slide 2 / 249 7th grade math number...

/ 249

7th Grade Math

Number System

www.njctl.org

2015-06-11

Slide 2 / 249

Number System Unit Topics

· Number System, Opposites & Absolute Value· Comparing and Ordering Rational Numbers· Adding Rational Numbers· Turning Subtraction Into Addition· Adding and Subtracting Rational Numbers Review· Multiplying Rational Numbers· Dividing Rational Numbers· Operations with Rational Numbers

Click on the topic to go to that section

· Converting Rational Numbers to Decimals· Glossary

Slide 3 / 249

http://www.njctl.org




page211svg

page50svg

page14svg

page51svg

page81svg

page90svg

page235svg

page100svg

page177svg

page155svg

Links to PARCC sample questions

Non-Calculator #10

Non-Calculator #6

Non-Calculator #14

Calculator #5

End of Year

Performance Based Assessment

Non-Calculator #12

Non-Calculator #4

Slide 4 / 249

Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number.

How many thirds are in 1 whole?

How many fifths are in 1 whole?

How many ninths are in 1 whole?

Vocabulary words are identified with a dotted underline.

The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall.

(Click on the dotted underline.)

Slide 5 / 249

Back to

Instruction

FactorA whole number that can divide into another number with no remainder.

15 3 5

3 is a factor of 153 x 5 = 15

3 and 5 are factors of 15

1635 .1R

3 is not a factor of 16

A whole number that multiplies with another number to make a third number.

The charts have 4 parts.

Vocab Word1

Its meaning 2

Examples/ Counterexamples

3Link to return to the instructional page.

4

(As it is used in the

lesson.)

Slide 6 / 249

page238svg

page243svg

page244svg

page246svg

page247svg

page250svg

Number System, Opposites &

Absolute Value

Return toTable ofContents

Slide 7 / 249

1 Do you know what an integer is?

Yes

No

Slide 8 / 249

Number System

0.22

Natural1,2,3...

Whole

0

Integer

...-4, -3, -2, -1

Rational

1/5

5/2

8.3

-2.756

-3/4

1/3

-1/11

Real

Irrational

Slide 9 / 249

page145svg

{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...}

Definition of Integer:

The set of whole numbers, their opposites and zero.

Define Integer

Examples of Integer:

Slide 10 / 249

Definition of Rational:

A number that can be written as a simple fraction

(Set of integers and decimals that repeat or terminate)

Define Rational

0, -5, 8, 0.44, -0.23,

Examples of rational numbers:

9 , ½

Slide 11 / 249

Slide 12 / 249

integer rational irrational

Classify each number as specific as possible:Integer, Rational or Irrational

5

-6

0

-21

-65

13.2

-6.329

2.34437 x 103½ ¾

3¾π5

Slide 13 / 249

-1 0-2-3-4-5 1 2 3 4 5

Rational Numbers on a Number Line

NegativeNumbers

PositiveNumbers

Numbers to the left of zero are less than zero

Numbers to the right of zero are greater than zero

Zero is neitherpositive or negative

`

Zero

Slide 14 / 249

-5 0

-3.212

12

4

5

-106

192

5.9

-1.12.9

16

Which of the following are examples of integers?(Touch all the integers)

Slide 15 / 249

2 Which of the following are examples of rational numbers?

A

B -3

C 10D 0.25E 75%

Slide 16 / 249

2 Which of the following are examples of rational numbers?

A

B -3

C 10D 0.25E 75%

[This object is a pull tab]

Ans

wer

A, B, C, D, E

Slide 16 (Answer) / 249

Numbers In Our World

Slide 17 / 249

You might hear "And the quarterback is sacked for a loss of 5 yards."

This can be represented as an integer: -5

Or, "The total snow fall this year has been 6 inches more than normal."

This can be represented as an integer: +6 or 6

Numbers can represent everyday situations

Slide 18 / 249

Can you think of anymore?

Words that represent positive and negative integers. gained

increased

up more

deposit

less

loss

underbelow

rose decreasewithdraw

Slide 19 / 249

Niko's grandmother put 20 dollars into his bank account. How would we show this integer?

A shark swims 30ft below sea level. How would we represent this as an integer?

20

-30

click

click

Slide 20 / 249

Niko's grandmother put 20 dollars into his bank account. How would we show this integer?

A shark swims 30ft below sea level. How would we represent this as an integer?

20

-30

click

click


Hin

t

REMEMBER:

When writing a positive integer it is not necessary to put a + in front of the integer.


1. Spending $6.75

2. Gain of 11 pounds

3. Depositing $700

4. 10 degrees below zero

5. 8 strokes under par (par = 0)

6. feet above sea level

Write a number to represent each situation:

Slide 21 / 249

1. Spending $6.75

2. Gain of 11 pounds

3. Depositing $700

4. 10 degrees below zero

5. 8 strokes under par (par = 0)

6. feet above sea level

Write a number to represent each situation:


Ans

wer

1. -6.752. 113. 7004. -105. -86. 350 2/3


3 Which of the following numbers best represents the following scenario:

The effect on your wallet when you spend $10.25.

