1. Solving problems using ratio tables
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 introduces ratio tables as a means of solving problems associated with proportional reasoning. The ratio table can be used in a variety of contexts, and the intention here is that students will come to see ‘the sameness’ of these questions rather than the differences. Teachers have suggested that the ratio table can be used to answer around 50% of Foundation Level GCSE Number questions, so it is undoubtedly an important tool.
The ratio table is really just a structure within which students can record their workings. Obviously, at post-16, some students will come to class with totally valid and reliable methods of their own. The intention in such cases is that ratio tables are seen as a way of representing student ideas and possibly developing them further. At no point, are ratio tables seen as a ‘different’ method that should replace what students can already do.

4. Teachers have found it important to allow students to ‘play’ with their ratio tables, even when this means finding a lot of redundant information. Finding the quickest route to an answer may be desirable eventually, but not if it is enforced and hence comes at the expense of understanding. Consequently, if a student is ‘stuck’ they are encouraged to ‘write down anything you know’, rather than being pushed down a particular route. Also, there is a large body of evidence which suggests that students find it much easier to see relationships within the same measure (i.e. working across the table) than to see relationships across measures (i.e. working down the table). Hence, only at the very end, is the ‘unitary method’ suggested as a general strategy.

5. Finally, the tables in this chapter are sometimes hand drawn and often without a ruler. It is not the intention that students spend time on the drawing of the tables, or on worrying how many columns they need, etc. In this respect, ratio tables are really for student ‘jottings’.
Some teachers like to encourage students to annotate their ratio tables, and these teachers often also do such annotations themselves. The following slide shows an example of this for the problem;
if 125g of cheese costs £1, what would be the cost of 600g?
Other examples of strategies are included in the slide notes

6. If 125g costs £1 and we
are trying to find the
cost of 600g, a ratio
table may look like;
Or, if fully annotated;

7. Can this be true? You need to sort that out. Mine’s only 44
My waist is size 80
Discussion of how this can be true. What are students’ own experiences in terms of using inches and/or cm. In trial classes with mature students this often led to a more general discussion about metric and imperial units and which units are in common usage
How can this be true?

8. Extending the Number Line
Unlike many countries, in the UK we still use two different units for measuring length. Both of these can be seen on a standard tape measure.
The tape shows that 6 inches is roughly the same as 15 cm.
What do you think 12 inches is in centimetres?
It could be useful to have tape measures available for students. Rulers also work, though the two scales are usually the reverse of each other which makes direct comparisons more difficult. Some students may know a rough conversion.

9. The Number bar
Daniel drew the number bar above to represent the tape measure
Make a copy of this bar and mark four other points on it that you know in both centimetres and inches.
The bar here is essentially a ‘model’ of the tape measure. In trials, teachers found it useful to refer back to the tape measure if students were struggling with the bar.
Question 1 should be accessible to all students, and the freedom to mark on any points is important at this stage. Four points are asked for, as the most obvious strategy of ‘halving’ is likely to only give two or at most 3 points. It is up to the teacher whether or not students at this stage are allowed to extend the bar (to allow, say, 60 cm and 24 in). Question 2 is considered again two slides further on, so not all students need to attempt this.
Mark is 5 ft and 3 inches. Alexandra is 1 metre and 72 centimetres. Use the 30cm to 12 inches rule of thumb to decide who do you think is the taller of the two?

10. Ratio Tables
A problem when you extend the number line is that it becomes rather difficult to fit on a page. Some people use a ratio table as a more flexible version of a number line. The idea is that you can now fill in other values.
Here is the ratio table for the first time, and is essentially the same as the number bar, but without scale. Teachers at this stage often draw a rough copy of this on a board and invite students to come and fill in more columns. By disallowing doubling and halving, strategies such as ‘adding columns’ can be stressed. If enough ideas come up here, it is possible to skip the next slide.
Teachers have found it important to keep referring back to the context to check the validity of entries. For example, “how do we know that we can double the numbers in one column to get another?”, “Could we multiply by 3?”, Could we add 5 to each of the numbers in a column?”, “Why not?”, etc.
Look at the example above of students having partly filled in a ratio table for the ruler. Can you see where the numbers in the 3rd column have come from?

