Show how many

Show how many

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about solving problems involving rates and ratios.
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.
 
 
 
 
 
 
 
 
Mara has _____ books
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.
 
 
 
 
 
 
 
 
It takes Sam _____ weeks
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.
 
 
 
 
 
 
 
 
The team has _____ boys
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Description of task: 
Students to show how to calculate problems that involve rates and ratios.
Curriculum Links: 
This resource can be used to help to identify students' understanding of ratio. 
A possible progression for understanding ratio could involve
  • 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.
Key competencies
This resource involves recording the strategies students use to solve rate problems involving multiplication. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y8 (10/2010)
a) 16
Describes any 1 of the following ratio strategies:

  • Unitises to a fractional unit ratio, e.g., Mara has 4/3 books per 1 of Chloe's so 12 × 4/3 = 16
  • Multiplies to derive a whole number unit ratio, e.g., 4 × 3 = 12; 4 × 4 = 16
  • Divides to derive a whole number unit ratio, e.g., 12 ÷ 3 = 4; 4 × 4 = 16
  • Uses a ratio table approach, e.g., 4:3 = 8:6 = 12:9 =16:12
  • Uses diagrams of sets of obj
easy
moderate
b) 20
Describes any 1 of the following strategies:

  • Unitises to a unit rate, e.g., Sam gets $15 ÷ 5 = $3 per week; $60 ÷ 3 = $20
  • Multiplies to derive a unit ratio, e.g., 4 × 15 = 60; 4 × 5 = 20
  • Divides to derive a unit ratio, e.g., 60 ÷ 15 = 4; 4 × 5 = 20
  • Uses a rate table approach, e.g., 15:5 = 30:10 = 45:15 = 60:20
  • Uses diagrams of sets of objects, sometimes linked with numbers
  • Other acceptable methods.
easy
easy
c) 15
Describes any 1 of the following strategies:

  • Unitises to a fractional unit ratio, e.g., 3/2 boys per girl ; 10 × 3/2 = 15
  • Multiplies to derive a unit ratio, e.g., 2.5 × 4 = 10; 2.5 × 6 = 15
  • Dividies to derive a unit ratio, e.g., 10 ÷ 4 = 2.5; 2.5 × 6 = 15
  • Uses a ratio table approach, e.g., 6:4 = 9:6 = 12:8 = 15:10
  • Uses diagrams of sets of objects, sometimes linked with numbers
  • Other acceptable methods.
difficult
difficult

Based on a representative sample of 187 Y8 students.

Teaching and learning: 

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).

Diagnostic and formative information: 
  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 ability1 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.

Next steps: 

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 re-read 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 re-do 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 = 11/3 books for every 1 of Chloe's.

Figure it out
  • Proportional Reasoning Book One, Level 3-4+ and
  • Proportional Reasoning Book Two, Level 3-4+. 
Both have numbers of relevant resources.