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### Number and algebra

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### Advanced mathematics

# Triangle Transformation

### Why do this problem?

When meeting area and considering methods for working out areas of triangles for the first time, it is tempting to present a formula and invite children to apply it. In this problem, learners are invited to consider that cutting and rearranging a shape does not change its area.

Along the way, there are opportunities to practise vocabulary associated with triangles, and apply knowledge of the properties of rectangles in order to generalise.

We hope children will be curious to explore the generality as they wonder whether this can be done with any triangle.

### Possible approach

For some students, you may wish to set the problem exactly as it appears in the task:

"Draw a triangle on a piece of paper."

"Can you find a way to cut your triangle into no more than four pieces, and reassemble the pieces to make a rectangle?"

Some students might require a little more scaffolding. This could involve providing some templates of particular triangles to begin with, or sharing this prompt from the problem:

*Start with an isosceles triangle. How could you make it into a rectangle?*

Is there a relationship between the base and height of your triangle, and the base and height of the rectangle?

While students are working, circulate and listen to any discussions they are having. Listen out for key vocabulary and insights about the properties of the shapes they are working with. After most students have created at least one "triangle-rectangle jigsaw", bring the class together to discuss what they have discovered so far. Invite students with useful insights to share them.

Next, introduce the following idea:

"I wonder if it's always possible, no matter what triangle I start with..."

During the next phase of the lesson, you could invite students to share any triangles they have not been able to dissect and rearrange into a rectangle, so that others can have a go.

This could provide a good opportunity to think about angles and side lengths, or a more informal approach considering "long, thin scalene triangles", "short and fat triangles" or other classifications suggested by students.

To develop a convincing argument of the generalisation that all triangles can be cut up and rearranged to make a rectangle, these two prompts from the problem might be useful:

*Draw any triangle, find the midpoints of two of the sides, and join them together. What do you notice?*

Once you have joined the midpoints, can you rearrange the pieces to make a parallelogram? Does that help you to create a rectangle?

To finish the lesson, bring the class together to share their reasoning and insights. It might be appropriate to leave this as a "simmering task", with space on a "working wall" for students to contribute their thoughts over time. The activity could then be revisited in a future lesson to draw some conclusions.

### Key questions

Here are some questions which might tease out students' curiosity while they are working on the task:
### Possible extension

For a challenging extension, students could work on Squaring the Rectangle, working towards a proof that any triangle can be cut up and reassembled to make a square with the same area.

### Possible support

Students could draw their triangles on square dotty paper (available on the Printable Resources page), and work out area by counting squares, and then explore possible rectangles with the same area.

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Age 7 to 14

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Along the way, there are opportunities to practise vocabulary associated with triangles, and apply knowledge of the properties of rectangles in order to generalise.

We hope children will be curious to explore the generality as they wonder whether this can be done with any triangle.

"Draw a triangle on a piece of paper."

"Can you find a way to cut your triangle into no more than four pieces, and reassemble the pieces to make a rectangle?"

Some students might require a little more scaffolding. This could involve providing some templates of particular triangles to begin with, or sharing this prompt from the problem:

Is there a relationship between the base and height of your triangle, and the base and height of the rectangle?

While students are working, circulate and listen to any discussions they are having. Listen out for key vocabulary and insights about the properties of the shapes they are working with. After most students have created at least one "triangle-rectangle jigsaw", bring the class together to discuss what they have discovered so far. Invite students with useful insights to share them.

Next, introduce the following idea:

"I wonder if it's always possible, no matter what triangle I start with..."

During the next phase of the lesson, you could invite students to share any triangles they have not been able to dissect and rearrange into a rectangle, so that others can have a go.

This could provide a good opportunity to think about angles and side lengths, or a more informal approach considering "long, thin scalene triangles", "short and fat triangles" or other classifications suggested by students.

To develop a convincing argument of the generalisation that all triangles can be cut up and rearranged to make a rectangle, these two prompts from the problem might be useful:

Once you have joined the midpoints, can you rearrange the pieces to make a parallelogram? Does that help you to create a rectangle?

To finish the lesson, bring the class together to share their reasoning and insights. It might be appropriate to leave this as a "simmering task", with space on a "working wall" for students to contribute their thoughts over time. The activity could then be revisited in a future lesson to draw some conclusions.

- Why did you choose that triangle? What else could you try?
- (For a particular example) Is it impossible? Or have you just not found a way to do it yet?
- Are there different methods for different types of triangle?
- Is there a method that works for lots of different triangles?
- To make a rectangle from your triangle, what do you need?
- How do you know that your rearranged shape is a rectangle?

For a challenging extension, students could work on Squaring the Rectangle, working towards a proof that any triangle can be cut up and reassembled to make a square with the same area.

Students could draw their triangles on square dotty paper (available on the Printable Resources page), and work out area by counting squares, and then explore possible rectangles with the same area.