Fractions Return to Table of Contents Slide 5 / 305 Slide 6 / - - PDF document

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7th Grade Math Review of 6th Grade

www.njctl.org 2012-07-31

Slide 2 / 305 Table of Contents

Fractions Number System Decimal Computation

Click on the topic to go to that section

Expressions Equations and Inequalities Ratios and Proportions Geometry Statistics

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Fractions

Return to Table of Contents

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List what you remember about fractions.

Hint

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We can use prime factorization to find the greatest common factor (GCF).

• 1. Factor the given numbers into primes.
• 2. Circle the factors that are common.
• 3. Multiply the common factors together to find the

greatest common factor. Greatest Common Factor

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1 Find the GCF of 18 and 44.

Pull Pull

SLIDE 2

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2 Find the GCF of 72 and 75.

Pull Pull

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3 Find the GCF of 52 and 78.

Pull Pull

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A multiple of a whole number is the product of the number and any nonzero whole number. A multiple that is shared by two or more numbers is a common multiple. Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ... Multiples of 14: 14, 28, 42, 56, 70, 84,... The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is 42.

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There are 2 ways to find the LCM:

• 1. List the multiples of each number until you find the first
• ne they have in common.
• 2. Write the prime factorization of each number. Multiply

all factors together. Use common factors only once (in

• ther words, use the highest exponent for a repeated

factor).

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EXAMPLE: 6 and 8 Multiples of 6: 6, 12, 18, 24, 30 Multiples of 8: 8, 16, 24 LCM = 24 Prime Factorization: 2 3 2 4 2 2 2 2 3 23 LCM: 23 3 = 8 3 = 24 6 8

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4 Find the least common multiple

• f 10 and 14.

A 2 B 20 C

70

D

140

Pull Pull

SLIDE 3

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5 Find the least common multiple

• f 6 and 14.

A 10 B 30 C

42

D

150

Pull Pull

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6 Find the LCM of 24 and 60. Pull Pull

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Which is easier to solve? 28 + 42 7(4 + 6) Do they both have the same answer? You can rewrite an expression by removing a common factor. This is called the Distributive Property.

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The Distributive Property allows you to:

• 1. Rewrite an expression by factoring out the GCF.
• 2. Rewrite an expression by multiplying by the GCF.

EXAMPLE Rewrite by factoring out the GCF: 45 + 80 28 + 63 5(9 + 16) 7(4 + 9) Rewrite by multiplying by the GCF: 3(12 + 7) 8(4 + 13) 36 + 21 32 + 101

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7 In order to rewrite this expression using the Distributive

Property, what GCF will you factor? 56 + 72

Pull Pull

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8 In order to rewrite this expression using the Distributive

Property, what GCF will you factor? 48 + 84

Pull Pull

SLIDE 4

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9 Use the distributive property to rewrite this expression:

36 + 84

A 3(12 + 28) B 4(9 + 21) C 2(18 + 42) D 12(3 + 7)

Pull Pull

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10 Use the distributive property to rewrite this expression:

88 + 32 A 4(22 + 8) B 8(11 + 4) C 2(44 + 16) D 11(8 + 3)

Pull Pull

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Adding Fractions...

• 1. Rewrite the fractions with a common denominator.
• 2. Add the numerators.
• 3. Leave the denominator the same.
• 4. Simplify your answer.

Adding Mixed Numbers...

• 1. Add the fractions (see above steps).
• 2. Add the whole numbers.
• 3. Simplify your answer.

(you may need to rename the fraction) Link Back to List

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11

3 10 2 10 +

Pull Pull

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12

5 8 1 8 +

Pull Pull

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13 Find the sum.

5 3

10 + 7 5 10

Pull Pull

SLIDE 5

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14 Is the equation below true or false?

True False

1 8

12 + 1 5 12

3 1

12

Pull Pull

Don't forget to regroup to the whole number if you end up with the numerator larger than the denominator.

Click For reminder

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A quick way to find LCDs... List multiples of the larger denominator and stop when you find a common multiple for the smaller denominator. Ex: and Multiples of 5: 5, 10, 15 Ex: and Multiples of 9: 9, 18, 27, 36 2 5 1 3 3 4 2 9

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Common Denominators Another way to find a common denominator is to multiply the two denominators together. Ex: and 3 x 5 = 15 = =

2 5 1 3 1 3

x 5 x 5

5 15 2 5 6 15

x 3 x 3

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15

2 5 1 3 +

Pull Pull

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16

3 10 2 5 +

Pull Pull

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17

5 8 3 5 +

Pull Pull

SLIDE 6

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18

A

5 3

4 + 2 7 12 =

7 16

12

B 8 4

12

C

7 5

8

D

8 1

3

Pull Pull

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19

A

2 3

8 + 5 5 12 =

7 19

24

7 8

20

B

7 8

12

C

8 7

12

D

Pull Pull

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20

5 2

10

5 5

12

A

3 1

4 + 2 1 6 =

B

5 1

2

C

6 5

12

D

Pull Pull

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Subtracting Fractions...

• 1. Rewrite the fractions with a common denominator.
• 2. Subtract the numerators.
• 3. Leave the denominator the same.
• 4. Simplify your answer.

Subtracting Mixed Numbers...

• 1. Subtract the fractions (see above steps..).

(you may need to borrow from the whole number)

• 2. Subtract the whole numbers.
• 3. Simplify your answer.

(you may need to simplify the fraction) Link Back to List

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21

7 8 4 8

Pull Pull

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22

6 7 4 5

Pull Pull

SLIDE 7

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23

2 3 1 5

Pull Pull

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24 Is the equation below true or false?

