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20 Word Problems For Year 3: Develop Their Problem Solving Skills Across Single and Mixed KS2 Topics

Emma Johnson

Word problems for Year 3 are an important tool for improving number fluency. The key focus in maths in Year 3 of primary school is on ensuring pupils are becoming more fluent with number facts and the concept of place value. Children are starting to develop more efficient written methods by this stage and are beginning to carry out calculations with increasingly larger whole numbers.

As children progress through school, they are exposed to a wider variety of problem solving questions covering a range of concepts. In Year 3 maths these include the four operations, fractions, measures and statistics.  

It is important that children are regularly exposed to reasoning and problem solving questions, alongside the fluency work each lesson. It is also important to remember that all children need exposure to reasoning and problem solving questions, not just the higher attaining pupils who finish quickest.

We have put together a collection of 20 word problems, aimed at Year 3 pupils.

Years 3 to 6 Rapid Reasoning (Weeks 1-6)

Download this free pack of word problems to improve your class' problem solving skills. Includes questions for Years 3 - 6.

Place value

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In Year 3, pupils focus on one-step problems, covering a range of topics across the National Curriculum. At this stage the majority of word problems pupils are tackling will have one-step, but they may also start to be introduced to simple two-step word problems. Here is a breakdown of topics that will be covered and expectations in Year 3. 

Solve number problems and practical problems involving recognising the place value of each digit of a 3-digit number; comparing and ordering numbers up to 1000 and identifying, representing and estimating numbers using different representations.

Solve problems, including missing number problems, using number facts, place value and more complex addition and subtraction word problems .

Solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems

 Solve problems involving counting in tenths; recognising unit and non-unit fractions; recognising equivalent fractions and adding/ subtracting fractions with the same denominator.

Solve problems involving length, volume and mass; adding and subtracting within money word problems ; perimeter and problems involving time.

Solve-step and two-step questions (For example, ‘How many more?’ and ‘How many fewer’?) using information presented in scaled bar charts, pictograms and tables.

By Year 3, children are starting to learn how to use some of the formal written methods of addition and subtraction. It is important that the link between maths in school and maths in real-life continues to be made. Word problems are a key element in helping pupils to make this link. 

When teaching maths problems to Year 3, it’s important to think of ways to make them fun, engaging and something the children are able to relate to. This might include acting out the problem, using concrete resources and providing visual images, to bring the problems to life. 

Benefits of pairs, groups and class discussion 

Children should have plenty of opportunity to talk in pairs, groups and as a whole class, to share their understanding of what is being asked and their strategies for solving the problem. As with Key Stage One, the use of manipulatives is important and all children should have access to a range of maths resources when solving problems like this.

Pupils need to be encouraged to read word problems carefully and to make sure they understand what is being asked, before attempting to tackle the problem. This is where the use of a partner and group discussion can really help children’s understanding. Students then need to think about what they already know and how they can use this to help them answer the question. Where appropriate, pupils should also be encouraged to draw diagrams and pictures to help them solve the question.

Here is an example:

Mason needs 4 apples to make an apple pie.

If he has 28 apples. How many apple pies can he make?

How to solve:

What do you already know?

• We know 4 apples are needed to make an apple pie.
• We are told how many apples Mason has in total, so we must have to divide the total number by 4.
• In year 3, children should be building confidence with multiplication and division facts for the 4 times table, so some will be able to quickly recall these facts to solve the problem.
• Children who aren’t apple to recall quickly could use counters to represent the apples, or draw a bar model to help solve it.

How can this be drawn/represented pictorially?

We can draw a bar model or counters to represent this problem:

• To calculate how pies the 28 apples will make, we can either use or draw 28 counters and put them into groups of 4. We can see that 28 counters will make 7 groups of 4.
• Using the bar model, we can keep adding 4 to the bar until we reach 28. From the bar we can see that 7 x 4 = 28.
• 7 apple pies can be made from the 28 apple pies that Mason has.

In Year 3, pupils are exposed to a range of addition word problems , including problems involving mental addition and addition of up to 3-digits using formal written methods.

See also: Mental maths Year 3

Addition question 1

A family driving on holiday, travel 146 miles from home to the first service station.

They then drive a further 175 miles to reach their destination.

