Number Sets 1,0,1,2,3,... } 3. Rational Numbers ( Q) 1. Natural Numbers ( N) A number is a rational number if. it can be written as where a and

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1 Number Sets 1. Natural Numbers ( N) N { 0,1,,,... } This set is often referred to as the counting numbers that include zero.. Integers ( Z) Z {...,,, 1,0,1,,,... }. Rational Numbers ( Q) A number is a rational number if a it can be written as where a and b b are integers and b 0. The rational numbers include N, Z fractions and all repeating decimals. This set is similar to N, but it has both positive and negative counting numbers.

2 4. Irrational Numbers ( Q' ) Any number having an infinite non repeating decimal representation is an irrational number The most famous irrational number is π. As a decimal π The common type of irrational numbers involve square roots. If a number a, is not a perfect square root then, a is an irrational number. Number Sets is an irrational number 9 is a rational number 5. Real Numbers ( R) The real numbers include N, Z, Q, Q '.

3 Intervals Any set of real numbers that is represented on the real number line by a segment is called an interval. Real intervals can be described by using inequalities, graphs (number lines), set builder notation or interval notation. Let s look at some eamples. is greater than 5 > { R > 5} 5 Note the open dot words inequality Set Builder Notation graph 5, Interval Notation

4 Intervals is greater than or equal to 4 but less than 7 words 4 < 7 inequality Note the dots graph { R 4 < 7} [ 4,7[ Set Builder Notation Interval Notation

5 Scientific Notation Scientific notation will always look like this: a 10 n where a is a real number greater than or equal to 1 and less than 10; n is an integer (any positive or negative whole number) is not in scientific notation is in scientific notation Note that we moved the decimal places to the left and increased the eponent by on the Note that we moved the decimal 4 places to the right and decreased the eponent by 4 on the 10.

6 Multiplication ( )( ) 5 4 ( )( 10 )( 10 ) 5 4 ( )( 10 ) 1 1 ( )( 10 ) Note that division calculations with scientific notation work the same way as multiplication. The Metric System Length km hm dam m dm cm mm or km hm dam m dm cm mm m 500cm 5dm.5m

7 The Metric System Area Volume km hm dam m dm cm mm km hm dam m dm cm mm or km hm dam m dm cm mm km,000,000m 14cm m or km hm dam m dm cm mm m 18,000dm 50cm m

8 1mL 1cm 5mL 5cm c a Capacity 1L 1dm 0.08L 0.08dm Pythagorean Theorem c a b Recall A Solid Geometry πr Circumference A A π r r radius ( Base)( Height ) ( Length)( Width) b c is always the longest side (directly across from 90 o angle). This formula only works for 90 o triangles.

9 Solid Geometry Prism Sphere h h r Lateral Area Total Surface Area Perimeter of Base Volume ( ABase )( Height ) ( Height ) ( )( ) Lateral Area A Base ( )( π )( ) Total Surface Area 4 r 4 Volume ( π )( r )

10 Solid Geometry Cylinder Cone h h s r ( )( π )( )( ) Lateral Area r h Total Surface Area ( )( ) Lateral Area A Base Volume ( ABase )( Height ) Lateral Area ( π )( r )( ) s Total Surface Lateral Area A Area Base ( ABase )( Height ) Volume s r h

11 Solid Geometry Pyramid Lateral Area ( Perimeter of Base)( s) h s Total Surface Area Lateral Area A Base Volume ( A )( Height ) Base

12 Similar Figures Similar figures (polygons) will always have proportional corresponding sides. 5cm A 10cm B 7cm E 6cm C 1cm 5cm A 10cm B 7cm E 6cm C 1cm 5cm 10cm 6cm 1cm 7cm 14cm D 14cm F D 14cm F

13 Similar Figures Similar figures (polygons) will always have proportional corresponding sides. 5cm D A 10cm B 7cm E 6cm 14cm C 1cm F 5cm 10cm 1 6cm 1cm 1 7cm 14cm The scale factor from DEF to 1 ABC is. Also, the scale factor from ABC to DEF is. Also note that the perimeter of DEF is double the perimeter of ABC. P 5cm 6cm 7cm 18cm ABC P 10cm 1cm 14cm 6cm DEF 1

14 Similar Solids When two solids are similar there is a relationship between the ratio of their sides, areas and volumes. A cm B 6cm If solid A is similar to solid B. Ratio of Sides : A B Ratio of Areas : A B Ratio of Volumes : 6 A B Note that we use the ratio of sides for 1-dimensional measurements, ratio of areas for -dimensional measurements and ratio of volumes for -dimensional measurements

15 Algebra y 5y 9y ( 5) ( 5) 6 15 ( 1)( 5) ( 1)( 5) ( ) ( )( )

16 Algebra ( 4 10 ) ( )

17 Algebra ( )( ) 5 To multiply when the bases are the same, you add the eponents ( ) ( )( ) 6 For a power of a power you multiply the eponents 1 y 5 y 1 y y 1 y To divide when the bases are the same, you subtract the eponents. ( ) 1 ( )( 1) 0 1 1

18 Algebra What is the algebraic epression that represents the perimeter of the triangle shown below. Perimeter Perform the following subtraction ( 5 ) ( 4 ) ( 5 ) ( 4 )

