NATIONAL MATHS YEAR 7. Jim Wade Jack Mock

Transcription

1 NATIONAL MATHS Jim Wade Jack Mock YEAR 7

2 01 First published 01 Private Bag 70 Marrickville NSW 17 Australia Tel: (0) Fax: (0) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of. ABN

3 Contents Contents Number and Algebra Number and Place Value Chapter 1 The Laws of Arithmetic 1 Getting started 1.1 Applying the associative and commutative laws 1. Operating with whole numbers 8 1. Solving number puzzles Using the number line to order numbers Integers Adding and subtracting integers 1.7 Order of operations and the distributive law The language of mathematics Miscellaneous extension exercise How much do you know? Chapter 1 Diagnostic test 7 Number and Algebra Number and Place Value Chapter Factors and Prime Numbers 9 Getting started 0.1 Special numbers 1. Factors and multiples of numbers 9. Prime numbers and composite numbers. Highest common factor and lowest 6 common multiple. Tests for divisibility 8.6 Extension: Exploring Fibonacci numbers 61 How much do you know? 6 Chapter Diagnostic test 66 Number and Algebra Fluency and Problem Solving Chapter Problem Solving 67 Getting started 68.1 Solve a simpler problem 69. The unitary method and make a table of values 7. Draw a diagram, look for a pattern, 76 invent your own numbers. Make a list and look for a pattern 80. Counting techniques and logic problems 86.6 Miscellaneous extension exercise 91 How much do you know? 9 Chapter Diagnostic test 96 Number and Algebra Real Numbers Chapter Fractions 97 Getting started 98.1 Comparing fractions using equivalence 99. Improper fractions and mixed numbers. Adding and subtracting fractions 7. Adding and subtracting mixed numbers 1. Multiplying and dividing fractions 11 and mixed numbers.6 Problem solving with fractions Miscellaneous extension exercise 117 How much do you know? 118 Chapter Diagnostic test 11 Number and Algebra Real Numbers Chapter Decimals, Percentages and Ratios 1 Getting started 1.1 Multiplying decimals 1. Dividing decimals 18. Rounding decimals to a given number 11 of places. Connecting fractions and percentages 1. Connecting decimals and percentages 18.6 Extension: Applying strategies to 11 solve unfamiliar problems.7 Simple ratios 1.8 Ratios with fractions, decimals 19 and percentages How much do you know? 1 Chapter Diagnostic test 1 Statistics and probability Chance Chapter 6 Probability 17 Getting started Ordering simple events from least likely 19 to most likely 6. Determining experimental probability Assigning probabilities to the outcomes 167 of events 6. Solving problems involving probability 171 and Venn diagrams How much do you know? 17 Chapter 6 Diagnostic test 177 Number and Algebra Patterns and Algebra Chapter 7 Algebra 179 Getting started Using letters to represent numbers Creating and evaluating algebraic expressions Applying the laws of arithmetic to simplify 189 algebraic expressions 7. Using the four arithmetic operations 19 with algebra 7. Using grouping symbols in algebra Adding and subtracting algebraic quantities Using algebraic symbols in other contexts Miscellaneous extension exercise 06 How much do you know? 07 Chapter 7 Diagnostic test 09 Revision Papers for Chapters 1 to 7 Chapter 8 Revision Papers for Chapters 1 to 7 11 Revision paper 1 1 Revision paper 1 Revision paper 18 Contents iii

