## Fractions: The Whole Thing

#### Various Teachers & Levels Catholic Diocese of Canberra Goulburn

Fractions make no sense without the whole. Now there's a common sense statement; but if it is self-evident why is so much traditional teaching of fractions, beyond the first colouring or cutting exercises, disconnected from the whole in favour of symbol manipulation?

The snapshots below are examples from an alternative approach guided by the question:

How do you know it's a ..th?
Which expects the responses:
• I know what the whole is?
• The whole is broken into equal parts by ...
• There are ... of them so each one is one ..th

Following Days 3 & 4 of Working Like A Mathematician presented by Mathematics Centre at the request of the Diocese teachers planned, executed and evaluated a unit on Fractions & Decimals. At every level teachers reported successful learning for all children and satisfying teaching for themselves. This article offers snapshots of their work. You will find more in depth samples of their inspired fraction teaching at these links:

#### Kindy (Year 0)

 Toni, Good Shepherd Primary School, Amaroo, created 'The Whole Story' Big Book above as the stimulus for fraction unit. She drew pictures and wrote about every 'whole thing' she could think of that the children might have experienced. Of course they loved it when she read it to them and because it was a Big Book groups could gather on the floor afterwards and enjoy it for themselves. She also created a QR code and fixed it to the back. The children used an App on their tablets to read the code. It led to the class web site, which had activities about the book. A little later, Toni created a half size Half Story book and the process began again.

 Melanie and Nickey at St. Mary's War Memorial School, West Wyalong, were creative in a different way. Melanie made lots of number cards and some Yes/No cards. Children chose a card, and guessed whether a collection of that number of Poly Plugs could be made into halves - Yes or No? Having guessed, they arranged their plugs to check whether the collection could be divided into two equal parts. Small numbers were easy, but what happens if we have a really big number of plugs, for example, 29? Another creative group of teachers, this time from Geeveston District High School, Tasmania, have created an activity called Fraction Game which also uses cards. It is for Years 3 - 6 and is stored in the Free Tour section of Calculating Changes.

#### Year 3

Lisa, Sacred Heart Primary School, Pearce built a rich program around several investigations which she married with encouraging students to keep a journal of their learning, just like a mathematician needs to keep a record of their investigations.
Students add to their Maths Journal after each lesson. After several lessons, they have accumulated many ideas, including diagrams and captions. The students are encouraged to record their personal reflections on concepts that they have learnt and understood.
Lisa contributed these sample pages from student journals to show us how important language is to mathematics learning. (Click an image to reveal its enlargement.) Notice that Lisa has frequently commented and encouraged in the journal. To what extent do you think this written interaction leads to better learning?

• Visit Exploring Fractions from another Year 3 class for a detailed program which also highlights the value of journal writing and includes assessment examples.
• In Fractions in Action you will find an extensive Year 4 program of fraction activities from Calculating Changes, Mathematics Task Centre and Maths300 presented as slide shows from the teacher.

#### Years 5 & 6

 Jess, Year 5, St. Patrick's Primary School, Gundagai asked her students to identify the strategies they used to place pegs along a tape in Peg & Tape Fractions, a task from Mathematics Task Centre with a companion lesson (Fraction Estimation) on Maths300. The companion lesson is supported by software. Damien, Sacred Heart Primary School, Pearce, had a WOW moment with Year 6 when using the Chocolate Cake activity described in Fractions in Action. He wrote: When I was teaching the chocolate cake game I asked the students to explain how we can calculate the fractions that each table is showing us. One girl decided to make the fraction by holding the blocks of chocolate over their heads. If three people are standing around two blocks of chocolate, we get the blocks and place them above their heads. We have then created the fraction two over three. Wow! A fantastic teaching moment. The Iceberg section of the Task Cameo for Making Fractions 3 also explores this visual way of calculating fractions. The first example is exactly the same as the one Damien describes here. Damien built an extensive unit of 6 to 8 lessons and has provided samples of student assessment responses.

Click an image to see its enlargement.

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