8. u). 9.8 v) w) 80.. x) y)..8 z). 9. zz). 9.8 page 0 Maths Mate /8 Skill Builder

45 Skill. Subtracting a decimal number from a whole number. Write a decimal point at the end of the whole number. Add as many zeros after the decimal point as there are decimal places in the decimal number. Work vertically, lining up the decimal points. Subtract from right to left. Whenever a digit cannot be subtracted from another digit to give a positive result then use either the decomposition or equal addition method. (see skill., page ) Q. 0.9 A tens units tenths decimal point hundredths Using the Decomposition method: Work from right to left until you reach a number you can borrow from, in this case the ten. Restructure the ten to 9 units, 9 tenths and 0 hundredths. Hundredths: 0 9 hundredths Tenths: Units: Tens: tenths 9 units tens a). b) 8. c) Use equal addition d). e).8 f) g). h) 9.8 i).. j) 0. k) l) Use decomposition 9.. page Maths Mate /8 Skill Builder

46 page Maths Mate /8 Skill Builder

47 8. [Decimal, ] Skill 8. Multiplying a whole number by a decimal number (). Multiply from right to left, disregarding the decimal point. Count the number of places to the right of the decimal point in the question. Position the decimal point the same number of places from the right in the answer. continues on page Q. 0. A write 8 carry, write 0 + carry write decimal places in question so move decimal point places from right in the answer a) 0.9 b) 0.8 c) d) 0. e) 0. f) g). h). i).... j).8 k).9 l) page Maths Mate /8 Skill Builder 8

48 Skill 8. Multiplying a whole number by a decimal number (). continued from page m) 0. n) 0. o) p).09 q). r) s).0 t) 0. u) v) w).008 x) 0. y).0 z). zz).0 page Maths Mate /8 Skill Builder 8

49 Skill 8. Dividing a decimal number by a whole number. Line up the decimal point in your answer with the decimal point in the question. Divide from left to right. Break down the division into smaller divisions. If any result is less than : Cross off the number being divided into. Carry over this amount to the next column. Add on the carry. Then try dividing again. Q.. 9 A ? (less than 0) carry, write 0 Line up the decimal point. 9 carry, write 9 write a). b). 8 c). from left Line up decimal points 0. d). e). f) g). 9 h). i) j). k) 0.9 l). 8 8 m) 0. n). o). 9 page Maths Mate /8 Skill Builder 8

50 Skill 8. Multiplying a decimal number by a decimal number. Multiply from right to left, disregarding the decimal point. Count the number of places to the right of the decimal point in the question. Position the decimal point the same number of decimal places from the right in the answer. Use zeros as place holders, if necessary. Example: If the result is less than, write a zero in the units place. Example: By convention 0. not. Q A carry, write carry 0 write 0 decimal places in question so move decimal point places from right in the answer a) b) c) d) e) f) g). 0. h). 0.9 i) j) k) l) < so write zero in units place remove unnecessary zero page Maths Mate /8 Skill Builder 8

51 Skill 8. Dividing a decimal number by a decimal number. Move the decimal point to the right in the divisor, as many places as you need to make it a whole number. Then move the decimal point the same number of places to the right in the dividend. Example: Add zeros as place holders, if necessary. Example: Line up the decimal point in your answer with the decimal point in the question. Divide from left to right. (see skill 8., page ) Q A ? (less than 0) carry, write 0 Line up the decimal point. 9 write 9 a). 0. place right makes a whole number... b) c) d) e) f) g) h) i) j) k) l) m) n) o) page Maths Mate /8 Skill Builder 8

52 Skill 8. Dividing a whole number by a decimal number. Move the decimal point to the right in the divisor, as many places as you need to make a whole number. Then move the decimal point the same number of places to the right in the dividend. Add zeros as place holders. Example: Divide the whole numbers. Q add zero as place holder A place right makes a whole number carry, write 0 write a) 0. b) 0. c) d) e) f) g) h) i) j) k) l) page 8 Maths Mate /8 Skill Builder 8

53 9. [Fraction +, ] Skill 9. Adding fractions with the same denominator (). continues on page 0 Simplifying a fraction Hint: If the numbers are both even then you can start with dividing by. numerator (even) denominator (even) Divide both the numerator and the denominator by the same number. 8 Changing an improper fraction to a mixed number numerator IMPROPER FRACTION denominator Divide the numerator by the denominator. remainder Write the result as the whole number and the remainder over the denominator. Changing a mixed number to an improper fraction MIXED NUMBER whole number proper fraction Multiply the whole number by the denominator and then add the result to the numerator. + + Rewrite the total over the denominator. Add the numerators (top numbers of the fractions). Do not change the denominators. Simplify the resulting fraction and/or change it to a mixed number if necessary. Q. + A. + + Add the numerators (top numbers) only remainder + + a) Add the numerators (top numbers) only... b) +... c) +... d) +... e) f) page 9 Maths Mate /8 Skill Builder 9

54 Skill 9. Adding fractions with the same denominator (). g) Change to mixed number h) i) continued from page j) 9 + k) 0 + l) m) Simplify Change to mixed number p) n) q) o) r) s) 0 + t) 0 + u) v) w) + x) page 0 Maths Mate /8 Skill Builder 9

55 Skill 9. Subtracting fractions with the same denominator. Subtract the numerators (top numbers of the fractions). Do not change the denominators. Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Q. A Subtract the numerators (top numbers) only Simplify (top numbers) only 8 9 a) b) c) Subtract the numerators d) 9 Change to mixed number e) f) g)... Simplify h) 8... i)... j) 9 k) l) m) n) o) 9... page Maths Mate /8 Skill Builder 9

56 Skill 9. Adding mixed numbers with the same denominator (). Add the whole numbers first. Add the fractions. (see skill 9., page 9) Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Write the result as a mixed number. 0 Q. + A Simplify Add the numerators (top numbers) only 0 continues on page a) b) +... c) d) +... e) + 9 f) g) Simplify +... h) i) j) + k) + l) page Maths Mate /8 Skill Builder 9

57 Skill 9. Adding mixed numbers with the same denominator (). continued from page m) + n) + o) + + Change to mixed number p) + q) 0 + r) s) Simplify Change to mixed number t) u) v) + w) + x) y) + z) + zz) page Maths Mate /8 Skill Builder 9

58 Skill 9. Subtracting mixed numbers with the same denominator (). continues on page Change mixed numbers to improper fractions before subtracting. (see skill 9., page 9) Subtract the fractions. (see skill 9., page ) Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Q. 9 A Change to improper fractions Subtract the numerators (top numbers) only Change to mixed number Simplify a) b) c) Subtract the numerators (top numbers) only d) e) 9 f) g) Change to mixed number h) i) page Maths Mate /8 Skill Builder 9

59 Skill 9. Subtracting mixed numbers with the same denominator (). Subtract the whole numbers first. Subtract the fractions. (see skill 9., page ) Simplify the resulting fraction if necessary. (see skill 9., page 9) Hint: For subtractions you may need to convert to an equivalent fraction. Example: whole circle numerator denominator continued from page Q. A and? can not be subtracted from and 8 8 give a positive answer, so borrow a from the. 8 (see hint) Simplify. 8 8 j) 9 9 k) 8 8 l) Subtract the numerators (top numbers) only m) + n) o) page Maths Mate /8 Skill Builder 9

60 Skill 9. Subtracting a fraction or a mixed number from a whole number (). Write the whole number as an improper fraction with the same denominator as the mixed number. Change the mixed number to an improper fraction before subtracting. (see skill 9., page 9) Subtract the fractions. (see skill 9., page ) Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Q. A. can be written 8 as or Subtract the numerators (top numbers) only a) b) 9 9 Change to improper fractions Subtract the numerators (top numbers) only Change to mixed number... can be written as: and c) continues on page d) e) f) g) h) i) page Maths Mate /8 Skill Builder 9

61 Skill 9. Subtracting a fraction or a mixed number from a whole number (). Subtract the whole numbers first. Borrow from the whole number and write it as a fraction with the same denominator. Subtract the fractions. (see skill 9., page ) Q. A j) l) and m) n) o) k) p) q) 9 r) continued from page page Maths Mate /8 Skill Builder 9

62 Skill 9. Adding fractions with different denominators - one denominator divides evenly into the other denominator (). Lowest Common Multiple (LCM) of two numbers Write in ascending order some multiples of the smaller number first. Write in ascending order some multiples of the bigger number and stop when you find a multiple that appears in the first list Lowest Common Multiple (LCM). Hint: The lowest common multiple is the smallest number that the two numbers divide into. Examples: One number divides evenly into the other number LCM of and LCM of and stop The numbers have NO common factors other than 8 9 stop continues on page 9 Hint: LCM is the largest number. Hint: LCM is the product of the numbers. Equivalent Fractions same area shaded 8 different numerators and denominators same value (equivalent fractions) 8 Equivalent fractions have the same value. Equivalent fractions are formed by multiplying the numerator and denominator by the same number. Find the lowest common denominator of the fractions, which is the Lowest Common Multiple (LCM) of the denominators. In this case the LCM is the largest denominator. Change the fractions to equivalent fractions with the lowest common denominator. Add the fractions with the same denominators. (see skill 9., page 9) Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Hint: If unsure which is the LCM of the denominators, use their product as the common denominator. Examples: (LCM of and is, because divides evenly into ) OR (common denominator of and is ) page 8 Maths Mate /8 Skill Builder 9

63 continued from page 8 Skill 9. Adding fractions with different denominators - one denominator divides evenly into the other denominator (). LCM of 0 and Q. + A. + To give the second 0 0 is 0 fraction a denominator of 0, multiply both the + 0 numerator and denominator by Add the fractions. 8 0 Simplify. 9 Change to a mixed number. a) d) LCM of 9 and is b) e) LCM of 8 and is c) f) g) h) i) j) +... k) +... l) page 9 Maths Mate /8 Skill Builder 9

64 Skill 9. Adding fractions with different denominators - the denominators have no common factors other than (e.g. and ). Find the lowest common denominator of the fractions, which is the Lowest Common Multiple (LCM) of the denominators. In this case the LCM is the product of the denominators. (see skill 9., page 8) Change the fractions to equivalent fractions with the lowest common denominator. Add the fractions with the same denominators. (see skill 9., page 9) Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Q. + A LCM of and 8 is Multiply the numerator and denominator of the first fraction by 8. Multiply the numerator and denominator of the second fraction by. Add the fractions. a) + + LCM of and is b) c) d) e) f) g) +... h) +... i) page 0 Maths Mate /8 Skill Builder 9

65 Skill 9.8 Subtracting fractions with different denominators - one denominator divides evenly into the other denominator. Find the lowest common denominator of the fractions, which is the Lowest Common Multiple (LCM) of the denominators. In this case the LCM is the largest denominator. (see skill 9., page 8) Change the fractions to equivalent fractions with the lowest common denominator. Subtract the fractions with the same denominators. (see skill 9., page ) Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Hint: If unsure which is the LCM of the denominators, use their product as the common denominator. Q. A LCM of and 0 is 0 To give the first fraction a denominator of 0, multiply both the numerator and denominator by. Subtract the fractions. Simplify. a) d) LCM of and is b) 0 0 e) c) f) g) h) i) page Maths Mate /8 Skill Builder 9

66 Skill 9.9 Subtracting fractions with different denominators - the denominators have no common factors other than (e.g. and ). Find the lowest common denominator of the fractions, which is the Lowest Common Multiple (LCM) of the denominators. In this case the LCM is the product of the denominators. (see skill 9., page 8) Change the fractions to equivalent fractions with the lowest common denominator. Subtract the fractions with the same denominators. (see skill 9., page ) Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) Q. A. 0 LCM of and is Multiply the numerator and denominator of the first fraction by. Multiply the numerator and denominator of the second fraction by. Subtract the fractions. a) 9 9 d) LCM of and 9 is b) e) c) f) g) 9... h)... i) j) 8... k)... l) page Maths Mate /8 Skill Builder 9

67 0. [Fraction, ] Skill 0. Multiplying a fraction by a whole number (). Highest Common Factor (HCF) of two numbers Write all the factors of each number (the factors must divide exactly into the number). Find the largest number that appears on both lists. Hint: The Highest Common Factor is the largest number that divides evenly into both numbers. Examples: Identical numbers One number divides evenly into the other number HCF of and Hint: is the HCF of and because is the largest number that divides into and. continues on page HCF of and Hint: is the HCF of and because is the largest number that divides into and. Changing an improper fraction to a mixed number numerator IMPROPER FRACTION denominator Divide the numerator by the denominator. remainder Write the result as the whole number and the remainder over the denominator. Cross simplifying a fraction and a whole number Simplify the denominator of the fraction and the whole number. This means to divide them by the same number, usually by their Highest Common Factor. Cross out the denominator of the fraction and the whole number. Write the result of the division next to each crossed number. Multiply the top numbers together. remainder 0 0 Divide and 0 by 0 0 Multiply the numerator of the fraction by the whole number. Do not change the denominator. Simplify the resulting fraction and/or change it to a mixed number if necessary. EITHER Cross simplify where possible before multiplying. OR Simplify at the end. page Maths Mate /8 Skill Builder 0

68 Skill 0. Multiplying a fraction by a whole number (). Q. A. OR A. Divide and by Change to mixed number Simplify Multiply by + + continued from page a) 9 9 d) 8 Change to mixed number g) 8 8 j) m) Divide 8 and by... b) e) h) k) n) c) f) i) l) 0 o) page Maths Mate /8 Skill Builder 0

69 Skill 0. Finding a fraction of a quantity. Replace the word of with the multiplication symbol. Multiply the fraction by the whole number. (see skill 0., page ) Write the unit of measurement in the result. Hint: To find a fraction of a whole number divide that number by the denominator of the fraction, and then multiply the result by the numerator. Q. of $80 A. of $ $00 Divide 9 and 80 by 9 Add the $ sign OR A. To find of $80: $00 a) of ml... Divide and by... ml b) of 0 kg c) of $ d) of 0 L e) of 000 m f) of $ g) of 00 L... h) of 0 cm... i) of 0 m j) of 0 g 9... k) of 0 ml... l) 8 of 80 kg page Maths Mate /8 Skill Builder 0

70 Skill 0. Dividing a whole number by a fraction (). continues on page Copy the whole number and change divide by ( ) into times ( ). Invert the fraction. Multiply the whole number by the numerator of the fraction. Do not change the denominator. To simplify: EITHER Cross simplify where possible before multiplying. (see skill 0., page ) OR Simplify at the end. Q. A. 8 Change the sign to Invert fraction How many eighths are there in two wholes? There are eighths in two wholes a) b) c) Invert fraction 8... Divide and by d) 0 e) 9 f) g) Invert fraction h) i) page Maths Mate /8 Skill Builder 0

71 Skill 0. Dividing a whole number by a fraction (). j) m) 8 k) Invert fraction n) l)... o) 8 continued from page p) Divide and by... 9 Invert fraction... q) r) s) t) u) 0 v) w) 9 x) page Maths Mate /8 Skill Builder 0

72 Skill 0. Multiplying two fractions (). continues on page 9 Cross simplifying two fractions Simplify the numbers in the fractions diagonally (in a cross). This means to divide top and bottom numbers by the same number, usually by their Highest Common Factor. (see skill 0., page ) Cross out the numbers in the fractions diagonally (in a cross). Write the result of the division next to each crossed number. Multiply the top results together. Multiply the bottom results together. Divide and 9 by 8 9 Divide and 8 by Multiply the numerators of the fractions. Multiply the denominators of the fractions. To simplify: EITHER Cross simplify where possible before multiplying. Q. A Divide and 9 by Divide and by OR Simplify at the end. OR A. 9 9 Simplify a)... 8 b)... c) 8... d) 0 e) 9 f) g) h) i) page 8 Maths Mate /8 Skill Builder 0

73 Skill 0. Multiplying two fractions (). j)... Simplify... k) l) continued from page m) p) s) n) q) t) o) r) u) v) w) 0 x) y) 0 9 z) 8 zz) page 9 Maths Mate /8 Skill Builder 0

74 Skill 0. Dividing a fraction by a whole number (). Copy the fraction and write the whole number as an improper fraction with denominator. Change divide by ( ) into times ( ). Invert the second fraction. Multiply the fractions. (see skill 0., page 8) To simplify: EITHER Cross simplify where possible before dividing. continues on page OR Simplify at the end. Q. A. Invert second fraction What is one third divided into equal parts? This can also be thought of as one half of a third. of a) b) c) Divide and by d) e) f) page 0 Maths Mate /8 Skill Builder 0

75 Skill 0. Dividing a fraction by a whole number (). g) h) i) continued from page j) 8 k) 0 l) m) 9 n) o) p) q) r) page Maths Mate /8 Skill Builder 0

76 Skill 0. Dividing two fractions (). Copy the first fraction and change divide by ( ) into times ( ). Invert the second fraction. Multiply the fractions. (see skill 0., page 8) To simplify: EITHER Cross simplify where possible before multiplying. (see skill 0., page 8) continues on page OR Simplify at the end. Q. 9 A. 9 Invert second fraction OR A Divide 9 and by Simplify a) b) c) d) 8 e) 9 f) g) h) 8 i) page Maths Mate /8 Skill Builder 0

77 Skill 0. Dividing two fractions (). j) Divide 0 and by k) l) continued from page m) n) o) p) q) 8 r) s) 0 t) u) page Maths Mate /8 Skill Builder 0

78 page Maths Mate /8 Skill Builder 0

79 . [Percentages] Skill. Writing a number out of 00 as a percentage. Write the number followed by the percent symbol % Hint: Percentage means per hundred or of each hundred. Q. Write as a percentage: 8 out of 00. A. 8 out of 00 8% a) Write as a percentage: 0 out of 00. 0% b) Write as a percentage: out of 00. c) Write as a percentage: out of 00. d) Write as a percentage: out of 00. e) Write as a percentage: out of 00. f) Write as a percentage: 9 out of 00. g) Write as a percentage: out of 00. h) Write as a percentage: out of 00. i) Write as a percentage: out of 00. j) Write as a percentage: out of 00. k) Write as a percentage: 9 out of 00. l) Write as a percentage: 9 out of 00. m) Write as a percentage: 8 out of 00. n) Write as a percentage: 9 out of 00. page Maths Mate /8 Skill Builder

80 Skill. Finding the remaining percentage. Subtract the given percentages from 00%, to find the remaining percentage. Q. According to a projection for 00, 9% of the U.S. population will be aged between 0-9 and % between 0-9. What percentage of the population will be aged 0 or more? a) Approximately 9% of the athletes at the 000 Sydney Olympics were male. What percentage of the athletes were female? 00% 9%... c) The green-yellow 8-carat gold is % gold and the rest is silver. What percentage is silver?... e) If 8% of the Supreme Court justices are male, what percentage are females?... g) In Mali % of people earn less than $ a day. What percentage of people earn more than $ a day?... % A. 00% 9% % 00% % % b) School is approximately 0% of the calendar year in the Russian Federation. What percentage do holidays account for? 00% 0%... d) If 89% of the West Point military academy graduates are male, what percentage are females?... f) If the cucumber is 9% water, what percentage do the other components equal?... h) If.% of the adult teeth are incisors and canines, what percentage is formed by molars and pre-molars?... i) Approximately 0.% of the world population lives in Asia and.% lives in North and South America. What percentage of the population lives in the rest of the world? 00% 0.%.%... k) If England occupies % and Scotland occupies % of Great Britain (the main island of the United Kingdom), what percentage is occupied by Wales?... j) Approximately.% of the world population is aged between 0 and years and.% between and years. What percentage of the population is aged years and over?... l) At the 008 Beijing Olympics, 9% of the medals won by Germany were gold, and % were silver. What percentage of the medals were bronze?... page Maths Mate /8 Skill Builder

81 Skill. Finding a percentage of multiples of 00 (). Change the percentage to a fraction out of 00. Example: 0% 0 00 Rewrite the question as a multiplication (change of to ). Change the whole number to a fraction over. Example: Cross simplify the fractions before multiplying. (see skill 0., page 8) continues on page 8 OR First find 0%. Then multiply by the amount needed to make the required percentage, i.e. multiply by to get 0%. Hint: To find 0% divide by 0 0 % half of 0% 0% divide by % divide by 0% divide by Q. 0% of $.00 A. 0% of $.00 0% of cents $.0 Convert $ to cents Simplify: 00 OR A cents 0 0 cents $.0 Find 0% Multiply by to get 0% a) % of Divide by 00 b) 8% of 00 c) 9% of 00 d) 9% of 00 e) % of 00 f) 0% of 00 g) % of j) 0% of h) 0% of 00 k) % of Divide 00 by 0 i) 0% of 00 l) 0% of Find 0% first... page Maths Mate /8 Skill Builder

82 Skill. Finding a percentage of multiples of 00 (). m) % of n) % of % is half of 0%... 0 Find 0% o) % of continued from page p) 0% of q) 0% of r) 0% of s) 80% of t) 90% of u) % of v) 0% of $.00 w) 0% of $.00 x) % of $ $ $ $ y) % of $.00 z) 0% of $.0 zz) 0% of $ page 8 Maths Mate /8 Skill Builder

83 Skill. Finding a percentage of any number (). Change the percentage to a fraction out of 00. Example: 0% 0 00 Rewrite the question as a multiplication (change of to ). Change the whole number to a fraction over. Example: Cross simplify the fractions before multiplying. (see skill 0., page 8) OR First find 0%. Then multiply by the amount needed to make the required percentage, i.e. multiply by to get 0%. Q. % of 0 A. of 0 % Simplify: continues on page 0 Hint: To find % divide by % divide by 8 8 % divide by % divide by multiply by Substitute % with Change of to 0 Change 0 to 0 Multiply by a) 0% of Simplify: 0, twice 0 b) 0% of Multiply by to get 0% Find 0% first c) 80% of 0... d) 0% of g) 0% of 0... j) % of 0 0% % is half of 0%... % % + Find 0% first e) 0% of 0... h) % of 0... k) % of 80 0% % % f) 0% of 0... i) % of 0... l) % of 0 0% % % page 9 Maths Mate /8 Skill Builder

84 Skill. Finding a percentage of any number (). m) % of Divide by Simplify: n) % of 0... o) % of 0... continued from page 9 p) % of Simplify: q) % of 0... r) 8% of s) % of t) % of 0... u) % of 0... v).% of Simplify: 8... w).% of x).% of 0... y) % of 0 0 Simplify:... z) % of A) % of 0... B) % of C) % of 0... D) % of 0... page 0 Maths Mate /8 Skill Builder

85 Skill. Working with percentages greater than 00%. Change the percentage to a fraction out of 00. Example: 0% 0 00 Rewrite the question as a multiplication (change of to ). Change the whole number to a fraction over. Example: Cross simplify the fractions before multiplying. (see skill 0., page 8) OR First find 00% or other multiples of 00%. Then find the remaining percentage. Add the results. Q. 0% of 0 A. 0% of 0 a) 00% of Simplify: 0, twice b) 00% of 0... Simplify: 0, twice Hint: To find 0% 0 divide by 0 0% divide by c) 00% of 0 00% multiply by 00% multiply by OR A. 00% of 0 is 0 So 00% is triple that, or 0 0% of 0 is 0 So 0% of 0 is d) 0% of 80 00% of Find 0% 0% of Add the results 80 + g) 0% of j) 0% of 0 Find 00% e) 0% of h) 0% of 0... k) 0% of 0 f) 0% of i) 0% of l) 0% of page Maths Mate /8 Skill Builder

86 Skill. Working with percentages to find discounts and sale prices. Calculate the percentage of the given amount. (see skill., page and skill., page 9) To find the sale price if a discount is applied: Subtract this result from the given amount. To find the total amount if a sales tax is applied: Add this result to the given amount. Q. If a sales tax rate of % is applied on a purchase of $00, what is the total amount that must be paid? A. Sales tax: % of Total: 00 + $ a) If a $0 T-shirt is reduced by %, what is the discount? discount: % of c) If a $000 laptop is reduced by 0%, what is the sale price? discount: 0% of Divide by sale price: $000 $00... e) If a sales tax rate of % is applied on a purchase of $00, what is the total amount that must be paid? sales tax: % of total: $ g) If a sales tax rate of % is applied on a restaurant bill of $80, what is the total amount that must be paid? sales tax: total: $ b) If a $0 bike is reduced by %, what is the discount? $.0 $ d) If a $00 dress is discounted by 0%, what is the sale price? discount:... $ sale price: $ $ $ $ $ page Maths Mate /8 Skill Builder f) If a sales tax rate of % is applied on a purchase of $0, what is the total amount that must be paid? sales tax: total:... h) If a sales tax rate of % is applied on a purchase of $0, what is the total amount that must be paid? sales tax:... discount: total:...

87 Skill. Writing one number as a percentage of another number. Form a fraction using the two numbers. EITHER Multiply this fraction by 00%: fraction fraction 00% Hint: 00% equals and does not change the value of the fraction. Simplify the resulting fraction and/or change it to a mixed number if necessary. (see skill 9., page 9) OR Find an equivalent fraction with the denominator 00, by multiplying or dividing both the numerator and denominator by the same number. Write this fraction as a percentage. (see skill.9, page 8) Hint: Both numbers must represent the same unit of measurement. Q. Write as a percentage: out of 0. A. out of 0 00% 0 00 % 0 % Simplify: 0 OR A. out of % a) Write as a percentage: out of Simplify: Find equivalent fraction % c) At the 00 Delhi Commonwealth Games, out of the medals won by Samoa were gold. What percentage is this? b) In Australia 88 out of every 00 people live in an urban area. What percentage is this? d) For every 0 Skype calls made, 8 calls are video to video. What percentage is this? e) A male lion weighs kg. It eats 9 kg of food each day. What percentage of its own weight does a lion eat each day? f) Of the billion cattle in the world, 00 million are in India. What percentage of the world s cattle are in India? page Maths Mate /8 Skill Builder

88 Skill.8 Calculating profit or loss as a percentage of the cost price. Calculate the profit or the loss, as the difference between the selling and the cost price. Express the profit or the loss as a percentage of the cost price. (see skill., page ) Q. A shop buys jackets in bulk for $0 each, then sells them for $9 each. Calculate the profit on each jacket as a percentage of the cost price. a) Lorien lost $0 on a ring costing $00. What was her loss as a percentage of the cost price? loss: $0... loss out of cost: $0 out of $ % 0 % 0%... c) John made $0 profit on a tool box costing $00. What was his profit as a percentage of the cost price? profit:... profit out of cost: e) Serena bought a car for $000. If she later sold it for $00, find the loss as a percentage of the cost price g) Amelia bought a table for $00. If she later sold it for $0, find the loss as a percentage of the cost price. A. profit: $9 $0 $ profit out of cost price: $ out of $ % 0 % 90% b) The Cycle Centre made $0 profit on a bicycle costing $0. What was the profit as a percentage of the cost price? profit:... profit out of cost: d) Jason lost $ on a book costing $0. What was his loss as a percentage of the cost price? loss:... loss out of cost: f) A shop buys school uniforms in bulk for $ each, then sells them for $00 each. Find the profit as a percentage of the cost price h) A painting was bought for $000. If it was later sold for $00, find the profit as a percentage of the cost price page Maths Mate /8 Skill Builder

89 . [Decimals / Fractions / Percentages] Skill. Illustrating fractions and percentages. To recognise a shaded fraction of a shape: Count the total number of equal parts in which the shape is divided. Use this number as the denominator of the fraction. Count the number of shaded parts. Use this number as the numerator of the fraction. Simplify the resulting fraction. (see skill 9., page 9) To recognise a shaded percentage of a shape: Count the shaded parts. Relate the amount shaded to out of 00, by dividing the number of total parts into 00. Hints: A percentage is a fraction out of 00. Compare to common fractions, like one half equals 0%, one quarter equals % or one tenth equals 0%. Q. What percentage of the shape is shaded? A. out of 0 parts 0 out of 00 parts 0% out of 0 parts are shaded. There are lots of 0 in 00 so multiply to get the percentage shaded. a) What fraction of the shape is shaded? b) What fraction of the shape is shaded? out of parts c) What fraction of the shape is shaded? d) What fraction of the shape is shaded?... e) What percentage of the shape is shaded?... f) What percentage of the shape is shaded?... g) What percentage of the shape is shaded?... h) What percentage of the shape is shaded? page Maths Mate /8 Skill Builder

90 Skill. Simplifying fractions (). continues on page Highest Common Factor (HCF) of two numbers Write all the factors of each number (the factors must divide exactly into the number). Find the largest number that appears on both lists. Hint: The Highest Common Factor is the largest number that divides evenly in both numbers. Examples: Identical numbers Hint: is the HCF of and because is the largest number that divides into and. GCF of and One number divides evenly into the other number HCF of and Hint: is the HCF of and because is the largest number that divides into and. Numbers have other common factors 0 Hint: is the HCF of and 0 because is the largest number that divides into and 0. Divide both the numerator and the denominator by their Highest Common Factor (HCF). OR Divide both the numerator and the denominator by any common factor. Divide again by another common factor, until the common factor of the numerator and the denominator is. Hints: The fraction is in simplest form when it cannot be simplified. If the numbers are both even then you can start with dividing by. 0 HCF of and 0 0 Q. Simplify A HCF of 0 and 0 is 0 Divide by 0 OR A Divide by Divide by a) Simplify HCF of and 0 is b) Simplify... c) Simplify... d) Simplify 9... e) Simplify 8... f) Simplify... page Maths Mate /8 Skill Builder

91 Skill. Simplifying fractions (). 9 g) Simplify h) Simplify i) Simplify continued from page... j) Simplify 8... k) Simplify... l) Simplify 0... m) Simplify... n) Simplify 0... o) Simplify p) Simplify... q) Simplify 8... r) Simplify... s) Simplify 9... t) Simplify u) Simplify 0... v) Simplify 9... w) Simplify x) Simplify... y) Simplify 0... z) Simplify zz) Simplify 0... page Maths Mate /8 Skill Builder

92 Skill. Finding equivalent fractions. Check if you need to multiply or divide the numerator or denominator of the first fraction to reach the numerator or denominator of the second fraction. Do the same operation to the top or the bottom of the fraction. Example: Q. Complete the equivalent fractions:? 8 90 A. and So and are equivalent fractions. 8 8? 0 8? a) Complete the equivalent fractions:???... b) Complete the equivalent fractions:? c) Complete the equivalent fractions:... d) Complete the equivalent fractions: e) Complete the equivalent fractions: f) Complete the equivalent fractions: g) Complete the equivalent fractions: h) Complete the equivalent fractions: i) Complete the equivalent fractions: and and and... page 8 Maths Mate /8 Skill Builder

93 Skill. Writing a decimal number as a percentage. Multiply the decimal number by 00, by moving the decimal point two places to the right. Add the percentage sign. Hint: Zeros can be added at the end of any decimal number: Q. Write 0.0 as a percentage. A % % zeros, places to the right a) Write 0. as a percentage. Add a zero %... c) Write 0. as a percentage.... 0% b) Write 0. as a percentage.... d) Write 0.9 as a percentage.... e) Write 0. as a percentage.... f) Write 0. as a percentage.... g) Write 0. as a percentage.... h) Write 0.8 as a percentage.... i) Write 0.9 as a percentage.... j) Write 0. as a percentage.... k) Write 0.0 as a percentage.... l) Write 0.0 as a percentage.... m) Write 0.0 as a percentage.... n) Write 0.8 as a percentage.... o) Write 0. as a percentage.... p) Write 0. as a percentage.... q) Write 0. as a percentage.... r) Write 0. as a percentage.... page 9 Maths Mate /8 Skill Builder

94 Skill. Writing a percentage as a decimal number. Write the percentage as a fraction out of 00. Divide the numerator of the fraction by 00, by moving the decimal point two places to the left. Hints: Fractions are divisions. There is a decimal point and zeros which are not written, at the end of any whole number:.00 Zeros can be used as place holders before any whole number: Q. Change 8.% to a decimal. A. 8.% zeros, places to the left a) Change % to a decimal. % c) Change 88% to a decimal.... e) Change 0% to a decimal.... g) Change 0.% to a decimal b) Change % to a decimal. %... d) Change % to a decimal.... f) Change 0% to a decimal.... h) Change.8% to a decimal.... i) In Mali % of people earn less than $ each day. Write this percentage as a decimal.... k) The percentage of Americans between and who play video games is 9%. Write this percentage as a decimal.... m) China has approximately 0% of the world s population. Write this percentage as a decimal.... j) In Oct 00 the unemployment figure for Australia was.%. Write this percentage as a decimal.... l) The Sun accounts for 99% of the mass of the solar system. Write this percentage as a decimal.... n) On average Australians spend.8% of their day on facebook. Write this percentage as a decimal.... page 80 Maths Mate /8 Skill Builder

