One Big Triangle
Here are nine triangles. Each one has three numbers on it.
Your challenge is to arrange these triangles to make one big triangle, so the numbers that touch add up to 10.
Let us know how you go about trying to make the big triangle.
How did you get started?
What did you do next?
You can print out this copy of the triangles and cut them out, then try arranging them.
If you prefer, here is a printable version of the triangles with the numbers represented on tens frames.
Alternatively, you might like to use this interactivity, which allows you to drag each triangle onto the large triangle.
Why do this problem?
This problem is useful for consolidating number bonds to 10 and the corresponding subtraction facts. The novel context is likely to appeal to learners and encourage them to persevere. Children who are fluent with number bonds to 10 will still be challenged as the approach is not obvious, but logical reasoning
will help them become more efficient in their search for a solution. The task will also draw on learners' geometrical reasoning too as they visualise triangles in different positions and orientations in order to solve the problem.
You could show the images of the small triangles on the board and ask children to look in silence for a few minutes and think about what they can see. Then encourage them to share with a partner before taking some suggestions as a whole group. Welcome all contributions, even if they don't seem relevant to the mathematics - children will need to get all sorts of observations 'off their
Using some already cut out pieces, or the interactivity on the screen, place a few triangles in the big triangle, making sure that touching faces add to 10. Invite children to gather round to see and ask them to think about what they notice. Again, take some contributions, and then introduce the task itself.
Children can work in pairs on the problem with cards made from this sheet
so that they are able to talk through their ideas with a partner. (If these are printed onto thin card and laminated you will have a permanent set that can be used for other purposes as well.) If tablets/computers are
available, pairs could use the interactivity. Note that in the interactivity, all the triangles are in the correct orientation and cannot be rotated, which makes the challenge more accessible than having cut-out triangles.
You could facilitate as many mini plenaries as you feel is helpful in order to share ideas that are being worked on around the room, including possible ways of getting started on the task. Learners' suggestions might include, for example, working top down, or bottom up, or hexagon out, matching colours, making pairs first, noticing the three 5s and therefore reasoning that one of these will have
to be on an outer edge... If using the interactivity, they may realise that there are only three small triangles which are oriented so that they have a horizontal 'top' edge, so they may start with one of those to narrow down the possibilities.
During the final plenary, encourage the children to talk about the strategies they have used. Did they guess and then check? Or did they have a more systematic approach to the problem? Did they imagine what a triangle would look like in a particular position before placing it there? You might find learners have used visualisation to plan which piece will go where.
Discuss the solutions that have been found. Are they all the same? If not, have they found all the possiblities?
Tell me what you have done so far.
What do you need to put with ... to make 10?
Can you find a different card with that number on it?
What might be helpful to try next?
Children could be asked whether they can find more than one solution. How will they know whether another solution is the same or different to any they have already got? How will they know that they have found all the solutions?
Learners could also use the cards to make a shape (not necessarily a triangle) where the touching numbers add to 9 (or 8 or 11). Alternatively, they could add their own choice of numbers to blank triangular pieces to create their own activity.
Children could use the cards to make a different shape (not necessarily a triangle) where the touching numbers add to 10, and/or these cards could be used in which the numbers are represented by tens frames. Some children may find it difficult to
cope with matching more than one pair of numbers at a time, in which case a domino activity would be more accessible. A set of nine-spot dominoes would be useful for this and you can find one for printing here. The task could be to join the dominoes together so that the 'match' adds to 10 or any other number of the
children's choice. This will then give them plenty of practice in identifying number bonds.