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If you had 36 cubes, what different cuboids could you make?

### Making Boxes

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

### More Christmas Boxes

What size square should you cut out of each corner of a 10 x 10 grid to make the box that would hold the greatest number of cubes?

##### Stage: 2 Challenge Level:

I went to the cinema with some friends last week and we decided to buy some bags of popcorn for during the movie.

"Three for the small, five for the medium size and the large size is seven", the popcorn server replied quickly.

"Does the medium bag have twice as much in as the small size?" Aimee asked her. "Of course it does!" said the server, "The bag is double the size. See, it's taller, it's wider and it's deeper."

We estimated that the size of the small bag was 20cm by 10cm by 5cm.
I wonder if you can figure out which size belongs to which side.

You could start by building a model of the small bag.

Use paper that is 20cm by 30cm and sticky tape. Don't forget a ruler to draw the lines. Striped paper is a big help to you for measuring and for cutting straight.

Aimee thought that was a bargain! "We only need one medium sized bag and we can all share that", she said. What was Aimee thinking?

You could construct a model of the medium sized bag to help you explain.

Plan this out first to make sure you have enough paper for your bag.

We wondered what size the large bag would be. The popcorn server told us, "Why, the large size is double the size of the medium bag, of course".

"Do you mean each side is two times bigger?" asked Aimee.

"Naturally!" said the server.

Do you think she is correct?

How big would the large size bag be?

If the small bag contains one serving of popcorn, how many serving would the large bag contain?

Do you think the popcorn server was correct when she said that each bag was double the smaller size?

What do you think she should have said to better explain?

I wonder if a drum container that was 20 cm high and 10 cm across the base (the diameter) would hold more or less than the bag of popcorn.

Perhaps you could find out?

This problem is also available in French: Double Ton Popcorn, Augmente Ton Plaisir!

### Why do this problem?

This problem applies and extends skills and knowledge in several directions; estimation, measuring both linear and mass, construction, surface area, proportion, methods of calculating the volume of different shapes ... Part of the benefit of such a question is that it sends a loud message about how we use mathematical vocabulary and how precisely it describes concepts. What does it mean to double something? If we are referring to a three dimensional object, which surfaces do we have to double? What are the implications for the increase in size?

### Possible approach

Presenting this problem as it is on the site will be further helped by having all the likely resources for making containers readily available.

### Key questions

Tell me about the amount of popcorn this container will hold.

### Possible extension

The last part of the problem gives an idea for a possible extension.

### Possible support

Some pupils will benefit from someone to one work with using cuboids made from unit cubes and talking about the volumes.