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Kissing Triangles

Determine the total shaded area of the 'kissing triangles'.

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Trice

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

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Linkage

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Shapely Pairs

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

The game provides a powerful and engaging context for working on properties of triangles/quadrilaterals. Two other related problems are Quadrilaterals Game and Property Chart

Teachers may be interested in Gillian Hatch's article Using Games in the Classroom in which she analyses what goes on when geometrical mathematical games are used as a pedagogic device.

Possible approach

This game can easily be played in groups. The rules are clear: after a brief demonstration students could be left to play the game. To encourage discussion and peer support, ask students to play as pairs; both must agree on the "final answer" before it counts. Again, to spread ideas and strategies around the class, you could organise a rotation or two so that all pairs move on and play a new pair.

After a period of play, invite the class to share their thoughts on the game. Were there any particularly 'good' cards? any particularly 'bad' card? Are there any mathematical insights that could be discussed?
Allocate the three questions from the problem to different pairs/fours to work on and ask them to report back at the end of the lesson.

Key questions

  • How many different sorts of triangle can be used to fit a particular card?
  • Is your opponents' drawing clear, correct and convincing?

Possible extension

  • How many triples of cards lead to possible triangles?
  • What is the greatest number of cards which will all apply to the same triangle?
  • Make your own version of this game, deciding what to put on the cards.
As an alternative game, group the students into small teams, shuffle the cards, and play it like charades: the only way to give clues to the property on the card is to draw appropriate triangles for the members of your team. Each team could have a minute at a time, and the winning team is the one who gets through most cards.

Possible support

Share out two or three sets of the cards (or big A4 versions) among all the students in the class, show a triangle on the board and ask students to stand if they have a card that describes it. The duplication of cards should generate useful conflict if people with the same card disagree.

It might be useful to have a worksheet available with lots of different triangles as 'ideas' or to save some students having to draw the shapes.