ratio problem solving nrich

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Ratio and Proportion

This collection of resources supports the teaching of Ratio and Proportion in primary mathematics. These activities are linked to the year groups containing the corresponding content in the National Curriculum. 

Here are some favourite activities selected by the NRICH team.

Orange Drink (Y6) A 750 ml bottle of concentrated orange squash is enough to make fifteen 250 ml glasses of diluted orange drink. How much water is needed to make 10 litres of this drink?

Rectangle Tangle  (Y6) The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?

These are just a few of the activities on Ratio and Proportion that you can find on the  NRICH curriculum pages .

The activities below, taken from the STEM Learning website , complement the NRICH activities above.

Ratio Makes Sense

Quality Assured Category: Mathematics Publisher: SMILE

A series of 12 activities which investigate ratio. They are aimed at older primary and younger secondary learners and include:

Pencils  - recognising and describing simple ratios in the context of measurement

Walking to school  - developing understanding of ratio using the context of distance

Under a magnifying glass  - developing understanding of ratio by calculating the actual length of objects which have been enlarged by a scale factor

Cooking numbers  - developing understanding of ratio by solving problems in the context of cooking recipes

Whiteboard Numeracy: Ratio and Proportion

Quality Assured Category: Mathematics Publisher: Teachers TV

This programme shows one way of illustrating ratio using ICT, though it could be presented using multi-link cubes instead. In the second lesson, children look at photographs of themselves that are out of proportion. They then investigate Leonardo da Vinci’s proposition that the length of our face fits exactly eight times into the length of our body using images of themselves to test this idea.

Quality Assured Category: Mathematics Publisher: The Virtual Textbook

This series of interactive excel sheets explore ratio and proportion. The first sheet displays a series of rectangles shaded in two colours. For each diagram the given ratio and the simplified version can be revealed. The sheet also asks what fraction of the whole each colour makes up. The second sheet generates a series of four equivalent ratios which can be revealed one value at a time. Other sheets support methods for solving ratio questions.

Quality Assured Category: Mathematics Publisher: Colin Foster

This resource contains sixteen instant maths ideas covering a whole range of suggestions that may be used as starter questions, extension questions or probing questions to assess understanding. Areas covered include: scaling up recipes, investigating paper sizes, value for money, proportion questions, rates and what does '24 carat gold' mean.

National Curriculum Resource Tool

Materials to support teachers and schools in embedding the National Curriculum

• National Curriculum Tool

KS3 - Ratio and Proportion and Rates of Change

New curriculum.

Pupils should be taught to:

• change freely between related standard units (for example time, length, area, volume/capacity, mass)
• use scale factors, scale diagrams and maps
• express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
• use ratio notation, including reduction to simplest form
• divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio
• understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
• relate the language of ratios and the associated calculations to arithmetic of fractions and to linear functions
• solve problems involving percentage change, including: percentage increase, decrease and the original value problems and simple interest in financial mathematics
• solve problems involving direct and inverse proportion, including graphical and algebraic representations
• use compound units such as speed, unit pricing and density to solve problems

Non-Statutory Guidance

Links and resources.

• Making connections
• Exemplification

Making connections to other topics within Key Stage 3

Pupils will need to be conversant with a whole range of number topics to support their learning and development within this area. In particular they will need to be secure in multiplication and division operations with fractions, percentages and decimals. They will also need to understand and use inverse operation.

Some aspects of this area will require pupils to use and interpret mathematical relationships both algebraically and graphically. Connections should be made when teaching direct and inverse proportion.

Connections between this area of study should be made when working with similar triangles and other similar shapes as the concept of proportionality is the basis of similarity.

Making connections to this topic in Year 6

In year 6 pupils will have used ratio and proportion in a range of contexts:

• Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (e.g. similar shapes, recipes).
• Pupils link percentages of 360° to calculating angles of pie charts.
• Pupils consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.
• Pupils solve problems involving unequal quantities e.g. ‘for every egg you need three spoonfuls of flour’, ‘ 35 of the class are boys’. These problems are the foundation for later formal approaches to ratio and proportion.