95 Skill. Writing a decimal number as a fraction in simplest form. Write the decimal number as the numerator of the fraction. Ignore any zeros at the start the number. Use the place value of the last digit of the decimal number to determine the size of the denominator. Example: units tenths hundredths 0 0 hundredths Write the as the numerator in hundredths place, denominator Write the fraction in simplest form. This means to divide both the numerator and the denominator by the same number. Hint: For the denominator, write followed by a zero for each digit after the decimal point. Example: Q. Write 0. as a fraction in simplest form. A Write as the numerator zero for decimal place Simplify: a) Write 0.9 as a fraction. 0.9 nine tenths b) Write 0. as a fraction. 0. eleven hundredths... c) Write 0. as a fraction.... d) Write 0. as a fraction.... e) Write 0.0 as a fraction in simplest form.... f) Write 0.0 as a fraction in simplest form.... g) Write 0. as a fraction in simplest form.... h) Write 0.8 as a fraction in simplest form.... i) Write 0. as a fraction in simplest form.... j) Write 0.8 as a fraction in simplest form.... page 8 Maths Mate /8 Skill Builder

96 Skill. Writing a fraction as a terminating decimal. When the denominator is a power of 0: Divide the numerator by the power of 0 by moving the decimal point to the left. Example: Hints: Fractions are just divisions. There is a decimal point and zeros which are not written, at the end of any whole number:.00 Zeros can be used as place holders before any whole number: The number of zeros in the denominator shows the number of digits after the decimal point. Q. Change to a decimal. A. zeros, places to the left When the denominator is not a power of 0: EITHER Multiply both the numerator and denominator by the same number to make the denominator a power of 0. (e.g. 0, 00 or 000). Example: power of 0 OR Divide the numerator by the denominator. Example: OR Make denominator a power of 0 zeros, places to the left A a) Change to a decimal d) Change to a decimal. b) Change to a decimal e) Change to a decimal. 9 c) Change to a decimal f) Change to a decimal g) In 008 a quarter of the Australian wheat exports went to Indonesia. Write this fraction as a decimal. h) Approximately 9 out 0 Nigerians attend church regularly. Write this fraction as a decimal. i) People have the smelling ability of one-twentieth of that of a dog. Write this fraction as a decimal page 8 Maths Mate /8 Skill Builder

97 Skill.8 Writing a percentage as a fraction in simplest form. Write the percentage as a fraction with the denominator of 00. Simplify the fraction by dividing both the numerator and the denominator by the same number. Hints: Percent means per hundred or out of a hundred. A percentage is another way of writing a fraction out of one hundred. Example: % is said percent and means out of 00. Q. USA accounts for % of the European Union exports. Write this percentage as a fraction in simplest form. A. % 00 Simplify: a) Write % as a fraction. % b) Write 9% as a fraction.... c) Write % as a fraction in simplest form. % Simplify: e) Write % as a fraction in simplest form.... g) The common metal for medals is 8% copper. Write this percentage as a fraction in simplest form.... i) India is home to 0% of the world s poor. Write this percentage as a fraction in simplest form.... k) The average person s left hand does % of the typing. Write this percentage as a fraction in simplest form.... d) Write 0% as a fraction in simplest form.... f) Write % as a fraction in simplest form.... h) About percent of of all New Zealand males aged between 8 and served in WWII. Write this percentage as a fraction in simplest form.... j) In Belgium, % of government ministers are female. Write this percentage as a fraction in simplest form.... l) The pupil of the eye expands up to % when a person looks at something pleasing. Write this percentage as a fraction in simplest form.... page 8 Maths Mate /8 Skill Builder

98 Skill.9 Writing a fraction as a percentage. EITHER Find the equivalent fraction which has a denominator of 00. The numerator of this fraction is the equivalent percentage. Example: P 00 0 P% % OR 00 Multiply the fraction by % Example: 00 % 0 0 0% Simplify: 0 Fraction 00 % Percentage Q. Change to a 0 percentage. A % OR A. 0 % % 00 0 % Simplify: 0 a) Change to a 0 percentage. 9 b) Change to a 0 percentage. c) Change to a percentage % d) Change to a 00 percentage. e) Change to a percentage. f) Change to a percentage g) Change to a percentage. h) Change to a percentage. i) Change to a percentage j) Change to a 0 percentage. k) Change to a 00 percentage. l) Change to a percentage page 8 Maths Mate /8 Skill Builder

99 Skill.00 Ordering decimal numbers. Line up the decimal numbers at their decimal points. Compare digits in the same places, starting from the left, until you find the smallest digit. Hint: The number with the smallest digit will be the smallest number. Look for the second smallest number. Continue in this way until you find the largest number. Q. Place in ascending order: 0., 0.0, 0.0, 0.0 rd smallest st nd largest th units U. T HTh 0 0 tenths hundredths thousandths A. 0.0, 0.0, 0., 0.0 Find the smallest digits. Work from left to right. Units: all 0 Tenths: 0 < < so 0.0 is the smallest 0.0 is the largest either 0.0 or 0. is the nd smallest Hundredths: 0 < so 0.0 is the nd smallest 0. is the rd smallest a) Place in order from largest to smallest: 0.09, 0.9, 0.09, 0.09 U. T HTh the largest number the smallest number b) Place in ascending order: 0.0, 0., 0.0, 0.0 U. T HTh c) Place in ascending order: 0.08, 0.08, 0.08, 0.8 d) Place in descending order: 0., 0., 0.0, 0. e) Place in ascending order: 0.80, 0.0, 0.8, 0.08, 0.08 f) Place in order from smallest to largest: 0., 0.0, 0., 0., 0.0 g) Place in order from largest to smallest: 0.9, 0.09, 0.09, 0.0, 0. h) Place in ascending order: 0., 0.0, 0., 0.0, 0.0 page 8 Maths Mate /8 Skill Builder

100 Skill. Comparing and ordering fractions. Find the least common denominator of the fractions, which is the Lowest Common Multiple (LCM) of the denominators. Change the fractions to equivalent fractions with the lowest common denominator. Arrange the fractions in order of the numerators (the smallest fraction has the smallest numerator and so on). smallest numerator smallest fraction < < same denominator Hint: If unsure which is the LCM of the denominators, use their product as the common denominator. When the smaller denominators divide evenly into the biggest denominator, this biggest number becomes the common denominator. Q. Place in ascending order: A.,,,,,, LCM of, and is < 8 < 0, so < < or < < a) Which fraction has greater value? or LCM of 8 and is < b) Which fraction has greater value? or 8... c) Which fraction has greater value? d) Which fraction has greater value? or or e) Place in order from smallest to largest: LCM of, 8 and,, is f) Place in order from largest to smallest: 9,, page 8 Maths Mate /8 Skill Builder

101 Skill. Converting between decimals, fractions and percentages (). continues on page 88 Convert between decimals, fractions and percentages. (see skills. to.9, pages 9 to 8) numerator by denominator Decimals 00% make denominator 00 and simplify Percentages Fractions 00% 00 % For denominator put followed by one zero for each digit after the decimal point and simplify Q. Complete the table: Decimal Fraction Percentage 0 A % 0 0 Simplify: 0 Make denominator a power of 0 % % Decimal Fraction Percentage 0. 0 % Decimal Percentage a) Complete the table: Decimal Fraction Percentage % Simplify: % %... b) Complete the table: Decimal Fraction Percentage % c) Complete the table: Decimal Fraction Percentage 0. d) Complete the table: Decimal Fraction Percentage page 8 Maths Mate /8 Skill Builder

102 Skill. Converting between decimals, fractions and percentages (). e) Complete the table: Decimal Fraction Percentage f) Complete the table: continued from page 8 Decimal Fraction Percentage 0.0 0% g) Complete the table: Decimal Fraction Percentage 0. h) Complete the table: Decimal Fraction Percentage i) Complete the table: Decimal Fraction Percentage 0. j) Complete the table: Decimal Fraction Percentage % k) Complete the table: Decimal Fraction Percentage 90% l) Complete the table: Decimal Fraction Percentage page 88 Maths Mate /8 Skill Builder

103 Skill. Comparing decimals, fractions and percentages (). Convert the decimals, fractions and percentages to the same form, by writing all as decimals, or as fractions, or as percentages. (see skill., page 8) Compare the decimals, or the fractions, or the percentages. Hint: The most convenient form is the decimal form. Write the fractions and percentages as decimals. Q. Which is greater? A. or 0% % Write the fraction as a decimal Make denominator a power of 0 Write the percentage as a decimal 0. is greater than 0., so 0% > 0% is greater. continues on page 90 Fraction Percentage a) Which is greater? 0.0 or 0% 0 0% > % b) Which is greater? 0% or c) Which is greater? d) Which is greater? 9 or 9% or % e) Which is greater? f) Which is greater? or % or % g) Which is greater? h) Which is greater? 0. or 0. or page 89 Maths Mate /8 Skill Builder

104 Skill. Comparing decimals, fractions and percentages (). i) Which is greater? 0. or.% j) Which is greater? 0. or % continued from page k) Which is greater? l) Which is greater? or 0% or % m) Which is greater? n) Which is greater? 8 or 8% or 0% o) Which is greater? p) Which is greater? 0. or 0.9 or q) Which is greater? or %... r) Which is greater? or % s) Which is greater? 0. or 0... t) Which is greater? 0.0 or page 90 Maths Mate /8 Skill Builder

105 . [Integers] Skill. Comparing and ordering integers (). continues on page 9 Use a number line. Hint: Numbers decrease as you move to the left or down and increase as you move to the right or up. A negative number is always smaller than a positive number negative numbers positive numbers C An altitude is lower when further down, below sea level (BSL) and higher when further up, above sea level (ASL). Above Sea Level (ASL) 00 m 00 m 00 m 800 m 800 m 00 m 00 m 00 m Sea Level 0 Below Sea Level (BSL) Temperatures below zero are lower than temperatures above zero. 0 C C 0 C - C -0 C - C Q. Who won the 00 Women s British Open Golf Tournament? [Hint: In golf the lowest score wins.] A ) + K. Webb B ) 0 K. Hull C ) Y. Tseng A. C C B Find the lowest score to determine the winner. A a) Which of Saturn s moons has the highest temperature? A ) 0ºC Enceladus B ) 00ºC Mimas C ) 8ºC Tethys colder AB C C hotter C b) Which temperature for oxygen is higher? A ) 8ºC boiling point B ) 8ºC melting point C c) Who won the 00 British Open Golf Tournament? [Hint: In golf the lowest score wins.] A ) L. Oosthuizen B ) + P. Senior C ) R. Allenby d) Which body of water is at the lowest altitude? A ) 8 m Caspian Sea B ) 08 m Dead Sea C ) m Lake Eyre m page 9 Maths Mate /8 Skill Builder

106 Skill. Comparing and ordering integers (). e) Which location has the lowest altitude? A ) m above sea level Amsterdam (Netherlands) B ) m below sea level Qattara Depression (Egypt) C ) 0 m above sea level Machu Picchu (Peru) f) Which location has the highest altitude? A ) 0 m below sea level Laguna Salada (Mexico) B ) m below sea level Lammefjord (Denmark) C ) 9 m above sea level Vatican City (Italy) continued from page 9 m m g) Which location recorded the lowest temperature? A ).ºC Kabul B ) +.ºC Christmas Island C ).ºC La Paz h) Which continent has the lowest recorded temperature? A ) ºC North America B ) ºC Australia C ) ºC Europe C C i) Arrange in ascending order:,,,, j) Arrange in order from largest to smallest: 0, 8, 9,, 0,,,, k) Arrange in descending order: 0, 8,, 8, l) Arrange in order from smallest to largest:,, 0,, m) Arrange in order from coldest to warmest: ºC, ºC, ºC, ºC n) Arrange in order from warmest to coldest: ºC, ºC, ºC, ºC 0 C 0 C page 9 Maths Mate /8 Skill Builder

107 Skill. Comparing integers using less than and greater than. Use a number line. Hint: A negative number is always smaller than a positive number. The larger the negative number the lesser the value, e.g. 9 is less than (<) The smaller the negative number the greater the value, e.g. is greater than (>) Q. Use < or > to make a true statement. A. > 8 is greater than less than < 0 > greater than a) Use < or > to make a true statement. b) Use < or > to make a true statement. a negative number is less than a positive number < c) Use < or > to make a true statement. d) Use < or > to make a true statement e) Use < or > to make a true statement. f) Use < or > to make a true statement g) Use < or > to make a true statement. h) Use < or > to make a true statement i) Use < or > to make a true statement. j) Use < or > to make a true statement k) Use < or > to make a true statement. l) Use < or > to make a true statement page 9 Maths Mate /8 Skill Builder

108 Skill. Modelling integer subtraction on a number line. Determine the value of each mark on the number line. Count the number of spaces between the integers using the number line. Hint: Use short cuts such as: counting to zero, counting by tens. Q. How many units between and? A Start a) How much cooler is it in Geneva than Mombasa? -0 C -0 C Geneva 0 C +... Count to zero first c) How many units between and? 0 C 0 C 0 C Mombasa 0 C ºC b) If Karrie Webb scores a triple bogie and Greg Norman scores an eagle, what is the difference between their scores? albatross eagle d) How many units between and? 0 0 e) How many units between 9 and? f) How many units between and? g) What is the difference between the highest and the lowest temperatures recorded in Dunedin, New Zealand? h) What is the difference between the highest and the lowest temperatures recorded in Rome, Italy? -0 C -0 C 0 C 0 C 0 C 0 C 0 C -0 C -0 C 0 C 0 C 0 C 0 C 0 C Low... High low high ºC ºC... i) What is the time difference in hours between Denver and Cape Town? j) What is the time difference in hours between Karachi and New York? LA Denver Chicago New York Greenland Sao Paulo Azores London Paris Cape Town Moscow Tehran Karachi Calcutta Jakarta LA Denver Chicago New York Greenland Sao Paulo Azores London Paris Cape Town Moscow Tehran Karachi Calcutta Jakarta birdie par bogie double bogie triple bogie quadruple bogie h GMT h GMT page 9 Maths Mate /8 Skill Builder

109 Skill. Finding the difference between a positive and a negative integer. Visualise the position of the values on a number line. Translate the words to number sentences. Add the numbers ignoring their signs. Hint: Taking away negative is the same as adding positive. 0 ( ) + Q. In Vienna (Austria) the highest recorded temperature is 9 C and the lowest is C. What is the temperature difference? A. 9 ( ) 9 + C Instead of subtracting negative, add positive to temperature difference a) The Bay of Fundy, Canada has a high tide of 8. m and a low tide of 8. m. What is the tidal range for the Bay of Fundy? 8. ( 8.) m c) Sparrow Hills station is the highest station in the Russian metro rail system with an altitude of 0 m above sea level. Park Pobedy is the lowest station at 90 m below sea level. What is their height difference? e) In Luxembourg the highest recorded temperature is 8 C and the lowest is C. What is the temperature difference? m tidal range (m) m b) The lowest point in Japan is Lake Hachirogata at m and the highest point is Mt Fujiyama at m. What is the height difference? d) In Reykjavik (Iceland) the highest recorded temperature is C and the lowest is C. What is the temperature difference? m C f) In Shanghai (China) the highest recorded temperature is 0 C and the lowest is C. What is the temperature difference? C g) The lowest point on the African continent is m at Lake Assal and the highest is 89 m at Mt Kilimanjaro. What is the height difference? C h) The highest point in Europe is m at Mt Elbrus and the lowest is m in the Caspian Sea. What is the height difference in Europe? m m page 9 Maths Mate /8 Skill Builder

110 Skill. Modelling integer addition on a number line. Start at the given point on the number line. Count up or down the number of spaces as directed. Q. From homewares Marion rides the elevator down levels and up levels. At what level is Marion now? ladies homewares books/toys mens & childrens cosmetics/shoes 0 food court carpark (A) carpark (B) carpark (C) A. down levels (add ) up levels (add +) Cosmetics/shoes Start down up homewares cosmetics/shoes 0 a) From level Hutch rides the elevator up levels and down 8. At what level is Hutch now? up levels (add +)... down 8 levels (add 8)... down 8 up Start penthouse 0 ground carpark basement basement b) From the carpark Kwong rides the elevator down level and up levels. At what level is Kwong now? penthouse 0 ground carpark basement c) A nurse starts in cardiac ward, goes down levels and then up levels. Where does she finish? cardiac surgical orthopaedics oncology gyneacology 0 emergency cafeteria carpark basement d) A termite entered his tower via the central vent and went to the main nest. How far did the termite travel? All distances in metres central vent side vent 0 ground level main nest food storage queen s cell m e) From carpark (A) Todd rides the elevator down levels and up levels. At what level is Todd now?... ladies homewares books/toys mens & childrens cosmetics/shoes 0 food court carpark (A) carpark (B) carpark (C) f) A snorkeller at the reef surfaces for lunch on the pier and then goes back to the reef. How far does he travel? All distances in metres pier platform 0 sea level reef m page 9 Maths Mate /8 Skill Builder

111 Skill. Solving word problems involving two or more integers. Start at the given point. Work in the given order. Visualise the position of the values on a number line. Hint: Positive words: up, above, over, forward, advance, gained, earned, later Negative words: down, below, under, backward, retreat, lost, owed, earlier Q. During a football game the ball advanced m, retreated m and then advanced m. Where did the ball finish in relation to its starting point? A. Start: m 0 start advance retreat advance a) If Pip had $00 and spent $90, what is her bank balance? b) Harry owed $0. If he earned $0, how much does Harry now have? $0... spent $90 $0 c) Chan owes $0. If he earned $80, what is Chan s bank balance? 0 $00 d) Carbon dioxide boils at 8ºC. At ºC below this, carbon dioxide solidifies. At what temperature does carbon dioxide solidify? ºC e) The Persians destroyed the original Acropolis in 80 BC. Pericles rebuilt it years later. What year was that?... f) Tutenkhamun reigned for 9 years up until BC. What year did Tutenkhamun come to the throne?... g) Oxygen boils at 8ºC. At ºC below this, oxygen solidifies. What is the temperature of solid oxygen? h) Helium boils at 9ºC. At ºC below this, helium solidifies. At what temperature does helium solidify?... ºC... ºC i) You bought $000 worth of stock. After the first year you lost $80, but after the second year you gained $0. What is the current value of your stock? j) A bear weighs kg. During hibernation it loses 0 kg. After hibernation it gains kg. What does the bear weigh now? kg page 9 Maths Mate /8 Skill Builder

112 Skill. Adding integers. Hint: Every number has a sign attached to it, so if there is no sign, the number is positive. These signs should not be confused with the operations of addition and subtraction. Using a number line: Start at 0. When the number is + move that many to the right. When the number is move that many to the left. Using rules: If the numbers to be added have the Rule : same sign - add the numbers ignoring their signs and keep that sign. (N.B. Brackets added to help) (+) + (+) +( + ) + or simply + ( ) + ( 8) ( + 8) + 8 Rule : different signs - subtract the numbers ignoring their signs and keep the sign of the larger number. ( 9) + (+) (9 ) or simply 9 + ( ) + (+) +( ) + + Q. + A. + +( ) Starting at 0 go units to the left. From this point, move units right. You stop at positive. Negative Integers Zero Positive Integers a) + ( ) b) + c) 8 + ( + )... both negative same signs, add keep d) 8 + ( ) e) + ( ) f) + ( ) g) + h) 9 + ( ) i) + ( ) j) 8 + k) + ( ) l) + ( ) page 98 Maths Mate /8 Skill Builder

113 Skill.8 Subtracting integers. Hint: Every number has a sign attached to it, so if there is no sign, the number is positive. These signs should not be confused with the operations of addition and subtraction. Using a number line: Start at 0. When the number is + move that many to the right. When the number is move that many to the left. Using rules: Consider subtracting an integer as adding its opposite. Example: 0 (+) Follow the addition rules. (see skill., page 98) Examples: (+) ( ) (+) + (+) +( + ) + or simply + ( ) (+8) ( ) + ( 8) ( + 8) + 8 ( 9) ( ) ( 9) + (+) (9 ) or simply 9 + ( ) ( ) ( ) + (+) +( ) + + Q. start at, move backward Negative Integers A. + ( ) ( + ) 9 Negative take away positive is the same as negative plus negative. OR Using a number line: Starting at 0 go units to the left. From this point, move units left. You stop at negative 9. Positive Integers Zero a) + + ( )... ( )... d) subtract means add different signs, subtract b) e) 9 c) f) ( ) g) ( ) h) 8 ( ) i) ( ) j) k) 9 ( ) l) ( ) page 99 Maths Mate /8 Skill Builder

114 Skill.9 Multiplying integers. Multiply the integers ignoring the signs. Determine the sign of the result. If the numbers to be multiplied have the: same sign: positive result Example: 9 ( ) + different sign: + + negative result Example: 9 ( ) +9 ( ) + Q. ( ) A. ( ) Same signs, both negative positive result a) b) c) 8 d) ( ) e) ( 9) f) 8 ( 8) g) 8 ( ) h) 9 i) ( ) j) ( 8) k) l) m) ( 9) n) ( ) o) ( ) p) ( ) q) ( 9) r) 8 ( ) s) ( ) t) u) 9 9 page 00 Maths Mate /8 Skill Builder

115 Skill.00 Dividing integers. Divide the integers ignoring the signs. Determine the sign of the result. If the numbers to be divided have the: same sign: positive result Example: 9 ( ) + different sign: + + negative result Example: 9 ( ) +9 ( ) + Q. 0 A Different signs negative result. + a) ( ) b) ( ) c) ( 9) d) ( ) e) 9 f) 8 g) 8 h) ( 8) i) 9 j) ( ) k) ( ) l) m) n) 8 ( ) o) p) 8 ( ) q) ( ) r) 0 ( ) s) t) ( ) u) 9 page 0 Maths Mate /8 Skill Builder

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117 . [Rates / Ratios] Skill. Simplifying ratios by comparing two numbers. EITHER Find the largest number that divides evenly into each number of the ratio (Highest Common Factor). Divide each number by the HCF. Hint: : means fraction and is read as to. a : b a b OR Divide each number of the ratio by any factor until the ratio is reduced to simplest form. Ratio Q. Simplify the ratio : A. : a) Simplify the ratio : Simplify: :... : : OR A. : HCF of and is 8 so 8 : : 8 8 : 8 8 : 8 : : b) Simplify the ratio : Simplify: : : :... Simplify: Simplify: Simplify: c) Simplify the ratio 0 : 0... e) Simplify the ratio : Simplify: :... g) Simplify the ratio 00 : 0... i) Simplify the ratio : 9... k) Simplify the ratio :... m) Simplify the ratio 0 :... d) Simplify the ratio 0 : : :... f) Simplify the ratio 8 : : :... h) Simplify the ratio : 8 : :... j) Simplify the ratio 0 : : :... l) Simplify the ratio : : :... n) Simplify the ratio 90 : 0 : :... page 0 Maths Mate /8 Skill Builder

118 Skill. Simplifying ratios by comparing two quantities. Write the quantities of the ratio with the same unit of measurement. EITHER Find the largest number that divides evenly into each quantity of the ratio (Highest Common Factor). Divide each quantity by the HCF. Hints: The order of the quantities in a ratio matters. : means fraction and is read as to. Examples: The ratio of legs to ears in a cat is : : The ratio of ears to legs in a cat is : : OR Divide each quantity of the ratio by any factor until the ratio is reduced to simplest form. a : b a b Ratio Q. Simplify the ratio h : 0 min A. h 0 min 0 min h : 0 min 0 min : 0 min min : 0 min : Ignore the units h 0 min HCF of 0 and 0 is 0 so 0 a) Simplify the ratio 8 kg : 80 kg Simplify: 8 : b) Simplify the ratio 0 m : m Simplify: : : 0 :... c) Simplify the ratio 0 cm : cm... e) Simplify the ratio $.00 : 0 cents $ Simplify: 0 $ : 0... zeros, places right d) Simplify the ratio 0 g : g : :... f) Simplify the ratio 0 s : min... : :... g) Simplify the ratio m : 0 cm h) Simplify the ratio $.00 : cents : :... i) Simplify the ratio days : weeks j) Simplify the ratio min : 0 s : :... page 0 Maths Mate /8 Skill Builder

119 Skill. Solving questions involving distance, time and speed (). distance travelled (d ) speed (v) OR v d time taken (t ) t distance travelled (d) speed (v) time taken (t) OR d vt distance travelled (d) time taken (t ) OR t d speed (v) v continues on page 0 Write the formula for speed or distance or time taken to travel. Convert the given units into the required units if necessary. (see Maths Facts, page 9) Hints: If the speed must be calculated in km/h, convert the units for distance to km and the units for time to hours. Changing from smaller units into larger units, always divide by the conversion factor. Changing from larger units into smaller units, always multiply by the conversion factor. Substitute the known values into the formula. Simplify and evaluate. Q. A jet travels at an average speed of 900 km/h. At this rate how long would it take to travel 00 kilometres? distance travelled A. t speed 00 km 900 km/h 8 00 h h. h d v Substitute into the formula Simplify: 0 Simplify: 9 Use d rt a) How far will John walk in minutes if he walks at 0 km/h? t min 0. h (three quarters of an hour)... d 0 km/h 0. h... km c) A cyclist rides at an average speed of 8 km/h. At this rate how long would it take to travel km? d t v... km Simplify: 9 h... 8 km/h e) The X- rocket plane is the fastest aircraft with a maximum speed of km/h, reached in 9. At this speed how far could it travel in hours? d km b) How far will a salmon swim in minutes if it swims at km/h? t... h km d... d) A hot air balloon travels at a speed of km/h. At this rate how far will it travel in 0 minutes? t... km d... f) An airplane flew from Sydney to Cairns a distance of 000 km. If the plane travelled at an average speed of 800 km/h, how long did the trip take? t min page 0 Maths Mate /8 Skill Builder

120 Skill. Solving questions involving distance, time and speed (). g) An airplane flew from Melbourne to Adelaide, a distance of 0 km. If the plane travelled. hours, how fast did it travel? 0 Simplify: d 0 km v 0 0. km/h t. h km/h... km/h h) An airplane flew from Alice Springs to Darwin, a distance of 00 km. If the plane travelled. hours, how fast did it travel? v continued from page 0 km/h i) An emu can run 9 km in minutes. What is its average speed in kilometres per hour? v km/h j) Some species of dolphin can swim km at 0 km/h. How long would it take to swim this distance? t min k) A train travels at an average speed of km/h. What distance would it travel in one hour and minutes? d km l) A satellite orbits the earth at an average speed of 8 km/s. What distance does it travel in 0 minutes? d km m) Earth moves around the sun at an average speed of km/h. What distance does it move in a quarter of an hour? d km n) In 90 the first speeding ticket went to Harry Myers of Dayton, Ohio. Harry drove 0 km/h in town. At this speed how far could he travel in minutes? d km o) A rifle was fired at a target 00 m away. If the bullet travelled at an average speed of 800 m/s, how long did the bullet take to hit the target? t s p) It is 8 km from Wodonga Creek to Doctors Point. If a Murray cod travelled at an average speed of 8 km/h, how long would this trip take? t h min page 0 Maths Mate /8 Skill Builder

121 Skill. Simplifying ratios by comparing three numbers. EITHER Find the largest number that divides evenly into each number of the ratio (Highest Common Factor). Divide each number by the HCF. OR Divide each number of the ratio by any factor until the ratio is reduced to simplest form. Q. Simplify the ratio : : 0 A. : : 0 : : 0 : : OR A. : : 0 HCF of, and 0 Simplify: is so : : 0 : : Simplify: : : : : a) Simplify the ratio : : 0 9 Simplify: 8 : : 0... b) Simplify the ratio : 8 : 9: : : :... c) Simplify the ratio : 9 :... d) Simplify the ratio 0 : 0 : : : : :... e) Simplify the ratio : 8 :... f) Simplify the ratio 8 : : : : : :... g) Simplify the ratio 8 : 8 :... h) Simplify the ratio 0 : 00 : 0 : : : :... i) Simplify the ratio : : 0... j) Simplify the ratio 0 : : 90 : : : :... k) Simplify the ratio : :... l) Simplify the ratio 0 : 0 : 80 : : : :... page 0 Maths Mate /8 Skill Builder

122 Skill. Deciding if two ratios are equivalent. Write the two ratios as equal fractions side by side. Cross multiply the numerators and the denominators of the fractions. If the two products are equal, then the two ratios are equivalent (or form a proportion). Cross product a : b c : d a c b d a d b c ad bc ratios Q. Which ratio is equivalent to :? A ) 0 : B) : C) 0 : 8 Cross multiply 0 A. 0 0 false 0 8 The answer is C 0 0 false true A B C a) : is equivalent to : 0 True or false? b) : 9 is equivalent to : 8 True or false? c) Which ratio is equivalent to :? A ) 9 : B) 9 : C) 8 : A B 9 9 (F) C... d) Which ratio is equivalent to :? A ) 0 : 0 B) : C) : A B C... e) Which ratio is equivalent to :? A ) 8 : 8 B) 0 : 0 C) : 9 A B C... f) Which ratio is equivalent to : 9? A ) : B) 9 : 8 C) : A B C... page 08 Maths Mate /8 Skill Builder

123 Skill. Completing equivalent ratios (). Write the equivalent ratios as two equal fractions. Cross multiply the numerators and the denominators of the fractions. Equate the products. Solve the equation to find the missing number (x). Cross product continues on page 0 a : b c : d a c b d a d b c ad bc ratios Q. Complete the equivalent ratios: : 8 : A. x 8 x 8 x 8 x 8 x 8 x x 8 Cross multiply Simplify: a) Complete the equivalent ratios: b) Complete the equivalent ratios: : : x x... Simplify:, x x x... : : x x... x... c) Complete the equivalent ratios: : 0 : x... e) Complete the equivalent ratios: 9... x... d) Complete the equivalent ratios: : : 0... x... f) Complete the equivalent ratios: 8... x... page 09 Maths Mate /8 Skill Builder

124 Skill. Completing equivalent ratios (). g) Complete the equivalent ratios: h) Complete the equivalent ratios: continued from page x x... i) Complete the equivalent ratios: j) Complete the equivalent ratios: x x... k) Complete the equivalent ratios: x... l) Complete the equivalent ratios: x... m) Complete the equivalent ratios: 8... x... n) Complete the equivalent ratios:... x... page 0 Maths Mate /8 Skill Builder

125 Skill. Solving word problems involving equivalent ratios (). To decide which deal is cheaper: EITHER Find the unit price for each case, by dividing the cost price by the number of units. Compare the results. continues on page To solve word problems involving equivalent ratios: EITHER OR Find the unit rate, by dividing the amount Write two equivalent ratios using words. by the given number of units. Replace the words with numbers. Multiply the unit rate by the required Find the missing term of the equivalent number of units. ratios. (see skill., page 09) OR Use any other method to make the cost price the same or the number the units the same for both deals. Q. Which is cheaper per can? A ) $.0 for a -pack B ) $ for a -pack A. Deal A) $.0 cans $0. can unit price $0. Deal B) $.00 cans $0.0 can OR unit price $0.0 Deal A) is cheaper. A. Make the same number of cans: Deal A) double the quantity double the cost $.0 for cans $.00 for cans Deal B) $.00 for cans Deal A) is cheaper. a) There are 0 calories in 0 g of white bread. How many calories in 80 g? calories : bread calories : bread 0 : 0 x : x x x x 0... b) If m of curtain material costs $, how much would 9 m cost? m : dollars m : dollars x x cal $ c) If 80 concert tickets cost $0, how much would 0 tickets cost? 80 tickets for $ tickets for $0 8 $... 0 tickets for $ $ d) If 9 pens cost $.0, how much would pens cost? 9 pens for $ $ page Maths Mate /8 Skill Builder

