One Big Triangle

Make one big triangle so the numbers that touch on the small triangles add to 10.

Age

5 to 7

Challenge level

Primary

Number - Addition and Subtraction

Number Bonds

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

One Big Triangle printable sheet

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.

Once you've finished making the big triangle, think about these questions:

How did you get started?

What did you do next?

You can print out the sheet at the top of this page and cut the triangles out, then try arranging them.

If you prefer, you can also use a printable version of the triangles with numbers represented on tens frames.

Alternatively, you might like to use this interactivity, which allows you to drag each triangle onto the large triangle.

Student Solutions

Shawn from Lanesborough School sent in this solution:

He added:

Step 1/ Put all the triangles in pairs to ten

Step 2/ Put all the red numbers together to make 10, do the same with blue and yellow

Step 3/ Make a hexagon, then the last three put them on with the coloured pairs.

Step 4/ Celebrate and write your maths down.

Ryan from Hayton Primary, Cumbria sent the same solution and wrote:

I started working from the top to the bottom with a small triangle.

I worked downwards to test out which was the right triangle to go at the top.

It took me about five tries to get the right triangle at the top.

When I got the right triangle at the top, all the other triangles were easy to fit in, working top to bottom.

Rowling Class from Manor School sent in the same picture and wrote:

Some of our class found it easy to start with but harder as you had to match two sides of the triangles at once. One group got stuck and had to start again with a different triangle.

It was really important to know your number bonds to 10. We used 10 - 7 to find out that we needed a triangle with a 3 to match with 7.

We were not surprised by the fact we found lots of different solutions.

Sophie managed to match one with colours as well as number bonds!

We also received our first solution from China in Chinese! Ma Xueying sent in this image which is the same solution as Shawn, Ryan and Rowling Class above:

The translation of the Chinese shows a very interesting approach and can be found here.docx or here.pdf.

Cohen, Nate, Isla and Alex from Burke Ward Public School in Australia sent the following different solution:

We worked together and figured out how to make the numbers equal ten and make them into a triangle.

- we remembered our friends of 10

- we put the numbers together

- when it didn't make a triangle we pulled the triangles apart and started again.

When we finished we checked we had made a triangle and that the numbers equalled ten.

Cruz, Maddix, Lillah and Zak from Burke Ward Public School in Australia found the same solution and wrote:

We saw that they all make ten and it looks like a triangle. We didn't worry about the numbers, we focused on the triangle shape first. Then we sorted out the numbers by ten and making a triangle. We turned the little triangles around to make it. Sometimes we had to try again.

We knew when we were done because then we checked that we had a big triangle and the numbers touching were making their friends of ten.

Aidan, Alicia and Jimmie from Millfield Primary School found a third solution and said:

Aidan started at the top. I worked my way down until it made a big triangle. When you're making the triangle, if one piece has 8, you need to find another piece with a 2. Sometimes there are multiple pieces with 2 on them so I looked carefully at the other numbers on that piece to see which would fit best. I kept going until there were no pieces left. Jimmie helped to work out the number bonds using his fingers to make sure the number bonds were correct.

Alicia started at the bottom. I worked my way up and I found the right numbers for the number bonds. Sometimes I didn't have a piece that would fit, so I had to swap it with a piece already on the triangle.

Victoria, a teacher from Broadfield Academy in the United Kingdom, described her approach:

- First, I lined all the triangles in a long line playing with different combinations. I cut the line at various points to attempt to stack the triangle from bottom to top. Unsuccessful.

- Then, I looked for tiles with duplicate numbers and used those as a starting point in the middle of the triangle, then on the edge to exclude them from use. Unsuccessful.

- I noted which numbers appeared least frequently (e.g. 5). I used this starting point and moved tiles around the pair of fives. As it began to take shape the colour pattern confirmed I was on the right lines.

I found the solution after an hour of trial and error.

She felt it was great for revising number bonds to 10 and developing problem-solving and reasoning skills, and resilience!

Thank you all for sharing these different solutions. I wonder if there are more different solutions?

Teachers' Resources

One Big Triangle

One Big Triangle printable sheet

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.

Once you've finished making the big triangle, think about these questions:

How did you get started?

What did you do next?

You can print out the sheet at the top of this page and cut the triangles out, then try arranging them.

If you prefer, you can also use a printable version of the triangles with 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.

Possible approach

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 chests'!

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 the printable 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?

Key questions

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?

Possible extension

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.

Possible support

Children could use the cards to make a different shape (not necessarily a triangle) where the touching numbers add to 10, and/or the tens frame cards could be used to provide more support. 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 on our printable resources page. 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.

Noah

Noah saw 12 legs walk by into the Ark. How many creatures did he see?

Age

5 to 7

Challenge level

Primary

Number - Addition and Subtraction

Problem Solving

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

You might like to take a look at the poster for this task.

Noah saw 12 legs walk by into the ark.

How many creatures could he have seen?

How many different answers can you find?

Can you explain how you found out these answers?

Student Solutions

We had a small number of submissions but they contained some very interesting thoughts.

The year 2 children in Laurel Class at Colman Infant School Norfolk worked on this problem and the teacher sent in the following:-

We decided to personalise the problem to fit in with our recent trip to Hickling Broad nature reserve In Norfolk. The children were amused by the idea that Noah might have visited Hickling and were all eager to getstarted.