How far have they travelled altogether?

Answer (1 mark): 321 miles

146 + 175 = 321

Addition question 2

Evie is buying a bottle of drink from a vending machine. She has put in 40p. The vending machine shows she still needs to put in £1.25. 

How much is the bottle of drink?

Answer (1 mark): £1.65

125p + 40p = 165p

Addition question 3

Jamie scored 643 on his new online game.

Jared scored 468.

How many points did they score between them?

Answer (1 mark): 1111 points

643 + 468 = 1111

At Third Space Learning we often tie word problems into our one-to-one online tuition. With each programme personalised to the needs of each individual student, children are able to develop their problem solving skills, maths fluency and grow confidence in maths.

Subtraction word problems in Year 3 also need to comprise of a combination of mental calculation questions and those involving formal written subtraction up to 3-digits. Children should also be starting to estimate answers and check their calculations by using the inverse.

Subtraction question 1

Ahmed collects 374 stickers.

He needs 526 stickers to fill his sticker album.

How many more stickers does he need to collect?

Answer (1 mark): 152 stickers

526 – 374 = 152

Subtraction question 2

 A bag of carrots weigh 360g

A bag of tomatoes weighs 235g.

How much heavier is the bag of carrots?

Answer (1 mark): 125g

360 – 235 = 125

Subtraction question 3

 Ahmed buys a bag of sweets for £1.45.

He has a £2 coin. How much change will he get?

Answer (1 mark): 55p

200 – 145 or count up from £1.45 to £2

By the end of Year 3, pupils should be able to recall and use multiplication facts for the 3, 4 and 8 times table. They should also be starting to progress to using the formal written method for solving multiplication word problems involving 2 digits multiplied by a 1-digit number. 

Multiplication question 1

It costs £7 for a cinema ticket

Amber’s mum pays for Amber and her 3 friends. 

How much does she pay for all the tickets?

Answer (1 mark): £28

7 x 4 = 28 

Multiplication question 2

5 mini buses are used to take Year 3 pupils on a school trip.

Each minibus transports 15 children.

How many children go on the school trip?

Answer (1 mark): 75 children

15 x 5 = 75

Multiplication question 3

There are 24 questions on a word problems worksheet. 

A group of 8 children each complete the worksheet.

How many questions will the teacher be marking?

Answer (1 mark): 112 questions

24 x 8 = 112

In Year 3, pupils will need to learn the division facts for the 3, 4 and 8 multiplication tables. At this stage, pupils are required to learn the formal method for division, division word problems will involve mainly mental calculations.

Division question 1

6 children share 18 cookies between them. 

How many cookies does each child get?

 Answer (1 mark): 3 cookies

Division question 2

The school choir needs to travel to the concert hall.

There are 32 children in the choir and parents have been asked to help transport the children.

Each parent is able to take 4 children in their car. How many cars will be needed to get all the children to the concert?

Answer (1 mark): 8 cars

 32 ÷ 4 = 8

Division question 3

 Cakes come in packs of 4.

If Jessica needs 36 cakes for her party, how many packs does she need to buy?

Answer (1 mark): 9 packs

In Year 3 pupils are exposed to a range of fraction word problems , including questions involving counting up and down in tenths; equivalent fractions and adding/subtracting fractions with the same denominator.

Fraction question 1

Jude had 28 sweets.

He gave \frac{1}{4} of his sweets to his little sister.

How many did he have left?

Answer (1 mark): 21 sweets

\frac{1}{4} of 28 = 7

\frac{3}{4} of 28 = 21 or 28 – 7 = 21

Fraction question 2

Khalifa ate \frac{2}{8} of the chocolate bar and Abdulrahman ate \frac{3}{8} of it.

How much chocolate did they have left?

Answer (1 mark): \frac{3}{8}

\frac{2}{8} + \frac{3}{8} = \frac{5}{8} eaten

\frac{8}{8} – \frac{5}{8} = \frac{3}{8}

Fraction question 3

 Molly ate half of a pizza and Rosie ate \frac{3}{6} .

Who ate the most?

Answer (1 mark): They both ate the same amount

\frac{3}{6} is equivalent to \frac{1}{2} .

