## Module 10: Ratios, Rates, Proportions

Ratios & rates.

A ratio is the quotient of two numbers or the quotient of two quantities with the same units.

When writing a ratio as a fraction, the first quantity is the numerator and the second quantity is the denominator.

A rate is the quotient of two quantities with different units. You must include the units.

When writing a rate as a fraction, the first quantity is the numerator and the second quantity is the denominator. Simplify the fraction, if possible. Include the units in the fraction.

6. Which box has a lower unit price?

## Proportions

A proportion says that two ratios (or rates) are equal.

Determine whether each proportion is true or false by simplifying each fraction.

A common method of determining whether a proportion is true or false is called cross-multiplying or finding the cross products . We multiply diagonally across the equal sign. In a true proportion, the cross products are equal.

Determine whether each proportion is true or false by cross-multiplying.

As we saw in a previous module , we can use a variable to stand for a missing number. If a proportion has a missing number, we can use cross multiplication to solve for the missing number. This is as close to algebra as we get in this textbook.

To solve a proportion for a variable:

- Isolate the variable by rewriting the multiplication equation as a division equation.
- Check the solution by substituting the answer into the original proportion and finding the cross products.

You may discover slightly different methods that you prefer. [1] If you think “Hey, can’t I do this a different way?”, you may be correct.

Solve for the variable.

Problems that involve rates, ratios, scale models, etc. can be solved with proportions. When solving a real-world problem using a proportion, be consistent with the units.

- The steps in the box are designed to avoid mentioning the algebraic step of dividing both sides of the equation by a number. If you are comfortable with basic algebra, then you would phrase step 2 differently. ↵

Technical Mathematics Copyright © 2020 by Morgan Chase is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

## Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

More specific topics in use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations..

- Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
- Solve unit rate problems including those involving unit pricing and constant speed.
- Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
- Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

## Popular Tutorials in Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

## How Do You Solve a Word Problem Using Unit Rates?

Word problems allow you to see the real world uses of math! This tutorial shows you how to take a rate and convert it to a unit rate. Then, you can use that unit rate to calculate your answer. Watch this tutorial to learn all about it!

## How Do You Convert a Rate To A Unit Rate?

When you talk about the speed of a car, you usually say something in miles per hour. For example, you say, 'I drove 40 miles per hour.' Normally, you don't say, 'I drove 120 miles per 3 hours.' Figure out how to convert a rate like 120 miles per 3 hours to the unit rate of 40 miles per hour by watching this tutorial.

## How Do You Solve a Word Problem Using a Percent Proportion?

Word problems allow you to see the real world uses of math! This tutorial shows you how to take a words problem and turn it into a percent proportion. Then see how to solve for the answer using the mean extremes property of proportions. Take a look!

## What's a Percent?

Sales tax, tips at restaurants, grades on tests... no matter what you do, you can't run away from percents. So watch this tutorial and see once and for all what percents are all about!

## How Do You Turn a Percent into a Decimal?

Percents and Decimals are just tools that let us represent numbers, and that is why percents can be converted into decimals, and decimals can be converted into percents. In this tutorial you'll see how quickly you can convert percents into decimals!

## How Do You Set Up a Percent Proportion from a Word Problem?

Sometimes the hardest part of a word problem is figuring out how to turn the words into an equation you can solve. This tutorial let's you see the steps to take in order to do just that! Take a look! You'll be glad you did!

## What's a Percent Proportion?

Percents are important, and the reality is that percents are actually proportions in disguise. In this tutorial you'll see exactly how to connect percents with proportions, and you'll be happier for it :)

## What's a Ratio?

Ratios are everywhere! The scale on a map or blueprint is a ratio. Ingredients sometimes need to be mixed using ratios such as the ratio of water to cement mix when making cement. Watch this tutorial to learn about ratios. Then think of some ratios you've encountered before!

## How Do You Convert Cups to Pints?

It can be helpful to see a capacity given in different units. This tutorial shows you how to take a measurement in cups and convert it to pints!

## How Do You Convert Quarts to Gallons?

If you go to the store to buy a certain number of quarts of milk and the store only sells milk in gallons, what do you do? You need to convert those quarts into gallons! Follow along with this tutorial to learn how.

## How Do You Find Equivalent Ratios by Making a Table?

To master equivalent ratios, you need to practice. Follow along with this tutorial to practice filling in a table with equivalent ratios.

## How Do You Use a Table of Equivalent Ratios to Predict a Value?

Patterns are everywhere! In this tutorial, you'll see how to use the pattern in a table to find an answer to a word problem.

## How Do You Write a Ratio in Simplest Form?

In this tutorial, you'll see how to take the information from a given table and use it to find a ratio. You'll also see how to write the answer in simplest form!

## How Do You Estimate a Percent?

When something is on sale, it's good to know how much you're saving! This tutorial shows you how to estimate a percent using an original price and a coupon!

## How Do You Use Mental Math to Estimate a Percent?

Mental math is a powerful tool! See how to use mental math in this tutorial to estimate a percent!

