ILLUSTRATING ILLUSTRATING INTEGERSINTEGERS

Mrs. Nighs

INTRODUCTION TO INTRODUCTION TO INTEGERSINTEGERS

Mrs. Nighs

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.

Mrs. Nighs

-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.

Mrs. Nighs

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.

Mrs. Nighs

-3 -3

-5+2

Mrs. Nighs

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.

Mrs. Nighs

-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.Mrs. Nighs

ADDITION AND ADDITION AND SUBTRACTIONSUBTRACTION

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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.

Mrs. Nighs

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 + -2 + ++5 = 5 = ++33Mrs. Nighs

+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 = ?

Mrs. Nighs

ANSWER

+ = -5

-2 + -3 = -2 + -3 = --55

ADDING INTEGERSADDING INTEGERS

Mrs. Nighs

-6 + +2 = ?

+ = - 4

+ = +1

-3 + +4 = ?

ANSWER

ANSWER

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

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

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 = ?

Mrs. Nighs

+3 +3

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

Mrs. Nighs

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.

Mrs. Nighs

--2 - 2 - --4 = ?4 = ?

ANSWER

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

Mrs. Nighs

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

Mrs. Nighs

Work this problem.Work this problem.

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

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

Mrs. Nighs

• 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

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

Mrs. Nighs

Work this problem.Work this problem.

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

SUBTRACTING INTEGERSSUBTRACTING INTEGERS

Mrs. Nighs

•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

Mrs. Nighs

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

Mrs. Nighs

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 fact family:Use your red and yellow tiles to verify this fact family:--3 + 3 + ++8 = 8 = ++55++8 + 8 + --3 = 3 = ++55++5 - 5 - ++ 8 = 8 = --33++5 - 5 - - - 3 = 3 = ++88

INTEGERS AND INTEGERS AND MULTIPLICATIONMULTIPLICATION

Mrs. Nighs

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 = ?

Mrs. Nighs

2 groups of 3 tiles = 6 tiles

MULTIPLICATIONMULTIPLICATION

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

Mrs. Nighs

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:

Mrs. Nighs

++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 = ?

ANSWER2 groups of +4

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

MULTIPLICATIONMULTIPLICATION

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

Mrs. Nighs

ANSWER

3 groups of -3

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

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

Mrs. Nighs

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.Mrs. Nighs

++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

Mrs. Nighs

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

MULTIPLICATIONMULTIPLICATION

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

Mrs. Nighs

ANSWER

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

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

INTEGERS AND INTEGERS AND DIVISIONDIVISION

Mrs. Nighs

DIVISIONDIVISION

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

Mrs. Nighs

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 =?

Mrs. Nighs

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 = --88

Compare your answer.Compare your answer.++2 x 2 x --4 = 4 = --88--8 8 ÷ ÷ ++2 = 2 = --44--8 ÷ 8 ÷ --4 = 4 = ++22

Mrs. Nighs

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

Mrs. Nighs

ANSWER

DIVISIONDIVISION

Mrs. Nighs

1) -2 + -8 = -10

2) +6 + -4 = +2

DIVISIONDIVISION

Mrs. Nighs

3) -5 - +4 = -9

4) -6 - -7 = +1

DIVISIONDIVISION

Mrs. Nighs

5) +3 x -4 = -12

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

of -2

DIVISIONDIVISION

Mrs. Nighs

7) -12 ÷ +6 = -2

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

groups

-12 divided into 6 equal

groups

Rules for Adding and Subt. IntegersRules for Adding and Subt. Integers

When signs are the same add integers When signs are the same add integers keep sign.keep sign.

When signs are different subtract. Bigger When signs are different subtract. Bigger number gets the sign.number gets the sign.

Multiply IntegersMultiply Integers

Signs are the same answer is always Signs are the same answer is always positivepositive

Signs different the answer is always Signs different the answer is always negative. negative.