As a class we looked at how there could be 2 creatures with 12 legs. We quickly established that there could either be 2 ladybirds or a spider and a mouse. We decided that this meant there would probably be more than one way of having three, four or five creatures etc. From then on the children worked with a partner. Some children worked on 12 legs and others on a more ambitious 18. They began by drawing pictures of the animals and making a zig-zag book. This was a way into the problem that made sense to everyone. All the children worked enthusiastically. Having pictures also meant that the children didn't use the same combination of animals twice as is often the case when ust using numbers. (E.g. 6+4+2=12 and 2+6+4=12)

The idea was to work systematically to find as many different ideas as possible. Some children began to realise that they could use one answer to inform another. For example, if you had 1 Iadybird, 1 mouse and 1 duck you could work out 4 creatures by exchanging a mouse for 2 ducks or a duck for 2 snails.

Later in the week the children looked at their booklets and realised that there were still a lot more answers to find. They were encouraged to work in a more abstract way but could still refer to their numbers as spiders, ladybirds, mice, ducks or snails. This is where they really began to work systematically. They were keen to find as many answers as possible.

Thank you all at Laurel, sounds good

Rhea from Bablake Junior School sent in the following

There are lots of different answers for this. NRICH have put up a very open ended question!

Firstly, lets pretend that all the animals that Noah saw were all 4 legged animals. This would mean that Noah saw 3 animals. You do this by dividing 12 by 4, which is 3.

Secondly, lets pretend that all the animals saw were all 2 legged animals. This would mean that Noah saw 6 animals. You find this out by dividing 12 by 2, which is 6.

Thirdly, lets pretend that all the animals were 3 legged animals. You do this by dividing 12 by 3, which is 4.

Fourthly, lets pretend that that half of the animals Noah saw were 2 legged animals, and the other half were 3 legged animals. To find out the half that were 2 legged, you need to find out what half of 12 is, which is 6. You then have to divide 6 by 2, which is 3! 3 of the animals Noah saw where 2 legged animals. Then, you need to find out how many animals were 3

legged. You do this by dividing 6 (half of twelve: 12 divided by 2) by 3, which is 2. 2 of the animals Noah saw were 3 legged animals.

This is all I have got time for, so make sure you come up with some more solutions for NRICH! Hope you understood 4 of my solutions, thank you for reading this! Bye!

She entered a second submission saying:-

Well, I just entered a solution but I forgot to give examples of some animals Noah could have seen.

For a two legged, maybe birds, like: robins, pigeons, blackbirds, finches, blue tits, doves or swans.

For 4 legged: lion, dogs, cats, tiger, hedgehog, fox, wolf and more. These are examples, but there could be a lot more! Read my other solution on how to work on the 4 legged and two legged animals. It will be very helpful, hopefully you will read it and even make a solution your self! Good luck, hope you have fun on NRICH!

Olivia also from Bablake Junior sent in this solution:-

There are definitely lots of different ways to solve this! Lets start and pretend that all the animals Noah saw were 1 legged animals. In this very easy case, all you need to do is divide 12 by 1, which is of course 12. This means that if all the animals Noah saw were 1 legged animals, there would be 12 of them. I'm sorry about this, but maybe you know some 1 legged animals? Now lets pretend that all the animals Noah saw were 2 legged animals. In

this case, all you need to do is divide 12 by 2, which is 6. If all the animals Noah saw were 2 legged animals, there would be 6 of them. Now lets think what types of animals they might be. Probably birds- robins, blackbirds, pigeons- maybe even humans- there are lots more!

Now lets pretend that all the animals Noah saw were 3 legged animals (I know there might not be 3 legged animals, but lets pretend there is!) Now all you have to do is divide 12 by 3 which is 4. If all the animals Noah saw were 3 legged animals, there would be 4 of them. Now lets think what type of animals they might be. I'm sorry about this, but I don't know any 3 legged animals, maybe you do? Now lets pretend that all the animals Noah saw were 4 legged animals. In this case, all you need to do is divide 12 by 4, which is definitely 3. So,

if all the animals Noah saw were 4 legged animals, there would be 3 of those animals. Different types of 4 legged animals are dogs, cats, hedgehogs, horses, pigs, cows and lots, lots more! Now lets pretend that all the animals Noah saw were 6 legged animals. (I

know these are probably not real, but lets just pretend, okay?) All you have to do now is divide 12 by 6, which is 2. If all the animal Noah saw were 6 legged animals, there would be 2 of them. They are all the basic solutions, but see if you can mix 2 legged animals and 4 legged animals. There are loads of answers but these are the basics. Hope I've helped, thank you for reading this! Hope you have fun on NRICH!

The teacher of Year 1 at Bute House Prep School sent in the following:-

Year 1 had a lot of fun investigating how many animals could have gone into the Ark if Noah had counted 12 legs. They thought of their own line of inquiry within this problem and so some looked at whether they could have every number of animal from 1-12 with 12 legs, others looked at how many ways they could have groups of animals with the same number of legs making 12, while others investigated how they could best record their findings

systematically. Here are a few of their discoveries :-)

Amelia and Olivia found a number of ways of counting 2,3 and 6 animals, although they realised they couldn't have a rabbit and a fly as this wouldn't be enough legs

Annabelle and Julia found solutions for 3, 4 and 6 animals.

Claudia explored which combinations of the same number of legs would work to add to 12.

Beatrix and Mariana decided that a snail had one leg, a bird 2, a horse 4, an insect 6 and a spider 8 and worked from there. They found solutions for 2,3,4,5,6,11 and 12 animals.

Sophie realised that if you used snails it was possible to make up lots of animal numbers eg 4 birds and 4 snails = 8 animals with 12 legs. 6 snails and 3 birds would be 9 animals and 12 legs.

The girls thought 1 animal would be impossible!

Children at Thorpe Primary in Peterborough created a display of their work. Thank you to their teacher Mr Condon for sending in these photos of the display:

Thank you all for these very detailed accounts of your work. maybe some would go further and consider a larger number of legs seen by Noah. What is there would be only two types of animals and the same numbers of each, which Rhea used.

Teachers' Resources