## How Do You Find Equivalent Ratios?

Ratios are used to compare numbers. When you're working with ratios, it's sometimes easier to work with an equivalent ratio. Equivalent ratios have different numbers but represent the same relationship. In this tutorial, you'll see how to find equivalent ratios by first writing the given ratio as a fraction. Take a look!

## How Do You Know If Two Ratios are Proportional?

Ratios are proportional if they represent the same relationship. One way to see if two ratios are proportional is to write them as fractions and then reduce them. If the reduced fractions are the same, your ratios are proportional. To see this process in action, check out this tutorial!

## How Do You Use Unit Rates to Compare Rates?

Word problems allow you to see the real world uses of math! This tutorial shows you how to use ratios to figure out which store has a better deal on cupcakes. Take a look!

## How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate?

Word problems are a great way to see math in action! In this tutorial, learn how to use the information given in a word problem to create a rate. Then, find and use a conversion factor to convert a unit in the rate. Take a look!

## How Do You Convert Miles to Kilometers?

The world is full of different units of measure, and it's important to know how to convert from one unit to another. This tutorial shows you how to convert from miles to kilometers. Check it out!

## How Do You Convert Yards to Meters?

## What are Equivalent Ratios?

Equivalent ratios are just like equivalent fractions. If two ratios have the same value, then they are equivalent, even though they may look very different! In this tutorial, take a look at equivalent ratios and learn how to tell if you have equivalent ratios.

## What is Dimensional, or Unit Analysis?

If youâ€™re solving a math problem or word problem that contains units, you need to remember to include your units in your answer. By using dimensional analysis or unit analysis, you can include those units as you solve! Watch this tutorial and take a look at dimensional analysis!

## How Do You Put Fractions, Decimals, and Percents in Order?

Ordering numbers from least to greatest? Are the numbers in different forms? To make comparing easier, convert all the numbers to decimals. Then, plot those decimals on a number line and compare them! This tutorial shows you how!

## How Do You Turn a Percent Into a Fraction?

Looking for some practice converting percents to fractions? Then this tutorial was made for you! Follow along as this tutorial shows you how to convert a percent to a fraction. Then, reduce the fraction to put it in simplest form. Check it out!

## How Do You Turn a Fraction Into a Percent?

Looking for some practice converting fractions to percents? Then this tutorial was made for you! Follow along as this tutorial shows you how to convert a fraction to a percent.Take a look!

## How Do You Turn a Decimal into a Percent?

Converting decimals into percents is easier than you may think! To convert a decimal to a percent, just move the decimal point to places to the right and put a percent sign at the end! To see it done, check out this tutorial!

## How Do You Use a Proportion to Find a Whole?

Taking a percent of a number? Trying to figure out the result? Use a percent proportion to solve! This tutorial will show you how!

## How Do You Use an Equation to Find a Whole?

Taking a percent of a number? Trying to figure out the result? Use a percent equation to solve! This tutorial will show you how!

## How Do You Use Compatible Numbers to Estimate a Part of a Whole?

If you're trying to find the percent of a number, it may be helpful to use compatible numbers to find an estimated answer. Follow along with this tutorial to see how to use compatible numbers to estimate the percent of a number!

## Related Topics

Other topics in understand ratio concepts and use ratio reasoning to solve problems. :.

- Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
- Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.
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## Ratio Problem Solving

Here we will learn about ratio problem solving, including how to set up and solve problems. We will also look at real life ratio problems.

There are also ratio problem solving worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is ratio problem solving?

Ratio problem solving is a collection of word problems that link together aspects of ratio and proportion into more real life questions. This requires you to be able to take key information from a question and use your knowledge of ratios (and other areas of the curriculum) to solve the problem.

A ratio is a relationship between two or more quantities . They are usually written in the form a:b where a and b are two quantities. When problem solving with a ratio, the key facts that you need to know are,

- What is the ratio involved?
- What order are the quantities in the ratio?
- What is the total amount / what is the part of the total amount known?
- What are you trying to calculate ?

As with all problem solving, there is not one unique method to solve a problem. However, this does not mean that there aren’t similarities between different problems that we can use to help us find an answer.

The key to any problem solving is being able to draw from prior knowledge and use the correct piece of information to allow you to get to the next step and then the solution.

Let’s look at a couple of methods we can use when given certain pieces of information.

When solving ratio problems it is very important that you are able to use ratios. This includes being able to use ratio notation.

For example, Charlie and David share some sweets in the ratio of 3:5. This means that for every 3 sweets Charlie gets, David receives 5 sweets.

Charlie and David share 40 sweets, how many sweets do they each get?

We use the ratio to divide 40 sweets into 8 equal parts.

Then we multiply each part of the ratio by 5.

3 x 5:5 x 5 = 15:25

This means that Charlie will get 15 sweets and David will get 25 sweets.

- Dividing ratios

Step-by-step guide: Dividing ratios (coming soon)

## Ratios and fractions (proportion problems)

We also need to consider problems involving fractions. These are usually proportion questions where we are stating the proportion of the total amount as a fraction.

