Math Detours · Grade 5 · Sharpen · 2

What lives between 0.3 and 0.4?

Zoom in on the number line and there is always room for more. Tenths, hundredths, thousandths, and why "0.35 is bigger than 0.4" is false.

Name ____________________Date ____________

Tonight's idea

Cut the number line from 0 to 1 into ten equal parts. Each part is a tenth, and the marks are 0.1, 0.2, 0.3 and so on.

Now zoom in on the gap from 0.3 to 0.4 and cut it into ten equal parts again. Each small part is a tenth of a tenth, called a hundredth. The new marks are 0.31, 0.32, up to 0.39.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4
Zoom in on the gap from 0.3 to 0.4. Nine new marks appear.

Zoom in again between 0.35 and 0.36 and you find thousandths: 0.351, 0.352 and so on. You can keep zooming for ever. There is always room.

In the picture 0.35 sits to the left of 0.4, so it is smaller. Why? 0.4 is the same as 0.40, which is 40 hundredths. 0.35 is only 35 hundredths.

More digits does not mean bigger. To compare decimals, write both with the same number of places.

Watch one: 0.6 and 0.58

Which is bigger? Write both in hundredths. 0.6 is 0.60, which is 60 hundredths. 0.58 is 58 hundredths. So 0.6 is bigger.

What lies between them? 59 hundredths does: 0.59.

What lies between 0.58 and 0.59? Write them in thousandths: 0.580 and 0.590. Now there are nine marks to choose from, such as 0.585.

Your turn

  1. Write the number that is half way between (a) 0.3 and 0.4; (b) 0.7 and 0.8; (c) 0.35 and 0.36.
  2. Put these in order, smallest first: 0.5,   0.45,   0.409,   0.54,   0.054.
  3. Write a number that lies between (a) 0.8 and 0.9; (b) 0.42 and 0.43; (c) 0.9 and 1; (d) 0.29 and 0.3.
  4. (a) Mia says "0.35 is bigger than 0.4, because 35 is bigger than 4". Explain her mistake. (b) There are 9 hundredths marks between 0.3 and 0.4, not counting the two ends. How many thousandths marks are there between 0.3 and 0.4?

Stretch: the next number

Is there a smallest number that is bigger than 0.3? Suppose a friend says they have found it. Explain how you could always find a number that is smaller than theirs and still bigger than 0.3.

Answers

  1. (a) 0.35. (b) 0.75. (c) 0.355.
  2. 0.054, 0.409, 0.45, 0.5, 0.54. In thousandths they are 54, 409, 450, 500 and 540.
  3. Many answers are right. For example: (a) 0.85; (b) 0.425; (c) 0.95; (d) 0.295. For (d), write 0.3 as 0.300 and 0.29 as 0.290 to see the gap.
  4. (a) Mia compared 35 hundredths with 4 tenths. They are different sizes of piece. In hundredths, 0.4 is 40 hundredths, and 40 beats 35. (b) 99. In thousandths the ends are 300 and 400. The marks between them are 301 up to 399, which is 99 marks.

Stretch. No. Whatever number your friend picks, take the number half way between 0.3 and theirs. It is bigger than 0.3 and smaller than their number. If they say 0.301, you say 0.3005. So no number can be "the next one" after 0.3.

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