Cross-curricular and real life connections

• Science: science can provide the context for many basic ratio problems such as NRICH’s Oh Harry!
• Geography: statistics on populations in different parts of Canada at different periods, given as percentages and also allowing calculations of average rates of change
• Art: see proportion wheel for creating art cards of different sizes by reducing pictures in different proportions
• Ratio and Proportional Reasoning A. Watson, K. Jones, D. Pratt, 2013 The authors use research about students’ understanding of ratio and proportional reasoning to shed light on difficulties which seriously hinder success across mathematics. They examine the strengths and limitations of various teaching approaches and make suggestions about how teachers might work with colleagues on teaching ratio and proportional reasoning.
• The struggle to achieve multiplicative reasoning 11-14 , Margaret Brown, Dietmar Küchemann and Jeremy Hodgen, 2010. Multiplicative reasoning is a key competence for many areas of employment and everyday life, and for further mathematical study. It is however a complex conceptual field. The ICCAMS (Improving Competence and Confidence in Algebra and Multiplicative Structures) project, with multiplicative reasoning as one of its two focus themes, has in Phase 1 conducted a broadly representative survey of attainment which suggests that standards in this area have not risen since the 1970s and that relatively few students are achieving competence in relevant areas. Student difficulties are illustrated by evidence from group interviews in Phase 2 of the project.
• Ratio or Proportion? , Nrich Team, Discussion of the similarities and differences between ‘ratio’ and ‘proportion’ and how the word ‘proportion’ is used in two different ways.
• Misconceptions with the Key Objectives, has a section (from page 60) identifying common misconceptions in ratio and proportion and suggesting effective questions to tackle them.

Ratio (mathematical moments)

This is a collaborative task to support pupils to develop concepts about equivalent ratios. (You need to be logged in to the National STEM Centre website to access this document)

Understanding ratio (Learning mathematics in context):

This is a collaborative task on combining quantities in a given ratio. Pupils will learn to: identify the ratio of two quantities and use the notation. Understanding of the meaning of ratio and equivalent ratios will be developed. Pupils combine quantities and divide quantities in a given ratio.

Proportion (mathematical moments)

This activity is intended to give pupils the opportunity to explore the proportional relationships between numbers using a relatively familiar resource (a multiplication grid) and make links with other areas of mathematics. (You need to be logged in to the National STEM Centre website to access this document)

Toad in the whole:

This activity uses proportion and the understanding of the multiplicative relationships between quantities to solve a problem that superficially appears a simple question of adding amounts of money.

Mixing Lemonade:

This activity uses ratio and proportional reasoning to determine the relative strength of mixing cordial with water. There are a range of ways of determining the answer using fractions, percentages or pie charts to illustrate the answer.

Calculating scales for scale drawing (Learning Mathematics in Context):

This activity develops pupils understanding of scales written in the form 1:x. It develops the ability to carry out calculations involving scales using a variety of units.

Developing Proportional Reasoning (Improving Learning in Mathematics N6):

Pupils are given four direct proportion problems to solve, taken from different areas of the mathematics curriculum. They then compare their methods for solving these with methods produced by other pupils.

Using Percentages to Increase Quantities (Improving Learning in Mathematics N7):

Pupils make links between percentages, decimals and fractions, represent percentage increase and decrease as multiplication and recognise the inverse relationship between increases and decreases using proportional reasoning.

Four on the road:

This activity involves using graphical representations to solve speed, time and distance problems.

Examples of what children should be able to do, in relation to each (boxed) Programme of Study statement

Examples of what children should be able to do, to show a broad grasp of the content of this topic

Simplify the ratios expressed below in different units.

2 parts of blue paint mixed with 3 parts of yellow paint makes green. A boy has 50ml of blue paint and 100ml of yellow.

What is the maximum amount of green he can make?

The ratios of Lycra to other materials in two stretch fabrics are 2 : 25 and 3 : 40.

Which fabric has the greater proportion of Lycra?

The cost of 4 kg of apples is £3.36

The total cost of 3 kg of apples and 2.5 kg of pears is £4.12

Work out the cost of 1 kg of pears.

A Large tub of popcorn costs £3.80 and holds 200 g.

A Regular tub of popcorn costs £3.50 and holds 175 g.

Mike says that the 200 g Large tub is the better value for money.

Lucy says that the 175 g Regular tub is the better value for money.

Who is correct?

Explain the reasons for your answer. You must show all your working.

This is a recipe for fruit squash for six people.

Tracy made fruit squash for ten people.

How many litres of lemonade did she use?

John used 2 litres of orange juice for the same recipe.

How many people was this enough for?

This map has a scale of 1cm to 6km.

The road from Ridlington to Carborough measured on

the map is 5.2cm long.

What is the length of the road in kilometres

The length of a coach is 15 metres.

Joe makes a model of the coach.

He uses a scale of 1:24

Work out the length, in centimetres, of the model coach.

A truck and van left from Paris in diametrically opposite directions. The truck travelled for 4

hours at 76 mph. The vehicles were 564 miles apart. Find the van's average speed.