A -10.25B 10.25C 0D +/- 10.25

Slide 22 / 249


The effect on your wallet when you spend $10.25.

A -10.25B 10.25C 0D +/- 10.25


Ans

wer

A


4 Which of the following integers best represents the following scenario:

Earning $50 shoveling snow.

A -50B 50C 0D +/- 50

Ans

wer

Slide 23 / 249


You dive feet to explore a sunken ship.

A

B

C 0

D

Ans

wer

Slide 24 / 249

Fractions and the Negative Sign

When we have a negative fraction, the negative sign can be in different places.

The following all are negative one-half.

Why are they all negative?

Slide 25 / 249

Fractions and the Negative Sign

These two fractions equal positive one-half.

Why are they both positive?

Slide 26 / 249

/ 249


Slide 28 / 249

(Answer) / 249

Slide 29 / 249


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

The numbers -4 and 4 are shown on the number line.

Both numbers are 4 units from 0, but 4 is to the right of 0 and -4 is to the left of zero.

The numbers -4 and 4 are opposites.

Opposites are two numbers which are the same distance from zero.

Opposites

Slide 30 / 249

9 What is the opposite of -7?A

nsw

er

Slide 31 / 249

10 What is the opposite of -3/2?

Ans

wer

Slide 32 / 249

11 What is the opposite of 18.2?

Ans

wer

Slide 33 / 249

What happens when you add two opposites?

Try it and see...

A number and its opposite have a sum of zero.

Numbers and their opposites are called additive inverses.

Click to Reveal

Slide 34 / 249

Jeopardy

Integers are used in game shows.

In the game of Jeopardy you:· gain points for a correct response· lose points for an incorrect response· can have a positive or negative score

Slide 35 / 249

When a contestant gets a $100 question correct: Score = $100

Then a $50 question incorrect: Score = $50

Then a $200 question incorrect: Score = -$150

How did the score become negative?

Let's take a look...

Slide 36 / 249

Let's organize our thoughts...

When a contestant gets a

$100 question correct

Then a $50 question incorrect

Then a $200 questionincorrect

Question Answered

Integer Representation

New Score

100 Correct

50Incorrect

200 Incorrect

-50

100 100

50

-150-200

Slide 37 / 249

Now you try...


$150 question incorrect

Then a $50 question incorrect

Then a $200 questioncorrect

Question Answered


New Score

150 Incorrect

50Incorrect

200 Correct

-50

-150 -150

-200

0200

Slide 38 / 249

Now you try...


$50 question incorrect

Then a $150 question correct

Then a $200 questionincorrect

Question Answered


New Score

Ans

wer

Slide 39 / 249

12 After the following 3 responses what would the contestants score be?

$100 incorrect $200 correct $50 incorrect A

nsw

er

Slide 40 / 249


$200 correct$50 correct $300 incorrect A

nsw

er

Slide 41 / 249


$150 incorrect$50 correct $100 correct

Ans

wer

Slide 42 / 249


$50 incorrect$50 incorrect $100 incorrect

Ans

wer

Slide 43 / 249


$200 correct$50 correct $100 incorrect

Ans

wer

Slide 44 / 249

· An integer is a whole number, zero or its opposite.

· A rational number is a number that can be written as a simple fraction.

· An irrational number is a number that cannot be written as a simple fraction.

· Number lines have negative numbers to the left of zero and then positive numbers to the right.

· Zero is neither positive nor negative.

· Numbers can represent real life situations.

To Review

Slide 45 / 249

Absolute Value of NumbersThe 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).

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

What is the distance from 0 to 5?

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

What is the distance from 0 to -5?

Slide 46 / 249

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

Absolute value is symbolized by two vertical bars

4

What is the 4 ?

This is read, "the absolute value of 4"

Slide 47 / 249

Remember

1 2 3 4 50-1-2-3-4-5

A number and its opposite have sum of zero. So ....

On a number line, a number and its opposite have the same distance from 0.

(Opposite numbers are on opposite sides from 0).

12341 2

34

-4 is 4 "jumps" from 0 4 is 4 "jumps" from zero

Both -4 and 4 are the same distance from zero

click

Slide 48 / 249

-4 = 4

-9 = 9

= 9.6|9.6|

Use the number line to find absolute value.

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

Moveto

check

Move to

check

Moveto

check

Slide 49 / 249

17 Find

Ans

wer

Slide 50 / 249

-818 Find

Ans

wer

Slide 51 / 249

19 What is A

nsw

er

Slide 52 / 249

20 What is

Ans

wer

Slide 53 / 249

21 Find

Ans

wer

Slide 54 / 249

22 What is the absolute value of the number shown in the generator?

Slide 55 / 249

23 Which numbers have 15 as their absolute value?

A -30B -15C 0

D 15

E 30

Ans

wer

Slide 56 / 249

24 Which numbers have 100 as their absolute value?

A -100B -50C 0D 50E 100

Ans

wer

Slide 57 / 249

Comparing and Ordering Rational

Numbers


Slide 58 / 249

To compare rational numbers, plot points on the number line.

