Carrying cards

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

Age

7 to 11

Challenge level

Primary

Number - Multiplication, Division and Ratio

Multiplication and Division Facts

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

Being curious Being resourceful Being resilient Being collaborative

Problem

Carrying Cards printable sheet

Sixteen children are standing in four rows of four, one behind the other. They are each holding a card with a number on it. In each column, all four children are wearing clothes of the same colour.

Red

Blue

Yellow

Green

Each child in blue is holding a number which is four more than the child next to them who is wearing red.

The children in yellow each have a number which is double the number of the child in the same row wearing red.

Some of the numbers that the children in red, blue and yellow are holding are missing. What should these numbers be?

How have the numbers of the children in green been worked out? What are the two missing numbers?

If there was another row of four children standing at the back, behind the fourth row, what numbers would they be holding?

Student Solutions

A Maths Club at Beacon School Amersham wrote:

First we worked out all the missing numbers for the red, blue and yellow children. This was easy because we were told how a number could be made from the number on the card before it in the same row.

Then we spotted that the children in green shirts have a number which is the sum of the yellow and blue shirts in the same row.

We then worked out the fifth row by looking for a pattern. The red shirt numbers went up in ones, the blue shirt numbers went up in ones, the yellow shirt numbers went up in 2s and the green shirts went up in 3s.

Well done! They sent in a table to represent the children's numbers:

5	9	10	19
4	8	8	16
3	7	6	13
2	6	4	10
1	5	2	7

Fantastic solutions were also sent in by lots of pupils from Crosshall Junior School; Jun and Colin from the Canadian Academy; Karnan from Stag Lane Middle School; Richard and Jacob from St Thomas More's School and Ellie from West Bridgford Juniors.

Trang from Central Foundation Girls' School, used symbols to help her write out what she needed to do:

First, I wrote the short letter for the colours: Red: $r$ ; Blue: $b$ ; Yellow: $y$; Green: $g$

Then I wrote out the fomula: $b = r + 4y = 2r$

Therefore: For the 1st row I have: $y = 4 \times 2 = 8$;

2nd row: $r = 6 \div 2 = 3$; $b = 3 + 4 = 7$

3rd row: $r = 4 \div 2 = 2$

4th row: $r = 2\div2 = 1$

Looking at the first and the last rows, I noticed that the blue and the yellow in the same rows added together equal the green.

So green in the 2nd row $= 7 + 6 = 13$ and the 3rd row $= 4 + 6 = 10$

I also noticed that the numbers in the red column are consecutive numbers ($1, 2, 3, 4$). It's also similar to the blue column ($5, 6, 7, 8$) and the yellow column is the multiples of $2$ ($2, 4, 6, 8$). So if there is one more row - the red colour and the blue colour need to add $1$ for each and the yellow need to times $2$ and the green is made by adding the blue and yellow.

Teachers' Resources

Carrying Cards

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it.

Each child in blue is holding a number which is four more than the child in the same row wearing red.

The children in yellow shirts each have a number which is double the number of the child in the same row wearing red.

Some of the numbers that the children in red, blue or yellow shirts are holding have got rubbed off. What should the numbers be?

Can you work out how the numbers that the children in green are holding have been worked out? What are the two missing numbers?

If there was another row of four children standing behind the fourth row, what numbers would they be holding?

Why do this problem?

This problem will give pupils experience of looking for, and explaining, number patterns, and it could lead into algebra. It would also make a good introduction to spreadsheet use.

Possible approach

You could introduce this problem practically in the classroom with children holding small whiteboards, for example. Rather than wearing coloured shirts, the children could have particular coloured pens or they might wear 'bibs' or ribbons usually used for sports matches. Arrange the sixteen children in four rows of four as in the picture and write the given numbers on the appropriate whiteboards. Once the problem has been introduced in this way, the sixteen children can return to their seats and the whole class can discuss the challenge. Learners should be given time to talk to each other in pairs or small groups as well as discussing the problem as a whole class.

In the plenary, you can invite the pupils to stand in the grid formation again and everyone can participate in building up the numbers on the whiteboards. Encourage learners to explain how they know what the number on each board is, and draw attention to the fact that each number might be worked out in several different ways. Invite four more children to stand behind the back row so that the last question can be answered.

You could end with a final challenge for the class to solve: Invite four more children to come up and stand in a row some way behind the fifth row. On the board which is held by the child standing behind numbers 1, 2, 3, 4 and 5, draw a shape or write 'any number' or write 'a number'. Explain that this is a number, any number - you don't know what it is. Can the group write expressions on the other three whiteboards?

Key questions

What do you know about the red/blue/yellow numbers?

How will that help you work out what this number should be?

What do you notice about the green numbers?

Can you make the green numbers from any of the other numbers?

Possible extension

Some children may enjoy the challenge of creating a spreadsheet which represents the children.

Possible support

It might help to have some cards or pieces of paper available which state the relationships between the different coloured numbers so that children can refer to them as they work.

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