126 Skill. Solving word problems involving equivalent ratios (). e) There are 00 calories in 00 g of pork chops. How many calories in grams? 00 g have 00 cal... g has... g... cal f) If kg of minced beef costs $, how much would kg cost? kg for... kg continued from page $ g) If a car travels 00 km on 0 L of petrol, how far does it travel on L at the same rate? h) There are 0 calories in 0 grams of fillet steak. How many calories in grams? km cal i) Which is cheaper per card? A ) $ for cards B ) $ for cards A) $ for cards $ for cards... make the same cost B) $ for cards $ for 0 cards... A j) Which is cheaper per pen? A ) $ for pens B ) $ for 8 pens make the same cost A)... B)... k) Which is cheaper per kilogram? A ) $0 for 0 kg B ) $0 for kg l) Which is cheaper per kilogram? A ) $ for kg B ) $ for kg A)... B)... A)... B)... m) Which is cheaper per apple? A ) $.80 for apples B ) $.0 for apples n) Which is cheaper per metre? A ) $ for m B ) $0 for m A)... B)... A)... B)... page Maths Mate /8 Skill Builder

127 Skill.8 Finding the ratio of two quantities (). Write the ratio in words. Replace the words with numbers. Simplify the ratio: EITHER Find the largest number that divides evenly into each quantity of the ratio (Highest Common Factor) and divide each quantity by the HCF. Hint: The order of the quantities in a ratio matters. continues on page OR Divide each quantity of the ratio by any factor until the ratio is reduced to simplest form. Q. The common metal for medals is 8% copper, and the rest is zinc. Find the ratio of zinc to copper. A. zinc 00% 8% % zinc : copper % : 8% : 8 : Ignore the % sign Simplify: a) The length of the school year in Egypt is weeks and in Indonesia is weeks. Find the ratio of the school year duration in Indonesia compared to Egypt. Indonesia : Egypt Simplify: :... 9 :... : 9 b) A computer screen with a diagonal of 0 cm has a length of 0 cm. Find the ratio of the length to the diagonal. length : diagonal 0 : : c) The alloy platinum is 90% platinum and 0% iridium. Find the ratio of iridium to platinum in the alloy. iridium : platinum : : d) In 98 only 8% of U.S. households had microwave ovens. As of 00 over 80% have them. Find the ratio of microwave oven ownership in 00 to 98. : : : e) The 8-carat gold is % gold, 0% silver and the rest is copper. Find the ratio of silver to other components. : : : f) For children aged to years, an airfare is % of the full adult airfare. Find the ratio of child to adult airfares. : : : page Maths Mate /8 Skill Builder

128 Skill.8 Finding the ratio of two quantities (). g) The Southern Star Observation Wheel (Melbourne) has a capacity of 0 passengers per capsule, and the London Eye has a capacity of. Find the ratio of the London Eye passengers per capsule to the Southern Star. : : : h) The London Eye has capsules, and the Singapore Flyer observation wheel has 8 capsules. Find the ratio of capsules in the Singapore Flyer to capsules in the London Eye. : : continued from page : i) In 009, of the seats in the Federal Parliament, 8 are held by women. What is the ratio of women to men in the Federal Parliament? : : j) Find the ratio of the height of the Statue of Liberty (9 m including the pedestal) to the height of the Eureka Tower, Melbourne (00 m). : :... : :... k) A soccer field is 0 metres long and 80 metres wide. Find the ratio of width to length. : : : l) The highest temperature recorded in Africa is C and in South America is 8 C. Find the ratio of the highest temperature in Africa compared to South America. : : : m) The lowest temperature recorded in Europe is C and in Antarctica is 90 C. Find the ratio of the lowest temperature in Europe compared to Antarctica. : : : n) The sensory, language and memory centres are located in the temporal lobe, which is % of the total cerebral cortex volume in the brain. Find the ratio of the temporal lobe to the rest of the cortex. : : : page Maths Mate /8 Skill Builder

129 Skill.9 Finding other rates. rate amount time Rate of change Divide the amount by the time taken. Example: A 00 L bathtub can be filled in 0 minutes. 00 L Rate 0 L/min 0 min Q. Some species of bamboo can grow up to 0 metres per year. At this rate how long will they grow in a month? a) A Mini Cooper Diesel with a. L engine emits 0 g/km of the greenhouse gas carbon dioxide (CO ). How many grams of CO will be emitted during a 00 km trip? amount (g) rate (g/km) distance (km)... 0 g/km 00 km g c) The Kudzu climbing plant can grow up to 0 m per year. What is this rate in metres per week? year weeks... rate/wk... e) A Holden Cruze has a fuel consumption of L of petrol per 00 km. How much petrol does it need for a 0 km trip? amount g) A Honda Civic Hybrid automatic has a highway consumption of L of petrol per 000 km. How much petrol does it need for a 00 km trip? amount amount rate time Amount Multiply the rate by the time taken. Example: Sam worked h at a rate of $/h. Amount (pay) $ L A. rate 0 m per year year months rate per month 0 m. m b) Most of the Lambert Glacier (Antarctica) moves around 0 metres in months. At this rate how much will it move in months? amount d) It takes minutes to fill a 00 litre swimming pool. What is the average rate in litres per minute? L $ page Maths Mate /8 Skill Builder m rate f) Every glass bottle recycled saves enough energy to light a 00-watt light bulb for hours. How many bottles are needed to light the same bulb for a week? week... bottles... h) The annual fuel cost for a Lamborghini Coupe is around $90. How much is the cost per month? year... time amount rate Time taken Divide the amount by the rate. Example: A Lexmark E prints 990 pages at a rate of pages/min (ppm). 990 p Time min ppm...

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131 . [Indices / Square Roots] Skill. Expressing powers as products and products as powers. To write a product as a power: Write the factor as the base. Count how many times the factor is multiplied by itself and make the result the index. To write a power as a product: Multiply the base by itself the same number of times as indicated by the index. power Read as: to the power of exponent base multiplied by itself times Q. Write the power as a product: A. multiplied by itself times a) Write the product as a power: c) Write the product as a power: factors of is the base the exponent b) Write the product as a power: d) Write the product as a power: e) Write the power as a product: 8 f) Write the power as a product: g) Write the power as a product: h) Write the power as a product: 9 i) Write the product as a power: j) Write the product as a power: k) Write the power as a product: l) Write the product as a power: page Maths Mate /8 Skill Builder

132 Skill. Squaring whole numbers. Multiply the number by itself. one squared two squared three squared four squared square squares 9 squares squares 9 Q. 90 A multiplied by itself times a) multiplied by itself times 9 b) c) d) 0 e) f) g) h) i) 0 j) k) 9 l) 0 m) 0 n) 0 o) 0 p) 80 q) 0 r) 0 page 8 Maths Mate /8 Skill Builder

133 Skill. Calculating powers of 0. Put the same number of zeros in the answer as the index shows. Example: 0 index is so the answer ends in zeros Q. 0 A. 0 Index is Answer ends in zeros a) multiplied by itself 9 times b) c) 0 d) e) 0 f) g) 0 h) i) 0 8 j) page 9 Maths Mate /8 Skill Builder

134 Skill. Finding square roots of whole numbers. Hint: Finding the square root of a number is the reverse of the procedure for squaring a number. EITHER Use trial and error to find the number that, when multiplied by itself, equals the original number. Example: The square root of the number that when multiplied by itself equals so OR Arrange that number of tiles in a square. Count the number of tiles along one side length Q. 9 A. 9 The square root of 9 means: what number multiplied by itself equals a) multiplied by itself b) 9 c) d) e) f) 00 g) h) i) j) 900 k) 900 l) 00 m) 800 n) 00 o) 00 page 0 Maths Mate /8 Skill Builder

135 Skill. Evaluating powers of whole numbers. Observe the index. Multiply the number (base) the same number of times by itself as the index shows. (see skill., page ) Hints: Any number raised to the power of zero (except 0) equals. Example 0 Any number raised to the power of one equals the number itself. Example Q. A. multiplied by itself times raised to the power of means lots of in the equation. a) multiplied by itself times b) c) d)... e)... f)... g)... h) 0... i)... j) 0... k)... l)... m)... n) 8... o)... p) 0... q) 8... r) page Maths Mate /8 Skill Builder

136 Skill. Finding powers of negative whole numbers. Observe the index. Multiply the number (base) the same number of times by itself as the index shows. (see skill., page ) Give the result a sign: even index ( ) ( ) + positive result odd index ( ) ( ) ( ) + ( ) negative result odd index Q. ( ) A. ( ) ( ) ( ) ( ) () negative result raised to the power of means lots of in the equation. a) ( ) even index b) ( ) c) ( ) ( ) ( ) ( )... positive result d) ( ) e) ( ) f) ( ) g) ( ) h) ( ) i) ( ) j) ( ) k) ( ) 9 l) ( ) m) ( ) n) ( ) o) ( 0) page Maths Mate /8 Skill Builder

137 . [Order of Operations] Skill. Using order of operations mixing only and/or,, or + and/or Use the order of operations rules: Multiply ( ) and/or divide ( ) in order from left to right. Add ( + ) and/or subtract ( ) in order from left to right. Q. A. 8 work from left to right divide first a) 9 + add first b) 9 + c) + 8 d) + e) 9 f) g) + h) 8 i) 9 j) 9 k) l) m) 9 + n) 8 + o) 0 p) q) r) page Maths Mate /8 Skill Builder

138 Skill. Using order of operations mixing,, + and/or Use the order of operations rules: First multiply ( ) or divide ( ). Finally add ( + ) or subtract ( ). Q. + A work from left to right divide first a) + multiply first + b) + 9 c) d) + e) f) + 8 g) 8 h) i) j) + k) + 9 l) m) + divide first n) 9... o) p) q) r) s) t) 0 u) page Maths Mate /8 Skill Builder

139 Skill. Using order of operations mixing ( ) with + and/or Use the order of operations rules: First evaluate inside the brackets. Finally add ( + ) and/or subtract ( ) from left to right. Q. + (8 9) + A. + (8 9) simplify inside the brackets work from left to right a) + ( + 9) b) + ( )... c) 9 ( + ) +... d) (9 ) + e) ( + 8) f) 8 + ( + ) g) ( + ) + h) ( ) i) + (8 ) j) + 9 ( + ) k) ( ) + 8 l) + 9 ( + ) m) + ( 8) + n) 8 (0 ) o) ( + ) p) 9 ( + 9) q) 9 + (8 + ) r) ( 9) page Maths Mate /8 Skill Builder

140 Skill. Using order of operations mixing ( ),,, + and/or Use the order of operations rules: First evaluate inside the brackets. Then multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Q. + ( + 9) A. + ( + 9) then multiply simplify inside the brackets a) ( + ) 0 brackets first 0 b) ( ) c) 8 ( + ) d) 8 ( ) e) ( ) f) ( ) 9 g) ( ) h) (8 ) i) ( + 8) j) ( + ) k) 8 ( ) l) 9 ( + ) m) 9 ( + ) n) 8 (8 ) o) 8 ( + ) p) + (8 ) q) + ( + ) r) + ( ) s) (9 ) t) ( ) (9 ) u) ( + 8) page Maths Mate /8 Skill Builder

141 Skill. Using order of operations mixing powers, ( ),,, + and/or Use the order of operations rules: First evaluate inside the brackets. Secondly evaluate the powers. Then multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Q. 8 A. 8 8 evaluate the power then divide work from left to right a) d) + (9 ) 0 b) e) 9 + c) f) ( ).... g) (8 0).... h) ( ).... i) (8 ).... j) k) l) 0 ( ) m) ( ).... n) + (9 ).... o) p) q) (9 + ).... r) ( 9 ) page Maths Mate /8 Skill Builder

142 Skill. Using order of operations involving negative numbers and mixing powers, ( ),,, + and/or Use the order of operations rules: First evaluate inside the brackets. Secondly evaluate the powers. Then multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Q. 8 + ( ) ( ) A. 8 + ( ) ( ) 8 + ( ) 8 + ( ) 8 + evaluate the bracket evaluate the power evaluate the division a) 0 + b) ( ) ( ) c) (8 + ) ( ) work from left to right d) ( ) + e) + ( ) f) ( + ) ( ) g) ( 8) h) 0 + ( ) i) j) ( )... k) + ( ) ( )... l) page 8 Maths Mate /8 Skill Builder

143 Skill. Using order of operations mixing square roots, powers,,, + and/or Use the order of operations rules: First evaluate inside the brackets. Secondly evaluate the powers. Then multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Q. A. 8 0 evaluate the square root evaluate the power a) + b) + 9 c) d) + e) + f) g) 0 h) 8 i) j) k) 9 + l) page 9 Maths Mate /8 Skill Builder

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145 . [Exploring Number] Skill. Comparing whole numbers. Compare the size of the digits in the same place, one at a time. Work from left to right across each number. Q. Which number is the largest? A ) 0 B ) 0 C ) 0 A. C Tens of thousands and thousands: All numbers have the same digit in the tens of thousands place (), and the same digit in the thousands place (). Hundreds: In the hundreds place is greater than. So A and C are greater than B. Tens: In the tens place is greater than 0. So 0 is greater than 0. a) 80 > 80 True or false? 8 >... > means is greater than < means is less than compare the hundreds place true b) 0 < 0 True or false?... c) 0 98 > 980 True or false?... d) 9 < 9 True or false?... e) 0 > 0 True or false?... f) 8 > 8 True or false?... g) Which number is the largest? A ) 80 B ) 80 C ) 800 > 0... compare the digits in the tens place B h) Which number is the largest? A ) 0 9 B ) 0 9 C ) i) Which number is the largest? A ) 9 0 B ) 9 0 C ) j) Which number is the largest? A ) 80 B ) 08 C ) k) Which number is the largest? A ) 0 B ) 0 C ) 0... l) Which number is the largest? A ) 09 B ) 90 C ) page Maths Mate /8 Skill Builder

146 Skill. Understanding and finding the place value of a digit in a number (). Compare the position of the digit to the position of the decimal point. Hint: There is a decimal point which is not written, at the end of any whole number. continues on page Place value Value tens of thousands thousands 000 hundreds tens units tenths 0 hundredths 00 thousandths Decimal point Q. What is the value of the underlined digit in the number 8.? A Consider the position of the digit to that of the decimal point. is five places to the left so it is in the tens of thousands place. The represents tens of thousands or a) In the number 08 which digit is in the tens place? b) In the number 90 which digit is in the units place? c) In the number 8 which digit is in the hundreds place? d) In the number 08 which digit is in the thousands place? e) In the number. which digit is in the units place? f) In the number 0.0 which digit is in the hundredths place? g) In the number.809 which digit is in the tenths place? h) In the number 0.08 which digit is in the thousandths place? i) What is the value of the underlined digit in the number 9? tens... 0 j) What is the value of the underlined digit in the number 0?... k) What is the value of the underlined digit in the number 09?... l) What is the value of the underlined digit in the number 8 9?... page Maths Mate /8 Skill Builder

147 Skill. Understanding and finding the place value of a digit in a number (). m) What is the value of the underlined digit in the number? continued from page n) What is the value of the underlined digit in the number 0? o) What is the value of the underlined digit in the number 9 0?... p) What is the value of the underlined digit in the number 8?... q) What is the value of the underlined digit in the number.0? hundredths r) What is the value of the underlined digit in the number.08? thousandths... s) What is the value of the underlined digit in the number 0.98?... t) What is the value of the underlined digit in the number.0?... u) In which number does the digit have greater value? A ). B ) 0. A) value... B) value 0. > A v) In which number does the digit have greater value? A ) 0 0 B ) 0 A)... B)... w) In which number does the digit have greater value? A ). B ).98 A)... B)... x) In which number does the digit 9 have greater value? A ) 90 B ) 0 89 A)... B)... y) In which number does the digit have greater value? A ) 8.9 B ). A)... B)... z) In which number does the digit have greater value? A ) 0. B ) 0. A)... B)... page Maths Mate /8 Skill Builder

148 Skill. Writing word numbers as numerals. Write the digits in order. Leave a space between the thousands and the hundreds, and between the millions and the hundreds of thousands. Write a zero in any place that is left empty between other digits. Q. Express in numerals: fifty thousand, six hundred and nine A Tens of Th. Th. 0 H T U 0 9 First write 0 for the words fifty thousand, then write a comma. Write the digit for the hundreds, then write the digit 0, because there are no tens. Finally write the digit 9 for the units. a) Express in numerals: two hundred and fifteen c) Express in numerals: six thousand and eighty-two e) Express in numerals: nine hundred and two g) Express in numerals: two hundred and ninety-eight i) Express in numerals: five hundred and thirty k) Express in numerals: seven hundred and fourteen m) Express in numerals: sixty thousand, five hundred and forty o) Express in numerals: four hundred and three thousand, two hundred q) Express in numerals: one million, nine hundred thousand and twenty-six b) Express in numerals: four thousand, one hundred and fifty d) Express in numerals: eight thousand, one hundred and seventeen f) Express in numerals: three thousand, four hundred h) Express in numerals: seven thousand, three hundred and nine j) Express in numerals: twelve thousand, six hundred l) Express in numerals: fourteen thousand and sixty-three n) Express in numerals: thirty-one thousand and seven p) Express in numerals: eight hundred thousand and fifty r) Express in numerals: seven million, six hundred thousand and forty page Maths Mate /8 Skill Builder

149 Skill. Writing whole numbers in words (). Start from left and write the word for each digit (unless it is a 0), followed by its place value. Don t write anything for any 0 s. 00 two hundred To write -digit numbers in words: Use a hyphen (-) to separate the word for the tens from the word for the units, for all numbers from to 99; e.g. is written as sixty-seven. Hint: Some -digit numbers have names that do not follow the usual rules. Use the following: 0 ten 0 fifty 90 ninety fourteen 8 eighteen 0 twenty 0 sixty eleven fifteen 9 nineteen 0 thirty 0 seventy twelve sixteen 0 forty 80 eighty thirteen seventeen To write -digit numbers in words: Describe the number of hundreds first. Always write hundred not hundreds. Write and after the word hundred, if other values follow. To write -digit numbers in words: Describe the number of thousands first. Always write thousand not thousands. Write and between the word thousand and the following numerals when hundreds are missing. To write -digit numbers in words: Describe the number of thousands by following the rules for -digit numbers. To write -digit numbers in words: Describe the number of thousands by following the rules for -digit numbers. continued on page word first! place next Q. Write the number 09 in words. A. seven thousand and sixty-nine Th. H T U 0 9 thousands, 0 hundreds, tens and 9 units become in words: seven thousand and sixty-nine a) Write the number 8 in words. b) Write the number in words. three hundred and eighteen c) Write the number 90 in words. d) Write the number in words. e) Write the number 0 in words. f) Write the number 0 in words. page Maths Mate /8 Skill Builder

150 Skill. Writing whole numbers in words (). g) Write the number 800 in words. h) Write the number 09 in words. continued from page i) Write the number 0 in words. j) Write the number 00 in words. k) Write the number 00 in words. l) Write the number 00 in words. m) Write the number 00 in words. n) Write the number 0 in words. o) Write the number 800 in words. p) Write the number 000 in words. q) Write the number in words. r) Write the number in words. s) Write the number 0 00 in words. t) Write the number 00 in words. u) Write the number 090 in words. v) Write the number 008 in words. w) Write the number in words. x) Write the number in words. y) Write the number in words. z) Write the number in words. page Maths Mate /8 Skill Builder

151 Skill. Rounding whole numbers to a given place. Circle the digit to the right of the requested place. If this digit is 0,,, or (< ) - round down - keep the digit in the requested place the same.,,, 8 or 9 ( ) - round up - add to the digit in the requested place. Keep the number of digits in the answer the same as in the question by using zeros to fill the vacated spaces. Q. Round 0 to the nearest hundred. A. 00 Th. H T U 0 Th. H T U 0 0 The digit to the right of the hundreds place is. so round up. Add to the 0 in the hundreds place to make. Put zeros in the tens and units places. a) Round 0 to the nearest thousand. < round down 0 by keeping... c) Round to the nearest hundred b) Round to the nearest ten.... d) Round 80 to the nearest ten.... e) Round to the nearest ten.... f) Round 0 to the nearest thousand.... g) Round to the nearest hundred.... h) Round to the nearest thousand.... i) Round 90 to the nearest ten.... k) Round 0 0 to the nearest thousand.... j) Round 9 to the nearest hundred.... l) Round 0 to the nearest hundred.... m) Round 8 to the nearest ten.... n) Round 0 to the nearest hundred.... page Maths Mate /8 Skill Builder

152 Skill. Rounding decimal numbers to a given place. To round a decimal number to the nearest whole number: Circle the first digit after the decimal point. If this digit is: 0,,, or (< ) - round down - keep the unit digit unchanged and drop all the digits after the decimal point.,,, 8 or 9 ( ) - round up - add to the unit digit and drop all the digits after the decimal point. To round a decimal number to a given place (one decimal place means tenths, two decimal places means hundredths and three decimal places means thousandths): Circle the digit to the right of the requested place. If this digit is: 0,,, or (< ) - round down - keep the digit in the requested place unchanged and drop all following digits.,,, 8 or 9 ( ) - round up - add to the digit in the requested place and drop all following digits. Q. Round. to the nearest whole number. A. Units Tenths Hundredths Units Tenths Hundredths 0 0 a) Round. to the nearest whole number. < round down. by keeping... c) Round.8 to the nearest whole number.... The first digit after the decimal point is. so round up. Add to the in the units place to make. Omit the digits after the decimal point. b) Round.9 to the nearest whole number.... d) Round.8 to the nearest whole number.... e) Round.8 to three decimal places. round up by.8 adding to... g) Round 0.9 to one decimal place.... f) Round 8.8 to two decimal places.... h) Round 9.8 to one decimal place.... i) Round.8 to two decimal places.... j) Round 0.08 to three decimal places.... k) Round 0. to one decimal place.... l) Round 0.98 to three decimal places.... page 8 Maths Mate /8 Skill Builder

153 Skill. Recognising whole numbers and integers. INTEGERS Negative integers Zero Positive integers WHOLE NUMBERS Decide if a number is a whole number or an integer, based on their definitions and the hints below. (see Glossary) Hints: Negative integers, fractions and decimals are not whole numbers. Any positive fraction whose numerator is divisible by the denominator is a whole number: Any positive decimal with only zeros after the decimal point is a whole number: Fractions and decimals are not integers. Any fraction whose numerator is divisible by the denominator is an integer: Any decimal with only zeros after the decimal point is an integer:.00 Q. Choose the whole numbers from this list: 9, 8.,,,, 0 a) Choose the whole numbers from this list: 9 8,,, 0., 8, c) Choose the whole numbers from this list:, 9, 9,.8, 0 e) Choose the integers from this list:,,., 8, 0 A. 9 is negative, so not a whole number 8. is a decimal, so not a whole number is a whole number is a fraction, so not a whole number So,, 0 are whole numbers. b) Choose the whole numbers from this list:,, 00,., 98 d) Choose the whole numbers from this list: ,, 0,, f) Choose the integers from this list:,,,., 00 9 g) Choose the integers from this list:,, 0., 0, 8 h) Choose the integers from this list:,.8,,, 0 i) Choose the integers from this list: 0.8,,,, 09 j) Choose the integers from this list: 0 0,,,., 000 page 9 Maths Mate /8 Skill Builder

154 Skill.8 Using inequality and equality signs to compare decimal numbers. Compare digits in the same places, starting from the left. Hint: The number with the greater digit in the same place will be greater. Use the inequality sign < (is less than) when the number on the left is less than the number on the right. Use the inequality sign > (is greater than) when the number on the left is greater than the number on the right. Use the equality sign (is equal to) when the numbers are equal in value. Q. Use <, or > to complete the statement...9 A >. >.9 Units: both Tenths: both Hundredths: is greater than so. is greater than.9 Use the symbol for greater than which is > a) Use <, or > to complete the statement. < < 8 b) Use <, or > to complete the statement c) Use <, or > to complete the statement...00 d) Use <, or > to complete the statement e) Use <, or > to complete the statement f) Use <, or > to complete the statement..0. g) Use <, or > to complete the statement h) Use <, or > to complete the statement i) Use <, or > to complete the statement... j) Use <, or > to complete the statement...08 k) Use <, or > to complete the statement l) Use <, or > to complete the statement m) Use <, or > to complete the statement n) Use <, or > to complete the statement o) Use <, or > to complete the statement..0.0 page 0 Maths Mate /8 Skill Builder

155 8. [Multiples / Factors / Primes] Skill 8. Finding the multiples of a number. Count by the number i.e. add the number to itself continuously. OR Multiply the number by, then,,,, etc. to get the multiples in order. Q. List all the multiples of up to. A. 0 0, 0,, 0, a) List all the multiples of 8 up to , + 8, ,,, keep adding 8 b) List all the multiples of up to.... c) List all the multiples of 0 up to 0. d) List all the multiples of up to e) List all the multiples of up to. f) List all the multiples of up to g) List all the multiples of 8 up to 0. h) List all the multiples of 9 up to i) List all the multiples of up to. j) List all the multiples of up to page Maths Mate /8 Skill Builder 8

156 Skill 8. Finding the common multiples of two numbers. List the multiples of each number. Compare the lists to find any numbers the same (common multiples). Q. List the common multiples of and up to 0. A. Multiples of :, 8,,, 0,, 8,,, 0,, 8, Multiples of :, 0,, 0,, 0,, 0,, 0 Common multiples of and up to 0: 0, 0 a) List the common multiples of and up to 0.,, 9,,, 8... multiples of multiples of,, 8...,, 8 b) List the common multiples of and up to c) List the common multiples of and 9 up to 0. d) List the common multiples of and 8 up to e) List the common multiples of and up to. f) List the common multiples of and 8 up to g) List the common multiples of and 8 up to 90. h) List the common multiples of and 9 up to page Maths Mate /8 Skill Builder 8

157 Skill 8. Finding the lowest common multiple (LCM) of two numbers. List the multiples of each number. Compare the lists and find the lowest matching number (Lowest Common Multiple, LCM). Hints: If one number divides evenly into the other number then the LCM is the larger number. If two numbers have as their only common factor then the LCM is their product. Q. What is the lowest common multiple (LCM) of 0 and? a) What is the lowest common multiple (LCM) of and 8?,, 9,,, 8,,,... multiples of 8 8,,,... multiples of A. Multiples of 0: 0, 0, 0, 0, 0, 0, 0, 80 Multiples of :,,, 8, 0,, 8 Lowest Common Multiple (LCM): 0 b) What is the lowest common multiple (LCM) of and? c) What is the lowest common multiple (LCM) of and? d) What is the lowest common multiple (LCM) of and 9? e) What is the lowest common multiple (LCM) of and 8? f) What is the lowest common multiple (LCM) of and 0? g) What is the lowest common multiple (LCM) of and? h) What is the lowest common multiple (LCM) of and 8? i) What is the lowest common multiple (LCM) of 8 and? j) What is the lowest common multiple (LCM) of 9 and? page Maths Mate /8 Skill Builder 8

158 Skill 8. Finding the factors of a number. To decide if a number is a factor of another number the first number must divide evenly into the second number, with no remainder. Hint: A number always has at least factors, and the number itself. Use trial and error. Be systematic. Divide into the number. If divides evenly then and the result are factors of the number. Divide into the number. If divides evenly then and the result are factors of the number. Divide into the number. If divides evenly then and the result are factors of the number. Continue until all possibilities are exhausted. Q. List all the factors of 0 in ascending order. a) Is a factor of? remainder... no A remainder 0 remainder 0,,, 0 b) Is a factor of 8? Back to & so possibilities exhausted 8... c) Is a factor of? d) Is a factor of? e) List all the factors of in ascending order. f) List all the factors of 8 in ascending order g) What is the smallest positive integer that has exactly three factors? h) What is the smallest positive integer that has exactly nine factors? i) The number has exactly three factors:,,. Find the next number after that has exactly three factors.... j) The number has exactly six factors:,,,, and. Find the next number after that has exactly six factors.... page Maths Mate /8 Skill Builder 8

159 Skill 8. Finding the common factors of two numbers. List the factors of each number. Compare the lists and find any matching numbers (common factors). Q. List all the common factors of 8 and. a) List all the common factors of 8 and. factors of 8,,, 8...,,,,, 9,, 8,...,, factors of A. Factors of 8:,,,, 9, 8 Factors of :,,,,,,, Common factors of 8 and :,,, b) List all the common factors of and c) List all the common factors of 0 and. d) List all the common factors of 0 and e) List all the common factors of and. f) List all the common factors of and g) List all the common factors of 8 and. h) List all the common factors of 8 and page Maths Mate /8 Skill Builder 8

160 Skill 8. Finding the highest common factor (HCF) of two numbers. List the factors of each number. Compare the lists and find the highest matching number (Highest Common Factor, HCF). Q. What is the highest common factor (HCF) of and 0? a) What is the highest common factor (HCF) of and? factors of,,,,, 8,,... factors of,,, 8,,... 8 A. Factors of :,,,,, 8,, Factors of 0:,,,,,, 0,,, 0, 0, 0 Highest common factor (HCF): b) What is the highest common factor (HCF) of and? c) What is the highest common factor (HCF) of 0 and? d) What is the highest common factor (HCF) of 0 and 0? e) What is the highest common factor (HCF) of and 8? f) What is the highest common factor (HCF) of and? g) What is the highest common factor (HCF) of 8 and? h) What is the highest common factor (HCF) of and? i) What is the highest common factor (HCF) of and? j) What is the highest common factor (HCF) of 0 and? page Maths Mate /8 Skill Builder 8

161 Skill 8. Recognising prime and composite numbers. To decide if a number is prime, find all the factors of the number to determine if it has exactly factors, and itself. (see skill 8., page ) Hint: 0 and are not prime or composite numbers. To decide if a number is composite, find all the factors of the number to determine if it has more than factors. Q. List all the prime numbers between and. a) Choose the composite numbers: 0,,,,,,, 0 & are not composite;, & are prime... is the only even prime; & are even..., A. List the factors of each number: (,) (,) 8 (,8), (,) (, ), (,), (,) 9 (,9), (,) (, ) 0 (, 0), (,) (,), (,) Prime numbers (only factors):,, b) Choose the composite numbers: 8, 9, 0,,,,, c) What is the prime number just before?... d) What is the next prime number after 00?... e) What is the next prime number after? f) What is the next prime number after 9? g) List all the prime numbers between 0 and 0 h) Choose the composite numbers:,, 8, 9, 0,,, i) What is the prime number just before 88? j) What is the next prime number after 90? page Maths Mate /8 Skill Builder 8

162 Skill 8.8 Expressing a number as a product of its prime factors using a factor tree (). Write the number as a product of any two factors excluding (not necessarily prime numbers). Then write each of these two numbers as a product of any two factors excluding. Continue in this way until only prime factors remain. continued on page 9 Q. Express as a product of prime numbers by completing the factor tree. A. only prime factors remaining and a) Express 0 as a product of prime numbers by completing the factor tree b) Express 0 as a product of prime numbers by completing the factor tree c) Express 8 as a product of prime numbers by completing the factor tree d) Express as a product of prime numbers by completing the factor tree e) Express as a product of prime numbers by completing the factor tree. f) Express 00 as a product of prime numbers by completing the factor tree page 8 Maths Mate /8 Skill Builder 8

163 Skill 8.8 Expressing a number as a product of its prime factors using a factor tree (). g) Express 90 as a product of prime numbers by completing the factor tree h) Express 0 as a product of prime numbers by completing the factor tree continued from page i) Express as a product of prime numbers by completing the factor tree. j) Express as a product of prime numbers by completing the factor tree k) Express 9 as a product of prime numbers by completing the factor tree. 9 l) Express as a product of prime numbers by completing the factor tree. m) Express as a product of prime numbers by completing the factor tree. 9 n) Express 80 as a product of prime numbers by completing the factor tree. 80 page 9 Maths Mate /8 Skill Builder 8

164 Skill 8.9 Expressing a number as a product of its prime factors using consecutive divisions. Find a prime number that divides evenly into the given number. Write this prime number next to the given number. Divide and write the result under the given number. Continue in this way until the result of the last division equals. Show all the resulting prime numbers as factors of the original number. EITHER Use divisibility tests. (see Glossary, page ) Hints: All even numbers are divisible by All numbers ending in 0 are divisible by 0 ( ) OR Use a factor tree. (see skill 8.8, page 8) Q. Express 8 as a product of its prime factors. A. 8 8 Even numbers divide by divides evenly by 8 a) Express 0 as a product of its prime factors c) List the prime factors of b) Express as a product of its prime factors d) List the prime factors of e) Express as a product of its prime factors. f) Express 98 as a product of its prime factors g) Express 8 as a product of its prime factors h) Express 0 as a product of its prime factors page 0 Maths Mate /8 Skill Builder 8

