Multiplicative Strategies progression, 6th step
The purpose of the activity
In this activity, the learners use estimation strategies and calculators to solve division problems involving decimals.
The teaching points
- The division of two numbers results in the same digits regardless of the decimal point. For example, 754 ÷ 13 = 58, 75.4 ÷ 1.3 = 58, 7.54 ÷ 1.3 = 5.8, 0.754 ÷ 0.13 = 5.8, 7.54 ÷ 13. = 0.58. Consequently, there is no need to develop new approaches for dividing decimals as 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 754 ÷ 13 = 58 still does not make it straightforward to work out the answer to 0.000754 ÷ 0.013.
- When precision is important and the computation is difficult, then calculators and spreadsheets should be used.
- 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 unit 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 that 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 of decimals is used (for example, rate problems).
The guided teaching and learning sequence
1. Pose the following problem to the class: The distance from Dunedin to Queenstown is 287 kilometres. If it takes 3.5 hours to travel between the two cities, what is the average speed in kilometres per hour?
2. Discuss with the learners how they might estimate the answer to the problem. Share and record possibilities on the board.
- 3.5 x 100 = 350, so maybe about 80 or 90 km/hr.
- 4 x 80 = 320, so somewhere close to 80 km/hr.
- 300 ÷ 3 = 100 and 300 ÷ 4 is a bit more than 70 km/hr.
3. Once somewhere between 80 and 90 km/hr has been established as an estimate, ask the learners to compute the exact answer. Discuss with the learners the options they have for the computation:
- using a calculator
- using written method
- using a strategy.
4. Ask the learners to share their solution and how they solved it with a learner who used a different approach.
The remainder of this sequence focuses on using estimation to work out where to place the decimal point in an answer.
5. Write 287 ÷ 3.5 = 82 on the board.
6. Below this write the following problems, telling the learners they are only able to use that earlier result to work out the exact answer to the problems.
7. 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, 28.7 ÷ 35 is going to give an answer less than 1 because 35 is larger than 28. The learners could also reason that it makes sense to put the decimal point before 82 (0.82) as there are not 8 lots of 35 in 28 and 0.082 lots of 35 is not close to 28.
8. Choose a couple of the problems to discuss as a class.
9. Give pairs of learners a calculator to compute the following quotients. Ask them to write a rationale for why the answer makes sense in terms of the numbers that were divided.
- 63 ÷ 4.2 =
- 7.446 ÷ 14.6 =
- 27 ÷ 0.45 =
Ask the learners to use 156 ÷ 8 = 19.5 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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