# Fraction Estimation (formerly Peg & Tape Fractions)

### Task 205 ... Years 4 - 8

#### Summary

Estimation and checking are used to involve students in demonstrating their understanding of the concept of a fraction as a part of a whole. Students begin be creating the whole then have to estimate and mark the position of a given fraction using a peg in their own colour. Who is closest? How do we know? Can we check it another way?

Teachers wishing to build a curriculum unit around Fractions are likely to find Nichola Brandon's article Fractions In Action worth researching.

#### Materials

• Two different lengths of cloth tape in two colours
• Four pegs (2 in each of 2 colours)

#### Content

• decimals, number line
• estimating fractions
• estimating number
• fractions, whole & parts
• measurement, length
• mental arithmetic
• number line
• ratio & proportion
• recording mathematics #### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Whether it's teachers in workshops...  Or kids in classrooms...  (Nichola Brandon's Year 4 at St. Benedict's, Narrabundah)

this task seems to hook everyone in.

St. Thomas More's School
At first we thought it was a bit boring but once we worked out what to do it was OK. The task does help/teach people how to estimate fractions mathematically to get close to the target. Jess O. & Jess W, Year 6
There are at least three ways ways to check the estimated fraction by folding:
• Trial, Record & Improve: Hold one end up then loop the tape into a sequence of 'waves' with the right number of sections. 'Jiggle' the sections until they are all near enough to equal. For 2/5 that would mean making five sections and checking if the peg was close to the second peak or trough.
• If/then from the start: The two in 2/5 has to be counted from a particular end. If the peg is correct, the tape can be folded from the chosen end to touch the peg, thus showing 1/5 twice. This defines one fifth which can they be folded over and over the remaining tape towards the end. If the peg was placed correctly in the first place there will be no left over tape when there are five layers of 1/5.
• If/then from the end: If instead of two fifths the requirement was to estimate one fifth, the complement could be used to define the length of one fifth. If the peg is placed correctly, the remaining tape must be 4/5. Folding from the complementary end to touch the peg would create 2/5 twice and folding this again would create 1/5 four times. Now fold the estimated section over the peg. If it is accurate, it will be near enough to the length of the four thicknesses.
Students should make an entry in their journal (writing and drawing) to explain how they folded to check their estimates.

It is expected that students will try estimating several different fractions using the two whole lengths provided.

#### Extension

Provide two tapes the same length to be wholes. Place a peg half way along one whole first, then estimate and clip mini-pegs to show tenths. This will be a reference whole, so the accuracy of the estimate can be checked by measurement so the pegs are as correctly placed as possible.

(Alternatively use a metre ruler as one whole and accurately mark tenths on the blank side guided by the decimetres on the measurement side. Provide another tape that is exactly one metre.) Without reference to the tenths tape, the second whole is pegged in (say) thirds, as in Fraction Estimation (Peg & Tape Fractions). The two wholes are now laid side by side to explore how thirds (or any other fraction) relate to tenths.

• Which fractions can easily be expressed as decimals?
• Which are more difficult?
• In either case, how does the calculator write the decimal equivalent to a fraction?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

With a few lengths of herring bone tape and a couple of packets of pegs (or mini-pegs) a school can easily produce a class set of material to turn this task into a whole class investigation. Use one tape first with the whole class and ask three volunteers to each place their colour of peg where they estimate 2/5 will be. Allow them to stand back and take a second look, altering their estimate if necessary. Briefly discuss strategies used to estimate.

Ask the class how these hypotheses could be checked and carry out any suggestions. (It is actually rare that anyone suggests measuring.)

Repeat the activity with another fraction and another set of three students, then separate the class into groups of 4 - 6 and hand out pegs and tape to each group. The tapes need not be the same length. Ask all groups to try the same task, say 3/7, and discuss how it is achieved and checked. Take a few minutes for students to record in words and pictures, then encourage groups to try their own fractions.

• For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 33, Fraction Estimation, which includes worksheets and companion software.
• Maths300 Lesson 182, Fractions to Decimals (on a rope!) uses the peg and tape idea to develop the concept of a decimal.
• For an extensive unit of work in fractions which includes Fraction Estimation (Peg & Tape Fractions) see Nichola Brandon's Fractions In Action story from the Calculating Changes Free Tour.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Fraction Estimation task is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 5 & 6

The Fraction Estimation lesson which is based on this task is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 5 & 6 