165 Skill 8.00 Expressing a number as a product of its prime factors using index notation. Express the number as a product of its prime factors. (see skill 8.8, page 8 and skill 8.9, page 0) Group like factors in ascending order. Use index notation to simplify like factors. (see skill., page ) Q. Express as a product of its prime factors using index notation. A. divides evenly by divides evenly by a) Express 0 as a product of its prime factors using index notation b) Express 00 as a product of its prime factors using index notation c) Express 0 as a product of its prime factors using index notation. d) Express as a product of its prime factors using index notation e) Express 900 as a product of its prime factors using index notation. f) Express as a product of its prime factors using index notation page Maths Mate /8 Skill Builder 8

166 page Maths Mate /8 Skill Builder 8

167 9. [Number Patterns] Skill 9. Completing number patterns by adding the same number. Look at consecutive terms of the pattern. Find the number and operation (in this case addition) used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next two terms of the pattern. Q. Complete the pattern:,, 0, 9,, A.,, 0, 9,, Rule: Add 9 to each term ,, 0, 9, 8, First note that each term in the pattern is increasing. Then find by how much. a) Complete the pattern: 0,, 8,, b) Complete the pattern: 0,,,, 0,,, , c) Complete the pattern:, 8,, 8,, d) Complete the pattern:,,,, 9,,, e) Complete the pattern:,, 8,, f) Complete the pattern:,,,,,, g) Complete the pattern:,, 9,, h) Complete the pattern:,, 9,,,, i) Complete the pattern:, 8,, 0, j) Complete the pattern:,,,,,, page Maths Mate /8 Skill Builder 9

168 Skill 9. Completing number patterns by subtracting the same number. Look at consecutive terms of the pattern. Find the number and operation (in this case subtraction) used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next two terms of the pattern. Q. Complete the pattern:,,, 8,, A.,,, 8,, Rule: Subtract 9 from each term ,,, 8, 9, 0 First note that each term in the pattern is decreasing. Then find by how much. a) Complete the pattern: 8,,, 9, b) Complete the pattern:,,,, 0, 8,, 9, c) Complete the pattern: 0,,,, 8, d) Complete the pattern:,, 0,, 0,,, e) Complete the pattern: 0,,, 8,, f) Complete the pattern:, 8,,, 0,, g) Complete the pattern: 98, 88, 8, 8, h) Complete the pattern:, 8,,,,, i) Complete the pattern:,,, 8, j) Complete the pattern:, 0,,,,, page Maths Mate /8 Skill Builder 9

169 Skill 9. Completing number patterns by adding or subtracting decimal numbers. Look at consecutive terms of the pattern. Find the number and operation used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next two terms of the pattern. Q. Complete the pattern: 0.8,,.,.,, A. 0.8,,.,.,, Rule: Subtract 9 from each term ,,.,.,.,.8 First note that each term in the pattern is increasing. Then find by how much. a) Complete the pattern: 0., 0.8,.,, b) Complete the pattern:.,..8,.,., 0.9,, , c) Complete the pattern:.,.,.9,.,.,, d) Complete the pattern:,.,,.,, e) Complete the pattern:,.,.8,.,.,, f) Complete the pattern:.,.9,.,.,, g) Complete the pattern:.9,.,.,, h) Complete the pattern:,,.,.,.,, i) Complete the pattern: 0.8,.,,., j) Complete the pattern:,.9,.,.,.,, page Maths Mate /8 Skill Builder 9

170 Skill 9. Completing number patterns in table format by adding the same number. Look at consecutive terms of the pattern. Find the number and operation used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next term of the pattern. Q. Complete the table: A.,?, 0,,? First note High-rise buildings Number of floors Number of rooms 0 + Rule: Add to each term. + +,, 0,, High-rise buildings Number of floors Number of rooms 0 that each term in the pattern is increasing. Then find by how much. a) Complete the table: Growth (mm) b) Complete the table: Bouquets fingernail 8 toenail White roses Red roses , c) Complete the table: Food Intake of a baby robin No. of days Length of worms (ft) 8 d) Complete the table: children (9-) day Calories consumed (hundreds) e) Complete the table: Rent Number of bedrooms Cost per week ($) 00 0 f) Complete the table: Shark teeth regeneration (thousands) No. of days Teeth regenerated g) Complete the table: Exercise program h) Complete the table: Equilateral triangle Time (min) Side length Energy (cal) Perimeter page Maths Mate /8 Skill Builder 9

171 Skill 9. Completing number patterns by multiplying by the same number. Look at consecutive terms of the pattern. Find the number and operation (in this case multiplication) used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next two terms of the pattern. Q. Complete the pattern:,,,,, A.,,,,, Rule: Multiply each term by,,,,, First note that each term in the pattern is increasing. Then find by how much. a) Complete the pattern:,, 8,, b) Complete the pattern:, 8,,, 8,,, c) Complete the pattern:,,, 08, d) Complete the pattern:,,,,,, e) Complete the pattern: 0., 0.,,,, f) Complete the pattern:,,, 8,, g) Complete the pattern:,,,, 8, h) Complete the pattern:,,,, 9, i) Complete the pattern: 0.0, 0., 0.,.,, j) Complete the pattern:,,,, , page Maths Mate /8 Skill Builder 9

172 Skill 9. Completing number patterns by dividing by the same number. Look at consecutive terms of the pattern. Find the number and operation (in this case division) used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next two terms of the pattern. Q. Complete the pattern: 0, 0, 0, 80,, A. 0, 0, 0, 80,, Rule: Divide each term by , 0, 0, 80, 0, 0 First note that each term in the pattern is decreasing. Then find by how much. a) Complete the pattern: 9, 8,,, b) Complete the pattern:, 8,,,,,, c) Complete the pattern: 0, 0, 0, 0, d) Complete the pattern:, 0, 0, 0, 80, 0,, e) Complete the pattern: f) Complete the pattern: 000, 00, 0,, 0.,, 9,, 8,,, g) Complete the pattern: h) Complete the pattern:.,., 0.8, 0.,,.,.,.,.,, i) Complete the pattern: j) Complete the pattern: 0 000, 000, 00, 0,,, 8,, 8,, page 8 Maths Mate /8 Skill Builder 9

173 Skill 9. Completing number patterns by using changing values in the rule. Look at consecutive terms of the pattern. Find the number and operation used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next two terms of the pattern. Q. Complete the pattern:,,,,,, A.,,,,,, Rule: Add then then 9 etc. to each term. (i.e. consecutive multiples of ) ,,,,, 8, First note that each term in the pattern is increasing. Then find by how much. a) Complete the pattern: 8, 0,, 0, 8, b) Complete the pattern: 8, 0,,,,,, ? +? , c) Complete the pattern: 9,, 0,, d) Complete the pattern:,,,,, 9,, e) Complete the pattern:,, 0, 9,, f) Complete the pattern:,,,, 9,,, g) Complete the pattern:,,,, 9, h) Complete the pattern:,, 00,,,,..., 00 0, 8... i) Complete the pattern:, 9,, 9, j) Complete the pattern:,,,,,, page 9 Maths Mate /8 Skill Builder 9

174 Skill 9.8 Completing number patterns involving negative integers by adding or subtracting the same integer. Look at consecutive terms of the pattern. Find the number and operation used to get from one term to the next. Define the rule of the pattern. Apply this rule to the last given term and find the next two terms of the pattern. Q. Complete the pattern:,,, 9,, A.,,, 9,, Rule: Subtract from each term. 9,,, 9,, First note that each term in the pattern is decreasing. Then find by how much. a) Complete the pattern: 9,,,,, b) Complete the pattern:, 9,,, 8,,, 8, c) Complete the pattern:,,,,, d) Complete the pattern:,,,,,,, e) Complete the pattern: 0,,,, f) Complete the pattern:,,,,,, g) Complete the pattern:, 8,, 0, h) Complete the pattern:,,, 9,,, i) Complete the pattern:,,,, j) Complete the pattern:,,,,,, page 0 Maths Mate /8 Skill Builder 9

175 Skill 9.9 Finding a term in a number pattern (). continued on page EITHER Find the terms in order until you get to the required term. OR Draw up a table and match the term numbers with the given terms in the pattern. Use observation and trial and error to find a relationship between the term number and its value in the pattern. Based on this relationship, find the requested term in the pattern. Q. Find the 8th term in the pattern: 8,, 0,,... A. 8,, 0,... OR Rule: Add to each term ,, 0,,..., 0 term number pattern relationship 8 First note that each term in the pattern is increasing. Then find by how much. Count on ? 8 + Relationship: times the term number + The 8th term of the pattern is a) Find the th term in the pattern:,,,,... b) Find the th term in the pattern:,,,,... term number... term number... pattern pattern? relationship relationship + Relationship: times the term number... The th term of the pattern is... Relationship:... The th term of the pattern is... c) Find the 0th term in the pattern:,,, 8, 0,... d) Find the th term in the pattern:, 0,, 0,,... term number... 0 term number... pattern 8? pattern 0 0? relationship relationship Relationship:... The 0th term of the pattern is... Relationship:... The th term of the pattern is... page Maths Mate /8 Skill Builder 9

176 Skill 9.9 Finding a term in a number pattern (). e) Find the 8th term in the pattern:,,,,,... f) Find the 0th term in the pattern:, 8,,,... continued from page term number... 8 term number... 0 pattern? pattern 8? relationship relationship Relationship:... The 8th term of the pattern is... Relationship:... The 0th term of the pattern is... g) Find the th term in the pattern:,, 9,,,... h) Find the th term in the pattern:,, 8,,,... term number... term number... pattern 9? pattern 8? relationship relationship i) Find the th term in the pattern:,,,, 9,... j) Find the 0th term in the pattern:,,, 8,... term number... term number... 0 pattern? pattern 8? relationship relationship k) Find the 0th term in the pattern:,,,,... 8 l) Find the 8th term in the pattern:,,,, page Maths Mate /8 Skill Builder 9

177 Skill 9.00 Finding a particular term of a sequence given its general rule. Identify the value of n for the requested term of the sequence. Hint: If the 0th term needs to be found, the value of n is 0. Substitute the value of n in the formula for the general rule of the pattern. Calculate the value of the particular term of the sequence. Q. If the general rule of a pattern is + n find the th term (n ). A. + n + 0 substitute n a) If the general rule of a pattern is n find the 0th term (n 0). n b) If the general rule of a pattern is n + find the 0th term (n 0). n c) If the general rule of a pattern is n 8 find the th term (n ). d) If the general rule of a pattern is n + 8 find the th term (n ) e) If the general rule of a pattern is n + find the 0th term (n 0). f) If the general rule of a pattern is 0 n find the th term (n ) g) If the general rule of a pattern is n + find the 9th term (n 9). h) If the general rule of a pattern is n find the th term (n ) i) If the general rule of a pattern is n + find the 0th term (n 0). j) If the general rule of a pattern is n + find the 8th term (n 8) page Maths Mate /8 Skill Builder 9

178 page Maths Mate /8 Skill Builder 9

179 0. [Expressions] Skill 0. Simplifying expressions by adding and subtracting like terms (coefficient ). Add or subtract, as instructed, all like terms. (see Glossary, page ) In your answer, write the coefficient (number) first follwed by the variable (letter) (see glossary, pages and ) Hint: In the term m, is the coefficient: m m Q. Simplify kl + kl + kl kl + kl A. kl + kl + kl kl + kl kl cancel first coefficient first a) Simplify n + n + n + n n b) Simplify a + a c) Simplify u + u d) Simplify t + t + t e) Simplify w + w + w + w f) Simplify z + z + z + z + z g) Simplify x x + x h) Simplify b + b + b b i) Simplify e + e e + e j) Simplify k + k + k + k k k k) Simplify p + p p p + p l) Simplify c c + c c + c + c m) Simplify ab + ab n) Simplify hi + hi + hi o) Simplify fg + fg + fg + fg p) Simplify op + op + op + op q) Simplify tu + tu + tu + tu + tu r) Simplify uv + uv uv + uv s) Simplify ab ab + ab + ab ab t) Simplify wx + wx wx + wx + wx u) Simplify de + de de + de de + de page Maths Mate /8 Skill Builder 0

180 Skill 0. Simplifying expressions by adding and subtracting like terms (coefficient ). Add or subtract the coefficients (numbers) first. Write the variable (letters) next. Hint: In the term m, is the coefficient: m m Q. Simplify b b + b a) Simplify m + m coefficient first m A. b b + b b + b b b) Simplify h + h c) Simplify g + g d) Simplify j + j e) Simplify z + z f) Simplify e e g) Simplify q q h) Simplify a a i) Simplify k k j) Simplify r + r + r r + r m) Simplify y + y + y p) Simplify j j + j s) Simplify op + op v) Simplify mn + mn k) Simplify f + f + f n) Simplify m + m + m q) Simplify c + c c t) Simplify ab ab w) Simplify ij ij l) Simplify a + a + a o) Simplify h + h + h r) Simplify k + k k u) Simplify kl kl x) Simplify de de page Maths Mate /8 Skill Builder 0

181 Skill 0. Finding like terms. Look at the combination of letters in all terms. EITHER Find the like terms, which use the same combination of letters. Example: c and c gh and gh like terms OR Find the unlike terms, which do not use the same combination of letters. Example: k and w uv and vw unlike terms Hint: The order of the letters in a term does not matter. gh hg Q. Choose the like terms: y, z, z A. y and z - are terms using different letters z and z - are terms using the same combination of letters z, z a) Choose the like terms: f, e, f f, f b) Choose the like terms: c,, c c) Choose the like terms: h, i, h d) Choose the like terms: b, d, b e) Choose the like terms: f, e, f f) Choose the like terms: m, n, n g) Choose the like terms: r, r, s h) Choose the like terms: l, m, m i) Choose the like terms: w, x, x j) Choose the like terms: k, jk, j, jk k) Choose the like terms: ab, ab, b, a l) Choose the like terms: w, x, x, wx m) Choose the like terms: h, hi, i, hi n) Choose the like terms: d, de, d, e o) Choose the like terms: uv, v, v, u p) Choose the like terms: n, o, no, no q) Choose the like terms: a, b, ab, a r) Choose the like terms: st, s, t, st page Maths Mate /8 Skill Builder 0

182 Skill 0. Simplifying expressions by first grouping like terms. Group like terms. (see skill 0., page ) Read the sign in front of each term. Add and/or subtract only the like terms. Hint: Unlike terms cannot be added or subtracted. Q. Simplify p + p + q + p + q A. p + p + q + p + q p + p + p + q + q p + q group like terms a) Simplify s + r + s b) Simplify d + e + d c) Simplify h + i + h s + s + r... s + r d) Simplify a + b + b + a e) Simplify l + m + l + m f) Simplify r + r + r + s g) Simplify p + p + q + p h) Simplify d + e + e + d i) Simplify y + z + y + z j) Simplify y + x + x + y + y... k) Simplify e + f + e + f + e... l) Simplify m + m + n m + n... m) Simplify t + u + u t + t... n) Simplify j + k j k + k... o) Simplify rs rs + qr + qr + rs... p) Simplify cd de + de + de + cd... q) Simplify h i + h + i... r) Simplify j + k j + k... page 8 Maths Mate /8 Skill Builder 0

183 Skill 0. Writing expressions to represent word problems. Write the expression using the variables and/or the numbers mentioned in the word problem. Decide about the operation or operations needed in the expression. Example: a + b (sum of a and b), n (product of and n), m 0 (0 less than m) Hint: Sum, altogether, in total, more than addition + Difference, less than, change subtraction Product, times, lots of multiplication A fraction (half, third, quarter) of division Q. Write as an expression: The number less than c A. less than c a) Write as an expression: The sum of n and n and +... n b) Write as an expression: The sum of b and b... c) Write as an expression: The sum of e and f... d) Write as an expression: A number more than j... e) Write as an expression: A number less than z... f) Write as an expression: A number less than v... g) Write as an expression: A number times m... h) Write as an expression: A number times d... i) Write as an expression: A number that is equal to twice as much as h... j) Write as an expression: A number that is equal to three times as much as m... k) A person grows cm every year for y years. How much did he grow?... l) Write as an expression: A number that is equal to seven lots of z... page 9 Maths Mate /8 Skill Builder 0

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185 . [Substitution] Skill. Substituting one value into expressions involving + and Replace the letter (variable) with the given value. Use the order of operations rule: Add ( + ) and/or subtract ( ) from left to right. Q. If a, find the value of a A. a 8 substitute a a) If p, find the value of + p b) If f, find the value of + f c) If c, find the value of + c + d) If m, find the value of m + e) If g, find the value of g + f) If z, find the value of z + g) If x, find the value of x + x h) If v, find the value of v + v i) If q, find the value of q + q j) If t, find the value of t + t + t k) If e, find the value of e + e + e l) If p 8, find the value of p + p + p m) If j 9, find the value of j + j 8 n) If k, find the value of k + k + o) If h 8, find the value of + h + h p) If m 8, find the value of m + m 9 q) If s, find the value of 9 + s + s r) If n, find the value of 8 + n + n page Maths Mate /8 Skill Builder

186 Skill. Substituting one value into expressions involving and Replace the letter (variable) with the given value. Use the order of operations rule: Multiply ( ) and/or divide ( ) from left to right. Q. If m, find the value of m A. m substitute m a) If a, find the value of 9 a b) If n, find the value of n c) If y, find the value of y d) If w, find the value of w e) If p 8, find the value of p f) If z, find the value of z g) If a, find the value of 8a h) If h, find the value of 9h i) If n, find the value of n j) If m, find the value of m k) If n, find the value of n l) If k, find the value of k m) If d 9, find the value of 8 d n) If p 8, find the value of p o) If i, find the value of i p) If m, find the value of m... q) If e 0, find the e value of... r) If w 9, find the 08 value of w... page Maths Mate /8 Skill Builder

187 Skill. Substituting one value into expressions involving +,, and Replace the letter (variable) with the given value. Use the order of operations rules: First multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Q. If q 8, find the value of q + A. q substitute q 8 a) If w, find the value of 0 w Do first b) If x, find the value of x +... c) If m, find the value of + m... d) If x, find the value of + x e) If a, find the value of + a f) If b, find the value of b g) If s, find the value of + s h) If v, find the value of 9v 8 i) If h, find the value of h j) If k, find the value of k k) If w, find the value of 8w l) If u, find the value of u m) If e 9, find the value of e + 8 n) If s, find the value of s + o) If c, find the value of 9 c page Maths Mate /8 Skill Builder

188 Skill. Substituting negative values into expressions. Replace the letter (variable) with the given value. Use the order of operations rules: First multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Determine the sign of the result. (see skills. to.0, pages 98 to 0) Q. If z, find the value of z 9 A. z 9 9 substitute z a) If e, find the value of 9 + e 9 + ( ) b) If y, find the value of 9y c) If r, find the value of r d) If n, find the value of n + 8 e) If z 9, find the value of z f) If h, find the value of 8 + h g) If j, find the value of 8 j h) If v 8, find the value of v i) If b, find the value of + b j) If b 9, find the value of b k) If f, find the value of f l) If i, find the value of i m) If a, find the a value of n) If e, find the e value of o) If c, find the c value of 8 p) If s, find the value of + s q) If q, find the value of q r) If x 9, find the value of x page Maths Mate /8 Skill Builder

189 Skill. Substituting two values into expressions involving + and Substitute the letters (variables) with the given values. Use the order of operations rule: Add ( + ) and/or subtract ( ) from left to right. Determine the sign of the result. (see skills. to.0, pages 98 to 0) Q. If h and i, find the value of h + i A. h + i + ( ) substitute h and i a) If s 9 and t 8, find the value of s + t b) If m and n, find the value of m + n c) If i 0 and j, find the value of i + j d) If y 0 and z, find the value of y + z e) If k and l, find the value of k l f) If g and h, find the value of g h g) If p and q, find the value of p + q h) If n and o 8, find the value of n o i) If a and b, find the value of a b j) If h and i, find the value of h + i k) If v 8 and w 9, find the value of v w l) If f and g, find the value of f g m) If r and s, find the value of r s n) If a and b, find the value of a b o) If q and r, find the value of q + r p) If t 0 and u, find the value of t u q) If v and w 8, find the value of v + w r) If w and x 9, find the value of w x page Maths Mate /8 Skill Builder

190 Skill. Substituting two values into expressions involving and Substitute the letters (variables) with the given values. Use the order of operations rules: Multiply ( ) and/or divide ( ) from left to right. Determine the sign of the result. (see skills. to.0, pages 98 to 0) Q. If q and r 8, find the value of qr A. qr 8 8 substitute q and r 8 a) If e and f, find the value of e f b) If n and o, find the value of n o c) If b 0 and c, find the value of b c d) If y and z 9, find the value of yz e) If g and h, find the value of gh f) If l and m, find the value of lm g) If s and t, find the value of st h) If w and x 8, find the value of wx i) If d and e 0, find the value of de j) If w 0 and x, find the value of w x k) If v and w 9, find the value of v w l) If u and v, find the value of u v m) If a and b, find a the value of b n) If c and d 9, find c the value of d o) If k and l, find k the value of l p) If l 0 and m, find the value of 9lm q) If k and l, find the value of 8kl r) If d and e, find the value of de page Maths Mate /8 Skill Builder

191 Skill. Substituting two values into expressions involving +,, and Substitute the letters (variables) with the given values. Use the order of operations rules: First multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Determine the sign of the result. (see skills. to.0, pages 98 to 0) Q. If m 8 and n 9, find the value of m n A. m n substitute m 8 and n 9 a) If t and u, find the value of t + u +... Do first + 9 b) If d 8 and e, find the value of d + e... c) If h and i, find the value of + h i... d) If i and j, find the value of ij e) If a and b 0, find the value of 8ab f) If m and n, find the value of m n g) If m and n, find the value of m n h) If b and c, find the value of bc + 0 i) If g and h 9, find the value of gh + h j) If a and b, find the value of a + b... k) If y and z, find the value of 9 y z l) If g and h, find the value of h g page Maths Mate /8 Skill Builder

192 Skill.8 Substituting into expressions involving powers. Substitute the letters (variables) with the given values. Use the order of operations rules: First evaluate all powers. Then multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Determine the sign of the result. (see skills. to.0, pages 98 to 0) Q. If i, find the value of i i A. i i ( ) 8 substitute i a) If x, find the value of 0 x d) If c, find the value of c multiply first b) If j 8, find the value of j e) If d, find the value of d 9 c) If m, find the value of 8 + m... f) If k, find the value of k... g) If z, find the value of z... h) If y 0, find the value of y + y... i) If b, find the value of b j) If t, find the value of t + t k) If e, find the value of e l) If n, find the value of n page 8 Maths Mate /8 Skill Builder

193 Skill.9 Substituting into expressions with brackets. Substitute the letters (variables) with the given values. Use the order of operations rules: First evaluate inside the brackets. Then multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Determine the sign of the result. (see skills. to.0, pages 98 to 0) Q. If r, find the value of (r ) A. (r ) ( ) substitute r a) If h, find the value of ( + h) ( + ) Do ( ) first... b) If z, find the value of ( z)... c) If s, find the value of s( + s)... d) If a, find the value of (a + ) e) If r, find the value of (r ) f) If r 9, find the value of r( + r) g) If q, find the value of 9(q + 8) h) If k, find the value of k(k 8) i) If h, find the value of (h ) j) If f 9, find the value of (f + ) k) If p, find the value of p( p) l) If m, find the value of m(m ) m) If g, find the value of (g + ) n) If h, find the value of h( h) o) If e, find the value of e(e ) page 9 Maths Mate /8 Skill Builder

194 Skill.00 Substituting into formulae. Substitute the letters (variables) with the given values. Use the order of operations rules: First evaluate all powers. Then multiply ( ) and/or divide ( ) from left to right. Finally add ( + ) and/or subtract ( ) from left to right. Determine the sign of the result. (see skill., page 98 to skill.0, page 0) Q. Use A lw to find the area (A) of a rectangle where l and w A. A lw substitute l and w a) Use F ma to find the force (F) where m and m F b) Use P l to find the perimeter (P) of a regular pentagon where l c) Use V Bh to find the volume (V) of a prism where B and h d) Use A l to find the area (A) of a square where l 9 ab e) Use A to find the area (A) of a kite where a 8 and b.... f) Use d vt to find the distance (d) where v 9 and t g) Use A bh to find the area (A) of a parallelogram where b. and h h) Use P 8l to find the perimeter (P) of an octagon where l. i) Use A h(a + b) to find the area (A) of a trapezium where h, a and b.... j) Use V l h to find the volume (V) of a square prism where l and h k) Use V l to find the volume (V ) of a cube where l l) Use A πr to find the area (A) of a circle where π. and r 0 page 80 Maths Mate /8 Skill Builder

195 . [Equations] Skill. Finding the missing number in equations involving + and (). EITHER Use trial and error: Guess the value of the missing number that will make the equation true (both sides of the equation are equal). Substitute this value in the equation. Check if the equation is true. Write the guessed value as the solution of the equation. Example: +? + 8 (true) The equation is true, so 8 is the solution. OR Use inverse operations: Consider the operation used to construct the sum or the difference. Get the missing number alone on one side of the equation, by performing the inverse operation to both sides of the equation. Evaluate the other side of the equation. Hints: Addition and subtraction are inverse operations. Adding and then subtracting leaves a number unchanged. Example: +? + 8?? continues on page 8 Q. 9 A.? 9 a) 9 What number subtracted from gives 9? Guess? Use trial and error OR (true) The solution is. b) +? 9? 9?? If was added to the missing number, then do the inverse operation and subtract from both sides of the equation. Finally, reverse the signs on both sides. c) + 0? 9...?... d) +?...?... e)? ?... f) ?......?......?... g) h) + i) 8...?......?......?... page 8 Maths Mate /8 Skill Builder

196 Skill. Finding the missing number in equations involving + and (). continued from page 8 Operation: j) 8 + Use inverse operations Inverse of + 8 is 8 8 +? ? 9... k) + 0? ?... l) ?... m) + n) + 0 o) p) + 9 q) + r) s) 8 t) 8 u) v) w) x) y) 9 z) 8 zz) page 8 Maths Mate /8 Skill Builder

197 Skill. Finding the missing number in equations involving (). EITHER Use trial and error: Guess the value of the missing number that will make the equation true (both sides of the equation are equal). Substitute this value in the equation. Check if the equation is true. Write the guessed value as the solution of the equation. Example:? (true) The equation is true, so is the solution. Q A.? 0 00 What number multiplied by 0 gives 00? Guess? (true) The solution is. OR Use inverse operations: Consider the operation used to construct the multiplication or the division. Get the missing number alone on one side of the equation, by performing the inverse operation to both sides of the equation. Evaluate the other side of the equation. Hints: Multiplication and division are inverse operations. Multiplying by and then dividing by leaves a number unchanged. Example:? OR??? 0 00? ? continues on page 8 If 0 was multiplied by the missing number, then do the inverse operation and divide by 0 both sides of the equation. a) 9 Use trial and error b) 0 0 c) 8 9?...?... 0? 0...?......?... d) e) 0 f)...?......?......?... g) h) i) 8...?......?......?... page 8 Maths Mate /8 Skill Builder

198 Skill. Finding the missing number in equations involving (). continued from page 8 Operation: j) 0 Use inverse operations Inverse of is? 0...?... k) 0? 0...?... l)...?... m) n) 9 o) p) 8 q) 8 r) s) 0 0 t) 9 8 u) v) 9 8 w) x) y) ( ) z) ( 8) zz) page 8 Maths Mate /8 Skill Builder

199 Skill. Finding the missing number in equations involving fractions (). EITHER Use trial and error: Guess the value of the missing number that will make the equation true (both sides of the equation are equal). Substitute this value in the equation. Check if the equation is true. Write the guessed value as the solution of the equation. Hint: of means multiplication, so use Example: of? (true) The equation is true, so is the solution. OR Use inverse operations: Consider the operation used to construct the division. Get the missing number alone on one side of the equation, by performing the inverse operation to both sides of the equation. Evaluate the other side of the equation. Hints: Multiplication and division are inverse operations. Multiplying by (which is the same as dividing by ) and then multiplying by leaves a number unchanged. Example:??? continues on page 8 Q. of A. of? OR What number multiplied 0 by gives? 8 (false) Guess? 0 (true) Guess? The solution is. of???? 0? 0? If the missing number has been divided by and then multiplied by, then do the inverse operations and multiply by and then divide by both sides of the equation. Use trial and error a) of 8 8? Guess? ? 8... b) of c) of 9?? ? ?... d) 9 e) 0 9 f) 0...?......?......?... page 8 Maths Mate /8 Skill Builder

200 Skill. Finding the missing number in equations involving fractions (). Operation: 8 g) 8 8? Use inverse operations Inverse of 8 is 8...?... h) 8...?... i) 0 continued from page 8...?... j) of 0 k) of l) of m) 0 n) 0 o) p) q) r) s) t) 8 u) page 8 Maths Mate /8 Skill Builder

201 Skill. Finding the missing number in equations involving +,, and/or brackets (). EITHER Use trial and error: Guess the value of the missing number that will make the equation true (both sides of the equation are equal). Substitute this value in the equation. Check if the equation is true. Write the guessed value as the solution of the equation. Example:? What number minus gives?? (true) The equation is true, so is the solution. OR Use inverse operations: Consider the operation used to construct the equation. Get the missing number alone on one side of the equation, by performing the inverse operation to both sides of the equation. Evaluate the other side of the equation. Hints: For simplicity consider the equation inside the brackets, as one number. Example:?? + + 8?? continues on page 88 OR? (true) The solution is. Q. ( ) 0 A. (?) 0 What number multiplied by gives 0? Guess? (?) 0???? If the bracket has been multiplied by, then do the inverse operation and divide by both sides of the equation. Then subtract from both sides. Finally reverse the signs. 0 Use trial and error a) ? 8...? 0...? 0... d) (9 ) (9?)... 9?...?... b) + +?...?...?... e) (8 )......?... c) ?... f) (9 )......?... page 8 Maths Mate /8 Skill Builder

202 Skill. Finding the missing number in equations involving +,, and/or brackets (). continued from page 8 Operation: + g) + Use inverse operations Inverse of + is +?...? 0...?... h) ?... i) ?... j) 8 ( ) k) ( ) l) 8 (0 ) m) 0 n) o) p) + q) + 8 r) s) t) 0 u) page 88 Maths Mate /8 Skill Builder

203 Skill. Finding the missing number in equations involving decimals. Use trial and error or inverse operation to find the missing number. (see skill., page 8 and skill., page 8) Q. +.. A.? +.. OR? +.. What number added ? to. gives.?.. (true)? 0. Guess? 0. The solution is 0. If. was added to the missing number, then do the inverse operation and subtract. from both sides of the equation. a)..?.. Guess?...?... d).8 Use trial and error b) ?....?... e).9 0. c) ?... f)..9...?......?......?... Operation: +.. g).. Use inverse operations Inverse of +. is..?......?....?.... h) i) j) +. k). + l) m).. n).. o) page 89 Maths Mate /8 Skill Builder

204 Skill. Solving one-step equations by using the inverse operations of + and (). Consider the operation used to construct the sum or the difference involving the variable. Get the variable alone on one side of the equation, by performing the inverse operation to both sides of the equation. Evaluate the other side of the equation. Hint: Remember that you must do the same operation to both sides of the equation. Operation + Inverse Operation Operation continues on page 9 + x + x x + x + + x x 9 Inverse Operation Q. Solve for p: p A. + p Simplify: 0 p p p Operation: + Inverse of + is Reverse sign both sides Operation: + a) Solve for t: t + Inverse of + is t +... b) Solve for y: y + y +... c) Solve for r: + r... t 9 y r d) Solve for a: a e) Solve for x: 8 + x f) Solve for m: + m a x m g) Solve for e: e + 9 h) Solve for g: g + i) Solve for s: + s e g s j) Solve for t: t k) Solve for y: y 9 l) Solve for z: z t y z page 90 Maths Mate /8 Skill Builder

205 Skill. Solving one-step equations by using the inverse operations of + and (). m) Solve for x: x 0 n) Solve for b: b 8 continued from page 90 o) Solve for s: s x b s p) Solve for a: a q) Solve for z: z 0 r) Solve for s: 8 s a... a a 8 z s s) Solve for j: 0 j t) Solve for c: c u) Solve for e: e j c e v) Solve for d: + d 9 w) Solve for v: + v x) Solve for n: 8 + n d v n y) Solve for h: 9 + h... h z) Solve for k: + k... k zz) Solve for m: + m... m page 9 Maths Mate /8 Skill Builder