11. Ratio Tables
The following ratio table was used to work out what 5ft 3 inches is in centimetres.
It is important now that students can see where each column has come from, and that there is generally more than one way to arrive at an answer.
For example, in trials, “45cm and 18in” was a common answer to Q1. Common strategies for justifying this were ‘15+30’, or ‘3x15’, or ‘halfway between 30 and 60’, or ‘60-15’, but there were others as well! This flexibility is a crucial element of ratio tables.
Give two other combinations of cm and ins you could find from the table
From the ratio table above, write down answers to (i) 225 cm in inches and (ii) 21 inches in cm

12. Recipes
Helen decides to use the ratio table below to work out the ingredients needed for different numbers of pancakes.
Pancakes
Makes 8 pancakes
Ingredients
125 g plain flour
1 medium size egg, beaten
300 ml milk
a little oil for frying
An important element in students making sense of their mathematics is that they make sense of the context. They need to enter and imagine the context, not simply take the mathematics from it. This question is intended to do this. Students may read the recipe themselves, or someone may read it aloud, or the teacher may ‘tell a story’. Or students could be asked to describe in their own words how to make pancakes, or there could be links made with Food Technology
Copy the table and fill in four more columns showing the ingredients needed for different numbers of pancakes

13. Recipes
There are 28 people in Helen’s college group and she wants to make pancakes for all of them.
Decide how much of each ingredient she will need.
2. As part of her Food Technology course, Helen plans to make 100 pancakes on Tuesday 4th March.
a) Use a ratio table to work out how much milk she will need.
b) If you were shopping for Helen, how much milk would you buy?
This slide could be done as a whole class, with students contributing ideas or coming to the board to fill in parts of the ratio table. If individual work, teachers have at this stage sometimes given students a prepared ratio table. If students are drawing their own, teachers have found it important to try and remove issues such as ‘how many columns do I need’? Obviously strategies will vary, and there is no desire at this stage to be ‘efficient’. Indeed, a variety of strategies is actually beneficial at this stage so that students can compare and discus the different ideas.
Many students are likely to begin by doubling, and halving is also useful here. So, for example, a typical table may be
Number of pancakes
Plain flour (grams)
Milk (ml)
Eggs
If students are really thinking about the context, there could also be some discussion here about how we deal with half an egg!
For Q2, some more able students may wish to use a ‘unitary’ method here, using 1, 2, or 4 as the ‘unit’. While this could be a useful discussion point if around in the class, teachers are discouraged at this point from introducing such strategies themselves.

14. One of the strengths of ratio tables is the wide variety of contexts in which they can be used. The following slides (13-15) and worksheet N9 are intended to stress this and to encourage students to see the mathematical ‘sameness’ of these problems rather than the contextual differences.

15. More ratio table problems
Elaine likes to keep fit by taking daily walks. On average she walks at a steady pace and can cover 3 km in 40 minutes.
What else do you know? Put 3 more entries into the ratio table

16. More ratio table problems
Ratio tables can be used to help to answer many types of questions. For example, here is a question, and also Helen’s solution.
The local supermarket sells multipacks of coke with 14 cans in one pack. Helen buys 17 multipacks, how many cans of coke is that?
Explain carefully how Helen has found her answer

17. More ratio table problems
For the next questions, draw your ratio table and fill in three other sets of values that you know to be true. Try to be as creative as possible with your values.
miles is roughly 8 km. What else do you know?
One multipack of crisps contains 15 bags. What else do you know?
metres of ribbon costs £4.80. What else do you know?
In trials, teachers sometimes used these questions as an opportunity to get groups of students to show their tables to the rest of the class, or teachers ‘pooled’ answers from different groups.
Teachers also asked follow up questions such as “What is 20km in miles?”, “How long would it take the printing machine to produce 60 copies?”, “How many bags of crisps there will be in 17 multipacks?”, etc.
Some teachers also felt that this was the place to consider a question where direct proportion (and hence a ratio table) was not valid. For example, in the problem
A taxi journey of 8 miles costs £18. What else do you know?
Some teachers introduced this separately, whilst others simply gave it with the four others on the slide above and then at the end asked “In one of these questions, it is not possible to deduce other information. Which one is it?”
Worksheet N9 can now be used here as a means of ‘practicing’ using ratio tables.
A printing machine can produce 25 copies in 30 seconds.
What else do you know?

18. Slides 17-22
This next set of slides are all concerned with the idea of ‘best buy’. Again, this may need some discussion. Buying a larger quantity is not always ‘best’, though it does usually represent ‘better value’ in a theoretical sense. It feels important that students are allowed to explore this.
These questions are also more difficult numerically, with more sophisticated strategies needed in order to arrive at quantities that can be directly compared. Annotating ratio tables may be particularly useful here.

19. Best buy
In the supermarket, Helen and Nisha are buying ingredients for other pancakes and trying to make sure that they get the ‘best buy’
This is really just asking for students’ ‘instincts’, and also gives an opportunity for the teacher to assess students’ prior knowledge of such problems.
Which do you think is the best buy?