In Year 3 time word problems may be incorporated into maths problem solving. Students are expected to know time vocabulary, be able to compare time in terms of seconds, minutes and hours and know how many seconds are in a minute and minutes are in an hour. 

Time question 1

The Smith family are driving to the beach.

They leave at 9:05am and arrive at 9:50am.

How long does the journey take them?

Answer (1 mark): 45 minutes

50 – 5 = 45    or     count up from 9:05 to 9:50 = 45 

Time question 2

Holly puts her cake in the oven at 4:22pm and takes it out again at 4:47. 

How long is the cake baking for?

Answer (1 mark): 25 minutes

47 – 22 = 25    or   count up from 4:22 to 4:47

When children first move into lower Key Stage 2, word problems are predominantly one-step. As they become more confident they can be exposed to more word problems, requiring a second step or multi-step word problems .

Multi-step question 1

Oliver had 3 bags of sweets.

Each bag contained 15 sweets.

If he shared the sweets between him and 4 friends, how many sweets would they all 5 of them get?

Answer (2 marks): 9 sweets

3 x 15 = 45 

45  ÷ 5 = 9

Multi-step question 2

A teacher photocopies 95 maths worksheets and 80 English worksheets in one week.

Teachers can print a maximum of 300 worksheets per week. 

How many can the teacher print for other subjects.

Answer (2 marks): 125 worksheets

 95 + 80 = 175

300 – 175 = 125

Multi-step question 3

Three friends go trick or treating. They add all their sweets together and share them out, so they all have an equal number of sweets.

If Ben gets 34 sweets, Sophie gets 28 and Maisie gets 22 sweets. How many will they each get, once they have put them together and shared them out?

Answer (2 marks): 28 sweets each

34 + 28 + 22 = 84 sweets

84  ÷ 3 = 28

We hope that this collection of word problems for Year 3 becomes a useful resource in your Year 3 maths classroom. For more Year 3 maths resources, take a look at our library of Year 3 maths worksheets .

Third Space Learning also offers a wide array of maths and word problems resources for other year groups such as word problems for year 6 , word problems for Year 5 and word problems for year 4 . Our practice word problems cover all four operations and include more specific topics such as percentage word problems and ratio word problems .

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Problem-solving maths investigations for year 3.

Hamilton provide an extensive suite of problem-solving maths investigations for Year 3 to facilitate mathematical confidence, investigative inquiry and the development of maths meta skills in 'low floor – high ceiling' activities for all.

Explore all our in-depth problem solving investigations for Year 3 .

Use problem-solving investigations within every unit to encourage children to develop and exercise their ability to reason mathematically and think creatively.

Investigations provide challenges that offer opportunities for the development of the key mathematical skills while deepening conceptual understanding. They are designed to be accessible in different ways to all children. An added bonus is the substantial amount of extra calculation practice they often incorporate! The problems are designed to help children identify patterns, to explore lines of thinking and to reason and communicate about properties of numbers, shapes and measures.

Hamilton provide a mix of our own specially commissioned investigations, that include guidance for teachers together with a child-friendly sheet to guide your pupils through the investigation, as well as links to investigations on other highly regarded websites.

I am very grateful for Hamilton Trust resources, particularly the maths investigations. Julia, teacher in Wiltshire

You can find Hamilton's investigations for Year 3:

• Individually, they are incorporated into every unit in our Year 3 flexible maths blocks .
• Collectively, they appear on our resources page where you can explore all our in-depth problem solving investigations for Year 3 .

Do read our extensive range of advice for more information about the investigations and for tips on how to use them effectively.

Hamilton’s problem-solving investigations are 'low floor, high ceiling' activities that give all children opportunities to develop mastery and mathematical meta-skills. Explore a set for a whole year group.

Hamilton’s Problem-solving Investigations provide school-wide solutions to the challenges of building investigative skills from Early Years to Year 6.

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2. Compare and order numbers

When comparing numbers up to 1000, your child should look at the digit with the largest value first. For example, if your child is comparing the numbers 765 and 276, they would first need to look at the digit with the largest value, i.e. the hundreds digit:

276 has 2 hundreds, and 765 has 7 hundreds, so 276 is less than 765.

