illustrating integers. introduction to integers - objectives you will be able to 1. state the...

ILLUSTRATING ILLUSTRATING INTEGERSINTEGERS

INTRODUCTION TO INTRODUCTION TO INTEGERS - ObjectivesINTEGERS - Objectives

You will be able to

1. State the coordinate of a point on a number line.

2. Graph integers on a number line.

3. Add and subtract integers.

INTRODUCTION TO INTEGERSINTRODUCTION TO INTEGERS

Integers are positive and negative numbers.Integers are positive and negative numbers.

……, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, …, -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, …

Each negative number is paired with a positive Each negative number is paired with a positive number the same distance from 0 on a number number the same distance from 0 on a number line.line.

-3 -2 -1 20 1 3

INTRODUCTION TO INTEGERSINTRODUCTION TO INTEGERS

We can represent integers using red and We can represent integers using red and yellow counters.yellow counters.

Red tiles will represent negative integers, Red tiles will represent negative integers, and yellow tiles will represent positive and yellow tiles will represent positive integers.integers.

Negative integer Positive integer

INTRODUCTION TO INTEGERSINTRODUCTION TO INTEGERS

The diagrams below show 2 ways to The diagrams below show 2 ways to represent -3.represent -3.

Represent -3 in 2 more ways.Represent -3 in 2 more ways.

-3 -3

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INTRODUCTION TO INTEGERSINTRODUCTION TO INTEGERS

Students may represent -3 by using 3 red Students may represent -3 by using 3 red tiles and any number of groups of +1 and -1 tiles and any number of groups of +1 and -1 tilestiles

INTRODUCTION TO INTEGERSINTRODUCTION TO INTEGERS

Tell which integer each group of tiles represents.Tell which integer each group of tiles represents.

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-2 +1

ANSWER+2

INTRODUCTION TO INTEGERSINTRODUCTION TO INTEGERS

If there are the same number of red tiles as If there are the same number of red tiles as yellow tiles, what number is represented?yellow tiles, what number is represented?

It represents 0.It represents 0.

ADDITION AND ADDITION AND SUBTRACTIONSUBTRACTION

ADDING INTEGERSADDING INTEGERS

We can model integer addition with tiles.We can model integer addition with tiles.

Represent -2 with the fewest number of Represent -2 with the fewest number of tilestiles

Represent +5 with the fewest number of Represent +5 with the fewest number of tiles.tiles.

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ADDING INTEGERSADDING INTEGERS

What number is represented by combining What number is represented by combining the 2 groups of tiles?the 2 groups of tiles?

Write the number sentence that is Write the number sentence that is illustrated.illustrated.

-2 + 5 = 3-2 + 5 = 3

+3

ADDING INTEGERSADDING INTEGERS

Use your red and yellow tiles to find each sum.Use your red and yellow tiles to find each sum.

-2 + (-3) = ?-2 + (-3) = ?

ANSWER

+ = -5

-2 + (-3) = -2 + (-3) = --55

ADDING INTEGERSADDING INTEGERS -6 + 2 = ?

+ = - 4

+ = +1

-3 + 4 = ?

ANSWER

ANSWER

--6 + 2 = 6 + 2 = --44

--3 + 4 = 3 + 4 = ++11

Shortcut for Addition

How about adding the large numbers?

Ex. 100 + (-27) =

100 + 27

73 + 27 + 27 = 73

What is the rule for Addition?

More or more ?

More positive

More negative

Rules for addition

(+) + (+) = (+)

(-) + (-) = (-)

(+) + (-) = (+) or (-)

(-) + (+) = (+) or (-)

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

We often think of subtraction as a “take away” We often think of subtraction as a “take away” operation.operation.

Which diagram could be used to computeWhich diagram could be used to compute

++3 - 3 - ++5 = ?5 = ?

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+3 +3

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

We can’t take away 5

yellow tiles from this

diagram. There is not

enough tiles to take

away!!

This diagram also represents

+3, and we can take away +5.

• When we take 5 yellow tiles

away, we have 2 red tiles left.