The numbers farther to the right are larger.

The numbers farther to the left are smaller.

Use the Number Line

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

Slide 59 / 249

page145svg

Place the number tiles in the correct places on the number line.

4

-45

-3

-2

30

2 -5

-1

1

Slide 60 / 249

4-4 5-3 -2 30 2-5 -1 1

Now, can you see:

Which integer is largest?

Which is smallest?

Slide 61 / 249

4-4 5-3 -2 30 2-5 -1 1

Where do rational numbers go on the number line?

Go to the board and write in the following numbers:

Slide 62 / 249

Put these numbers on the number line.

-3

Which number is the largest? The smallest?

Slide 63 / 249

Comparing Positive Numbers

Numbers can be equal to; less than; or more than another number.

The symbols that we use are:

Equals "=" Less than "<" Greater than ">"

For example:

4 = 4 4 < 6 4 > 2

When using < or >, remember that the smaller side points at the smaller number.

Slide 64 / 249

25 10.5 is ______ 15.2.

A =B <C >

Ans

wer

Slide 65 / 249

26 7.5 is ______ 7.5

A =B <C > A

nsw

er

Slide 66 / 249

27 3.2 is ______ 5.7

A =B <C >

Ans

wer

Slide 67 / 249

Comparing Negative Numbers

The larger the absolute value of a negative number, the smaller the number. That's because it is farther from zero, but in the negative direction.

For example:

-4 = -4 -4 > -6 -4 < -2

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

Remember, the number farther to the right on a number line is larger.

Slide 68 / 249

Comparing Negative Numbers

One way to think of this is in terms of money. You'd rather have $20 than $10.

But you'd rather owe someone $10 than $20.

Owing money can be thought of as having a negative amount of money, since you need to get that much money back just to get to zero.

So owing $10 can be thought of as -$10.

Slide 69 / 249

28 -4.75 ______ -4.75

A =B <C >

Ans

wer

Slide 70 / 249

29 -4 ______ -5

A =B <C >

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

Ans

wer

Slide 71 / 249

30

A =B <C >

Ans

wer

Slide 72 / 249

31 -14.75 ______ -6.2

A =B <C >

Ans

wer

Slide 73 / 249

32 -14.2 ______ -14.3

A =B <C >

Ans

wer

Slide 74 / 249

Comparing All Numbers

Any positive number is greater than zero or any negative number.

Any negative number is less than zero or any positive number.

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

Slide 75 / 249

Drag the appropriate inequality symbol between the following pairs of numbers:

1) -3.2 5.8

3) 63 36

5) -6.7 -3.9

7) -24 -17

9) -8.75 -8.25

2) -237 -259

4) -10.2 -15.4

6) 127 172

8)

10) -10 -7

Slide 76 / 249

33 -17 ______ 12

A =B <C >

Ans

wer

Slide 77 / 249

34 -5.5 ______ 0

A =B <C >

Ans

wer

Slide 78 / 249

35 5.32 ______ 0

A =B <C >

Ans

wer

Slide 79 / 249

36 -4 ______ -9

A =B <C >

Ans

wer

Slide 80 / 249

37 1 ______ -54

A =B <C >

Ans

wer

Slide 81 / 249

38 -355.5 ______ 0

A =B <C >

Ans

wer

Slide 82 / 249

A thermometer can be thought of as a vertical number line. Positive numbers are above zero and negative numbers are below zero.

Slide 83 / 249

0

Ans

wer

If the temperature reading on a thermometer

is 90, what will the new reading be if the temperature:

falls 3 degrees?

rises 2 degrees?

falls 9 degrees?

Slide 84 / 249

39 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature rises 5 degrees?

Slide 85 / 249

39 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature rises 5 degrees?


Ans

wer

15ºC


40 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature falls 12 degrees?

Slide 86 / 249

40 If the temperature reading on a thermometer is 10º, what will the new reading be if the temperature falls 12 degrees?


Ans

wer

-2ºC


41 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature falls 3 degrees?

Slide 87 / 249



Ans

wer

-6ºC


42 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature rises 5 degrees?

Slide 88 / 249

42 If the temperature reading on a thermometer is -3º, what will the new reading be if the temperature rises 5 degrees?


Ans

wer

2ºC



Slide 89 / 249



Ans

wer

-15ºC


Adding Rational Numbers


Slide 90 / 249

page145svg

Symbols

We will use "+" to indicate addition and "-" for subtraction.

Parentheses will also be used to show things more clearly. For instance, if we want to add -3 to 4 we will write: 4 + (-3), which is clearer than 4 + -3.

Or if we want to subtract -4 from -5 we will write:-5 - (-4), which is clearer than -5 - -4.

Slide 91 / 249

While this section is titled "Addition", we're going to learn how to both add and subtract using the number line.

