Math Detours · Grade 5 · Sharpen · 10

Where does the decimal point go?

0.3 × 0.2 is not 0.6. Tenths of tenths are hundredths, and one square picture shows it.

Name ____________________Date ____________

Tonight's idea

Is \(0.3 \times 0.2\) equal to 0.6? It looks like \(3 \times 2\). But \(0.3 \times 0.2\) means 0.3 of 0.2, which is only a part of 0.2. A part of 0.2 must be smaller than 0.2, and 0.6 is bigger.

0.3 0.2 1 2 rows 3 columns
The whole square is 1. The orange rectangle is 0.3 of 0.2.

The big square is 1. Cut it into ten strips across and ten strips up. That makes 100 small squares. Each small square is one hundredth, written 0.01.

The bottom two rows are 0.2 of the big square. Now take 0.3 of those rows: three columns out of ten. That rectangle has \(3 \times 2 = 6\) small squares. Six hundredths is 0.06.

Tenths times tenths are hundredths. Multiply the digits, \(3 \times 2 = 6\), then write the answer as hundredths: \(0.3 \times 0.2 = 0.06\).

Watch one: \(0.4 \times 0.7\), then \(1.2 \times 0.3\)

0.4 is 4 tenths and 0.7 is 7 tenths. \(4 \times 7 = 28\), and tenths times tenths are hundredths. 28 hundredths is 0.28.

Does that make sense? 0.4 of 0.7 is a bit less than half of 0.7. Yes.

1.2 is 12 tenths and 0.3 is 3 tenths. \(12 \times 3 = 36\), so the answer is 36 hundredths, which is 0.36. Check: about a third of 1.2 is 0.4. Close enough.

Your turn

  1. Look at the square in the picture. Think of a rectangle 0.5 wide and 0.3 tall in its corner. (a) How many small squares does it cover? (b) Write the answer to \(0.5 \times 0.3\).
  2. Work out: (a) \(0.6 \times 0.2\) (b) \(0.9 \times 0.4\) (c) \(0.5 \times 0.8\) (d) \(0.1 \times 0.1\)
  3. Write each number as tenths first, then multiply. (a) \(1.5 \times 0.6\) (b) \(2.4 \times 0.3\) (c) \(3 \times 0.4\). Careful with (c): 3 is a whole number, so what are the pieces of the answer?
  4. The digits are right but the decimal point is missing. Put it in by thinking about size, not by calculating. (a) \(3.2 \times 0.5 = 160\) (b) \(0.7 \times 0.9 = 63\) (c) \(6.1 \times 1.9 = 1159\). Say how you decided each one.

Stretch: three tenths in a row

A tenth of a tenth is a hundredth. What is a tenth of a tenth of a tenth? Write \(0.1 \times 0.1 \times 0.1\) as a decimal. Then work out \(0.2 \times 0.3 \times 0.5\).

Answers

  1. (a) \(5 \times 3 = 15\) small squares. (b) 15 hundredths: \(0.5 \times 0.3 = 0.15\).
  2. (a) 12 hundredths, 0.12. (b) 36 hundredths, 0.36. (c) 40 hundredths, 0.40, which is 0.4. (d) 1 hundredth, 0.01.
  3. (a) \(15 \times 6 = 90\) hundredths, 0.9. (b) \(24 \times 3 = 72\) hundredths, 0.72. (c) 3 lots of 4 tenths is 12 tenths, 1.2. Whole numbers times tenths stay tenths.
  4. (a) 1.60, which is 1.6: half of 3.2. (b) 0.63: a bit less than 0.7, because 0.9 is nearly 1. (c) 11.59: close to \(6 \times 2 = 12\).

Stretch. A tenth of a hundredth is a thousandth: \(0.1 \times 0.1 \times 0.1 = 0.001\). Then \(2 \times 3 \times 5 = 30\), and 30 thousandths is 0.030, which is 0.03.

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