However, if we compare the numbers 765 and 754, they both have the same number of hundreds. Therefore, we now need to look at the tens digit:

765 has 6 tens, and 754 only has 5 tens, so 765 is more than 754.

Try this game to practise comparing numbers. Write twenty two- and three-digit numbers and the ‘>’ and ‘<’ symbols on separate pieces of paper. Deal your child two numbers, face down. Ask them to turn over the pieces of paper and to use the ‘>’ and ‘<’ symbols to show which number is bigger or smaller.

Activity: Number statements

3. Practise counting

Your child should now use the word  multiples to describe counting up in steps from zero, securing their understanding of multiplication.

They will be expected to count in multiples of 2, 3, 4, 5, 8, 10, 50, and 100. You could help your child practise by taking it in turns to say the multiples of a number. For example:

You : 4 Your child : 8 You : 12 …  and so on.

Set a timer and see what number you can get to before a minute is up! Be sure to note any interesting patterns, like how multiples of five always end in a 5 or a 0.

4. Learn multiplication facts

In Year 3, it’s important that your child is able to recall multiplication facts. They will be likely to focus on the 3, 4, and 8 times tables. They will already be familiar with the 2, 5, and 10 times tables, but they will still practise them.

Video: What are multiples?

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Years 3 & 4: Number and place value

This list consists of activities, games and videos designed to support the new curriculum programme of study in Years Three and Four. Containing tips on using the resources and suggestions for further use, it covers: Year 3: Count from 0 in multiples of 4, 8, 50 and 100; find 10 or 100 more or less than a given number, recognise the place value of each digit in a three-digit number, compare and order numbers up to 1000, identify, represent and estimate numbers using different representations, read and write numbers up to 1000 in numerals and in words, solve number problems and practical problems involving these ideas. Year 4: Count in multiples of 6, 7, 9, 25 and 1000, find 1000 more or less than a given number, count backwards through zero to include negative numbers, recognise the place value of each digit in a four-digit number, order and compare numbers beyond 1000, identify, represent and estimate numbers using different representations, round any number to the nearest 10, 100 or 1000, solve number and practical problems that involve all of the above, read Roman numerals to 100 (I to C) and know that, over time, the numeral system changed to include the concept of zero and place value. Visit the primary mathematics webpage to access all lists.

The Deca Tree

This NRICH maths problem will help children to apply and deepen their understanding of  10, 100, 1000 up to 10,000 and could easily be extended to allow children to explore number patterns further. The visual representation of a tree shows what the extra zeros do to a number when multiplying by 10 or 100 by looking at the number of trunks, branches, twigs and leaves.

Place Value Charts

Use this interactive resource to help children understand more about place value. A target number is given and children select which hundred, ten and one is required to make the number. This activity is a great visual demonstration and also offers an opportunity to looks at tenths.

Rounding Numbers

A range of worksheets, interactive resoureces and games for both teacher planning and direct student use. These are all a very useful way to practise rounding numbers to the nearest 10, 100 or 1000. The interactive resources provide children with a range of methods to practice rounding pictorally - using numberlines, mazes, and sports themed activities.

Place Value Games for Students Aged 7-9

Quality Assured Category: Mathematics Publisher: Nelson Thornes

Published by BEAM, these games provide opportunities for students to apply their knowledge of place value, number recognition and ordering. *Aim for 500 - in this game both players roll the dice and use the numbers to make two three-digit numbers, then add them up. Whoever gets a total closest to 1500 wins the game. *Making numbers - students create two four-digit numbers from digits generated by rolling a dice before  decidng  which is the highest. *Mastermind - using a two-digit number one player tries to discover the number by being given clues relating to the place value of the digits.

Maths Chest 6 - Number: Pack 1

Aimed at lower Key Stage Two, this resource provides games and activities, including  photocopiable  worksheets. These could be used with the whole class or with smaller groups  practising  specific topics.

Reasoning/Problem Solving Maths Worksheets for Year 3 (age 7-8)

Money problems and challenges.

A variety of problem solving activities involving money.

50p to spend, but can you make sure you get the correct change?

Watch out when writing pence as pounds and remember to always have two digits after the decimal point.

The hardest part of these money problems is to read the question and work out what to do.

Put three items in your basket that you would like to buy and then use the money cards to work out the sum.

