# Biscuit Decorations

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

## Problem

*Biscuit Decorations printable sheet*

Andrew decorated $20$ biscuits to take to a party.

He lined them up and put icing on every second biscuit.

Then he put a cherry on every third biscuit.

Then he put a chocolate button on every fourth biscuit.

So there was nothing on the first biscuit.

How many other biscuits had no decoration? Did any biscuits get all three decorations?

Image

Printable NRICH Roadshow resource.

## Getting Started

Perhaps you could sketch the biscuits?

The second biscuit has icing on it. Which other biscuits have icing on?

Which biscuits have a cherry on them as well as the third one?

What about the biscuits with a chocolate button on them? Which ones are they?

## Student Solutions

We had a good number of solutions sent in for this challenge. There were also some good descriptions as to how you did it and some recording ideas for everyone to see.

First from Ellie at Stanley Park Junior School:

There are 20 biscuits.

0= Biscuits

O= Cherry

\= Chocolate Buttons

D= Icing

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

D D D D D D D D D D

O O O O O O

\ \ \ \ \

Seven have nothing on.

Next, from Brampton Primary School, we had two emails sent in.

First from Toby;

Secondly from Brampton we have Georgia and Joe:

Two Alfies at Christ the King Catholic Primary School wrote:

There is one biscuit with all three decorations on....it is biscuit 12

This is because it is in the 2, 3 and 4 times table.

Miss Martin bought us biscuits, cherries and buttons and we put 20 biscuits in a line.

We followed the instructions on the problem. We noticed it was a number

pattern.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

3, 6, 9, 12, 15, 18

4, 8, 12, 16, 20

There are some biscuits with nothing on. 1, 5, 7, 11, 13, 17, 19

Next JJ, JU, IA and DM at Ardley Hill Academy, Dunstable, sent in the following creative account of what happened:

Cookie problem

JU said “We stuck 20 pieces of paper in our books and numbered them.”

IA said “Then we drew a circle on every 2nd one for the icing.”

JJ said “Then we drew a red dot on every 3rd one for the cherries.”

“What about the brown dot on every 4th one for the chocolate button,” said DM.

IA said, “We found that only odd numbers have no icing on.”

“Even numbers all have some decoration on,” said JJ.

“There was nothing on 1, 5, 7, 11, 13, 17 and 19,” they read out.

“They are all odd numbers,” said JU”.

“But some are missing” added DM.

“Number 12 got all decorations.” They all agreed.

Then Mrs Clarke helped us to draw a Venn Diagram.

After this we tried it with real biscuits using “randoms” instead of cherries!!!

Lastly, we had the following from Aymen at Arnhem Wharf School:

I drew pictures of the biscuits. I used a key to then show all the decorations. I then counted all the biscuits that did not have decorations.

Well done all of you and thanks for sending in your recordings and explanations. We look forward to seeing other solutions you send in in the future.

First from Ellie at Stanley Park Junior School:

There are 20 biscuits.

0= Biscuits

O= Cherry

\= Chocolate Buttons

D= Icing

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

D D D D D D D D D D

O O O O O O

\ \ \ \ \

Seven have nothing on.

Next, from Brampton Primary School, we had two emails sent in.

First from Toby;

Image

Secondly from Brampton we have Georgia and Joe:

Image

Two Alfies at Christ the King Catholic Primary School wrote:

There is one biscuit with all three decorations on....it is biscuit 12

This is because it is in the 2, 3 and 4 times table.

Miss Martin bought us biscuits, cherries and buttons and we put 20 biscuits in a line.

We followed the instructions on the problem. We noticed it was a number

pattern.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

3, 6, 9, 12, 15, 18

4, 8, 12, 16, 20

There are some biscuits with nothing on. 1, 5, 7, 11, 13, 17, 19

Next JJ, JU, IA and DM at Ardley Hill Academy, Dunstable, sent in the following creative account of what happened:

Cookie problem

JU said “We stuck 20 pieces of paper in our books and numbered them.”

IA said “Then we drew a circle on every 2nd one for the icing.”

JJ said “Then we drew a red dot on every 3rd one for the cherries.”

“What about the brown dot on every 4th one for the chocolate button,” said DM.

IA said, “We found that only odd numbers have no icing on.”

“Even numbers all have some decoration on,” said JJ.

“There was nothing on 1, 5, 7, 11, 13, 17 and 19,” they read out.

“They are all odd numbers,” said JU”.

“But some are missing” added DM.

“Number 12 got all decorations.” They all agreed.

Then Mrs Clarke helped us to draw a Venn Diagram.

After this we tried it with real biscuits using “randoms” instead of cherries!!!

Lastly, we had the following from Aymen at Arnhem Wharf School:

I drew pictures of the biscuits. I used a key to then show all the decorations. I then counted all the biscuits that did not have decorations.

Image

Well done all of you and thanks for sending in your recordings and explanations. We look forward to seeing other solutions you send in in the future.

## Teachers' Resources

**Why do this problem?**

This problem fits in well with counting and skip-counting (counting by twos etc.) and can be solved by physically modelling the biscuits and decorations with whatever objects are convenient. It is a good opportunity for children to choose the way they represent the problem in order to solve it. It may also be
appropriate to introduce vocabulary such as "multiple".

### Possible approach

An important element in understanding the problem is the language of ordinal numbers, so 'warm-up' activities which involve using the concepts of first, second, third and fourth would be worthwhile for young children.

Invite learners to work on the problem using whatever they find most helpful - have paper, pens, pencils, cubes, counters etc. easily available. You may like to stop them part way through to share some different representations with the whole group. Some children might have made models with differently-coloured cubes for the decorations, some may have drawn pictures, some may have used
symbols. Invite the children to comment on the different ways of recording - what are the advantages of each way? You may find that some learners adopt a different representation following the discussion and it would be interesting to know why this was.

For those children who are more mathematically experienced, consider linking this problem with the idea of common multiples through the multiplication tables and the hundred square.

### Key questions

Which other biscuits have icing on?

Which biscuits have a cherry on them as well as the third one?

What about the biscuits with a chocolate button on them? Which ones are they?

Tell me about the biscuits that have no decorations on them.

### Possible extension

Generate your own similar problems using a greater number of biscuits and different combinations of skip counting, or encourage investigation of the various possibilities. Can children find a combination of skip-counting that allows every biscuit to be decorated?

### Possible support

With practical equipment available to model the problem, it should be accessible to most learners.

Handouts for teachers are available here (Biscuit Decorations.doc, Biscuit Decorations.pdf), with the problem on one side and the notes on the other.