Recognise and solve problems involving simple ratios - Maths Curriculum Companion - Department of Education & Training

## Teaching Context

### Victorian Curriculum

Recognise and solve problems involving simple ratios (VCMNA249).

VCAA Sample Program: A set of sample programs covering the Victorian Curriculum Mathematics.

VCAA Mathematics glossary: A glossary compiled from subject-specific terminology found within the content descriptions of the Victorian Curriculum Mathematics.

##### Achievement standards

Students solve problems involving the order, addition and subtraction of integers. They make the connections between whole numbers and index notation and the relationship between perfect squares and square roots. They solve problems involving all four operations with fractions, decimals, percentages and their equivalences, and express fractions in their simplest form.

Students compare the cost of items to make financial decisions, with and without the use of digital technology. They make simple estimates to judge the reasonableness of results.

Students use variables to represent arbitrary numbers and connect the laws and properties of number to algebra and substitute numbers into algebraic expressions. They assign ordered pairs to given points on the Cartesian plane and interpret and analyse graphs of relations from real data.

Students develop simple linear models for situations, make predictions based on these models, solve related equations and check their solutions

## Teaching ideas

### Diagnostic and Key stages of development

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Detailed explanations of developmental stages and common misconceptions

The SMART tests website includes a diagnostic tool to help teachers identify student level of understanding of ratios prior to beginning the unit. You can access the SMART test website here.

Stage 0 students are below stage 1. These students are unable to compare simple ratios in a problem situation.

Stage 1 students can identify equivalent ratios involving doubles and triples. For example, they would recognise that 2 parts of cordial to 7 parts of water would make the same strength drink as 4 parts of cordial to 14 parts of water.

Stage 2 students can compare ratios involving doubles, triples, halves and thirds. For example, they would recognise that 2 parts of cordial to 7 parts of water would make a stronger drink than 4 parts of cordial to 15 parts of water.

Stage 3 students can compare ratios of many types. For example, they would recognise that 3 parts of cordial to 8 parts of water would make a stronger drink than 4 parts of cordial to 13 parts of water.

Misconception

Adder - This misconception is to reason additively, instead of multiplicatively (i.e. using simple differences instead of proportional reasoning). For example students with this misconception would say incorrectly that the ratio 3 : 7 is the same as the ratio 5 : 9, giving the "reason" that 5 is 2 more than 3 and 9 is 2 more than 7.

### Teaching suggestions

Stage 0 students are unable to compare simple ratios in a problem situation. They should benefit from activities like enlarging a recipe for multiples of the original number of people, or working out the shopping list for a 30 person barbecue where there needs to be, for example, 3 sausages per person, 2 pieces of bread per person, 1 bottle of drink for every 3 people and one bottle of sauce for every 15 people.
This could be followed by considering different ratios of sausages for given numbers of people, for example 6 sausages for every 2 people, 9 sausages for every 3 people, 8 sausages for every 4 people,.. and deciding which ratios mean that each person gets more than 2 each or less than 2 each or exactly 2 each.

Stage 1 students can identify equivalent ratios involving doubles and triples, and are ready to progress to those involving halves and thirds. For example, a recipe for making a fruit drink for 30 people could be scaled down for 15 people and then 10 people.

Stage 2 students can compare ratios involving doubles, triples, halves and thirds. They are ready to progress to comparing a full rage of many types of ratios.
Activity 3 in this part of the Mathematics Developmental Continuum:
http://smartvic.com/teacher/mdc/number/N55004P.html introduces the use of a dual number line which is helpful with students who need to generalise a method for comparing, and later calculating with, proportions.

Stage 3 students can correctly compare a varied range of ratios and can now be introduced to calculating with proportions. The SMART test Calculating with proportions could give useful information about a student's understanding in this section of mathematics.

For those with the "adder" misconception it is helpful to provoke cognitive conflict . Activity 4 on the page of the Mathematics Developmental Continuum reached by the above link does this, so that a student's misconception can be exposed to him or her, and then be resolved.

### References

Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. Handbook of research on mathematics teaching and learning, 296-333.

Hart, K. (1988). Ratio and proportion. Number concepts and operations in the middle grades, 2, 198-219.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. Second handbook of research on mathematics teaching and learning, 1, 629-667.

