Show how many


a) 
Mara has 4 books for every 3 books Chloe has. Chloe has 12 books. Show how to work out how many books Mara has.





b) 
Sam gets $15 every 5 weeks for doing work around the house. Show how to work out how many weeks it will take Sam to save $60.





c) 
There are 6 boys for every 4 girls in a hockey team. There are 10 girls in the team. Show how to work out how many boys are in the team.



 Creating a ratio or rate table diagrammatically
 Creating a ratio or rate table using repeated addition.
 Creating a ratio or rate table using multiplication or division
 Solving by unitising to a whole number unit rate or ratio using multiplication facts.
 Solving by unitising to a fractional unit rate or ratio using multiplication or division.
Y8 (10/2010)  
a) 
16 Describes any 1 of the following ratio strategies:

easy moderate 
b) 
20 Describes any 1 of the following strategies:

easy easy 
c) 
15 Describes any 1 of the following strategies:

difficult difficult 
Based on a representative sample of 187 Y8 students.
This resource is about using proportional reasoning to find a missing amount when two ratios are compared. Students need to be advanced multiplicative thinkers or early proportional reasoners (Number Framework, Stage 7) for this resource. In this resource, students can reason with either rates, or ratios.
 A rate is used when comparing two different types of amounts, e.g., dollars per week in part b).
 A ratio is used when comparing two similar types of amounts, e.g., books:books.
In parts a) and c) all comparisons are ratios, either people:people or books:books. However, some ratios compare more similar items than others.
 In part c), the ratio could be of boys to girls or the ratio of girls to girls. The latter compares more similar entities.
 Likewise, in part a) Chloe's books after:Chloe's books before (12:3) compares more similar entities than the ratio of Mara's books:Chloe's books (^{4}/_{3}).
Common response  Likely calculation  Likely misconception  
a) b) c) 
13 50 12 
4 – 3 = 1; 12 + 1 = 13 15 – 5 = 10; 60 – 10 = 50 6 – 4 = 2; 10 + 2 = 12 
Uses an additive relationship Subtracts elements of the given rate (^{4}/_{3} ; ^{15}/_{5} or ^{6}/_{4}) instead of seeing the division relationship between them. 
a) b) c) 
12 75 24 
4 × 3 = 12 15 × 5 = 75 6 × 4 = 24 
Confuses multiplication and division with a rate Multiplies elements of the given rate (^{4}/_{3} ; ^{15}/_{5} or ^{6}/_{4}) instead of seeing the division relationship between them 
b)  12  60 ÷ 5 = 12 
Uses wrong elements to create rate Divides final number of dollars (60) by the initial number of weeks (5). 
a) b) c) 
48 or 4 4 60 
12 × 4 = 48 or 12 ÷ 3 = 4 60 ÷ 15 = 4 10 × 6 = 60 
Uses wrong unit rate Assumes that a given number is a unit rate, e.g., a) Mara has 4 books for every 1 of Chloe's or Mara has 1 book for every 3 Chloe has; b) $15 per week; or c) 6 boys for every 1 girl. 
c)  14  10 ÷ 4 = 2r2; so 6 × 2 + 2  Misinterpret the remainder  Treats it additively 
See Student work samples [pdf] for examples of the different strategies students use.
Patterns within the strategies used
The mean ability^{1} of students using each strategy broadly followed where they were according to the
National Standard table above.
 Unitising on a rate or on the ratio of similar entities gives almost equally high success rates and attracts the students with the highest (and virtually identical) mean abilities as those of students who unitise on the ratio of similar entities. This indicates that either utilising on the rate or the ratios may be equally sophisticated.
 Consistently exploiting the rate or ratio with the simplest relationship shows good flexibility of strategy choice.
 Using either a rate/ratio table or doubling had high success rates and attracted students with the next highest (and virtually identical) mean abilities as unitising.
 The use of diagrams was the least successful strategy, and was used by students with lower mean abilities than other acceptable strategies.
 In part c), students who misinterpreted the remainder when looking at the ratio of boys:girls had reasonably strong mean abilities, indicating that they had reasonable understanding of rate or ratio problems with rates or ratios where the relationships are whole numbers.
Click on the link Analysis of student responses [pdf] for more detailed information.
^{1}.The mean ability is a measure of the level of sophistication of different student responses. Strategies or answers with high mean ability indicate that the more able students responded this way, while answers with low mean ability indicates that less able students tended to give this type of answer.
Uses an additive relationship
Adding on the difference between two elements of a rate or ratio indicates that the student does not understand these concepts. Students need to recognise that this is a sharing problem that is multiplicative rather than additive. They could work with physical objects such as coloured counters, making them into sets. For example make a set of 4 red counters to represent Mara's books, and 3 yellow to represent Chloe's. Make a second set and then they can see two lots of 4 red and 3 yellow (i.e. 4:3) gives 8 red and 6 yellow (i.e. = 8:6). Lay out three lots of red and yellow counters, covering the second and third sets of yellow counters. Ask "For each 4 red counters there are 3 yellow. Here are 12 red counters. How many yellow are there altogether?"
Confuses multiplication and division within a rate or ratio OR Uses wrong elements to create a rate or ratio
These students are aware that problems of this sort are multiplicative rather than additive, but have trouble distinguishing how to proceed and so arbitrarily multiply or divide numbers. These students need to look more carefully at the relationship between the two sets of values (one of which has a missing element). Getting them to draw rate or ratio tables will help them see the relationship.
Uses the wrong unit rate
These students may not have read the problem carefully enough. Get them to reread each problem and prompt them with questions such as "For every book of Mara's, how many does Chloe have?"; "How much money does Sam get each week?" or "How many girls are there if there are 6 boys?" Some students will immediately spot that they have not read the question carefully enough. Get them to answer the three questions again.
For students who still do not know how to proceed, rephrase the first sentence in part a) so that it uses a unit fraction, e.g., "Mara has 4 books for every 1 book Chloe has." If they can correctly answer "48", then ask them the original question again. Ask them "Will Mara now have more books, the same number of books, or fewer books." If they say "fewer", get them to redo the original question.
Treats the remainder additively
These students can correctly see that the answer is 6 × (^{10}/_{4}), but evaluate ^{10}/_{4} as 2r2, and do not know how to process the remainder of 2. Many of them just add it on. Ask them "You multiplied by 2. Where did that come from?" Their likely response is that 4 goes into 10 two whole times. Ask "What does the remainder of 2 represent?" and even prompt them with "It is 2 out of how many parts?" If they can see it is 2 out of 4, then it is a short step to see this as a half!
Unitises the rate or ratio to get a correct answer
Get these students to discuss and compare their methods. The aim is to get them to see that either way that the rate or ratio is created will work. Generally students will work with the one where the relationship is easier. For example the ratio that Chloe has 4 times as many books is easier than the ratio that Mara has ^{4}/_{3} = 1^{1}/_{3} books for every 1 of Chloe's.
 Proportional Reasoning Book One, Level 34+ and
 Proportional Reasoning Book Two, Level 34+.