# Rod Mats

### Task 202 ... Years 2 - 8

#### Summary

Any of the rods in the collection can be the whole, but the starting point example uses the brown one. A mat is made of other rods that fit the whole and are all the same colour. Now the fraction relationship of each colour rod to the whole can be worked out and from that a huge number of fraction stories (equations) can be derived. Students quite naturally find equivalent fractions and operate using fractions. Recording and explaining is essential.

Rod Mats is the starting point for all of the slide shows on the Picture Puzzles Number & Computation C menu where the problem is presented using one screen, two learners, concrete materials and a challenge. The five puzzles use five different wholes which between them include families of fractions based around 8ths, 9ths, 10ths, 15ths and 18ths. You can also read the story of how one teacher used the 10ths Picture Puzzle (Orange is whole) in a Year 6 class.

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

#### Materials

• A collection of rods as described on the card

#### Content

• concept of a fraction
• recording fractions in words and symbols
• equivalent fractions
• addition and subtraction of fractions
• multiplication of fractions by a whole number #### Iceberg

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

The thrust of this task is the generation of mathematical conversation about fractions in a visual and kinaesthetic context. Spoken words are converted to written words and written words are converted to symbols. The task first highlights that in order to use fraction language in a situation you must:

• know what the whole is
• be able to show that the whole is partitioned into equal parts
Once these criteria have been established, deciding which fraction language is based on counting the number of equal parts to make the whole. In the photo:
• pink rods show two halves make the whole
• red rods show four fourths (quarters) make the whole
• white rods show eight eighths make the whole
Student responses to Q. 2 need to indicate that they recognise all of these aspects.
Note: Over time, encouraging the writing sequence of:
• two thirds
• 2 thirds
• 2/3
helps students to understand the purpose of the different messages given by the two numbers that make each fraction.
Since fractions make no sense without the whole, in Question 3 students would need to either state that brown is the whole or include it in their drawings, something like: which shows:
one eighth + three eighths = four eighths ...or two quarters ...or one half

One way to work out subtraction is to place the 'take away rods' on top of the 'starting rods', then measure the uncovered space. So this diagram shows:

1 half + 1 quarter - 3 eighths = ... (3 eighths) Another way is to convert all the 'starting rods' to the same type as the 'take away rods', then do the subtraction. This is the introduction to common denominators. So 1 half + 1 quarter can be converted to 6 eighths. Take away three eighths from that and the answer is again 3 eighths.

The answers to Questions 4 and 5 depend on the students' choices, so will vary. Questions 6 and 7 open the iceberg of this task. Any rod (or collection of rods, for example blue joined to dark green) can be the whole. Each new whole produces a new rod mat and a new discussion and recording session. Over time, students' understanding of fractions and facility with fraction operations develops soundly.

#### 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.

This task can only be converted to a whole class investigation if you have a rod set for each pair. Once you have the resources it may be best to use the activity for a few minutes each day two or three times a week over several weeks. Just choose a different whole each day then: Threading the activity through the curriculum in this way is more reflective both of the history of how fractions developed and were understood and the view that learners need time and challenging experiences to construct their own learning.

The activity can work well in groups of 3 or 4 with one set of rods, poster paper and felt markers. Teams choose a whole and write it on their poster (eg: Dark Green is whole). Then they make the rod mat and if there is enough time can also sketch the rod mat - no colouring in, just hatching, or initials for the colours. Teams are then given say 10 minutes to record all the equations they can find. Groups swap posters and check and discuss each others' work. Posters can be displayed and used for small group or whole class discussions with the teacher.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 144, Rod Mats. Also Maths300 Lesson 191, Fractions & Fraction Charts, is concrete, visual and language-based. In particular, the software presents symbolic challenges which can be worked out with materials or with a 'brain picture', by seeking the whole in the way experienced both in Rod Mats and Lesson 191.

#### 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 Rod Mats task is an integral part of:

• MWA Number & Computation Years 3 & 4
• MWA Number & Computation Years 7 & 8

The Rod Mats lesson is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 7 & 8 