Possible Pairs
For this challenge, you will need to print out a set of triangle cards, or scroll down to use the interactivity.
The first part is a game for two (or more) players; then there is a question you can think about.
This type of game is played with lots of different sorts of cards; you might have heard it called Matching Pairs.
To play this game:
- Shuffle the cards and then lay them face down on the table, arranged in rows.
- Players take it in turns to turn over two cards. If the player can draw a triangle with the two properties shown, then s/he takes the cards. If not, once all the players have had a chance to look at the two cards and see where they are placed, the cards are turned back over.
- The game finishes when no matter which two cards are turned over, there is no triangle with both of those properties.
- The winner is the person with the most cards at the end of the game. Of course it will help you if you can remember where the cards are!
Play the game several times to get a feel for it.
Are there some cards that are particularly 'good'? Why?
Are there some cards that are particularly 'bad'? Why?
Now, suppose instead of having the cards face down we have them all face up.
If it's your turn first, how many possible pairs of cards are there that you could choose and win (that is, in how many ways could you choose a pair so that there is a triangle with both the properties)?
Can you list all the possible pairs? How do you know you have found them all?
If you would prefer not to print out cards, here is an interactivity.
Player 1 clicks on a card to turn it over, then clicks on another card. If the player can draw a triangle with the two properties shown, then s/he can press 'Accept'. The two cards will remain face-up.
If a triangle cannot be drawn, once players have had a chance to look at the two cards and see where they are placed, s/he presses 'Don't accept'.
You will need to keep track of the number of pairs you have 'collected'.
You may like to explore alternative versions of the interactivity by clicking on the 'Settings' icon (the purple cog) in the top right-hand corner.
You may like to have a go at Name That Triangle! as a follow-up to this task.
This problem is based on the Triangle Property Game from "Geometry Games", a photocopiable resource produced by Gillian Hatch and available from the Association of Teachers of Mathematics.
Elizabeth wrote to us to say:
I first was a bit stuck. Then I made a table of the grid. I made it so I would be able to mix all the facts up and try new combinations. I used symbols such as ticks and crosses. I was also surprised at how much time it took.
I discovered that every card on its own can be made into a triangle.
Thank you, Elizabeth, for sharing your method. It is a shame you didn't send us a picture of your grid. We would love to see it if you still have it!
If you have thought about whether there are some cards that are particularly 'good'/'bad' and why, please send us your conclusions, along with your reasoning. Equally, if you have been working on the number of possible pairs using any two of the cards, please get in touch.
Why play this game?
The game provides an engaging and purposeful context for working on properties of triangles. It will help develop learners' conceptual understanding alongside their fluency with geometrical ideas.
There are plenty of opportunities for encouraging pupils to articulate their reasoning, one of the five key ingredients that characterise successful mathematicians. For example if they believe that a triangle cannot be drawn which has both properties of the turned-over cards. They may also need to convince an opponent that the triangle they have drawn does indeed satisfy both properties. If followed up with Name That Triangle!, as a pair in fact they offer the chance to focus on any of the five key ingredients.
(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. Although written with a secondary classroom in mind, the ideas are very applicable to primary settings too.)
Possible approach
You could use the interactivity to introduce the game. Click on two cards and give learners a chance to talk to a partner about whether a triangle with these two properties can be drawn. If pairs have access to a mini whiteboard and pen, or paper and pencil, they can try out some ideas as they talk. Gather feedback and as a class come to an agreement on whether it is possible to draw a triangle which has both properties or not. This might mean several pairs coming to the front to sketch ideas and explain their reasoning before everyone is convinced. You could try another pair of cards as a whole class using the interactivity to give another opportunity for pairs to come to a decision and rehearse their reasoning.
You can then set learners off playing the game, either using the interactivity or a set of cards. To encourage discussion and peer support, ask pupils to play as one pair against another pair; both pairs must agree on the 'final answer' before it counts. 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.
Key questions
- How many different sorts of triangle can be used to fit a particular card?
- Is your opponents' drawing clear, correct and convincing?
- How could you convince someone else about what you think is or isn't possible?
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.