206 Skill. Solving one-step equations by using the inverse operations of and (). Consider the operation used to construct the expression involving the variable. Get the variable alone on one side of the equation, by performing the inverse operation on both sides of the equation. Evaluate the other side of the equation. Hint: Remember that you must do the same operation to both sides of the equation. Operation Inverse Operation Operation continues on page 9 x x x x x x 8 Inverse Operation x Q. Solve for x: A. 8 Simplify: 8 8 x 8 x x 8 Operation: 8 Inverse of 8 is 8 Operation: a) Solve for a: a b) Solve for m: m 0 c) Solve for c: c a 9 Simplify: Inverse of is a 9 m c d) Solve for h: h e) Solve for n: 9 n 8 f) Solve for p: 8 p h n p g) Solve for b: 8b h) Solve for z: z 8 i) Solve for l: 9l b z l j) Solve for r: 0r 0 k) Solve for y: y l) Solve for u: u r y u page 9 Maths Mate /8 Skill Builder

207 Skill. Solving one-step equations by using the inverse operations of and (). m) Solve for g: g 0 g 0... n) Solve for a: 0a continued from page 9 o) Solve for s: s... g a s p) Solve for m: m 0 q) Solve for n: n 8 r) Solve for g: g m n g s) Solve for d: 0d 80 t) Solve for p: p u) Solve for h: 9h d p h x v) Solve for x: 9 x 9 Inverse of is Operation:... w) Solve for c:... c x) Solve for q: q 8... x c q y) Solve for n: n z) Solve for r: r A) Solve for j: 8 j n r j B) Solve for b: b C) Solve for e: e D) Solve for k: 9 0 k b e k page 9 Maths Mate /8 Skill Builder

208 Skill.8 Solving two-step equations by using the inverse operations of +,, and (). continues on page 9 Get the variable alone on one side of the equation, by performing the inverse operations, in order, to both sides of the equation. (see skill., page 90 and skill., page 9). Evaluate the other side of the equation. Hint: Remember that you must do the same operation to both sides of the equation. Q. Solve for v: 9v 0 A. 9v 0 Simplify: + 0 9v v 8 9v 8 Simplify: v Operation: Inverse of is + Operation: 9 Inverse of 9 is 9 Operation: + 8 a) Solve for x: x x x Inverse of is... Simplify: x Inverse of + 8 is 8... b) Solve for y: y c) Solve for a: a x y a d) Solve for d: d + 9 e) Solve for e: e f) Solve for u: 8u d e u g) Solve for x: x h) Solve for t: t i) Solve for h: h x t h page 9 Maths Mate /8 Skill Builder

209 Skill.8 Solving two-step equations by using the inverse operations of +,, and (). j) Solve for i: i 9 k) Solve for q: q continued from page 9 l) Solve for s: 8s i q s m) Solve for i: i + 0 n) Solve for j: j + 0 o) Solve for l: 0l i j l p) Solve for x: 9x + 0 q) Solve for z: z + 9 r) Solve for c: c x z c s) Solve for g: g + 8 t) Solve for m: 9m + 0 u) Solve for p: p g m p page 9 Maths Mate /8 Skill Builder

210 page 9 Maths Mate /8 Skill Builder

211 . [Coordinates] Skill. Describing the position of ordered pairs on a Cartesian plane. Start at the origin (0,0). Move left or right by the number of given units. This first number becomes the x-coordinate. Use a + sign if you moved to the right and use a if you moved to the left. From that point, move up or down by the number of given units. This second number becomes the y-coordinate. Use a + sign if you moved up and use a if you moved down. Plot the final point on the Cartesian plane. Q. Start at the origin. Move units to the left along the x-axis and then down units. Plot a point. What are the coordinates of the point? Y 0 X A. units down units left (, ) Y 0 The first coordinate is ( units left) The second coordinate is ( units down) The answer is (, ) X a) Start at the origin. Move units to the left along the x-axis and then up units. Plot a point. What are the coordinates of the point? Y 0 X b) Start at the origin. Move 0 units to the right along the x-axis and then down units. Plot a point. What are the coordinates of the point? Y X (, ) (, ) c) Start at the origin. Move units to the left along the x-axis and then up units. Plot a point. What are the coordinates of the point? Y 0 X d) Start at the origin. Move units to the right along the x-axis and then up units. Plot a point. What are the coordinates of the point? Y (, ) (, ) X page 9 Maths Mate /8 Skill Builder

212 H I J K L A B C D E F G M Skill. Using grid references to describe location on a map (). continues on page 99 Locate the object on the grid. Starting from the corner of the grid, first read across the horizontal axis to find the letter that matches the column of the object. Then read along the vertical axis to find the number that matches the row of the object. Write the letter followed by the number to specify the grid reference. Q. In this AFL starting line up, what is the grid reference of the Ruck (Ru)? a) In which country would you be in if you were located at A? A FB A B C D E F G H I J K BP BP EGYPT CHB Backs MIDDLE EAST HBF HBF Centres SYRIA LEBANON IRAQ ISRAEL JORDAN W RR Ro Ru C W KUWAIT SAUDI QATAR ARABIA UAE YEMEN Forwards IRAN OMAN HFF HFF B C D E F G H CHF FP FP FF A. The grid reference is F. A B C D E F G H I J K W BP HBF HFF FP FB CHB RR Ru Ro CHF FF C BP HBF HFF FP W Backs Centres Forwards b) Above which continent would you be if your airplane is flying at O? 8 A B C D E F G H I J K L M N O P c) Which occupation is listed at C? d) On which island would you be in if you were located at G8? FLORIST OPTICIAN CLERK EDITOR BANKER LOCKSMITH SECRETARY CLEANER SCIENTIST COURIER ATTORNEY PRINTER BUTCHER AUTO MECHANIC TAILOR DOCTOR A B C D LANDSCAPER BAKER ENGINEER ACCOUNTANT E 8 A ISABELA FERNANDINA THE GALÁPAGOS ISLANDS B C D E F G H I PINTA SANTIAGO MARCHENA PINZÓN RÁBIDA GENOVESA SANTA CRUZ FLOREANA SANTA FÉ Equator SAN CRISTÓBAL ESPAÑOLA page 98 Maths Mate /8 Skill Builder

213 Skill. Using grid references to describe location on a map (). e) On which mountain would you be if you were located at C? A B C D E F G H I J K South Pacific Ocean Mount Tuutapu 0 m Hanga Piko Mataveri Volcano Rana Kao Mount Vaka Kipo m Mount Orito 0 m Volcano Terevaka 0 m Mount Pui 0 m Mount O Tu u 00 m Volcano Rano Raraku Mount Kote Miro Oone 9 m Mount Ana Marama m Volcano Puakatike 0 m EASTER ISLAND (CHILE) continued from page 98 f) In this grid iron starting line up, what is the grid reference of the Tight End (TE)? A WR CB OT FS DE OG LB DL TB FB QB C OG OT TE LB DL LB B C D E F G H I J K DE SS WR CB Offense Defense g) What is the grid reference of an enemy hit on a battleship? h) What is the grid reference of Grass Lake? A B C D E F G H I J K A B C D E F G H I J K MINNEAPOLIS Meadowbrook Lake Mirror Lake 00 Cedar Lake Lake Calhoun Lake Harvey 9 Grass Lake 9 Lake of the Isles LYNDALE PARK Lake Harriet W POWDERHORN PARK HIAWATHA PARK Lake Nokomis Diamond Lake Mother Lake Enemy hit Battleship i) What is the grid reference of the Olympic flag? j) What is the grid reference of hydrogen? A B C D E A B C D E page 99 Maths Mate /8 Skill Builder

214 Skill. Using coordinates to describe location on a map. Locate the object on the coordinate plane. Move vertically from the object until you intersect the horizontal axis (x-axis). Write the number you find on the horizontal axis as the x-coordinate of the point (x, ). Move horizontally from the object until you intersect the vertical axis (y-axis). Write the number you find on the vertical axis as the y-coordinate of the point (,y). Read the coordinate on the horizontal axis first, then on the vertical axis. Hint: x before y in the alphabet is one way to remember this order. Q. What are the coordinates of Dampier? A. Locate Dampier on the map. Y Follow the vertical line that Dampier is on, Arafura Sea AUSTRALIA down to where it meets the horizontal axis. Timor Sea Darwin The x-coordinate is. Follow the horizontal line that Dampier is on, back to where it meets the vertical axis. The y-coordinate is. The coordinates that describe the location of Dampier are (,). a) What are the coordinates of Amritsar? b) Which city is located at the coordinates (,)? Ahmadabad Srinagar Amritsar Jaipur Bombay Arabian Sea Mormugao Indian Ocean Dampier Perth Calicut Alice Springs Simla New Delhi Agra Jabalpur Raipur Nagpur Hyderabad Madurai Madras Cairns Coral Sea Rockhampton Whyalla Broken Hill Canberra Sydney Indian Ocean Melbourne Tasmania Brisbane Launceston INDIA Ganglok Ledo Dispur Patna Kohima Agartala Imphal Ajal Calcutta Bay of Bengal Puri X 0 0 NIIHAU KAUAI HAWAII OAHU Kailua MOLOKAI Honolulu Kahului LANAI KAHOOLAWE MAUI HAWAII Hilo c) What are the coordinates of Johannesburg? 0 0 SOUTH AFRICA Atlantic Ocean Okiep Calvinia Cape Town Victoria West Queenstown Messina TRANSVAAL Pretoria Mbabane Johannesburg SWAZILAND Kroonstad NATAL Kimberley ORANGE FREE Maseru CAPE STATE LESOTHO Durban Port Elizabeth Port Shepstone Indian Ocean d) What are the coordinates of Christchurch? Y NEW ZEALAND North Island Auckland Greymouth Wellington Christchurch South Island Stewart Island X page 00 Maths Mate /8 Skill Builder

215 Skill. Finding the coordinates of a point on a Cartesian plane (). continues on page 0 Locate the point on the coordinate plane. Move vertically from the object until you intersect the horizontal axis (x-axis). Write the number you find on the horizontal axis as the x-coordinate of the point (x, ). Move horizontally from the object until you intersect the vertical axis (y-axis). Write the number you find on the vertical axis as the y-coordinate of the point (,y). Hints: Always write the x-coordinate first. The coordinates of the origin O are (0,0). Q. What are the coordinates of the pear and the strawberry? A. Y Y X coordinate plane 0 pear (9,) strawberry (,) X write horizontal coordinate first Read as: strawberry of coordinates and a) What are the coordinates of the points M and N on this Cartesian plane? M Y 0 N X b) What are the coordinates of the sun and the moon? Y X M(, ) N(, ) sun (, ) moon (, ) c) Which point lies on the line graphed below? M(,) N(0,) P(,) d) Which point lies on the line graphed below? E(, ) F(,) G(, ) H(0,8) Y N(0,) P(,) Y M(,) X 0 X page 0 Maths Mate /8 Skill Builder

216 Skill. Finding the coordinates of a point on a Cartesian plane (). e) These dots, if joined, would form a line. A point on this line has an x-coordinate of. What is the y-coordinate of this point? Y X continued from page 0 f) These dots, if joined, would form a line. A point on this line has a y-coordinate of. What is the x-coordinate of this point? Y X (, ) (, ) g) These dots, if joined, would form a line. A point on this line has an x-coordinate of. What is the y-coordinate of this point? Y 0 X h) What are the coordinates of point R that will make OPQR a square? Y 0 P Q X (, ) i) What are the coordinates of point F that will make CDEF a parallelogram? C Y 0 D E X j) What are the coordinates of point J that will make GHIJ a rectangle? H G I Y 0 X page 0 Maths Mate /8 Skill Builder

217 Skill. Plotting ordered pairs on a Cartesian plane. Start at the origin (0,0) of the Cartesian plane. Move across the x-axis by the number of units equal to the first coordinate (move to the right if the coordinate is positive and to the left if the coordinate is negative). Draw a vertical line passing through this point. From the origin, move along the y-axis by the number of units equal to the second coordinate (move up if the coordinate is positive and down if the coordinate is negative). Draw a horizontal line passing through this point. Plot the point at the intersection of these two lines. Q. Draw crosses at the following points: (, ), (, ), (,0), (9,), (,), (,) Y A. Y Plot all points (,) (,) (,0) (9,) X X (, ) (, ) a) Starting at H, draw a line to H then continue to D, D, H, D, F, H and D. What shape have you drawn? 0 A B H C D E F G H I J K b) Starting at E, draw a line to E then continue to J, J, H, H, G, G and E. What letter have you drawn? 0 A B C D E F G H I J K c) Plot the following points on this Cartesian plane: P at coordinates (,) Q at coordinates (, ) R at coordinates (, ) Y 0 X d) Which set of ordered pairs lie within this pentagon? A ) (,), (, ) B ) (,), (, ) C ) (, ), (,) 0 y x page 0 Maths Mate /8 Skill Builder

218 Skill. Writing linear expressions to represent real-life situations (). Use addition to represent a total or a sum. Use subtraction to represent a difference or a remainder. Use multiplication to represent a quantity which is a number of times greater or smaller than another quantity. Simplify the expression. Q. A school canteen sells cups of soup for $.0 and sandwiches for $. Which expression represents the cost of cups of soup and sandwiches? A ) (. + ) B ). + C ) (. + ) continues on page 0 A. cups of $.0 each. $.00 each total cost is. + The answer is B a) A printer can print 0 pages per minute. Which expression represents the total number of pages printed in minutes? A ) 0 B ) + 0 C ) 0 total pages 0... A b) Suzie bought a bike for $90, and then sold it for $0 less. Which expression represents the selling price? A ) B ) 90 0 C ) 90 0 selling price... c) A Boeing has a typical cruise speed of 9 km/h. Which expression represents the distance travelled in hours? A ) 9 + B ) 9 C ) 9 distance... d) A bike can travel km/h. Which expression represents the time, in hours, taken to travel km? A ) B ) C ) time... e) To send a fax costs $ for the first page and then $ for each of the following pages. Which expression represents the total cost to send a -page fax? A ) + B ) ( + ) C ) + cost first page $... cost remaining pages $... total cost +... f) The first minutes of a phone call cost $, and then each minute costs 0 cents. Which expression represents the total cost of a minute call? A ) + 0. B ) + 0. C ) + 0 cost first min... cost remaining minutes... total cost... page 0 Maths Mate /8 Skill Builder

219 Skill. Writing linear expressions to represent real-life situations (). g) Aquatic Centre Tickets type of ticket price ($) adult child ( - ) student pensioner Which expression represents the total cost for adults and children aged six? A ) + B ) ( + ) C ) ( + ) cost adult tickets... cost child tickets... total cost... h) A bus company spends $0 per hour to run a bus. Fifty students paid $0 each to travel by bus. If the excursion lasted hours, which expression represents the profit made by the bus company? A ) B ) C ) profit continued from page 0... i) Population - July 00 continent population (millions) The Americas & the Caribbean 9 Europe Asia 0 Oceania x Which expression represents the total population living in these four continents? A ) x B ) x C ) x j) Space Shuttle Missions up to August 009 shuttle missions Columbia 8 Challenger 0 Endeavour Discovery 9 Atlantis x Which expression represents the total number of missions flown by all shuttles? A ) + x B ) 0 + x C ) 0 x... total population total missions... k) Di has x one-dollar coins and y two-dollar coins in her bag. Which expression represents the total amount in dollars she has in her bag? A ) (x + y) B ) (x + y) C ) x + y... l) Nico drove x kilometres, and Mia drove y kilometres. If Nico drove z kilometres more than Mia, which equation expresses this? A ) x y + z B ) y x + z C ) z x + y... page 0 Maths Mate /8 Skill Builder

220 Skill. Completing a table of values for a linear rule (). Substitute the variable x with the given values. Calculate the values of y. continues on pages 0 Q. Complete the table of values for the linear rule y + x x 0 + x + 0 y A. y + x x y + x y + 0 x y + x y + x y + x 0 + x y 0 Substitute x a) Complete the table for this rule: Houses sold (x) Earnings (000x) b) Complete the table for this rule: Number of guests (x) Dinner cost in dollars (x) c) Complete the table for this rule: No. of days (x) Records entered (90x) d) Complete the table for this rule: No. of days (x) Number of T-shirts sold (x) e) Complete the table for this rule: No. of hours worked (x) Pay in dollars (8x) f) Complete the table for this rule: No. of seconds (x) Distance travelled in metres (8x) page 0 Maths Mate /8 Skill Builder

221 Skill. Completing a table of values for a linear rule (). g) Complete the table of values for the linear rule y x + continued from page 0 h) Complete the table of values for the linear rule y 8 x x x + y x 8 x y i) Complete the table of values for the linear function y + x j) Complete the table of values for the linear function y x x x + 0 y x 0 x 0 y k) Complete the table of values for the linear function y x l) Complete the table of values for the linear function y x x x y x x y 0 m) Complete the table of values for the linear function y 00 x n) Complete the table of values for the linear function y x x x 00 0 y 0 x 0 x 0 y page 0 Maths Mate /8 Skill Builder

222 Skill.8 Graphing linear functions on a Cartesian plane (). To determine the correct equation of a given line: EITHER Choose two points lying on the linear graph. Substitute the coordinates of these points in the equation of the line. Check if they are both true statements. OR Check for special properties of the x-coordinates or the y-coordinates. Example: All the points where x means that all points are lying on a vertical line passing through the point (,0). To draw the graph of a given equation: Choose two different pairs of numbers (x,y) which satisfy the equation. Plot these two pairs of coordinates. Join the points. continues on page 09 Q. Draw a line connecting all the points where the x-coordinate and the y-coordinate add to 8. A. x + y 8 Choose x 8 and y 0 the point (8,0) Choose x and y the point (,) Y Y 8 8 (,) (8,0) X X a) Y b) Y 0 X 0 (, ) (0, ) X The line above shows: A ) All points where x y B ) All points where x C ) All points where y x 0 and y x y A) 0 ( ) or (true)... x and y x y ( ) or (false)... B) 0 (false) and (false)... C) (true)... (true)... The line above shows: A ) All points where y x B ) All points where x C ) All points where y A) B)... C) page 08 Maths Mate /8 Skill Builder

223 Skill.8 Graphing linear functions on a Cartesian plane (). c) Y d) Y continued from page 08 0 X 0 X The line above shows: A ) All points where y B ) All points where x C ) All points where x + y... The line above shows: A ) All points where x 0 B ) All points where y x C ) All points where y 0... e) Draw a line through all the points where the x-coordinate and the y-coordinate add to 0. f) Draw a line through all the points where the x-coordinate and the y-coordinate add to. 8 Y Y 0 X X g) Draw a line through all the points where the y-coordinate is more than the x-coordinate. h) Draw a line through all the points where the x-coordinate is more than the y-coordinate. Y Y 0 X X page 09 Maths Mate /8 Skill Builder

224 Skill.9 Using coordinates to visualise and draw transformations of two-dimensional shapes on a Cartesian plane. Calculate the new coordinates of every vertex of the shape. Plot the new points. Draw and label the new shape. Q. Redraw the parallelogram ABCD after subtracting from the x-coordinate and to from the y-coordinate of every point on the parallelogram. Y B A C D A. A(0,) 0 and the new position of A becomes A (,) Y B' A'(,) 0 C' B A(0,) D' C D X X a) Redraw the square ABCD after adding to both the x and y coordinates of every point on the square. b) The value of the coordinates of all points A, B and C are doubled. Redraw the triangle. Y Y D'(0,) B C B + C A D + A X X c) Redraw the rectangle ABCD after subtracting from both the x and y coordinates of every point on the rectangle. d) The value of the coordinates of all points A, B, C and D are halved. Redraw the rhombus. Y B A C D Y D A C B X X page 0 Maths Mate /8 Skill Builder

225 Skill.0 Plotting points from a table of values on a Cartesian plane. For each point read the x-coordinate and the y-coordinate from the table of values. Plot and label each point on the Cartesian plane. Q. Using the table of values, plot the points on the Cartesian plane. x y 0 0 Y 0 X A. x-coordinate, y-coordinate point (, ) Continue reading the ordered pairs: points (, ) (0, ) (,0) (,) (,) Y 0 (, ) (, ) (,0) (0, ) (,) (,) X a) Using the table of values, plot the points on the Cartesian plane. b) Using the table of values, plot the points on the Cartesian plane. x y 0 x y 9 Y Y 0 X X c) Using the table of values, plot the points on the Cartesian plane. x y d) Using the table of values, plot the points on the Cartesian plane. x y 8 0 Y Y X 0 X page Maths Mate /8 Skill Builder

226 page Maths Mate /8 Skill Builder

227 . [Units of Measurement / Time] Skill. Converting units of time (). Use these conversion factors for units of time. continues on page week days 8 h min s day h 0 min 8 00 s h 0 min 00 s min 0 s To change from smaller units to larger units Divide by the conversion factor (because you need less). Example: To change s to min by 0 To change from larger units to smaller units Multiply by the conversion factor (because you need more). Example: To change h to min by 0 Q. min 0 s s A. min 0 s 0 s + 0 s 0 s + 0 s 80 s min to s: 0 a) 00 seconds minutes s to min: 0 00 s 00 0 min 0 min... b) hours minutes h to min: 0... c) minutes seconds d) 80 s min e) 0 h min f) 0 min h g) 00 min h h) min s i) days h j) weeks days page Maths Mate /8 Skill Builder

228 Skill. Converting units of time (). continued from page k) 0 years months l) days h m) 90 min h n) 0 min h o) h min min p) min 0 s s q) weeks, days days r) h 0 min min s) min s s t) h 0 min min u) days h day to h:... v) h min... w) day h x) h min page Maths Mate /8 Skill Builder

229 Skill. Converting units of length (). Use these conversion factors for metric units of length. continues on page km 000 m cm mm m 00 cm 000 mm cm 0 mm To change from smaller units to larger units Divide by the conversion factor (because you need less). Example: To change mm to cm by 0 mm cm To change from larger units to smaller units Multiply by the conversion factor (because you need more). Example: To change cm to mm by 0 Q. 800 cm m A. 800 cm m 8 m cm to m: 00 0 a) cm mm b) 0 mm cm cm to mm: 0 mm to cm: c) 0 cm mm d) 8 km m e) 000 m km f) m cm g) 9 m mm h) 0 mm cm i) km m j) 000 m km page Maths Mate /8 Skill Builder

230 Skill. Converting units of length (). continued from page k) m mm l) m cm m) 000 m km n) 000 cm m o) 0 m cm p) 9 cm mm q) 0 cm m r) 0 m cm s) 00 mm m t). km m u).8 m mm v) 00 m km w) 0. km m x). m mm page Maths Mate /8 Skill Builder

231 Skill. Converting units of mass. Use these conversion factors for metric units of mass. tonne 000 kg g kg 000 g To change from smaller units to larger units Divide by the conversion factor (because you need less). Example: To change g to kg by 000 To change from larger units to smaller units Multiply by the conversion factor (because you need more). Example: To change tonnes (t) to kg by 000 Q.. kg g A.. kg. 000 g g kg to g: 000 zeros, places to the right a) t kg t to kg: 000 zeros, places to the right b) 9000 g kg g to kg: c) 000 kg tonnes d). kg g e) 000 g kg f) g kg g) 8 tonnes kg h).9 kg g i) g kg j) kg t page Maths Mate /8 Skill Builder

232 Skill. Converting units of capacity. Use these conversion factors for metric units of capacity. kl 000 L ml L 000 ml or 000 cm To change from smaller units to larger units Divide by the conversion factor (because you need less). Example: To change ml to L by 000 To change from larger units to smaller units Multiply by the conversion factor (because you need more). Example: To change kl to L by 000 Q. 00 ml L A. 00 ml L L ml to L: 000 zeros, places to the left a). L ml L to ml: 000 b) L ml L to ml: c) L ml d) 8000 ml L e) 0 L ml f) 9. L ml g) 0. L ml h). L ml i) ml L j). L ml k) 00 ml L l) 00 ml L page 8 Maths Mate /8 Skill Builder

233 Skill. Converting units of time, length, mass and capacity by using real-life facts. Use the conversion factors to convert the units of time, length, mass and capacity. (see skills. to., pages to 8) Q. The ruby throated hummingbird can beat its wings at a rate of 00 beats per minute. Is this more or less than beats per hour? A. 00 beats in a minute h 0 min 00 0 min 000 beats in an hour 000 > The answer is more. a) The longest river in the world is the Nile (North-East Africa). It is km long. Express this in metres. km to m: m 000 m... c) The average weight of an adult blue whale is 0 tonnes. Express this in kilograms. m... kg e) The first athlete to run a mile in under four minutes was Australian distance champion John Landy, who ran it in minutes and 8 seconds. Express this in seconds. b) While brushing your teeth, a running tap wastes litres of water. Express this in millilitres.... d) Bamboo can grow up to metre in a day. How many centimetres is this? ml... cm f) The newborns average respiratory rate is breaths per minute. Is this more than or less than 000 breaths per hour? s g) An astronaut weighs kg on the moon. Express this weight in grams. h) Our bodies lose on average. litres of water a day. Express this in millilitres.... i) The average weight of an elephant at birth is about 0 kilograms. Express this in grams.... g g... ml j) Your heart pumps about 000 ml of blood every minute. How many litres will it pump in a day?... L page 9 Maths Mate /8 Skill Builder

234 Skill. Finding the elapsed time between two events. Calculate the time until the next closest hour. am to pm pm to am Add the time to midday. Add the time to midnight. Then add the remaining time. Then add the remaining time. Q. School starts at 8:0 am and ends at :0 pm. How long is a school day in hours and minutes? A. 8:0 to 9:00 0 min 9:00 to :00 h :00 to :0 h 0 min 0 min + h + h + 0 min h 0 min a) Find the time in hours and minutes between 8:0 am and :00 pm the same day. b) The movie begins at : pm and ends at :00 pm. How long is the movie in hours and minutes? 8:0 to 9:00 0 min, 9:00 to :00 h... :00 to :00 h... 0 min + h + h c) Mum started cooking at :0 pm and finished at : pm. How long did she cook in hours and minutes? d) Find the time in hours and minutes between :0 pm and :0 am the next day e) Find the time in hours and minutes between :00 am and : pm on the same day. f) Find the time in hours and minutes between 09:0 and :0 on the same day page 0 Maths Mate /8 Skill Builder

235 Skill. Using time zones to calculate durations. To calculate the time ahead: Add the time difference to the given time (count forward on the clock). To calculate the time behind: Subtract the time difference from the given time (count backward on the clock). To calculate the time difference: Subtract the two given times. Q. A Virgin Blue flight departs Gold Coast at :0 pm and arrives in Perth the same day at :0 pm. If Perth time is hours behind Gold Coast time how long was the flight? A. Gold Coast departure time :0 pm (Perth time :0 less h 0:0 am Perth arrival time :0 pm Flight time (using Perth time) 0:0 am to :0 pm h a) It is 0: pm in Sydney. If London time is 9 hours behind Sydney time, what time is it in London? subtract the time difference London time 0: pm less 9 h... b) You live in Canberra and want to call Grandma in Darwin, at noon, on Christmas day, Darwin time. If Darwin time is. h behind Canberra time, at what time should you call? c) Roger is in Brisbane and wants to ring Alina in Los Angeles at midnight on New Year, LA time. If Los Angeles time is hours behind Brisbane time, at what time in Brisbane should he call? d) Sven is in Melbourne and wants to ring Oscar in London at 9:00 am London time. If London time is h behind Melbourne time, at what time in Melbourne should he call? e) It is Sunday, 8 hours in Shanghai, China, and Sunday, hours in Sydney. By how many hours is Shanghai time behind Sydney time? f) A Qantas flight departs Sydney on Friday at :0 pm and arrives in Singapore on Friday at 0:0 pm. If Singapore time is hours behind Sydney time, how long is the flight? h min page Maths Mate /8 Skill Builder

236 page Maths Mate /8 Skill Builder

237 . [Perimeter] Skill. Finding the perimeter of polygons by measuring their side lengths. Measure each side length of the shape. Add together the side lengths. Q. Use a ruler to find the perimeter of the scalene triangle in millimetres. a) Use a ruler to find the perimeter of the square in centimetres. A. mm + mm + 0 mm 0 mm Measure the side lengths. Write down the lengths next to each side. mm 0 mm mm Start here b) Use a ruler to find the perimeter of the rectangle in milllimetres 8 cm mm c) Use a ruler to find the perimeter of the right-angled triangle in centimetres. d) Use a ruler to find the perimeter of the trapezium in centimetres. cm cm e) Use a ruler to find the perimeter of the parallelogram in millimetres. f) Use a ruler to find the perimeter of the polygon in millimetres. mm mm page Maths Mate /8 Skill Builder

238 Skill. Calculating the perimeter of polygons when all side lengths are given (). Add together the side lengths. continues on page Q. Calculate the perimeter of the quadrilateral. mm 0 mm mm 8 mm A. 0 mm + mm + mm + 8 mm mm mm 0 mm mm Start here 8 mm a) Calculate the perimeter of the parallelogram. mm mm b) Calculate the perimeter of the kite. cm cm mm mm cm cm 0 mm cm c) Calculate the perimeter of the quadrilateral. cm cm d) Calculate the perimeter of the scalene triangle. 0 mm cm 9 mm 0 mm cm cm mm e) Calculate the perimeter of the quadrilateral. 0 mm f) Calculate the perimeter of the trapezium. cm mm mm cm cm. cm 0 mm mm cm page Maths Mate /8 Skill Builder

239 Skill. Calculating the perimeter of polygons when all side lengths are given (). g) Calculate the perimeter of the parallelogram.. cm cm. cm cm h) Calculate the perimeter of the trapezium. 8 mm mm 0 mm continued from page 9 mm cm mm i) Calculate the perimeter of the quadrilateral.. cm cm cm j) Calculate the perimeter of the quadrilateral. mm mm 0 mm 8 mm. cm cm mm k) Calculate the perimeter of the quadrilateral.. cm l) Calculate the perimeter of the right-angled triangle. cm mm 8 mm. cm cm mm cm mm m) Calculate the perimeter of the polygon. n) Calculate the perimeter of the polygon. 0 mm mm mm 0 mm mm 0 mm mm mm mm 0 mm mm mm page Maths Mate /8 Skill Builder

240 Skill. Calculating the perimeter of polygons by recognising congruent sides. Determine and label all side lengths. Hint: Sides marked with a dash ( ) are of equal length. Sides marked with two dashes ( ) are of equal length etc. Add together the side lengths. Q. Calculate the perimeter of the isosceles triangle. A. mm + mm + mm 8 mm mm mm isosceles triangle has congruent sides mm mm mm label all sides a) Calculate the perimeter of the regular pentagon. b) Calculate the perimeter of the rhombus. mm mm mm mm mm mm mm mm pentagon - sides mm c) Calculate the perimeter of the regular hexagon. d) Calculate the perimeter of the isosceles triangle. 0 mm mm mm... mm... mm e) Calculate the perimeter of the trapezium.. cm f) Calculate the perimeter of the parallelogram. mm. cm 0 mm cm cm mm page Maths Mate /8 Skill Builder

241 Skill. Calculating the perimeter of polygons using real-life examples. Determine and label all side lengths. Hint: Sides marked with a dash ( ) are of equal length. Sides marked with two dashes ( ) are of equal length etc. Add together the side lengths. Q. Calculate the perimeter of the regular octagonal table top. A cm cm a) What is the perimeter of the gymnastics floor? b) What is the perimeter of the rectangular Luxio TV screen? m cm m cm cm c) What is the perimeter of this Romanian stamp valued at 0 bani? d) What is the perimeter of the upper surface of this regular hexagonal column of basalt seen at the Giant s Causeway in Ireland? mm 8 mm 0 mm mm mm e) What is the perimeter of the rectangular ceiling of the Sistine Chapel? f) What is the perimeter of the eye of the pentagonal vortex of hurricane Isabel (00)? m km m... m... km page Maths Mate /8 Skill Builder

