Download High School Mathematics Preparation Booklet

Table of Contents 1 Numeracy Skills & Arithmetic ................................................................................................................ 3 1-­‐1 Place Value: .......................................................................................................................................................................................................... 3 1-­‐2 Reason for Rounding:....................................................................................................................................................................................... 3 1-­‐3 How to Round: .................................................................................................................................................................................................... 3 1-­‐4 Estimating by Rounding: ................................................................................................................................................................................ 4 1-­‐5 Arithmetic Facts: ................................................................................................................................................................................................ 5 2 Working with Integers ............................................................................................................................ 6 2-­‐1 Adding Integers: ................................................................................................................................................................................................. 6 2-­‐2 Subtracting Integers ......................................................................................................................................................................................... 7 2-­‐3 Multiplying and Dividing Integers.............................................................................................................................................................. 7 2-­‐4 Evaluating Powers with Integer Bases..................................................................................................................................................... 8 2-­‐5 Order of Operations for Evaluating Expressions ................................................................................................................................. 9 3 Working with Fractions ........................................................................................................................ 12 3-­‐1 Equivalent Fractions ......................................................................................................................................................................................12 3-­‐2 Lowest Common Denominator (LCD) ....................................................................................................................................................13 3-­‐3 Comparing Fractions......................................................................................................................................................................................13 3-­‐4 Ordering From Lowest to Greatest ..........................................................................................................................................................14 3-­‐5 Adding and Subtracting Fractions............................................................................................................................................................14 3-­‐6 Multiplying and Dividing Fractions .........................................................................................................................................................15 3-­‐7 Powers with Fractional Bases ....................................................................................................................................................................16 3-­‐8 Order of Operations with Fractions.........................................................................................................................................................17 4. Percentages, Ratios & Rates ................................................................................................................ 19 4-­‐1 Common Fraction-­‐Decimal Equivalents ................................................................................................................................................19 4-­‐2 Converting Percentages into Fractions in their Lowest Terms ...................................................................................................19 4-­‐3 Converting Decimals into Percentages...................................................................................................................................................20 4-­‐4 Converting Fractions into Percentages ..................................................................................................................................................20 4-­‐5 Calculating Percentage of a Quantity ......................................................................................................................................................20 4-­‐6 Increasing or Decreasing Quantities by a Given Percentage ........................................................................................................21 4-­‐7 Solving Problems with Percentages ........................................................................................................................................................22 4-­‐8 Determining and using unit rates .............................................................................................................................................................22 4-­‐9 Working with ratios........................................................................................................................................................................................22 5 Word Problems...................................................................................................................................... 