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.
You could try putting one piece in to start with and then seeing whether you can place another piece next to it.
The interactivity gives all the pieces in the correct orientation so you might like to use it to try out your ideas. If you would prefer to use cut-out pieces, you could check which way up each small triangle goes by looking at the interactivity.
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 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.
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:
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?
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
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.
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.
Discuss the solutions that have been found. Are they all the same? If not, have they found all the possiblities?
Key questions
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
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?
How many legs do they each have?
What's the greatest number of creatures he could have seen?
What's the smallest number of creatures he could have seen?
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.
Noah
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?
Click here for a poster of this problem.
Why do this problem?
Cross curricular links can be made by using the story of Noah and the ark as an introduction. Considering creatures with one 'leg' such as slugs and snails, no legs such as snakes and all the other possibilities is great fun and provokes interesting responses from the children. You may be surprised by their ingenuity.
Possible approach
- Ask for suggestions and encourage the children to come to the board and draw their animals. Count the number of creatures each child draws and discuss who has drawn the least, the most, the same as ... etc.
- Give the children playdough and 12 equal lengths of straw each so that they can make the creatures - and put them into groups of 12 legs for a display.
- Let each child draw a picture, write the number sentence below it (eg $6 \: + \:4 \; + \; 2 \; = \; 12$) and make them all into a class zig-zag book - Our Story of Twelve.
- Give each group of children a different number of legs to investigate and ask them to make their own group zig-zag book.
Key questions
Possible extension
What if there were more legs... 18, 24, 19 ... or fewer?
Possible support
Giving children 12 straws for legs and modelling clay will support them to think about the ways they might use the 12 legs to make a range of different creatures that use up their legs.Ladybirds in the Garden
Ladybirds in the Garden printable sheet
In Sam and Jill's garden there are two types of ladybirds. There are red Seven-Spot ladybirds with 7 black spots and black Four-Spot ladybirds with 4 red spots.
Sam and Jill looked at a leaf with three ladybirds on it.
"One Seven-Spot ladybird," said Sam, "and two Four-Spot ones."
"That's 15 spots altogether!" said Jill.
"I wonder if we could find ladybirds whose spots add to other numbers. I know how to do 16."
"And 14 is easy too," added Sam.
How would you make 16 and 14 spots with the Seven-Spot and Four-Spot ladybirds?
What other numbers can you make with adding 4s and 7s?
Can you get lots of numbers from 4 to 35?
Are there some numbers you can't get?
We were sent this solution by Jacob, Luc and William at The Hall School in London:
We looked at the number of spots that could be produced when you have a four spotted ladybird or a seven spotted ladybird or a combination of both.
We were asked if it was possible to make 16 and we found it is, by using four of the four spotted ladybirds. We discovered that the smallest number of spots we could produce was 4.
We were then asked what number of spots between 4 and 35 could be produced. We started off by writing a list of those numbers that did and didn't work. Then we made a table to show how we made the numbers of spots that were possible.
We found seventeen different numbers could be made and three that could be made in two different ways. These were 28, 32 and 35.
We have drawn a table to show the only number of spots that can be made between 4 and 35. For each number of spots that it is possible to make we have shown the number of four spotted and seven spotted ladybirds that make up the number of spots.
Thank you all for sharing those ideas with us! I wonder if any of the numbers that are missing from that table are possible or if they're all impossible? If you find any more solutions to add to this table, please email us to let us know.
We also had some ideas sent in from Christian at Heronsgate School in England and Olivia from Risley Lower Primary School in England. From Australia we had a solution sent in from Maths Group 2 at Brunswick South Primary School.
Well done everyone!
Ladybirds in the Garden
Ladybirds in the Garden printable sheet
In Sam and Jill's garden there are two types of ladybirds. There are red Seven-Spot ladybirds with 7 black spots and black Four-Spot ladybirds with 4 red spots.
Sam and Jill looked at a leaf with three ladybirds on it.
"One Seven-Spot ladybird," said Sam, "and two Four-Spot ones."
"That's 15 spots altogether!" said Jill.
"I wonder if we could find ladybirds whose spots add to other numbers. I know how to do 16."
"And 14 is easy too," added Sam.
How would you make 16 and 14 spots with the Seven-Spot and Four-Spot ladybirds?
What other numbers can you make with adding 4s and 7s?
Can you get lots of numbers from 4 to 35?