True False

4 5

9 3 9

3 2

9

Pull Pull

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25 Is the equation below true or false?

True False

2 7

9 1 9

1 2

3

1

Pull Pull

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26 Find the difference.

4 7

8

2 3

8

Pull Pull

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27

6 7 3 5

Pull Pull

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A Regrouping Review When you regroup for subtracting, you take

• ne of your whole numbers and change it into

a fraction with the same denominator as the fraction in the mixed number.

3 3

5 = 2 5 5 3 5 = 2 8 5 Don't forget to add the fraction you regrouped from your whole number to the fraction already given in the problem.

SLIDE 8

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4

3 7

12

5 3

12

3 7

12

4 12

12

3 7

12 3 12

4 15

12

3 7

12

1 8

12

1 2

3

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28 Do you need to regroup in order

to complete this problem? Yes

• r

No

3 1

2 1 4

Pull Pull

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29 Do you need to regroup in order

to complete this problem?

7 2

3 3 4

6

Pull Pull

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30

What does 17 become when regrouping?

3 10

Pull Pull

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31

What does 21 become when regrouping?

5 8

Pull Pull

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32

2 1

12

A

1 22

24

B

4 1

6

2 1

4 =

1 11

12

C

1 1

12

D

Pull Pull

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33

A

3 13

21

B

6 2

7

3 2

3 =

3 8

21

2 2

3

C

2 13

21

D

Pull Pull

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34

A

6 1

6

B

15 8 10

12 =

7 5

6

7 1

6

C

6 2

12

D

Pull Pull

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Multiplying Fractions...

• 1. Multiply the numerators.
• 2. Multiply the denominators.
• 3. Simplify your answer.

Multiplying Mixed Numbers...

• 1. Rewrite the Mixed Number(s) as an improper fraction.

(write whole numbers / 1)

• 2. Multiply the fractions.
• 3. Simplify your answer.

Link Back to List

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35

1 5 x 2 3 =

Pull Pull

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36

2 3 x 3 7 =

Pull Pull

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37

= 4 9 3 8

( )

Pull Pull

SLIDE 10

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38

True False x 1 2 =

5

5 1 x 1 2

Pull Pull

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39

A

x 4 7

3

B C

3 5

7

D

12 21 12 7

1 5

7

Pull Pull

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40

True False x =

2 1

4

3 1

8

6 3

8

Pull Pull

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41

15 1

4

A

18 1

8

B

20 3

8

C

19 1

8

D

5 8

( )

5

2 5

(3 )

Pull Pull

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Dividing Fractions...

• 1. Leave the first fraction the same.
• 2. Multiply the first fraction by the reciprocal of the second

fraction.

• 3. Simplify your answer.

Dividing Mixed Numbers...

• 1. Rewrite the Mixed Number(s) as an improper fraction(s).

(write whole numbers / 1)

• 2. Divide the fractions.
• 3. Simplify your answer.

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To divide fractions, multiply the first fraction by the reciprocal of the second fraction. Make sure you simplify your answer! Some people use the saying "Keep Change Flip" to help them remember the process. 3 5 x 8 7 = 3 x 8 5 x 7 = 24 35 3 5 7 8 = 1 5 x 2 1 = 1 x 2 5 x 1 = 2 5 1 5 1 2 =

SLIDE 11

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42

True False 8 10 = 5 4 x 8 10 4 5

Pull Pull

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43

True False 2 7 = 3 4

2 7

8

Pull Pull

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44

1

A

39 40

B C

8 10 = 4 5 40 42

Pull Pull

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45 Pull Pull

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To divide fractions with whole or mixed numbers, write the numbers as an improper

• fractions. Then divide the two fractions by

using the rule (multiply the first fraction by the reciprocal of the second). Make sure you write your answer in simplest form. 5 3 x 2 7 = 10 21 2 3 =

1

1 2

3

5 3 7 2 = 6 1 x 2 3 = 12 3 =

6

1 2

1

6 1 3 2 = = 4

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46

= 1 2

2 2

3

1

Pull Pull

SLIDE 12

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47

= 1 2

2 2

3

1

Pull Pull

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48

= 1 2

5 2

Pull Pull

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Decimal Computation

Return to Table of Contents

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List what you remember about decimals.

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Some division terms to remember.... · The number to be divided into is known as the dividend · The number which divides the other number is known as the divisor · The answer to a division problem is called the quotient

divisor 5 20 dividend

4 quotient

20 ÷ 5 = 4 20

__

5 = 4

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When we are dividing, we are breaking apart into equal groups EXAMPLE 1 Find 132 3 Step 1: Can 3 go into 1, no so can 3 go into 13, yes 4

• 12

1 3 x 4 = 12 13 - 12 = 1 Compare 1 < 3 3 132 3 x 4 = 12 12 - 12 = 0 Compare 0 < 3

• 12

2 Step 2: Bring down the 2. Can 3 go into 12, yes 4

Click for step 1

Click for step 2

SLIDE 13

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EXAMPLE 2 (change pages to see each step) Step 1: Can 15 go into 3, no so can 15 go into 35, yes 2

• 30

5 15 x 2 = 30 35 - 30 = 5 Compare 5 < 15 15 357

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2

• 30

5 15 357 15 x 3 = 45 57 - 45 =12 Compare 12 < 15 7

• 45

12 Step 2: Bring down the 7. Can 25 go into 207, yes 3 EXAMPLE 2 (change pages to see each step)

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2

• 30

5 15 357.0 7

• 45

120

• 120

3 Step 3: You need to add a decimal and a zero since the division is not complete. Bring the zero down and continue the long division. 15 x 8 = 120 120 - 120 = 0 Compare 0 < 15 .8 EXAMPLE 2 (change pages to see each step)

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49 Compute. Pull Pull

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50 Compute. Pull Pull

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51 Compute. Pull Pull

SLIDE 14

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If you know how to add whole numbers then you can add

• decimals. Just follow these few steps.