23 APPENDICES .............................................................................................................................................. 25 A1 Glossary.................................................................................................................................................................................................................25 A2 Answers.................................................................................................................................................................................................................26 A3 Sources of Resources for Extra Practice .................................................................................................................................................28 A4 References............................................................................................................................................................................................................28 A5 Credits....................................................................................................................................................................................................................28 PREFACE The teachers in the mathematics department at AB Lucas in 2011 created this booklet in response to current students’ difficulty mastering high school concepts due to gaps in numeracy understanding and arithmetic mastery. It is imperative that students come out of elementary school with a solid grasp on the relationship between numbers and operations, and the with a mastery of arithmetic facts. Throughout this book you will be guided by examples demonstrating the skills that are necessary to succeed at the high school level. These skills should be practiced without the use of a calculator. Students must master units 1-­3 and have good working knowledge of units 4-­6. At the start of each chapter there is a list of what students should know (this refers to the skills which students should be able to do on the spot without think time), and a list of what students should be able to do (this refers to the skills students should be able to work out) 2 © Jen Brown et al. AB Lucas Mathematics Department 2011 1 Numeracy Skills & Arithmetic 1-­‐1 Place Value Students should know: • The names of all the place values • Basic arithmetic facts: o Multiplication tables to 12 o Division facts up to 12 • Round to different place values • Perfect Squares Students should be able to: • Estimate calculations using rounding • Use mental mathematic strategies to quickly calculate sums, differences, products and quotients The current notation for recording numbers is to leave a space between each set of three digits. This helps in quickly identifying place value. Below is a visual of the common place values. 1-­‐2 Reason for Rounding We use rounding to express numbers as approximate simpler values. Often rounding is used to indicate the accuracy of a calculated value. Rounding can also be used to estimate an answer to a more complex problem. 1-­‐3 How to Round When rounding a value we are asking if a number is closer to the next higher value or to the value just before? Look at the number line below and consider rounding to the nearest tens. The number 48 is closer to 50 than it is to 40, the number 63 is closer to 60 than 70, therefore the number 48 rounded to the nearest tens is 50 and the number 63 rounded to the nearest tens is 60. We can see this relationship without drawing a number line by looking at the next smallest place value (the digit to the right). If it is less than 5 we “round down”, if it is five or greater we “round up”. Ex 1.) Round to place value specified: a) tens: 4 is in t he tens place so we look at therefore we “round up”. the 8, this is greater than 5 b) thousands: means nearly equal to (use this symbol whenever you round) 6 is in the thousands place so we look at the 2, this is less than 5 therefore we “round down”. © Jen Brown et al. AB Lucas Mathematics Department 2011 3 Ex 2.) Round to the number of decimal places specified: a) 2 decimal places: b) 1 decimal place: Notice in these examples the number of decimal places refer to how many digits after the decimal we are going to keep. 1-­3 Practice: 1. Round to the place value specified: a) tenths: 56.89 b) ten thousands: 356 461 c) units: 56.47 d) hundreds: 4 624 e) hundredths: 0.842 37 f) tens: 497.7 2. Round to number of decimal places specified in parenthesis: a) (2) 56.389 b) (1) 658.36 c) (2) 0.983 d) (3) 0.9847 e) (1) 57.335 f) (2) 43.582 7 1-­‐4 Estimating by Rounding When numbers are too complex to perform simple operations mentally we can use rounding to make the values easier. This will give us an estimation of the answer. 4.76 ! 12.3 " 5 ! 12
Ex. 1.) If we round 4.76 & 12.3 to the nearest unit we are left with a " 60
computation that we know mentally. The actual value is 58.548. Ex. 2.) 786 ! 354 " 800 ! 400
" 320 000
Here we round to the nearest hundreds. The more digits you keep the more accurate your answer will be. The actual value is 278 244. 1-­4 Practice: 1. Estimate the following by rounding to the place value specified: b) 457 + 984 (tens) c) 34.6 ! 16.8 (tens) d) 1354 ! 475 (hundreds) e) 6.54 ! 3.86 (unit) f.) 32.4 ÷ 7.88 (unit)
a) 3.6 ! 8.6 (unit) 4 © Jen Brown et al. AB Lucas Mathematics Department 2011 1-­‐5 Arithmetic Facts In order to be successful in mathematics student must master the arithmetic facts. These facts include addition, subtraction, multiplication facts up to 12, and division facts up to 12. Here is a mixture of arithmetic facts to practice. © Jen Brown et al. AB Lucas Mathematics Department 2011 5 2 Working with Integers 2-­‐1 Adding Integers Students should know: • The relationship between positive and negative numbers • The order of operations • That a power is repetitive multiplication Student should be able to: • Add integers • Subtract integers by adding the opposite • Multiply & Divide integers • Evaluate powers with inter bases • Evaluate expressions using the correct order of operations A number line can be used to add integers. Ex 1. Find each sum. a) 3 + (–5) = –2 b) –4 + (–6) = –10 • on the number line, place a dot at the first integer. • draw an arrow in the direction and with the magnitude of the second integer (Note: right is positive, left is negative) • the sum is the integer where the arrow stops 2-­1 Practice: 6 1. Use a number line to model each sum. a) 5 + (–8) b) –7 + 7 2. Find each sum a) 2 + (–7) b) –5 + (–2) c) 5 + (–5) d) –12 + (–8) 3. Find each sum a) –2 + (–8) + 4 b) 1 + (–3) + 2 c) 2 + (–6) + (–5) d) –12 +10 + (–6) 4. The temperature in London starts at –6oC, rises 8 oC, then falls 12 oC. What is the final temperature? 