# Word Problems Explained For Primary School Parents: Maths Word Problem Examples and How To Solve Them

In this blog primary maths teacher Sophie Bartlett breaks down word problems for primary school parents, and tells you how you can help your child tackle this important area of the primary curriculum.

This blog is part of our series of blogs designed for parents supporting home learning and looking for home learning resources during the Covid-19 epidemic.

### What is a word problem?

Mathematical ‘word problems’ can be defined as a few sentences requiring children to apply their maths knowledge to a ‘real-life’ scenario.

This means that children must be familiar with the vocabulary associated with the mathematical symbols they are used to, in order to make sense of the word problem.

For example:

### My child’s arithmetic is brilliant – isn’t that enough?

The National Curriculum states that its mathematics curriculum “aims to ensure that all pupils:

• become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately;
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language;
• can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.”

#### To support this schools are adopting a ‘mastery’ approach to maths

The National Centre for Excellence in the Teaching of Mathematics (NCETM) have defined “teaching for mastery”, with some aspects of this definition being:

• Maths teaching for mastery rejects the idea that a large proportion of people ‘just can’t do maths’.
• All pupils are encouraged by the belief that by working hard at maths they can succeed.
• Procedural fluency and conceptual understanding are developed in tandem because each supports the development of the other.
• Significant time is spent developing deep knowledge of the key ideas that are needed to underpin future learning. The structure and connections within the mathematics are emphasised, so that pupils develop deep learning that can be sustained.

(The Essence of Maths Teaching for Mastery, 2016)

### Mastery helps children to explore maths in greater depth

One of NCETM’s Five Big Ideas in Teaching for Mastery (2017) isMathematical Thinking: if taught ideas are to be understood deeply, they must not merely be passively received but must be worked on by the student: thought about, reasoned with and discussed with others”.

In other words – yes, fluency in arithmetic is important; however, with this often lies the common misconception that once a child has learnt the number skills appropriate to their level/age, they should be progressed to the next level/age of number skills.

The mastery approach encourages exploring the breadth and depth of these concepts (once fluency is secure) through reasoning and problem solving.

See the following example:

Year 6 objectiveFluencyReasoningProblem solving
Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why.7,208 females attended a concert as well as 8,963 males. There were originally 20,000 seats on sale. How many empty seats were there at the concert?Abdul says, “If I add any two 4-digit numbers together, it will make a 5-digit number.” Do you agree? Explain why.Three pandas are eating bamboo sticks. There are 51 altogether. They all eat an odd number of sticks. How many bamboo sticks did they each eat? How many different ways can you do it?

### What sort of word problems might my child encounter at school?

In Key Stage 2, there are nine ‘strands’ of maths – these are then further split into ‘sub-strands’.

For example, ‘number and place value’ is the first strand: a Year 3 sub-strand of this is to “find 10 or 100 more or less than a given number”; a Year 6 sub-strand of this is to “determine the value of each digit in numbers up to 10 million”. The table below shows how the ‘sub-strands’ are distributed across each strand and year group in KS2.

StrandYear 3Year 4Year 5Year 6Total
Number and place value697527
Calculations7815939
Fractions, decimals and percentages710121140
Ratio and proportion00044
Algebra00055
Measurement17910844
Geometry: properties of shape546722
Geometry: position and direction03126
Statistics22228

As well as varying in content (sometimes by using a combination of strands in one problem, e.g. shape and calculations), word problems will also vary in complexity, from one-step to multi-step problems (where children are required to complete more than one ‘operation’ to achieve the answer).

Different word problems will provide a different level of cognitive demand as an alternative method of adapting the level of difficulty.

The STA mathematics test framework (2015) sets these out below:

Strand1234
Depth of understandingrecall of facts or application of proceduresuse facts and procedures to solve simple problemsuse facts and procedures to solve more complex problemsunderstand and use facts and procedures creatively to solve complex or unfamiliar problems
Computational complexityno numeric stepsone, or a small number of numeric stepsa larger number of numeric steps all steps are simplea larger number of numeric steps, at least one of which is more complex
Spatial reasoningno spatial reasoning requiredmanipulation of the geometric information is requiredcomplex manipulation of the geometric information is requiredinterpret, infer or generate new geometric information
Data interpretation no data interpretation requiredselect and retrieve informationselect and interpret informationgenerate or infer new information from data
Response strategy select one or more responses or construct a simple responseconstruct a small set of responsesconstruct a straightforward explanation shows evidence of a methodconstruct a complex explanation

The rating scale for this table is 1=low, 4=high

### Word problem examples for Years 1 to 6

The more children learn about maths as the go through primary school, the tricky the word problems they will be facing will become.

Below you will find some information about the types of word problems your child will be coming up against on a year by year basis, and how word problems apply to each primary year group.

#### Word problems in Year 1

Throughout Year 1 your child is likely to be introduced to word problems with the help of concrete resources (pieces of physical apparatus like coins, cards, counters or number lines) to help them understand the problem.

An example of a word problem for Year 1 would be:

Chris is going to buy a cake for his mum which costs 80p. How many 20p coins would he need to do this?

#### Word problems in Year 2

Year 2 is a continuation of Year 1 when it comes to word problems, with children still using concrete resources to help them understand and visualise the problems they are working on.

An example of a word problem for Year 2 would be:

A class of 10 children each have 5 pencils in their pencil cases. How many pencils are there in total?

#### Word problems in Year 3

In Year 3, children will  move away from using concrete resources when solving word problems, and move towards using written methods. Teachers will begin to demonstrate the four operations (addition, subtraction, multiplication and division) too.

