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Fair Shares

Achievement Objectives:

Achievement Objective: NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
AO elaboration and other teaching resources

Specific Learning Outcomes: 

Find halves and quarters of sets, regions and objects by sharing.

Find simple fractions of regions.

Find fractions of sets by sharing.

Description of mathematics: 

Number Framework Stages 3 and 4

Required Resource Materials: 
Plastic glasses
Play dough
Paper circles
A length of paper
Sliced bread (to make sandwiches)
Paper bags of 20 cubes
Plastic jars (for example, peanut butter jars)
Plastic teddies

The students must come to know common vocabulary for fractions, particularly halves, thirds, quarters (fourths), fifths, and tenths.Initially, the emphasis is on unit fractions, like 1/2 and 1/4, that have one as the numerator. However, it is important to introduce non-unit fractions, like 3/4 and 2/5, when learning opportunities permit.

It is important to expose the students to both continuous models, such as lengths and regions, and discrete models, using sets of objects. A significant development for the students is to use their whole number strategies to anticipate the result of equal sharing. This is easier with halves and quarters than with thirds and fifths, as halving is linked to doubles addition and subtraction facts.

Using Materials

Problem: Show the students the length of paper and tell them that it is a strap of liquorice.“I have this big liquorice strap, and I’m going to cut it in half to share it between Tyler and Briar (two students from the group). I’m going to run my scissors along the strip, and Tyler and Briar will tell me when to stop.”

Run the scissors slowly along the strip until one student calls, “Stop.”

Cut the strip at that place and give the caller the passed over part and the other student the remainder.

Discuss whether the sharing has been fair (it may not be). Ask for other ways to cut the strip in half so that each half is the same length. Folding will be a likely reply. The two pieces can be placed beside or on top of each other to check that they are equal.

Record one-half using both words and the symbol, 1/2 . Discuss why the number two is the denominator. The denominator gives the number of equal parts that one (whole) is divided into.

Tell the students that you want them to find half of some other things.

Set up the following examples:

“Cut the blob of play dough in half.”

“Pour half of the water into one glass, leaving half in the other.” (Start with two glasses, one full of water.)

“Cut this pizza in half.” (Start with a paper circle.)

“Give half of these lollies to Tyler and the other half to Briar.” (Start with a bag of cubes.)

“Cut this sandwich in half.”

Let the students attempt the challenges in small groups. Discuss the ways they shared each item fairly.

Broaden the discussion by posing each of the previous challenges but requiring the students to partition into quarters. For example:

Put one-quarter of the water in each glass. (four plastic glasses)

Some students may discover that halving one-half gives one-quarter.

Record one-quarter using words and the symbol, ¼.

Using Imaging


Fill a plastic jar to the top with equal-sized plastic teddies. Count out the teddies to see how many the jar holds when full. Provide each group of students with a plastic jar, a thick rubber band, and one teddy. Tell them that they are to work out how far up the jar half of the teddies would come and mark where that level would be.

Putting the rubber band around the jar and sliding it up and down is a good way to show the level.

Discuss their predictions, looking for the use of number knowledge, like simple doubles, to estimate half of the teddies and for equalising in their prediction of the level.

The jar can be filled and the teddies shared into two equal sets, counted, and put into the jar to confirm the students’ predictions. Pose the same problem requiring quarters, thirds, or fifths to broaden the students’ knowledge of fractions. Note that finding thirds and fifths of shapes and lengths can be difficult as it involves concepts of angle, area, and volume. Students’ inability to equally share may be due to these factors rather than the fraction concept. Encourage students to work from part to whole, that is, given a fraction they have to recreate the whole. For example:



“This is one-quarter of a shape. “This is one-third of a set.

What is the shape?” How many objects are in the set?”

Using Number Properties

The students’ lack of strategies for division will limit their ability to apply numberproperties. Examples provided should include simple halving and quartering. For example: half of six (1/2 of 6), one-quarter of eight (1/4 of 8).