Shapes ABCD and EFGH are mathematically similar.

• Calculate the length of BC .
• Calculate the length of EF .

The graph above shows the relationship between gallons and litres.

Does the graph illustrate direct proportion? How do you know?

• Change 13 gallons to litres.
• Change 500 litres to gallons.

Explain how you worked out your answers.

• Ratios and proportions video from Teachers’ Media (free registration)
• American POP video illustrating proportion by comparing movie ticket price to monthly income at two different dates:

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Ratio problem solving

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

Students will first learn about ratio problem solving as part of ratio and proportion in 6 th grade and 7 th grade.

What is ratio problem solving?

Ratio problem solving is a collection of ratio and proportion 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 you can use to help you 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 you can use when given certain pieces of information.

When solving ratio word 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?

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

40 \div 8=5

Then you multiply each part of the ratio by 5.

3\times 5:5\times 5=15 : 25

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

There can be ratio word problems involving different operations and types of numbers.

Here are some examples of different types of ratio word problems:

Common Core State Standards

How does this relate to 6 th and 7 th grade math?

• Grade 6 – Ratios and Proportional Relationships (6.RP.A.3) Use ratio and rate reasoning to solve real-world and mathematical problems, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• Grade 7 – Ratio and Proportional Relationships (7.RP.A.2) Recognize and represent proportional relationships between quantities.

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.

[FREE] Ratio Check for Understanding Quiz (Grade 6 and 7)

Use this quiz to check your 6th and 7th grade students’ understanding of ratios. 10+ questions with answers covering a range of 6th and 7th grade ratio topics to identify areas of strength and support!

Ratio problem solving examples

Example 1: part:part ratio.

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

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

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

You could write this as:

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

You know that 465 students have school lunch.

2 Know what you are trying to calculate.

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

You need to find the value of p.

3 Use prior knowledge to structure a solution.

You are looking for an equivalent ratio to 5 : 7. So you need to calculate the multiplier.

You do this by dividing the known values on the same side of the ratio by each other.

465\div 5 = 93

This means to create an equivalent ratio, you can multiply both sides by 93.

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} \textbf{ (GBP)} to Euros \textbf{€ } \textbf{(EUR)}.

The two values in the table that are important are \text{GBP} and EUR. Writing this as a ratio, you can state,

You know that you have £520.

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

To get from 1 to 520, you multiply by 520 and so to calculate the number of Euros for £520, you 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 500 \, ml of concentrated plant food must be diluted into 2 \, l 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{500 \, ml} of concentrated plant food must be diluted into \bf{2 \, l} 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, you can now state the ratio of plant food to water as 500 \, ml : 2 \, l. As you can convert liters into milliliters, you could convert 2 \, l into milliliters by multiplying it by 1000.

2 \, l=2000 \, ml

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

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

You need to find an equivalent ratio where the first part of the ratio is equal to 1. You 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 an allowance each week proportional to their age. Kieran is 3 years older than Josh. Luke is twice Josh’s age. If Josh receives \$ 8 allowance, how much money do the three siblings receive in total?

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

You 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, you have:

You also know that Luke receives \$ 8.

You want to calculate the total amount of allowance for the three siblings.

You need to find the value of x first. As Luke receives \$ 8, you can state the equation 2x=8 and so x=4.

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

The total amount of allowance is therefore 4+7+8=\$ 19.

Example 5: simplifying ratios

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

Express this data as a ratio in its simplest form.

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

You need to simplify this ratio.

To simplify a ratio, you 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, you 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.

You know the two ratios

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

Using equivalent ratios you can say that the ratio of Silica:Soda is equivalent to 15 : 3 by multiplying the ratio by 3.

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

Example 7: using bar modeling

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

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

You want to find the original quantity.

Drawing a bar model of this problem, you have:

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

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

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

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

You can fill in each share to be 25 \, g.

Adding up each share, you get

India=5 \times 25=125 \, g

Beau=2 \times 25=50 \, g

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

Teaching tips for ratio problem solving

• Continue to remind students that when solving ratio word problems, it’s important to identify the quantities being compared and express the ratio in its simplest form.
• Create practice problems for students using the information in your classroom. For example, ask students to find the ratio of boys to the ratio of girls using the total number of students in your classroom, then the school.
• To find more practice questions, utilize educational websites and apps instead of worksheets. Some of these may also provide tutorials for struggling students. These can also be helpful for test prep as they are more engaging for students.
• Use a variety of numbers in your ratio word problems – whole numbers, fractions, decimals, and mixed numbers – to give students a variety of practice.
• Provide students with a step-by-step process for problem solving, like the one shown above, that can be applied to every ratio word problem.