20. Best Buy
To work out which is the best buy, they decide to use a ratio table. They begin by doubling to see if this helps them to compare the two buys. After a few minutes, they both have exactly the same thing on their notepads. It looks like this:  
This is still really about students’ own ideas, though there is now more information available. It is possible that incorrect additive strategies may be aired here. Some students may immediately see that the top table suggests that 625g costs £5 and so this is better value while others may conclude that ‘the bigger pack is usually better value’.
Can you decide from here which is the ‘best buy’?

21. From the original table on the right, Helen and Nisha then
work in slightly different ways to make a comparison easier. Below is their working.
Class discussion.
The approaches shown here are ones used by many students for determining ‘best value’; comparing different quantities for the same price or same quantity for different prices.
Explain carefully what each of them has done and how you can now see which is the best buy.

22. Best Buy-Ham
Here are some other ingredients we might use in pancakes. Use ratio tables to work out which pack of ham is the best buy (the tables have been started for you below)
One of the strengths of ratio tables is that they allow a variety of strategies to be used within the same structure. A number of such strategies, all of which were initially produced by students in class, are shown for each of the next three problems.
Here, the 120g pack is better value.
A possible strategy is
Weight (gram)
Price (£)
Weight (gram)
Price (£)

23. Packs of cheese
Use ratio tables to work out which pack of cheese is the best buy
These come out to be the same
Weight(g)
Price(£) £2.40 £4.80 £7.20
Weight (g)
Price(£) £1.80 £3.60 £7.20 OR
Weight (g)
Price(£) £2.40 £ p £1.80
Weight (g) 240
Price(£) £1.80
Weight (gram)
Price (£)
Weight (gram)
Price (£)

24. Best Buy-Mushrooms
Mushrooms can either be bought in a 150g pack for p, or a 250g pack for £1.40.
Which pack would be the ‘best buy’?
250g pack is better value
A possible strategy is
Weight (gram)
Price (£)
Weight (gram)
Price (£)

25. Comparing Prices Here is part of an online shopping page
The next four slides could be preceded by a discussion about how shops help us to determine ‘best buy’.
For example, most supermarkets now include information on their labels to help customers determine the best value. This is usually in the form of price per 100g or 100ml, but may also be price per kg, per litre, etc. Students could easily find such examples.
This information is usually also available when shopping online; slides 23 & 25 show a couple of typical web pages. But again, these are very easy for students to find themselves.
What information has been given to help you decide which is the best buy?
The 500g tin of coffee costs £ Explain carefully how you can work out that this is £2.30 per 100g

26. Comparing Prices
You could have used a ratio table for this. For example or
This slide is important, as it stresses division as an important strategy. This is known to be more difficult for students than doubling, halving, adding, etc.
Which method do you prefer?

27. Costs per 100g
Aimee has started work at the supermarket and has to work out the 100g costs for the website
£2.25
£4.40
The information above shows that tea bags are £2.25 for 250g. Use a ratio table to work out the cost of 100g
Which of the three sizes shown here is the cheapest per 100g?

28. Practice
Work out the cost per 100g of the following products, in each case drawing a ratio table to show how you did this.
250g of tea bags priced at £1.50
A 400g tin of soup priced at 92p
1kg box of cornflakes priced at £3.30
750g of cornflakes priced at £2.70
A 150g bar of chocolate priced at £1.00
This is ‘practice’ at using ratio tables to find cost per 100g.
Worksheet N10 can also be used here. This is a review of all the work covered so far. It is hoped that by the end of this, students are relatively confident and fluent in their use of ratio tables. Worksheet N10 could also be delayed until after the next two slides.

29. The Unitary method
Ratio tables can be used to help with many different kinds of problems. For example;
To make 12 scones, Dave uses 150g of flour. How much flour would he use to make 20 scones?
On the right are two possible ways of approaching this problem. Explain carefully what has been done in each case
The unitary method is introduced as just one more way of solving problems, and as such is simply an extension of the work done so far. It does, of course, rely on the use of division, which may be why students often find it conceptually difficult.

30. Another example
With 8 litres of petrol, a motor bike can travel 144 miles. How far would it travel with 10 litres of petrol?
Worksheet N11 contains a variety of questions that can all be solved using ratio tables. Some of these questions are quite challenging, and consequently some teachers have used worksheet N10 here as a review of ratio tables, with worksheet N11 as extension work for some rather than all students.

31. Summary
Ratio tables can be used to answer many questions which you find on GCSE papers.
It is important to ensure that the ratio table makes sense in the context you are using it in
Doubling and halving, multiplying by a number, dividing by a number, adding or subtracting columns are all strategies you can use in ratio tables

32. So, for example, if
125g costs £1 and
we are trying to find the cost of 600g, a ratio table may look like
Or, if fully annotated