Step 1: Put the numbers in a vertical column, aligning the decimal points. Step 2: Add each column of digits, starting on the right and working to the left. Step 3: Place the decimal point in the answer directly below the decimal points that you lined up in Step 1.

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C

52 Add the following:

0.6 + 0.55 = A 6.1 B 0.115

click

C 1.15 D 0.16

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53 Find the sum

1.025 + 0.03 + 14.0001 =

15.0551

click

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54 Find the sum:

5 + 100.145 + 57.8962 + 2.312 =

165.3532

click

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What do we do if there aren't enough decimal places when we subtract? 4.3 - 2.05 Don't forget...Line 'em Up! 4.3 2.05 What goes here? 4.30 2.05 2.25 2 1

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55

5 - 0.238 =

4.762

click

SLIDE 15

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56

12.809 - 4 =

8.809

click

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57

4.1 - 0.094 =

4.006

click

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58

17 - 13.008 =

3.992

click

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If you know how to multiply whole numbers then you can multiply decimals. Just follow these few steps. Step 1: Ignore the decimal points. Step 2: Multiply the numbers using the same rules as whole numbers. Step 3: Count the total number of digits to the right of the decimal points in both numbers. Put that many digits to the right of the decimal point in your answer.

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23.2 x 4.04 928 92800

0000

93.728

}

There are a total of three digits to the right of the decimal points. There must be three digits to the right of the decimal point in the answer. EXAMPLE

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59 Multiply 0.42 x 0.032

0.1344

click

SLIDE 16

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60 Multiply 3.452 x 2.1

7.2492

click

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4.73836

61 Multiply 53.24 x 0.089 click

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Dividend Divisor Step 1: Change the divisor to a whole number by multiplying by a power of 10. Step 2: Multiply the dividend by the same power of 10. Step 3: Use long division. Step 4: Bring the decimal point up into the quotient.

Divide by Decimals

Quotient

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15.6 6.24 Multiply by 10, so that 15.6 becomes 156 6.24 must also be multiplied by 10 156 62.4 .234 23.4 Multiply by 1000, so that .234 becomes 234 23.4 must also be multiplied by 1000 234 23400 Try rewriting these problems so you are ready to divide!

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62 Divide

0.78 ÷ 0.02 =

39

click

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63

10 divided by 0.25 =

40

click

SLIDE 17

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64

12.03 ÷ 0.04 =

300.75

click

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There are two types of decimals - terminating and repeating. A terminating decimal is a decimal that ends. All of the examples we have completed so far are terminating. A repeating decimal is a decimal that continues forever with

• ne or more digits repeating in a pattern.

To denote a repeating decimal, a line is drawn above the numbers that repeat. However, with a calculator, the last digit is rounded.

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Examples:

6600 2342 2200 14200 13200 10000 8800 12000 11000 10000 8800 12000 11000 63 48 45 39 36 32 27 51 45 60 54 6

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65

click

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66

click

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67

click

SLIDE 18

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Statistics

Return to Table of Contents

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List what you remember about statistics.

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Measures of Center Vocabulary:

· Mean - The sum of the data values divided by the number

• f items; average

· Median - The middle data value when the values are written in numerical order · Mode - The data value that occurs the most often

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68 Find the mean

14, 17, 9, 2, 4,10, 5, 3 Pull Pull

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69 Find the median: 5, 9, 2, 6, 10, 4

A 5 B 5.5 C 6 D 7.5 Pull Pull

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70 Find the mode(s): 3, 4, 4, 5, 5, 6, 7, 8, 9

A

4

B

5

C

9

D

No mode

Pull Pull

SLIDE 19

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71 Consider the data set: 78, 82, 85, 88, 90. Identify the data

values that remain the same if "79" is added to the set. A mean B median C mode D range E minimum Pull Pull

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Measures of Variation Vocabulary: Minimum - The smallest value in a set of data Maximum - The largest value in a set of data Range - The difference between the greatest data value and the least data value Quartiles - are the values that divide the data in four equal parts. Lower (1st) Quartile (Q1) - The median of the lower half of the data Upper (3rd) Quartile (Q3) - The median of the upper half of the data. Interquartile Range - The difference of the upper quartile and the lower quartile. (Q3 - Q1) Outliers - Numbers that are significantly larger or much smaller than the rest of the data

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72

Find the range: 4, 2, 6, 5, 10, 9 A 5 B 8 C 9 D 10 Pull Pull

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73

Find the range for the given set of data: 13, 17, 12, 28, 35 Pull Pull

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Quartiles There are three quartiles for every set of data. Lower Half Upper Half 10, 14, 17, 18, 21, 25, 27, 28

Q1 Q2 Q3

The lower quartile (Q1) is the median of the lower half of the data which is 15.5. The upper quartile (Q3) is the median of the upper half of the data which is 26. The second quartile (Q2) is the median of the entire data set which is 19.5. The interquartile range is Q3 - Q1 which is equal to 10.5.

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74 The median (Q2) of the following data set is 5.

3, 4, 4, 5, 6, 8, 8 True False Pull Pull

SLIDE 20

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75 What are the lower and upper quartiles of the data set

3, 4, 4, 5, 6, 8, 8? A Q1: 3 and Q3: 8 B Q1: 3.5 and Q3: 7 C Q1: 4 and Q3: 7 D Q1: 4 and Q3: 8 Pull Pull

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76 What is the interquartile range of the data set

3, 4, 4, 5, 6, 8, 8?

