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Primary Maths (Grades 4 and 5) - Free Questions and Problems With Answers

Grade 4 and 5 maths questions and problems to test the understanding of maths concepts and procedures are presented. Answers to the questions are provided and located at the end of each page. Online calculators to check your answers are provided at the bottom of this page.

Online Calculators and Solvers

• Mental Math Calculator For Basic Operations
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• Problems - Grade 4 Maths Questions With Answers . Also Solutions and explanations to are included.
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• Numbers - Grade 4 Maths Questions With Answers
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• Fractions - Grade 4 Maths Questions With Answers . Also Solutions and explanations are included.
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• Grade 5 Math Practice Test Questions and their Solutions .
• Maths Word Problems with Answers for Grade 5 . Also Solutions and explanations are included.
• Grade 5 Number Patterns Quiz .
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• Add Fractions and Mixed Numbers - Examples and Questions with Answers (Grade 5)
• Rectangular Coordinate Systems
• Find the LCM and the GCF of Integers - Examples and Questions with Answers (Grade 5)
• Grade 5 Maths Questions and Problems With Answers on Lowest Common Multiple
• Fractions - Grade 5 Maths Questions With Answers . Also Solutions and explanations are included.
• interactive tutorial on fractions Explore fractions interactively using an applet.
• interactive tutorial on equivalent fractions Explore equivalent fractions interactively using an applet.
• Exponents - Grade 5 Maths Questions With Answers
• Grade 5 Questions with Answers on Prime and Composite Numbers
• How to Find the Perimeter of a Rectangle?
• How to Find the Area of a Rectangle?
• Perimeter - Grade 5 Maths Questions With Answers . Also Solutions and explanations are included.
• Convert Mixed Numbers to Fractions - Examples and Questions with Answers (Grade 5)
• Ratios - Grade 5 Maths Questions With Answers

Online Calculators

• Common Factors Calculator. An online calculator that computes the common factors and the GCF of two or more positive integers.
• Divisibility Test Calculator. An online calculator that tests whole numbers for visibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13.
• Add, Subtract and Multiply Integers Calculators. Three separate online calculators to add, subtract and multiply integers. Quotient and Remainder Calculator. An online calculator that computes the quotient and remainder of the division of two whole numbers.
• Lowest Common Multiple (lcm) Calculator. Calculate the lowest common multiple of two positive integers.
• Greatest Common Factor (gcf) Calculator. Calculate the greatest common factor of two positive integers.
• Prime Factors Calculator. Factor a positive integer into prime factors.
• Fractions Addition Calculator. Add 2 or 3 fractions and reduce the final answer.
• Multiply Fractions Calculator. Multiply 2 fractions and reduce the answer.
• Divide Fractions Calculator. Divide 2 fractions and reduce the answer.
• Reduce Fractions Calculator. Rewrite Fractions in reduced form.

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Problem Solving

 A selection of resources containing a wide range of open-ended tasks, practical tasks, investigations and real life problems, to support investigative work and problem solving in primary mathematics.

Problem Solving in Primary Maths - the Session

Quality Assured Category: Mathematics Publisher: Teachers TV

In this programme shows a group of four upper Key Stage Two children working on a challenging problem; looking at the interior and exterior angles of polygons and how they relate to the number of sides. The problem requires the children to listen to each other and to work together co-operatively. The two boys and two girls are closely observed as they consider how to tackle the problem, make mistakes, get stuck and arrive at the "eureka" moment. They organise the data they collect and are then able to spot patterns and relate them to the original problem to find a formula to work out the exterior angle of any polygon. At the end of the session the children report back to Mark, explaining how they arrived at the solution, an important part of the problem solving process.

In a  second video  two maths experts discuss some of the challenges of teaching problem solving. This includes how and at what stage to introduce problem solving strategies and the appropriate moment to intervene when children find tasks difficult. They also discuss how problem solving in the curriculum also helps to develop life skills.