Addition and subtraction are inverse operations (they have the opposite effect). If you add a number and then subtract the same number you haven't changed anything.

Addition undoes subtraction, and vice versa.

Addition: A walk on the number line.

Slide 92 / 249

1. Start at zero 2. Walk the number of steps indicated by the first number.3. Walk the number of steps given by the second number. 4. Look down, you're standing on the answer.

Addition: A walk on the number line.

Rules for directions· Go to the right for positive numbers· Go to the left for negative numbers· Go in the opposite direction when subtracting, rather than adding, the second number· Subtracting a negative number means you move to the right: since that's the opposite of moving to the left

Here's how it works.

Slide 93 / 249

Let's do 3 + 4 on the number line.

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

1. Start at zero 2. Walk the number of steps indicated by the first number.3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer.

Slide 94 / 249


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


· Go to the right for positive numbers

Slide 95 / 249


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



Slide 96 / 249

Let's do -4 + (-5) on the number line.

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


Slide 97 / 249

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



· Go to the left for negative numbers

Slide 98 / 249

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




Slide 99 / 249

Let's do 5 + -7 on the number line.

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


Slide 100 / 249

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




Slide 101 / 249

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




Slide 102 / 249

Let's do -4 + 9.5 on the number line.

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


Slide 103 / 249

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




Slide 104 / 249

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




Slide 105 / 249

Addition: Using Absolute Values

You can always add using the number line.

But if we study our results, we can see how to get the same answers without having to draw the number line.

We'll get the same answers, but more easily.

Slide 106 / 249

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

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

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

3 + 4 = 7

-4 + 9.5 = 5.5

5 + (-7) = -2

-4 + (-5) = -9 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

We can see some patterns here that allow us to create rules to get these answers without drawing.


Slide 107 / 249

To add rational numbers with the same sign 1. Add the absolute value of the rational numbers. 2. The sign stays the same.

(Same sign, find the sum)

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-103 + 4 = 7

-4 + (-5) = -9 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

3 + 4 = 7; both signs are positive; so 3 + 4 = 7

4 + 5 = 9; both signs are negative; so -4 + (-5) = -9


Slide 108 / 249

Interpreting the Absolute Value Approach

The reason the absolute value approach works, if the signs of the numbers are the same, is:

The absolute value is the distance you travel in a direction, positive or negative.

If both numbers have the same sign, the distances will add together, since they're both asking you to travel in the same direction.

If you walk one mile west and then two miles west, you'll be three miles west of where you started.

Slide 109 / 249

To add rational numbers with different signs 1. Find the difference of the absolute values of the rational numbers. 2. Keep the sign of the integer with the greater absolute value.

(Different signs, find the difference)

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10-4 + 9.5 = 5.5

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-105 + (-7) = -2

9.5 - 4 = 5.5; 9.5 > 4, and 9.5 is positive; so -4 + 9.5 = 5.5

7 - 5 = 2; 7 > 5 and 7 is negative; so 5 + (-7) = -2


Slide 110 / 249

Interpreting the Absolute Value Approach

If the signs of the rational numbers are different:

For the 2nd leg of your trip you're traveling in the opposite direction of the 1st leg, undoing some of your original travel. The total distance you are from your starting point will be the difference between the two distances.

The sign of the answer must be the same as that of the larger number, since that's the direction you traveled farther.

If you walk one mile west and then two miles east, you'll be one mile east of where you started.

Slide 111 / 249

Adding Rational Numbers:

To add rational numbers with the same sign 1. Add the absolute value of the rational numbers. 2. The sign stays the same.

(Same sign, find the sum)

To add rational numbers with different signs 1. Find the difference of the absolute values of the rational numbers. 2. Keep the sign of the number with the greater absolute value.

(Different signs, find the difference)

Slide 112 / 249

44

11 + (-4) =A

nsw

er

Slide 113 / 249

45

-4 + (-4) =

Ans

wer

Slide 114 / 249

46

17 + (-20) =

Ans

wer

Slide 115 / 249

47

-15 + (-30) =

Ans

wer

Slide 116 / 249

48

-5 + 10 =

Ans

wer

Slide 117 / 249

49

-9 + (-2.3) =

Ans

wer

Slide 118 / 249

50

5.3 + (-8.4) =

Ans

wer

Slide 119 / 249

51

4.8 + (12.6) =

Ans

wer

Slide 120 / 249

52

-14.3 + 7.93 =

Ans

wer

Slide 121 / 249

53A

nsw

er

Slide 122 / 249

54

Ans

wer

Slide 123 / 249

55

-7.34 + (-8.21) =

Ans

wer

Slide 124 / 249

Turning Subtraction Into Addition


Slide 125 / 249

Subtracting Numbers

Subtracting a number is the same as adding it's opposite.

(Add a line,change the sign of the second number)

Slide 126 / 249

page145svg

Subtracting NumbersSubtracting a number is the same as adding its opposite.