5. In a magic square, each row, column, and diagonal has the same sum. Find the integers that complete this magic square. -­‐2 -­‐6 4 10 © Jen Brown et al. AB Lucas Mathematics Department 2011 2-­‐2 Subtracting Integers The Zero Principle: opposite integers add to zero. For example, –5 + 5 = 0. Subtracting an integer is identical to adding the opposite. Ex 1. Subtract. a) –3 – (–5) = –3 + 5 = 2 b) –10 – (–6) = –10 + 6 = –4 c) -­‐2 – 5 = -­‐7 2-­2 Practice: 1. Subtract. a) 5 – 8 b) –4 – (–5) c) –7 – 7 d) –2 – (–1) e) 3 – (–5) 2. Fill in each with the correct integer a) 2 – = –1 b) – (–3) = 5 c) –2 – = –6 d) – 5 = -­‐8 3. You have a positive balance of $120 in your account. You use your debit card to buy an Xbox for $310. How much do you have left in your account (assume you have overdraft protection which allows your account to go into the negatives)? 4. The temperature is 2 oC. With wind chill it feels like –10 oC. How many degrees does the temperature change because of the wind chill? 2-­‐3 Multiplying and Dividing Integers The product or quotient of two integers with the same sign gives a positive answer. For two integers with the opposite sign the answer is negative. If there are more than two integer values the product or quotient is positive if there is an even number negative values, and negative if there are an odd number of negative values. Ex 1. Multiply. a) –2 X 5 b) –10 X (–6) c) (–3)(–4) = –10 = 60 = 12 © Jen Brown et al. AB Lucas Mathematics Department 2011 7 Ex 2. Divide. a) –10 ÷ 5 b) –24 ÷ (–6) c) = –2 = 4 = –3 2-­3 Practice: 1. Find the product. a) 2 X 15 b) –2 X (–4) c) (–3)(–4) d) 3(–6) e) –4 X 7 2. Find each quotient. a) 12÷ (–6) b) –18 ÷ (–3) c)
d)
3. Multiply. a) –2 X (–4) X 3 b) 5 X (–6) X (–2) c) –2 X (–8) X (–2) d) –7 X 2 X (–3) 4. Write a multiplication statement and a division statement that would give result shown. a) –14 b) –60 5. Determine the pattern of the sequences below, then determine the next two terms a) 2, 8, 32,… b) -­‐324, 108, -­‐36,… 2-­‐4 Evaluating Powers with Integer Bases A power is a product of identical factors and consists of two parts: a base which is raised to an exponent. The expression below is a power: exponent 6 base 5 The exponent tells you how many times the base will be multiplied by itself. Examples base exponent power expanded product 2
2 2 2 2 x 2 4 3 4 34 3 x 3 x 3 x 3 81 4 5 45 4 x 4 x 4 x 4 x 4 1,024 7
5 7 5 5x5x5x5x5x5x5x5 78,125 Ex 1. Write in expanded form, then evaluate: a) 25 = 2 x 2 x 2 x 2 x 2 = 32 There is an even number of negative factors, so the b) (-­‐3)3 answer is positive = (-­‐3) x (-­‐3) x (-­‐3) = -­‐27 8 © Jen Brown et al. AB Lucas Mathematics Department 2011 c) (-­‐3)4 = (-­‐3) x (-­‐3) x (-­‐3) x (-­‐3) = 81 d) -­‐34 = -­‐(3) x (3) x (3) x (3) = -­‐81 e) 3.53 = 3.5 x 3.5 x 3.5 = 42.875 The base of this power is 3, not -­‐3. The negative in front makes the result negative. 2-­4 Practice: 1. Which is 6 x 6 x 6 x 6 written as a power? a) 64 b) 64 c) 46 d) 1296 2. Which is 35 written in expanded form? a) 3 x 5 b) 5 x 5 x 5 c) 3x3x3x3x3 d) 243 3. Write each expression as a power. a) (-­‐5) x (-­‐5) x(-­‐5) b) 1.05 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05 4. Write each power in expanded form. Then, evaluate the expression. a) (-­‐4)3 b) 0.82 5. Evaluate. a) 93 b) (-­‐7)2 c) -­‐24 d) 1.22 e) 18 f) (-­‐1)55 g) 0.53 2-­‐5 Order of Operations for Evaluating Expressions In order to evaluate an expression, students follow the order of operations: 1. Simplify inside the parenthesis or brackets. 2. Simplify powers. 3. Multiply and divide. (In the order which they occur) 4. Then add and subtract. (In the order which they occur) Students can remember these steps by one of the acronyms below: © Jen Brown et al. AB Lucas Mathematics Department 2011 9 Note: It is important that students have good communication and show all their steps in multi-­‐step questions. Each line must be equivalent to the line above. Ex 1: Evaluate a) !2 + ( !6 ) ( 4 )
b) ( 3 + 2 ) ! 5 c) 4 + 2 ! 6 ÷ 3
d) ( !5 ) + 4 " ( !2 )
= 4 + 3÷ 3
= !2 + ( !24 )
= ( 5 ) ! 5
= ( !5 ) + ( !8 )
= 4 +1
= !2 ! 24
= 25
= !5 ! 8
=
5
= !26
= !13
e) !3( 2 ! 4 ) ! ( !2 + 4 )
f) ( !2 ) 2 + 3 ! 2 " 4 ! ( !3
) (1) $ g) !"18 + ( 3) ( 6 ) ÷ 2 #$ ÷ 3 + 8 ( 4 % 1)
#
%
2
= !3( !2 ) ! ( 2 )
= [18 + 18 ÷ 2 ] ÷ 3 + 8 ( 3)
= ( !2 ) + 3 ! 2 "# 4 ! ( !3) $%
2
= [18 + 9 ] ÷ 3 + 24
= 6 ! 2
= ( !2 ) + 3 ! 2 [ 4 + 3]
= 4
= 27 ÷ 3 + 24
2
= ( !2 ) + 3 ! 2 [ 7 ]
= 9 + 24
= 4 + 3 ! 2 ( 7)
= 33
= 4 + 3 ! 14
= 7 ! 14
= !7
h) !5 + ( !3) ( !6
)
i) Add (-­‐5)2 to the product of -­‐3 and 2. ( !2 )2 + ( !3 )2
( !3) ( 2 ) + ( !5 )2
= ( !3) ( 2 ) + 25
= !5 + ( !3) ( !6 )
= !6 + 25
4+9
= 19
!5 + 18
=
4+9
13
=
13
Simplify the numerator and denominator completely = 1
before dividing Notice that each line is equivalent to the line above it. The expression is rewritten, not just the portions that are being evaluated. Incorrect Form: e.) ! 3( 2 ! 4 ) ! ( !2 + 4 )
( 2 ! 4 ) = !2
!3 " ( !2 ) = 6
6!2= 4
10 ( !2 + 4 ) = 2
While this will arrive at the correct answer if no errors are made (which are difficult to avoid in this form), the mathematical justification is not valid. © Jen Brown et al. AB Lucas Mathematics Department 2011 2-­5 Practice: 1. Simplify. a) 2. Simplify. a) e) 3. Evaluate a) 4. Evaluate a) b) b) b) c) b) 2
c) d) d) e) !4 "#( !4 + 1) ÷ 3$% ! ( !3
)
g) h) 2
c) c) d) 2
f) "# 2 ( !3 + 1) + 1$% ! 3 ( 5 ! 8 ÷ 2 )
2