242 Skill. Calculating the perimeter of polygons using unit conversions. Convert all measurements to the same unit. (see skill., page ) Determine and label all side lengths. Hint: Sides marked with a dash ( ) are of equal length. Sides marked with two dashes ( ) are of equal length etc. Add together the side lengths. Q. Calculate the perimeter of the equilateral triangle in centimetres. A. 0 mm 0 0 cm cm P cm 0 mm cm cm mm to cm: 0 label all sides cm a) Calculate the perimeter of the trapezoid in millimetres. 8 mm b) Express all measurements in centimetres and then find the perimeter of the parallelogram. mm 9 mm 0 mm cm. cm... P... mm c) Express all measurements in millimetres and then find the perimeter of the right-angled triangle.... P... d) Calculate the perimeter of the trapezium in centimetres.. cm cm 9 mm cm mm mm cm... P... mm e) Calculate the perimeter of the kite in millimetres.... P... cm f) Calculate the perimeter of this polygon in centimetres. mm 0 mm mm cm. cm. cm... P... mm... P... cm page 8 Maths Mate /8 Skill Builder

243 Skill. Calculating an unknown side length when the perimeter of a polygon is given. Add together all the given side lengths. Subtract the total from the perimeter to find the unknown side length. OR Use algebra. Q. The perimeter of this regular heptagon is 0 mm. What is the length of a side? A. If? represents the length of a side: P 0 mm P? 0?? 0 mm? mm a) The perimeter of this right-angled triangle is 80 mm. Find the missing side length.? mm b) The perimeter of this regular pentagon is. cm. What is the length of a side? mm mm? cm P + +? Guess? ? so?... mm P... so?... cm c) The perimeter of this scalene triangle is 08 mm. Find the missing side length. d) The perimeter of this polygon is 0 mm. Find the missing side length.? mm mm? mm 8 mm mm mm mm mm P... so?... mm P... so?... mm page 9 Maths Mate /8 Skill Builder

244 Skill. Calculating the circumference of circles (). Substitute the known values into the formula for the circumference of a circle. Hint: You need the radius which is half the diameter. continues on page Q. Using C πr where π, calculate the circumference of the circle. mm A. C πr where d and r Simplify: mm d r a) Using C πr where π., calculate the circumference of the circle. b) Using C πr where π., calculate the circumference of the circle. cm cm C cm C πr cm c) Using C πr where π, calculate the circumference of the circle. d) Using C πr where π, calculate the circumference of the circle. mm mm C mm C mm page 0 Maths Mate /8 Skill Builder

245 Skill.8 Calculating the perimeter of composite shapes. Determine and label all side lengths. Hint: Sides marked with a dash ( ) are of equal length. Sides marked with two dashes ( ) are of equal length etc. Add together the side lengths. Q. Calculate the perimeter of the polygon. A mm mm OR (0 ) + (8 ) mm mm 8 mm mm 8 mm 8 mm 8 mm a) Calculate the perimeter of the polygon. b) Calculate the perimeter of the polygon.. cm Start here cm cm c) Calculate the perimeter of the polygon. d) Calculate the perimeter of the polygon. 0 mm cm 0 mm. cm cm cm. cm cm 8 mm cm cm 0 mm 0 mm. cm mm cm e) Calculate the perimeter of the polygon. f) Calculate the perimeter of the polygon. mm mm 9 mm mm mm 0 mm mm mm mm page Maths Mate /8 Skill Builder

246 Skill.8 Calculating the perimeter of composite shapes. Determine and label all side lengths. Hint: Sides marked with a dash ( ) are of equal length. Sides marked with two dashes ( ) are of equal length etc. Add together the side lengths. Q. Calculate the perimeter of the polygon. A mm mm OR (0 ) + (8 ) mm mm 8 mm mm 8 mm 8 mm 8 mm a) Calculate the perimeter of the polygon. b) Calculate the perimeter of the polygon.. cm Start here cm cm c) Calculate the perimeter of the polygon. d) Calculate the perimeter of the polygon. 0 mm cm 0 mm. cm cm cm. cm cm 8 mm cm cm 0 mm 0 mm. cm mm cm e) Calculate the perimeter of the polygon. f) Calculate the perimeter of the polygon. mm mm 9 mm mm mm 0 mm mm mm mm page Maths Mate /8 Skill Builder

247 . [Area / Volume] Skill. Calculating the area of polygons by counting squares and triangles on a square grid (). Count the number of fully shaded squares on the grid. If necessary add on the number of half shaded squares (or triangles). Look for shortcuts in your counting. Hint: To calculate the area of a rectangular shape it is possible to count the number of squares in a row and then multiply by the number of squares in a column. Q. Find the area of the rectangle. A. There are squares in a row cm and squares in a column. Area cm continues on page Area cm a) Find the area of the triangle. b) Find the area of the rectangle. Area cm Area cm cm cm c) Find the area of the rectangle. d) Find the area of the rectangle. Area cm Area cm cm cm e) Find the area of the rectangle. f) Find the area of the rectangle. Area cm Area cm cm cm page Maths Mate /8 Skill Builder

248 Skill. Calculating the area of polygons by counting squares and triangles on a square grid (). g) Find the area of the rectangle. h) Find the area of the polygon. continued from page Area cm Area cm cm cm i) Find the area of the triangle. j) Find the area of the polygon. Area cm Area cm cm cm k) Find the area of the trapezium. Area cm l) Find the area of the parallelogram. Area cm cm cm m) Find the area of the triangle. Area cm n) Find the area of the trapezium. Area cm cm cm page Maths Mate /8 Skill Builder

249 Skill. Comparing the area of polygons on a square grid (). continues on page Break the shape up into rectangles and triangles if necessary. Calculate the area of any rectangle by: Counting the squares OR Multiplying the number of squares in a row by the number of squares in a column. Calculate the area of any triangle by halving the area of the rectangle that would enclose it. Compare your results. Q. Do the rectangle and the triangle have the same area? A. Area A Area B No Area A sq. units Area A enclosing rectangle Area B 8 9 sq. units Area B a) Do these rectangles have the same area? b) Do these rectangles have the same area? A B Area A 0... Area B 0... yes Area A... Area B... c) Do these polygons have the same area? d) Do these triangles have the same area? Area A... Area B... Area A... Area B... page Maths Mate /8 Skill Builder

250 Skill. Comparing the area of polygons on a square grid (). continued from page e) Do these triangles have the same area? f) Do these triangles have the same area? Area A... Area B... Area A... Area B... g) Do these triangles have the same area? h) Do these triangles have the same area? i) Do these polygons have the same area? j) Do these polygons have the same area? k) Do the square and the parallelogram have the same area?... l) Do the parallelogram and the kite have the same area? page Maths Mate /8 Skill Builder

251 Skill. Estimating the area of irregular shapes on a square grid. Break the shape up into workable parts (rectangles/triangles/curved shapes). Calculate the area of any rectangle by: Counting the squares OR Multiplying the number of squares in a row by the number of squares in a column. Calculate the area of any triangle by halving the area of the rectangle that would enclose it. Estimate the area of any partly curved shape by making up whole squares from the shaded region. Add the results. Q. Find the area of the shaded shape. [Round to the nearest whole number.] A. Area A + Area B + 9 sq. units Area A whole units Area A Area B 9 units (made up from part units) 8 Area B 9 0 a) Find the area of the shaded shape. [Round to the nearest whole number.] b) Find the area of the shaded shape. [Round to the nearest whole number.] A B Area A and Area B... Area A + B +... sq. units Area A and Area B... Area A + B... sq. units c) Find the area of the shaded shape. [Round to the nearest whole number.] d) Find the area of the shaded shape. [Round to the nearest whole number.] sq. units sq. units page Maths Mate /8 Skill Builder

252 Skill. Calculating the area of squares, rectangles and parallelograms (). continues on page 9 Use the appropriate formula. square A l l l l length rectangle A l w lw Q. Using A lw find the area of the rectangle.. cm l length w width parallelogram A b h bh b base A. A lw where l and w.. 9 cm h height cm a) Using Area length width find the area of this rectangle. b) Using Area length width find the area of this rectangle. cm cm cm cm A... cm A... cm c) Using Area base height find the area of this parallelogram. d) Using Area base height find the area of this parallelogram. cm cm cm cm A... cm A... cm e) Using Area length width find the area of this rectangle. f) Using A lw find the area of the rectangle. cm 0 mm mm cm A... cm A... mm page 8 Maths Mate /8 Skill Builder

253 Skill. Calculating the area of squares, rectangles and parallelograms (). g) Using Area length length find the area of the square. 0 mm h) Using Area l find the area of the square.. cm continued from page 8 mm A A cm i) Find the area of the square. j) Using Area base height find the area of the parallelogram. cm 0 mm b 0 mm cm A A mm k) Using A lw find the area of the rectangle. l) Using Area base height find the area of the parallelogram. mm. cm. cm 0 mm A... mm A... cm m) Using A bh find the area of the parallelogram. mm 0 mm n) Find the area of the parallelogram. 0 mm mm mm A A mm page 9 Maths Mate /8 Skill Builder

254 Skill. Calculating the area of triangles (). Use the formula. Q. Using Area base height find the area of the triangle. cm cm triangle b base h height A b h bh A. A bh where b and h cm continues on page a) Using Area base height find the area of the triangle. b) Using Area base height find the area of the triangle. cm cm cm cm A cm c) Using Area base height find the area of the triangle. A d) Using A bh find the area of the right-angled triangle. 0 mm cm cm mm cm A... A cm... mm page 0 Maths Mate /8 Skill Builder

255 Skill. Calculating the area of triangles (). e) Using Area base height find the area of the triangle. continued from page 0 f) Using Area bh find the area of the triangle.. cm mm 0 mm cm A... A cm... mm g) Using Area base height find the area of the triangle. h) Using Area bh find the area of the triangle. cm 0 mm A. cm... A 0 mm cm... mm i) Using Area bh find the area of the triangle. 0 mm j) Using Area bh find the area of the triangle. mm mm 0 mm A... A mm... mm page Maths Mate /8 Skill Builder

256 Skill. Calculating the volume of rectangular prisms by counting cubes (). continues on page Count the cubes. Hint: Count the cubes in one layer and then multiply the result by the total number of layers. Q. Find the volume of the rectangular prism. A. V 80 cm cubes in top layer layers all together cm Vol cm 8 cm cm a) Using Volume length width height, find the volume of the rectangular prism. cm cm V... cm c) Using Volume length width height, find the volume of the rectangular prism. cm cm Vol cm Vol cm b) If boxes can fit inside these shelves, find the total volume of the boxes. box Vol 0 cm V... cm d) Using Volume length width height, find the volume of the rectangular prism. cm cm cm cm 0 cm Vol cm V... cm V... cm e) Using Volume length width height, find the volume of the rectangular prism. f) Using Volume length width height, find the volume of the rectangular prism. cm Vol cm cm Vol cm cm cm 9 cm V... cm cm V... cm page Maths Mate /8 Skill Builder

257 Skill. Calculating the volume of rectangular prisms by counting cubes (). g) Using Volume length width height, find the volume of the rectangular prism. continued from page h) Using V lwh find the volume of the rectangular prism. cm cm cm Vol cm cm cm 8 cm Vol cm V... cm V... cm i) Using V lwh find the volume of the rectangular prism. j) Using V lwh find the volume of the rectangular prism. cm cm Vol cm 9 cm cm Vol cm 0 cm cm V... cm V... cm k) Using V lwh find the volume of the rectangular prism. l) Using V lwh find the volume of the rectangular prism. cm cm cm Vol cm 9 cm Vol cm cm cm V... cm V... cm page Maths Mate /8 Skill Builder

258 Skill. Calculating the volume of square and rectangular prisms (). Use the appropriate formula. cube V l l l l square prism l length V l l h l h h height l length rectangular prism V l w h lwh l length continues on page h height w width Q. Using V l h find the volume of the square prism.. cm A. V. 9.. cm cm a) Using V lwh find the volume of the rectangular prism. b) Using V l h find the volume of the square prism. 0 mm 0 mm mm l cm h cm V mm V cm c) Using V l h find the volume of the square prism. d) Using V l find the volume of the cube. l 0 mm. cm cm V V... cm mm... page Maths Mate /8 Skill Builder

259 Skill. Calculating the volume of square and rectangular prisms (). e) Using V lwh find the volume of the bank of lockers that is a rectangular prism. m m 0. m V m f) Using V lwh find the volume of the rectangular prism. 0 mm V continued from page mm mm mm g) Using V l find the volume of the cube. h) Using V lwh find the volume of the rectangular prism. mm V mm. cm cm V cm cm i) Using V l h find the volume of the square prism. j) Using V l h find the volume of the square prism. l cm h. cm l 0 mm V cm h 0 mm V mm page Maths Mate /8 Skill Builder

260 Skill.8 Calculating the area of composite shapes (). Break the shape up into workable parts (rectangles/triangles). Calculate the area of each part. (see skills. to., pages 8 to 0) Add the results. continues on page Q. Find the area of the shaded polygon. A. A lw where l and w 0 A bh where b and h. A bh where b and h. A A + A + A sq. units A A A a) Find the area of the shaded polygon. b) Find the area of the shaded polygon.. cm cm cm. cm cm cm cm A A A.. A A cm A A A... sq. units c) Find the area of the shaded polygon. d) Find the area of the shaded polygon. A A A... sq. units A A A... sq. units page Maths Mate /8 Skill Builder

261 Skill.8 Calculating the area of composite shapes (). continued from page e) Find the area of the shaded polygon. f) Find the area of the shaded polygon. A A A A A... sq. units... A... sq. units g) Find the area of the bowtie. h) Find the area of the polygon. 8 cm 0 cm cm cm A A A... cm A A A... cm i) Find the area of the shaded polygon. j) Find the area of the polygon. 0 mm cm mm mm cm cm mm A A A A A... mm A cm... page Maths Mate /8 Skill Builder

262 Skill.9 Calculating the area of trapeziums and rhombii. Use the appropriate formula. trapezium b h a A (base a + base b) height h (a + b)h Q. Using A (a + b)h find the area of the trapezium. cm cm rhombus A diagonal a diagonal b ab A. A (a + b)h where a, b and h ( + ) 9 9 cm a b l cm a) Using A ab find the area of the rhombus. mm A mm mm b) Using Area (base a + base b) height find the area of the trapezium. a 0 mm 0 mm b mm A mm c) Using A (a + b)h find the area of the trapezium. cm a cm b cm A cm d) Using Area diagonal a diagonal b find the area of the rhombus. a 0 mm A b mm mm page 8 Maths Mate /8 Skill Builder

263 Skill.0 Calculating the area of circles and composite circular shapes. Use the formula. circle radius A π radius radius where π... πr or Hint: If you are given the diameter then halve to find the radius: Q. Using A πr and π., find the area of the semi-circle. 0 mm d r A. Area of circle πr where d 0 and r Area of semi-circle mm d r a) Using A πr and π., find the area of the shaded shape. cm cm b) Using A πr and π, find the area of the circle. cm A cm mm A If d then r A..... A and using A A + A... A for the semi-circle cm A mm d) Using A πr and π., find the area of the shaded shape. c) Using A πr and π., find the area of the semi-circle. cm cm cm A A... A... cm cm A... page 9 Maths Mate /8 Skill Builder

264 Skill. Calculating the volume of any prism. Use the general formula. prism h V Area of base height of prism Bh B Q. Using V Bh find the volume of the prism. A. A Bh where B 0 and h mm B 0 mm 0 mm a) Using V Bh find the volume of the triangular prism. b) Using Volume area of the base height of the prism find the volume of the pentagonal prism. h 0 mm mm 0 mm h cm B B 8. cm V mm cm V... c) Using V Bh find the volume of the hexagonal prism. d) Using V Bh find the volume of the prism. B 0. m h m B 0 mm mm V... m mm V... e) Using V Bh find the volume of the triangular prism.. cm cm B... V... h cm f) Using V Bh find the volume of the triangular prism. 0 mm 0 mm 0 mm B... cm mm V... page 0 Maths Mate /8 Skill Builder

265 [Shapes] Skill. Measuring angles using a protractor (). Place the centre of the protractor at the vertex (corner) of the angle. Align one of the lines forming the angle to pass through 0 on either the inside or outside scale. Read the measurement where the other line of the angle crosses the scale on the protractor. Q. Use a protractor to measure this angle. A. 0 continues on page count the degrees on the inside scale start at a) Use a protractor to measure this angle. b) Use a protractor to measure this angle. start at count the degrees on the outside scale c) Use a protractor to measure this angle. d) Use a protractor to measure this angle. page Maths Mate /8 Skill Builder

266 Skill. Measuring angles using a protractor (). continued from page e) Use a protractor to measure this angle. f) Use a protractor to measure this angle. g) Use a protractor to measure this angle. h) Use a protractor to measure this angle. i) Use a protractor to measure this angle. j) Use a protractor to measure this angle. k) Use a protractor to measure this angle. l) Use a protractor to measure this angle. page Maths Mate /8 Skill Builder

267 continues on page Skill. Estimating the size of angles (). To estimate the size of an acute angle: Draw a right angle (90 ) overlapping one line of the given angle. Divide the right angle into smaller divisions, e.g. halves or thirds To estimate the size of an obtuse angle: Draw a straight angle (80 ) overlapping one line of the given angle. Divide the straight angle into smaller divisions, e.g. quarters or sixths Q. Without measuring, would you estimate that the size of this angle is closer to or to 0? A the angle is slightly greater than 0 a) Without measuring, would you estimate that the size of this angle is closer to 0 or to 0? 90 b) Without measuring, would you estimate that the size of this angle is closer to or to 90? the angle is slightly smaller than 0 0 c) Without measuring, would you estimate that the size of this angle is closer to 0 or to 80? d) Without measuring, would you estimate that the size of this angle is closer to or to 0? page Maths Mate /8 Skill Builder

268 Skill. Estimating the size of angles (). e) Without measuring, would you estimate that the size of this angle is closer to 9 or to 0? continued from page f) Without measuring, would you estimate that the size of this angle is closer to or to? g) Without measuring, would you estimate that the size of this angle is closer to or to 0? h) Without measuring, would you estimate that the size of this angle is closer to 0 or to? i) Without measuring, would you estimate that the size of this angle is closer to 9 or to 0? j) Without measuring, would you estimate that the size of this angle is closer to or to 0? k) Without measuring, would you estimate that the size of this angle is closer to 0 or to 0? l) Without measuring, would you estimate that the size of this angle is closer to or to? page Maths Mate /8 Skill Builder

269 Skill. Recognising polygons and quadrilaterals. Consider the definition of a polygon. (see Glossary, page ) Consider the definition of a quadrilateral. (see Glossary, page ) Q. Circle the shapes that are not polygons. A. st shape - closed shape with all sides line segments (polygon) nd shape - not a closed shape (not a polygon) rd shape - closed shape with two sides line segments and a curved line (not a polygon) th shape - closed shape with all sides line segments (polygon) a) Circle the shapes that are polygons. b) Circle the shapes that are not polygons. not a closed shape curved line c) Circle the shapes that are polygons. d) Circle the shapes that are not polygons. e) Circle the shapes that are quadrilaterals. f) Circle the shapes that are quadrilaterals. g) Circle the shapes that are quadrilaterals. h) Circle the shapes that are not quadrilaterals. page Maths Mate /8 Skill Builder

270 Skill. Classifying and describing the properties of quadrilaterals. Consider the properties of squares, rectangles, rhombi, parallelograms, kites and trapeziums. (see Glossary, page ) Q. I am a quadrilateral with no parallel sides. I have one pair of opposite angles equal, and my diagonals intersect at right angles. What am I? A ) a rhombus B ) a trapezium C ) a kite D ) a square A. A) a rhombus has opposite sides parallel A false B) a trapezium has one pair of opposite sides parallel B false C) a kite has a pair of opposite angles equal and diagonals intersecting at right angles C true D) a square has opposite sides parallel D false The answer is C. a) Match each quadrilateral to its name: b) Match each quadrilateral to its name: square trapezium rectangle rhombus parallelogram rectangle c) I am a -dimensional shape with four sides. Both my pairs of opposite sides are parallel. All angles are equal to 90. What am I? A ) a trapezium B ) a rectangle C ) a rhombus D ) a parallelogram d) I am a quadrilateral with all my sides equal in length. My diagonals intersect at right angles, but are not equal in length. What am I? A ) a kite B ) a rectangle C ) a rhombus D ) a parallelogram e) I am a -dimensional shape with four sides. My diagonals are equal, and all my sides are equal. What am I? A ) a rhombus B ) a rectangle C ) a parallelogram D ) a square f) I am a quadrilateral with all my angles equal to 90. My diagonals are equal in length. What am I? A ) a trapezium B ) a parallelogram C ) a rectangle D ) a rhombus g) Circle the shape that is not a parallelogram. h) Circle the shape that is a rhombus. page Maths Mate /8 Skill Builder

271 Skill. Drawing lines and polygons. Consider the definitions of triangles, squares, rectangles, rhombi, parallelograms, kites, trapeziums and regular polygons. (see Glossary) Mark: Right angles with a corner ( ). Congruent angles with similar curved lines ( ) (the second pair of congruent angles takes on a pair of curved lines). Congruent sides with a dash ( ) (the second pair of congruent lines takes on a pair of dashes). Parallel lines with an arrow ( ) (the second pair of parallel lines takes on a second pair of arrows). Q. Draw an isosceles right-angled triangle marking the congruent sides and congruent angles. A. One corner marking the right angle (90 ) One dash marking each of the congruent sides 90 One curved line marking each of the congruent angles ( ) a) Draw a regular hexagon marking the congruent sides and congruent angles. regular the same size b) Draw a rectangle marking all congruent sides and diagonals. six equal sides six equal angles c) Draw a rhombus marking all congruent sides and perpendicular diagonals. d) Draw an isosceles obtuse-angled triangle marking the congruent sides and congruent angles. e) Draw a regular pentagon marking the congruent sides and congruent angles. f) Use arrows to show the pair of parallel lines in this diagram. g) Use arrows to show the pair of parallel lines in this diagram. h) Use arrows to show the pair of parallel lines in this diagram. page Maths Mate /8 Skill Builder

272 Skill. Classifying and describing the properties of D shapes. Count the number of: - faces - edges - vertices (points/corners) vertex face edge lateral face Q. How many vertices are there in a hexagonal prism? base A. Count the vertices, or corners in the prism: six vertices in one base and six vertices in the other base base The answer is. base a) The base of a rectangular pyramid is a rectangle. What shape are the lateral faces? base lateral face b) The base of a pentagonal prism is a pentagon. What shape are the lateral faces? c) What is the name of this solid? A ) triangular prism B ) square prism C ) square pyramid D ) rectangular prism d) What is the name of this solid? A ) triangular pyramid B ) square pyramid C ) rectangular pyramid D ) triangular prism e) Circle the shapes that are not prisms. f) Circle the shapes that are not pyramids. g) How many edges are there on a triangular prism? h) How many edges are there on a cube? i) How many vertices are there on a pentagonal pyramid? j) How many faces are there on a square pyramid? k) Sketch and name the three-dimensional shape that has two square faces and four rectangular faces. l) Sketch and name the three-dimensional shape that has one rectangular face and four triangular faces. page 8 Maths Mate /8 Skill Builder

273 Skill. Classifying angles. Consider the definitions and properties of a variety of angles. (see Glossary and Maths Facts, page ) Hints: An angle can be classified according to its size (acute, right, obtuse, straight and reflex). Two angles can be classified according to their position in relation to one another (adjacent, supplementary, complementary or vertically opposite). Q. Circle the obtuse angle. A. right angle obtuse angle greater than 90 less than 80 acute angle less than 90 a) Match each angle to its description: b) Match each angle to its description: acute right obtuse obtuse acute straight c) Match each angle to its description: d) Match each angle to its description: obtuse reflex straight acute reflex right e) Circle the right angle. f) Circle the reflex angle. g) Circle the acute angle. h) Circle the obtuse angle. page 9 Maths Mate /8 Skill Builder

274 Skill.8 Classifying and describing the properties of triangles. Look for equal sides or equal angles. Look at the types of angles inside the triangle. Sides and angles no equal sides/angles two equal sides/angles three equal sides/angles Triangle type scalene isosceles equilateral Angles all acute angles one right angle one obtuse angle Triangle type acute-angled right-angled obtuse-angled Q. Match each triangle to its description: A. no equal sides scalene equilateral isosceles scalene two equal sides isosceles three equal sides equilateral equilateral isosceles scalene a) Circle the triangle that is not isosceles. b) Circle the triangle that is obtuse-angled. no equal sides c) Match each triangle to its description: d) Match each triangle to its description: scalene isosceles equilateral acute-angled obtuse-angled right-angled e) I am a -dimensional shape with three sides. I have two of my sides of equal length. What am I? A ) a square B ) a right-angled triangle C ) an isosceles triangle D ) an equilateral triangle f) I am a -dimensional shape with three sides. I have an obtuse angle. What am I? A ) an acute-angled triangle B ) a right-angled triangle C ) an equilateral triangle D ) an obtuse-angled triangle page 0 Maths Mate /8 Skill Builder

275 Skill.9 Working with vertically opposite angles and complementary angles. Use the properties: - the sum of complementary angles is 90. (see Glossary, page ) - two vertically opposite angles are congruent. (see Glossary, page ) To find the size of an angle when its complementary angle/angles are given: EITHER Subtract the given angles from 90. OR Write an equation involving the unknown angle x. Solve the equation for x. Q. Find the value of x. A. x and are complementary: x 90 x OR x + 90 x + 90 x a) Find the value of x. x same size angles b) Find the value of x. x c) Find the value of x. d) Find the value of x. x x x... 9 x... x... e) Find the value of x. f) Find the value of x. x... x... x 0... x... g) Find the value of x. 8 x x x x x... h) Find the value of x. x x... page Maths Mate /8 Skill Builder

276 Skill.0 Working with supplementary angles. Use the property: - the sum of supplementary angles is 80. (see Glossary, page ) To find the size of an angle when its supplementary angle/angles are given: EITHER Subtract the given angles from 80. OR Write an equation involving the unknown angle x. Solve the equation for x. Q. Find the value of x. A. x and are supplementary: x 80 x OR x + 80 x + 80 x a) Find the value of x. b) Find the value of x. 8 x x x x... x... c) Find the value of x. d) Find the value of x. x x... x x... e) Find the value of x. f) Find the value of x. x... x... x... x... g) Find the values of x and y. 0 y x x x h) Find the values of x and y. 0 y x x x... y... y page Maths Mate /8 Skill Builder

277 Skill. Finding the size of angles inside a triangle. Use the property: - the sum of the interior angles of any triangle is 80. (see Maths Facts, page ) To find the size of an angle of a triangle when the other two angles are given: EITHER Subtract the sum of the given angles from 80. OR Write an equation involving the unknown angle x. Solve the equation for x. Q. Find the value of x. A. Isosceles triangle base angles are equal: x 80 ( + ) 80 8 x x OR x x x a) Find the value of x. b) Find the value of x. 08 x x 80 ( + 08 ) x 0 x x... c) Find the value of x. 0 x... d) Find the value of x. x... x... x x... e) Find the value of x. f) Find the value of x. x x... x x... 0 x... g) Find the value of x. x x... h) Find the value of x. 0 x x... page Maths Mate /8 Skill Builder

278 Skill. Finding the size of angles inside a quadrilateral. Use the property: - the sum of the interior angles of any quadrilateral is 0. (see Maths Facts, page ) To find the size of an angle of a quadrilateral when the other two angles are given: EITHER OR Subtract the sum of the given angles Write an equation involving the unknown from 0. angle x. Solve the equation for x. Q. Find the value of x. A. x 0 ( ) x 0 0 OR x x + 0 x x a) Find the value of x. x 0 80 x 0 ( ) b) Find the value of x. 9 x x... c) Find the value of x. x d) Find the value of x. 0 0 x 80 x x e) Find the value of x. 0 x 0 x f) Find the value of x. 0 x... x... page Maths Mate /8 Skill Builder

279 Skill. Describing the properties of circles. Consider the definitions and properties of radius (plural radii), chord, diameter, tangent and circumference of a circle. (see Glossary, and Math Facts, page ) Radius joins the centre with any point on the circle Chord joins any two points on the circle Diameter a chord passing through the centre Tangent a line touching the circle in one point Circumference the distance around the circle A D C O B M O E N A O B O T O OA OB OC AB OA Q. What is EF in this diagram? A ) diameter B ) tangent O C ) chord D ) radius E F A. EF joins two points on the circle & does not pass through the centre chord a) Match each diagram to its description: b) Match each diagram to its description: diameter circumference radius tangent circumference radius diameter area c) What is CD in this diagram? A ) tangent B ) diameter C C ) radius O D ) circumference D d) What is OT in this diagram? A ) chord B ) tangent C ) diameter O D ) radius T e) Draw the diameter passing through P. f) Draw the radius passing through S. O O S P page Maths Mate /8 Skill Builder

280 page Maths Mate /8 Skill Builder

281 ] [ ] [ 8. [Exploring Geometry] Skill 8. Following directions and using compass bearings to describe location on a map. Follow the directions one at a time. Hints: A compass showing North will allow you to find your bearings. Clockwise from North, Never Eat Sea Weed is one way to remember the point compass. Q. At Homebush, in which direction is the Olympic Stadium from the Golf Driving Range? a) From where you are, travel east until you reach David Street. Then walk north. If you take the second turn left, what street are you in? Albury - Australia Stanley Street You are here Wodonga Place Townsend Street To Melbourne Kiewa Street Clive Street Dean Street Smollett Street Hume Street Swift Street To Sydney c) From the northern most bridge over Rio Kusichaca you travel south east on the Inca Trail until the T intersection. Then you turn right and follow the Inca Trail to the Inca Steps. How many more bridges do you cross? Aguas Calientes train station Piaza bus to ruins ] [ Winay Wayna 00 m ] [ Phuyupatamarca 00 m Machu Picchu 00 m Intu Punku 00 m Inca Steps Sayaqmarca 00 m START day trek ] [ Runkuracay 800 m David Street INCA TRAIL to Machu Picchu Chachabamba Pacaymayo 00 m Rio Urubamba Rio Pacamayo Edwin Flack Ave Young Street N Warmiwanusca Pass 00 m Llulluchapampa N Olympic Boulevard Ave Dawn Fraser Ave ] [ Albury - Australia Stanley Street Wodonga Place You are here Homebush - Sydney Townsend Street To Melbourne START day trek ] [ Llactapata Rio Kusichaca Herb Elliot Ave Kiewa Street 0 Sarah Durack Ave. ] [ ] [ Clive Street Dean Street Smollett Street Hume Street Australia Ave Swift Street David Street LEGEND River Rail line Inca ruin Camp Inca Trail Bridge Cusco 8 9 A. NW Focus on the relevant information. b) From Montrésor castle, which direction do you have to drive to reach Loches castle? To Sydney Young Street N Bennelong Rd Shirley Strickland Ave Homebush Bay Drv LOIRE VALLEY CASTLES Loire East - FRANCE Tours Bicentennial Park Indre Olympic Stadium Athletic Centre Warm-up Arena Aquatic Centre Hockey Centre Baseball Stadium Railway Station 8 Golf Driving Range 9 Tennis Centre 0 Sports Centre N N Montbazon Montlouis Cormery Loches Olympic Stadium Blois Amboise Chaumont Loire Pontlevoy Chenonceau Montpoupon Montrésor NW W SW Cheverny Ouchamps Chambord Cher 8 Golf Driving Range d) Using the closest tunnel entrance to building 8, take the first turn right, then turn left. Turn right and walk to the end of the tunnel. If you turn left again, which building are you facing? N S NE E SE Northeastern University - Boston, MA Tunnel Map 0 Barletta Natatorium HUNTINGTON AVENUE 0 Tunnel Entrance Tunnel Entrances Cabot Physical Education Centre Richards Hall Dodge Hall 8 Mugar Life Sciences Building 0 Curry Student Centre Blackman Auditorium Ell Hall Hayden Hall Forsyth Building Dana Research Centre 8 Snell Engineering Centre 9 Snell Library page Maths Mate /8 Skill Builder 8

282 Skill 8. Identifying and classifying symmetry in two-dimensional shapes. Imagine a line along which the shape can be folded to have one part fit exactly over the other part. Q. Draw the axes of symmetry for these shapes. Circle the shapes that are both horizontally and vertically symmetrical. A. vertical & horizontal oblique vertical vertical & horizontal a) Draw all the axes of symmetry for this shape. How many axes of symmetry does this shape have? b) Draw all the axes of symmetry for this shape. How many axes of symmetry does this shape have? c) Draw all the axes of symmetry for this shape. How many axes of symmetry does this shape have? d) Draw all the axes of symmetry for this shape. How many axes of symmetry does this shape have? e) Draw the axes of symmetry for these shapes. Circle the shapes that have horizontal symmetry. f) Draw the axes of symmetry for these shapes. Circle the shapes that are both horizontally and vertically symmetrical. g) Draw the axes of symmetry for these shapes. Circle the shapes that have vertical symmetry. h) Draw the axes of symmetry for these shapes. Circle the shapes that are both horizontally and vertically symmetrical. page 8 Maths Mate /8 Skill Builder 8

