# Picture Your Method

*Picture Your Method printable sheet*

For the first part of this task, you're going to try to answer a calculation in your head, without using pencil or paper. When you're ready, click below to see the calculation.

What is 18 x 5?

Jot down your answer.

The answer is definitely not the most interesting part of this problem! Much more interesting is thinking about the way you arrived at your answer.

Below you can read what five learners said when they were asked how they worked out their answer:

Bryan:

First I doubled 18 to get 36.

Then I doubled that to get 72.

Then I added 18 again.

Neil:

I took 18 and I halved that, which is 9.

9x5 is 45, 9x5 is 45.

Then I added 45 and 45 together.

Sammi:

I separated 18 into 8 and 10.

8x5 is 40. 10x5 is 50.

I then added 40 and 50 together.

Ricardo:

I did 9x10 instead of 18x5 because that's the same thing.

Jaime:

I did 20x5, which is 100.

Then I took away 2x5, which is 10.

Was your method the same as any of these? If not, describe what you did.

We can also draw each of these ways of working out 18x5. (We might say we can represent each one visually.)

Can you match each drawing below to one of the methods described above? (We've labelled each of the drawings with a letter to make it easier to refer to a particular one.)

For accessibility, clicking on the 'show' button below will show a description of this image.

A is a rectangle with one side of length 5 and the other side of length 18 + 2, with the '2 by 5' section greyed out.

B is a rectangle with one side of length 18, split into 9 + 9, and the other side of length 5.

C is a rectangle with one side of length 18, split into 10 + 8, and the other side of length 5.

D is an L-shape - at the top is a rectangle with one side of length 18, split in half, and the other side of length 5. Underneath is another rectangle, half as long as the top one, with the other side of length 5.

E is a rectangle with one side of length 18 and the other side of length 5, split into 2, 2 and 1.

You may like to print off a sheet of all five descriptions and all five drawings, which can be cut up into ten cards.

How did you decide on the pairings?

If you used a different method, create a drawing of your method too.

*This task is inspired by a YouCubed resource and is used with permission.*

It might help to tell someone else what you see in each drawing.

Thank you to everyone who sent in their ideas for calculating 18 x 5.

Clearly, the answer is definitely not the most interesting part of this problem! Much more interesting is thinking about the way you arrived at your answer. In the problem we suggested five possible ways to find the answer, and we challenged you to match each of those ways with their correct diagram.

Simran, from Maurice Hawk School in the USA, shared the following list of matched calculation methods and diagrams:

A Jamie

B Neil

C Sammi

D Ricardo

E Bryan

Simran also shared her preferred method:

My method was the same as Sammi's method. I used the distributive property to solve this problem. I separated 18 into 8 and 10, found the products of 5 X 8 and 5 X 10 and added the two products to get 90 like Sammi did.

Rachel, from Burrough Green School, sent in this video explaining her reasoning for each pair of cards:

Well done, Rachel!

We shared our five diagrams with you, but there's other ways to explain your reasoning too. Rachel decided to use cubes to explain why one of the methods works:

We also challenged you to think about other ways you could calculate 18 x 5.

Lachlan, from Full Spectrum Education in Australia, shared his method and diagram:

I split 18 into 12 and 6, then I multiplied 12 x 5 (=60) and 6 x 5 (=30) and then I added the answers together.

My diagram:

Krishna, from the CS Academy in India, shared this photo of another method for calculating 18 x 5:

First I took 5 and partitioned it into 3 and 2.

Then I took 18 and multiplied it by 3, and 18 x 3 = 54.

Then I took 18 again and mulitplied it by 2, and 18 x 2 =36.

Then I took 54 and added it to 36, which is equivalent to 90.

Do you think you could draw a sketch to represent Krishna's method visually?

Krishna also shared another way for calculating 18 x 5:

First I took 18 and partitioned it into 6, 6 and 6.

Then I took 5 and multiplied it three times with 6.

5 x 6 = 30 so 30 + 30 + 30 = 90.

Simran also thought about different ways to partition 18:

We can also separate 18 in different ways, one way is shown here:

X |
5 |
5 |
5 |
3 |

5 |
25 |
25 |
25 |
15 |

Very well done to you all!

### Why do this problem?

Offering learners visual representations of calculation methods helps them develop number sense by encouraging them to break numbers apart and re-group them again. This task offers learners an experience of maths as a creative, flexible subject at the same time as developing their fluency and their conceptual understanding of number.

Read more about the benefits of having a flexible approach to calculation in our Let's Get Flexible! article.

### Possible approach

Invite everyone in the class to work out 18x5 mentally, and ask them to jot down their answer. Reassure the group that this is not about speed, but you would like them not to write anything down, other than the answer. Give them time to compare their answer with a partner, and at this stage ask them to explain to their partner how they arrived at their answer.

Bring everyone together again and invite a pupil to share their partner's method. Jot down what is said (preferably verbatim as far as you can) and give the whole group chance to ask questions for clarification. (Try not to answer these yourself, but encourage the pair to respond, or others in the class.) You might ask who else used that method. Repeat this process so you that you have a variety of methods written on the board reflecting the range of approaches in the room. Label each one with the pupil's name so you can easily refer to them e.g. Oscar's method.

At this point, give out a set of the description cards and their visual representations (printed from this sheet) to each pair or group of four. You may like to say that the cards show just five ways of approaching the calculation 18x5. Challenge the class to match each description to its visual representation, and explain that you will be keen to know how they decided on the pairings.

Circulate round the room, listening out for learners who are justifying their choices. In the final plenary, you could call on some of the pairs/groups you overheard to share their reasoning with the whole class. You could also invite learners to say which method/s they prefer and why. This has the potential to provoke some interesting discussions, which might reveal the influence of wider society (e.g. parents or tutors) for 'preferred methods/algorithms' whatever the calculation. It would be interesting to learn whether pupils found it difficult to work mentally when they may feel the 'more sophisticated' method is written. We know from research that once learners have been introduced to written methods they become the default approach for many, even those who can perform a given calculation mentally.

If time allows, or in subsequent lessons, you could pose a new calculation and ask learners to use a particular method (perhaps by referring to it using the name of the pupil who first described it), or to use a different method from the one they used originally.

*An alternative way to approach this task could be to share all the images, without the descriptions, and invite learners to talk in pairs about what calculation they represent. You could invite learners to create a 'description card' for each image themselves.*

### Key questions

Tell me what you see in this drawing.

What does this part of the drawing show?

### Possible support

Some learners will benefit from an adult or peer reading aloud the descriptions.

### Possible extension

You could invite learners to create a visual representation of a method that has been shared by a member of the class, which isn't included in the set of five. (It is important to capture the different ways of thinking, so even if the resulting visual looks similar to one that has already been given, if the method is slightly different, it is worth including it as an alternative.)

*This task and teachers' resources are based on an activity in this extract from the Number Talks online course, created by the YouCubed team.*