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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Rearranged Rectangle

## Rearranged Rectangles

Talk to someone else about what you see in this picture:

Click below to read what some other children said:

Meena:

"It's a rectangle! Or is it a square?"

Seb:

"There are lots of different coloured rectangles. They are fitted together with no gaps."

Anna:

"There is more than one of each colour rectangle."

Do you agree with each of them? Why or why not?

In fact, the picture has been made using Cuisenaire rods.

How many other, different, rectangles can you make using all of the rods that were used in the picture?

How do you know you have found them all?

You could use the interactivity below to try out your ideas if you haven't got any rods.

### Why do this problem?

This task gives children the chance to work systematically in a spatial and/or numerical context.

### Possible approach

If your children are not already familiar with Cuisenaire rods, it is essential to give them time to 'play' with the rods before having a go at this activity.

Show the group the picture and invite them to talk in pairs about what they see. Take some feedback, writing up everything that is said on the board without responding, but encouraging other children to comment. Then reveal what Meena, Seb and Anna say in turn, giving time for learners to talk to their partner about whether they agree or not, and why. Use this discussion to clarify that ten 'rods' have been used to make this arrangement with no gaps, some of which are the same and some of which are different from each other. In this case, the rods form a square (which is a rectangle with equal length sides).

At this point, introduce the fact that the image was created with Cuisenaire rods and make sure that pairs of children have access to 'real' rods, or to the interactivity on a tablet or computer. Pose the challenge, "How many other, different, rectangles can you make using all the rods that were used in the picture?". Try not to give any further information at this stage and invite pairs to have a go.

As you move around the room, watch out for the different ways that learners are beginning the task. You may find that some children work in a purely geometric way, moving rods around and trying out their ideas spatially. However, others might approach the task in a numerical way by assuming that the white rod has a length of 1 unit, meaning that the area of the original square is 36 square units. They may therefore look at factor pairs of 36 and then use the rods. Neither of these approaches is better than the other (and you may well find some learners use a different method again), but you may want to stop the group and draw attention to these different ways of working. It would also be useful to have a conversation about how each pair is defining 'different'. Does 'different' in this context just mean rectangles of different dimensions? How about two rectangles of the same dimensions but made by arranging the rods in different ways and/or in different orientations?

Allow more time for pairs to work together, encouraging them to consider how they will know that they have found all the possible rectangles. In the final plenary, focus on the different ways of working systematically you have observed, whether it is in a geometrical or numerical sense, or both. You could ask learners to create a poster outlining what they have done and the conclusions they have reached to contribute to a classroom display.

Key questions

What have you done so far?

How are you recording your thinking and/or your solutions?

What do we mean by 'different'?

How are you making sure you don't miss out any possibilities?

### Possible support

Having access to Cuisenaire rods or the interactivity will mean that learners can try out their ideas freely, and refine their approaches, before committing anything to paper should they wish.

### Possible extension

Encourage learners to ask their own 'what if...?' questions and to choose one to investigate. Examples might include:
## You may also like

### Geoboards

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Age 7 to 11

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Talk to someone else about what you see in this picture:

Click below to read what some other children said:

Meena:

"It's a rectangle! Or is it a square?"

Seb:

"There are lots of different coloured rectangles. They are fitted together with no gaps."

Anna:

"There is more than one of each colour rectangle."

Do you agree with each of them? Why or why not?

In fact, the picture has been made using Cuisenaire rods.

How many other, different, rectangles can you make using all of the rods that were used in the picture?

How do you know you have found them all?

You could use the interactivity below to try out your ideas if you haven't got any rods.

Show the group the picture and invite them to talk in pairs about what they see. Take some feedback, writing up everything that is said on the board without responding, but encouraging other children to comment. Then reveal what Meena, Seb and Anna say in turn, giving time for learners to talk to their partner about whether they agree or not, and why. Use this discussion to clarify that ten 'rods' have been used to make this arrangement with no gaps, some of which are the same and some of which are different from each other. In this case, the rods form a square (which is a rectangle with equal length sides).

At this point, introduce the fact that the image was created with Cuisenaire rods and make sure that pairs of children have access to 'real' rods, or to the interactivity on a tablet or computer. Pose the challenge, "How many other, different, rectangles can you make using all the rods that were used in the picture?". Try not to give any further information at this stage and invite pairs to have a go.

As you move around the room, watch out for the different ways that learners are beginning the task. You may find that some children work in a purely geometric way, moving rods around and trying out their ideas spatially. However, others might approach the task in a numerical way by assuming that the white rod has a length of 1 unit, meaning that the area of the original square is 36 square units. They may therefore look at factor pairs of 36 and then use the rods. Neither of these approaches is better than the other (and you may well find some learners use a different method again), but you may want to stop the group and draw attention to these different ways of working. It would also be useful to have a conversation about how each pair is defining 'different'. Does 'different' in this context just mean rectangles of different dimensions? How about two rectangles of the same dimensions but made by arranging the rods in different ways and/or in different orientations?

Allow more time for pairs to work together, encouraging them to consider how they will know that they have found all the possible rectangles. In the final plenary, focus on the different ways of working systematically you have observed, whether it is in a geometrical or numerical sense, or both. You could ask learners to create a poster outlining what they have done and the conclusions they have reached to contribute to a classroom display.

Key questions

How are you recording your thinking and/or your solutions?

What do we mean by 'different'?

How are you making sure you don't miss out any possibilities?

- What if I take out one/two/three... of the rods, will I still be able to make rectangles?
- What if I have to have rods that are both horizontal and vertical in my rectangle?
- What if I can only arrange rods horizontally/vertically?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.