We can see this from the number line, remembering our rules for directions. Compare these two problems: 8 - 5 and 8 + (-5).

For "8 - 5" we move 8 steps to the right and then move 5 steps to the left, since the subtraction sign tells us to move in the opposite direction that we would for +5.

For "8 + (-5)" we move 8 steps to the right, and then 5 steps to the left since we're adding -5.

Either way, we end up at +3.

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

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

Slide 127 / 249

Subtracting Negative NumbersCompare these two problems: 8 - (-2) and 8 + 2.

For "8 - (-2)" we move 8 steps to the right and then move 2 steps to the right, since the negative sign tells us to move in the opposite direction that we would for -2.

For "8 + 2" we move 8 steps to the right, and then 2 steps to the right since we're adding 2.

Either way, we end up at +10.

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

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

Slide 128 / 249

Subtracting Numbers

Any subtraction can be turned into addition by:

· Changing the subtraction sign to addition.

· Changing the integer after the subtraction sign to its opposite.

EXAMPLES:

5 - (-3) is the same as 5 + 3

-12 - 17 is the same as -12 + (-17)

Slide 129 / 249

56 Convert the subtraction problem into an addition problem.

8 – 4

A -8 + 4B 8 + (-4)

C -8 + (-4)D 8 + 4

Ans

wer

Slide 130 / 249


-3.7 - (-10.1)

A -3.7 + 10.1B 3.7 + (-10.1)C -3.7 + (-10.1)D 3.7 + 10.1

Ans

wer

Slide 131 / 249


A

B

C

D

Ans

wer

Slide 132 / 249


A

B

C

D

Ans

wer

Slide 133 / 249


1 - 9

A -1 + 9

B 1 + (-9)

C -1 + (-9)

D 1 + 9

Ans

wer

Slide 134 / 249

61 Which expressions are equivalent to

Select all that apply.

A

B

C

D

E

F From PARCC sample test

Slide 135 / 249

http://practice.parcc.testnav.com/#

61 Which expressions are equivalent to

Select all that apply.

A

B

C

D

E

F From PARCC sample test


Ans

wer

B & D


Adding and Subtracting Rational Numbers

Review


Slide 136 / 249

62 Calculate the sum or difference.

-6 – 2

Ans

wer

Slide 137 / 249


page145svg


5 + (-5)

Ans

wer

Slide 138 / 249


-10.5 + 6.2

Ans

wer

Slide 139 / 249


7.3 – (-4)

Ans

wer

Slide 140 / 249


Ans

wer

Slide 141 / 249


9.27 + (-8.38)A

nsw

er

Slide 142 / 249


-4.2 + (-5.9)

Ans

wer

Slide 143 / 249


-2 – (-3.95)

Ans

wer

Slide 144 / 249


5 - 6 + (-7)

Ans

wer

Slide 145 / 249


19 + (-12) - 11

Ans

wer

Slide 146 / 249


-2.3 + 4.1 + (-12.7)

Ans

wer

Slide 147 / 249


-8.3 - (-3.7) + 5.2

Ans

wer

Slide 148 / 249


Ans

wer

Slide 149 / 249

75 Two numbers, n and p are plotted on the number line shown.

The numbers n - p, n + p, and p - n will be plotted on the number line. Select an expression from each group to make this statement true.

The number with the least value is _____,

and the number with the greatest value is _____.

A n - p B n + p C p - n

D n - p E n + p F p - n

From PARCC sample test

Slide 150 / 249

75 Two numbers, n and p are plotted on the number line shown.

The numbers n - p, n + p, and p - n will be plotted on the number line. Select an expression from each group to make this statement true.

The number with the least value is _____,

and the number with the greatest value is _____.

A n - p B n + p C p - n

D n - p E n + p F p - n



Ans

wer A n - p is the least value

F p - n is the greatest value


Multiplying Rational Numbers


Slide 151 / 249



page145svg

Symbols

In the past, you may have used "x" to indicate multiplication. For example "3 times 4" would have been written as 3 x 4.

However, that will be a problem in the future since the letter "x" is used in algebra as a variable.

There are two ways we will indicate multiplication: 3 times 4 will be written as either 3∙4 or 3(4).

Slide 152 / 249

Parentheses

The second method of showing multiplication, 3(4), is to put the second number in parentheses.

Parentheses have also been used for other purposes. When we want to add -3 to 4 we will write that as 4 + (-3), which is clearer than 4 + -3.

Also, whatever operation is in parentheses is done first. The way to write that we want to subtract 4 from 6 and then divide by 2 would be (6 - 4) ÷ 2 = 1. Removing the parentheses would yield 6 - 4 ÷ 2 = 4, since we work from left to right.

Slide 153 / 249


Multiplication is just a quick way of writing repeated additions.

These are all equivalent:

3·43 +3 + 3 + 3 4 + 4 + 4

they all equal 12.

Slide 154 / 249

We know how to add with a number line.

Let's just do the same thing with multiplication by just doing repeated addition.

