illustrating integers
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ILLUSTRATING ILLUSTRATING INTEGERSINTEGERS
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INTRODUCTION TO INTRODUCTION TO INTEGERSINTEGERS
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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.
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-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.
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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.The University of Texas at Dallas
-3 -3
NOTE TO TUTOR
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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.The University of Texas at Dallas
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.
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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 + -2 + ++5 = 5 = ++33The University of Texas at Dallas
+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 = ?
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ANSWER
+ = -5
-2 + -3 = -2 + -3 = --55
ADDING INTEGERSADDING INTEGERS
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-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 = ?
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+3 +3
SUBTRACTING INTEGERSSUBTRACTING INTEGERS
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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
SUBTRACTING INTEGERSSUBTRACTING INTEGERS
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Work this problem.Work this problem.
--3 - 3 - ++2 = ?2 = ?ANSWER
SUBTRACTING INTEGERSSUBTRACTING INTEGERS
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•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
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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 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
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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 = ?
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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
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 = --88
Compare 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