Integers with Manipulatives

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. ADDING INTEGERS

9. What is addition? Addition is combining one or more addends (collections of counters).

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

13. Build the following addition problems: • -7 + 2 = • 2) 8 + -4 = • 4 + 5 = • -6 + (-3) = -5 4 9 -9

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

15. SUBTRACTING INTEGERS

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

17. 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

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

19. Subtract : –11 – (–5) = –6

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

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

22. Build –6. Now try to subtract +5. Can’t do it? Think back to building collections in different ways.

23. Remember? +5 = or or

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

25. –6 – 5 = –11

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

27. Subtract: 8 – 9 = –1

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

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

30. 1) 8 – (–3) = 11 = 8 + 3

31. –4 – 3 = –7 = –4 + (–3)

32. 3) –7 – 1 = –8 = –7 + (–1)

33. 4) 9 – (–3) = 12 = 9 + 3

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

35. –8 Subtract: –3 – 5 = –5 –3 +

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

37. Write an addition problem to solve the following: • –8 – 14 2) –24 – (–8) • 3) 11 – 15 4) –19 – 3 • 5) –4 – (–8) 6) 18 – 5 • 7) 12 – (–4) 8) –5 – (–16)

38. MULTIPLYING INTEGERS

39. What is multiplication? Repeated addition!

40. 3 × 4 means 3 groups of 4: + + 3 × 4 = 12

41. 3 × (–2) means 3 groups of –2: + + 3 × (–2) = –6

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

43. 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.

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

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

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

47. Write a “rule” for multiplying integers.

48. DIVIDING INTEGERS

49. 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.

50. 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