4 Contents Measurement and Geometry Using Units of Measurement Chapter 9 Perimeter and Area 1 Getting started 9.1 Length 9. Perimeter 0 9. Area 9. Areas of rectangles, triangles and 7 parallelograms 9. Area conversions 9.6 Problem solving with areas and perimeters 9.7 Miscellaneous extension exercise 7 How much do you know? 0 Chapter 9 Diagnostic test Number and Algebra Money and Financial Mathematics Chapter Money, GST and Best Buys Getting started 6.1 Reviewing percentages 7. The goods and services tax (GST) 8. Best buys 6. Miscellaneous extension exercise 6 How much do you know? 66 Chapter Diagnostic test 69 Measurement and Geometry Location and Transformations Chapter 11 Transformations 71 Getting started Translation Reflection Rotation Enlargement and reduction Identifying line symmetry and 9 rotational symmetry 11.6 Solving geometrical problems 99 with transformations How much do you know? 01 Chapter 11 Diagnostic test 0 Number and Algebra Linear and Non-linear relationships Chapter 1 Equations 0 Getting started Solving equations by inspection Solving equations by the balance method Using backtracking to solve equations 1. Solving problems by using an equation 1 1. Miscellaneous extension exercise 1 How much do you know? 1 Chapter 1 Diagnostic test 17 Measurement and Geometry Using Units of Measurement Chapter 1 Volume 19 Getting started Identifying -dimensional shapes 1 1. Drawing nets of solids 1. Drawing different views of solids 8 1. Drawing solids from different perspectives 1. Calculating volumes of rectangular prisms 1.6 Capacity Miscellaneous extension exercise 0 How much do you know? Chapter 1 Diagnostic test Measurement and Geometry Geometric Reasoning Chapter 1 Geometry Getting started Identifying points, lines, rays and angles 7 1. Measuring angles 1 1. Calculating the angle sum of triangles, 6 quadrilaterals and polygons 1. Classifying triangles according to their 6 side and angle properties 1. Using compasses to bisect intervals 66 and angles 1.6 Constructing shapes from given 70 specifications 1.7 Identifying corresponding, alternate 71 and co-interior angles associated with parallel lines 1.8 Describing special quadrilaterals Miscellaneous extension exercise 8 How much do you know? 8 Chapter 1 Diagnostic test 90 Statistics and Probability Data Representation and Interpretation Chapter 1 Data 9 Getting started Collecting and classifying data 9 1. Displaying data in tables Displaying secondary data in column 0 graphs, pie charts and bar graphs 1. Displaying data in stem-and-leaf plots, 08 dot plots and frequency histograms 1. Calculating and interpreting the mean, 1 median, mode and range 1.6 Using spreadsheets to tabulate and graph data 1.7 Miscellaneous extension exercise 6 How much do you know? 8 Chapter 1 Diagnostic test 0 Revision Papers for Chapters 1 to 1 Chapter 16 Revision Papers for Chapters 1 to 1 Revision paper Revision paper 8 Revision paper 6 Appendices 7 ACARA syllabus map 8 Learning program 9 How to use Geogebra Grid paper 8 Answers 9 Glossary 9 Index 97 iv Contents

5 Chapter 1 The Laws of Arithmetic Syllabus Apply the associative, commutative and distributive laws to aid mental and written computation. (ACMNA11) Compare, order, add and subtract integers. (ACMNA80) KEY SKILLS AND KNOWLEDGE By the end of this chapter you should be able to: Use the associative and commutative laws for addition. (1.1) Use the associative and commutative laws for multiplication to aid calculation. (1.1) Use an appropriate non-calculator method to divide -digit and -digit numbers by a -digit number. (1.) Check estimates of answers obtained by written methods with a calculator. (1.) Solve number puzzles such as magic squares and arithmagons. (1.) Compare different numbers for size and plot them on a number line. (1.) Plot both positive and negative numbers on a number line. (1.) Add and subtract both positive and negative numbers. (1.6) Perform mathematical operations in the correct order and use the distributive law. (1.7) Use basic mathematical symbols. (1.8) Chapter 1 The Laws of Arithmetic 1

6 Number and Algebra Number and Place Value WHO AM I? I am an integer between and 99. The sum of my digits is 7. I am even. My units digit is one greater than my tens digit. Who am I? 1. Solving number puzzles Magic squares A magic square is one in which all the rows, columns and diagonals have the same sum. For example, the magic square shown is of the order ( columns, rows) and has the magic number 1. The Lo-Shu (scroll from the River Lo) square first appeared on an ancient Chinese tablet in 00 BCE, in the reign of Emperor Yu, and is the earliest record of a magic square. In legend, it is said to have appeared in dots on a tortoise s shell following a flood EXERCISE 1. Solving number puzzles 1. Complete these magic squares. (a) 8 (b) (c) 0 1 (d) 8 1 Chapter 1 The Laws of Arithmetic 11

7 Number and Algebra Number and Place Value. Use each of the numbers 1,,, once only to complete this magic square. Hint: What is the sum of all the integers from 1 to 16? What is the sum of one row or column? Here is a method for completing odd-sided magic squares. 1. Start at the top in the middle with 1.. Move diagonally up to the right for the next number.. If you move outside the square, slide to the far end of the adjacent column or row.. If rules and don t apply then go below the current square. 1. Have a go at completing this magic square with the numbers from 1 to using the above method. Try a 7 7 square now you know how easy it is. Arithmagons Example: Solution: Find the missing numbers, given that all the numbers on the vertices of the triangle add up to the number given on the line joining them. We could use the guess and check strategy. Now 9 = or 7 + or 6 + or + or + 7 = or + or + = or 8 + or 7 + or 6 +? 9 7 +?? Let us try 9 = Try 9 = ¹ 6 + = 1 Chapter 1 The Laws of Arithmetic

8 Chapter Fractions Syllabus Compare fractions using equivalence. Locate and represent fractions and mixed numerals on a number line. (ACMNA1) Solve problems involving addition and subtraction of fractions, including those with unrelated denominators. (ACMNA1) Multiply and divide fractions using efficient written strategies and digital technologies. (ACMNA1) Express one quantity as a fraction of another, with and without the use of digital technologies. (ACMNA1) KEY SKILLS AND KNOWLEDGE By the end of this chapter you should be able to: Find equivalent fractions. (.1) Write fractions with the same denominator to enable addition and subtraction. (.1) Reduce a fraction to its simplest equivalent form by cancelling. (.1) Express improper fractions as mixed numbers and vice versa. (.) Locate fractions and mixed numbers on a number line. (.) Express one quantity as a fraction of another. (.) Add and subtract fractions. (.) Add and subtract mixed numbers using written and calculator methods. (.) Subtract a fraction from a whole number. (.) Solve problems involving addition and subtraction of fractions. (.) Multiply and divide fractions and mixed numbers. (.) Solve problems involving fractions. (.6) Chapter Fractions 97

9 Number and Algebra Real Numbers INVESTIGATION Take a simple sum of fractions, say = = 1. Now investigate the possibility of finding two other fractions in their lowest terms (not twelfths) which sum to 7 1. GROUP ACTIVITY Ask two people in a group to nominate one fraction each between zero and one. Find any other fraction that lies between the two nominated fractions (see Exercise. Question 9). You can use a calculator with a fraction key to help you. Now take the new (middle) fraction and find another fraction which lies between this one and the previous lower fraction. Keep passing the answer around the group, always finding a fraction between the new one and the smaller of the original two fractions. Answer this question: How many fractions lie in between any given pair of fractions?. Multiplying and dividing fractions and mixed numbers Example 1: Example : Example : Finding a fraction of a quantity. Find of 1. This means 1 Multiplying fractions (finding a fraction of a fraction). = 8 = 1 Multiplying with cancelling. 1 = = 8 So of means multiply eh? Isn't this just like cancelling a fraction? goes into both and 1. And goes into both and Chapter Fractions

10 Number and Algebra Real Numbers Using a calculator We can use a fraction calculator to multiply 11. Using the fraction key enter 1 ab c 1 ab c ab c ab c = Answer: 1 8 Similarly to divide 1: Enter ab c ab c ab c 1 ab c = Answer: EXERCISE. Multiplying and dividing fractions and mixed numbers 1. Calculate these quantities. (a) of $16 (b) of hours (c) of 60 seconds (d) of 0 litres (e) of 600K (f) 6 of one dozen (g) 8 of 1 hours (h) of $18 (i) of 00 metres (j) 1 of 1 minutes 6. Multiply these fractions by selecting the required portion of the squares. (a) (i) What fraction of the rectangle is coloured in? (ii) What fraction of the coloured part is pink? (iii) What fraction of the whole rectangle is pink? (iv) Explain from the diagram the answer to 1 of 1. (b) (i) What fraction of the rectangle is coloured in? (ii) What fraction of the coloured part is green? (iii) What fraction of the whole rectangle is green? (iv) Explain from the diagram the answer to 1 of. (c) (i) What fraction of the rectangle is coloured in? (ii) What fraction of the coloured part is purple? (iii) What fraction of the whole rectangle is purple? (iv) Explain from the diagram the answer to 1 of. (v) Explain from the diagram the answer to of. (d) (i) What fraction of the rectangle is coloured in? (ii) What fraction of the coloured part is yellow? (iii) What fraction of the rectangle is yellow? (iv) Explain from the diagram the answer to of 6.. Verify your answers to Question by calculating these products on a fraction calculator. (a) 1 1 (b) 1 (c) (d) 6 Chapter Fractions 11