Easy mistakes to make

• 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 milliliters. You did not mix ml and l in the ratio.
• Writing ratios 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.

• 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 : \textbf{n}} The assumption can be incorrectly made that n must be greater than 1, but n can be any number, including a decimal.

Related ratio lessons

• Unit rate math
• Simplifying ratios
• Ratio to fraction
• How to calculate exchange rates
• Ratio to percent
• How to write a ratio
• Dividing ratios
• How to find the unit rate
• Ratio scale
• Constant of proportionality

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

You need to simplify the ratio so that the first number is 1. That means you need to divide each number in the ratio by 45.

45 \div 45=1

155\div{45}=3\cfrac{4}{9}

3. 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.

3 \times S : C=6 : 9

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

You should know that the 3 angles in a triangle always equal 180^{\circ}.

\begin{aligned} & x+2 x+3 x=180 \\\\ & 6 x=180 \\\\ & x=30^{\circ} \\\\ & 2 x=60^{\circ} \\\\ & 3 x=90^{\circ} \end{aligned}

5. A clothing company has a sale on tops, dresses and shoes. \cfrac{1}{3} of sales were for tops, \cfrac{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.

\cfrac{1}{3}+\cfrac{1}{5}=\cfrac{5+3}{15}=\cfrac{8}{15}

1-\cfrac{8}{15}=\cfrac{7}{15}

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

The given ratio in the word problem is 2. 75 \mathrm{~L}: 18^{\circ} \mathrm{K}

Divide 45 by 18 to see the relationship between the two temperatures.

45 \div 18=2.5

45 is 2.5 times greater than 18. So we multiply 2.75 by 2.5 to get the amount of gas.

2.75 \times 2.5=6.875 \mathrm{~l}

Ratio problem solving FAQs

A ratio is a comparison of two or more quantities. It shows how much one quantity is related to another.

A recipe calls for 2 cups of flour and 1 cup of sugar. What is the ratio of flour to sugar? (2 : 1)

In middle school ( 7 th grade and 8 th grade), students transition from understanding basic ratios to working with more complex and real-life applications of ratios and proportions. They gain a deeper understanding of how ratios relate to different mathematical concepts, making them more prepared for higher-level math topics in high school.

The next lessons are

• Converting fractions, decimals and percentages

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Sharing in a Ratio: Rich Tasks

You cannot beat a good rich task! For me, a rich task is one that both stimulates and challenges students of all ages and abilities. Here is a selection of some of my favourites.

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There are no words to express how much I love Don Steward and his amazing resources on his Median Blog . They allow for purposeful practise, to enable students to gain fluency in key skills. But they also have more than a sprinkling of richness, that leads students merrily along the path towards hypotheses and generalisations. Amazing.

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NRICH is simply one of the best websites for rich maths problems in the whole wide world.

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A selection of some of my favourite rich tasks, including – I must confess – some of my own. For my complete collection, including my thoughts on what makes a good rich task, please visit the Rich Tasks page .

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Resources to develop mathematical reasoning and problem solving. Nrich aims to enrich the mathematical experiences of all learners and embeds rich mathematical tasks into everyday classroom practice.

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Activities and games can be explored independently or together with families.

The website covers topic such as:

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The 'Thinking Mathematically' sections provide additional educational activities.

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Teacher notes.

The resources are grouped into levels. Level 1 aligns with early years, while level 4 aligns with lower secondary. Teachers can use the search functionality to find tasks that match the topic that they are teaching.

The tasks are not mapped to the Australian Curriculum but align well with the mathematical concepts.

Page last updated: 12 Oct 2022

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What Is Problem Solving?

The Joint Mathematical Council of the United Kingdom

Addressing the five ‘big questions’ in problem-solving with NRICH

The importance of ensuring learners acquire the problem-solving skills which will enable them to thrive both socially and economically in their increasingly automated world is widely recognised (Luckin et al., 2017). Nevertheless, government inspectors have reported serious concerns about the quality and quantity of problem-solving in our schools (Ofsted, 2015). This summer schools were challenged to reflect on ‘Five big questions for problem-solving’ (EEF, 2021). In this blog, we will consider each of those five questions and explore the ways that the NRICH team is supporting schools to address them.

Question one: Do teachers in your school select genuine problem-solving tasks for which pupils do not already have a ready-made method available?