5. Given the following sentences i) Write an expression to represent the sentence. ii) Evaluate. a) Increase the sum of -­‐3 and 5 by -­‐6. b) -­‐8 is added to the product of -­‐3 and -­‐2 c) Divide the sum of 7 and -­‐16 by -­‐3. d) By how much is the sum of -­‐8 and 6 more than -­‐4? e) Increase the product of -­‐7 and -­‐3 by -­‐18. f) Decrease the quotient of -­‐8 and -­‐2 by -­‐3. g) By how much is more than ? © Jen Brown et al. AB Lucas Mathematics Department 2011 11 3 Working with Fractions Students should know: • Key terms: numerator, denominator, lowest terms, improper fraction, mixed number, reciprocal • How to convert between improper fractions & mixed numbers Student should be able to: • Convert between equivalent fractions • Add & Subtract fractions using common denominators • Multiply & Divide fractions by reducing to make numbers easier to work with • Evaluating Expressions using proper order of operations with fraction 3-­‐1 Equivalent Fractions We can see that one-­‐half or of the square is shaded. If we divide the square into 4 smaller squares, we can see that the fraction of the shaded square is The fractions and are called equivalent because they represent the same shaded region. Likewise, the shaded region on the right can be represented by three different but equivalent fractions: , , and 3-­1 Practice: 1. Write four equivalent fractions to represent the shaded region: a) b) 2. Write three equivalent fractions to represent the fraction of the pizza already eaten. 3. Write three equivalent fractions for each: a) 12 b) c) © Jen Brown et al. AB Lucas Mathematics Department 2011 3-­‐2 Lowest Common Denominator (LCD) When you find the lowest common denominator, you are finding the smallest number that is divisible by each of the denominators. One way to do this is to find the least common multiple. Ex. Find the lowest common denominator for and . Look at the list of multiples. You can see that 12 is the smallest number that appears in each list. List the multiples of 3 and 4 (the denominators) Multiples of 3: 3, 6, 9, 12, 15, 18,… Multiples of 4: 4, 8, 12, 16, 20, 24, … Therefore, 12 is the lowest common denominator. 3-­2 Practice: 1. Use multiples to find the LCD for each pair or set of fractions. a) b) c) d) e) f) g) h) i) 3-­‐3 Comparing Fractions Which fraction is bigger: or ? One way to compare fractions is by getting common denominators and finding equivalent fractions 2 8
3 9
=
3 12
& 4 = 12
Since 9 is larger than 8: Therefore 3 2
>
4 3
3-­3 Practice: 1. For each pair of fractions, state which one is bigger by finding a common denominator. Express your answer using a greater than or less than sign between the original two fractions. a) b) c) © Jen Brown et al. AB Lucas Mathematics Department 2011 13 3-­‐4 Ordering From Lowest to Greatest Ex. Place the fractions in order from smallest to largest by finding a common denominator. 1 4
1 6
1 3
1 2
=
=
=
=
a) 3 12
& 2 12
& 4 12
& 6 12
< < < < < < Find the LCD & Convert to equivalent fractions Order them from Lowest to Greatest using the numerators. Use these to reorder the original fractions. 3-­4 Practice: 1. Place the fraction in order from smallest to largest a) b) c) 3-­‐5 Adding and Subtracting Fractions To add or subtract fractions: 1)
2)
3)
4)
Fractions can be added or subtracted only if they have the same denominator Convert any mixed fractions to improper fractions. Find the lowest common denominator (LCD). Add or subtract the numerators only. The denominator stays the same. Write in lowest terms. Ex 1. Add or subtract. a) 2 + 1
5 5
3 =
5
b) 3 1
+
4 2
3 2
= +
4 4
5
=
4
c) 3 1
!
8 6
9
4
=
!
24 24
5
=
24
e.) 3 2 ! 2 1
5
4
17 9
=
!
5 4
68 24
=
!
5 20
23
=
20
d) 1 1
+
2 6
6
2
=
+
12 12
8
=
12
2
=
3
14 Express your answers as an improper fraction in lowest terms . Convert to improper fraction before performing any operation. © Jen Brown et al. AB Lucas Mathematics Department 2011 3-­5 Practice: Find each sum or difference. Express answers in lowest terms. 4
9
8
9 b) 3 7
+
8 8
c) 7 2
!
15 15
d) 9 7
!
4 4
b) 3 4
+
10 3
c) 3 1
!
5 3
d) 11 7
!
9 6
13
10
b) 2 4
+ 6 5
c) 1 1
! 2 6
d) 4 ! 2
1. a) +
1
1
2. a) +
2 8
1
2
3. a) +
9
3
3-­‐6 Multiplying and Dividing Fractions •
•
•
To multiply fractions, reduce the numerators and denominator by any common factor. Multiply the numerators and multiply the denominators. Any mixed fractions should be first converted to improper fractions. Ex 1. Multiply 2
1
a) b) 2 1
1 !1
8
3
3
4
!
9
4
5 5
3
2
= !
3 4
2 1
= !
25
3 2
=
12
2
=
3
Ex 2. Divide a) 2 4
÷
5 9
1
2
9
= !
5
4
2
1 9
!
5 2
9
=
10
=
We can only reduce when the expression is written as a multiplication. b) 1 6
÷
2 7
7 6
= ÷
2 7
7 7
= !
2 6
49
=
12
3
Ex 3. Divide ! 2$
#" &%
9
! 5$
#" &%
9
Notice that when we divide two fractions with the same denominator the answer is the numerator of the top divided by the numerator of the bottom. 2 9
'
9 5
2
=
5
=
© Jen Brown et al. AB Lucas Mathematics Department 2011 15 3-­6 Practice: 1. Multiply: a) 2. Multiply: a) 3. Multiply: a) 4. Divide: a) 5. Divide: a) 6. Divide: a) b) c) b) c) b) c) d) b) c) d) b) c) d) b) c) d) b) of 28 c) of 15 d) of 20 7. Calculate: a) of 10 3-­‐7 Powers with Fractional Bases Ex. Evaluate a) ! 3 $
#" &%
4
2
! 3$ ! 3$
=# &# &
" 4% " 4%
=
! 2$
b) #" 3 &%
Notice a quicker way than writing repetitive multiplication is to apply the exponent to the numerator and the denominator 3
23
33
8
=
27
=
9
16
3-­7 Practice: 1. Evaluate a) ! 1 $ 2 #" &%
3
16 b) ! 3 $ 3
#" &%
2
c) ! 2 $ 2 #" &%
5
d) ! 1 $ 4
#" &%
2
© Jen Brown et al. AB Lucas Mathematics Department 2011 3-­‐8 Order of Operations with Fractions When evaluating an expression with fractions you must still follow the order of operations as defined by BEDMAS. B Brackets first! E Exponents You do not need common denominators! M Multiply D Divide A Add You must have common denominators! S Subtract Exponent before multiplication, Multiplication and division are first in both the numerator and the order that they appear. Ex 1. Evaluate 2
denominator were squared 5 " 3%
a) b) ! $ '
7 5 3 1
#
&
7
2
+ ÷ !
Division is multiplication of the reciprocal 3 2 2 2
(notice only the fraction immediately 5 9 = !
after the division is flipped) 7
5
2
1
7 4 denominators are Common = + ! !
not needed to multiply. 3 2
3 2
45
Cancellations should always be made ASAP. =
This o
ccurs i
n m
ultiplication i
f t
here i
s a
7 5 1
28
common factor in the numerator and = + !
denominator. This will make the numbers 3 3 2
more manageable. = 7 + 5
3 6
Addition & Subtraction require 14 5
common denominators =
+
6 6
19
Convert all mixed numbers to improper =
fractions first. Negative signs always go 6
with the numerators 1
3
1
!1 ! ÷ 2
4 4
3
c) d) !5 3 7
=
! ÷
4 4 3
!5 3 3
=
! "
4 4 7
!5 9
=
!