Are there some numbers you can't get?
Why do this problem?
Ladybirds in the Garden is a good problem for children to practise addition, subtraction and possibly multiplication. It is a useful context in which to encourage learners to "have a go" and "play" with numbers, but then you can focus on having a system to find all the possible totals and giving reasons why some can't be made.
Possible approach
It would be good to have the two pictures of the ladybirds in the problem displayed for the children to see, either printed off or on screen. You could get them started simply by asking how many spots there are altogether on the two, then how many on the picture of the three. You might say that you're picturing some ladybirds on a leaf whose spots add to $16$ and ask children in pairs to think about which ladybirds they are.
You could bring their ideas together by asking them to write each different total, and how they made it, on a strip of paper. Gather these strips on the board or a wall and ask the children to arrange them in numerical order. Draw out a list of numbers which haven't been made and ask all pairs to check that they're sure they are impossible. The important point here is for pupils to try to explain why these totals can't be made. Ask pairs to convince each other and then invite some explanations for all to hear.
Key questions
Which totals have you found so far? Which ladybirds made these totals?
Possible extension
The article, Opening Out, suggests possible routes for further investigation in general, but this problem is also mentioned specifically. Children could be asked what combination of spots would be needed to be able to generate all the numbers under $30$. Would it be possible to do this without a one-spot ladybird? If we couldn't use a one-spot ladybird, what totals would be possible?
Possible support
Some children might find it useful to have a sheet with consecutive numbers from $1$ to $29$ written on so that they can fill in ways of making each total as they go along. Calculators might be available, or other apparatus to support the arithmetic, for example a hundred square, numberline, Cuisenaire rods or counters/cubes. You might want to print out a sheet of ladybirds and cut these into cards for children to choose from. To make the task even more accessible, you could choose and create ladybirds with $2$ or $3$ spots instead.
Strike it Out
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Watch the video below which shows two people playing the first few turns of a game.
What do you notice?
What do you want to ask?
If you can't access YouTube, here is a direct link to the video.
Watch the video a second time.
Can you work out how to play the game?
What do you think the rules might be?
How might someone win the game?
If you are unable to view the video, you could click through this powerpoint presentation, which also demonstrates how to play. Alternatively, the rules of the game are hidden below.
The aim of the game:
The player who stops their opponent from being able to go wins the game.
How to play:
- Start by drawing a number line from 0 to 20 like this, or print off a sheet of number lines:
- The first player chooses two numbers on the line and crosses them out. Then they circle the sum or difference of the two numbers and they write down the calculation.
For example, the first player's turn could look like this:
- The second player must start by crossing off the number that Player 1 has just circled. They then choose another number to cross out and then circle a third number which is the sum or difference of the two crossed-off numbers. Player 2 also writes down their calculation.
For example, once the second player has had a turn, the game could look like this:
- Play continues in this way with each player starting with the number that has just been circled.
- Once a number has been used in a calculation, it cannot be used again.
- The game ends when one player cannot make a calculation. The other player is the winner.
It's your turn!
Try playing the game against someone else a few times to get a feel for it.
Do you have any good ways of winning?
Now it's time to work together with a partner, rather than against them.
Try to create a string of calculations that uses as many numbers as possible on the 0-20 number line.
Is it possible to create a string of number sentences that uses all the numbers on the 0-20 number line? Why or why not? How would you convince a mathematician?
Once you've had a good think about it, you may like to look at this proof that has been scrambled up.
Can you rearrange it into its original order?
If you would prefer to work away from a screen, you could print off, cut up and rearrange the statements. (This sheet includes three copies of each statement.)
What about the 1-20 number line? Is it possible to create a string of number sentences that uses all the numbers on the 1-20 number line? Why or why not? How would you convince a mathematician?
Again, once you have thought about it, you may like to look at this proof that has been scrambled up.
Can you rearrange it into its original order?
If you would prefer to work away from a screen, you could print off, cut up and rearrange the statements. (This sheet includes two copies of each statement.)
It's a good idea to play the game lots of times to start with to get the feel of it.
You could keep the lines you use for each game and compare them afterwards.
Are there any numbers that you haven't crossed off in any of the games you've played on the 0-20 number line? Will it ever be possible to cross them off?
When you come to consider the 1-20 number line, have a think about the total number of numbers you use in a game.
Thank you to those of you who submitted comments about this game.
