Compatible multiples

This activity uses a variety of strategies involving decimal and fraction compatibles to solve and write problems. Students also need good recall of multiplication facts to 10 x 10 and the corresponding division facts. Note that some students may be confused about multiplication answers being smaller than the multiplier, when the reverse is usually the case. As in earlier activities, a reminder that “multiplied by” can be thought of as “of” will help. (See the notes for pages 16–17, "Using mates".)

This activity emphasises the development of efficient strategies to solve problems involving decimals and fractions. Encourage the students to share and justify the strategies they used to solve each problem. The activity assumes that students know the commutative property of multiplication (that the order of the factors does not affect the product). For example, for question 1a, 5 x 7 x 0.2, the students may know that 0.2 is the same as 1/5, so 5 x 1/5 = 1, 1 x 7 = 7. Alternatively, they may know 5 x 0.2 = 1.0. Some strategies may not be based on compatible numbers but may be derived from known facts such as 5 x 7 = 35, so 1/5 of 35 is 7.

Question 2 requires the students to use both fractions and decimals when coming up with their compatible numbers. Make sure they do so, as reinforcement and as good preparation for question 3.

In question 3, the students create four of their own problems based on compatible numbers, which they then ask a classmate to solve. You need to reinforce the compatible number patterns from question 1 by asking why particular combinations in that question were considered “compatible”.

Often it is because they multiply to 1 or another whole number such as 2, 3, 5, or 10. The students may come up with other compatible number combinations.

In question 4, the students are asked to write a decimal multiplication and a fraction multiplication that use compatible number combinations and have a product of 20. For example, 0.2 x 5 x 20 = 20 and 10 x 1/2 x 4 = 20. Get the students to explore several solutions and record these as a poster, Compatible Number Names for 20. The students can present their poster to the group and justify their number sentences by explaining which combinations are considered compatible numbers and why.

1.

a. 7. 5 x 0.2 = 1, so 5 x 7 x 0.2 = 1 x 7 = 7

b. 62. 0.5 x 4 = 2, so 0.5 x 31 x 4 = 2 x 31 = 62

c. 54. 1.5 x 2 = 3, 3 x 18 = 6 x 9 (double and halve) = 54

d. 7. 0.25 x 4 = 1, so 7 x 0.25 x 4 = 1 x 7 = 7

e. 7. x 3 = 1, so x 7 x 3 = 1 x 7 = 7

f. 370. x 20 = 10, so x 37 x 20 = 10 x 37 = 370

g. 68. 18 x = 2, so 18 x 34 x = 2 x 34 = 68

h. 250. 8 x = 2, so 8 x 125 x = 2 x 125 = 250

i. 26. x 14 = 2, so 13 x x 14 = 2 x 13 = 26

j. 69. x 18 = 3, so 23 x x 18 = 3 x 23 = 69

k. 6. 15 x 0.2 = 3 and 0.25 x 8 = 2, so 15 x 0.25 x 0.2 x 8 = 3 x 2 = 6

l. 10. 1/8 x 16 = 2 and 25 x 1/5 = 5, so 1/8 x 25 x 16 x 1/5 = 2 x 5 = 10

2.

There are many possibilities. Here are some examples:

• 8 x 0.25
• 12 x 1/3 x 1/2
• 9 x 2 x 1/9
• 20 x 0.1
• 4 x 25 x 0.2 x 0.5 x 0.2
• 1/12 x 6 x 4

3.

There are many possibilities. Here are some examples:

• 28 x 0.25 x 27 x 1/9
• 120 x 0.1 x 0.5 x 8
• 0.3 x 16 x 40 x 1/8

4.

Answers will vary.  For example, 8 x 5 x 0.5 or 30 x 4 x 1/6.