## Simplifying and equivalent ratios

- Simplifying ratios

Equivalent ratios

## Units and conversions ratio questions

Units and conversions are usually equivalent ratio problems (see above).

- If £1:\$1.37 and we wanted to convert £10 into dollars, we would multiply both sides of the ratio by 10 to get £10 is equivalent to \$13.70.
- The scale on a map is 1:25,000. I measure 12cm on the map. How far is this in real life, in kilometres? After multiplying both parts of the ratio by 12 you must then convert 12 \times 25000=300000 \ cm to km by dividing the solution by 100 \ 000 to get 3km.

Notice that for all three of these examples, the units are important. For example if we write the mapping example as the ratio 4cm:1km, this means that 4cm on the map is 1km in real life.

Top tip: if you are converting units, always write the units in your ratio.

Usually with ratio problem solving questions, the problems are quite wordy . They can involve missing values , calculating ratios , graphs , equivalent fractions , negative numbers , decimals and percentages .

Highlight the important pieces of information from the question, know what you are trying to find or calculate , and use the steps above to help you start practising how to solve problems involving ratios.

## How to do ratio problem solving

In order to solve problems including ratios:

Identify key information within the question.

Know what you are trying to calculate.

Use prior knowledge to structure a solution.

## Explain how to do ratio problem solving

## Ratio problem solving worksheet

Get your free ratio problem solving worksheet of 20+ questions and answers. Includes reasoning and applied questions.

## Related lessons on ratio

Ratio problem solving is part of our series of lessons to support revision on ratio . You may find it helpful to start with the main ratio lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

- How to work out ratio
- Ratio to fraction
- Ratio scale
- Ratio to percentage

## Ratio problem solving examples

Example 1: part:part ratio.

Within a school, the number of students who have school dinners to packed lunches is 5:7. If 465 students have a school dinner, how many students have a packed lunch?

Within a school, the number of students who have school dinners to packed lunches is \bf{5:7.} If \bf{465} students have a school dinner , how many students have a packed lunch ?

Here we can see that the ratio is 5:7 where the first part of the ratio represents school dinners (S) and the second part of the ratio represents packed lunches (P).

We could write this as

Where the letter above each part of the ratio links to the question.

We know that 465 students have school dinner.

2 Know what you are trying to calculate.

From the question, we need to calculate the number of students that have a packed lunch, so we can now write a ratio below the ratio 5:7 that shows that we have 465 students who have school dinners, and p students who have a packed lunch.

We need to find the value of p.

3 Use prior knowledge to structure a solution.

We are looking for an equivalent ratio to 5:7. So we need to calculate the multiplier. We do this by dividing the known values on the same side of the ratio by each other.

So the value of p is equal to 7 \times 93=651.

There are 651 students that have a packed lunch.

## Example 2: unit conversions

The table below shows the currency conversions on one day.

Use the table above to convert £520 (GBP) to Euros € (EUR).

Use the table above to convert \bf{£520} (GBP) to Euros \bf{€} (EUR).

The two values in the table that are important are GBP and EUR. Writing this as a ratio, we can state

We know that we have £520.

We need to convert GBP to EUR and so we are looking for an equivalent ratio with GBP = £520 and EUR = E.

To get from 1 to 520, we multiply by 520 and so to calculate the number of Euros for £520, we need to multiply 1.17 by 520.

1.17 \times 520=608.4

So £520 = €608.40.

## Example 3: writing a ratio 1:n

Liquid plant food is sold in concentrated bottles. The instructions on the bottle state that the 500ml of concentrated plant food must be diluted into 2l of water. Express the ratio of plant food to water respectively in the ratio 1:n.

Liquid plant food is sold in concentrated bottles. The instructions on the bottle state that the \bf{500ml} of concentrated plant food must be diluted into \bf{2l} of water . Express the ratio of plant food to water respectively as a ratio in the form 1:n.

Using the information in the question, we can now state the ratio of plant food to water as 500ml:2l. As we can convert litres into millilitres, we could convert 2l into millilitres by multiplying it by 1000.

2l = 2000ml

So we can also express the ratio as 500:2000 which will help us in later steps.

We want to simplify the ratio 500:2000 into the form 1:n.

We need to find an equivalent ratio where the first part of the ratio is equal to 1. We can only do this by dividing both parts of the ratio by 500 (as 500 \div 500=1 ).

So the ratio of plant food to water in the form 1:n is 1:4.

## Example 4: forming and solving an equation

Three siblings, Josh, Kieran and Luke, receive pocket money per week proportional to their age. Kieran is 3 years older than Josh. Luke is twice Josh’s age. If Josh receives £8 pocket money, how much money do the three siblings receive in total?

Three siblings, Josh, Kieran and Luke, receive pocket money per week proportional to their ages. Kieran is \bf{3} years older than Josh . Luke is twice Josh’s age. If Luke receives \bf{£8} pocket money, how much money do the three siblings receive in total ?

We can represent the ages of the three siblings as a ratio. Taking Josh as x years old, Kieran would therefore be x+3 years old, and Luke would be 2x years old. As a ratio, we have

We also know that Luke receives £8.