4 28
!35 9
=
!
28 28
!24
=
28
!6
=
7
© Jen Brown et al. AB Lucas Mathematics Department 2011 17 Ex 2. Evaluate Compare the two solutions: a) Both solutions come to the same final answer. 2 4 8 6
! ÷
"
5 7 15 5
= 2 ! 4 " 15 " 6
Solution A has not done any cancelling or reduction of fractions to lowest 5 7 8 5
terms until the very end of the 2 60 6
question. This solution requires performing calculations with large = 5 ! 56 " 5
numbers. 2 360
= !
5 280
= 112 ! 360
Solution B has performed all possible cancellations before performing the 280 280
multiplication. This solution only !248
requires simple arithmetic facts. = 280
!31
=
35
2 4 8 6
"
b) ! ÷
5 7 15 5
1
3
2 4 15 6
= ! "
"
5 7
8
5
2
1
3
=
2 1 3 6
! " "
5 7 2 1
1
2 1 3 3
! " "
5 7 1 1
2 9
= !
5 7
14 45
=
!
25 25
!31
=
25
=
3-­8 Practice: 1. Evaluate. a) b) c) d) 2. Evaluate. a) b) c) 3. Evaluate. a) b) c) d) 4. Evaluate. a) b) c) d) f) g) h) e) 18 © Jen Brown et al. AB Lucas Mathematics Department 2011 4. Percentages, Ratios & Rates Students should know: • Percent is a fraction of 100 • Common percent, fraction, and decimal equivalents Student should be able to: • Convert between fractions, percent and decimals • Calculate percentages of a quantity without a calculator using multiplication and reductions of fractions • Increase or decrease a quantity by a percentage • Solve problems with percentages • Solve equivalent ratios • Calculate unit rates • Use proportional reasoning to solve problems 4-­‐1 Common Fraction-­‐Decimal Equivalents Ex 1. Convert to a decimal a) 1 = 0.5
2
b) 2 = 0.66
3
above digits The line after the decimal mean those d igits repeat forever. Ex 2. Convert to a fraction a) 1 0.25 =
4
b) 0.7 =
7
10
4-­1 Practice: 1. Convert to a decimal a) 1 3
b) 3 4
c) 1
4
d) 8 10
e) 2
5
2. Convert to a fraction in lowest terms a) 0.5 b) 0.6 c) 0.1 d) 0.05 e) 0.33 4-­‐2 Converting Percentages into Fractions in their Lowest Terms Ex. Write as a fraction of 100 then convert to lowest terms 12.5
30
=
1.) 30%
= 2.) 12.5%
100
100
125
Divide numerator and 3 =
denominator by 125. =
1000
10
1
=
8
4-­2 Practice: Convert to a fraction in lowest terms a) 45% b) 80% c) 35% d) 96% e) 48% f) 55% g) 12% h) 160% i) 84% j) 5% k)16% l) 24% m) 38% n) 95% o) 150% p) 67.5% © Jen Brown et al. AB Lucas Mathematics Department 2011 19 4-­‐3 Converting Decimals into Percentages Ex. Convert to a percentage (3 different methods are demonstrated) A. Fraction approach B. Multiply by 100 6
10 60
=
100
= 60%
0.02 = 0.02 ! 100
= 2%
0.6 =
C. Recognition 0.26 = 26%
4-­3 Practice: Convert the decimal to a percent a) 0.5 b) 0.6 c) 0.25 d) 0.55 e) 0.02 f) 2.35 g) 0.48 h) 1.93 i) 0.005 j) 0.16 k) 0.075 l) 94.36 4-­‐4 Converting Fractions into Percentages Ex. Convert to a percent (3 different methods are demonstrated) A. Equivalent fraction B. Recognition 30
6
=
20 100
= 30%
3 = 3%
100
C. Division 3
= 0.25
12
= 25%
0.25
12 3.00
4-­4 Practice: a) 1 5
Convert the fraction to a percent. b) 2 10
3
8
d) 9 h) 3 i) 3 4
5
c) f) 4 g) 8 9
12
20
e) 4
50
j) 7
21
4-­‐5 Calculating Percentage of a Quantity Look for common factors between the numerators and denominators to make the calculation simpler. Ex. 1 Evaluate 3
12
300
12%
of 300 = ! 1
100
a) Common factor of 100. 20 1
3
13
15
260
15% of 260 =
!
1
100
b) 20
1
Common factors of 5, 12 3
3 13
and 20. =
!
= !
1 1
1 1
= 39 Department 2011 = 36
© Jen Brown et al. AB Lucas Mathematics 4-­5 Practice: Calculate the following a) 10% of $20 b) 40% of $450 c) 45% of 300cm d) 75% of 900 nails e) 36% of 12litres f) 35%of 500g g) 96% of 6kg h) 225% of 2km 4-­‐6 Increasing or Decreasing Quantities by a Given Percentage Ex 1. Increase 350 by 20% Method 1: Find full percent of the value Method 2: Find percent of the value then add to original value 100% + 20% = 120%
12 0
! 35 0
10 0
= 12 ! 35
= 420
20
! 35 0
10 0
= 2 ! 35
20% of 350 =
120% of 350 =
= 70
350 + 70 = 420
Method 2: Find the percent of the value then subtract from the original Ex 2. Decrease 390 by 35% Method 1: Find full percent of the value 100% ! 35% = 65%
7
35
39 0
35% of 390 =
!
1
10 0
13
65
39 0
65% of 390 =
"
1
10 0
2
7 39
!
