1. Material developed by Paul Dickinson, Steve Gough & Sue Hough at MMU
Fish and Chips
Material developed by Paul Dickinson, Steve Gough & Sue Hough at MMU

2. Thank you
Sue, Steve and Paul would like to thank all the teachers and students who have been involved in the trials of these materials
Some of the materials are closely linked to the ‘Making Sense of Maths’ series of books and are reproduced by the kind permission of Hodder Education

3. Note to Teacher
This section uses the context of a chip shop to introduce simplifying, expanding, and factorising. This is a context that should be recognisable to students, and the intention is that they get involved in the context and make sense of the mathematics through this. Students may wish to drop the context as they come up with ‘quicker ways’ or ‘rules’, but the teacher’s role is to take them back to the context to check if their methods ‘work’. The basic idea is that an order can be asked for in different ways, and the cost can be worked out in different ways. So, for example, ‘fish and chips 3 times’ can be worked out by doing 3(f+c), or by 3f+3c (‘three fish and three lots of chips’).

4. Note to Teacher
Students are asked to imagine themselves working at the shop, writing down phone orders, and then ‘bagging’ the orders. This leads to simplifying, expanding, and factorising expressions. The cost of different chip shop orders is used throughout as the mechanism for verifying the validity of individual solutions and more general methods. It is important that the teacher keeps reminding students that letters stand for the price of items and not the items themselves In trials of the materials, the biggest problems occurred if the context was dropped too early. The switch from ‘f’, ‘c’, etc to ‘x’ and ‘y’ is far from seamless, and teachers need to keep referring back to the context so that students can make sense of the more formal mathematics. The final part of this section looks at expanding and factorising, though the context is still there for checking answers.

5. At the Fish shop Fish £3.20 Sausage £1.30 Chips £1.40 Peas £0.80
Gravy £0.50
It is useful initially if the teacher encourages the students to become involved in the context. This can be done by asking students about the context, their own experiences, what they like to eat, how they order, how the prices compare to their local fish shop, etc
Then the first question is asking students how they would order if three of them all wanted a fish and some chips. Listen to responses here, and then ask for the total cost. The total cost is £13.80, but the important thing at this stage is how students worked this out. The next slide shows the two most common methods, and these are the important ones from now on.
Mathematically, the issue here is the equivalence of 3(f+c) and 3f + 3c

6. At the Fish shop
I do the price of 3 fish, then the price of 3 lots of chips and add them up.
I do the price
of 1 fish and 1 lot of chips, then times that by 3.
If both these methods emerge in the class, they can be identified by the names of the students in your class, for example ‘Paul’s method’. Otherwise, it is useful to give names to the characters in the slide. These two equivalent ways of calculating cost are the focus for most of the work from here.
Once the two different methods have been established, ask for the cost of ‘sausage and chips 3 times and two portions of peas’ (answer £9.70) to check on both ideas.

7. Lunchtime orders
At lunchtime, people sometimes come in with big orders.
Why do you think this is?
One lunchtime, a full order is fish and chips 3 times, sausage and chips twice, fish and peas twice, and 2 extra portions of chips
Presumably at lunchtime, one person may order for a group of people. Some teachers preferred to read out the order to see how students wrote it down.
Students work out the total cost of this order, which is £ What is of interest now is how students have done their calculation and whether they have recorded this in a particular way. Teachers have found it useful here for the class to see and discuss different students’ methods
In reality, orders are often taken over the phone and need to be written down. The price in then worked out. This is the issue in the next slide

8. 3(f+c) + 4(s+p) + 1c + 2p + 2(f+p).
Orders over the phone
3(f+c) + 4(s+p) + 1c + 2p + 2(f+p).
These two phone orders were written down by someone in the shop.
Some teachers initially asked students to read these out, or asked how the order would ‘have sounded’ over the phone.
The question then is whether students think they are the same order written down twice, or two different orders.
Perhaps half the class work out the cost of 1, the other half the cost of 2 (the actual cost is £33.20)
In trials, although cost was used to decide if the two orders were the same, some students began to see the equivalence by looking (in context) at the ‘algebraic’ expressions.
5f + 4s + 4c + 8p

9. 3(f+c) + 4(s+p) + 1c + 2p + 2(f+p). Made simpler,
Orders over the phone
3(f+c) + 4(s+p) + 1c + 2p + 2(f+p).
Made simpler,
3(f+c) + 4(s+p) + 1c + 2p + 2(f+p)
= 3f + 3c + 4s + 4p +c +2p +2f + 2p
=5f + 4s + 4c + 8p
There is no question here, this is a statement, more mathematically, of what the students have been doing. Students need to be able to make sense of this, and ask any questions, as the next slide consists of similar problems.