Anika from Holy Family Catholic Primary School in Boothstown, Manchester suggested the following rules:
- The answer from the previous calculation should be the answer to the next one
- You can only use each number once
- Player A uses addition and Player B uses subtraction, and this pattern follows throughout the game
- The aim of the game is to stop the other player from getting a calculation from the remaining numbers
Thank you, Anika! Jake said:
- Player A does a calculation using numbers 0-20. They cross off the numbers they use but circle the answer.
- Player B then uses the answer of player A's calculation to start their number sentence.
- Each player can decide whether they would like to add or subtract.
- The player who cannot make a calculation from the numbers remaining loses the game.
So, it seems that Anika and Jake were largely in agreement. Anika, it does look like the order of calculations goes addition, subtraction, addition, subtraction etc from what we see in the video, you're right. However, I would suggest that the order doesn't matter. (You can read our suggested rules by clicking on the 'Show' button underneath the video on the problem page itself.)
Eleanor from Cottenham Primary School told us:
Good ways to win are to try and use up the numbers 1, 2, 3, 4 and possibly 5 and 6, because you can get to many different numbers by adding those whereas with the larger numbers you cannot get to so many because you can only go up to the number 20.
Eleanor also went on to explore the cooperative challenges. She says:
We can't use all of the numbers because:
For the number line going from 0 to 20, you cannot use zero as you wouldn't be adding or taking away anything so you would stay on the same number.
Yes, Eleanor, if you add zero, or take away zero, from a number, the answer to your calculation is the same number, so you would need to be able to use a number more than once and the rules don't allow this.
For the number line going from 1 to 20, you cannot use all the numbers because there will always be one number left over.
Each new calculation uses two new numbers apart from the first calculation which uses three so if you keep adding two to three you will always be on odd numbers and 20 is an even number, not odd.
Thank you, Eleanor, what great reasoning. Can you follow Eleanor's argument?
Strike it Out
Watch the video below which shows two people playing the first few turns of a game.
What do you notice?
What do you want to ask?
If you can't access YouTube, here is a direct link to the video.
Watch the video a second time.
Can you work out how to play the game?
What do you think the rules might be?
How might someone win the game?
If you are unable to view the video, you could click through this powerpoint presentation, which also demonstrates how to play. Alternatively, the rules of the game are hidden below.
The aim of the game:
The player who stops their opponent from being able to go wins the game.
How to play:
- Start by drawing a number line from 0 to 20 like this, or print off a sheet of number lines:
- The first player chooses two numbers on the line and crosses them out. Then they circle the sum or difference of the two numbers and they write down the calculation.
For example, the first player's turn could look like this:
- The second player must start by crossing off the number that Player 1 has just circled. They then choose another number to cross out and then circle a third number which is the sum or difference of the two crossed-off numbers. Player 2 also writes down their calculation.
For example, once the second player has had a turn, the game could look like this:
- Play continues in this way with each player starting with the number that has just been circled.
- Once a number has been used in a calculation, it cannot be used again.
- The game ends when one player cannot make a calculation. The other player is the winner.
It's your turn!
Try playing the game against someone else a few times to get a feel for it.
Do you have any good ways of winning?
Now it's time to work together with a partner, rather than against them.
Try to create a string of calculations that uses as many numbers as possible on the 0-20 number line.
Is it possible to create a string of number sentences that uses all the numbers on the 0-20 number line? Why or why not? How would you convince a mathematician?
Once you've had a good think about it, you may like to look at this proof that has been scrambled up.
Can you rearrange it into its original order?
If you would prefer to work away from a screen, you could print off, cut up and rearrange the statements. (This sheet includes three copies of each statement.)
What about the 1-20 number line? Is it possible to create a string of number sentences that uses all the numbers on the 1-20 number line? Why or why not? How would you convince a mathematician?
Again, once you have thought about it, you may like to look at this proof that has been scrambled up.
Can you rearrange it into its original order?
If you would prefer to work away from a screen, you could print off, cut up and rearrange the statements. (This sheet includes two copies of each statement.)
Why play this game?
Possible approach
Key questions
Possible support
Possible extension
The possibilities are endless but do make sure they try out their new version of the game to check it is a 'good' game.
The Highland Numeracy Team in Scotland kindly shared with us some ideas for adapting Strike it Out. Some variations include the use of concrete materials suited to the version of the game, such as tens frames, Dienes, or place value counters.