We want to calculate the total amount of pocket money for the three siblings.

We need to find the value of x first. As Luke receives £8, we can state the equation 2x=8 and so x=4.

Now we know the value of x, we can substitute this value into the other parts of the ratio to obtain how much money the siblings each receive.

The total amount of pocket money is therefore 4+7+8=£19.

## Example 5: simplifying ratios

Below is a bar chart showing the results for the colours of counters in a bag.

Express this data as a ratio in its simplest form.

From the bar chart, we can read the frequencies to create the ratio.

We need to simplify this ratio.

To simplify a ratio, we need to find the highest common factor of all the parts of the ratio. By listing the factors of each number, you can quickly see that the highest common factor is 2.

\begin{aligned} &12 = 1, {\color{red} 2}, 3, 4, 6, 12 \\\\ &16 = 1, {\color{red} 2}, 4, 8, 16 \\\\ &10 = 1, {\color{red} 2}, 5, 10 \end{aligned}

HCF (12,16,10) = 2

Dividing all the parts of the ratio by 2 , we get

Our solution is 6:8:5 .

## Example 6: combining two ratios

Glass is made from silica, lime and soda. The ratio of silica to lime is 15:2. The ratio of silica to soda is 5:1. State the ratio of silica:lime:soda.

Glass is made from silica, lime and soda. The ratio of silica to lime is \bf{15:2.} The ratio of silica to soda is \bf{5:1.} State the ratio of silica:lime:soda .

We know the two ratios

We are trying to find the ratio of all 3 components: silica, lime and soda.

Using equivalent ratios we can say that the ratio of silica:soda is equivalent to 15:3 by multiplying the ratio by 3.

We now have the same amount of silica in both ratios and so we can now combine them to get the ratio 15:2:3.

## Example 7: using bar modelling

India and Beau share some popcorn in the ratio of 5:2. If India has 75g more popcorn than Beau, what was the original quantity?

India and Beau share some popcorn in the ratio of \bf{5:2.} If India has \bf{75g} more popcorn than Beau , what was the original quantity?

We know that the initial ratio is 5:2 and that India has three more parts than Beau.

We want to find the original quantity.

Drawing a bar model of this problem, we have

Where India has 5 equal shares, and Beau has 2 equal shares.

Each share is the same value and so if we can find out this value, we can then find the total quantity.

From the question, India’s share is 75g more than Beau’s share so we can write this on the bar model.

We can find the value of one share by working out 75 \div 3=25g.

We can fill in each share to be 25g.

Adding up each share, we get

India = 5 \times 25=125g

Beau = 2 \times 25=50g

The total amount of popcorn was 125+50=175g.

## Common misconceptions

- Mixing units

Make sure that all the units in the ratio are the same. For example, in example 6 , all the units in the ratio were in millilitres. We did not mix ml and l in the ratio.

- Ratio written in the wrong order

For example the number of dogs to cats is given as the ratio 12:13 but the solution is written as 13:12.

- Ratios and fractions confusion

Take care when writing ratios as fractions and vice-versa. Most ratios we come across are part:part. The ratio here of red:yellow is 1:2. So the fraction which is red is \frac{1}{3} (not \frac{1}{2} ).

- Counting the number of parts in the ratio, not the total number of shares

For example, the ratio 5:4 has 9 shares, and 2 parts. This is because the ratio contains 2 numbers but the sum of these parts (the number of shares) is 5+4=9. You need to find the value per share, so you need to use the 9 shares in your next line of working.

- Ratios of the form \bf{1:n}

The assumption can be incorrectly made that n must be greater than 1 , but n can be any number, including a decimal.

## Practice ratio problem solving questions

1. An online shop sells board games and computer games. The ratio of board games to the total number of games sold in one month is 3:8. What is the ratio of board games to computer games?

8-3=5 computer games sold for every 3 board games.

2. The volume of gas is directly proportional to the temperature (in degrees Kelvin). A balloon contains 2.75l of gas and has a temperature of 18^{\circ}K. What is the volume of gas if the temperature increases to 45^{\circ}K?

3. The ratio of prime numbers to non-prime numbers from 1-200 is 45:155. Express this as a ratio in the form 1:n.

4. The angles in a triangle are written as the ratio x:2x:3x. Calculate the size of each angle.

5. A clothing company has a sale on tops, dresses and shoes. \frac{1}{3} of sales were for tops, \frac{1}{5} of sales were for dresses, and the rest were for shoes. Write a ratio of tops to dresses to shoes sold in its simplest form.

6. During one month, the weather was recorded into 3 categories: sunshine, cloud and rain. The ratio of sunshine to cloud was 2:3 and the ratio of cloud to rain was 9:11. State the ratio that compares sunshine:cloud:rain for the month.

## Ratio problem solving GCSE questions

1. One mole of water weighs 18 grams and contains 6.02 \times 10^{23} water molecules.

Write this in the form 1gram:n where n represents the number of water molecules in standard form.

2. A plank of wood is sawn into three pieces in the ratio 3:2:5. The first piece is 36cm shorter than the third piece.

Calculate the length of the plank of wood.

