Triangle in a Square

Why do this problem?

The idea of this task is to give children the opportunity to critique a chain of reasoning. Having this experience will help learners to appreciate what constitutes watertight mathematical reasoning, so they can create their own proofs using words (and images, where appropriate).

This particular example of flawed reasoning also gives learners the chance to deepen their understanding of properties of 2D shapes.

The task An Easy Way to Multiply by 10? offers an identical structure but in the context of place value and calculation.

Possible approach

You may like to begin with a non-mathematical example of flawed reasoning:

Penguins are black and white.
Some old TV shows are black and white.
Therefore some penguins are old TV shows.

Give the class chance to talk in pairs about what is wrong with this, and use it as a springboard to introduce the task. Explain that being able to reason logically is a key skill for a mathematician, and the interactivity is going to give them the chance to 'unpick' someone else's reasoning.

Show the interactivity on the screen or whiteboard, with the Badger's first statement 'A triangle has 3 sides' showing. Invite learners to talk in pairs about whether they think that is true, and crucially, how they would use their mathematical knowledge to convince a mathematician that it was true, or not. Make sure everyone has access to a range of resources but try not to steer them in their choice.

Draw the whole class together to share ideas. If they are struggling to offer suggestions that convince you, click on the 'Pause' button and then reveal some other children's thinking. Do any of these examples give them a starting point? Which examples are not so helpful and why? Clicking on a particular example of children's thinking reveals Badger's response.

Continue in this way, giving everyone chance to consider each statement in turn. Only move on to the next statement when the class has a watertight mathematical argument to support or refute the previous one. In some cases, learners will come up with different chains of reasoning, not necessarily those in the examples, and that is fantastic. The key point is that the logic must be interrogated so that the class is satisfied it is correct.

Key questions

How do we know that statement is true (or not true)?
How could we check whether that statement is true (or not true)?
Are you sure that [this] follows on from [that]? 

Possible support

Having a range of resources available, including access to pencils/paper/whiteboards/pens, will support all learners to access this task. Sharing the children's thinking built into the interactivity will help those who are struggling.

Possible extension

Diagonally Square offers learners the chance to create their own proof, but also includes a proof sorter, which will scaffold their thinking and help them appreciate the key features of a proof.