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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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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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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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Lesson objectives Teachers' notes
Adding Integers
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
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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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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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
The Positive , the Negative , the SUM
Add positive and negative integers to make 0.
1 1
3 + -5 = _____5) -2
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The Positive , the Negative , the SUM
Add positive and negative integers to make 0.
1 1
-6 + 2 = _____6) -4
The Positive , the Negative , the SUM
Add positive and negative integers to make 0.
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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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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Lesson objectives Teachers' notes
Subtracting Integers
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
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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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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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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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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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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3-5 Multiplying and Dividing Integers
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Lesson objectives Teachers' notes
Multiplying Integers
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
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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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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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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Lesson objectives Teachers' notes
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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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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Lesson objectives Teachers' notes
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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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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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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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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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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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