5-3=2 \ parts = 36cm so 1 \ part = 18cm

3. (a) Jenny is x years old. Sally is 4 years older than Jenny. Kim is twice Jenny’s age. Write their ages in a ratio J:S:K.

(b) Sally is 16 years younger than Kim. Calculate the sum of their ages.

## Learning checklist

You have now learned how to:

- Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions
- Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems
- Make and use connections between different parts of mathematics to solve problems

## The next lessons are

- Compound measures
- Best buy maths

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## Ratios and Rates

The purpose of this unit is to synthesise students’ fraction and decimal place value knowledge, thus enabling them to work with rates and ratios with understanding and competence.

- Understand what a ratio is and how it is written and read.
- Identify, describe, generate, and record equivalent part-to-part ratios.
- Create ratio tables.
- Identify, describe, generate, and record part-to-whole ratios.
- Recognise and record fractional relationships in ratios.
- Express ratios as fractions, decimal fractions and percentages.
- Apply ratio understanding to problem solving.
- Create and solve ratio problems.
- Understand what rates are and how they differ from ratios.
- Calculate and compare rates.
- Use graphs to show the linear relationship of a constant rate.

Implicit within students’ understanding of fractions and percentages is relational thinking, or the ability to see and use possibilities of variation between numbers in an equation. Just as fractions and percentages express part-whole relationships, a ratio expresses a relationship between two numbers of the same kind. It is usually expressed as ‘a to b’ or ‘a : b’, meaning that for every amount of one thing, there is so much more of another thing of the same kind. Ratios can express a part-to-part relationship, such as the number of boys to the number of girls in a group of children or a part-to-whole relationship, such as the number of boys to the number of children in the whole group.

It is not a significant conceptual shift for students to work with ratios, as their proportional nature is familiar, and the students’ ability to understand and work easily with multiplicative relationships is fundamental to any proportional thinking. This thinking has been consolidated in their fractional and percentage work, so the shift in their work with ratios is rather in the contexts in which ratios are most often used, and the interpretation of these contexts. However, students do encounter for the first time, the colon notation used in the expression of a ratio.

Students need to understand that, as with fractions, ratios can be reduced by common factors of the quantities, to make the simplest form. For example, a ratio of 20:60 can be represented in its simplest form as a ratio of 1:3. This can also be expressed as a fraction. The first amount is one third of the second amount. However, in this 1:3 part-to-part ratio, we can also see that there are 4 parts altogether (1 + 3) so the relationship of the first quantity (1) to the whole (4) can be expressed as 1/4. Students may not have recognised before that every fraction is in fact a ratio. It is important to model well with equipment both the part-to-part and part-to-whole relationships that make up ratios.

The key idea that students need to develop is that a ratio compares two amounts of the same kind of thing (eg. people: girls to boys, drink mix, odds) whilst a rate is a special kind of ratio that compares different kinds of measures such as dollars per kilogram. A distinguishing feature of a rate is that it uses the word per and the symbol /. Students therefore need to be able to co-ordinate pairs of numbers, and be competent in working with fractions, decimals and percentages, to calculate the multiplicative relationship between the number pairs and to make comparisons of rates.

A rate is a very important kind of ratio because many practical tasks in our lives involve some kind of rate: speed (kilometres per hour), remuneration (dollars paid per hour of work), health (pulse, heartbeats per minute), or the price of something (cost per unit bought), to name a few.

Constant rates are ordered pairs that result in a straight line when plotted on a number plane. The slope of the line is the unit rate, and the relationship between the numbers on the line does not change. This co-linear representation of a constant rate captures visually this multiplicative relationship ‘in action’.

Proportional thinking underpins one’s ability to make sense of and use easily ratios and rates in our daily lives. These ideas are presented in five sessions however, as they include complex concepts that are fundamental to a student’s success with ratios and rates, these sessions can be extended over a longer period of time.

Links to the Number Framework

Stages 7- 8 (Advanced Multiplicative to Advanced Proportional)

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

- beginning with simple ratios, e.g., 1:2, 1:3, and 2:3, and using objects like connecting cubes to model the ratios
- using symbols for ratios and fractions alongside physical and diagrammatic models
- progressing to working with the symbols alone, with the option of folding back to materials
- applying focused variation, particularly varying the number of objects while keeping the ratio the same
- asking predictive questions to encourage students to think beyond what is visible, e.g., If you make three copies of this ratio, 2:3, what ratio do you have in total?
- validating students' use of multiplication strategies, with the aim of moving them towards the use of more efficient strategies
- providing opportunities for students to work in a range of flexible groupings to encourage peer learning, scaffolding, and extension
- constructing a class chart of key ideas, models, and expressions to be used by students as a point of reference
- modelling and providing explicit teaching around the construction of tables, diagrams and graphs, and around the mathematical thinking involved in each stage of the unit. Gradually releasing your level of responsibility allows you to scaffold students towards working independently
- allowing the use of calculators for making predictions and confirming calculations, and to ease the mental load associated with calculation.

