Multiplicative Strategies progression, 5th step
The purpose of the activity
In this activity, the learners use strategies, traditional written methods and calculators to solve division problems. The aim of exploring the three computation approaches is to encourage the learners to anticipate from the complexity and structure of a problem the approach that best suits them.
The learners need to be familiar with the concepts addressed in the “Division strategies” activity before starting this activity. This activity could follow, or be completed in combination with, the “Multiplying options” activity.
The teaching points
- Different division problems lend themselves to different mental strategies, and competent learners have a range of strategies to choose from.
- There are three main computation approaches: using strategies to calculate a problem mentally, using traditional written methods (algorithms) and using calculators. The complexity of the problem determines the most effective and efficient computation approach.
- There are significant differences between mental strategies and traditional algorithms.
- Mental strategies are number oriented rather than digit oriented. This means that the value of the number is considered rather than just the digit. Using the traditional algorithm, the learner solves 639 ÷ 3 by thinking of 6 ÷ 3 rather than 600 ÷ 3.
- Mental strategies are flexible and change with the numbers involved in order to make the computation easier. With the traditional algorithm, the same rule is used on every problem.
- Irrespective of the computation approach used, it is important that the learners are able to judge the reasonableness of their answer in relation to the problem posed.
- Discuss with the learners relevant or authentic situations where division is used (for example, sharing a restaurant account).
The guided teaching and learning sequence
1. Write the following problem on the board: 534 ÷ 6 =
2. Discuss with the learners possible situations where they might need to solve a calculation like that. For example: The accommodation account for the weekend for 6 people at the Lazy Days Hotel came to $534. How much does each person need to pay (assuming they are paying equal amounts)?
3. Discuss with the learners the three computation approaches that can be used to solve the problem:
- using a calculator
- using a written method
- using a strategy.
4. Ask the learners to choose one of the options and solve the problem giving them time to do this. Then ask them to share their solution and how they solved it with another learner. Tell them they also have to be able to explain why the answer they obtained is reasonable (or makes sense) in relation to the problem.
5. Ask the learners to indicate (with a show of hands) which of the three approaches they used.
6. Ask for a volunteer who used a calculator to solve the problem.
“Why did you choose to use a calculator?”
“Why is your answer reasonable?”
For example, a learner could explain that 89 is reasonable as 90 x 6 = 540.
7. Ask for a volunteer who used a written method to solve the problem.
“Why did you use that method?”
“Show us what you did on the board.”
Notice if the learner has explained the algorithm using digits (the first example) or the value of the numbers (the second example).
“How did you know that your answer was reasonable?”
If the learner used the value of the numbers, the explanation of the algorithm might include an explanation of the reasonableness of the answer. If they used a digits-only explanation of the algorithm, encourage them to explain why 89 is reasonable as an answer to 534 ÷ 6.
8. Ask for a volunteer who used a mental strategy to solve the problem.
“Why did you decide to use a strategy?”
“Explain what you did to work out the answer.”
“How did you know that your answer was reasonable?”
9. Pose another problem: 12,342 ÷ 44 =
10. Discuss with the learners which of the three approaches they consider most efficient for solving this problem. Hopefully the learners will agree that using a calculator is the most efficient as it is quick and likely to be the most accurate. Try to avoid getting into a debate with the learners about the availability of calculators, explaining that in most situations where you need to calculate complex, accurate problems, both pen and paper and calculators will generally be available.
11. Ask the learners to work in pairs, using a calculator to answer the problem (12,320 ÷ 44 = 280). Also ask them to decide how they know if the answer obtained is reasonable.
12. Ask the learners to share the ways they considered that the answer was reasonable.
Possible explanations include:
Using rounding: 12,000 ÷ 40 = 300, which is close to 280.
Using doubling: 280 x 100 = 28,000, which is 2-and-a-bit times larger than 12,340.
13. Give the learners the following problems on a sheet of paper. Without actually solving the problems, ask the learners to look at each problem and to write down which approach they would choose to solve the problem.
14. As a class, discuss which problems are ones that seem best suited to mental strategies and which the learners would prefer to solve with a calculator or written method.
Give the learners cards with one division problem on each card. Have the learners work in pairs to select a computation approach, solve the problem and explain the reasonableness of their answer.
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