### Using proportional reasoning

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This activity is from the Mathematics Assessment Project and a detailed PDF lesson plan can be accessed here

Mathematical goals

This lesson unit is intended to help you assess how well students are able to reason proportionally when comparing the relationship between two quantities expressed as unit rates and/or part-to-part ratios. In particular, it will help you assess how well students are able to:

Describe a ratio relationship between two quantities.

Compare ratios expressed in different ways.

Use proportional reasoning to solve a real-world problem.

Introduction

The lesson unit is structured in the following way:

Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You then review their solutions and create questions for students to consider, in order to improve their work.

After a whole-class introduction, students work in groups, putting diagrams and descriptions of orange and soda mixtures into strength order. Students then compare their work with their peers.

Next, in a whole-class discussion, students critique some sample work stating reasons why two mixtures would or wouldn’t taste the same. Students then revise and correct any misplaced cards.

After a final whole-class discussion, students work individually either on a new assessment task, or return to the original task and try to improve their responses.

Materials Required

Each student will need a mini-whiteboard, pen, and eraser, and a copy of Mixing Drinks and Mixing Drinks (revisited).

Each small group of students will need the cut-up Card Set: Orange and Soda Mixtures and Card Set: Blank Cards, a sheet of poster paper and a glue stick.

You may wish to have some orange juice and soda for mixing/tasting but this is not essential.

Time needed

15 minutes before the lesson, a 100-minute lesson (or two 55-minute lessons), and 15 minutes in a follow-up lesson. Timings given are approximate and will depend on the needs of your class.

### Can you make?

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This activity is used as an introduction to ratios to support student understanding of using the word ‘to’ in ratios (2 to 3, 4 to 7 etc).

Each student is given 10 red, 10 blue and 10 yellow counters/blocks.

Ask various ‘Can you make’ ratio questions to see if students can place the correct number of counters for each group. For example, can you make a group where the number of yellow counters to red counters is in the ratio 1 to 3? Note: If students are experiencing difficulty assist them by saying ‘that means 1 yellow counter for every 3 red counters.

Can you make a group where the number of red counters to blue counters to yellow counters is in the ratio 5 to 2 to 4?

### Writing ratios

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This activity gives students practise at writing ratios and which quantity to write first.

Present students with an image like the example shown below.

If students are experiencing difficulty getting started, prompt them to think about the number of triangles to squares, squares to total shapes, blue shapes to yellow shapes etc.

Note: Alternatively write out 5 different ratios and have students work out what each ratio is comparing.

### Dividing ratios practice

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These scenarios can be given to groups of students, used as a whole class activity or to individual students.

Scenario 1: Three people decide to split $500 into the ratio 2:3:5. How much would each person get? Scenario 2: Two friends Mia and Harper bought a$10 lottery ticket. How would they share first prize of $500,000 if Mia paid$3 and Harper paid $7? Scenario 3: Liam gets$4.50 pocket money this week. His parents split the pocket money between their three children the ratio 1:2:3 (Liam: Sarah: Hunter). How much more money does Ryan get compared to Liam?

Scenario 4: In a middle school (Yr. 7-9) the ratio of students in Years 7, 8 and 9 is 2:4:3. How many students are there in the middle school? Prompt: if there are 200 year 8 students, can you find the ratio?

### Learning from home

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Activity 1: Writing ratios

This activity gives children practise with writing ratios. They will have to determine which quantity to write first in the ratio.

Show your child the diagram below which asks them to write as many ratios as they can using the shapes. They should always simplify ratios if possible. Simplifying a ratio is the same as simplifying a fraction. Look for a common factor for each number and divide.

If your child is experiencing difficulty getting started, prompt them to think about the number of triangles to squares, squares to total shapes, blue shapes to yellow shapes etc.

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Activity 2: Dividing ratios

Scenario 1: Three people decide to split $500 into the ratio 2:3:5. How much would each person get? Scenario 2: Two friends Mia and Harper bought a$10 lottery ticket. How would they share first prize of $500,000 if Mia paid$3 and Harper paid $7? Scenario 3: Liam gets$4.50 pocket money this week. His parents split the pocket money between their three children the ratio 1:2:3 (Liam: Sarah: Hunter). How much more money does Ryan get compared to Liam?