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

Use your red and yellow tiles to model each Use your red and yellow tiles to model each subtraction problem.subtraction problem.

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--2 – (2 – (--4) = ?4) = ?

ANSWER

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

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This representation

of -2 doesn’t have

enough tiles to

take away -4.

Now if you add 2 more reds

tiles and 2 more yellow tiles

(adding zero) you would

have a total of 4 red tiles and

the tiles still represent -2.

Now you can take away 4 red

tiles.

-2 - -4 = 2

2 yellow tiles are left, so the

answer is…

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

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Work this problem.Work this problem.

++3 – (3 – (--5) = ?5) = ?ANSWER

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

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• Add enough red and yellow pairs so you can take away 5 red tiles.

• Take away 5 red tiles, you have 8 yellow tiles left.

3 – (-5) = +8

Shortcut

-4 – (-7) =

5 – (-1) =

-3 – (-6) =

What pattern do you see?

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

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Work this problem.Work this problem.

--3 - 2 = ?3 - 2 = ?ANSWER

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

•Add two pairs of red and yellow tiles so you can take away 2 yellow tiles.

• Take away 2 yellow tiles, you have 5 red tiles left.

-3 - 2 = -5

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

A fact family gives 4 true equations using A fact family gives 4 true equations using the same 3 numbers.the same 3 numbers.

For example: For example:

7 + 8 = 157 + 8 = 158 + 7 = 158 + 7 = 1515 – 7 = 815 – 7 = 815 – 8 = 715 – 8 = 7

All of these statements are true.

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

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We can also use fact family with integers.We can also use fact family with integers.

Use your red and yellow tiles to verify this Use your red and yellow tiles to verify this fact family:fact family:--3 + 8 = 53 + 8 = 5

8 + (8 + (--3) = 53) = 5

5 - 8 = 5 - 8 = --33

5 – (5 – (- - 3) = 83) = 8

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

How would you explain to another student how to subtract positive and negative integers? (Use words/pictures in your explanation. Make sure to label your pictures!)

Ex. 4 – 7 = -3

Ex. 2 – (-6) = 8

Rules for subtraction

Adding the opposite! -2 – (-5)= -2 + 5 (follow the addition rules)

Ex. 3 – 9 =Ex. 7 – (-10) = Ex. -6 – 5 = Ex. -8 – 2 =

More on SubtractionMore on Subtraction4 – 9 =4 – 9 =

3 – 10 = 3 – 10 =

7 – 14 =7 – 14 =

1 – 5 =1 – 5 =

-4 – (-2) =-4 – (-2) =

-5 – (-6) = -5 – (-6) =

-9 – (-1) =-9 – (-1) =

-2 – (-5) =-2 – (-5) =

-3 – 6 =-3 – 6 =

-7 – 3 = -7 – 3 =

-8 – 2 =-8 – 2 =

-1 – 5 =-1 – 5 =

9 – 4 =9 – 4 =

10 - 3 = 10 - 3 =

14 -7 =14 -7 =

5 - 1 =5 - 1 =

INTEGERS AND INTEGERS AND MULTIPLICATIONMULTIPLICATION

MULTIPLICATIONMULTIPLICATION

Red and yellow tiles can be used to model Red and yellow tiles can be used to model multiplication. multiplication.

Remember that multiplication can be described Remember that multiplication can be described as repeated addition.as repeated addition.

So 2 x 3 = ?So 2 x 3 = ?

2 groups of 3 tiles = 6 tiles

MULTIPLICATIONMULTIPLICATION

2 x 2 x --3 means 2 groups of 3 means 2 groups of --33

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2 x x --33 = -6

MULTIPLICATIONMULTIPLICATION

Since 2 x 3 = 6 and 3 x 2 = 6, does it make Since 2 x 3 = 6 and 3 x 2 = 6, does it make sense that sense that --3 x 2 = 3 x 2 = --6 ?6 ?

