Ratio reasoning in year 2: Jelly Babies
What follows is an example of the episode structure of one LTTM! lesson, taken from the Year 2 to 4 materials.
Episode 1: Comparing pictures, focusing on heights of parts
First the teacher tells a story about a factory wanting to make jelly sweets in two flavours. Pupils discuss whether it is a good idea and vote on the flavours they prefer (concrete preparation). Then they handle a picture of a jelly sweet shape that has an equal-sized head and body in two colours indicating flavours. They compare that with another picture where the head is only a quarter of the size of the body. They choose names for these, normally ‘jelly baby’ and ‘jelly daddy’ or ‘jelly man’, then they address the question: ‘What is different about them?’. This is the start of the cognitive conflict.
Pupils notice that the two figures are the same height, but that in one the head is as big as the body, while in the other it is much smaller (see Figure 5.2). The teacher asks them to check if the head and body are indeed the same height in one and smaller than the other by using their fingers and/or sticks or strips of paper that they can tear up and compare.
Figure 5.2: Checking heights of jelly figures
Some children notice the stacked boxes and sticks at the side of the jelly figures and start using numbers to describe the relationship. The teacher asks these pupils to explain themselves further and asks other pupils what they understand by it (social construction). By now the ratio of the head to the body is clearly understood to be equal or ‘one-to-one’ in the first and ‘one-tofour’ in the second.
The teacher then introduces a third picture (see Figure 5.3): ‘How different is this?’, leading to naming it, normally as ‘jelly boy’ or ‘jelly girl’.
Figure 5.3: Ratios of head to body in jelly figures
It was noticeable in trials that children easily order the pictures 1:1, 1:2 and 1:4 without using the convention. At least one 6-year-old girl came out with: ‘You can also say the head is one fifth’! It seems clear that external visual ratio, i.e. comparing lengths of two visible things, is an accessible thinking step for most children at this age. It is also the case that some can differentiate this from fractions, where the ratio is a part to the whole, which includes the part. The plenary act in this episode is focused on the agreement of the differences between the three pictures and the use of numbers in their descriptions.
Episode 2: Deciding between three ratio types when the size changes
(As you read this next episode, try to identify the relevant ‘pillars’ of cognitive acceleration.)
Here children apply the newly acquired concepts in a new situation, using the same story context. Working in pairs or small groups, pupils look at nine pictures of different-sized jelly sweets (see Figure 5.4). They decide for each whether it is a ‘jelly baby’, ‘jelly daddy’ or ‘jelly girl’ (if these are the names they had used in episode 1). To help, pupils fold or tear strips of paper or make pencil marks using the head as a measure. They must give reasons for their decisions which may lead to a dicussion of ratios, recorded in various ways, as well as of fractions. Most pupils succeed in this task, and this can be checked in a plenary class sharing.
Figure 5.4: Categorising jelly figures
Then a further question is put to them: ‘How is it that these three lassoed different-sized sweets are all “jelly daddies”?’ (see Figure 5.5). The pupils have to explain, in their own different ways, that size is not important and that what we look at is how the head compares with the body. Whether a child manages to make up a coherent sentence or not, all children would have by now experienced thinking challenges on the relations between lengths or heights, using numbers. Older children may go further.
Figure 5.5: Ratios when size differs
Episode 3: Making a jelly giant — a further ratio
Here we ask pupils to invent yet one more different relation between size of head and size of body. For a jelly giant the head would be even smaller in relation to the body: it might be one-to-six. The pupils again use the height of the head as a unit of measure to find the height of the body.
There are many further possible extensions:
• Give pupils pictures of heads alone or bodies alone in different sizes, with the label of what they are, i.e. jelly baby, jelly daddy, etc, and ask them to complete the figure, using rulers or strips of paper.
• Suggest that the sweet factory is thinking of adding small chocolate buttons to the jelly sweets somewhere half-way down the body. How can they find that position? What about if the button is halfway from top of head to bottom of feet? What is the difference between the two positions?
• Write the ratios for each type of jelly sweet and how to write these as fractions. Then, for the still more able, look at how to explain the difference between a ratio and a fraction. Hopefully some interesting natural language descriptions can be aired to the effect that ratios are when things are compared that are separate, while a fraction is when one is part of the whole. In both cases there is a unit that is used for both parts. The potential for confusion can then be explored, since a fraction is really a special kind of ratio, where one of the numbers includes the other in itself.
The structure of free-standing episodes linked in a progression towards higher order thinking is used in all LTTM! and PCAME lessons, and implicitly in all cognitive acceleration lessons. Using the language of conventional maths teaching, these can be described as lessons with a multiple three-part structure, with two or many plenary slots. It is a function of responsive teaching that, whenever the teacher finds some fruitful focus for the particular class’s attention and discussion, he or she would raise a question on which the class can work in pairs or small groups to produce ideas for the whole class to share later. That is a mini-episode. The guidance for the thinking lessons, based on successful trials, allows teachers to retain the direction of the lesson and be aware of the potentials in insights, connections and misconceptions. In the Jelly Babies lesson with an older class there is potential for exploring the link between ratio and fractions, and the notations used in each.