operations with Integers.notebooknreevesmath.weebly.com › uploads › 1 › 4 › 2 › 8 ›...

operations with Integers.notebook

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September 17, 2013

Operations with Integers

Lesson objectives Teachers' notes

Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between. CC.7.EE.3

Integers and Absolute Values


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4 ft4 ft/sec

4 ft/sec

positive


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120 ft

120 ft/sec

-120 ft/sec

negative

14 20 2 0 25 15

−16 ft/sec and 16 ft/sec

3

The object with the velocity of −4 feet per

second, because 4 > 3.


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September 17, 2013

Speed tells how fast an object is moving.

Velocity tells how fast it is moving and in

what direction.

How many units away from zero is -10 ?Answer10 units

DEFINITION: The number of units, or distance, between an integer and zero is called the absolute value. The absolute value of a number a is written as |a|.

0 +10 +20 +30102030

How many units away from zero is +10 ?Answer10 units


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The absolute value of an integer is never negative because it asks, "How far away is it from zero?"

0 5 15 2551525

Click on the arrow to help you decide if negative 15 and positive 15 are the same distance from zero.

Yes! No!

Finding Absolute Value

Use a number line to find each absolute value.

The absolute value of an integer is its distance from 0 on a number line.

3)

4)

Touch above the number line for help.


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September 17, 2013

What is the absolute value of each number . Drag it through the tunnel and see!

12| 12|

| - 15 | 15

| - 2.5 | 2.5

| -3| 3

| 10 | 10

| 0 | 0

Match pairs of EQUIVALENT absolute values.


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September 17, 2013


Adding Integers


Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts. CC.7.NS.1b

Apply properties of operations as strategies to add and subtract rational numbers. CC.7.NS.1d


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September 17, 2013

Modeling Integer Addition using a Number Line.

Add. Use a number line.

Start at 0. Move left 7 spaces. Then move left 4 more spaces.

Start at 0. Move left 12 spaces. Then move right 19 spaces.

1)

2)

Step 1Step 2

Step 1

Step 2

Start at 0. Move right 9 spaces. Then move right 13 spaces.

3)

4)

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

2 + (-5) = -3 Plus a negative 5

Positive 2

Start at 0. Move right 2 spaces. Then move 5 spots in the negative direction.

4 + (-4) = 0

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

positive 4

negative 4

5)

Start at 0. Move right 4 spaces. Then move 4 spots in the negative direction.

54321-5 -4 -3 -2 0-1

Plus a negative 5

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 10987654320 1

9 + 13 = 22Step 1 Step 2

Step 1

Step 2

Step 1Step 1

Step 1Step 2


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September 17, 2013

The Positive , the Negative , the SUM

obj: to add positive and negative integers

Place or to represent the first number. 1

2 All numbers added together represent 0.

3 Add or remove chips as indicated by each expression to verify that they are equivalent.

1 1

1 1

1 1

1 1

1 11 11 11 11 1


Add positive and negative integers to make 0.

1 1

3 + -5 = _____5) -2


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1 1

-6 + 2 = _____6) -4



1 1

-4 + -4 = _____7) -8


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Two numbers that are the same distance from 0, but on opposite sides of 0, are called opposites. For example, −3 and 3 are opposites.

Adding Integers Using Absolute Values

The signs are different.

Find the difference of the absolute values. Think: 8 4 = 4.

Use the sign of the integer with the greater absolute value (positive).

-4 + 8

4

Find the difference of the absolute values.

Think: 35 23 = 12.

23 + (-35)

-12

The signs are different.

Use the sign of the integer with the greater absolute value (positive).

9)

10)

To add integers, first determine the sign of each number. If the signs are the same, then add the numbers and keep the sign. If the signs are different, then subtract the absolute value of the numbers and keep the sign of the number with the greatest absolute value.


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September 17, 2013

Another way to think of it

7 + 9 = 16

4 + 2 = 6

5 + 3 = 8

Lose $7 Lose $9

Lose $4 Lose $2

Win $5 Win $3

Lose $16 overall

Lose $6 overall

Win $8 overall

The list shows four bank account transactions in July. Find the change C in the account balance.


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September 17, 2013


Subtracting Integers


Understand subtraction of rational numbers as adding the additive inverse, p q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts. CC.7.NS.1c

Apply properties of operations as strategies to add and subtract rational numbers. CC.7.NS.1d

3-4 Subtracting Integers

SubtractingIntegers

Add the Opposite


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September 17, 2013

1) 4 - 1 = 3

Modeling Integer Subtraction using a number line.

Use a number line to find each difference.

Start at 0. Move right 4 spaces. To subtract 1, move to the left.

