Integers_1 Bridge Bridge courses course.ppt

Integers

1. INTRODUCTION TO INTRODUCTION TO
INTEGERSINTEGERS

2. 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 -2-1 201 3

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

4. 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

5. 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

6. INTRODUCTION TO INTEGERSINTRODUCTION TO INTEGERS
Tell which integer each group of tiles represents.Tell which integer each group of tiles represents.
-2
+1
ANSWER
+2

7. 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.

8. ADDITION AND ADDITION AND
SUBTRACTIONSUBTRACTION

9. 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.

10. 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 =
++
33
+
3

11. 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

12. ADDING INTEGERSADDING INTEGERS
-
6 +
+
2 = ?
+ =- 4
+ =
+
1
-
3 +
+
4 = ?
ANSWER
ANSWER

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

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

13. 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

14. 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.

15. 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.

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

16. SUBTRACTING INTEGERSSUBTRACTING INTEGERS
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…

17. SUBTRACTING INTEGERSSUBTRACTING INTEGERS
Work this problem.Work this problem.
++
3 - 3 -
--
5 = ?5 = ?
ANSWER

18. SUBTRACTING INTEGERSSUBTRACTING INTEGERS
• 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

19. SUBTRACTING INTEGERSSUBTRACTING INTEGERS
Work this problem.Work this problem.
--
3 - 3 -
++
2 = ?2 = ?
ANSWER

20. 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

21. 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 = 15
8 + 7 = 158 + 7 = 15
15 – 7 = 815 – 7 = 8
15 – 8 = 715 – 8 = 7
All of these statements are true.

22. SUBTRACTING INTEGERSSUBTRACTING INTEGERS
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

23. INTEGERS AND INTEGERS AND
MULTIPLICATIONMULTIPLICATION

24. 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

25. MULTIPLICATIONMULTIPLICATION
2 x 2 x
--
3 means 2 groups of 3 means 2 groups of
--
33
2 x x
--
33 =
-
6

26. 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:
++
2 x 2 x
--
3 = 3 =
--
66
--
3 x 3 x
++
2 = 2 =
--
66
--
6 6 ÷ ÷
++
2 = 2 =
--
33
--
6 6 ÷÷
-
3 =
+
2

27. 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

28. MULTIPLICATIONMULTIPLICATION
++
3 x 3 x
--
3 = ?3 = ?
ANSWER
3 groups of
-
3
++
3 x 3 x
--
3 = 3 =
--
99

29. 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

30. 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.
++
1 x 1 x
++
3 = 3 =
++
33
--
1 x 1 x
++
3 = 3 =
--
33

31. 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
--
2 x 2 x
--
3 = 3 =
++
66

32. MULTIPLICATIONMULTIPLICATION
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

33. INTEGERS AND INTEGERS AND
DIVISIONDIVISION

34. 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.

35. DIVISIONDIVISION
Use tiles to model Use tiles to model
--
15 15 ÷ ÷
++
5 =?5 =?
Divide
-
15 into 5
equal groups
--
15 ÷15 ÷
++
5 = 5 =
--
33

36. 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

37. 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

38. DIVISIONDIVISION
1)
-
2 +
-
8=
-
10
2)
+
6 +
-
4=
+
2

39. DIVISIONDIVISION
The University of Texas at Dallas
3)
-
5 -
+
4=
-
9
4)
-
6 -
-
7=
+
1

40. DIVISIONDIVISION
5)
+
3 x
-
4 =
-
12
6)
-
5 x
-
2 =
+
10
5 groups of
the opposite
of
-
2

41. DIVISIONDIVISION
7)
-
12 ÷
+
6 =
-
2
8)
-
8 ÷
+
2 =
-
4 -8 divided
into 2 equal
groups
-12 divided
into 6 equal
groups

42. THANK YOU THANK YOU