Pull Pull

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77 What is the median of the data set

1, 3, 3, 4, 5, 6, 6, 7, 8, 8? A 5 B 5.5 C 6 D No median Pull Pull

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78 What is the interquartile range of the data set

1, 3, 3, 4, 5, 6, 6, 7, 8, 8?

Pull Pull

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Outliers - Numbers that are relatively much larger or much smaller than the data Which of the following data sets have outlier(s)?

• A. 1, 13, 18, 22, 25
• B. 17, 52, 63, 74, 79, 83, 120
• C. 13, 15, 17, 21, 26, 29, 31
• D. 25, 32, 35, 39, 40, 41

Pull Pull

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79

The data set: 1, 20, 30, 40, 50, 60, 70 has an outlier which is ________ than the rest of the data. A higher B lower C neither Pull Pull

SLIDE 21

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80 In the following data what number is the outlier?

{ 1, 2, 2, 4, 5, 5, 5, 13}

Pull Pull

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81

Find the maximum value: 15, 10, 32, 13, 2 A 2 B 15 C 13 D 32 Pull Pull

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The mean absolute deviation of a set of data is the average distance between each data value and the mean. Steps

• 1. Find the mean.
• 2. Find the distance between each data value and the mean.

That is, find the absolute value of the difference between each data value and the mean.

• 3. Find the average of those differences.

*HINT: Use a table to help you organize your data.

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Let's continue with the "Phone Usage" example. Step 1 - We already found the mean of the data is 56. Step 2 - Now create a table to find the differences.

48 8 52 4 54 2 55 1 58 2 59 3 60 4 62 6

Data Value Absolute Value of the Difference |Data Value - Mean|

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Step 3 - Find the average of those differences. 8 + 4 + 2 + 1 + 2 + 3 + 4 + 6 = 3.75 8 The mean absolute deviation is 3.75. The average distance between each data value and the mean is 3.75 minutes. This means that the number of minutes each friend talks on the phone varies 3.75 minutes from the mean of 56 minutes.

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82 Find the mean absolute deviation of the given set of data.

Zoo Admission Prices $9.50 $9.00 $8.25 $9.25 $8.00 $8.50 A $0.50 B $8.75 C $3.00 D $9.00 Pull Pull

SLIDE 22

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83 Find the mean absolute deviation for the given set

• f data.

Number of Daily Visitors to a Web Site 112 145 108 160 122

Pull Pull

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F R E Q U E N C Y

8 6 4 2 0 30-

40- 50- 60- 70- 80- 90- 39 49 59 69 79 89 99 GRADE

Grade Tally Frequency 30-39 I 1 40-49 50-59 60-69 I 1 70-79 IIII 4 80-89 IIII III 8 90-99 III 3

TEST SCORES 95 85 93 77 97 71 84 63 87 39 88 89 71 79 83 82 85

SAMPLES: Data

TEST SCORES 87 53 95 85 89 59 86 82 87 40 90 72 48 68 57 64 85

F R E Q U E N C Y

8 6 4 2 0 40-

50- 60- 70- 80- 90- 49 59 69 79 89 99 GRADE

Grade Tally Frequency 40-49 II 2 50-59 III 3 60-69 II 2 70-79 I 1 80-89 IIII II 7 90-99 II 2

Frequency Table Histogram

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A box and whisker plot is a data display that organizes data into four groups

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

The median divides the data into an upper and lower half The median of the lower half is the lower quartile. The median of the upper half is the upper quartile. The least data value is the minimum. The greatest data value is the maximum.

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1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

median

25% 25% 25% 25% The entire box represents 50% of the data. 25% of the data lie in the box on each side of the median Each whisker represents 25% of the data

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84 The minimum is

A 87 B 104 C 122 D 134

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

Pull Pull

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85 The median is

A 87 B 104 C 122 D 134

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

Pull Pull

SLIDE 23

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86 The lower quartile is

A 87 B 104 C 122 D 134

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

Pull Pull

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87 The upper quartile is

A 87 B 104 C 122 D 134

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

Pull Pull

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88 In a box and whisker plot, 75% of the data is between

A the minimum and median B the minimum and maximum C the lower quartile and maximum D the minimum and the upper quartile

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

Pull Pull

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89 In a box and whisker plot, 50% of the data is between

A the minimum and median B the minimum and maximum C the lower quartile and upper quartile D the median and maximum

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10 80

90 100 110 120 130 140 150

Pull Pull

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A dot plot (line plot) is a number line with marks that show the frequency of data. A dot plot helps you see where data cluster. Example:

35 40 45 50 30

x x x x x x x x x x x x x x x x x x x x x x x x x x Test Scores The count of "x" marks above each score represents the number

• f students who received that score.

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90 How many more students scored 75 than scored

85?

Pull Pull

SLIDE 24

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91 What is the median score?

Pull Pull

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92 What are the mode(s) of the data set?

A 75 B 80 C 85 D 90 E 95 F 100 Pull Pull

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93 Which measure of center appropriately represents the data?

A Mean B Median C Mode

Paper Plane Competition Distance (ft) F R E Q U E N C Y

4 3 2 1 0 0-4 5-9 10-14 15-19 20-24

Pull Pull

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Number System

Return to Table of Contents

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List what you remember about the number system.

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{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...} Definition of Integer: The set of natural numbers, their opposites, and zero.

Define Integer

Examples of Integers:

SLIDE 25

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• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

Integers on the number line

Negative Integers Positive Integers Numbers to the left of zero are less than zero Numbers to the right of zero are greater than zero Zero is neither positive or negative

`

Zero

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94 Which of the following are examples of integers?