283 Skill 8. Using a scale to calculate distance on a map. Place a piece of paper against the scale matching the starting points. Slide the paper across the length of the scale marking the start and end points as you go. Add together the scale lengths covered. Q. You walk from the Inspiration Point to A Grand View, along the marked path.. km What distance did you travel in kilometres? Canyon Village Visitor Centre Post Office Yellowstone National Park Upper Falls View Lookout Point Uncle Tom s Trail N Amphitheatre Showers - Laundry Canyon Lodge Grand View Inspiration Point 0 0. km Services & Facilities Ranger station Campground Lodging Food service Picnic area Store Gas station Self-guiding trail Horse rental 0. km Grand View Inspiration Point There are distances to be measured. Mark the start of the first distance and the turning point on paper. Rotate the paper to match the second distance and then mark the end. Check the paper against the scale. Slide the paper along the scale as necessary. a) How far is it from Central Station, along Hope St. to the Glasgow Royal Concert Hall? Central Station 0 0 m North Queen Street Station GRCH 0 Glascow Royal Concert Hall (GRCH) m c) Using the scale, what is the marked distance from the University via the High Court to the Homiman Circle Gardens? MUMBAI - India High Court m 0 00 University of Mumbai Mahatma Gandhi Rd Veer Nariman Rd Dalai St St Thomas Cathedral Ambalal Doshi Rd Homiman Circle Gardens Shahid Bhagat Singh Rd m b) Using the scale, what is the marked distance on this map of Antarctica? ATLANTIC OCEAN Bellingshausen (Russia) Palmer (U.S.) Rothera (U.K.) PACIFIC OCEAN Year-round research station 0 00 km d) Using the scale, what is the marked distance from the ranger station closest to Lake Hotel to the Fishing bridge? Fishing Bridge, Lake Village & Bridge Bay Yellowstone National Park km 0. mi Lake Village Ice Marina Antarctic Circle Bridge Bay Fishing Bridge Post Office Lake Hotel Gull Point Halley (U.K.) SOUTHERN OCEAN SOUTHERN OCEAN Syowa (Japan) Visitor Centre Lake Lodge YELLOWSTONE LAKE N Mawson (Australia) RONNE ICE AMERY ICE SHELF SHELF Davis (Australia) South Pole Amundsen-Scott (U.S.) Vostok (Russia) ROSS ICE Casey (Australia) SHELF McMurdo (U.S.) Scott (New Zealand) Dumont d'urville (France) Services & Facilities INDIAN OCEAN km Ranger station Lodging Food service Picnic area Store Gas station Boat launch km page 9 Maths Mate /8 Skill Builder 8

284 Skill 8. Recognising basic transformations of two-dimensional shapes. Q. According to the compass, you are facing north-west. How many degrees clockwise must you turn to walk north? W SW NW N NE SE E A. Find the North NW direction. clockwise W SW NW N S NE SE E N Calculate the number of degrees by picturing a circle. S a) Which transformation (translation, rotation, reflection) has moved the original shape to its new position? Original Image Original Image b) Which transformation (translation, rotation, reflection) has moved this comet to its new position? Original Image reflection c) Which transformation (translation, rotation, reflection) has moved the original shape to its new position? d) Which transformation (translation, rotation, reflection) has moved the original shape to its new position? Original Image Original Image e) Which transformation (translation, rotation, reflection) has moved the original shape to its new position? f) Which transformation (translation, rotation, reflection) has moved the original shape to its new position? Original Image Original Image g) How many degrees must the big hand of this clock turn to show exactly 9:? h) According to the compass, you are facing south-east. How many degrees clockwise must you turn to walk west? W SW NW N NE SE E S page 0 Maths Mate /8 Skill Builder 8

285 Skill 8. Drawing translations, reflections and rotations of objects on a grid (). continues on page Translation (slide) Move the shape up (positive, vertically), down (negative, vertically), left (negative, horizontally) or right (positive, horizontally) on the grid, without flipping, turning or changing its size. Reflection (like in a mirror) Draw a perpendicular line to the mirror line from each vertex of the shape. Extend the perpendicular line beyond the mirror line by the same distance. Plot and join the reflected points. Rotation (turning about a point or centre of rotation) Rotate each vertex by the given angle, in the given direction. Plot and join the rotated points. Q. Redraw this shape reflected in the horizontal line. A. a) Translate this shape 0 units left and units down. b) Redraw this shape reflected in the vertical line. 0 c) Translate this shape units up and units right. d) Redraw this shape rotated 80 about the point O. O page Maths Mate /8 Skill Builder 8

286 Skill 8. Drawing translations, reflections and rotations of objects on a grid (). e) Redraw this diagram reflected in the horizontal line. continued from page f) Redraw this diagram reflected in the horizontal line. g) Draw and label the reflection of the quadrilateral ABCD in the vertical line. C h) Draw and label the reflection of the rhombus MNOP in the horizontal line. M B P O N A D i) Redraw this diagram reflected in the horizontal line. j) Redraw this diagram reflected in the horizontal line. k) Redraw this diagram reflected in the horizontal line. l) Redraw this shape rotated 80 about the point O. O m) Redraw this shape rotated 80 about the point O. n) Redraw this shape rotated 80 about the point O. O O page Maths Mate /8 Skill Builder 8

287 Skill 8. Recognising nets of three-dimensional shapes. Identify the shapes in the net. Imagine the shape folded. OR Make a model by tracing, cutting out and folding the net. base Q. What -dimensional shape can this net be used to make? Prism Pyramid Cube base lateral face base A. regular octohedron lateral face folding line a) What -dimensional shape can this net be used to make? b) What -dimensional shape can this net be used to make? folding line c) What -dimensional shape can this net be used to make? d) What -dimensional shape can this net be used to make? folding line folding line folding line trapezoidal prism e) What -dimensional shape can this net be used to make? f) What -dimensional shape can this net be used to make? folding line folding line page Maths Mate /8 Skill Builder 8

288 Skill 8. Drawing the top, side and front views of three-dimensional shapes. Imagine what you would see from the stated direction OR Make a model and observe the view. Q. Draw the top view of this solid. A. top view a) Which solid has the top, front and side views shown? b) Which solid has the top, front and side views shown? top front side top front side A) B) C) A) B) C) T F S T F S T F S as above C c) Draw the top view of this solid. d) Draw the side view of this solid. side view e) Draw the top view of this solid. f) Draw the side view of this solid. top view top view side view page Maths Mate /8 Skill Builder 8

289 Skill 8.8 Recognising the shapes of cross sections through three-dimensional shapes (). continues on page To name the shape of a cross section through a D shape, imagine that you cut the solid through that section, then separate the two parts and look at the shape of the slice. Q. What shape is the cross section drawn through this cube? A. hexagon a) What shape is the cross section drawn through this pyramid? b) What shape is the cross section drawn through this cube? triangle c) What shape is the cross section drawn through this prism? d) What shape is the cross section drawn through this prism? e) What shape is the cross section drawn through this cone? f) What shape is the cross section drawn through this cone? g) What shape is the cross section drawn through this prism? h) What shape is the cross section drawn through this sphere? page Maths Mate /8 Skill Builder 8

290 Skill 8.8 Recognising the shapes of cross sections through three-dimensional shapes (). i) What shape is the cross section drawn through this cube? continued from page j) What shape is the cross section drawn through this prism? k) What shape is the cross section drawn through this cylinder? l) What shape is the cross section drawn through this pyramid? m) What shape is the cross section drawn through this pyramid? n) What shape is the cross section drawn through this cube? o) What shape is the cross section drawn through this prism? p) What shape is the cross section drawn through this prism? q) What shape is the cross section drawn through this pyramid? r) What shape is the cross section drawn through this prism? page Maths Mate /8 Skill Builder 8

291 Skill 8.9 Recognising congruence in two-dimensional shapes. Check which shapes are the same shape. Check which shapes are the same size. Q. Circle the pair of congruent shapes. A. a) Circle the triangle that is not congruent with ΔABC. C A B B M C N P S T b) Circle the pair of congruent shapes. A U c) Circle the pair of congruent triangles. d) Circle the pair of congruent triangles. e) Find the pair of congruent hexagons. f) Find the pair of congruent triangles. and and g) Which triangle is not congruent with ΔCDE? O I h) Which triangle is not congruent with ΔABC? E C Q D P D C E G H A B C G H I A B C D E F page Maths Mate /8 Skill Builder 8

292 Skill 8.0 Recognising rotational symmetry in two-dimensional shapes. Try to visualise the shape during a full turn of 0 and make sure that the shape could cover itself at least once before the full turn is completed. Q. Circle the shapes which have rotational symmetry. A. original position original position Sample rotations of 90 show only the first and last shape get back to their original position before the full, 0 turn. a) Circle the shapes which have rotational symmetry. b) Circle the shapes which have rotational symmetry. c) Circle the shapes which have rotational symmetry. d) Circle the shapes which have rotational symmetry. e) Circle the shapes which have rotational symmetry. f) Circle the shapes which have rotational symmetry. g) Circle the shape which does not have rotational symmetry. h) Circle the shape which does not have rotational symmetry. page 8 Maths Mate /8 Skill Builder 8

293 9. [Statistics] Skill 9. Interpreting dot plots. Hint: Each dot ( ), cross (x) or picture shows the position of a sample of the data above a number line. Q. How many European countries have or more official languages? () X X X X X X X X X X X X X X X X X X X X X Official languages spoken Official languages of European countries (adapted for language which ) X X X A. + + () X X X X X X X X X X X X X X X X X X X X X Official languages spoken Official languages of European countries (adapted for language which ) X X X Count the crosses a) How many countries scored more than goals in the 00 soccer world cup? Number of countries Number of countries Soccer World Cup 00 - goals scored Goals scored Soccer World Cup 00 - goals scored Goals scored c) Hey Jude was The Beatles single that held the number one position on the US Billboard for the longest amount of time. For how many weeks was Hey Jude at number one? BEATLES SINGLES at number position on the US Billboard b) In 00, how many films received four Oscar nominations each? Number of films... d) Between 90 and 000, how many musicians composed more than symphonies? Number of composers (0) 00 Oscar nominations 8 Number of nominations Symphonies Composed (90-000) Number of weeks Number of symphonies page 9 Maths Mate /8 Skill Builder 9

294 Skill 9. Interpreting pictograms. Q. Which country had closest to % annual growth in 009? Sth Africa Namibia Botswana Zimbabwe Zambia Angola Mozambique Malawi African countries - GDP annual growth (%) A. Mozambique African countries - GDP 009 Sth Africa Namibia Botswana Zimbabwe Zambia Angola Mozambique Malawi 0 9 annual growth (%) a) Name the planet with the greatest diameter. Diameter (km) OUR SOLAR SYSTEM (Diameters of planets) Mercury Venus Neptune Mars Jupiter Saturn Uranus Earth Planet b) Which river is closest to 000 km in length? LONGEST RIVERS (by continent) Nile (Africa) ---- (Antarctica) Yangtze (Asia) Murray (Australasia) Volga (Europe) Mississippi (North America) Amazon (South America) 000 km in length c) Which country has won the US Open half as often as South Africa? Countries to win the US Open (Golf) (except USA) d) Which source of power generates approximately gigawatt hours of electricity each year in the USA? Coal Petroleum USA - Annual electric power generation by source Natural Gas Argentina Australia New Zealand South Africa Scotland England Northern Ireland Nuclear Hydro-electric Gigawatt hours (GWH) page 80 Maths Mate /8 Skill Builder 9

295 Skill 9. Interpreting tables. Read the title and sub-headings. Check what each row (across) and column (down) represents. To find the information you need, follow a row across to where it meets the relevant column. Using the information gathered, perform any calculations necessary. Q. What percentage of girls in grade in Vermont ride a high rise bike with a sissy bar? Percentage of girls owning bicycles of different styles by school grade - Vermont School Grade Bike style K Total% Girls Standard High rise High rise, sissy bar Standard and high rise A..% Percentage of girls owning bicycles of different styles by school grade - Vermont School Grade Bike style K Total% Girls Standard High rise High rise, sissy bar Standard and high rise a) Of the birds listed, which bird has a wingspan of. metres? b) Which type of coal has the highest percentage of carbon? barn owl wedge tailed kookaburra eagle 0.9. Wing Spans (m) swan. white pelican condor stork albatross.. Coal Type moisture lignite % sub-bituminous coal 0% bituminous coal % carbon 0% % 80% other % % % anthracite % 9% % c) Which activity expends one third of the amount of kj per minute as swimming? sleeping writing 0 Energy needed (kj per minute) house work tennis swimming (doubles) 0 cycling sprinting d) How many chemicals in sea water have a content of more than one part per thousand? Principal constituents of seawater Chemical Constituent Calcium (Ca) Magnesium (Mg) Sodium (Na) Pottasium (K) Bicarbonate (HCO) Sulfate (SO) Chloride (Cl) Bromide (Br) Total dissolved solids (Salinity) Content (parts per thousand) e) Which ski field listed below has the second most number of lifts? Australian snow field Perisha Blue Thredbo Selwyn Charlotte Pass Mt Buller Mt Hotham Falls Creek Mt Baw Baw Skiable area (hectares) Number of Lifts 9 f) Which animal has the greatest jump height ratio? weight How high can they jump? Animal Weight (g) Jump height (cm) Antelope Human Cat 00 0 Galago 00 Cuban tree frog.9 Locust Flea page 8 Maths Mate /8 Skill Builder 9

296 Skill 9. Interpreting bar graphs (). Q. Which vertebrate class has closest to twice as many species as mammals? Class mammals birds reptiles amphibians fishes Total number of vertebrate species in the world Number of species A. birds Class mammals birds reptiles amphibians continues on page 8 Find the mammals bar. Double the bar length and compare to bar lengths. Total number of vertebrate species in the world fishes Number of species a) How many actors have played James Bond more often than Pierce Brosnan? Sean Connery George Lazenby Roger Moore Timothy Dalton Pierce Brosnan Daniel Craig Sean Connery George Lazenby Roger Moore Timothy Dalton Pierce Brosnan Daniel Craig James Bond Actor (inc. 008) 0 James Bond Actor (inc. 008) 0 8 Number of roles 8 Number of roles b) Using the graph below, how many countries have a life expectancy of less than 0 years? Angola Botswana Ghana Madagascar Mozambique Niger Somalia Life Expectancy Zimbabwe 0 0 years 0 c) Which creature listed below has a brain weighing 0.% of their body weight? d) Using the graph below, which animal has a tusk closest to twice the length of a rhinocerus tusk?.0 % sperm whale Brain size as a percentage of body weight bottle-nosed dolphin human (adult) dog (beagle) alligator rat Length of tusk (m) 0 Narwhal Animals Tusks Straight-tusked elephant (extinct) Mammoth Rhinocerus Elephant page 8 Maths Mate /8 Skill Builder 9

297 Skill 9. Interpreting bar graphs (). e) In which years did the average thickness of the world s glaciers increase? average thickness change (mm) Average Thickness Change - World s Glaciers Time (years) continued from page 8 f) Which type of cancer was responsible for closest to 00 deaths in Colorado in 00? Bowel Breast Leukemia Lung Prostate Number of deaths ( 000) Colorado New York Cancer deaths by state, US, 00 g) Which team had the highest percentage of shots on goal actually converted to goals? h) Are male or female facebook users more likely to disclose their telephone number? 00 FIFA World Cup MP matches played GS goals scored GA goals against G shots on goal Spain Netherlands Harvard MIT NYU Oklahoma U Facebook users who disclose their phone number percent Male Female i) Which of the months shown has a total energy usage (electricity and gas) of closest to 00 kwh? 000 kwh 8 0 Billed energy use May 009 Jul Sept Nov Jan 00 March Electricity Gas j) Which cost of living item is twice as expensive in Hong Kong as it is in Australia? L Milk kg Cheese way transport ticket L Petrol Internet Mbps ADSL Cheap Restaurant Meal Taxi km City Cost of Living AU$ Australia HongKong page 8 Maths Mate /8 Skill Builder 9

298 Skill 9. Interpreting stack graphs (). Q. Which of the mammals shown has the highest protein content in their milk? Cow Dog Man Sheep Domestic mammals - Gross composition of milk, % of total Wolf % Fat Protein Lactose Cow Dog Man Sheep Wolf A. wolf Domestic mammals - Gross composition of milk, % of total % Fat Protein Lactose continues on page 8 The width of the wolf bar for protein represents 9%. No other bar for protein content is as long in any other mammal shown. a) Who had the best ratio of wins to losses in their grand slam singles finals? b) Who had the best ratio of wins to losses in their grand slam singles finals? Number of slam finals Agassi Win GRAND SLAM TITLES - MENS SINGLES Borg Connors Edberg Emerson Laver McEnroe Loss Sampras Number of slam finals Win GRAND SLAM TITLES - WOMENS SINGLES Loss Court Evert Graf King Seles Henin Navratilova Emerson c) What percentage of the world s population lives on less than $0 a day? d) In which age group did McCain and Obama split the votes evenly? World population (%) 00% 0% World Poverty Levels Age group (yr) 008 US presedential election result > % $.00 $. $. $.00 $.0 $0.00 purchasing power/day (USD) % McCain Obama Above the purchasing power Below the purchasing power page 8 Maths Mate /8 Skill Builder 9

299 Skill 9. Interpreting stack graphs (). e) Which of the countries shown below has approximately % of their population over 0 years of age? continued from page 8 f) For which of the literary awards shown did males make up closest to three fifths of the recipients? Population structure Male Literary Awards Female Australia UK USA 0 0 Under % Over 0 years Pulitzer Prize (fiction) Nobel Prize (literature) National Book Critics Circle Award PEN/Faulkner Award Booker Prize Percent g) In which of the wine regions shown does red wine production account for two fifths of total production? h) Of all the years in which Australia has won more than medals, which year produced the least gold medals? Victoria s Wine Regions Yarra Valley Murray Darling Goulburn Valley Heathcote Grampians % White wine production Red wine production Medals won AUSTRALIAN OLYMPIC MEDALS (Since 98) 9 98 BRONZE SILVER GOLD Olympic Year i) Which of the mammals shown has the lowest protein content in their milk? Harp Seal Polar Bear Reindeer Arctic Fox Fur Seal Arctic mammals - Gross composition of milk, % of total Fat Protein Lactose % j) Which country has the highest percentage of people living on their own? percent (%) HOUSEHOLD MAKEUP - Number of dwellers France Japan USA Germany page 8 Maths Mate /8 Skill Builder 9

300 Skill 9. Calculating the mean and median of sets of data (). Mean (or average) Add all the values in the set. Divide the total by the number of values in the set. continues on page 8 Set of data:,,,,,,, Mean values so 8 Median (middle value) Write all the values in order. Odd numbered set - middle value. Even numbered set - average of the middle values. Set of data (even):,,,,,,, Ordered set:,,,,,,, Median +. Q. This table shows the number of ski runs at selected resorts in Colorado. Find the mean and median of the data. Resorts in Colorado - ski runs 8 A. Mean Median,,,, 8, + 8 find middle value values in the set, so divide by order values average middle values a) This table shows the number of stations on some of the monorails in the USA. Find the median of the data. Monorails of USA (Number of Stations) 8 8 8,,,,,,, 8, 8, 8... average middle values middle values median b) This table shows the NASCAR sprint finishes for Carl Edwards between 00 and 009 at Pocono raceway. Find the median of the data. Pocono Raceway NASCAR sprint car series 0-09 finishes - Carl Edwards median c) This table shows the atomic number of the non-metals in the periodic table of elements. Find the median of the data. d) This table shows a -day temperature forecast for Geelong. Find the median high temperature for the interval. Non-metals Periodic table of elements Carbon (C) Nitrogen (N) Oxygen (O) 8 Phosphorus (P) Sulphur (S) Selenium (Se) High Low Sydney -day Forecast Mar 9 00 WED THU FRI SAT SUN 0 C C C C C C 8 C C 8 C C MON 0 C C TUE C 8 C order values median... median page 8 Maths Mate /8 Skill Builder 9

301 Skill 9. Calculating the mean and median of sets of data (). e) This table shows the average lifespan of some animals. Find the mean of the data. cat ANIMAL LIFESPANS - years camel Mean... rhinoceros grizzly bear values in the set, so divide by f) Ada selects scrabble letters that spell the letter word quartzy. Find the mean value of her tiles. A... mean mean Mean U R... T continued from page 8 Value of Scrabble tile selection Y Q 0 Z 0 g) This table shows the number of calories per serving of some carbohydrates. Find the mean of the data. Carbohydrate Calories per 00 g potatoes 0 macaroni 9 spaghetti 00 pasta 0 rice 0 bread 0 h) This table shows the number of stations on some of Europe s monorails. Find the mean of the data. Europe s monorails (Number of Stations) 0... Mean mean mean Mean... i) This table shows the number of petals on some flower species. Find the mean and median of the data. Lily Iris Fuschia Buttercup Amaryllis Number of petals Narcissus Delphinium 8 j) This table shows the total number of medals won by the USA at the winter Olympics from 99 to 00. Find the mean and median of the data. [Round to the nearest integer.] United States Winter Olympics Medals year medals... Mean... Median... mean median... Mean... Median... mean median page 8 Maths Mate /8 Skill Builder 9

302 Skill 9. Calculating the mode and range of sets of data. Mode (most common value) Set of data:,,,,,,, Range Ordered set:,,,,,,, Write all the values in order. Mode Subtract the lowest value from the highest value. Range Hint: A set of data can have more than one mode if two or more values repeat the same number of times. Q. The number of scrabble tiles available for each letter is shown below. Find the mode and range of the data. Z Q X J K Y W V H F P M C B G D U L S Numbers of Scrabble tiles R T N O I A E A. Mode Range Z Q X J K Y W V H F P M C B G D U L S R T N O I A E difference between highest and lowest Numbers of Scrabble tiles 9 The value is in the set 9 times a) This table shows the NASCAR sprint finishes for Jeff Gordon between 00 and 009 at Pocono raceway. Find the mode and range of the data. Pocono Raceway NASCAR sprint car series 0-09 finishes - Jeff Gordon A b) The values of scrabble tiles are shown below. Find the mode and range of the data. Values of Scrabble tiles E I L N O R S T U D G B C M P F H V W Y K J X Q Z The value is in the set times Range Range mode range mode range c) This table shows the total number of gold medals won by Great Britain at the summer Olympics. Find the mode and range of the data. Great Britain Olympic Gold Medals year medals d) This table shows the number of calories per serving of some fruits. Find the mode and range of the data. Fruit Calories per 00 g Apple Orange 0 Banana 9 Grapes 0 Kiwi fruit 9 Pear 0 Range Range mode range mode range page 88 Maths Mate /8 Skill Builder 9

303 Skill 9.8 Interpreting line graphs (). Q. In Brisbane, between which days did the maximum temperature increase and the minimum temperature decrease? Temperature ( C) Wed Brisbane Weather - Sept 8th - th Thur Fri Sat Sun Daily maximum temperature Daily minimum temperature Mon Tue A. Friday & Saturday Temperature ( C) Brisbane Weather - Sept 8th - th Wed Thur Fri Sat Sun Mon Tue Daily maximum temperature Daily minimum temperature continues on page 90 Find the daily maximum temperature graph and mark where the line increases between days. Then find the minimum graph and mark where the line decreases. They match between Friday and Saturday. a) Using the graph below, in March 008 what was the GDP growth rate for Australia? % growth % growth c) In which decade was the biggest percentage decrease in the number of slaves? Slaves (%) Jan / Jan / 00 Australian GDP (Growth Rate) Jan / 00 Jan / 008 Australian GDP Growth Rate Jan / 00 Jan / 008 Jan / 009 Jan / 00 Jan / 009 Jan / 00 % Slaves as a percentage of total US population b) According to the graph below, at what time of day is your body temperature the lowest? [Give your answer to the nearest hour.] temperature ( C) 8 0 Body temperature changes during a day exercise vigilance after meal alertness dip sleep midday midnight time - hours d) For how many hours would this torch have been in use if the battery had. volts remaining? Voltage (volts) How torches drain a battery 0 President Ulysses S. Grant Battery use (h) h page 89 Maths Mate /8 Skill Builder 9

304 Skill 9.8 Interpreting line graphs (). e) Which day of Melbourne s week shown below had a difference between the daily maximum and daily minimum temperatures of 9 C? Temperature ( C) 0 0 Melbourne Weather - Sept 8th - th continued from page 89 f) Find the difference between the lowest price of a chicken and the lowest price of a dozen eggs. price ($) The chicken & the egg 0 Wed Thur Fri Sat Sun Daily maximum temperature Daily minimum temperature Mon Tue price/dozen eggs price/chicken cents 00 g) In which decade did both the USA and Colorado median house price first exceed $00 000? $ (000 s) USA Colorado Median House prices i) In which of the years shown was the total number of wholesale digital and physical CD singles sales the greatest? $ s Australian Wholesale Music Sales 00 Digital - Single Physical CD - Single h) In which year were the unemployment rates in Japan and New Zealand the same? unemployment (%) UK Japan NZ Unemployment Rates Year j) In which of the years shown was there the greatest difference between the average crowds at ANZ stadium and the SCG? Average attendance AFL - Average crowds in Sydney 0 '0 '0 SCG ANZ Stadium '08 '09 '0 ' Year page 90 Maths Mate /8 Skill Builder 9

305 Skill 9.9 Interpreting pie charts. Q. From 009 to 00, which community platform had a percentage growth of closest to %? % Growth from 09 to 0 - top community platforms twitter A. twitter 0% zimbio face book % multiply wikea a) Which region almost halved its share of hard coal production between 9 and 00? Hard coal production 9 00 China former USSR Asia Africa OECD b) Which element makes up about 80% of Neptune s atmosphere? Atmospheric composition of Neptune Hydrogen (H ) Helium (He) Ammonia (CH ) c) Which age group most uses the internet? US Internet usage 00 by ages (% of internet-using population) d) Which two animals account for 0% of all pets? Pet owners choice dogs cats birds horses other e) Approximately what percentage of indoor water usage is accounted for by toilets? A ) % B ) % C ) % D ) 0% other leaks Indoor water use (% of daily total) toilets f) Which response, to accepting a stranger as a friend on facebook, is most similar between males and females? Accepting strangers as friends on facebook? No Yes Sometimes taps clothes washing Male Female showers page 9 Maths Mate /8 Skill Builder 9

306 Skill 9.0 Interpreting stem-and-leaf plots (). continues on page 9 To complete a stem-and-leaf plot from a given set of data: Write the values from the data set - each unit digit is a leaf beside its corresponding tens (or hundreds) digit, which is a stem. Hint: tens value STEM 0 units value LEAF and hundreds & tens values units value STEM LEAF To calculate values from a stem-and-leaf plot: Median (middle value) Count the number of leaves. If an odd number of leaves: Count from the top left leaf until you reach the middle leaf. This digit is the unit and must be put with the corresponding stem. If an even number of leaves: Count from the top left leaf until you reach the two middle leaves. Read the digits with their corresponding stems. Find the average of the middle numbers. Range Subtract the lowest number (top left leaf) from the highest number (bottom right leaf). stem Data set of elements: {, 8, 8, 9, 0,,,,,, 9, 0, } leaves median (th element) range lowest value median highest value range high low 8 Q. This stem-and-leaf plot shows the number of winners of the Tour de France by country. Find the median of the data. Stem 0 Leaf A. countries have Tour de France winners median th score Stem 0 Leaf values above values below middle leaf page 9 Maths Mate /8 Skill Builder 9

307 Skill 9.0 Interpreting stem-and-leaf plots (). a) Complete the stem-and-leaf plot for the data: 8,,,, 8,,, 9,,, 9, 0,, Stem Leaf Key continued from page 9 b) Complete the stem-and-leaf plot for the data showing the results of the women s high jump at the Olympics: 0, 0, 0, 0, 0, 0, 0, 9, 9, 9, 8 Stem 8 Leaf cm c) Find the median and range of the monthly average high temperatures for Adelaide, SA. Stem Leaf lowest score scores so middle score halfway between score () and score () C highest score 9 d) This stem-and-leaf plot shows the results of the men s pole vault jump at the Olympics. Find the median and range of the data. Stem Leaf cm median... range... median... range... median range median range e) Find the median and range for these American states that have the lowest number of counties. Stem 0 Leaf counties median... range... f) This stem-and-leaf plot shows the water consumption rates Stem Leaf per kilogram of body 0 weight for selected livestock. Find the 9 median and range 0 0 of the data. 8 9 ml median... range... median range median range page 9 Maths Mate /8 Skill Builder 9

308 Skill 9. Interpreting step graphs, histograms and scatter plots (). Q. Of the items shown which has the longest breakdown time span? A. plastic bag continues on page 9 Check the length of each line segment. Breakdown of household waste Breakdown of household waste paper paper orange peel orange peel milk carton milk carton cigarette butt cigarette butt plastic bag years plastic bag years a) If you were a female with % body fat into which catgory would you fall? Essential fat Athlete Fit Acceptable fat ranges of body fat each mark % b) A sample of coal contains 8% carbon. To which category does it belong? Anthracite Coal ranks Bituminous sub - bituminous lignite Find woman Obese women percentage men carbon % c) Apart from styrofoam which product has the longest breakdown time span? disposable nappy glass bottle aluminum can plastic drink bottle Breakdown of household waste styrofoam Time (years) NEVER d) For what age is the target heart rate zone between 8 and beats per minute? Target HR (beats /min) Target Heart Rate Zones (HR) by age Age (years) e) Of the liquids shown below which could have a ph that is neutral? Ammonia Vinegar Sea water Blood Urine Lemon 0 ph ranges of common liquids Acid Neutral Base f) Which animal sleeps the most hours each day? Cat Dog Bat Giraffe Human The range of daily sleep requirements for selected animals Sleep (h/day) page 9 Maths Mate /8 Skill Builder 9

309 Skill 9. Interpreting step graphs, histograms and scatter plots (). g) Approximately what percentage of facebook users login 0 to 0 times per week? frequency 0% 0% 0% 0% 0% facebook users login/week continued from page 9 h) Did most shoppers spend more or less than $0? Amount spent in a grocery store by 0 consecutive shoppers Number of shoppers Amount ($) i) Are more cuckoo eggs longer than or shorter than mm? No. of cuckoo eggs Length of Cuckoo eggs length (mm) j) Based on this data what is a reasonable exam result for a student who sleeps for hours on the night before their exam? A) 0% B) 0% C) 0% D) 00% Sleep and exam performance Exam result (%) Night s sleep (hours) k) An inclined plane is propped at one end to a certain height and a ball is rolled down. Which best describes the trend line for the time taken for a ball to roll the length of the inclined plane? A ) no height, faster rolling ball B ) greater height, faster rolling ball C ) lesser height, faster rolling ball height (cm) Ball rolling on an incline plane time (seconds) l) Which best describes the sample? A ) increased price, decreased sales B ) increased price, increased sales C ) decreased price, decreased sales Sales volume Average sales price of $0 books by month average price ($) page 9 Maths Mate /8 Skill Builder 9

310 page 9 Maths Mate /8 Skill Builder 9

311 0. [Probability] Skill 0. Describing the degree of likelihood of an event. Q. There are marbles in a bag and 8 of them are green. How many marbles do you have to select to make sure you have a green marble? a) There are red, purple and white rulers in a drawer. What is the largest number of rulers you could select from the drawer without taking a white ruler? red, purple, white... red + purple... A. Chances of selecting a green marble 8 Chances of selecting a different color marble It is possible to select all 9 other coloured marbles before you choose a green marble. b) There are 8 chocolate, 0 milk and cream biscuits in a box. How many biscuits do you have to pick to make sure you have a chocolate biscuit? c) Linda has 8 malteses and coolmints in her bag. How many lollies does she need to take out from her bag to make sure she has coolmints? d) There are orange, 9 red and white jellybeans in a jar. How many jellybeans do you have to select without looking in order to pick a white jellybean? e) There are twenty different pairs of socks in the drawer. How many socks need to be picked up without looking in order to have a pair of matching socks? f) Of the 00 songs on an ipod, 8 are pop songs. How many songs do you need to play on a random setting to be sure you will hear a pop song? g) Of the 8 movies on Tom s computer, are comedies. How many movies does Tom have to watch on a random setting to be sure he watches a comedy? h) How many people do you need to gather to be sure that at least two of them were born on the same day of the week? page 9 Maths Mate /8 Skill Builder 0

