## Sizing Them Up

Arrange these shapes in order of size. Put the smallest first.

How are you deciding on the order?

You may like to use the interactivity below to keep track of the order and/or to record your final solution.

Once you've had a chance to think about it, click below to see how four different pupils began working on the task.

Kelsey said:

"I printed the shapes and then measured the length of each shape at the longest point."

Louise and Rosie said:

"We observed the area of each and tried to rearrange the shapes in our heads to compare them."

Thomas said:

"I cut out the shapes then cut each up into little pieces and laid them on top of each other to see which was bigger. I also put them on a grid with small squares and counted the number of squares for each."

Can you take each of these starting ideas and develop it into a solution?

*You may be interested in the other problems in our Celebrating Solutions Feature.*

**Why do this problem?**

This low threshold high ceiling activity is designed to help children begin to understand the meaning of area as a measurement of surface. It gives them a chance to choose and then justify a way of measuring. It can be solved in many different ways and the sample approaches offer a basis for discussion of possible different methods.

### Possible approach

*This problem featured in an NRICH Primary webinar in November 2021.*

This activity is deliberately open to encourage children to try to define "smallest" for themselves. At this level, the important point is to be able to explain and justify a particular order, rather than there being any right or wrong way to do it. Children might use criteria such as length, height or perimeter, for example. The activity could lead into the introduction of the concept of
area, (even if the word "area" itself is not used).

You could introduce the shapes in the problem using

this sheet or by projecting the image or interactivity onto an interactive whiteboard. Try not to direct learners too much at this stage but make sure they understand that they can use any resources or equipment that they might find helpful. Children could work in pairs with
the shapes.

Once you feel that most learners have made progress and understand the problem well (this does not necessarily mean that they have found a 'final' solution), give out

this Word document or

this pdf. Suggest to the class that when they've finished or can't make any
further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?". It might be appropriate to read through each method as a whole class before giving pairs time to work on each one. Alternatively, you may prefer to allocate a
particular starting point to each pair.

Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued, using the interactivity to keep track of the order that the pairs describe. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You
can then facilitate a discussion about the advantages and disadvantages of each. Which way would they choose to use if they were presented with a similar task in the future? Why?

There is no reason why you should not make your own irregular shapes using the activity as an idea rather than a problem to be solved.

### Key questions

Why do you think this shape is bigger/smaller?

How are you going to decide which is smallest?

### Possible extension

Children could be asked to cut out shapes which they think are 'the same size' but which are very different shapes from those given.

### Possible support

Some children might benefit from cutting out the shapes from this sheet and putting them one on top of another to aid comparison (laminating them may help prolong their life!).