A 0 B -8 C -4.5 D 7 E

Pull Pull

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95 Which of the following are examples of integers?

A B 6 C -4 D 0.75 E

25%

Pull Pull

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96 What is the opposite of -5?

Pull Pull

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97 What is the opposite of 0?

Pull Pull

Slide 150 / 305 Absolute Value of Integers

The absolute value is the distance a number is from zero on the number line, regardless of direction. Distance and absolute value are always non- negative (positive or zero).

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• 2
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• 10

What is the distance from 0 to 5?

SLIDE 26

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• 7

98 Find

Pull Pull

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• 28

99 Find

Pull Pull

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3

100 Find

Pull Pull

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101 Which numbers have 50 as their absolute value?

A -50 B -25 C 0 D 25 E 50

Pull Pull

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To compare integers, plot points on the number line. The numbers farther to the right are greater. The numbers farther to the left are smaller.

Use the Number Line

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• 1
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• 6
• 7
• 8
• 9
• 10

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102 The integer 7 is ______ 7.

A = B < C >

Pull Pull

SLIDE 27

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103 The integer -20 is ______ -14.

A = B < C >

Pull Pull

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104 The integer -4 is ______ 6.

A = B < C >

Pull Pull

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105

A B C

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

What is the position of the dot on the number line below?

Pull Pull

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106

A -5.5 B -6.5 C -5.2

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

What is the position of the dot on the number line below?

Pull Pull

Slide 161 / 305 Comparing Rational Numbers

Sometimes you will be given fractions and decimals that you need to compare. It is usually easier to convert all fractions to decimals in order to compare them on a number line. To convert a fraction to a decimal, divide the numerator by the denominator. 4 3.00

• 28

020

• 20

0.75

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107

A = B < C >

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

Pull Pull

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108

A = B < C >

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

Pull Pull

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109

A = B < C >

1 2 3 4 5 6 7 8 9 10

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10

Pull Pull

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The coordinate plane is divided into four sections called quadrants. The quadrants are formed by two intersecting number lines called axes. The horizontal line is the x-axis. The vertical line is the y-axis. The point of intersection is called the origin. (0,0)

x - axis y - axis

• rigin

(+, -) (-, +) (-, -) (+, +)

Slide 166 / 305

To graph an ordered pair, such as (3,2): · start at the origin (0,0) · move left or right on the x-axis depending on the first number · then move up or down from there depending on the second number · plot the point (3,2)

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110

The point (-5, 4) is located in quadrant_____.

A I B II C III D IV

Pull Pull

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111

The point (7, -2) is located in quadrant _____.

A I B II C III D IV

Pull Pull

SLIDE 29

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112

The quadrant where the x & y coordinates are both negative is quadrant ___.

A I B II C III D IV

Pull Pull

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113

When plotting points in the Cartesian Plane, you always start at ____.

A the x - axis B the origin C the y-axis D the Coordinate Plane E (0,0)

Pull Pull

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114

Point A is located at (-5, 1) True False Pull Pull

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115

Point A is located at (-2, 3) True False

A

Pull Pull

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Expressions

Return to Table of Contents

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List what you remember about expressions.

SLIDE 30

Slide 175 / 305 Exponents

Exponents, or Powers, are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition. These are all equivalent: 24 Exponential Form 2∙2∙2∙2 Expanded Form 16 Standard Form In this example 2 is raised to the 4th power. That means that 2 is multiplied by itself 4 times.

Slide 176 / 305 Powers of Integers

Bases and Exponents When "raising a number to a power", The number we start with is called the base, the number we raise it to is called the exponent. The entire expression is called a power. You read this as "two to the fourth power."

24

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116 What is the base in this expression?

32

Pull Pull

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117 What is the exponent in this expression?

32

Pull Pull

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118 Evaluate 32.

Pull Pull

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119 Evaluate 43.

Pull Pull

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120 Evaluate 24.

Pull Pull

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What does "Order of Operations" mean? The Order of Operations is an agreed upon set

• f rules that tells us in which "order" to solve a

problem.

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The P stands for Parentheses: Usually represented by ( ). Other grouping symbols are [ ] and { }. Examples: (5 + 6); [5 + 6]; {5 + 6}/2 The E stands for Exponents: The small raised number next to the larger number. Exponents mean to the ___ power (2nd, 3rd, 4th, etc.) Example: 23 means 2 to the third power or 2(2)(2) The M/D stands for Multiplication or Division: From left to right. Example: 4(3) or 12 ÷ 3 The A/S stands for Addition or Subtraction: From left to right. Example: 4 + 3 or 4 - 3 What does P E M/D A/S stand for?

Slide 184 / 305 Watch Out!

When you have a problem that looks like a fraction but has an

• peration in the numerator, denominator, or both, you must solve

everything in the numerator or denominator before dividing. 45 3(7-2) 45 3(5) 45 15

3 Slide 185 / 305

121

1 + 5 x 7

Pull Pull

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122 40 ÷ 5 x 9

Pull Pull

SLIDE 32

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123

6 - 5 + 2

Pull Pull

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124 18 ÷ 9 x 2

Pull Pull

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125

5(32)

Pull Pull

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[ 6 + ( 2 8 ) + ( 42 - 9 ) ÷ 7 ] 3 Let's try another problem. What happens if there is more than

• ne set of grouping symbols?

