Multiplicative Strategies progression, 5th step
The purpose of the activity
In this activity, the learners use strategies, traditional written methods and calculators to solve multiplication 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 “Multiplication strategies” activity before starting this activity.
The teaching points

Different multiplication 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 45 x 6 by thinking of 6 x 5 and 6 x 4 rather than 40 x 6 and 5 x 6.

Mental strategies tend to consider the largest value of the number first, while traditional algorithms start on the rightmost digits, which are the smallest in value.

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 multiplication is used (for example, area measurements).
Resources

Empty number lines.

Calculators.
The guided teaching and learning sequence
1. Write the following problem on the board: 39 x 21 =
2. Discuss with the learners possible situations where they might need to do a calculation like that. For example: A set menu at the Blue Cheese Restaurant costs $39 per head. How much will it cost to take 21 people to dinner at the restaurant?
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 computation 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 819 is reasonable as 39 x 21 is similar to 40 x 20, which equals 800.
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 or the value of the numbers (for example, 2 x 3 = 6 or 20 x 30 =600).
“How did you know that your answer was reasonable?”
If the learner used the value of the numbers, their explanation of the algorithm might include an explanation of the reasonableness of the answer. If they used a digitsonly explanation of the algorithm, encourage them to explain why 819 is reasonable as an answer to 39 x 21.
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: 120 x 225 =
10. This time, ask the learners to work in pairs to solve the problem, using the three different approaches. As the learners solve the problem, discuss with them their preferred approach.
11. 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 write down which approach they think they would prefer to use to solve the problem.
12. As a group discuss which problems seem best suited to mental strategies and which they would prefer to solve with a calculator or written method. The learners may suggest that 8,576 x 99 could be solved by 8,576 x 100 – 8,576. However, it is important then to ensure that the learners consider whether 857,600 – 8,576 is a computation that they can do mentally.
This activity could be followed by or combined with the “Dividing options” activity.
Followup activity
Give the learners cards with one multiplication 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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