Too often, learners are presented with routine word problems which merely require the application of a known algorithm. ‘Genuine’ problems enable them to make their own problem-solving decisions by choosing their own strategies and enabling them to compare their approach with those of other learners, thus developing their problem-solving efficiency and flexibility. At NRICH , our award-winning activities allow learners to develop these key skills alongside the confidence to tackle genuine problems. Moreover, our ‘ low threshold, high ceiling ‘ approach enables everyone to get started on the problem while ensuring a suitable level of challenge too, making them ideal for whole-class teaching.

Question two: Are pupils given the opportunity to see – through multiple worked examples – to use, and to compare different approaches to solving a problem?

Many problems can be explored in more than one way. Working flexibly, making connections between different areas of the curriculum and reflecting on various problem-solving approaches are key steps towards becoming a more fluent mathematician. NRICH encourages learners to develop these skills in these two ways:

Our primary , secondary and post-16 Live Problems invite learners to explore and submit their ideas to the team. We review each submission that we receive and publish a selection on our website showcasing different approaches and the reasoning behind them.

Our NRIC H online activities sometimes feature ‘hide and reveal’ buttons showcasing different starting points towards a solution for learners to explore further for themselves. This approach enables learners to widen their range of strategies for solving unfamiliar problems and develop alternative approaches to explore when they get stuck using their first-choice strategy.

Question three: Are pupils encouraged to use visual representations to support them to solve a problem?

One of the most important approaches towards solving an unfamiliar problem is drawing a good diagram. Learning to draw diagrams is a skill which we encourage learners of all ages to develop alongside their other mathematical skills and knowledge. From sketching graphs to drawing a bar model, good diagrams can help learners clarify their understanding and identify possible ways forward.

Our four steps towards problem-solving feature highlights the importance of drawing a diagram to enable young learners to get started on a problem. We often highlight a useful diagram, table or sketch graph in the solutions chosen for publication. As learners progress through their learning, the team model more specific drawing skills, such as sketching a graph to help solve a STEP problem.

Question four: Are pupils supported to monitor, reflect on, and communicate their reasoning and choice of strategies, possibly through the use of prompt questions?

NRICH  encourages learners to reflect on their learning using this approach inspired by the Strands of Mathematical Proficiency model introduced by Kilpatrick et al. (2001).

Our approach uses child-friendly language that teachers and parents can share with students five key ingredients that characterise successful mathematicians. At NRICH , we believe that learning mathematics is about much more than just learning topics and routines. Successful mathematicians understand the curriculum content and are fluent in mathematical skills and procedures, but they can also solve problems, explain their thinking and have a positive attitude about themselves as learners of mathematics.

With this in mind, we have created  this self assessment tool  to help learners recognise where their mathematical strengths and weaknesses lie. We hope learners will explore NRICH activities and then take time to reflect on their own mathematical capabilities using our model.

Question 5: Is professional development time allocated to develop teachers’ pedagogical understanding of problem-solving, with particular support for early career teachers?

NRICH supports teachers to maximise the potential of our activities by offering free, regular professional development for teachers .  Each session is delivered online, enabling teachers to access the support wherever they are based, reducing teacher travel and cover costs for schools. We also record the sessions and upload them to our website so that schools can access them for future professional development days or staff/department meetings in their settings.

The live sessions are led by NRICH team members and they link directly to our latest primary , secondary and post-16 Live Problems. This approach enables teachers to consider the possibilities of the activities with the NRICH team before exploring them the next day with their own classes. Later, they are invited to share their classwork with our team for possible publication on the NRICH website.

The five ‘big questions’ provide excellent starting points for evaluating the teaching and learning of problem-solving in different settings. I hope that this blog shares an insight into the different ways that NRICH can support schools to address the five questions for themselves by engaging with our activities, Live Problems and teacher webinars.

Dr Ems Lord FCCT

Director of NRICH

Centre for Mathematical Sciences

University of Cambridge

Selected references

EEF. (2021). EEF Blog: Integrating evidence into maths teaching – guiding problem-solving. Accessed from https://educationendowmentfoundation.org.uk/news/eef-blog-integrating-evidence-into-mathematics-guiding-problem-solving /

Kilpatrick, J. Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics (Vol. 2101). J. Kilpatrick, & National research council (Eds.). Washington, DC: National Academy Press.

Luckin, R., Baines, E., Cukurova, M., Holmes, W., & Mann, M. (2017). Solved! Making the case for collaborative problem-solving. Accessed from http://oro.open.ac.uk/50105/1/solved-making-case-collaborative-problem-solving.pdf

Ofsted. (2015). Better Maths Conference Spring Keynote 2015. Accessed here https://www.slideshare.net/Ofstednews/better-mathematics-keynote-spring-2015