Solve money problems using the information provided.

Solve money problems at the skating rink.

Plenty of monkey business here!

It's a trip to the zoo to find the very best value.

Some tricky money problems to solve.

More tricky money problems to solve.

Finding different amounts from a given set of coins.

Number and calculating problems

Solve number problems and calculating using addition, subtraction, multiplication and division.

Find the 2-digit numbers that can be made using these digit cards.

Make 2-digit numbers and find the smallest and largest numbers.

More on finding 2-digit numbers from three digit cards.

Tricky little problems involving monsters' legs.

Use logic, addition and subtraction to find out how many strawberries are eaten.

Bar Model: addition and subtraction facts.

Use bar models to add and subtract.

Encourage children to make addition and subtraction come to life by writing short number stories.

What numbers can you make with digit cards?

What are the largest and smallest numbers that can be made with 3 cards?

More number stories to write: this time all about multiplication and division.

All the numbers are here, but the signs are missing! Can you work out what the signs should be?

Investigating odd and even numbers and what happens when you add them together.

Use knowledge of place value to find all possible answers.

Finding the numbers and missing digits to make number sentences correct.

Investigate statements involving odd and even numbers.

Investigate statements about multiplication and times tables.

I?m good at thinking of numbers, but can you work out what number I am thinking about ? I do give a clue!

Work out the missing value using division and addition.

Use reasoning to find the missing values.

More on finding the missing values - an early introduction to algebra.

Use division and addition to find the missing values.

Find the missing values from the information given.

More on finding the value using reasoning.

Real life and word problems

A selection of real life problems and word problems.

Tricky word problems.

More tricky word problems.

Tricky questions, but you only have to carry out one maths step to answer them.

Even trickier questions, and you have to carry out at least two steps to work them out!

More word problems, from the library to shopping and on to flying around the world.

Here we have four pages of questions all on time, including a trip to Alicante!

Solving tricky fraction problems.

More challenges and activities

A great selection of activities requiring logical thinking.

Investigation looking at possibilities when adding the digits from 1 to 5.

This time you can decide the total for the sides of the triangle.

Can you put the numbers 1 to 9 in the diagram so that the difference between each pair of joined numbers is odd?

This challenge is to find as many ways as possible of making 12 using three number cards and the add, subtract and multiply signs.

A development of the 'Caterpillar investigation' but using multiples of 10. Great for practising addition.

3 dinosaurs laid some eggs. They laid 19 altogether. How many did they each lay?

How many ways can three runners cover a distance of 19 miles? They all have to run an odd number of miles.

A book challenge here. How accurately can you guess the number of pages in books?

2D card shapes and a 3 x 3 pinboard are useful for these shape activities.

A calculator is needed to find how many different answers can be made from the numbers given. Good for working in an organised, logical way.

A brief summary of some of the most important maths concepts to be taught in Year 3 by way of challenges and investigations.

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Year 3 Block A Problem Solving

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

19 July 2019

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Year 3 Money Challenges Solving Money Problems

Welcome to our Year 3 Money Challenges page. Here you will find our selection of printable money problem worksheets to help your child learn to use their money skills to solve a range of problems.

These challenges are a great resource as an extension for more able pupils, or to consolidate and extend learning.

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Year 3 Money Learning

Knowing how to handle and calculate with money is a very important life-skill.

During Year 3, children build on their knowledge and skills they have achieved during Year 2.

By the time children reach the end of Year 3, they should be able to:-

• use decimal notation for amounts of money, e.g. £3.27
• change amounts in £ to amounts in pence and back;
• count money up to £10;
• calculate change for amounts of money up to £1 mentally;
• solve simple problems involving money;
• use informal methods to add and subtract money amounts in £ and pence.

Our worksheets will support your child with these objectives.

Year 3 Money Challenges Information

The following worksheets involve solving challenges involving money amounts. They are a great way to consolidate and extend money learning once children are confident with counting money amounts.

The challenges start off at a fairly easy level with problems that involve lower value coins and smaller money amounts. The challenges later on involve more complex problems with larger values to calculate.

If you are teaching a class or group of children, then these sheets should give you some great differentiated learning activities with a money theme.

An answer sheet is available for each challenge provided.