The contexts for this unit involve collections of discrete objects (e.g. people, scoops of mochaccino mix). T hese contexts might be supplemented by, or adapted to, better reflect students' interests, cultural backgrounds, or to make connections to learning from other curriculum areas. Examples might include ratios in cooking, ratios of positions in sports teams or genders in a class, or in the dimensions of human faces. Consider how you can utilise these ratio and rate problems as a way to make connections between mathematics and your students' 'real-world' contexts.

Te reo Māori kupu such as ōwehenga (ratio), hautanga (fraction), whakarea (multiplication, multiply), and pāpātanga (rate) could be introduced in this unit and used throughout other mathematical learning.

- Paper or small whiteboards
- Copymaster 1
- Copymaster 2
- Copymaster 3
- Copymaster 4
- Brown and white Unifix cubes (or a digital representation)
- Begin the session by sharing this problem (or an adapted version) with the class. Ask them to try use multiplication (as opposed to addition) strategies in their solution: 1/4 or 25% of the class are absent. There are 18 students present. How many students are in the whole class?
- Give students time to solve the problem and have several share their solutions. Highlight the fact that they were using their multiplicative thinking and looking for relationships between the numbers.
- Point out that this session is about exploring relationships between numbers but that these will be presented in a slightly different way.
- Distribute the shuffled picture cards only of Copymaster 1 (Class Ratios) to student pairs. Pose this problem: Look closely at the numbers of girls and boys on each card. Can you group any of these cards together? If so, can you explain why you have grouped them as you have? Give students time to explore, discuss and group. (a. 1:3, 2:6, 3:9 and 4:12, b. 2:3, 4:6, c. 1:2, 2:4, d. 4:4, 3:3. Two do not have an equivalent: 3:4 and 2:5)
- Have student pairs share their groupings. Roam and listen as they explain. Summarise student findings, accepting all ideas. Emphasise that students are looking at part-to-part ratios (e.g. the number of boys to the number of girls in a group of children). You might write this on the board to refer back to.

- Have students read several of the ratios aloud: eg. (“one to three”, “two to six”…).
- Have student pairs match the white ratio cards with the picture cards. If same ratio groupings have not already been established, encourage students to look for these by looking closely at the numbers.

- Explore the two cards that do not have given equivalents: (3:4 and 2:5). Emphasise that each of these is presented in its simplest form.
- Together generate a ratio table for 3:4.
- Discuss what is happening to the numbers of girls and the number of boys. Emphasise that the ratio of 3:4 remains the same. In each case there are 3/4 as many girls as there are boys.
- Ensure students have paper/small whiteboards available. Have student pairs discuss and create a ratio table for 2:5 and ‘draw’ what the first three would look like using simple girl/boy ‘pictures (modeled on the cards of Copymaster 1 ).
- Have student pairs suggest at least one more different girl:boy ratio of their own, and generate a table for this.
- Conclude this session by collecting the Copymaster 1 ratio cards from this session and summarising, on the class chart, what the students have learned about part-to-part ratios.
- Write part-to-part ratio on the class chart. Distribute the shuffled picture cards only of Copymaster 1 (Class Ratios) to student pairs. Point out that so far the class has explored the ratio between two parts of a group, the number of girls to the number of boys. Emphasise that this kind of ratio is known as a part to part ratio .
- Explain that this session will introduce a different kind of ratio. Distribute the shuffled green cards to student pairs and have the students match these with the picture cards. Tell them that they need to be prepared to explain what each of the ratio cards expresses (i.e. the ratio of girls to the whole class).
- Have student pairs share their results with another pair. Select some students to share their thinking with a whole class, and confirm what the ratio cards express.
- As a group, ask the students what this kind of ratio describes: this is a part to whole ratio . Ask if the students notice anything about the ratio (each can be expressed as a fraction: what fraction are the girls of the whole group?)
- Explore a part-to-whole fractional relationship by using the ratio table generated in Session 1. (Refer to the picture card and ratio card showing 3:7, number of girls to number in the whole group).
- Add the group total line to the ratio table and ask a student to complete this for the class, with the help of the rest of the class.
- As a class, identify what fraction of the class the girls are, in each instance, and record these: (3/7, 6/14, 9/21, 12/28, 15/35, 18/42, 21/49)
- Write these fractions on the class chart: 1/1, 2/2, 3/3, 4/4, 5/5, 6/6 and 7/7 Ask students what they notice about these fractions. Emphasise that they are all ways of writing 1.
- Explore and record these fractional relationships. 3/7 x 2/2 = 6/14 3/7 x 3/3 = 9/21 3/7 x 4/4 = 12/28 3/7 x 5/5 = 15/35 3/7 x 6/6 = 18/42 3/7 x 7/7 = 21/49 Elicit from the students that all fractions are an expression of 3/7 or a 3:7 part to whole ratio.
- Distribute the blank pale blue cards from Copymaster 1 to each student pair and have them write for each of the picture cards, the ratio of boys to the whole group.
- Have student pairs use the ratio table generated in Session 1, Activity 2, Step 6 for the 2:5 ratio of girls to boys. They should: 1. Add a group total line and complete this. 2. Explore the fractional relationships as in Activity 2, Step 2 above between the boys and the whole group. 3. Confirm that the decimal fraction for each is the same. 4. State what percentage the boys are of the whole group. For example:

The fraction of boys in each group: 5/7, 10/14, 15/21, 20/28, 25/35, 30/42, 35/49

These can be explored this way: 5/7 x 1/1 = 5/7 5/7 x 2/2 = 10/14, 5/7 x 3/3 = 15/21, 5/7 x 4/4 = 20/28, 5/7 x 5/5 = 25/35, 5/7 x 6/6 = 30/42, 5/7 x 7/7 = 35/49

- Discuss and agree that all are expressions of a 5:7 part-to-whole ratio which can be expressed as a fraction. Therefore, the boys are always 5/7 of the whole class according to these expressions.
- Discuss and agree that this can be expressed as a decimal fraction: 5 ÷ 7 = 0.714 (check each fraction is equivalent to the same decimal fraction eg. 35 ÷ 49 = 0.714).
- Discuss and agree that the 5:7 part-to-whole ratio can also be expressed as a percentage. In this case the percentage of boys in the group class is 71.4%.
- Have students pair share and check their results.
- Conclude the session by recording students’ reflections about part-to-whole ratio learning and the decimal and percentage calculations.
- Distribute Copymaster 2 asking students to complete this on their own or in pairs.
- Discuss. Ask students to share their responses to the question in Copymaster 2 , “Is there anything else you know?” Notice students’ explanations.
- Ask for, and develop, a list of contexts in the students’ lives in which ratios are used in a very practical sense. (For example: in recipes for food and drink, scaling up and down models and maps, ratio of kinds of people such as left handed to right handed people).
- Pose this problem ( Copymaster 3 ) and make brown and white Unifix cubes (or a digital representation) available :

Mochaccino Mix Millie and Maxwell are creating their own homemade ‘mochaccinos’ with scoops of mochaccino mix (MM) and scoops of milk (m).

Which cup has the stronger mochaccino flavour? How do you know?

- Have student pairs model the ratio with cubes, discuss how they would solve it and agree on a solution.
- Have students share their thinking and solution with another pair. Choose some pairs to share their thinking with the rest of the class. Confirm the answer and model the solution, as necessary. Answer: They should include (some of) these ideas. Millie: 2:6, MM is 2/8 (1/4) of the mix (this is the same as 4:16). MM is 25% of the mix. Maxwell 5:11, MM is 5/16 of the mix. MM is 31.25% of the mix. Maxwell’s mochaccino has a slightly stronger flavour. It has 1/16 (6.25%) more mochaccino mix.
- Tell the students that it is their turn to make a mochaccino. Pose the following questions (included on the Copymaster) and allow students to work in groups to find a solution. Roam and provide support, questioning, and guidance. You decide to make some mochaccinos using Millie’s recipe. If you use 18 scoops of milk, how many scoops of MM will you use? If you use 1 1/2 scoops of MM, how many scoops of milk will you use? If Millie’s recipe makes a small drink for one person, what quantities will you use to make a small drink for eight people using Millie’s recipe? You decide to make your own mix. You use a 2:8 ratio in your first drink. What percentage is mochaccino mix? Your second drink mix is stronger. You use a 3:7 ratio. What percentage is milk? Your third drink mix is really thick. It is made with 40% mochaccino mix. What ratio did you use? Answers: 6, 4 1/2 , 16MM:48m 20%, 70%, 4:6 or 2:3
- Pose this problem (from Copymaster 3 ): Odds It’s winter in the south. Four children are predicting the likelihood of school having to close early because of the snowfall that is predicted. They state their odds as Yes will close (Y) to No , won’t close (N). Here are their odds. Y:N
- If you wanted the odds of school closing to be greater, whose odds would you pick and why?
- Whose odds show the greatest chance of school remaining open? Answers: 1. Toni. 2. Arapeta. (Toni 7:9 is a 7/16 chance of closing. Filipo 3:5 is a 3/8 or 6/16 chance of closing. Mona 6:10 is a 6/16 chance of closing. Arapeta 5:11 is a 5/16 chance of closing.)
- Refer to the contexts brainstormed in Activity 2 of this session.
- Have student pairs write (at least) two ratio problems and work out the answers.
- Have them give their ratio problems to another pair to solve.
- Bring the class together, as necessary, to discuss the mathematics involved in the new ratio problems. Identify the problems that were particularly challenging/successful/enjoyable.
- Investigate conjectures involving ratios.
- Remind students of the mochaccino ratios from Session 3.

Mochaccino Mix Millie and Maxwell are creating their own homemade ‘mochaccinos’ with scoops of mochaccino mix (MM) and scoops of milk (m).

