Fractions and Decimals

This week a year 5 teacher asked me to run a mental routine around decimals because some of her class was having difficulty 'getting it'.

I decided to take John Van De Walle's advice when he says it is important that the students learn to count by fractions and that they see the numerator as a noun in the same way as apples are nouns. What this means is that the students count, one tenth, two tenths, three tenths and so on in the same way as we can count, one apple, two apples. Van De Walle also suggests that students should not stop at one whole but should keep counting on eventually making links between fractions and their equivalent whole numbers, such as 4 halves is equivalent to 2..  What follows is the Mental Routine that I used with the class and a few comments on how they went with this approach.

A Kind of Mental Routine

The students worked with laminated 100 squares for this activity.

Closed Questions

How could you divide the board into ten equal shares?

What different ways could you write one tenth (in words, as a faction as a decimal)?

What does the fraction one tenth really mean?

We stopped at this point to review the way fractions are written and that the numerator tells how many shares the whole has been divided into.

I then repeated the process asking the students to mark five tenths of the boards and show as many ways as they could of writing five tenths.

This seemed like the moment to start counting by tenths. We counted once to 23 tenths and this seemed like a lot of fun to the class. I then asked the students to combine boards with a partner and to point at the tenths as we counted. When we got to ten tenths I had to keep the momentum going as everyone stopped counting when they got to the end of their board. This is very much in line with John Van De Walle's comments. We teach students to find a half or a quarter but we always work with only one whole. After years of practicing students find it hard to use mixed fractions. Hence the idea of learning to count by fractions. We repeated the process of counting and pointing a couple of times and then I switched from counting by tenths to counting by whole numbers and tenths, so the count went nine tenths, one whole, one whole and one tenth. The process was not at all boring for the class who asked to count again after one child suggested that we should also use the equivalent half when we got to five or fifteen tenths. We also used this opportunity to introduce a place value grid and to make links between the counting and the number of whole grids and part grids to formalise the method of writing decimals. It was easy for the students to begin to see the role that the decimal point has when they translated their counting and models to the writing of decimals.

Open Ended Questions

Choose a number of tenths that are less than one half and represent it in as many forms as you can. This included using tenths, decimals and a place value chart as well as drawings and helped students to make connections.

This question was repeated using other numbers of tenths including 23 tenths which the students tackled well.

I think we had hoped to go further but we were happy with the number of light bulbs that seemed to go on. Often less is more. A few students moved themselves on to working with hundredths.

Flip Questions

Your job is to guess the number of tenths I am thinking of. Questions such as:

"Do your tenths make more than a half/ a whole one?"

"Do you have more than half but less than seven tenths?"

are to be encouraged and allow time for counting the tenths and eliminating possibilities.

I didn't have time to try this but you might and let me know how it goes.