# Number & Computation C Years 2 - 8

A Mathematics Centre Resource
You are encouraged to contribute to these notes. Comments from teachers or students, photos and student work samples will enrich the professional experience for all of us.

Introduction

### Materials

Picture Puzzles have multiple levels of success. They do not have to be 'finished'. They can be revisited and continued.

### Introduction

Fractions make no sense without the whole. To be able to succeed at any fraction exercise I have to know what the whole is, or I have to be able to discover what the whole is, or I have to create a whole that suits my need. The alternative is to know a procedure but not know why I am doing it - something illustrated quite clearly in one student's work using one of these Rod Mats puzzles as reported by Madonna Pianegonda in her article Picture Puzzles - Rod Mats.

It is possible to replace 'Fractions. Oh no!' with 'Fractions. Aha!'.

Rod Mats challenges involve choosing a Cuisenaire Rod as the whole, then creating rows of equal parts parts to match that whole. The rod mat created is used to explore the concept of a fraction, equivalence of fractions and operations with fractions. This is the visual and kinaesthetic imperative that helps to create understanding. But there's more...

Fractions have their own language - not a symbolic language, a natural language which gives the symbols sense. Consider a whole which can be split into eighteen equal parts. How do I know what name to give to each part? What needs to go on in my head is:

• I know what the whole is.
• Pieces this size split the whole into equal parts.
• There are eighteen parts so I can say eighteenth.
• So one of these parts is one eighteenth.
Apart from halves and quarters (and perhaps the Americans are right to use fourths) every fraction name makes linguistic sense. (Pity that twoths wouldn't work.) So isn't it even more sensible to begin by recording fractions in words, as a bridge to the symbols, by creating a link between what is seen, touched and said, before introducing one number above a bar that is above another number? Working this way, even quite young children begin to shortcut the words and use, or are ready to be shown, the appropriate symbols. Then the symbols start to make sense too.

This menu of Rod Mat Picture Puzzles is built around visual, kinaesthetic and language based learning in co-operation with a partner and supported by technology. These puzzles are not the 'be all and end all' of investigating fractions, but they are, as Madonna's article confirms, a valuable component.

### The Puzzles

There are five Picture Puzzles in this menu.

 They are all structured in the same way. In each case Slide 3 informs students they will be asked to: Make a Rod Mat from a whole. Name the parts of the whole shown by the mat. Find more than one name for some parts. Create and record equations using your names.

 What changes is the fractions shown by each chosen whole. For example if Orange is whole the smallest fraction explored is tenths and for this whole rods can also be found to show halves and fifths. Change the whole and the fractions available also change. The smallest fraction in each slide show is shown on the title slide. In addition halves, for example, appear when Brown is whole and fifths appear when Brown joined with Black is whole. Therefore, exploring more than one of these Picture Puzzles helps students realise that fractions aren't about absolute size (a half can be smaller than a quarter), but rather, size relative to a whole. Slides 4 - 9 in each puzzle lead to type of request shown here. Slide 10 in each puzzle confirms (or corrects) what the students have made.

 Now there is a visual, kinaesthetic starting point for talking about the fractions displayed in any rod mat. The next step is for students to tell each other the fraction name they would choose for each colour of rod in a mat.

 Students do come to this exercise 'knowing stuff'. They don't have to be 'taught' before using the slide show. Most will be able to work out the appropriate name by drawing on language used in their environment (quarters in football for example) and the explanations on the slides to date. Those who are unsure will ask and the teacher will be able to join them to see, touch and talk. As Madonna points out in her article linked in the Introduction above, working with Picture Puzzles creates time for the teacher to assist those who need it. Telling each other how they know is also expected. Slides 13 - 17 develop and refine this metacognition into a language 'script' that becomes common across the class. This sequence models a conversation between two other students which the current students learn to have with each other. It leads to Slide 18 in each puzzle.

 Yes, in the first one it could be said that we know pink is one half because 'there's two of them and they're the same', but this 'script' is not transferable to other fractions. It begins with a piece of common knowledge rather than a principle applicable to all cases. The modelled conversations are actually teaching students what they can say to themselves whenever they need to decide 'what the fraction name is'.

 Eventually this sequence leads to recording a conversation without the help of the students named in the slides. Madonna's article has a photo of a student journal at this stage. Madonna has captioned the photo Easily scaffolded into the task. This same linguistic scaffolding is now used to tackle problems involving equivalence and operations.

 Note that the students are not being asked to operate on fractions. They are being asked to find another fraction language name for a rod, or combination of rods. The operators are being used as a natural way of combining the rods. (Yes they imply some previous knowledge, or require the students to ask at their point of need.)

Also note:

• the gradual, non-threatening, introduction of symbols
• recording that encourages further metacognition
and all of this is still happening in this visual, kinaesthetic, conversational context.

Some students will carry out these calculations without reference to the rods because they 'already know'. Madonna provides an example of a child in her class who had an aha moment about why the rules work when challenged to picture their symbolic processing with the rods.

There may be more than one answer to the problems on the slides - an opportunity to ask the students' opinion of what they think might be the simplest answer?

 Each Rod Mats Picture Puzzle also explores the concept of the complement of a fraction. This concept is sometimes useful in solving problems. For example anyone who has used Task 205, Peg & Tape Fractions, or Maths300 Lesson 33, Fraction Estimation, will know that placing a peg on a length of cord at what is estimated to be one third, can be easily checked by looking at the complement. If the peg is correctly at one third, the complement will be two thirds, which when folded in half would be two lots of one third, thus defining one third. If this folded distance is the same as the distance from the start to the peg, then the peg must have been correctly placed at one third.

 The next section of the slide show encourages students to use first two, then three, then any number of rods from the mat to make and record their own equations. Expect interesting outcomes here. As students become more confident they will create constructions that are bigger than the whole and consequently enter the world of mixed numbers. Sometimes you will also hear insights such as this one which came from using Brown as whole. Three eighths + one half + one quarter + three eighths = one whole and a half ...and that's three quarters of two wholes. Whenever possible sit with groups from a moment, ask the students to explain some of their recording and look for questions to challenge and extend.
• What happens if you add another ___rod to that?
• What happens if you subtract ___rod from what you have?
• Could make the same fraction using rods that are all the same? ... What equation would you write now?

 The final challenge in each Rod Mat puzzle is to think outside the available mat. This is the basis of carrying out symbolic addition and subtraction. What is the whole that will suit my purpose? If I need to add fifths and fourths then I need a whole that will show both simultaneously. Using graph paper here allows students to make an hypothesis by sketching a whole rectangle 'rod' and seeing if it can be split into both fifths and fourths.

Once the right length is discovered the number of cells in the whole will be the common conversion element - the 'white rod'. With appropriate experience, it is quite easy to move from here to the more traditional algorithm, if necessary, and make sense of the term 'common denominator'.

### Extensions

1. Turn Rod Mats into a game along the lines of Task 203, Make The Whole.
2. Encourage students to draw their own wholes on graph paper and explore their parts.
3. Ask students to create a word problem for which the final slide in each puzzle is the equation.
4. Encourage students to choose their own fractions to add and subtract and operation on in other ways if they wish.
5. Maths300 members could integrate this menu with Lesson191, Fractions & Fraction Charts, which is also 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 these Picture Puzzles and Lesson 191.