2) -3 - 1 = -4

Start at 0. Move left 3 spaces. To subtract 1, move to the left.

Step 1

Step 2

Step 1Step 2

3) -2 - (-4) = 2

Start at 0. Move left 2 spaces. To subtract -4, move to the right.

Step 1

Step 2


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September 17, 2013

Using Tiles to Add & Subtract Integers

We can use Algebra Tiles to represent positive and negative numbers. Yellow = Positive, Red = Negative

What do we get when we have one positive and one negative?

+

HINT

HINT

Hint: It's called a Zero Pair Why do you think that is?+

What is 3 5 ?

+

+ Need a Zero Pair?

-8


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What is 6 (8) ?

+

+ Need a Zero Pair?

14

What is 4 7 ?

+

+ Need a Zero Pair?

-11


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What is 5 (1) ?

+

+ Need a Zero Pair?

6


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September 17, 2013

IN YOUR OWN WORDS How are adding integers and subtracting integers related?

Write a general rule for subtracting integers.

Answer

The Rule is...

Steps to solve:

1. Leave the first number alone.

2. Change the minus sign to a plus

3. Use the OPPOSITE of the second number

200 20

200 + 20

200 + (20)

SAME

CHANGE

OPPOSITE


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September 17, 2013

What's the Rule?

4 -5 = -1 4 + (- 5) = -1

-2 - (-3) = 1 -2 + 3 = 1

-14 - 5 = -19 -14 + (-5) = -19

12 - (-3) = 15 12 + 3 = 15


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September 17, 2013

3-5 Multiplying and Dividing Integers


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Multiplying Integers


Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts. CC.7.NS.2a

Apply properties of operations as strategies to multiply and divide rational numbers. CC.7.NS.2c

Lesson


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September 17, 2013

Multiplying Integers by Skip Counting

Find each product. Use a number line to model repeated addition and skip counting.

1)

2)


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Check Point


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September 17, 2013

The bar graph shows the number of taxis a company has in service. The number of taxis decreases by the same amount each year for four years. Find the total change in the number of taxis.

The bar graph shows that the number of taxis in service decreases by 50 each year. Use a model to solve the problem.


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September 17, 2013


Dividing Integers

Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts. CC.7.NS.2b

Apply properties of operations as strategies to multiply and divide rational numbers. CC.7.NS.2c

Solve realworld and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) CC.7.NS.3

-5

Use integer counters to find 15 3.

+


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September 17, 2013


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September 17, 2013

IN YOUR OWN WORDS Is the quotient of two integers positive, negative, orzero? How can you tell?

Write general rules for dividing (a) two integers with the same sign and(b) two integers with different signs.


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September 17, 2013


The Coordinate Plane

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts. CC.7.NS.1b


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September 17, 2013

Coordinate Grids

Coordinate grids are just 2 number lines put together.

10 2 3 4 5 6 7 8 9 1012345678910

10 2 3 4 5 6 7 8 9 1012345678910


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September 17, 2013

Find and touch 3 on the xaxis. P

ull

Pull

1 2 3 4 5 6 701234567

1

2

3

4

5

0

1

2

3

4

5

The xaxis is horizontal.

the xaxis


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September 17, 2013

Find and touch 2 on the yaxis. P

ull

Pull

1 2 3 4 5 6 701234567

1

2

3

4

5

0

1

2

3

4

5

The yaxis is vertical.

the yaxis

1 2 3 4 5 6 701234567

1

2

3

4

5

0

1

2

3

4

5

We can find any point on a coordinate plane using ordered pairs (also called coordinates).

The point on the yaxis always comes second.

The point on the xaxis always comes first.

(3,2)

Find and touch the ordered pair (4,3) P

ull

Pull


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September 17, 2013

1 2 3 4 5 6 701234567

1

2

3

4

5

0

1

2

3

4

5

A coordinate plane is divided into 4 quadrants.

Quadrant I

Quadrant III

Quadrant II

Quadrant IVClick here to go to explore this further. P

ull

Pull What patterns would you see in points in:

• Quadrant I? • Quadrant II? • Quadrant III? • Quadrant IV?

Locate the Aliens

Billy Bug and His All New Quest for Grub

Check out these websites for more practice plotting points on a coordinate plane.


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September 17, 2013

You Sank My Battleship!

yaxis

xaxis

2 1

43

yaxis

xaxis

2 1

43

yaxis

xaxis

2 1

43

yaxis

xaxis

2 1

43

yaxis

xaxis

2 1

43

,( (

,( (

,( (

,( (

,( (

x

y

5

5

5

5

1. Randomely choose x and y coordinates.2. Slide a to the quadrant or axis where the point would lie.3. Graph the point using a red dot.

(x, y) Coordinates Quadrant

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