2 1
273
=
2
= 136.5
=
2
13 39
"
2
1
507
=
2
= 253.5
=
390 " 136.5 = 253.5
4-­6 Practice: 1. Increase the following by the given percent: a) $20 by 10% b) $450 by 40% c) 300cm by 45% d) 200 by 75% 2. Decrease the following by the given percent: a) $20 by 10% b) $450 by 40% c) 400cm by 45% d) 900 by 75% © Jen Brown et al. AB Lucas Mathematics Department 2011 21 4-­‐7 Solving Problems with Percentages 4-­7 Practice: 1. A TV costs $420 but is marked with a sale tag saying 30% off. How much will the TV cost in the sale? 2. A retailer bought 52 CD’s at $8.50 each. He breaks two CD’s and sold the rest at a 30% profit. Calculate his total profit. 3. In Lucas Secondary school 328 students are in grade 9, 72% of the students are not in grade 9. Determine the population of the school. 4-­‐8 Determining and using unit rates Ex. A car travelled 348 km in 4 h. Write a unit rate that describes how fast the car was travelling 348km 87km
=
4h
1h The car was travelling 87 km/h 4-­8 Practice: 1. Calculate each unit rate a) a bus travelled 50km in 2hrs b) a person swam 40m in 5 seconds 2. Calculate each unit cost a) five pencils cost $2 b) a 420g box of cereal costs $3.50 3. Use unit costs to determine which brand of butter is a better buy. Brand A: 325g for $2.75 Brand B: 454g for $3.80 4-­‐9 Working with ratios What is a ratio? A ratio is a comparison of quantities measured in the same units. A ratio can be written in ratio form as 3:6 or in a fraction form as 3
6
Similar to fractions, ratios can be written with terms that have no common factors, or in simplest form. A proportion is a statement that two ratios are equal. Ex. 3 : 6 = 1 : 2 or 3 1
=
6 2
4-­9 Practice: Write each ratio in simplest form a) 6:12 b) 15:5 c) 16:40 d) 100:30 e) 24:9 f) 150:10 g) 33:162 h) 80:256 22 © Jen Brown et al. AB Lucas Mathematics Department 2011 Students should know: • Key terms: total, difference, percent, sale, discount, proportions, etc. Student should be able to: • Interpret the meaning of the values in a word problem and their relationship to each other 5 Word Problems Strategies: Highlight keywords and values Underline what the question is asking for Think of the relationship not just the numbers Create “Let statements” using variables to represent unknown values Always think about whether or not your answer makes sense Always finish with a full statement regarding what you have calculated When finished reread the question to ensure it is answered fully Ex 1. Jeremiah has raised $450. He has decided to give an equal amount to 3 charities. How much will each charity get? Let a represent the amount ($) each charity will receive 450
3
= 150 RELATIONSHIP: a=
We are splitting $450 into 3 equal groups. Splitting into groups means division. Therefore each charity will receive $150. Ex 2. JLC has a maximum capacity of 9 000 seats for a concert. A ticket to a general admission concert is $30. The night of the event there are still 1 200 tickets not sold. What is the total amount of revenue made from the concert? Let r represent the total revenue ($) earned from the concert. Let t represent the number of tickets sold r = ( 30 ) ( t )
t = 9000 ! 1200
= 7800
RELATIONSHIP: = ( 30 ) ( 7800 )
Revenue is total cost, we have a cost per ticket, so we need to use multiplication = 234000
Revenue = cost per ticket x # of tickets Therefore the concert earned a revenue of $234 000. Ex 3. Lisa Lilly was the best runner in the eighth grade. One day she ran 100m in 40 seconds, 200m in 1 minute and 10 seconds, and 200m over low hurdles in one and a half minutes. How many more seconds did it take her to run the 200m over low hurdles then it did to run the 200m dash? Convert to the same units first = 1min
30 sec t
t 200m hurdles
difference = t 200m hurdles ! t 200m dash
= 90 sec
= 90 ! 70
t 200m dash = 1min10 sec
= 20
= 70 sec
Don’t get distracted by extra information. Always read the question fully before highlighting so you know what you need. Therefore the 200 m dash took her 20 seconds less than the 200 m hurdles. © Jen Brown et al. AB Lucas Mathematics Department 2011 23 8-­1 Practice: 1.) Silver's Cleaners decided to raise the price of dry cleaning a sports coat from $4.00 to $5.00. The same percentage increase was applied to dry cleaning a jacket. The old cost of dry cleaning a jacket was $10.00. What is the new cost of dry cleaning a jacket? 2.) A Christmas gift is tied with ribbon as shown. The bow requires 47cm of ribbon. What is the total length of the ribbon in metres? 3.) Matthew is at a zoo. He takes a picture of a one-­‐
