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Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

Square It

Stage: 3 and 4 Challenge Level: Challenge Level:1

Why play this game?

This game offers an excellent opportunity to practise visualising squares and angles on grids and also encourages students to look at strategies using systematic approaches. Describing strategies to others is always a good way to focus and clarify mathematical thought.

Working with tilted squares provides an opportunity to examine the properties of gradients of parallel and perpendicular lines. This can lead on to Square Coordinates and Opposite Vertices, or to Tilted Squares, for students who are going on to work on Pythagoras' Theorem.

Possible approach

You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.

Start with a demonstration playing against a student rather than against the computer. Students are often surprised when the winning square isn't aligned with the grid. This leads to discussions about what makes a square a square.
After a demonstration of the game, students could be left to play for a while in pairs, either on the computer or on a paper grid. Give them the option of reducing the size of the board if they seem overwhelmed!

Bring the class together for a discussion of their thoughts on the game. Did anyone consistently win or lose? Can anyone think of any good strategies which might help them win?

Once ideas have been shared the group can return to playing in pairs, or they can play a game together against the computer, trying, as a class to decide on the best move at each stage. Ask each student to explain the reasoning behind the moves they choose.

One aspect of developing a winning strategy that could be considered is the number of distinctly different starting points ($6$ on a $5 \times 5$ board) and the number of different squares that can be drawn that include each of those points. That is, "Is there a good place to start and why?". This is a great investigation, with the capacity to expand by changing the sizes of the starting grid, and which leads back into the game itself.

Working on the properties of a square offers an opportunity to look at gradients to establish whether a square is a square.

With classes who never arrive together or on time, this and other interactive games can be used on the Interactive White Board to engage the early arrivers and set up a relaxed mathematical atmosphere.

Key questions

  • Is your move a good one? Why did you make it?
  • Why do you think the computer made that move? Was it a good one?
  • How do you know this is a square?

Possible extension

The computer follows an algorithm (which may or may not be random) to place its pieces.
By studying the moves over a series of games can you work out the computer's strategy?
Do you think that it is random or deterministic (i.e. the computer will always play in a certain position given a certain configuration of pieces)?

Suitable follow up problems are Square Coordinates and Opposite Vertices, which encourage exploration of the relationship between coordinates of the vertices of squares.

Possible support

The problem can be built up gradually from a $16$ dot board and a $25$ dot board to the $36$ dot board in the question.
Students could experiment making different squares using the interactivity in Square Coordinates

The group could be asked to draw examples of the different squares on their specific board size, and to compare notes to check for wrong or omitted solutions.

Some students might find 'believing' in the tilted squares difficult. On paper they could use a set-square to convince themselves that the angles in a shape are $90^\circ$, or they could be encouraged to cut the shapes out and move them around to see if the cut-out really looks square.