To do that, we'll start at zero and then just keep adding: either 3+3+3+3 or 4+4+4.

We should get the same result either way, 12.


Slide 155 / 249

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

Let's do 4 x 3 on the number line.

11 1312 14 16 1715

We'll do it as 3+3+3+3 and as 4+4+4

Slide 156 / 249

10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715

Try 5 x 2 on the number line.

Try it as 5+5 and as 2+2+2+2+2

Slide 157 / 249

Multiplying Negative Numbers

Let's use the same approach to determine rules for multiplying negative numbers.

If we have 4 x (-3) we know we can think of that as (-3) added to itself 4 times. But we don't know how to think of adding 4 to itself -3 times, so let's just get our answer this way:

4 x (-3) = (-3)+(-3)+(-3)+(-3)

Slide 158 / 249

10 2 3-17 -1-2-3-4-5-6-7-8-9-10-11-13-15-16 -14 -12

4 x (-3) On the Number Line

4 x (-3) = (-3)(-3)(-3)(-3)

So we can see that 4 x (-3) = -12

Slide 159 / 249

4∙34 + 4 + 412

4(-3)(-3) + (-3) + (-3)-12

Multiplying positive numbers has a positive value.

Multiplying a negative number and a positive number has a negative value.

What about multiplying together two negative numbers: what is the sign of (-4)(-3)

Sign Rules for Multiplying Rational Numbers

?

Slide 160 / 249

Multiplying Negative Numbers

We can't add something to itself a negative number of time; we don't know what that means.

But we can think of our rule from earlier, where a (-) sign tells us to reverse direction.

So if we think of (-4)(-3) as -(4)(-3) we can then see that the answer will be the opposite of (-12):12

Each negative sign makes us reverse direction once, so two multiplied together gets us back to the positive direction.

Slide 161 / 249

4∙34 + 4 + 412

4(-3)(-4) + (-4) + (-4)-12

Multiplying positive numbers yields a positive result.

Multiplying a negative number and a positive number yields a negative result.

Multiplying two negative numbers together yields a positive result.

Sign Rules forMultiplying Rational Numbers

(-4)(-3)-((-4) + (-4) + (-4))-(-12)12

Slide 162 / 249

Every time you multiply by a negative number you change the sign.

Multiplying with one negative number makes the answer negative.

Multiplying with a second negative change the answer back to positive.

1(-3) = -3 -3(-4) = 12


Slide 163 / 249

When multiplying two numbers with the same sign (+ or -), the product is positive.

When multiplying two numbers with different signs, the product is negative.

When multiplying several numbers with different signs, count the number of negatives. An even amount of negatives = positive product An odd amount of negatives = negative product


Slide 164 / 249

We can also see these rules when we look at the patterns below:

3(3) = 9 -5(3) = -153(2) = 6 -5(2) = -103(1) = 3 -5(1) = -53(0) = 0 -5(0) = 03(-1) = -3 -5(-1) = 53(-2) = -6 -5(-2) = 103(-3) = -9 -5(-3) = 15

2.5(3) = 7.5 -3.1(3)(-2) = 18.62.5(2) = 5 -3.1(2)(-2) = 12.42.5(1) = 2.5 -3.1(1)(-2) = 6.22.5(0) = 0 -3.1(0)(-2) = 02.5(-1) = -2.5 -3.1(-1)(-2) = -6.22.5(-2) = -5 -3.1(-2)(-2) = -12.42.5(-3) = -7.5 -3.1(-3)(-2) = -18.6


Slide 165 / 249

76 What is the value of (-3)(-9)?

Ans

wer

Slide 166 / 249

77 What is the value of 5(-4.82)?

Ans

wer

Slide 167 / 249

78 What is the value of (-3.2)(-6.4)? A

nsw

er

Slide 168 / 249

79 What is the value of (-5.12)(-9)?

Ans

wer

Slide 169 / 249

80 What is the value of -2(-7)(-4)?

Ans

wer

Slide 170 / 249

81 What is the value of: A

nsw

er

Slide 171 / 249

82 What is the value of:

Ans

wer

Slide 172 / 249

83 Jane has entered a baking contest. Jane uses 3.1 ounces of flour to make one cinnamon roll. How many ounces of flour does Jane need to make 7 cinnamon rolls? A

nsw

er

Slide 173 / 249

84 Timmy is shipping 4 boxes of shirts. Each box weighs 6.3 pounds. If cost 5.20 per pound to ship. How much does Timmy have to spend?

Ans

wer

Slide 174 / 249

Dividing Rational Numbers


Slide 175 / 249

page145svg

Division Symbols

You may have mostly used the "÷ " symbol to show division.

We will also represent division as a fraction. Remember: 9 9÷ 3 = 33

are both ways to show division.

= 3

Slide 176 / 249


Division is the opposite of multiplication, just like subtraction is the opposite of addition.

When you divide a number, by another number, you are finding out how many of that second number would have to add together to get the first number.