metre snake beside a brick wall. When he developed his pictures, the one-­‐metre snake is 2cm long and the wall is 4.5cm high. What was the actual height of the brick wall in cm. 4.) Julie and Murray work at different coffee shops. Julie works at Starnite and earns $35 for 4 h of work. Murray works at Brew On and earns $49.50 for 6 h of work. Which coffee shop offers better pay? 5.) On Monday the price of a companies stock is $25 per share. On Tuesday the price drops $2, on Wednesday it rises $6, on Thursday it rises $3, and on Friday it drops $4. What was the price at the end of the week? 6.) During one week, it rained for 2.5 h on Monday, 1 3/4 h on Tuesday, and 2 5/6 h on Wednesday. a) Find the total period of rainfall for this week. b) How much longer did it rain on Wednesday than on Tuesday 7.) Kyle had four bags of candy that he had bought for $1.50 per bag. Each bag has six pieces of candy in it. How many more bags does he need to buy to give each of his twenty-­‐five classmates one piece? How much will it cost altogether. 8.) Jasmine had $4.00. She bought four lollipops that sold at two for 18 cents, two candy bars for 65 cents each, and a notebook for $1.46. How much money did she spend? How much money did she get back? 9.) Fifteen students were in a classroom as well as three teachers. Each student and teacher had four pens. How many pens are in the classroom? 10.) The distance to the next town is 90 km. The car that will be used gets 10km/L. How many litres will the car need to reach the next town? 11.) Three students have to write a make up test. Mark scored 24/60 on his first test and 32/40 on his make up test. Jake scored 35/70 on his first test and 54/60 on his makeup test. Marilyn scored 27/90 on her first test and 45/50 on her second test. A) Which student improved the most and by what percentage? B) If the teacher gives each student a final grade using 70% of the makeup test mark and 30% of the first test mark, what mark would each student receive? Who did the best overall? 24 © Jen Brown et al. AB Lucas Mathematics Department 2011 APPENDICES A-­‐1 Glossary Base: The number that is being repetitively multiplied in a power 3
Ex. The 4 in 4 Denominator: The value below the divisor (fraction bar) in a rational number. 2
Ex: the 3 in 3
Difference: The answer to a subtraction question Divisor: A number by which another number is to be divided Exponent: The number indicating how many times a base is multiplied by itself in a power 3
Ex. The 3 in 4 Factor: A number or quantity that when multiplied with another produces a given number or expression Ex: the number 24 has factors 1, 2, 3, 4, 6, 8, 12, & 24 Greatest Common Factor (GCF): the highest whole number that divides into 2 or more numbers without a remainder. Ex. The GCF for 12, 18, & 24 is 6. Integer: A number which is either positive or negative or zero and has only zeroes after the decimal place. {…-­‐3,-­‐2,-­‐1,0,1,2,3…}. They can be denoted with an I or a Z. Improper Fraction: A fraction that has a numerator that is greater than the denominator 4
Ex. 3
Lowest Common Multiple (LCM): The smallest whole number that is a multiple of 2 or more numbers Ex. The LCM for 3. 4, & 6 is 12 Lowest Terms: A fraction with no common factors between the numerator and the denominator. 12
2
Ex. The lowest terms of is 18
3
Natural Numbers: The counting numbers {1,2,3…}, they are denoted by N. Mixed Number: A number written as a combination of a whole number and a fraction Ex. 1
3
4
Multiple: A number that can be divided by another number without a remainder Ex: the number 15 & 20 are multiples of 5 Numerator: The value above the divisor (fraction bar) in a rational number Ex: the 2 in 2
3
Perfect Square: A number made by squaring a whole number. Power: A number with an exponent Prime Factor: The factors of a number that are prime numbers Ex: The number 24 has prime factors: 1, 2, & 3 Prime Number: A number that has exactly 2 factors, 1 and the number itself. Product: The answer to a multiplication question Proportion: a set of equivalent ratios Ex. 3 : 4 = 9 : 12 Proportional Reasoning: using equivalent ratios to solve problems. Quotient: The answer to a division question © Jen Brown et al. AB Lucas Mathematics Department 2011 25 Rate: a comparison of 2 of more values Ex. 14 km per 6 minutes Ratio: a comparison of 2 or more values with the same type of units Ex. A scale on a map, map distance : actual distance 5 cm : 15 km Rational Number: any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Reciprocal: When a fraction is flipped, the numerator becomes the denominator and the denominator becomes the numerator 4
5
Ex. The reciprocal of 5 is 4
Remainder: The amount that is left after division. 7 R2
Ex. 37 divided by 5 is 7 with remainder 2 ( 5 37
) Square Root: A number that when multiplied by itself will give the original number. Ex. The square root of 25 is 5 ( 25
