We got out the popsicle sticks this week to help C(9) understand the meaning of fractions like ⁵/₈ of 24.

Memorising Algorithms v. Understanding

C had no trouble calculating one eighth of twenty-four. There’s an easy-to-understand algorithm:

¹/₈  of  24  =  24  ÷ 8  =  3

But algorithms don’t facilitate real understanding.

This became apparent when we changed the numerator of the fraction. C is a bright girl and it wouldn’t have taken her long to memorise the extra step in the algorithm:

3/₈  of  24  =  (24  ÷ 8)  x  3  =  9

But I wanted her to understand what was actually happening here.

Bring on the popsicle sticks!

The equation C was working on at this point was 3/₁₀ of 70 (because of the numbers involved, the answer “30” kept jumping in to her head). This was going to take a lot of popsicle sticks!

A Maths Story

We gathered 70 plain popsicle sticks.  These, we decided, were 70 new children starting out in kindergarten. (Don’t ask me how this metaphor got into our homeschooling house.)

Then we found 10 coloured popsicle sticks. These were helpful grade 5 children who were looking after the kindergarteners for the day.

One by one, we distributed the seventy plain sticks equally between the ten coloured sticks, until each “older child” had seven “kindergarteners” to look after:

¹/₁₀  of  70  =  7

This step, naturally, involved many imaginary conversations between the children, some of whom were rather wilful!

Next step – different numbers of groups of “children” were taken “to see different parts of the school”. Of course it was important to count the number of children at each point, to make sure we didn’t lose any!

So:

• Two of the groups went to see the science room. “Now then, how many children do we have here? Are you all going to fit in the science room? Let’s count. One, two, three … fourteen. Yes, you should be all right. Off you go.”

• Seven of the groups went to see the gym. “How many children is that? Goodness – forty-nine!”

7/₁₀  of  70  =  49

• Three of the groups visited the art studio. “How many paintbrushes do we need for you all? One, two, three … twenty-one.”

3/₁₀  of  70  =  21

And so on. So much fun!

From Concrete to Abstract

C went on to complete a page of abstract problems with no trouble. As she practised, she came to recognise the algorithm as a convenient shortcut.

3/₈  of  24  =  (24  ÷ 8)  x  3  =  9

But thanks to ten minutes of popsicle stick fun, she now understood what the numbers represented.

Do you use stories in your maths lessons?

For more maths ideas, visit the Math Monday Blog Hop at love2learn2day, and the Math Teachers at Play monthly carnival at Learners in Bloom.