Fractions To Decimals

Years 2 - 6

This page would be more informative
with a photo from your classroom.
doug@blackdouglas.com.au

Summary

Children begin to use fraction language when they are quite young. One half and one quarter, for example, are part of common language. To help them understand what these terms mean, teachers look for a moment to ask the challenging questions:
  • How would the calculator show...?
  • Could you check that another way?
 

Materials

  • One calculator per child
  • Materials that can be used to represent wholes and parts

Acknowledgement

Stories in this activity are from pages 21 & 22 of the C.A.N Report. For more from this section of the report see Decimals & Fractions. This paper includes several investigation sheets and an assessment sheet added by Calculating Changes members. Both Fractions To Decimals and the paper Decimals & Fractions emphasise the use of calculators to explore number patterns that lead into decimals. If your school is a member of Maths300, Lesson 182, Fractions to Decimals (on a rope!) will help build a conceptual understanding of where these decimals fit on a number line.

Procedure

Listen for occasions when your children use fraction language then consider a direction something like this story in the CAN Report:

A teacher asked a group of first-year junior children (aged seven to eight):

How could we show fractions on a calculator?
None of them knew, so she took an example, and wrote down the fraction 1/4 . She said:
The way we write it - it contains the numbers one and four. What can you do with one and four on the calculator?
Gus recorded what he could do in a very systematic way; he wrote:
4 + 1 = 5
4 x 1 = 4
4 - 1 = 3
4 ÷ 1 = 4 		1 + 4 = 5
1 x 4 = 4
1 - 4 = -3
1 ÷ 4 = 0·25
Then he said: I think a quarter is 0·25. And he checked it in two ways:
0·25 + 0·25 + 0·25 + 0·25 = 1
0·25 x 4 = 1
 

Content

  • decimal calculations
  • decimal interpretation
  • decimal representation of a fraction
  • division
  • fraction calculations
  • fractions as a partition of a whole
  • multiplication
  • number line - ordering, operations
  • operations - whole number
  • pattern generalisation
  • pattern interpretation
  • pattern recognition
  • place value
  • recording - calculator
  • recording - written
  • visual representation of fractions
This short story offers much more than expected. For example:
  1. The children were very young - younger than curriculum documents would expect them to be when learning this mathematical content - but they were interested and understood.
  2. The open-endedness of the questions:
    How could we show fractions on a calculator?
    What can you do with one and four on the calculator?
    suggests a learning adventure together, with a teacher who expects the children to learn for themselves. It also encourages children to express their insights. Such insights can guide the teacher's next teaching step.
  3. The teacher's presentation is best practice from this point of view, but also best practice because it encourages children to work like a mathematician. These elements of the activity:
    • An interesting problem.
    • Time to play with it, collect and organise data and make hypotheses.
    • Application of the mathematician's question: Can I check this another way?
    are identified by mathematicians as key elements in their work.
  4. Is one experience like this sufficient to establish learning? Could we investigate a 'fraction a day' three times a week for three weeks - ie: threading.
  5. Manipulating digits to produce 0·25 is one thing, but where does this number fit in the big picture of 'all' numbers? Would it help to support this activity by asking students to position 0·25 on the number line. Can you use Making Number Lines to develop this idea further?
  6. Did you notice that Gus didn't seem to be perturbed by negative numbers? Where would they go on the number line??
The CAN Report offers another example with slightly older children that verifies the importance of personal exploration and the intercession of the teacher.
One group became very interested in the remainders in division, and on their calculator checks. They had:
43 ÷ 2 = 21 rem 1 ... Check: 21·5
41 ÷ 3 = 13 rem 2 ... Check: 13·666666
I asked them about 40 ÷ 3. That was easy: 13 rem 1. On the calculator? 'Thirteen point something.' They did the following calculations mentally and checked them on the calculator:
40 ÷ 3 = 13 rem 1 ... Check: 13·333333
39 ÷ 3 = 13 rem 0 ... Check: 13
38 ÷ 3 = 12 rem 2 ... Check: 12·666666
They were able to predict the next calculation:
37 ÷ 3 = 12 rem 1 ... Check: 12·333333
and they continued the pattern down to 27 ÷ 3 = 9.

I returned and asked: 'Why ·333333 and ·666666?' They were not sure. We recalled:

49 ÷ 2 = 24 rem 1 ... Check: 24·5
They said that point five is half, and remainder one when dividing by two is a half. One of them then suggested that ·333333 was three quarters. 'No,' the other one said. 'Point seven five is three quarters and point two five is one quarter.'

So we got out the fraction cakes and shared them between three people. they realised that 4 ÷ 3 = 1 rem 1, and in dividing by three, remainder one is one third. We returned to the calculator. and recorded:

40 ÷ 3 = 13 rem 1 ... Check: 13·333333
The children said: '·333333 is one out of three, or a third.'
41 ÷ 3 = 13 rem 2 ... Check: 13·666666
The children said: '·666666 is two out of three, or two thirds.'
Again, one experience will not be enough to cement this learning, but it does suggest that the calculator can be used to successfully support exploration of 'advanced' mathematics at a young age through the power of pattern, another key element in a mathematician's work. This second story also confirms the importance of the interplay between calculator exploration and concrete material confirmation.


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