[ 6 + ( 2 8 ) + ( 42 - 9 ) ÷ 7 ] 3 When there are more than 1 set of grouping symbols, start inside and work out following the Order of Operations. [ 6 + ( 16 ) + ( 16 - 9 ) ÷ 7 ] 3 [ 6 + ( 16 ) + ( 7 ) ÷ 7 ] 3 [ 6 + ( 16 ) + 1 ] 3 [ 22 + 1 ] 3 [ 23 ] 3 69

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126

4 - 2[5 + 3] + 7 Pull Pull

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127

42 + 9 + 3[2 + 5] Pull Pull

SLIDE 33

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128

62 ÷ 3 + (15 - 7) Pull Pull

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129 Which expression with parenthesis added in

changes the value of: 5 + 4 - 7

A (5 + 4) - 7 B

5 + (4 - 7)

C

(5 + 4 - 7)

D

none of the above change the value

Pull Pull

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130 Which expression with parenthesis added in

changes the value of: 36 ÷ 2 + 7 + 1

A (36 ÷ 2) + 7 + 1 B

36 ÷ (2 + 7) + 1

C

(36 ÷ 2 + 7 + 1)

D

none of the above change the value

Pull Pull

Slide 196 / 305 What is a Constant?

A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative.

Example: 4x + 2

In this expression 2 is a constant.

click to reveal

Example: 11m - 7

In this expression -7 is a constant.

click to reveal

Slide 197 / 305 What is a Variable?

A variable is any letter or symbol that represents a changeable or unknown value. Example: 4x + 2 In this expression x is a variable.

click to reveal

Slide 198 / 305 What is a Coefficient?

A coefficient is the number multiplied by the variable. It is located in front of the variable. Example: 4x + 2 In this expression 4 is a coefficient.

click to reveal

SLIDE 34

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If a variable contains no visible coefficient, the coefficient is 1. Example 1: x + 4 is the same as 1x + 4

• x + 4

is the same as

• 1x + 4

Example 2: x + 2 has a coefficient of Example 3:

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131 In 3x - 7, the variable is "x"

True False Pull Pull

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132 In 4y + 28, the variable is "y"

True False Pull Pull

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133 In 4x + 2, the coefficient is 2

True False Pull Pull

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134 What is the constant in 6x - 8?

A 6 B x C 8 D

• 8

Pull Pull

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135 What is the coefficient in - x + 5?

A none B 1 C

• 1

D 5 Pull Pull

SLIDE 35

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136 Evaluate 3h + 2 for h = 3

Pull Pull

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137 Evaluate 2x2 for x = 3

Pull Pull

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138 Evaluate 4a + a for a = 8, c = 2

c Pull Pull

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139 Use the distributive property to rewrite the expression

without parentheses (x + 6)3 A 3x + 6 B 3x + 18 C x + 18 D 21x Pull Pull

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140 Use the distributive property to rewrite the expression

without parentheses 3(x - 4) A 3x - 4 B x - 12 C 3x - 12 D 9x Pull Pull

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141 Use the distributive property to rewrite the

expression without parentheses 2(w - 6) A 2w - 6 B w - 12 C 2w - 12 D 10w Pull Pull

SLIDE 36

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Equations and Inequalities

Return to Table of Contents

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List what you remember about equations and inequalities.

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A solution to an equation is a number that makes the equation true. In order to determine if a number is a solution, replace the variable with the number and evaluate the equation. If the number makes the equation true, it is a solution. If the number makes the equation false, it is not a solution.

Determining the Solutions of Equations

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142

Which of the following is a solution to the equation: x + 17 = 21 {2, 3, 4, 5} Pull Pull

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143

Which of the following is a solution to the equation: m - 13 = 28 {39, 40, 41, 42} Pull Pull

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144

Which of the following is a solution to the equation: 3x + 5 = 32 {7, 8, 9, 10} Pull Pull

SLIDE 37

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Why are we moving on to Solving Equations? First we evaluated expressions where we were given the value

• f the variable and had which solution made the equation true.

Now, we are told what the expression equals and we need to find the value of the variable. When solving equations, the goal is to isolate the variable on

• ne side of the equation in order to determine its value (the

value that makes the equation true). This will eliminate the guess & check of testing possible solutions.

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To solve for "x" in the following equation... x + 7 = 32 Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides. x + 7 = 32

• 7 - 7

x = 25 To check your value of "x"... In the original equation, replace x with 25 and see if it makes the equation true. x + 7 = 32 25 + 7 = 32 32 = 32

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145

What is the inverse operation needed to solve this equation? 7x = 49 A Addition B Subtraction C Multiplication D Division Pull Pull

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146

What is the inverse operation needed to solve this equation? x - 3 = 12 A Addition B Subtraction C Multiplication D Division Pull Pull

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147

What is the inverse operation needed to solve this equation? A Addition B Subtraction C Multiplication D Division Pull Pull

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148

What is the inverse operation needed to solve this equation? A Addition B Subtraction C Multiplication D Division Pull Pull

SLIDE 38

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To solve equations, you must use inverse operations in order to isolate the variable on one side of the equation. Whatever you do to one side of an equation, you MUST do to the other side! +5 +5

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149

Solve. x + 6 = 11 Pull Pull

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150 Solve.

x - 13 = 54 Pull Pull

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151

Solve. j + 15 = 27 Pull Pull

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152 Solve.

x - 9 = 67 Pull Pull

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153

Solve. 115 = 5x Pull Pull

SLIDE 39

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154 Solve.

33 = 11m Pull Pull

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155

Solve. 48 = 12y Pull Pull

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156 Solve.

n = 13 6 Pull Pull

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An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:

Symbol Expression

Words

< A < B

A is less than B

> A > B

A is greater than B

< A < B

A is less than or equal to B

> A > B

A is greater than or equal to B

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157 Write an inequality for the sentence:

m is greater than 9 A m < 9 B m < 9 C m > 9 D m > 9 Pull Pull

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158 Write an inequality for the sentence:

12 is less than or equal to y A 12 < y B 12 < y C 12 > y D 12 > y Pull Pull

SLIDE 40

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159 Write an inequality for the sentence:

The grade, g, on your test must exceed 80% A g < 80 B g < 80 C g > 80 D g > 80 Pull Pull

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160 Write an inequality for the sentence:

y is not more than 25 A y < 25 B y < 25 C y > 25 D y > 25 Pull Pull

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Remember: Equations have one solution. Solutions to inequalities are NOT single numbers. Instead, inequalities will have more than one value for a solution.