19 The linear function is a straight line. y -1-5 (-1, -5) The Linear Function y m b (0, -) (1, -1) 1 (, 1) (, ) (4, 5) 4 5 The rule y m slope y intercept To find the rule, use two coordinates from the table of values or that are on the line in the Cartesian Plane. (, 1) (4, 5) 1 y 1 y y y m m 4 4 m m b (4, 5) m y y m b y b 5 ( 4) b 5 8 b 5 8 b b y

20 Solving Inequalities These can be solved the same way as an equation. Remember that if you multiply or divide both sides of the equation by a negative, the inequality symbol must be reversed

21 System of Equations The Comparison Method y We use a system of equations to determine the point of intersection of two straight lines. Point of Intersection y The comparison method allows us to find the point of intersection without drawing the graph. y y y y 6 6

22 System of Equations The Comparison Method y Point of Intersection y Let in both equations to make sure you get the same y value. y y y ( ) y y 4 y y 1 y 1 ( ) Point of Intersection (, 1)

23 Rational Function Odds and Probability The rule of the basic rational function is ( )( y ) a or (1, 8) (, 4) y (4, ) 8 y (8, 1) a y Odds of an Event Happening Chances For : Chances Against Odds of an Event Not Happening Chances Against : Chances For Probability of an Event Happening P Chances For ( Chances For ) ( Chances Against ) Probability of an Event Not Happening P Chances Against ( Chances For ) ( Chances Against )

24 Geometric Probability Here we calculate the probability when we are dealing with either number lines or -dimensional geometric figures. We will use eamples to illustrate these probabilities. 1. Given the number line below, what is the probability of a cursor that moves back and forth being in the bold region? Probability of Bold Region Cursor Length of Bold Segments Total Length of the Line 5. Find the probability of a pointer pointing at the bolded arc AB after someone spins the wheel in the diagram below. Pointer Probability of m AOB Bold Arc

25 Geometric Probability. A game involves throwing a dart at the target below and we know that the dart hits the target. What is the probability that the dart lands in the dark region? Probability of Dark Region Total Area of Dark Region Total Area of the Target A A A A

26 The following table shows a break down of the boys and girls in Grade 9 and whether they are left or right handed. Boys Girls Total Left Right Total For eample, the table tells us that there are 10 students in Grade 9. There are 48 boys and 7 girls. There are 1 boys who are left handed. Statistics Samples from a Table If we wanted to take a sample of 40 students, how many right handed boys are there? original # of right handed boys original total new total 6 Solve for ( 6)( 40) ( 10)( ) There should be 1 right-handed boys in a sample of 40 students.

27 Weighted Mean (Average) The most typical type of situation that uses a weighted mean is when a student is interested in calculating their final grade for a term mark. When we use a weighted mean, it is because not all of the data values will have the same value. For eample, let s say that a teacher makes their student s homework mark worth 0%, their quizzes results mark 0% and their test results worth 50%. Ron s results are summarized in the table shown below: Homework (0%) Quiz (0%) Test (50%) Ron Final Mark The formula for the weighted mean in this case is: Weighted Homework Homework Mean Weight % Result Quiz Quiz Weight % Result Test Test Weight % Result ( 0% )( 75) ( 0% )( 80) ( 50% )( 70) Ron s final mark is 74.

28 Calculating an Approimate Mean Given the histogram and its corresponding interval rate, we can calculate an approimation for the mean. frequency Class Frequency [ 4, 8 [ [ 8, 1 [ 4 [ 1, 16 [ classes

29 Calculating an Approimate Mean In order to do this calculation, we need to find the center of each class and add this column to the table of values. Class Frequency [ 4, 8 [ [ 8, 1 [ 4 [ 1, 16 [ Center Net, we need to add a column that multiplies the frequency (f) by the center (c). We will also add up and total the values in the (f)(c) column and the frequency (f) column. Class Frequency [ 4, 8 [ [ 8, 1 [ 4 [ 1, 16 [ Total Center (f)(c)

30 Calculating an Approimate Mean frequency Class Frequency Center (f)(c) [ 4, 8 [ 6 1 [ 8, 1 [ [ 1, 16 [ 14 4 Total 9 94 classes To calculate an approimation for the mean, we will use the sum of the frequency column ( f ) and the sum m of the (f)(c) column ( ( f )( c) ). ( f )( c) f ( ) Formula for the approimate mean.

31 Bo and Whisker - Plots A bo and whisker plot looks like this: Min Q 1 Q Q Ma The number line must have a constant scale Min : Minimum Value 10 Q 1 : 1st Quartile 0 Median of the First half Q : nd Quartile (the Median) 50 Q : rd Quartile 60 Median of the Second half Ma : Maimum Value 90 Range Ma Min Interquartile Range Q Q

32 Bo and Whisker - Plots 5% 5% 5% 5% Min Q 1 Q Q Ma st Quarter : Min Q 1 5% of the data nd Quarter : Q 1 Q 5% of the data rd Quarter : Q Q 5% of the data 4th Quarter : Q Ma 5% of the data

33 1. Min 5. Q Bo and Whisker - Plots Let s create a bo and whisker plot from the set of data values given below Note the order in which we identify the five markers 4. Ma 0. Q Q Min Q 1 Q Q Ma 4 1 Min Q 1 Q Q Ma Use a scale of