This is also the year in which 2-step problems will be introduced. This is a problem which requires two individual calculations to be completed.

2 examples of word problems for Year 3 would be:

Geometry: properties of shape

Shaun is making 3-D shapes out of plastic straws.

At the vertices where the straws meet, he uses blobs of modelling clay to fix them together.
Here are some of the shapes he makes:

ShapeNumber of strawsNumber of blobs of modelling clay
A85
B128
C64

One of Sean’s shapes is a cuboid. Which is it? Explain your answer.

Answer: shape B as a cuboid has 12 edges (straws) and 8 vertices (clay)

Statistics

Year 3 are collecting pebbles. This pictogram shows the different numbers of pebbles each group finds.

Answer: a) 9   b) 3 pebbles drawn

#### Word problems in Year 4

At this stage of their primary school career, children should feel confident using the written method for each of the four operations.

This year children will be presented with a variety of problems, including 2-step problems and be expected to work out the appropriate method required to solve each one.

An example of a word problem for Year 4 would be:

Number and place value

My number has four digits and has a 7 in the hundreds place.

The digit which has the highest value in my number is 2.

The digit which has the lowest value in my number is 6.

My number has 3 fewer tens than hundreds.

What is my number?

#### Word problems in Year 5

One and 2-step word problems continue in Year 5, but this is also the year that children will be introduced to word problems containing decimals.

2 examples of word problems for Year 5 would be:

Calculations

This is a multi-step problem.

Mara is in a bookshop.
She buys one book for £6.99 and another that costs £3.40 more than the first book.
She pays using a £20 note.
What change does Mara get?

Fractions, decimals and percentages

Stan, Frank and Norm are washing their cars outside their houses.

Stan has washed 0.5 of his car.

Frank has washed 1/5 of his car.

Norm has washed 5% of his car.

Who has washed the most?

Answer: Stan (he has washed 0.5 whereas Frank has only washed 0.2 and Norm 0.05)

#### Word problems for Year 6

In Year 6 children move on from 2-step word problems to multi-step word problems. These will include fractions, decimals and percentages.

3 examples of word problems for Year 6 would be:

Ratio and proportion

This question is from the 2018 KS2 SATs paper. It is worth 1 mark.

The Angel of the North is a large statue in England. It is 20 metres tall and 54 metres wide.

Ally makes a scale model of the Angel of the North. Her model is 40 centimetres tall. How wide is her model?

Algebra

This question is from the 2018 KS2 SATs paper. It is worth 2 marks as there are 2 parts to the answer.

Amina is making designs with two different shapes.

She gives each shape a value.

Calculate the value of each shape.

Answer: 36 (hexagon) and 25

Measurement

This question is from the 2018 KS2 SATs paper. It is worth 3 marks as it is a multi-step problem.

There are 28 pupils in a class.
The teacher has 8 litres of orange juice.
She pours 225 millilitres of orange juice for every pupil.
How much orange juice is left over?

Answer: 1.7 litres or 1,700ml

### How important are word problems when it comes to the SATs?

In the KS1 SATs, 58% (35/60 marks) of the test is comprised of maths ‘reasoning’ (word problems).

In KS2, this increases to 64% (70/110 marks) spread over two reasoning papers, each worth 35 marks. Considering children have, in the past, needed approximately 55-60% to reach the ‘expected standard’, it’s clear that children need regular exposure to and a solid understanding of how to solve a variety of word problems.

### How can I help my child solve word problems at home?

Here are two simple strategies that can be applied to many word problems before solving them.

1. What do you already know?
2. How can this problem be drawn/represented pictorially?

Let’s see how these can be applied to a few of the previously mentioned word problems to help achieve the answer.

There are 28 pupils in a class.
The teacher has 8 litres of orange juice.
She pours 225 millilitres of orange juice for every pupil.
How much orange juice is left over?

1. What do you already know?

• There are 1,000ml in 1 litre
• Pours = liquid leaving the bottle = subtraction
• For every = multiply
• Left over = requires subtraction at some point

2. How can this problem be drawn/represented pictorially?

Bar modelling is always a brilliant way of representing problems, but if you are not familiar with this, there are always other ways of drawing it out.

For example, for this question, you could draw 28 pupils (or stick man x 28) with ‘225 ml’ above each one and then a half-empty bottle with ‘8 litres’ marked at the top.

Now to put the maths to work. This is a Year 6 multi-step problem, so we need to use what we already know and what we’ve drawn to break down the steps.

This approach can also be applied to this Year 5 multi-step problem.

1. What do you already know?

• More than = add
• Using decimals means I will have to line up the decimal points correctly in calculations
• Change from money = subtract

2. How can this problem be drawn/represented pictorially?

See this example of bar modelling for this question:

 First book Second book £6.99 £6.99 £3.40 Change? £20

Now to put the maths to work using what we already know and what we’ve drawn to break down the steps.

Mara is in a bookshop.

She buys one book for £6.99 and another that costs £3.40 more than the first book. 1) £6.99 + (£6.99 + £3.40) = £17.38

She pays using a £20 note.
What change does Mara get? 2) £20 – £17.38 = £2.62

### Remember: The word problems can change but the maths won’t

It can be easy for children to get overwhelmed when they first come across word problems, but it is important that you remind them that whilst the context of the problem may be presented in a different way, the maths behind it remains the same.

Word problems are a good way to bring maths into the real world and make it more relevant for your child, so help them practice, or even ask them to turn the tables and make up some word problems for you to solve.

Practice makes perfect when it comes to word problems, so start now!

##### Ellie Williams
With a love for all things KS2 maths, Ellie is a part of the content team that helps all of the Third Space Learning blogs and resources reach teachers!