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. Now let’s look at models for operations with integers.

8. Warm Up
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Notes on
Modeling
Integers

9. ADDING INTEGERS

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

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

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

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

14. SUBTRACTING INTEGERS

15. What is subtraction?
There are different models for subtraction, but when using the two-colored counters you will be using the “take-away” model.

16. 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

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

18. Subtract : –11 – (–5) = –6

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

20. We can also use fact family with integers.
Use your red and yellow tiles to verify this fact family:
= +5
= +5
= -3
= +8

21. 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.

22. Remember?
+5 = or or

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

24. –6 – 5 = –11

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

26. Subtract: 8 – 9
= –1

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

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

29. 1) 8 – (–3) = 11
= 8 + 3

30. –4 – 3 = –7
= –4 + (–3)

31. 3) –7 – 1 = –8
= –7 + (–1)

32. 4) 9 – (–3) = 12
= 9 + 3

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

34. Subtract: –3 – 5 = –8 –3 + –5

35. 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

36. 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)

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. Division cannot be modeled easily using two-colored counters, but since division is the inverse of multiplication you can apply what you learned about multiplying to division.

49. does it make sense that -3 × 2 = -6 ?
Since 2 × 3 = 6 and 3 × 2 = 6,
does it make sense that -3 × 2 = -6 ?
Yes
+2 × -3 = -6 and -3 × +2 = -6 belong to a fact family:
+2 × -3 = -6
-3 × +2 = -6
-6 ÷ +2 = -3
-6 ÷ -3 = +2

50. If 3 × (–5) = –15, then –15 ÷ –5 = ? 3
and –15 ÷ 3 = ? –5
If –2 × –4 = 8, then 8 ÷ (–4) = ? and 8 ÷ (–2) = ? –2 –4

51. Write a “rule” for dividing integers.