11 Number and Algebra Real Numbers HOW MUCH DO YOU KNOW? You are nearly at the end of the chapter. Check that you are able to do the following. Find equivalent fractions (.1) Find fractions equivalent to Solution: 6 = = 1 = 0 with denominators 8, 1, 0. Write fractions with the same denominator to enable addition and subtraction (.1) Write the fractions 8, with the same denominator. Solution: Find the LCM of the denominators 8 and = 0 Change fractions to denominator 0: 8 =, = 0 Reduce a fraction to its simplest equivalent form by cancelling (.1) Cancel 18 to its simplest form. 0 Solution: Find the HCF of 18 and 0 = 6. Cancel = = 6 Express improper fractions as mixed numbers and vice versa (.) (a) Change to a mixed number. (b) Change to an improper fraction. Solutions: (a) divides into times with remainder. = (b) is 0. So = 0 + = Locate fractions and mixed numbers on a number line (.) Locate the numbers 1, 1, 1 on a number line. 0 1 Express one quantity as a fraction of another (.) Express 1 kg as a fraction of 8 kg and write the fraction in its simplest terms. Solution: 1 8 = 7 = 7 Note: Both quantities must be in the same units (kg). 118 Chapter Fractions

12 Number and Algebra Real Numbers CHAPTER DIAGNOSTIC TEST 1. What fraction is the green part of the whole? (a) (b) (c).1. Cancel these fraction to their lowest terms. (a) 6 (b) What fraction in its lowest terms is the part of the whole described? (a) Luke birdied 1 of the 18 holes on the golf course. (b) Australia won 16 gold medals from a total of 8 medals. (c) Emma spent $ of her $0 birthday gift on a CD.. Change these mixed numbers to improper fractions. (a) 1 (b) 8. Change these improper fractions to mixed numbers. (a) (b) 1 6. State true or false for each of these. (a) < 9 (b) 1 (c) 0 6 (c) 9 (c) 0 6 > 1 (c) (a) + = 1 (b) = (c) + = (a) + = 8 (b) 7 = (c) = Arrange these fractions in ascending order (increasing from smallest to largest) (a) (a) = 8 (c) + = (a) = (a) of = (b) = 9 (c) = (a) = (a) = (b) = 8 8 (c) = (a) = Calculate the required fraction of these quantities. (a) of $0 (b) of minutes (c) 7 8 of 1 km 17. Calculate the required fraction of these quantities. (a) of $60 (b) of 0 metres (c) 8 of dozen eggs = Chapter Fractions 11

13 Chapter 7 Algebra Syllabus Introduce the concept of variables as a way of representing numbers using letters. (ACMNA17) Create algebraic expressions and evaluate them by substituting a given value for each variable. (ACMNA176) Extend and apply the laws and properties of arithmetic to algebraic terms and expressions. (ACMNA177) KEY SKILLS AND KNOWLEDGE By the end of this chapter you should be able to: Use letters to represent numbers. (7.1) Create and evaluate algebraic expressions. (7.) Apply the laws of arithmetic to simplify algebraic expressions. (7.) Use the four arithmetic operations with algebra. (7.) Recognise the role of grouping symbols and the different meanings of expressions. (7.) Connect algebra with the commutative and associative properties of arithmetic. (7.6) Interpret statements involving algebraic symbols in other contexts. (7.7) Chapter 7 Algebra 179

14 Number and Algebra Patterns and Algebra GETTING STARTED 1. In algebra, if x stands for an unknown number, then one more than x is: (A) y (B) x + 1 (C) x 1 (D) Depends on what x is.. In algebra, one less than x is written: (A) w (B) x 1 (C) 0x (D) 1 x. If x = then x + 1 equals: (A) y (B) (C) 6 (D) 1. If y = then y = 1 equals: (A) (B) y (C) x (D). When a letter stands for a number that could be many different values it is called a: (A) Variable. (B) Changeable. (C) Alterable. (D) Valuable. 6. If x = 7 then x equals: (A) 1 (B) 7 (C) 9 (D) 1x 7. If x = 1 then x + equals: (A) (B) 6 (C) 6x (D) None of these. 8. If x = 8 then y equals: (A) 9 (B) x + 1 (C) x (D) Unknown. 9. If x + 1 = then x must be: (A) (B) (C) 1 (D) One before y.. If I had x bottles of lemonade in the fridge and I drank one, then the number of bottles left in the fridge would be: (A) Still x. (B) x 1 (C) It depends how many you had in the first place. (D) w 180 Chapter 7 Algebra

15 Number and Algebra Patterns and Algebra 7.7 Using algebraic symbols in other contexts Sometimes algebra is used to represent numbers in a different way to what we have seen so far. For example, in a computer spreadsheet calculator there are different symbols for operations and formulas are constructed from their position in the grid. Spreadsheets Arithmetic symbol Spreadsheet symbol Example * * / 0/ ^ ^ In a computer spreadsheet the value of a variable is found by referring to a cell in the spreadsheet. The cell has a name according to the column and row it belongs to. A B C D 1 Length Breadth Perimeter Area 7 = *(A + B) = A*B A refers to the length cell with value. B refers to the breadth cell with value 7. A in this case doesn t mean A times. The result is 8, not AB. Computer programming The instruction The instruction X = Y + means: Fetch the value of Y. () Add to it. (8) Store the result in X. Y = Y + means: Fetch the value of Y. () Add to it. (8) Store this in Y. (Note the original value of Y is lost.) Y X 8 Y 8 This statement would be meaningless in the context of ordinary algebra. It would say that there is some number X when multiplied by (X) and added to (X+) gives the same number as when you multiplied it by (X), i.e. adding the makes no difference. Class property languages Modern computer languages have variables that are subclasses of other variables. For example, TABLEA may have columns X and Y while TABLEB also has columns X and Y. To refer to the different Xs and Ys they are separated from their table name by a dot. This allows the programmer to use the same variables X and Y in different contexts without getting them mixed up. Table A Table B X Y X Y So TABLEA.X() + TABLEB.X() = 9 + = 0 Chapter 7 Algebra

16 Number and Algebra Patterns and Algebra EXERCISE 7.7 Using algebraic symbols in other contexts 1. Calculate the result in column D when these values and formulas are placed in a spreadsheet. Note that the normal order of operations applies, and brackets take precedence with powers being done before multiplication and division while addition and subtraction are last. A B C D 1 First Second Third Result 8 =A*B*C 1 =A*C/B 6 =B^-A*C 9 8 =A/C*B =(A6+C6)*B =(C7+B6-A)*B7. Calculate the final values of x and y given the initial values of x = and y = 6. (a) y = x^ + (b) y = x^ + y^ (c) x = x*y + x (d) x = x*(x + y) (e) y = (*x + *y)/7. Write a formula for the spreadsheet that will calculate the area of a triangle with given base and height. Place the formula in cell C. A B C 1 Base Height Area of triangle 7 Hint 1: All spreadsheet formulas begin with =. Hint : The area of a rectangle is found by multiplying height by width. Hint : Since a triangle may be considered to be half a rectangle, the area of a triangle is found by multiplying the height by the width (base) and halving the result.. Work out the values of these variables from the given table of class variables. Rectangle Triangle Height Width Height Width (a) Rectangle.Height(1)*Rectangle.Width(1) (b) Rectangle.Height()*Rectangle.Width() (c) Triangle.Height()*Triangle.Width()/ (d) Triangle.Height()*Triangle.Width()/ (e) (Rectangle.Height(1)+Rectangle.Width(1))* Chapter 7 Algebra 0

17 Measurement and Geometry Using Units of Measurement 1. Drawing solids from different perspectives As we approach someone s house we see the front view. If we look at the house from the neighbour s view it would be the side view and if we approached the house from the rear it would be the back view. An architect s floor plan of a house could be considered as the top view of the house (i.e. looking at the house from above). We can put all the views together to describe fully what the solid looks like. Example 1: Draw the top, left side, right side and front views of this house. Left side Right side Solution: Left side Right side Example : Draw the top, the two sides and front views of this solid. Right side Left side Solution: view Left view view Right view Chapter 1 Volume

18 Measurement and Geometry Using Units of Measurement Example : Given the various views of a solid, draw a diagram of the solid on isometric paper. Right view Left view view Solution: EXERCISE 1. Drawing solids from different perspectives 1. For each of the solids shown, first build the shape using centicubes and then draw the diagrams from the front, top, right and left views. (a) (b) (c) (d) (e) (f) Chapter 1 Volume

19 Measurement and Geometry Using Units of Measurement. Given these solids draw the front, top, right and left views. (a) (b) Right side Right side (c) (d) Right side Right side (e) (f) Right side Right side. Draw the top and side views for: (a) A cone. (b) A sphere.. Name the solid, given these front, top, and side views. (a) (b) (c) Side Side Side (d) (e) (f) Side Side Side (g) (h) (i) Side Side Side Chapter 1 Volume