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Multiplying 0.2 by 0.3: visualise by area

Having marked a few score short written diagnostic tests, I am beginning to believe that 0.2 × 0.3 = 0.6.

The test is marked out of 27, and scores range from 5 to 25, but only two people so far have remembered how to multiply decimals without a calculator. I've seen attempts at the standard multiplication algorithm written out with 0.3 on the top, 0.2 underneath, and the decimal point in the same column in the answer line. I've seen 00.6 written more than a few times.

My strategy for rectifying this is to introduce percentages early, and to ask students to estimate the percentage of the grid below that is coloured salmon. Then I'll ask students to show me how they write that percentage as a decimal fraction of one. Might use the small whiteboards.

"picture of a 10 by 10 grid drawn on white with black lines. At the bottom left is a rectangle of 3 columns by 2 rows coloured salmon. There is no labelling."

Then, I'll suggest that the whole grid represents a 1 by 1 square. How long in decimals is 3 columns? How high as a decimal is two rows?

Finally, I'll ask what calculation gives the area of the small rectangle. We should then reach 0.2 × 0.3 = 0.06. Along the way I'll visit place notation, decimals, percentages, dividing by 100, the analogue between pence and pounds and percentages and wholes. Plenty of 'drill' worksheets available at each stage just in case. If I find I'm pushing against a door that is open with some people, I can move sideways into area and perimeter pretty quickly.

"picture of a 10 by 10 grid with 3 columns coloured yellow and 2 rows coloured blue. The intersection is coloured salmon. The two rows are labelled 0.2 at the right hand side and the three columns are labelled 0.3 at the top."

Keith Burnett, Last update: Sun Sep 04 2011