The bar can be a valuable representation to enable students to represent problems in such a way that the mathematical structure is exposed. This enables students to ‘see’ the problem clearly and to then recognise the strategy they need to solve the problem. NCETM
Avoid the common mistakes!
Do not jump straight into the pictorial representation.
It is important to go through all the stages from concrete to pictorial to abstract at the start so that students can make sense of the problem and build up from something concrete to an abstract method that they can use fluently.
Choosing the right model
The part whole model is used to see parts as a proportion of the whole.
The comparison model allows you to see the difference and directly compare part to part rather than part to whole.
In the examples below red : yellow = 2:5
What fraction of the whole is red?
There are 35 sweets in the bag. How many are red?
What is red as a fraction of yellow?
There are 15 more yellow sweets. How many red sweets are there?
It is important to consider which model is most appropriate for the problem. I often show both models and get students to decide.
Introducing the bar model using CPA
Alfie and Billy go crabbing. Each bucket holds the same number of crabs. Alfie has one bucket and Billy has 3 buckets. If Alfie has 3 crabs, how many crabs does Billy have?
Try these questions using CPA.
Billy has 15 crabs. How many crabs does Alfie have?
There are 28 crabs in total. How many crabs does Billy get?
Billy has 20 more crabs than Alfie. How many crabs are there in total?
In my experience, students see the word ratio and share the number they are given into the given ratio, regardless of the question.
I try to avoid this by using SSDD questions in the first few lessons.
The number and the ratio stay the same but the question varies.
I follow this with a blank grid and ask students to write 4 different questions using the same numbers. See this article for more details.
Try the following NCETM (adapted) problems using bar models.
Year 6 Problems
1. Three quarters of a number is 54. What is the number?
2. There are 36 packets of biscuits. One half are chocolate, a ninth are digestive and a third are wafer biscuits. The rest are ginger nuts. How many biscuits are ginger nuts?
3. There is 20% off in a sale. How much would a track suit cost, if the normal price was £44.50?
4. There is 20% off in a sale. The reduced price of the jeans is £36. What was the original price?
5. At a dance there are 4 girls to every 3 boys. There are 63 children altogether? How many girls are there?
6. Seven in every nine packets of crisps in a box are salt and vinegar. The rest are plain. There are 63 packets of salt and vinegar crisps. How many packets of plain crisps are there?
Key Stage 3 Problems
1. Ralph posts 40 letters, some of which are first class, and some of which are second class. He posts four times as many second class letters as first. How many of each class of letter does he post? (This question appeared on a GCSE higher tier paper.)
2. A computer game was reduced in a sale by 20% and it now costs £55 What was the original cost?
3. Sally had a bag of marbles. She gave one-third of them to Rebecca, and then one quarter of the remaining marbles to John. Sally then had 24 marbles left in the bag. How many marbles were in the bag to start with?
4. Sam bakes a variety of biscuits. 13 are peanut, 12 are raisin, the remaining 5 were oat. If you choose 1 biscuit at random, what is the probability that you will get an oat biscuit?
5. Tom spent 30% of his pocket money and put away 45% into his savings. He was left with £2.50. How much pocket money did he receive?
6. Two numbers are in the ratio 4:5. They both sum to 135. Identify both numbers.
7. Two numbers are in the ratio 5 : 7. The difference between the numbers is 12. Work out the two numbers.
8. A herbal skin treatment uses yoghurt and honey in the ratio 5 : 3. How much honey is needed to mix with 130 g of yoghurt?
GCSE Exam Questions
Comparison and part whole
Comparison and fraction/percentages
Comparison and equivalent fractions
Comparisons with changes
Compare a:b to b:c
1:n and n:1
x:y = a:b to fraction
x:y = a:b to equation
Percentage Question(s) - Bar Models and Ratio Tables
Find 20% of 160
Find 80% of 160
20% of a number is 160 . What is the number?
80% of a number is 160 . What is the number?
Dynamic Percentage Change Model
Checkpoints are diagnostic activities that will help teachers assess the understanding students have brought with them from primary school, and suggest ways to address any gaps that become evident.