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• 9
• 10

This would be read as, "The solution set is all numbers greater than or equal to negative 5."

Solution Sets Slide 238 / 305

Let's name the numbers that are solutions to the given inequality. r > 10 Which of the following are solutions? {5, 10, 15, 20} 5 > 10 is not true So, 5 is not a solution 10 > 10 is not true So, 10 is not a solution 15 > 10 is true So, 15 is a solution 20 > 10 is true So, 20 is a solution Answer: {15, 20} are solutions to the inequality r > 10

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161 Which of the following are solutions to the inequality:

x > 11 {9, 10, 11, 12} Select all that apply. A 9 B 10 C 11 D 12 Pull Pull

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162 Which of the following are solutions to the inequality:

m < 15 {13, 14, 15, 16} Select all that apply. A 13 B 14 C 15 D 16 Pull Pull

SLIDE 41

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163 Which of the following are solutions to the inequality:

x > 34 {32, 33, 34, 35} Select all that apply. A 32 B 33 C 34 D 35 Pull Pull

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Since inequalities have more than one solution, we show the solution two ways. The first is to write the inequality. The second is to graph the inequality on a number line. In order to graph an inequality, you need to do two things:

• 1. Draw a circle (open or closed) on the number that is your

boundary.

• 2. Extend the line in the proper direction.

Slide 243 / 305 Remember!

Closed circle means the solution set includes that number and is used to represent ≤ or ≥. Open circle means that number is not included in the solution set and is used to represent < or >. Extend your line to the right when your number is larger than the variable. # > variable variable < # Extend your line to the left when your number is smaller than the variable. # < variable variable > #

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164

This solution set graphed below is x > 4?

True

False

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Pull Pull

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• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

165

A

x > 3

B

x < 3

C

x < 3

D

x > 3 Pull Pull

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• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

166

A x > -1 B x < -1 C x < -1 D x > -1 Pull Pull

SLIDE 42

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167

A

x > 0

• 1
• 2
• 3
• 4
• 5

1 2 3 4 5

B x < 0 C x < 0 D x > 0 Pull Pull

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Geometry

Return to Table of Contents

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List what you remember about geometry.

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A = length(width) A = lw A = side(side) A = s2 The Area (A) of a rectangle is found by using the formula: The Area (A) of a square is found by using the formula:

Slide 251 / 305

168 What is the Area (A) of the figure?

13 ft 7 ft

Pull Pull

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169 Find the area of the figure below.

8

Pull Pull

SLIDE 43

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A = base(height) A = bh The Area (A) of a parallelogram is found by using the formula: Note: The base & height always form a right angle!

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170 Find the area.

10 ft 9 ft 11 ft

Pull Pull

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171 Find the area.

8 m 13 m 13 m 8 m 12 m

Pull Pull

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172 Find the area.

13 cm 12 cm 7 cm

Pull Pull

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The Area (A) of a triangle is found by using the formula: Note: The base & height always form a right angle!

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173 Find the area. 8 in

6 in 10 in 9 in

Pull Pull

SLIDE 44

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174 Find the area

14 m 9 m 10 m 12 m

Pull Pull

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The Area (A) of a trapezoid is also found by using the formula: Note: The base & height always form a right angle!

10 in 12 in 5 in

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175 Find the area of the trapezoid

by drawing a diagonal.

Pull Pull

9 m 11 m 8.5 m

Slide 262 / 305

176 Find the area of the trapezoid

using the formula. 20 cm 13 cm 12 cm

Pull Pull

Slide 263 / 305 Area of Irregular Figures

• 1. Divide the figure into smaller figures

(that you know how to find the area of)

• 2. Label each small figure and label the new lengths and

widths of each shape

• 3. Find the area of each shape
• 4. Add the areas
• 5. Label your answer

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Example: Find the area of the figure. 12 m 8 m 4 m 2 m 12 m 6 m 4 m 2 m

#1 #2

2 m

SLIDE 45

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Pull

177 Find the area.

4' 3'

1'

2' 10' 8' 5'

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178 Find the area.

12 10 13 20 25 10 Pull

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179 Find the area.

8 cm 18 cm 9 cm

Pull

Slide 268 / 305 Area of a Shaded Region

• 1. Find area of whole figure.
• 2. Find area of unshaded figure(s).
• 3. Subtract unshaded area from whole figure.
• 4. Label answer with units2.

Slide 269 / 305

Example Find the area of the shaded region. 8 ft 10 ft 3 ft 3 ft Area Whole Rectangle Area Unshaded Square Area Shaded Region

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180 Find the area of the shaded region.

11' 8' 3' 4' Pull

SLIDE 46

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181 Find the area of the shaded region.

