Mental calculation with decimals and reasonable answers — 11–13
Choose the shortest path to calculate decimals in your head, and use an estimate to see whether a result makes sense.
Three tools for mental calculation
For mental calculation with decimals you have three tools. The rule for 10, 100, 1000 and for 0.1, 0.01, 0.001: 2.6 × 100 = 260 and 2.6 ÷ 0.1 = 26. The properties of operations: split and swap factors, as in 6 × 3.5 = 6 × 3 + 6 × 0.5. And the link between multiplication and division: 64 × 0.25 = 64 ÷ 4, because 0.25 is 1/4. You pick the one that gives the shortest path.
Why compare paths and check results
Mental calculation is not about memorising: it makes you see what numbers are (2.8 is 28 tenths). Two people can reach the same result by different paths, and comparing them shows which is shorter and when each works. You also need to decide whether a result makes sense, whether it came from an algorithm, from your head or from a calculator, and an estimate is your test.
5 × 2.8 by two paths, and a check
Work out 5 × 2.8. Path A: 5 × 2.8 = 5 × 28 × 0.1; 5 × 28 = 5 × 30 − 5 × 2 = 150 − 10 = 140; 140 × 0.1 = 14. Path B: double one factor and halve the other: 5 becomes 10 and 2.8 becomes 1.4, so 5 × 2.8 = 10 × 1.4 = 14. Both give 14, but B is shorter. Check: 5 × 3 = 15, and 2.8 is a little less than 3, so 14 is reasonable.
The calculator is not a judge
A reasonable mistake is to trust a result just because it came from a calculator or from a careful calculation. If you type 42 × 6 instead of 4.2 × 6, the calculator shows 252, and it is right for what you typed. But estimating 4 × 6 = 24, you see that 252 is about ten times bigger; the right result is 25.2. Always estimate first: what size should the answer be?
Shopping and measuring
In shopping and measuring, mental calculation and estimating save you from mistakes: 3 notebooks at 2.95 € each cost close to 3 × 3 = 9 €, and the exact sum is 3 × 2.95 = 8.85 €. Cutting a 2.5 m ribbon into 0.5 m pieces gives 2.5 ÷ 0.5 = 25 ÷ 5 = 5 pieces. If the result is very different from your estimate, check the calculation again.
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