# Cover the tray

These practical challenges are all about making a 'tray' and covering it with paper.

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*Please note: this problem uses whole numbers of centimetres throughout.*

Start with a square piece of card.

Draw the same smaller square in each corner of the card.

Cut one edge of each of these squares so that you can fold the card up to make a tray.

(See the pictures above.)

We are going to cover the tray. The covering goes over all that can be seen, so it is just the square base underneath that remains uncovered.

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**First challenge**

If the card is 18cm x 18cm and the amount of wrapping paper used is 432cm$^2$, what are the dimensions of the base of this tray and what size are the square corners that need to be cut?

**Second challenge**

The amount of wrapping paper used is 420cm$^2$.

What possible squares of card can you start with and what size corners can you use?

**Third challenge**

The amount of wrapping paper used can be between 400cm$^2$ and 499cm$^2$.

What possible squares of card and corners can you find?

Can you find a system that helps you get them all?

*This problem featured in the final of the Young Mathematicians' Award.*

Maria and Rosalind from Maryvale USA were the only ones to come up with a solution to the first challenge. This is what they wrote:

First challenge:

To solve this problem, at first we did 18 x 18 which equals 324. The reason we did 18 x 18 is to find the area of the card.

Next, we did 432 - 324.The answer we got was 108. The 108 really represents 108 cm ². Also, 108 cm ² represents the difference between the amount of wrapping paper used and the card.

This problem was very hard, and we made many errors before solving it. At first we started with a drawing (model) of a square. We labeled the sides 18cm each. The first attempt, we thought, (18 x n) x 4 = the difference 108 cm ². We divided both sides of the equation by four and found 18n = 27. And that n = 1.5 which can't be true because it can only be whole centimeters.

Second attempt:

Sadly, the second try didn't work either. We tried solving the problem by realising we DON'T have to wrap the little square of n ². So the equation was 4 x (18 x n) - 4n ² = 108 cm ². But that did not give us a whole number either.

Final Try:

We realised that the outside panels aren't 18 cm long anymore because we need to subtract 1n from both ends. So the equation is: 4 x [(18 - 2n) x n] - 4n ² = 108 cm ². At the end we found 18n - 3n ² = 27. Therefore, our answer for n = 3cm.

The square corners will be 3cm by 3cm, and the dimensions of the base square is 12 cm x 12 cm .

Thank you for the explanation of your attempts and allowing us to see the errors that occurred along the way.

### Why do this problem?

This problem engages the pupils in both arithmetic relationships between numbers, multiplication facts and spatial awareness. It is a very challenging activitiy and probably should only be presented to pupils who feel confident about such investigations.### Possible approach

A bit of preparation beforehand will be useful - make a box following the idea in the instructions. It is best to have a square card that has had the corner pieces cut ready for gluing and another in which the corners have been glued and you have suitable covering paper cut to exactly the correct size as one sheet.

Perhaps rather like this:

Talk with the pupils about the three pieces you have and then gradually introduce them to the actual challenge, emphasising that they decide on both the size of the card to start with, and the size of the four corners to be used to fold round to make the "walls" secure.

They may need reminding that they are to find many (if not all) the possibilities.

Perhaps rather like this:

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Talk with the pupils about the three pieces you have and then gradually introduce them to the actual challenge, emphasising that they decide on both the size of the card to start with, and the size of the four corners to be used to fold round to make the "walls" secure.

They may need reminding that they are to find many (if not all) the possibilities.

### Key questions

What have you done to decide on the size of your starting square card?

How are you finding out how much paper you will need to cover it?

You've made some that were too big/small how are you adjusting your numbers to get it within the limits?

How are you finding out how much paper you will need to cover it?

You've made some that were too big/small how are you adjusting your numbers to get it within the limits?