--3 x 2 = 3 x 2 = --6 is true. 6 is true. ++2 x 2 x --3 = 3 = --6 and 6 and --3 x 3 x ++2 = 2 = --6 belong to a fact 6 belong to a fact family:family:

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++2 x 2 x --3 = 3 = --66--3 x 3 x ++2 = 2 = --66--6 6 ÷ ÷ ++2 = 2 = --33--6 6 ÷÷ -3 = +2

MULTIPLICATIONMULTIPLICATION

Use your tiles to model each multiplication Use your tiles to model each multiplication problem.problem.++2 x 2 x ++4 = ?4 = ?

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ANSWER2 groups of +4

++2 x 2 x ++4 = 4 = ++88

MULTIPLICATIONMULTIPLICATION

++3 x 3 x --3 = ?3 = ?

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ANSWER

3 groups of -3

++3 x 3 x --3 = 3 = --99

MULTIPLICATIONMULTIPLICATION--2 x 2 x ++4 = ?4 = ?

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ANSWER4 groups of -2

--2 x 2 x ++4 = 4 = --88

Use the fact family forUse the fact family for--2 x 2 x ++4 = ? 4 = ? We can’t show We can’t show --2 groups of 2 groups of ++44++4 x 4 x --2 = ?2 = ? we can show 4 groups of we can show 4 groups of --22

MULTIPLICATIONMULTIPLICATION

++1, 1, --1 are opposites1 are opposites

the products are the products are oppositeopposite

Since Since ++2 and 2 and --2 are opposites of each other,2 are opposites of each other,

++2 x 2 x --3 and 3 and --2 x 2 x --3 have opposite products.3 have opposite products.

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++1 x 1 x ++3 = 3 = ++33--1 x 1 x ++3 = 3 = --33

MULTIPLICATIONMULTIPLICATION

To model To model --2 x 2 x --3 use 2 groups of the 3 use 2 groups of the opposite of opposite of --33

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--2 x 2 x --3 = 3 = ++66

MULTIPLICATIONMULTIPLICATION

Use tiles to illustrate Use tiles to illustrate --3 x 3 x --4 =?4 =?

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ANSWER

3 groups of +4 (the opposite of -4)

--3 x 3 x --4 = 4 = ++1212

Rules for Multiplication

(+) (+) = (+)(-) (-) = (+)(+)(-) = (-)(-)(+) = (-)

INTEGERS AND INTEGERS AND DIVISIONDIVISION

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DIVISIONDIVISION

Use tiles to model Use tiles to model ++12 12 ÷ ÷ ++3 = ?3 = ?

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Divide 12 yellow tiles into 3 equal

groups

++12 ÷12 ÷ ++3 = 3 = ++44

4 yellow tiles in each group.

DIVISIONDIVISION

Use tiles to model Use tiles to model --15 15 ÷ ÷ ++5 =?5 =?

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Divide -15 into 5 equal groups

--15 ÷15 ÷ ++5 = 5 = --33

DIVISIONDIVISION

Write the fact familyWrite the fact family--4 x 4 x ++2 = 2 = --88Compare your answer.Compare your answer.++2 x 2 x --4 = 4 = --88--8 8 ÷ ÷ ++2 = 2 = --44--8 ÷ 8 ÷ --4 = 4 = ++22

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DIVISIONDIVISION

Now try these! Use red and yellow tiles to find Now try these! Use red and yellow tiles to find each answer.each answer.

1) 1) --2 + 2 + --88 5) 5) ++3 x 3 x --44

2) 2) ++6 + 6 + --44 6) 6) --5 x 5 x --22

3) 3) --5 - 5 - ++44 7) 7) --12 12 ÷ ÷ ++66

4) 4) --6 - 6 - --77 8) 8) --8 8 ÷ ÷ ++22

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ANSWER

DIVISIONDIVISION

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1) -2 + -8 = -10

2) +6 + -4 = +2

DIVISIONDIVISION

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3) -5 - +4 = -9

4) -6 - -7 = +1

DIVISIONDIVISION

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5) +3 x -4 = -12

6) -5 x -2 = +105 groups of the opposite

of -2

DIVISIONDIVISION

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7) -12 ÷ +6 = -2

8) -8 ÷ +2 = -4 -8 divided into 2 equal

groups

-12 divided into 6 equal

groups