For instance, since 5∙2 = 10, that means that I could divide 10 into 5 groups of 2's, or 2 groups of 5's.

This is just what we did on the number line for multiplication, but backwards.

Let's try 10 ÷ 2

Slide 177 / 249

10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715

Try 10 ÷ 2 on the number lineThis means how many lengths of 2 would be needed to add up to 10.

The answer is 5: the number of red arrows of length 2 that end to end give a total length of 10.

Slide 178 / 249

10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715

Try 10 ÷ 5 on the number lineThis means how many lengths of 5 would be needed to add up to 10.

The answer is 2: the number of green arrows of length 5 that, end to end, give a total length of 10.

Slide 179 / 249

10 2 3-17 -1-2-3-4-5-6-7-8-9-10-11-13-15-16 -14 -12

-12 ÷ 3 On the Number LineThis can be read as how many steps of 3 would it take to get to -12.

Each red arrow represents a step of 3, so we can see that -12 ÷ 3 = -4 (The answer is

negative because the steps are to the left.)

Slide 180 / 249

-15 ÷ 3 = -5

We know that -5(3) = -15, so it makes sense that -15 ÷ 3 = -5.

We also know 3(-5) = -15. So, what is the value of -15 ÷ -5

The value must be positive 3, because 3(-5) = -15

-153 = -5


Slide 181 / 249

The quotient of two positive numbers is positive

The quotient of a positive and negative number is negative.

The quotient of two negative numbers is positive.

When dividing several numbers with different signs, count the number of negatives. An even amount of negatives = positive quotient An odd amount of negatives = negative quotient


Slide 182 / 249

85 Find the value of 32 ÷ 4A

nsw

er

Slide 183 / 249

86 Find the value of:

Ans

wer

Slide 184 / 249


Ans

wer

Slide 185 / 249

88 Find the value of:A

nsw

er

Slide 186 / 249


Ans

wer

Slide 187 / 249


Ans

wer

Slide 188 / 249

91 Find the value of:A

nsw

er

Slide 189 / 249


Ans

wer

Slide 190 / 249

93 Kobe put 8 toy cars in a row. The line of cars was 16.4 meters long. How long was each car?

Ans

wer

Slide 191 / 249

94 Olivia squeezed 3/4 of a gallon of orange juice. She split the orange juice equally into 6 cups. How many gallons was in each cup?

Ans

wer

Slide 192 / 249

95 In which situation could the quotient of -24 3 be used to answer the question?A The temperature of a substance decreased by 24

degrees per minute for 3 minutes. What was the overall change of the temperature of the substance?

B A football team loses 24 yards on one play, then gains 3 yards on the next play. How many total yards did the team gain on the two plays?

C Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day?

D A cookie jar contains 24 cookies. Each child receives 3 cookies. How many children are there?


Slide 193 / 249


95 In which situation could the quotient of -24 3 be used to answer the question?A The temperature of a substance decreased by 24

degrees per minute for 3 minutes. What was the overall change of the temperature of the substance?

B A football team loses 24 yards on one play, then gains 3 yards on the next play. How many total yards did the team gain on the two plays?

C Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day?

D A cookie jar contains 24 cookies. Each child receives 3 cookies. How many children are there?



Ans

wer

C


Operations with Rational Numbers


Slide 194 / 249

When simplifying expressions with rational numbers,

you must follow the order of operations whileremembering your rules for

positive and negative numbers!

Slide 195 / 249


page145svg

Order of Operations

ParenthesesExponentsMultiplicationDivisionAdditionSubtraction

Complete at the same time...whichever comes first...from left to right

(ALL Grouping Symbols)

Slide 196 / 249

Let's simplify this step by step...

What should you do first?

5 - (-2) = 5 + 2 = 7

What should you do next?

(-3)(7) = -21

What is your last step?

-7 + (-21) = -28

-7 + (-3)[5 - (-2)]

Slide 197 / 249

Let's simplify this step by step...

What should you do first? What should you do second?

What should you do third? What should you do last?

Clickto

Reveal

Clickto

Reveal

Clickto

Reveal

Clickto

Reveal

Slide 198 / 249

96 Simplify the expression.

-12÷ 3(-4)

Ans

wer

Slide 199 / 249

97 Simplify the expression.[-1 - (-5)] + [7(3 - 8)]

Ans

wer

Slide 200 / 249

98 Simplify the expression.40 - (-5)(-9)(2)

Ans

wer

Slide 201 / 249

99 Simplify the expression.5.8 - 6.3 + 2.5

Ans

wer

Slide 202 / 249

100 Simplify the expression.-3(-4.7)(5-3.2)

Ans

wer

Slide 203 / 249


Ans

wer

Slide 204 / 249


Ans

wer

Slide 205 / 249

103 Complete the first step of simplifying. What is your answer?

[3.2 + (-15.6)] - 6[4.1 - (-5.3)]

Ans

wer

Slide 206 / 249

104 Complete the next step of simplifying. What is your answer?

[3.2 + (-15.6)] - 6[4.1 - (-5.3)]