The money challenges in each section are carefully graded, allowing you to introduce concepts at an easier level before introducing harder work, or differentiate within your class.

Using the sheets in this section will help your child to:

• count money in coins up to £1;
• develop their reasoning and thinking skills;
• solve problems involving money.

All the money worksheets in this section will help your child to become more confident with money, and develop better problem solving skills.

Year 3 Money Challenges Worksheets

There are 3 challenges - Tyger's Money Square Challenge, Row of Coins Challenge and How Much Money Challenge.

Each main challenge has several versions which start with the easiest level and progress onto harder levels of challenge.

Row of Coins Challenges

• Row of Coins Challenge 1
• PDF version
• Row of Coins Challenge 2
• Row of Coins Challenge 3
• Row of Coins Challenge 4

Tyger's Money Square Challenges

• Tyger's Money Square Challenge 1
• Tyger's Money Square Challenge 2
• Tyger's Money Square Challenge 3
• Tyger's Money Square Challenge 4

How Much Money Challenges

• How Much Money Challenge 1
• How Much Money Challenge 2
• How Much Money Challenge 3

Extension Activity Ideas

If you are looking for a way to extend learning with the How Much Money challenge, why not...

Try playing 'How Much Money' in pairs.

One person chooses 2 or 3 coins and the other person has to ask questions which involve 'yes' or 'no' answers.

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Overview Frequency Principle/Spectral Bias in Deep Learning

• Review Article
• Published: 04 September 2024

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• Zhi-Qin John Xu   ORCID: orcid.org/0000-0003-0627-3520 1 , 2 ,
• Yaoyu Zhang 1 , 2 &
• Tao Luo 1 , 2 , 3 , 4  

Understanding deep learning is increasingly emergent as it penetrates more and more into industry and science. In recent years, a research line from Fourier analysis sheds light on this magical “black box” by showing a Frequency principle (F-Principle or spectral bias) of the training behavior of deep neural networks (DNNs)—DNNs often fit functions from low to high frequencies during the training. The F-Principle is first demonstrated by one-dimensional (1D) synthetic data followed by the verification in high-dimensional real datasets. A series of works subsequently enhance the validity of the F-Principle. This low-frequency implicit bias reveals the strength of neural networks in learning low-frequency functions as well as its deficiency in learning high-frequency functions. Such understanding inspires the design of DNN-based algorithms in practical problems, explains experimental phenomena emerging in various scenarios, and further advances the study of deep learning from the frequency perspective. Although incomplete, we provide an overview of the F-Principle and propose some open problems for future research.

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Training Behavior of Deep Neural Network in Frequency Domain

Using Applicability to Quantifying Octave Resonance in Deep Neural Networks

A systematic review on overfitting control in shallow and deep neural networks

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Acknowledgements

This work is sponsored by the National Key R &D Program of China Grant No. 2022YFA1008200 (Z. X., Y. Z., T. L.), the National Natural Science Foundation of China Grant Nos. 92270001 (Z. X.), 12371511 (Z. X.), 12101402 (Y. Z.), 12101401 (T. L.), the Lingang Laboratory Grant No. LG-QS-202202-08 (Y. Z.), the Shanghai Municipal Science and Technology Key Project No. 22JC1401500 (T. L.), the Shanghai Municipal of Science and Technology Major Project No. 2021SHZDZX0102, and the HPC of School of Mathematical Sciences and the Student Innovation Center, and the Siyuan-1 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University.

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Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

Zhi-Qin John Xu, Yaoyu Zhang & Tao Luo

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, China

Shanghai Artificial Intelligence Laboratory, Shanghai, 200232, China

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This paper is dedicated to the memory of Professor Zhong-Ci Shi.

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Xu, ZQ.J., Zhang, Y. & Luo, T. Overview Frequency Principle/Spectral Bias in Deep Learning. Commun. Appl. Math. Comput. (2024). https://doi.org/10.1007/s42967-024-00398-7

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Received : 02 August 2023

Revised : 26 February 2024

Accepted : 04 March 2024

Published : 04 September 2024

DOI : https://doi.org/10.1007/s42967-024-00398-7

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• Neural network
• Frequency principle (F-Principle)
• Deep learning
• Generalization
• Optimization

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