Factor lines

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Age

7 to 14

Challenge level

Primary

Number - Multiplication, Division and Ratio

Properties of Numbers: Multiples, Factors, Primes, Square and Cube Numbers

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

Being curious Being resourceful Being resilient Being collaborative

Problem

Arrange the four number cards (1, 2, 3 and 21) on the grid above to make a diagonal, vertical or horizontal line.

You can put a number card on a square with

a multiple of that number
a factor of that number.

In how many different ways can you do it? Can you use a strategy that you haven't tried before?

You can use the interactivity above or this sheet. The interactivity keeps track of the number of different lines you have created and you can view the lines you've already made by clicking 'Show lines'.

This problem has been adapted from the book "Cards on the Table" by Fran Mosley, published by BEAM Education. This book is out of print but can still be found on Amazon.

Student Solutions

Thank you to everybody who shared their ideas about this task with us. Anyu from Bisspuxi in China had an idea about how to start this activity:

I started off listing factors and multiples of 1,2,3 and 21 up to 25 so it was simpler to find the answers.

Good idea, Anyu - those factors and multiples are going to be really important for helping us find solutions to this problem.

Om from St Leonard Primary School in Exeter, UK focused on the factors and multiples of 21. They sent in this picture of their six solutions:

Om has circled the numbers 1, 3, 7 and 21 because as these are the only factors or multiples of 21, every solution must pass through one of these. This is a good idea, Om. I wonder why Om chose to focus on the factors and multiples of 21 rather than 1, 2 or 3?

Amina from St Mary's Catholic Primary School in Bath, UK also began by thinking about the factors of 21:

First of all I noticed that 21 could only be placed on four squares: 7,3,1,21. The 1 could be placed on any square and the 2 can go on any even number. To create the lines I started with 21 then the 3 because these are the hardest numbers to place on the grid. So once I found a place for the 21 and 3, all I had to do is find an even number for the 2 and then the 1 can go on any square because any whole number can be divided by one. I managed to find 9 lines in total. All of my lines were vertical and horizontal because when I looked for diagonal lines I realized I couldn't make one because all the numbers were odd and there was nowhere to put the 2.

This is very clearly explained, Amina - well done.

We received quite a few solutions from children who had found nine possible factor lines. Dhruv from The Glasgow Academy, UK sent in this explanation and picture showing the nine possible factor lines:

When I tried all the ways these are the ones I got,

1 6 11 16

21 16 11 6

1 2 3 4

2 7 12 17

22 17 12 7

21 22 23 24

6 7 8 9

10 9 8 7

3 8 13 18.

The trick I used was I knew that 21 could not have had any multiples in a 5 x 5 table, so I worked out all the factors of 21 and started every line from there. The factors of 21 are 1, 3, 7 and 21, that’s why every line starts from these numbers.

For numbers like 6 and 12 I could have put 1, 2 or 3 because 6 and 12 are multiples of these numbers but I had to think carefully as some of the numbers (in the lines) could only be multiples of 3 or 2.

I ended up with 9 lines and could not make any more.

It sounds like you thought really carefully about this, Dhruv - well done!

Some children thought about the different possible ways of making each factor line, as well as how many factor lines were possible to make. The children at Our Lady's & St. Oswald's Catholic Primary School, UK sent us this picture:

They said:

We found 13 solutions, are there any more?

We worked systematically by focusing on factors and multiples of 21 first.

This seems to be a popular strategy! In the picture above, the children have put multiple lines on some of the solutions because there are different ways of making them using factors or multiples of the numbers 1, 2, 3 and 21. For example, in the factor line 1 2 3 4, either the 1 or the 3 can be used as a factor of 21.

There are actually more than 13 solutions if we consider each of the different ways of making a factor line to be a different solution. Anyu from Bisspuxi in China said:

I found 15 in total.

I also think there are 15 solutions in total, Anyu. We asked if anybody could send us the full list of 15 solutions, and we received these images from Dhruv describing the full solution:

Thank you, Dhruv, for your perseverance in finding all fifteen solutions!

Teachers' Resources

Factor Lines

Arrange the four number cards (1, 2, 3 and 21) on the grid above to make a diagonal, vertical or horizontal line.

You can put a number card on a square with

a multiple of that number
a factor of that number.

Why do this problem?

This problem will help children become more familiar with factors and multiples. Moreover, beacuse a systematic approach is required in order to find all the lines, it will also help children to build up their resilience and perseverance skills. It will encourage learners to talk to each other, and it provides a good context in which to discuss different methods of approach. It has lots of scope for being opened out and extended through 'what if ...?' questions. It is also a great problem to enable learners to build resilience and work with each other to overcome any difficulties they might experience along the way.

Possible approach

You could introduce this problem by showing the group the interactivity and telling the children that you are going to make lines of four, but they have to work out the rules for making a line. Make one line (e.g. the 21, 3, 1, 2 over the 1, 6, 11, 16 respectively) and invite the children to talk to each other about what they think the rules are at this stage. Then give them another line and ask them to talk further to clarify the rules. You could give them another line before having a whole-class discussion about the rules.

Once you have established the rules, you could set the challenge of finding lines of four with the 1 to 25 grid. Children could use the interactivity or a copy of this sheet. Give them time to work in pairs without saying too much more and then bring them all together for a mini-plenary at a suitable moment. Give learners a chance to share what they have done so far so that some different ways of approaching the task are revealed and different ways of recording are discussed. (They may find it helpful to use copies of this recording sheet to keep track of the lines they have found.) You may find some draw attention to the properties of some of the numbers in the grid, for example the way in which the prime numbers affect a line of four. If this does not come up naturally, you may like to bring it up later in the lesson.

After giving children more time to work on the problem, you could invite everyone to contribute to a whole-group solution, perhaps recording lines collectively on the board using the interactivity. At this point, encourage the pupils to scrutinise the grid carefully so that everyone is happy all the lines have been found. This will rely on having a systematic way of working, but there are many systems that could be adopted.

You can extend this challenge in many different ways (see below) and you may like to choose some of the suggestions to leave for children to mull over during the following week. You could set up part of your wall as a place for learners to pin up contributions and then you could plan to review the work altogether at a future date.

Of course, the interactivity in this problem can be returned to time after time, as it can be tailored to suit your objectives.

Key questions

Which of the cards can be multiples?

Which can be factors?

Can any be both?

How will you remember which lines you have found?

Are there any other ways to make that line?

Are there any numbers on the grid which are impossible to cover? Why or why not?

Which of the four numbers will be the most difficult to place do you think? Why?

Which of the four numbers will be the easiest to place do you think? Why?

Possible extension

Children could investigate whether by rearranging the grid numbers, they could increase/decrease the number of lines. Could they pick four different dragging numbers so that more/fewer lines could be made? Could they choose different grid numbers to increase/decrease the number of lines? What would happen if, rather than a line, the aim was to get a block of four, or an 'L' shape? Of course children will be able to come up with their own questions to investigate.

Possible support

You can use the interactivity to change the size of the grid and the numbers used. Some children might find it easier to start with a $4$ by $4$ grid and to identify lines of three, for example. Cutting out the numbers at the bottom of this sheet might be useful so that they can be moved around the grid. Alternatively, transparent counters with numbers stuck on them would be helpful.

Many thanks to all the teachers participating in the Enriching Mathematics in Devon Project who provided lots of ideas and inspiration for these notes.

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