-12.4 - 6[4.1 - (-5.3)]click to reveal step from previous slide

Ans

wer

Slide 207 / 249


[3.2 + (-15.6)] - 6[4.1 - (-5.3)]

-12.4 - 6[9.4]

-12.4 - 6[4.1 - (-5.3)]click to reveal steps from previous slides

Ans

wer

Slide 208 / 249


[3.2 + (-15.6)] - 6[4.1 - (-5.3)]

-12.4 - 56.4-12.4 - 6[9.4]

-12.4 - 6[4.1 - (-5.3)]click to reveal steps from previous slides

Ans

wer

Slide 209 / 249

Slide 210 / 249


Ans

wer

Slide 211 / 249

109A

nsw

er

Slide 212 / 249

110

Ans

wer

Slide 213 / 249

111(-4.75)(3) - (-8.3)

Ans

wer

Slide 214 / 249

Solve this one in your groups.A

nsw

er

Slide 215 / 249

How about this one?

Ans

wer

Slide 216 / 249

112

Ans

wer

Slide 217 / 249

113[(-3.2)(2) + (-5)(4)][4.5 + (-1.2)]

Ans

wer

Slide 218 / 249

114

Ans

wer

Slide 219 / 249

115

Ans

wer

Slide 220 / 249

116A

nsw

er

Slide 221 / 249

117 Select the correct number from each group of numbers to complete the equation.

A 2

B -2

C 3/4

D -4/3

E 2

F -2

G 4/3

H -3/4

_____ _____


Slide 222 / 249


117 Select the correct number from each group of numbers to complete the equation.

A 2

B -2

C 3/4

D -4/3

E 2

F -2

G 4/3

H -3/4

_____ _____



Ans

wer A 2

H -3/4


Slide 223 / 249



Converting Rational Numbers to

Decimals


Slide 224 / 249

Definition of Rational:

A number that can be written as a simple fraction

(Set of integers and decimals that repeat or terminate)

In order for a number to be rational, you should be able to divide the fraction and have the decimal either terminate or repeat.

Do you recall the definition of a Rational Number?

Slide 225 / 249

Use long division!

Divide the numerator by the denominator.

If the decimal terminates or repeats, then you have a rational number.

If the decimal continues forever, then you have an irrational number.

How can you convert Rational Numbers into Decimals?

Slide 226 / 249

page145svg

Long Division Review

Slide 227 / 249

119 Convert to a decimal.

(If the number is repeating, use bar notation in your notebook by entering the repeating number(s) 3 times on your responder.)

Ans

wer

Slide 228 / 249


(If the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder.) A

nsw

er

Slide 229 / 249


(If the number is repeating, use bar notation in your notebook but enter the repeating number(s) 3 times on your responder.)

Ans

wer

Slide 230 / 249



Ans

wer

Slide 231 / 249



nsw

er

Slide 232 / 249



nsw

er

Slide 233 / 249



Ans

wer

Slide 234 / 249



Ans

wer

Slide 235 / 249

127 Mike needed meters of fabric to fix his couch. How can this be written as a decimal?

Ans

wer

Slide 236 / 249

128 Hannah rode her bike miles to the neighborhood pool. What is the distance she rode her bike as a decimal?

Ans

wer

Slide 237 / 249

129 Kevin Durant made shots in the first quarter of the NBA finals, how is that written as a decimal?

Ans

wer

Slide 238 / 249

130 Coral's mother wants to know show as a decimal. How can that fraction be written as a decimal?

Ans

wer

Slide 239 / 249

Glossary


Slide 240 / 249

Back to

Instruction

Absolute Value

How far a number is from zero on the number line.

-2 = 22 = 2

0 1 2-2 -10 1 2-2 -1

0 = 0

0 1 2-2 -1

Slide 241 / 249

page145svg

page231svg

Back to

Instruction

IntegersZero, all whole numbers

and their opposites.

... -1, 0, 1... 35

2.3

Slide 242 / 249

Back to

Instruction

Irrational Numbers

Any number that cannot be made by dividing one

integer by another.

The decimal form of a number that goes on forever

without repeating.

= 3.14159...??

(no ratio).75 = 34

= 2.718281...??

(no ratio)

Slide 243 / 249

Back to

Instruction

Opposites

-3 and 3 -5 and 5

Two numbers which are the same distance from zero.

-(3) = -3 -(-3) = 3

-3 and 3

Slide 244 / 249

page158svg

page36svg

page138svg

Back to

Instruction

Rational Numbers

Any number that can be made by dividing one integer by another.

ab .75 = 34

= 3.14159...

= ??

(no ratio)

ratio

Slide 245 / 249

Back to

Instruction

Slide 246 / 249

Back to

Instruction

Slide 247 / 249

page179svg

page231svg

page231svg

Back to

Instruction

Slide 248 / 249

Back to

Instruction

Slide 249 / 249

page231svg

page231svg

slide 2 / 249 7th grade math number...

Documents