1. Integers with Manipulatives

2. Operations with integers can be modeled using two-colored counters.
Positive +1
Negative -1

3. The following collections of counters have a value of +5.
Build a different collection that has a value of +5.

4. What is the smallest collection of counters with a value of +5?
As you build collections of two-colored counters, use the smallest collection, but remember that there are other ways to build a collection.

5. The collections shown here are “zero pairs”.
They have a value of zero.

6. Describe a “zero pair”.

7. ADDING INTEGERS

8. When using two-colored counters to model addition, build each addend then find the value of the collection.
5 + (-3)
= 2
zero pairs

9. Modeling addition of integers:
8 + (–3)
= 5

10. (Notice that there are no zero pairs.)
Here is another example:
-4 + (-3)
= -7
(Notice that there are no zero pairs.)

11. Build the following addition problems:=
2) =
4 + 5 =
-6 + (-3) = -5 4 9 -9

12. Write a “rule”, in your own words, for adding integers.

13. Warm Up:
Get your folder and a set of integer disks.
Add the integers.
4 + (-2) = = =
6+2 = = =

14. SUBTRACTING INTEGERS

15. When using two-colored counters to model subtraction, build a collection then take away the value to be subtracted.
For example: 9 – 3
= 6
take away

16. Here is another example: –8 – (–2) = –6
take away

17. Subtract : –11 – (–5) = –6

18. Build the following:
–7 – (–3)
6 – 1
–5 – (–4)
8 – 3
= –4
= 5
= –1
= 5

19. Can’t do it? Think back to building collections in different ways.
Now try to subtract +5.
Can’t do it? Think back to building collections in different ways.

20. Remember?
+5 = or or

21. Now build –6, then add 5 zero pairs. It should look like this:
This collection still has a value of –6. Now subtract 5.

22. –6 – 5 = –11

23. Another example:
5 – (–2)
Build 5:
Add zero pairs:
Subtract –2:
5 – (–2) = 7

24. Subtract: 8 – 9
= –1

25. Try building the following:
1) 8 – (–3)
–4 – 3
–7 – 1
9 – (–3)
= 11
= –7
= –8
= 12

26. Look at the solutions. What addition problems are modeled?

27. 1) 8 – (–3) = 11
= 8 + 3

28. –4 – 3 = –7
= –4 + (–3)

29. 3) –7 – 1 = –8
= –7 + (–1)

30. 4) 9 – (–3) = 12
= 9 + 3

31. These examples model an alternative way to solve a subtraction problem.

32. Subtract: –3 – 5 = –8 –3 + –5

33. Any subtraction problem can be solved by adding the opposite of the number that is being subtracted.
11 – (–4) = = 15
–21 – 5 = –21 + (–5) = –26

34. Write an addition problem to solve the following:
–8 – ) –24 – (–8)
3) 11 – ) –19 – 3
5) –4 – (–8) 6) 18 – 5
7) 12 – (–4) 8) –5 – (–16)

35. Warm Up
Get folder and counters
Solve the following
a) = b) 6 – (-2)=
= d) -5 – (-3) =
e) = f) -24 – 6 =

36. (-3) + 9 (-5) + (-2) 16 + (-18) 7 – (-4) -5 – 6 -12 – (-3) 16 - 20
16 + (-18)
7 – (-4)
-5 – 6
-12 – (-3)

37. MULTIPLYING INTEGERS

38. What is multiplication?
Repeated addition!

39. 3 × 4 means 3 groups of 4: +
3 × 4 = 12

40. 3 × (–2) means 3 groups of –2: +
3 × (–2) = –6

41. If multiplying by a positive means to add groups, what doe it mean to multiply by a negative?
Subtract groups!

42. means to take away 2 groups of positive 3.
Example:
–2 × 3
means to take away 2 groups of positive 3.
But, you need a collection to subtract from, so build a collection of zero pairs.

43. What is the value of this collection?
Take away 2 groups of 3. What is the value of the remaining collection?
–2 × 3 = –6

44. Try this:
(–4) × (–2)
(–4) × (–2) = 8

45. Solve the following:
1) 5 × 6
2) –8 × 3
3) –7 × (–4)
4) 6 × (–2)
= 30
= –24
= 28
= –12

46. Write a “rule” for multiplying integers.

47. DIVIDING INTEGERS

48. 12÷3
It means take 12 and make 3 equal groups.

49. -10÷5 What does this mean?

50. Write a “rule” for dividing integers.

51. Try these problems using your rule:
20÷-4=
-24÷-3 =
36 ÷ -9 =
45 ÷ 5 =

52. Warm Up: =
7 – (-4) = – 6
(-5)(8) (-7)(-6) =
÷ - 8 = ÷ - 6 =

53. Order of Operations PEMDAS
1. Evaluate expressions inside of ___________
2. Evaluate _______
3. __________and _________from left to right
4. __________ and ________ from left to right

56. Warm Up: Use Order of Operations to solve
3 + 4* ÷4 * 2
12 – (4 – 5) – (-3)
30 – (52 – 4) –

58. You try: