illustrating integers the university of texas at dallas

ILLUSTRATING ILLUSTRATING INTEGERSINTEGERS

The University of Texas at Dallas

INTRODUCTION TO INTRODUCTION TO INTEGERSINTEGERS


Student ExpectationsStudent Expectations

66thth Grade: Grade:

6.1.1C6.1.1C Use integers to represent real-life situations. Use integers to represent real-life situations.

77thth Grade: Grade:

7.1.2C7.1.2C Use models, such as concrete objects, pictorial Use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to and divide integers and connect the actions to algorithms. algorithms.

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


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


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.The University of Texas at Dallas

-3 -3

NOTE TO TUTOR



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


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


-2 +1

ANSWER+2


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.The University of Texas at Dallas

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.



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 = ++33The University of Texas at Dallas

+3


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



-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 = ?


+3 +3



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.


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


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

ANSWER



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…



Work this problem.Work this problem.

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



• 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



Work this problem.Work this problem.

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



•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



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.



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 + 3 + ++8 = 8 = ++55++8 + 8 + --3 = 3 = ++55++5 - 5 - ++ 8 = 8 = --33++5 - 5 - - - 3 = 3 = ++88

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


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


2 x x --33 = -6


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:


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


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


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


ANSWER

3 groups of -3

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

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


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


++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.


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


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


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


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


ANSWER

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

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

INTEGERS AND INTEGERS AND DIVISIONDIVISION


DIVISIONDIVISION

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


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 =?


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


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


ANSWER

DIVISIONDIVISION


1) -2 + -8 = -10

2) +6 + -4 = +2

DIVISIONDIVISION


3) -5 - +4 = -9

4) -6 - -7 = +1

DIVISIONDIVISION


5) +3 x -4 = -12

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

of -2

DIVISIONDIVISION


7) -12 ÷ +6 = -2

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

groups

-12 divided into 6 equal

groups

illustrating integers the university of texas at dallas

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