= 5 ) Sum: The answer to an addition question Unit Rate: how many of one quantity corresponds to one of another quantity. Ex. $1.25/chocolate bar Whole Numbers: Zero and the counting numbers {0,1,2,3…}, they are denoted by W. A-­‐2 Answers 1-­3 1-­4 1-­5 2-­1 2-­2 2-­3 2-­4 2-­5 1. a) 56.9 2. a) 56.39 1. a) 36 1.) 12,1,9,8 7.) 14,22,27,88 13.) 6,8,56,32 1. a) -­‐3 3. a) -­‐6 1. a) -­‐3 2. a) 3 1. a) 30 2. a) -­‐2 c) -­‐32 1.) b 5.a) 729 1. a) 6 c) 6 4. a) 13 g) -­‐2 c) i) f) i) 3-­1 3-­2 1.a) 2
1.a) 6 g) 6 3-­3 1.a) 3-­4 1.a) 26 1 2 4
, , , etc
4 8
1 1
<
3 2 1 1 1 3
< < <
8 4 2 4 b) 360 000 b) 658.4 b) 1 440 2.) 3,28,7,40 8.) 30,0,5,6 14.) 7,4,48,42 b) 0 b) 0 b) 1 b) 2 b) 8 b) 6 d) 42 2.) c b) 49 b) 8 d) -­‐1 b) -­‐8 h) 3 ii) 3 ii) 7 1 2 6
, , , etc
12
b) 2 4
b) 8 h) 20 b) b) 4 5
<
5 6 1 2 4 3
< < <
3 5 6 4
c) 56 c) 0.98 c) 600 3.) 64,5,24,5 9.) 58,8,80,4 d) 4 600 d) 0.985 d) 900 4.) 4,5,56,9 10.) 96,24,56,6 2. a) -­‐5 b) -­‐7 c) -­‐9 d) -­‐8 c) -­‐14 d) -­‐1 c) 4 d) -­‐3; c) 12 d) -­‐18 c) -­‐4 d) 1 4. answers will vary 3.a) (-­‐5)3 b) 1.056 c) -­‐16 d) 1.44 c) 3 d) -­‐4 e) -­‐21 3. a)-­‐136 c) 4 d) -­‐15 5. a) i) ii) -­‐4 d) i) ii) 2 g) i) ii) 8 1 2 3
, , , etc
12
1 3 6
, , , etc
6 12
2.) 4 8
c) 12 i) 18 c) c) 1
<
5
3
<
9
2
9 1 2 5
< <
2 3 6
3.a) 2
d) 35 e) 0.84 e) 57.3 e) 28 5.) 132,20,6,23 11.) 11,5,38,0 f) 500 f) 43.58 f) 4 6.) 3,40,4,7 12.) 12,54,30,43
c) 0 d) -­‐20 4.) -­‐10oC 5.) e) 8; 3.) -­‐$190 4.) -­‐120C e) -­‐28 3. a) 24 b) 60 5. a) 128, 512 b) 12, -­‐4 4.) (-­‐4)(-­‐4)(-­‐4), -­‐64 b) (0.8)(0.8),0.64 e) 1 f) -­‐1 g) 0.125 2. a)-­‐10 b) -­‐50 b)13 c) -­‐11 e) -­‐21 f) 0 b) i) ii) -­‐2 e) i)
ii) 3 1 2 3
, , , etc
9
b) 3 6
e) 20 © Jen Brown et al. AB Lucas Mathematics Department 2011 2 3 4
, , , etc
16
c) 8 12
f) 30 3-­5 1.a) 4
3 b) 4
15 3-­6 c) 1a) 3a) 4
5 c) 6a) b) c) 10 d) 1
9 3-­7 1.a) 3-­8 1.a) 2 d) b) 5
4 1
18 25
16
c) 3
h) 4 b) b) d) e) 4-­2 1.a) d) 2.a) b) 2a) b) c) d) 4a) d)
5a) b) c) 27
8 !5
8 81
64 c) d) 5
8
b) 1
3
1
2 17
15 c) c) b) 49
30 3
4 d) c) 9 b) d) 7a) 5 b) 21 4
25
c) 7
c) 6 37
12
4.a) d) d) b) 1
16 1
2 31
15 2.a) c) 5
4
5
12 b) d) 22
9 121
40 c) 10 3a.) 4
9
e) 4
3 f) 18
25
b) 3
5 c) 1
10 h) 8
5
81
g) 1
15 3
2 d) 1
20 b) c) 0.25 d) 0.8 7
20
c) 6
l) 25 24
d) 25 19
m) 50 e) 0.4 2.a) 1
2
9
20 1
20 4-­3 j) 1.a) 50% h) 48% 4-­4 b) k) 4
5 4
25 e) n) 12
25 19
20 g) p) f) 2% l) 7.5% 3
25 27
40 i) c) 25% j) 0.5% 1.a) 20% b) 20% c) 37.5% d) 45% e) 8% f) 44.4 4-­5 4-­6 4-­7 4-­8 4-­9 g) 66.6 1.a) $2 g)5.76kg 1.a) $22 d) 225 1.) $294 1.a) 25km/h 1.a) 1:2 g) 11:54 h) 75% b) $180 h) 4.5km b) $630 i) 60% c) 135cm j) 33.3 d) 675nails e) 4.32l f)175g c) 435cm 2.a) $18 b) $270 c) 220cm 2.) $127.50 b) 8m/s b) 3:1 h) 5:16 3.) 1171 students 2. a) $0.40 b) 0.83cents per gram or 120g/$ c) 2:5 d) 10:3 e) 8:3 3. Brand B f) 15:1 5-­1 1.) $12.50 2.) 177cm 3.) 2.25m 1
1 h
12 b) 7.) $7.50 8.) $3.12, $0.88 b) Mark, 68%, Jake, 78%, Marlyn 72%; Jake did best e) 55% k) 16% 11
f) 20 3
o) 2 b) 60% i) 193% 4-­1 1.a) 0.33 b) 0.75 1
3
1
3 9
3.a) 5
c) g) 235% m) 9436% 7
1
h
12 21
25 4.) Julie 5.) $28 6.a) 9.) 72 pens 10.) 9L 11.a) Marlyn, 60% © Jen Brown et al. AB Lucas Mathematics Department 2011 27 A-­‐3 Sources of Resources for Extra Practice www.math-­‐drills.com many versions drill sheets including arithmetic, percent, order of operations, etc. www.mathworksheetwizard.com/arithmetic.html creates worksheets for practice of basic arithmetic facts www.superkids.com/aweb/tools/math creates worksheets for practice of basic arithmetic facts, order of operations, fractions, percent, rounding, exponents etc www.mathta.com creates worksheets for practice of basic arithmetic facts, fractions, order of operations and more www.mathplayground.com/math_worksheets.html creates worksheets for practice of basic arithmetic facts, can choose types of questions to include www.mathsisfun.com has worksheets and online practice www.thatquiz.org online quizzes on all levels of mathematics www.math.com/students/practice.html online lessons and practice on a variety of mathematical concepts www.19online.net/math interactive online games for arithmetic practice www.sheppardsoftware.com/math.htm online math games web.me.com/brown.jen click on teacher resources, login tvdsb password: iteach for more Mad Math Minutes, lessons, activities, worksheets, and links. A-­‐4 References C. Dearling et al. McGraw-­‐Hill Ryerson. Principles of Mathematics 9, Student Skills Book, 2006. D. Zimmer et al. Nelson, Mathematics 9, 1999. D. Zimmer et al. Nelson, Mathematics 10, 2001. A-­‐5 Credits Compiled By: AB Lucas Mathematics Department 2011 Thames Valley District School Board London, Ontario Jen Brown Paul Laxon Ian Charlton Peggy Slegers Rebecca Cober Anna Tran Jason Eichstedt Tara Wade Peter Gubbels Layout & Publishing By: Jen Brown Edited By: Jen Brown Peggy Slegers Anna Tran
A-­‐6 Reproduction Rights This document can be copied for educational purposes as long as proper credit is given to its creators. Please leave the © on the bottom of each page. 28 © Jen Brown et al. AB Lucas Mathematics Department 2011