# Tray Bake

## Problem

- How are you going to scale the recipe for different numbers of cakes?

Then you have to decide how much you will charge for the cakes. Clearly you need to cover the cost of ingredients, and you want to make a decent profit for the charity.

- From this set of cards, find a mauve card for 12 cakes, a turquoise card with the recipe for 12 cakes and a green card with the cost of ingredients of 12 cakes, and lay them out in a group in the centre of the table.
- Now sort the rest of the cards into groups - in each group you should have one turquoise recipe card, one mauve card for the number of cakes and one green card for the cost of ingredients. Put each group of cards separately on the table.
- Use the arrow cards to show the relationships between different group. You may also wish to use blank arrows for your own relationships between cards.

When you're satisfied you've sorted all the first set of cards and found connections between them, use these two sets of additional cards (set 1, set 2) and arrows to extend the activity.

## Student Solutions

12 cakes cost 65p.

For 24 cakes, double the cost is £1.30.

For 36 cakes, the cost of 12 cakes multiplied by 3 is £1.95.

For 6 cakes, half the cost is not 32.5p, since we no longer have half pence in the UK. The cost is therefore 33p.

For the ingredients:

Ingredients | 12 cakes | 24 cakes | 36 cakes | 6 cakes | 1 cake |

Caster sugar | 130g | 260g | 390g | 65g | 10.8g |

Margarine | 130g | 260g | 390g | 65g | 10.8g |

Self raising flour | 130g | 260g | 390g | 65g | 10.8g |

Eggs | 2 | 4 | 6 | 1 | 1/3 |

Baking powder (5ml spoons) | 1 | 2 | 3 | 1/2 | 1/12 |

To scale the recipe for other numbers of cakes which are not a simple multiple of 12 or 6, you will need to find the cost of 1 cake and the amount of ingredients required for 1 cake. Then multiply by the number of cakes required, rounding your answers sensibly and remembering that eggs come in whole numbers! (Professional kitchens weigh the eggs, rather than using whole numbers of eggs.)

If you are using the provided arrows, these facts may help you:

- use x2 (doubling) arrows going from 6 cakes to 12 and from 12 cakes to 24
- use divide by 2 or multiply by 1/2 (halving, same thing as division and multiplication are inverse processes,and 1/2 is the inverse of 2) arrows for 12 cakes to 6 or 24 cakes to 12
- use x 1.5 or x3/2 or half as much again cards going from 24 cakes to 36
- use divide by 1.5 or x2/3 cards going from 36 cards to 24

## Teachers' Resources

### Why do this problem?

This is one of a series of problems which students might encounter in Food Technology.The problem originated in a discussion with Food Technology PGCE students, who said that scaling recipes is something many students find difficult and which can be challenging for the FT teacher to help them with.

Mathematically, this problem gives practice in calculating proportions and ratios.

### Possible approach

Note: the basic recipe is for 12 cakes (which is why the cost of 6 cakes is 33p - no half pence any more).

Start by laying out the three cards below in a group (they come from this set of cards, one mauve for the number of cards, one turquoise for the recipe, and one green with the cost of the ingredients):

Explain that this is the basic recipe, the number of cakes it will provide and the cost of the ingredients, without any extras.

Now pull out any one card from the rest of the cards and lay it out by itself. Then lay out all the rest of the cards face up and ask the students to find the other two that go with your single card. You should end up with a second group of 3 cards, a mauve one for the number of cakes, a turquoise one for the recipe, and a green one for the cost of the ingredients. Point out that some of the quantities in the recipe are missing - can they supply the missing information?

Then lay out the arrow cards. Students should choose as many of these as they can to connect the two groups of cards, making sure the arrows are pointing the right way. Ask if they can think of other connections, using words or numbers, and get them to fill in blank arrows.

Students should then be ready to work in small groups, grouping the cards and using arrow cards (either those provided or their own ideas on the blank arrows) to connect the groups of cake cards. They should ensure that everyone in the group understands the placing of all cards and arrows and agrees them - if they don't all understand or agree, they should discuss the problem, not just
move on.

### Key questions

- Can you all explain how you've grouped the cards and connected them with arrows?
- Is there a number of cakes which acts as a unit or basis for the other calculations?
- Are there any arrows provided which you haven't used yet? Could you use them?
- Are there any arrows provided which you can't use at all? If so, why not?
- Could you put some of your arrows in different places between the groups and still have a correct arrangement?
- Are there arrows which are equivalent?