- Write on the class chart and explain that Millie has stated: If Maxwell and I combine our mochaccino mixtures to make one big cup of coffee, the combined ratio will be 7:17 (MM:m). You add two ratios together to give a combined ratio.
- Challenge the students in pairs to explore, argue and justify their position of agreement or disagreement with this conjecture. (Answer. This is so with a single combination, or where the number of cups are the same for each)
- Pose the following problem: Millie makes her mochaccinos (2:6MM:m) for 1 person and Maxwell makes his mochaccinos (5:11 MM:m) for 2 people. Maxwell makes this statement: The combined ratio of our combined mochaccino mixtures is 7:17 You add two ratios together to give a combined ratio.
- Challenge the students in pairs to explore, argue and justify their position of agreement or disagreement with this conjecture. (Answer: This is not correct. These represent different quantities. The combined ratio must be calculated from the combined quantities. 2:6 1 mochaccino made by Millie 10:22 2 mochaccinos made by Maxwell 12:28 the combined ratios of MM and m 6:14 or 3:7 the simplified combined ratio of MM and m)
- Have students pose a similar combination problem of their own for investigation, using different numbers of cups for Millie and Maxwell. Have them swap problems with a partner, and provide time for them to solve and discuss the problems they have been given.
- Gather as a class and discuss the problems, as well as any challenges students faced.
- Have pairs of students work together to formulate a general statement about ratios and how they work (conjecture). Share (e.g. digitally, on sticky notes) and discuss these as a class. Emphasise that just one counter example is sufficient to disprove the proposition (false conjecture).
- Have each student write down their definition of a ratio. Ask selected students to share what they have written. Eg. A ratio is a relationship between two amounts of the same kind (eg. people: girls to boys, drink mix, odds) usually expressed as ‘a to b’.
- Write Rate on the class chart. Have each student write their own definition and share what they have written. Elicit and record the difference and these key ideas about rates: A rate is a special kind of ratio: a rate compares different kinds of measures such as dollars per kilogram. A rate involves a multiplicative comparison and uses the word per and the symbol /.
- Have students brainstorm and record rates that they experience/know of in their own lives: For example: dollars per hour (pay rate), metres per second or kilometres per hour (rate of speed), heartbeats per minute (pulse rate).
- Have students suggest a rate of speed that they travel on their bicycles: for example 30 km/hour. Record this in full: 30 kilometres per hour and beside this write the abbreviation.
- Ask for a student to explain what this rate means: “In every hour I ride 30 kilometres”. Write on class chart:
- Pose: How long will it take to ride 10 km? Ask students to discuss this in pairs and be prepared to explain 2 ways of working out their response.
- Ask at least one student pair to explain their thinking.
- Elicit and demonstrate:
- the between relationship: 30 is half of 60 so 10 must be half of 20

- Explore other examples, each time highlighting the between and within relationships. How long to ride 60 km? How long to ride 40 km? How long to ride 15 km? How far in 30 minutes? How far in 45 minutes? How far in 10 minutes?
- Ask, what is meant by a constant rate ? (In this case the rate is 30km/hour and this doesn’t change regardless of distance.) Discuss whether this is realistic, for example, cycling over a long distance.
- Highlight the fact that in exploring the examples in the next task a constant rate is assumed.
- Explain that students are going to investigate Millie and Maxwell’s coffee making a little further.
- Distribute one set of coloured cards (3) from page 1 of Copymaster 4 to each student pair.
- Provide time for students to solve the problems posed, recording their ‘between’ or ‘within’ thinking as they do so.
- Emphasise that they need to be able to show and explain the multiplicative relationship between the numbers.
- Make the relevant coloured section of page 2, Copymaster 4 available, if required, to scaffold their calculations and recording.
- Have student pairs share their work with another student pair with the same coloured problem, explaining their results and the rationale for these.
- Have one person from the group of 4 collect a piece of graph paper for each member of the group. Have them discuss in a group how they would show this rate on a graph.
- Have each student create a graph showing the linear relationship. Compare.
- Have student pairs swap coloured problems, solve and repeat 2-4 above.
- Have student pairs write two more problem scenarios for Millie and Maxwell. Suggested contexts: speed that they ride home from school on their bikes, the pulse rate of the person who drinks the mochaccino.
- Solve each other’s problems in pairs or as a class.
- On the class chart, summarise key learning about rates.

Dear families and whānau,

We have been working with rates and ratios in class. Your child would like to tell you about what they have been learning and invite you to solve with them and discuss these problems that they have worked on in class.

Perhaps you could also discuss and explore rates that you know and use in your own lives.

We trust that you find this both challenging and enjoyable.

We hope that you learn together and enjoy the mathematics.

Ratios (Teaching unit) Mixing Colours from Book 7: Teaching Fractions, Decimals and Percentages Extending Mixing Colours from Book 7: Teaching Fractions, Decimals and Percentages Rates of Change from Book 7: Teaching Fractions, Decimals and Percentages Ratios with Whole Numbers from Book 8: Teaching Number Sense and Algebraic Thinking Comparing by Finding Rates from Book 8: Teaching Number Sense and Algebraic Thinking Sharing in Ratios from Book 8: Teaching Number Sense and Algebraic Thinking Inverse Ratios from Book 8: Teaching Number Sense and Algebraic Thinking

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