Fraction circles

Achievement objectives

NA2-5: Know simple fractions in everyday use.

Required materials

• paper circles or commercial fraction kits

Activity

Background maths

The students must realise that the size of a fraction involves coordinating both the "numerator" (top number) and the "denominator" (bottom number). This principle is developed strongly in situations where fractions are ordered by size with the support of materials.

Students also need to appreciate that fractions are both numbers and operators. It is vital to develop an understanding of the “home” of fractions among the whole numbers.

Using materials

Problem:

• Let’s pretend that the circle you have is a pizza. Show me how you could cut your pizza into halves.

Get the students to cut their circles into halves in any way they wish.

Many will choose to fold the circle. Show them some circles that have been folded into two pieces unequally.

Ask,

• Are these pizzas cut in half?
• How do you know?

Look for the students to note that the halves must be equal or the cutting is unfair.

Record the symbol 1/2.

Say,

• This is a way to write one half. Where do you think the two on the bottom come from?

The students may make the connection that “2” is the number of pieces that one circle is divided into. Get the students to label their halves using symbols and to cut out the pieces (across the circles).

Ask,

• What do you think would happen if I put three halves together?

Invite the students to give their ideas and then confirm them by joining halves.

Ask how they think the symbols for this new fraction could be written.

Ask the students what the three and the two mean in the fraction 3/2 and relate this to the number of parts in one whole (pizza) and how many parts are chosen.

Using circles of the same size as models, get the students to cut another pizza into quarters. Label these, using the symbol for one quarter (1/4).

Ask them why four is the bottom number. Point out that quarters are sometimes called fourths.

Get the students to predict what three-quarters and seven-quarters will look like.

Check their predictions by putting together the circle pieces.

Record the symbols (3/4 , 7/4). Next, fold the circles into eighths. Get the students to fold quarters, and then predict how many parts will be formed if they fold the fractions in half again. Have the students record their answers using the symbols cut out the pieces, and form fractions with more than “1” as the top number.

Using imaging

Shielding: Model comparing the size of fractions by placing some pizza pieces under paper plates. For example, place one-half under one plate and three-quarters under another. Label the plates by writing the symbols 1/2 and 3/4 on top. Ask the students which plate has the most pizza under it.

Listen for the students to use benchmarks for comparison. Often these involve equivalent fractions, such as “Two-quarters is one-half, so three-quarters will have more.”

Using number properties

The students have a good understanding of coordinating the numerator and denominator of fractions when they demonstrate that they do not need materials or images to make comparisons. Their justifications will relate to relationships within the symbols. For example, “Three halves are one and a half. Five-quarters is one and a quarter. A half is bigger than a quarter.”

Note that halves, quarters, and eighths are particularly suited to the students at the early stages, as they easily relate to doubling and halving.

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