10. 2(s+c) + 3 (f+p) + 3(f+c) + 2p + 3c + 2s
Simplify
2(s+c) + 3 (f+p) + 3(f+c) + 2p + 3c + 2s
2. 2(f+c+g) + 2(f+c) + f + 3(s+c+p) + (s+c) + 2(c+g) + 3c
3. 5(s+c+g) + (f+c) +2(f+c+p+g) +3(f+c+g) +2s +2c
Three more orders to simplify. Some teachers like to keep asking students to find the cost of each order. This serves to emphasise that each letter stands for ‘cost’, and also keeps more in touch with the context.
Fully simplified expressions are 1. 4s +8c +6f + 5p, f + 10c + 4g + 4s + 3p s + 13c + 10g + 6f + 2p
The next slide is more practice with such expressions, and can be omitted if necessary.

11. 9s + 6c 2. 12f + 16c 3. 6f + 8c + 4s Simplified orders
Below are three ‘simplified’ orders. In each case, give two possible ways in which they may have been ordered over the phone
9s + 6c
2. 12f + 16c
3. 6f + 8c + 4s
This can be seen as very informal factorising. As such, teachers may wish to value equally all correct answers, or to highlight particular ones. For example, in Q1, while 6(s + c) +3s is likely in context, the teacher may wish to introduce the possibility of 3(3s + 2c). Factorising is dealt with in more detail from slide 22

12. On day, Jane looked at Azim’s notepad and saw 3(f+c)
Cancelling orders
Sometimes, people phone through an order, then ring up a bit later and change it.
On day, Jane looked at Azim’s notepad and saw 3(f+c)
She came back to check it a few minutes later and saw that now on the notepad was 3(f+c) - (f+c)
What do you think has happened here?
A little contrived, but a means of using the context to help with the difficult idea of subtracting brackets. The likely story here is that someone ordered 3 portions of fish & chips, then phoned back to cancel one of the portions

13. Cancelling orders The order was 3(f+c) - (f+c)
What should Jane wrap up for the customer?
On another occasion, Jane saw on Azim’s pad
3(f+c) – f + c
Is this the same?
The final attempt, in context, to address the issue of subtraction.
Students can now attempt the questions on Worksheet A1
What do you think 5(s+c+g) - 2(s+c) means?
What is the simplified order here?

14. Wrapping it up
People sometimes complain when they collect their orders if different bags have different things in them.
One order was for 12f + 16c
How could this be bagged so that each bag contains exactly the same?
This is obviously a more difficult question, and the teacher may have to introduce one possibility, for example 2(6f + 8c). Some teachers used questions such as “could we use just three bags and have the same in each bag?”, etc.
4(3f + 4c) is the preferred solution.
The next slide contains six other questions of this kind.

15. Try to do the same with these orders: 3s + 3c 4f + 2c + 2p
Wrapping it up
Try to do the same with these orders:
3s + 3c
4f + 2c + 2p
2c + 4s + 6f
3s + 3f + 6p + 9c
12s +9c
12f +8c +4g
Most teachers at this point were happy to accept any correct equivalences, though again, for more able students, there was discussion of which are ‘simpler’ etc. And once again, even at this stage, it is important that the context is referred to as a means for students to check and verify their answers
‘Fully factorised’ solutions would be 1. 3(s+c) 2. (2f+c+p) 3. 2(c+2s+3f) 4. 3(s+f+2p+3c)
5. 3(4s + 3c) (3f + 2c + g)

16. Doing the maths
Remember that when we say ‘3 fish’ we are actually talking about the cost of 3 fish
Expanded form
How to say it in expanded form
Factorised form
How to say it in factorised form
3f + 3c
3 fish and 3 chips
3(f + c)
3 lots of fish and chips (or fish and chips 3 times)
2f + 2c + 2p
2(f + c + p)
6f + 3c + 3p
3(3f + 2c)
2f + 4c + 2p
Here now is a more formal statement of what students have been doing in the previous slides. ‘How to say it’ keeps the context. This slide is available as Worksheet A2

17. Using different letters
Expanded form
Factorised form
3x + 3y
3(x + y)
2(p + q + 3r)
9x + 6c + 3y
5(3x + 2y)
6f + 4x + 2p
Could be titled ‘Fish and Chips at GCSE’. Very similar questions to the last slide, and some letters still the same.
This slide is available as worksheet A3.
Worksheet A4 gives more practice with simplifying, expanding and factorising.

18. Summary
The context of ‘fish and chips’ can help you to simplify, expand, and factorise algebraic expressions, and to see the equivalence of such expressions. It can help to keep thinking about the context of fish and chips when you are solving algebraic problems