312 Skill 0. Recognising the likelihood of an event. Hints: Probability ranges from 0 to. The closer the probability is to, the more likely the event is to happen. The closer the probability is to 0, the more unlikely the event is to happen. Impossible Unlikely Equally likely Likely Certain 0 0/0 Q. Mary s new baby will be a boy. Which expression best represents the probability of this event? A ) very unlikely B ) a 0/0 chance C ) very likely D ) certain A. Consider the possibilities. The new baby can either be a boy or a girl so there is a 0/0 chance B a) Einstein will be the oldest living person. Which expression best represents the probability of this event? A ) impossible B ) very unlikely C ) very likely D ) certain A b) It will rain in the Sahara desert tomorrow. Which expression best represents the probability of this event? A ) impossible B ) very unlikely C ) very likely D ) certain c) The football team will win the next game. Which expression best represents the probability of this event? A ) very unlikely B ) a 0/0 chance C ) very likely D ) certain d) The whole class will get 00% in the next Maths test. Which expression best represents the probability of this event? A ) very likely B ) a 0/0 chance C ) very unlikely D ) impossible e) There will be no school holidays next year. Which expression best represents the probability of this event? A ) very unlikely B ) a 0/0 chance C ) very likely D ) certain f) Wednesday comes before Thursday. Which expression best represents the probability of this event? A ) very unlikely B ) a 0/0 chance C ) very likely D ) certain page 98 Maths Mate /8 Skill Builder 0

313 Skill 0. Finding the possible outcomes (sample spaces) of an event by completing tables. Complete the table to reveal all the possible outcomes (PO) (sample space). Count the number of possible outcomes (PO) (sample space). Q. How many different outcomes are possible when a die is thrown and this spinner is spun? [Complete the table.] Possible outcomes A B C Spinner A B C Die A, B, C, A, a) A zoo has both male and female primates. There are gorillas and chimpanzees. Find the size of the sample space. [Complete the table.] Outcomes (sample space) male gorilla male female female chimpanzee gorilla chimpanzee A. PO 8 Possible outcomes Spinner b) How many different outcomes are possible choosing a primary colour (red, blue and green) and tossing a coin? [Complete the table.] Possible outcomes Coin A B C H T Die A, B, C, A, B, C, A, B, C, A, B, C, A, B, C, A, B, C, Primary colour R G B R,H Each space represents outcome c) How many different outcomes are possible when rolling a die and flipping a coin? [Complete the table.] Possible outcomes Coin H T Each row represents outcome Die H, T, H, d) How many different outcomes are possible when spinning a spinner labelled,,,, and flipping a coin? [Complete the table.] Possible outcomes Coin H T Spinner,H e) A car comes in silver, red or purple as a convertible or hardtop. Find the size of the sample space. [Complete the table.] Outcomes (sample space) silver convertible silver red f) A vendor sells vanilla and chocolate ice cream. Customers can have a waffle or sugar cone and either hot fudge or caramel topping. How many different outcomes are possible when ordering an ice cream in a cone with a single topping? [Complete the table.] Outcomes (sample space) vanilla waffle hot fudge vanilla caramel vanilla page 99 Maths Mate /8 Skill Builder 0

314 Skill 0. Finding the possible outcomes (sample spaces) of an event by completing tree diagrams (). continues on page 0 From the start use the first condition to list all the possible outcomes (PO) on the first set of branches. From each of the first outcomes create enough branches to list all the possible outcomes of the second condition. Continue in this way until the tree diagram is completed. Count the number of pathways from the start to the end of each branch line. The number of pathways equals the total number of possible outcomes (sample space). Q. How many different -digit numbers can be made using the digits, and if the digits can be used only once? [Complete the tree diagram.] First digit Second digit Third digit Start A. Possible outcomes (PO),,,,, First condition Second condition Third condition First digit Second digit Third digit Start Pathways a) How many different outcomes are possible when flipping a coin and spinning this spinner? [Complete the tree diagram.] Coin Spinner H 8 pathways 8 PO Start Coin T H Start C Spinner A B C D A B C D A T B D b) How many different outcomes are possible when flipping a coin and cutting a king from any of the suits in a pack of cards? [Complete the tree diagram.] coin (H,T) King (,,, ) K H Start PO HA, HB, HC, HD, TA, TB, TC, TD... c) How many different outcomes are possible when choosing a season of the year and one of the point compass directions? [Complete the tree diagram.] Start season direction Summer Autumn d) How many different outcomes are possible when flipping coins (assuming order matters)? [Complete the tree diagram.] First coin Second coin Third coin H H Start page 00 Maths Mate /8 Skill Builder 0

315 Skill 0. Finding the possible outcomes (sample spaces) of an event by completing tree diagrams (). e) How many different gendre combinations are possible if a couple have children and order matters? [Complete the tree diagram.] st child B Start continued from page 00 f) Photos can be printed in various sizes (, ) and finishes (matte, gloss) with single or double prints an option. How many choices are possible? [Complete the tree diagram.] Start nd child B size rd child finish m st child B Start G number nd child B G B G rd child BGBG BGBG g) Jeans come with fly types (zipper or button fly) and cuts (boot cut, stove pipe, straight leg, skinny, flared). How many choices of jeans are possible? [Complete the tree diagram.] Start h) How many different -digit numbers less than 00 can be made using the digits,, and 8 if the digits can be used only once? [Complete the tree diagram.] Start fly type zipper button fly First digit cut Second digit Third digit i) There are kinds of soup on the menu: chicken, vegetable and pumpkin. They may be served hot or cold and always with a condiment of parsley or basil. How many choices are possible? [Complete the tree diagram.] Start j) On a library visit Tara must decide whether to loan a fiction or non-fiction item in one of the available formats (book, movie, tape, large print) for weeks or weeks. How many different options does Tara have? [Complete the tree diagram.] Start soup type temperature herb chicken H category type time fiction non-fiction page 0 Maths Mate /8 Skill Builder 0

316 Skill 0. Calculating the probability of a simple event (). Find the number of favourable outcomes for the event. Find the total number of possible outcomes. Divide the number of favourable outcomes by the number of possible outcomes: number of favourable outcomes Pr(event) number of possible outcomes continues on pages 0 to 0 OR Example: Experiment throwing a standard die Event throwing a number greater than Possible outcomes (PO) (throwing a,,,, or a ) Favourable outcomes (FO) (throwing a or a ) Probability (Pr) out of (FO out of PO) Equally Impossible Unlikely Likely Certain likely 0 A B C D E F G FO PO Hints: Probability ranges from 0 to. The closer the probability is to, the more likely the event is to happen. The closer the probability is to 0, the more unlikely the event is to happen. Q. When a spinner is spun, what is the probability of spinning a number greater than 0? 9 0 A. FO (,, 9, ) PO 8 (,,, 0,,, 9, ) FO Pr(number > 0) PO 8 0. a) A hookey ring is thrown. What is the probability of hooking an even number? FO (,,, 8, 0, )... PO ( to )... FO Pr(even number) PO... b) A card deck of playing cards is shuffled and one card is dealt from the top of the deck. What is the probability that it will be a black card? FO... PO... FO Pr(black card) PO... page 0 Maths Mate /8 Skill Builder 0

317 Skill 0. Calculating the probability of a simple event (). c) When a die is rolled, what is the probability of rolling an even number? continued from page 0 and continues on pages 0 to 0 d) When a spinner is spun, what is the probability of spinning a G? G F H E A D B C FO... PO... FO Pr(even number) PO... e) A spinner is divided into 0 equal parts. When it is spun once, what is the probability of spinning an odd number? FO... PO... FO Pr(spinning a G) PO... f) If a letter tile is chosen at random, find the probability of choosing letter M. M R E E E M N S A T U FO... PO... FO Pr(odd number) PO... FO... PO... FO Pr(letter M) PO... g) A card deck of playing cards is shuffledand one card is dealt from the top of the deck. What is the probability that it will be a King? FO... PO... FO Pr(king) PO... h) A day is randomly selected from the month of November. What is the probability that it will be a holiday? November SUN MON TUE WED THU FRI SAT holiday FO... PO... FO Pr(holiday) PO... page 0 Maths Mate /8 Skill Builder 0

318 Skill 0. Calculating the probability of a simple event (). i) A bag contains 0 keys, one of which opens the door to the prize car. One key is randomly selected from the bag. What is the probability of selecting the winning key? continued from pages 0 and 0 and continues on pages 0 and 0 j) A card deck of playing cards is shuffled and one card is dealt from the top of the deck. What is the probability that it will be a club? FO... PO... FO Pr(winning key) PO... k) A card deck of playing cards is shuffled and one card is dealt from the top of the deck. What is the probability that it will be a red card? FO... PO... Pr(club)... l) A spinner is divided into 8 equal parts. When it is spun once, what is the probability of spinning an even number? 8 FO... PO... Pr(red card)... m) When the spinner is spun once, what is the probability of spinning a prime number? 9 FO... PO... Pr(prime number)... FO... PO... Pr(even number)... n) When the spinner is spun once, what is the probability of spinning a composite number? FO... PO... Pr(composite number)... page 0 Maths Mate /8 Skill Builder 0

319 Skill 0. Calculating the probability of a simple event (). o) There are 8 horses, 0 dogs, chickens and pigs in a yard. If an animal is selected at random, what is the probability that a chicken is chosen? continued from pages 0 to 0 and continues on page 0 p) Ten balls numbered to 0 are mixed together and then one ball is drawn. Find the probability that a number less than is drawn q) There are tomato soup cans, chicken soup cans, vegetable soup cans and pumpkin soup cans in the cupboard. If a can is chosen at random, what is the probability that it is a chicken soup can? r) Mia has a bag that contains blue, white, green and yellow marbles. If Mia is randomly selecting a marble, what is the probability that she chooses a green marble? Q. Which event is most unlikely to happen? A ) choosing a spade from a deck of playing cards B ) rolling a on a standard die C ) selecting a white marble from a bag of 8 black and white marbles A. Consider each alternative: a) spades in cards b) four on a sided die c) white marbles out of 0 0 B Simplify: Least likely Simplify: s) Which event is most unlikely to happen? A ) rolling a on a standard die B ) drawing a diamond from a deck of playing cards C ) predicting boy for an unborn baby t) Which event is most likely to happen? A ) choosing false as the answer B ) selecting the winner in a 0 horse race C ) scoring the only touchdown in a game of football page 0 Maths Mate /8 Skill Builder 0

320 Skill 0. Calculating the probability of a simple event (). u) Which event is most likely to happen? A ) winning the jackpot in a lottery B ) rolling an odd number on a die C ) selecting a consonant from the word GEOMETRY continued from pages 0 to 0 v) Which event is most likely to happen? A ) turning heads on a tossed coin B ) serving an ace ten times in a row C ) rolling a number greater than on a standard die w) Which event is most unlikely to happen? A ) marking a cross playing noughts and crosses B ) selecting an even number from the numbers to 8 C ) throwing a on a hookey board marked to x) Which event does not have a 0% chance of success? A ) drawing a red card from a deck of playing cards B ) throwing a bullseye on a dartboard C ) marking a nought in noughts and crosses y) Which event is most likely to happen? A ) selecting red from the colors of the rainbow B ) moving a pawn at the start of a chess game C ) randomly hitting a key on a keyboard and it being the tab key z) Which event is most likely to happen? A ) choosing a prime number from the numbers to B ) winning a car in a raffle C ) selecting a vowel from the word PROBABILITY page 0 Maths Mate /8 Skill Builder 0

321 Skill 0. Calculating the probability of a simple event using probability scales. Divide the number of favourable outcomes (FO) by the number of possible outcomes (PO). (see skill 0., page 0) Q. A coin is tossed and tails comes up. Which letter A to E best represents the probability of the event? Impossible Unlikely Equally likely Likely Certain A B C D E A. FO (tails) PO (heads or tails) FO Pr (event) possible outcomes PO 0. The answer is C a) A blue tile will be drawn from a box containing 8 black tiles and blue tiles. Which letter A to D best represents the probability of the event? Impossible 0 A Unlikely B Equally likely FO PO... simplify: FO PO... c) A caramel chocolate will be drawn from a box containing caramel and spearmint chocolates. Which letter A to E best represents the probability of the event? Impossible Unlikely Equally likely Likely Certain A B C D E FO PO e) A red marble will be drawn from a bag containing red and blue marbles. Which letter A to D best represents the probability of the event? Impossible 0 A Unlikely B Equally likely C C Likely Likely Certain FO PO D Certain D b) A die is rolled and a comes up. Which letter A to G best represents the probability of the event? Impossible 0 A Unlikely Equally likely Likely B C D E F Certain FO PO... FO PO... d) A -sided pencil is rolled and the logo, printed on side, comes up. Which letter A to F best represents the probability of the event? Impossible 0 A Unlikely B C Equally likely Likely FO PO f) A club is drawn from a pack of playing cards. Which letter A to E best represents the probability of the event? D E FO PO G Certain Impossible Unlikely Equally Likely likely Certain 0 A B C D E F page 0 Maths Mate /8 Skill Builder 0

322 Skill 0. Interpreting Venn diagrams. Count the total number of possible outcomes. Shade the areas inside the Venn diagram that fit the description for favourable outcomes. Use the formula for the probability of an event. Q. What is the probability that a student selected at random from the year 8 class did not watch any Sunday night TV? Sunday night TV - year 8 survey ABC a) What is the probability that a student chosen at random from Ms Bourke s class learns both piano and violin? Ms Bourke s Year Class learn piano ch 0 S B S learn violin Venn diagram A. FO PO FO Pr(event) PO Sunday night TV - Year 8 survey ABC Students who do not watch TV are shown in the shaded area. Students who do watch TV are shown in the white area. b) What is the probability that a surveyed river, visited at random, contained only phosphate pollutants? River pollutants survey crude oil ch 0 S B S Venn diagram phosphates possible outcomes Ms Bourke s th Grade Class learn piano learn violin Venn diagram Venn diagram 0 Venn diagram FO PO FO Pr(event) PO... c) What is the probability that a tennis player chosen at random named the serve as the stroke that needs improvement? tennis strokes for improvement serve backhand volley Venn diagram FO PO... Pr(event)... FO PO... Pr(event)... d) What is the probability that a person chosen at random did not wear sunscreen? sun glasses Sun wear 0 sunscreen FO PO... Pr(event)... hat Venn diagram page 08 Maths Mate /8 Skill Builder 0

323 Skill 0.8 Calculating the probability of complementary events. Identify and calculate the probability of the event. number of favourable outcomes Pr(event) number of possible outcomes Identify the complementary events. To calculate the probability of the complementary event, subtract the probability of the event from : Pr(complementary event) Pr(event) Hints: The complement of the event the plane will be on time is the plane will not be on time. Winning - not winning, voting yes - voting no are examples of complementary events. FO PO Q. A box contains 0 blue, green and white ribbons. If a ribbon is selected at random, find the probability that it is not a green ribbon. a) The probability of an earthquake of. magnitude occurring in San Francisco in any year is %. What is the probability of there being no earthquake in San Francisco next year? [Give the answer as a percentage.] Pr(earthquake) %... Pr(no earthquake) 00% %... c) A bag contains gold and silver discs. The probability of choosing a gold disc is. What is the probability of not choosing a gold disc? Pr(gold)... Pr(not gold)... e) A ballot box contains 0 liberal, green and 8 labour votes. If one vote is picked at random, what is the probability that it is not labour? A. Event green Complementary event not a green Pr(green) 8 Pr(not green) b) The cookie jar contains cookies of which are burnt. What is the probability of Leah choosing a cookie that is not burnt? Pr(burnt)... Pr(not burnt)... d) Ten balls numbered to 0 are mixed together and then one ball is drawn. Find the probability that the number drawn is not a perfect square (i.e., or 9). Pr(perfect square)... Pr(not perfect square)... f) Of the New Zealand families who have children, % have one child and % have two children. What is the probability that a New Zealand family with children, when selected at random, has more than two children? [Give the answer as a percentage.] page 09 Maths Mate /8 Skill Builder 0

324 Skill 0.9 Calculating the probability of mutually exclusive events. Find the probability of each event. number of favourable outcomes Pr(event) number of possible outcomes Add the probabilities of each event in order to find the probability of both events occurring. Pr(A and B) Pr(A) + Pr(B) FO PO Simplify the fraction where necessary. Hint: Mutually exclusive events cannot occur at the same time. Example: A card selected from a pack of playing cards Playing cards hearts clubs can either be diamonds spades red or black, but not both. red suits black suits Venn diagram Q. A stable contains mares, stallion and 8 geldings. If a horse is selected at random, find the probability that it is a mare or a gelding. A. Pr(M) 8 Pr(G) Pr(M or G) Pr(M) + Pr(G) 8 + a) What is the probability of drawing a red card or a club from a pack of cards? Pr(red) Pr(club)... Pr(red or club) Pr (red) + Pr (club) b) When a die is rolled, what is the probability of rolling a or a? Pr() Pr()... Pr( or ) Pr () + Pr () c) In the lucky dip box there are lolly bags, marble bags and sand bags. If a bag is selected at random, find the probability that it is a lolly or a marble bag. Pr(red) Pr(club) d) When a die is rolled, what is the probability of rolling an odd number or an even number? page 0 Maths Mate /8 Skill Builder 0

325 Skill 0.0 Finding the possible outcomes of an event by applying the counting principle. Multiply the number of possibilities in event by the number of possibilities in event. Hint: The counting principle can be extended to or more events. Q. Maria chose one chemistry class, one maths class, one history class and one English class. According to the schedule she has different chemistry classes, different maths classes, different history classes and different English classes to choose from. If no scheduling conflicts exist, how many different four-course selections can Maria make? A. Number of -course selections a) How many different -digit numbers can be made from the digits,,, and 8, if a digit can appear just once? N... digits in st position, digits in nd position, digits in rd position c) A coin and a six-sided die are tossed. How many results are possible? N... 0 b) In how many ways can a family of five stand in a line for a photograph? N... d) How many -digit numbers can be formed with the digits,, and if no digit can be used more than once? N... e) Two coins and one five-sided die are tossed. How many results are possible? N... f) In how many ways can six books be arranged on a shelf? N... g) How many possible outfits can be created with different dresses, different vests and different pairs of shoes? N... i) In how many ways can a coach select two emergencies from a total of five players? N... h) Using one of each kind of ingredient, how many hamburger combinations can be made with different kinds of bread, different fillings and different sauces? N... j) In how many ways can any of the vowels be grouped assuming they are not repeated? N... page Maths Mate /8 Skill Builder 0

326 page Maths Mate /8 Skill Builder Glossary

327 GLOSSARY TERMS DEFINITIONS EXAMPLES ac - an accuracy A measure of how close the result of a measuring comes to the true value.. is a fairly accurate estimation of π. acute angle An angle measuring less than acute-angled triangle A triangle in which every angle measures less than add (+) To join together. If you add black cow and white cows, there are + cows all together. addition The operation of finding the total or sum of two or more numbers to make one number. The result is called the sum or total. Adding and we reach a total (sum) of. + adjacent Immediately next to. The Daniher s live adjacent to the Bourke s. Church De Castella's house Daniher's house Bourke s house algebra A branch of Mathematics where numbers are represented by letters or symbols, called variables. x + x, so x equals, so equals am (ante meridiem) The time from midnight to midday (morning). analogue clock A clock or watch that has rotating hands and shows hour time. page Maths Mate /8 Skill Builder Glossary

328 angle The amount of turning between two straight lines that are fixed at a point. An angle is measured in degrees. an - ax annual Happening once a year. HAPPY NEW YEAR anticlockwise Moving in the opposite direction to the hands on a clock. approximate Very close to the actual size. To estimate by rounding off. If you have $.8 in your wallet, you can say you have approximately $.00 area ascending order The amount of surface covered by a D shape. Area is measured in square units, e.g. square centimetres (cm ) or square metres (m ). Arranged from smallest to largest. Becoming larger, greater or higher. The area of a rectangle is calculated by multiplying length (l) by width (w): A lw 8 Area 8 square units units units, and are in ascending order. associative property of addition and multiplication average Rule: When adding or multiplying, no matter how the numbers are grouped, the answers will always be the same. Or mean, is the total of all scores divided by how many scores there are. The number that could be used to replace every number in a set of numbers without changing the sum for the set. a + (b + c) (a + b) + c + ( + ) ( + ) a (b c) (a b) c ( ) ( ) The average of, and 9 is and So + + average speed axis of symmetry See speed. (pl. axes) See line of symmetry. Axis of symmetry page Maths Mate /8 Skill Builder Glossary

329 ba - bi backwards Away from your front. In reverse of the usual way. balance (money) The amount of money remaining in a bank account after all transactions have been completed. The bank account held $. After $ was withdrawn the balance of the account was $0. bar graph A graph using columns to show quantities or numbers so they can be easily compared. Camping is the favourite holiday. Number of people 00 0 Favourite Holiday 0 Beach Camping Skiing Caravaning Type base A line or surface on which a figure stands. b base base of a parallelogram The base (b) of a parallelogram is the length of any of its sides. h b h b base of a triangle The base (b) of a triangle is the length of any of its sides. h b h b b h between At a place bounded by two or more places. The child is between her parents. bi (or di) Prefix meaning two. A bicycle has wheels. page Maths Mate /8 Skill Builder Glossary

330 bisect To split into two equal parts. AM MB A M B bi - co brackets ( ) A pair of symbols used to enclose a mathematical expression. ( ) Brackets group take away. calculate calendar To find the exact value of mathematical operations. A time chart that tells us what day, week, month and year it is. + + calibration A mark on a scale. 0 cancel capacity To strike out an equal term on each side of an equation. Or volume, is the measure of the amount of liquid a container can hold. x cancel by adding to both sides of the equation x x 9 A jug has capacity because it can hold liquid, a brick does not. 000 ml 00 ml Cartesian plane See coordinate plane. chance The likelihood that a particular result or outcome will occur. The chance of rolling a with a standard die is in. chord A line segment on the interior of a circle. A chord has both end points on the circumference of the circle. chord closest Nearest to. The son is closest to the mother. coefficient The number which multiplies a variable. is the coefficient of x is the coefficient of y page Maths Mate /8 Skill Builder Glossary

331 co - co column A vertical line of data in a table. Medal Tally - Beijing Olympics 008 COUNTRY China United States Gold Silver Bronze 8 8 Russia 8 Great Britain 9 Germany 0 Australia combinations A selection of objects from a collection. Order is irrelevant. A class committee is a combination of boys and girls chosen from a total of boys and girls. common factor A whole number that is a factor of two or more non-zero whole numbers. The common factors of 8 and are,, and. common multiple A whole number that is a multiple of two or more non-zero whole numbers. The common multiples of and are 0, 0, 90,... compass An instrument that shows direction. N S complement of an angle An angle that, when added to the first angle, makes a right angle (or 90 in total). is the complement of, because + 90 complementary event composite number composite shapes The opposite of an event. All of the outcomes that are not included in the event. A positive integer that has factors other than just and the number itself. A combination of two or more D shapes into one figure. If the event is raining then the complementary event is not raining. is a composite number. The factors of are:,,,,, The above diagram is the composite of rectangular shapes. page Maths Mate /8 Skill Builder Glossary

332 composite space figures A combination of two or more D shapes into one object. co - co cone A solid with one circular base and one vertex. vertex base congruent shapes Have exactly the same size and shape. Triangles ABC and DEF are congruent. C F E A B D Sides Corresponding sides are congruent: AB DE, BC EF, AC DF Angles Corresponding angles are congruent: A D, B E, C F consecutive numbers Numbers that follow each other. and are consecutive numbers. constant term A term that has a fixed value and does not contain a variable. In y x + is constant x and y are variables. The speed of light in a vacuum (c) is a constant. c m/s conversion factor The amount by which you multiply or divide a number to change it to a different unit of measurement. m 00cm The conversion factor for changing metres to centimetres is 00 convert Change from a unit to another. kg can be converted to 000 g. page 8 Maths Mate /8 Skill Builder Glossary

333 co - cu coordinate plane A plane divided into four quadrants by a horizontal line called the x-axis and a vertical line called the y-axis. Quadrant Coordinate (,) Y Quadrant (0,0) Origin X Quadrant Quadrant coordinates An ordered pair of numbers that locate a point on a coordinate plane. The first number tells you how far from the origin to move along the x-axis. The second tells you how far from the origin to move along the y-axis. They are written in brackets with a comma in between. (,) are the coordinates of a point located units to the right and units upward from the origin (0,0). y-axis 0 0 (,) x-axis counting number cross multiply cross-section Any of the whole numbers from zero onwards. To simplify a proportion, written as two equal fractions OR To multiply each numerator by the denominator of the fraction across from it. The shape of the face that results when an object is cut through. 0,,,,,... are counting numbers. a : b c : d a c b d a d b c ad bc rectangle cross simplify cube To divide the diagonal numbers (when multiplying two fractions) by the same number to reduce their value before multiplying. A solid with six identical square faces cubed cubic unit A number cubed is the third power of the number. A unit of volume expressed in cubic form. cubed The volume of a solid is measured in the appropriate cubic units, e.g. cm or m. page 9 Maths Mate /8 Skill Builder Glossary

334 cylinder A solid with two parallel circular bases of the same size. bases cy - de data Collection of information that can include facts, numbers or measurements. Number of days HOSPITAL EMERGENCIES (MAY) 0 Number of emergencies day A unit of time equal to hours. A day starts and ends at midnight. daylight saving time Use of fictitious time in the summer months that prolongs light in the evening hours. During daylight saving clocks are one hour ahead of real time. deca Prefix meaning ten. Decathlon is an athletics contest with ten events. decade A unit of time equal to 0 years. 0 to 00 make a decade. decagon A shape with 0 sides. decimal number A number based on the ten place value system where a decimal point separates the units and tenths. The decimal number. represents: - ones - tenths OR and tenths. decimal place units tenths hundredths thousandths 0 is in the tenths place. is in the hundredths place. is in the thousandths place. decimal point (.) A point that separates the units and tenths in a decimal number.. is a decimal number where the and the are separated by a decimal point. decrease deduct To make smaller. To take away. 8 must decrease by to become. If you deduct from there are left. page 0 Maths Mate /8 Skill Builder Glossary

335 de - di degree ( ) A unit used to measure the amount of turn in an angle. Angle measures. degrees Celsius ( C) A unit used to measure temperature on the Celsius scale, used in the metric system. The Celsius (or centigrade) thermometer was invented by Swedish astronomer Anders Celsius in. 0 C freezing point of water 00 C boiling point of water C human body temperature 0 C C 0 C C 0 C C 0 C C denominator The number below the fraction bar in a fraction. The number of equal parts in one whole. Considering fifths, parts would make a whole. denominator deposit (money) To pay an amount of money into a bank account. A deposit of $ into a bank account with a balance of $ will increase the account balance to $0. descending order Arranged from largest to smallest. Becoming smaller, lesser or lower. 8, and are in descending order. diagonal A straight line inside a polygon joining any two vertices that are not next to each other. diagonal diameter of a circle A segment that passes through the centre of a circle and has both endpoints on the circle. The diameter of a circle is twice the length of its radius. diameter centre die (pl. dice) A numbered cube that is used in games. A standard die has to pips (dots) on each face with opposite faces adding to. difference The result when a number is subtracted from another number. The amount by which one number is bigger or smaller than another number. The difference between and is. digit Any of the first ten whole numbers from 0 to 9. There are 0 digits: 0,,,,,,,, 8 or 9 digit sum The sum of the digits in a number. has a digit sum of. + + page Maths Mate /8 Skill Builder Glossary

336 digital clock A clock that uses only numbers to show the time. (No hands!) di - di dimension A measure of size. A two-dimensional shape (D shape) has length and width. A three-dimensional shape (D shape) has length, width and height. D shape D shape l l w w h direction The way something is placed, pointing or moving. North, east, south, west, up, down, sideways, backwards and forwards. N W E discount (money) An amount subtracted from the regular price of an item to get the sale price. When $80 track shoes are on sale at % off discount % of $80 $0. S distance The length between two points. The distance between the fish is metres. m divide ( ) To share into groups. These cows are divided into groups. in each group dividend The first number written in a division. It is the number being divided. In a fraction the dividend is the numerator. In the division: 9 the number is called the dividend. divisible Can be divided without a remainder. 0 0 with 0 remainder. So 0 is divisible by and 0. page Maths Mate /8 Skill Builder Glossary

337 di - do divisibility tests Checks performed to help find the factors of a number. Divisibility tests (factor tricks) Examples is a factor of all even numbers. is a factor of all numbers with a digit sum that is divisible by. is a factor of all numbers where the last two digits are divisible by. is a factor of all numbers whose last digit is a or a 0. is a factor of all numbers that have both and as factors. 9 is a factor of all numbers with a digit sum that is divisible by 9. For to be a factor of a number, the difference between the sum of the even placed digits and the sum of the odd placed digits must be divisible by. For 0, 00, to be a factor of a number, that number must end in one 0 or two 0's or three 0's, etc. Numbers that end with 0,,, and 8, e.g., 0 has as a factor because and 9 is divisible by. has as a factor because is divisible by. 0 and 9 both have as a factor. 0 has as a factor because and are both factors. has 9 as a factor because and 8 is divisible by has as a factor because and which is divisible by. 0 has 0 as a factor, 00 has 00 as a factor etc. division The operation of sharing or grouping a number into equal parts. The division means: How many groups of can be divided into? OR How many groups of can be taken from before none remain? groups of. divisor The second number written in a division. It is the number that will divide the dividend. In a fraction the divisor is the denominator. 8 OR divisor 8 dodecahedron A regular solid in which all twelve faces are regular pentagons. dot plot A graph showing the frequency of data, using a number line. The number line has all the numbers in the sample. Each observation is marked with a point above the line. A a graph using dots to show how many hours are dedicated to sport by people. Hours playing sport page Maths Mate /8 Skill Builder Glossary

338 double Twice as much. Multiplied by two. Double is: + 8 OR 8. do - eq double bar graph A bar graph that shows two sets of data on the same graph. CHINESE SPANISH Officially Spoken Languages Speakers (millions) ARABIC ENGLISH FRENCH Countries east A compass direction. The sun rises in the east. E edges of a solid The segment (line part) where two faces of a solid meet. A rectangular prism has edges. face edge face eighth The position after seventh. st, nd, rd, th, th, th, th, 8th... elapsed time The amount of time between the start time and the finish time. The amount of elapsed time from : pm to :00 pm is minutes. ellipse A curved shape that looks like a squashed circle. The orbit of the Earth around the Sun is approximately an ellipse. enlargement To reproduce and make bigger. The second object is an enlargement of the first. equal () Exactly the same in value or size centimetres is equal to metre: 00 cm m equation A mathematical sentence formed by placing an equals sign () between two expressions. 9 + x 0 y 0 are examples of equations. equilateral triangle A triangle with all three sides of equal length. cm cm cm page Maths Mate /8 Skill Builder Glossary

339 eq - fa equivalent fractions Fractions that represent the same number. and 8 are equivalent fractions. They both equal 8. error The variation of a measurement. It may be contributed to by the precision of the instrument or the accuracy of the measured value. My measuring may be off by %! estimate To make a close guess based on rounding. 8 +? By rounding to 0 + 0, the estimation of the sum is 0. evaluate To work out the value. Evaluate: 0 0 even numbers A whole number that can be divided by two. Even numbers end with 0,,, or 8. is an even number. is not an even number. event Possible outcomes resulting from a particular experiment. Experiment: A die is rolled. Possible outcomes: Either a or a may result expense (money) The cost involved. You buy CDs at $ each. Your expense is $. experiment A controlled, repeatable process carried out in the study of probability. Tossing a coin is an experiment. expression A sequence of numbers and/or variables (letters) connected by operation signs. 0 x + y x are examples of expressions faces of a solid Polygons that join on their edges to form a solid. A rectangular prism has rectangular faces. face edge... face page Maths Mate /8 Skill Builder Glossary