16" 17" 15" 7" 5" Pull

Slide 272 / 305 3-Dimensional Solids

Categories & Characteristics of 3-D Solids: Prisms

• 1. Have 2 congruent, polygon bases which are parallel

to one another

• 2. Sides are rectangular (parallelograms)
• 3. Named by the shape of their base

Pyramids

• 1. Have 1 polygon base with a vertex opposite it
• 2. Sides are triangular
• 3. Named by the shape of their base

Cylinders

• 1. Have 2 congruent, circular bases which

are parallel to one another

• 2. Sides are curved

Cones

• 1. Have 1 circular bases with a vertex opposite it
• 2. Sides are curved

Slide 273 / 305

3-Dimensional Solids

Vocabulary Words for 3-D Solids: Polyhedron A 3-D figure whose faces are all polygons (Prisms & Pyramids) Face Flat surface of a Polyhedron Edge Line segment formed where 2 faces meet Vertex (Vertices) Point where 3 or more faces/edges meet Solid a 3-D figure Net a 2-D drawing of a 3-D figure (what a 3-D figure would look like if it were unfolded)

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182 Name the

figure.

A rectangular prism B

triangular prism

C

triangular pyramid

D

cylinder

E

cone

F square pyramid

Pull

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183 Name the figure.

A rectangular prism B

triangular prism

C

triangular pyramid

D

cylinder

E

cone

F square pyramid

Pull

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184 Name the figure.

A rectangular prism B

triangular prism

C

triangular pyramid

D

pentagonal prism

E

cone

F square pyramid

Pull

SLIDE 47

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185 Name the figure.

A rectangular prism B

triangular prism

C

triangular pyramid

D

pentagonal prism

E

cone

F square pyramid

Pull

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186 Name the figure.

A rectangular prism B

cylinder

C

triangular pyramid

D

pentagonal prism

E

cone

F square pyramid

Pull

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187 How many faces does a cube

have? Pull

Slide 280 / 305

188 How many vertices does a triangular prism

have? Pull

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189 How many edges does a square pyramid

have? Pull

Slide 282 / 305

6 in 2 in 7 in 7 in 2 in 2 in 6 in A net is helpful in calculating surface area. Simply label each section and find the area of each. #2 #4 6 in #1 #3 #5 #6

Surface Area

The sum of the areas of all outside faces of a 3-D figure. To find surface area, you must find the area of each face of the figure then add them together.

SLIDE 48

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7 in 2 in 2 in 6 in #2 #4 6 in #1 #3 #5 #6 #1 #2 #3 #4 #5 #6

Example

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190 Find the surface area of the figure given its net.

7 yd 7 yd 7 yd 7 yd

Pull Since all of the faces are the same, you can find the area

• f one face and multiply it

by 6 to calculate the surface area of a cube. What pattern did you notice while finding the surface area

• f a cube?

Slide 285 / 305

191 Find the surface area of the figure given its net.

9 cm 25 cm 12 cm Pull

Slide 286 / 305 Volume Formulas

Formula 1 V= lwh, where l = length, w = width, h = height Multiply the length, width, and height of the rectangular prism. Formula 2 V=Bh, where B = area of base, h = height Find the area of the rectangular prism's base and multiply it by the height.

Slide 287 / 305

Example Each of the small cubes in the prism shown have a length, width and height of 1/4 inch. The formula for volume is lwh. Therefore the volume of one of the small cubes is: Multiply the numerators together, then multiply the

• denominators. In other

words, multiply across. Forget how to multiply fractions?

Slide 288 / 305

192 Find the volume of the given figure.

Pull

SLIDE 49

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193 Find the volume of the given figure.

Pull

Slide 290 / 305

194 Find the volume of the given figure.

Pull

Slide 291 / 305

Ratios and Proportions

Return to Table of Contents

Slide 292 / 305

List what you remember about the ratios and proportions.

Slide 293 / 305

Ratio- A comparison of two numbers by division Ratios can be written three different ways: a to b a : b a b Each is read, "the ratio of a to b." Each ratio should be in simplest form.

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195 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and

the rest are strawberry. What is the ratio of vanilla cupcakes to strawberry cupcakes? A 7 : 9 B 7 27 C 7 11 D 1 : 3

SLIDE 50

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196

There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate & strawberry cupcakes to vanilla & chocolate cupcakes? A 20 16 B 11 7 C 5 4 D 16 20

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197

There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of total cupcakes to vanilla cupcakes? A 27 to 9 B 7 to 27 C 27 to 7 D 11 to 27

Slide 297 / 305

Equivalent ratios have the same value 3 : 2 is equivalent to 6: 4 1 to 3 is equivalent to 9 to 27 5 35 6 is equivalent to 42

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4 12 5 15 x 3 Since the numerator and denominator were multiplied by the same value, the ratios are equivalent There are two ways to determine if ratios are equivalent. 1. 4 12 5 15 x 3 4 12 5 15 Since the cross products are equal, the ratios are equivalent. 4 x 15 = 5 x 12 60 = 60

• 2. Cross Products

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198 4 is equivalent to 8

9 18 True False

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199 5 is equivalent to 30

9 54 True False

SLIDE 51

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Rate: a ratio of two quantities measured in different units Examples of rates: 4 participants/2 teams 5 gallons/3 rooms 8 burgers/2 tomatoes Unit rate: Rate with a denominator of one Often expressed with the word "per" Examples of unit rates: 34 miles/gallon 2 cookies per person 62 words/minute

Slide 302 / 305

Finding a Unit Rate

Six friends have pizza together. The bill is $63. What is the cost per person? Hint: Since the question asks for cost per person, the cost should be first, or in the numerator. $63 6 people Since unit rates always have a denominator of one, rewrite the rate so that the denominator is one. $63 6 6 people 6 $10.50 1 person The cost of pizza is $10.50 per person

Slide 303 / 305

200 Sixty cupcakes are at a party for twenty children. How many

cupcakes per person?

Slide 304 / 305

201 John's car can travel 94.5 miles on 3 gallons of gas. How

many miles per gallon can the car travel?

Slide 305 / 305

202 The snake can slither 24 feet in half a day. How many feet

can the snake move in an hour?