Multiplicative Strategies progression, 6th step
The purpose of the activity
In this activity, the learners use estimation strategies and calculators to solve multiplication problems involving decimals.
The teaching points
- The multiplication of two numbers results in the same digits regardless of the decimal point. For example, 24 x 59 = 1,416, 2.4 x 5.9 = 14.16, 24 x 0.59 = 14.16. Consequently there is no need to develop new approaches for multiplying decimals because the whole-number approaches apply and estimation can be used to work out where the decimal point should be placed. The use of estimation as a method for deciding on the position of the decimal is more difficult when the numbers are smaller. For example, knowing that 24 x 59 = 1,416 still does not make it straightforward to work out the answer to 0.0024 x 0.00059.
- It is as important to develop number sense with decimals as it is with whole numbers. While counting decimal points in multiplication problems works as a rule and has a conceptual rationale, it does not develop number sense.
- The decimal point is a convention that indicates the units, place. The role of the decimal point is to indicate the units, or ones, place in a number, and it does that by sitting immediately to the right of that place. Consequently, the decimal point also works to separate the unit (on the left) from parts of the unit (on the right).
- Irrespective of the computation approach used, it is important learners are able to judge the reasonableness of their answer in relation to the problem posed.
- When precision is important and the computation is difficult, then calculators and spreadsheets should be used.
- Discuss with the learners relevant or authentic situations where multiplication of decimals is used (for example, area measurements).
The guided teaching and learning sequence
1. Write the following problem on the board: 25 x 0.33 =
2. Discuss with the learners possible situations where they might need to do a calculation like that. For example: Each can of coke holds 0.33 litres (330 millilitres). How many litres of coke are there in 25 cans?
3. Write the following estimates (20, 10, 5, 8) on the board and ask the learners to think about which is a reasonable estimate and which aren’t reasonable. Discuss reasons for and against each estimate. Possible reasons include:
- 0.33 is 1/3 and 1/3 of 25 is a bit more than 8 so 8 is the best estimate.
- 25 x 3 is 75 so 25 x .3 = 7.5 so 8 is close to that.
4. Once 8 is established as the best estimate, tell the learners to use one of the following three computation approaches to determine the exact answer:
- using a calculator
- using written method
- using a strategy.
5. Ask the learners to share their solution and how they solved the problem with another learner.
6. Ask the learners to indicate (with a show of hands) which of the three approaches they used.
7. Ask for volunteers to share the approach they used and their reasons for selecting that approach. Ensure the volunteers are able to explain their answer in relation to the agreed estimate of 8.
“How did you know the decimal point went between the 8 and the 25?”
In the remainder of this activity, we focus on using estimation to work out where to place the decimal point in an answer.
8. Ask the learners to calculate the answer to 24 x 59. When they have agreed that 24 x 59 = 1,416, write the equation on the board.
9. Below this, write the following problems, telling the learners they are only able to use that first result of 1,416 to work out the exact answer to the problems.
- 0.24 x 59
- 2.4 x 5.9
- 24 x 0.59
- 0.24 x 0.59
- 2.4 x 0.059
10. Ask the learners to work in pairs to write down their rationale for each answer. As the learners work on the problems, encourage them to think about the ‘size’ of the numbers involved. For example, 24 x 0.59 is close to a half of 24 so it makes sense to put the decimal point after 14. The learners could also reason that 1.416 is too small and 141.6 is too large.
11. Choose a couple of the problems to discuss as a class.
12. Give pairs of learners a calculator to compute the following products. Ask them to write a rationale for why the answer makes sense in terms of the numbers that were multiplied together.
- 623.1 x 0.5 (311.55 makes sense as 300 is 1/2 or 0.5 of 600).
- 5.1666 x 5 = 25.833 (25.833 makes sense as 5 x 5 = 25).
Ask the learners to use 56 x 45 = 2,520 to give exact answers to the following problems. Ask them to write down or be prepared to explain the position of the decimal point in each answer.
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