Cards for Cubes: Problem Solving Activities for Young Children

Quality Assured Category: Mathematics Publisher: Claire Publications

This book provides a series of problem solving activities involving cubes. The tasks start simply and progress to more complicated activities so could be used for different ages within Key Stages One and Two depending on ability. The first task is a challenge to create a camel with 50 cubes that doesn't fall over. Different characters are introduced throughout the book and challenges set to create various animals, monsters and structures using different numbers of cubes. Problems are set to incorporate different areas of mathematical problem solving they are: using maths, number, algebra and measure.

Problem solving with EYFS, Key Stage One and Key Stage Two children

Quality Assured Category: Computing Publisher: Department for Education

These three resources, from the National Strategies, focus on solving problems.

  Logic problems and puzzles  identifies the strategies children may use and the learning approaches teachers can plan to teach problem solving. There are two lessons for each age group.

Finding all possibilities focuses on one particular strategy, finding all possibilities. Other resources that would enhance the problem solving process are listed, these include practical apparatus, the use of ICT and in particular Interactive Teaching Programs .

Finding rules and describing patterns focuses on problems that fall into the category 'patterns and relationships'. There are seven activities across the year groups. Each activity includes objectives, learning outcomes, resources, vocabulary and prior knowledge required. Each lesson is structured with a main teaching activity, drawing together and a plenary, including probing questions.

Primary mathematics classroom resources

Quality Assured Collection Category: Mathematics Publisher: Association of Teachers of Mathematics

This selection of 5 resources is a mixture of problem-solving tasks, open-ended tasks, games and puzzles designed to develop students' understanding and application of mathematics.

Thinking for Ourselves: These activities, from the Association of Teachers of Mathematics (ATM) publication 'Thinking for Ourselves’, provide a variety of contexts in which students are encouraged to think for themselves. Activity 1: In the bag – More or less requires students to record how many more or less cubes in total...

8 Days a Week: The resource consists of eight questions, one for each day of the week and one extra. The questions explore odd numbers, sequences, prime numbers, fractions, multiplication and division.

Number Picnic: The problems make ideal starter activities

Matchstick Problems: Contains two activities concentrating upon the process of counting and spotting patterns. Uses id eas about the properties of number and the use of knowledge and reasoning to work out the rules.

Colours: Use logic, thinking skills and organisational skills to decide which information is useful and which is irrelevant in order to find the solution.

GAIM Activities: Practical Problems

Quality Assured Category: Mathematics Publisher: Nelson Thornes

Designed for secondary learners, but could also be used to enrich the learning of upper primary children, looking for a challenge. These are open-ended tasks encourage children to apply and develop mathematical knowledge, skills and understanding and to integrate these in order to make decisions and draw conclusions.

Examples include:

*Every Second Counts - Using transport timetables, maps and knowledge of speeds to plan a route leading as far away from school as possible in one hour.

*Beach Guest House - Booking guests into appropriate rooms in a hotel.

*Cemetery Maths - Collecting relevant data from a visit to a local graveyard or a cemetery for testing a hypothesis.

*Design a Table - Involving diagrams, measurements, scale.

Go Further with Investigations

Quality Assured Category: Mathematics Publisher: Collins Educational

A collection of 40 investigations designed for use with the whole class or smaller groups. It is aimed at upper KS2 but some activities may be adapted for use with more able children in lower KS2. It covers different curriculum areas of mathematics.

Starting Investigations

The forty student investigations in this book are non-sequential and focus mainly on the mathematical topics of addition, subtraction, number, shape and colour patterns, and money.

The apparatus required for each investigation is given on the student sheets and generally include items such as dice, counters, number cards and rods. The sheets are written using as few words as possible in order to enable students to begin working with the minimum of reading.

NRICH Primary Activities

Explore the NRICH primary tasks which aim to enrich the mathematical experiences of all learners. Lots of whole class open ended investigations and problem solving tasks. These tasks really get children thinking!

Mathematical reasoning: activities for developing thinking skills

Quality Assured Category: Mathematics Publisher: SMILE

Problem Solving 2

Reasoning about numbers, with challenges and simplifications.

Quality Assured Category: Mathematics Publisher: Department for Education

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Home > Learning Resources

KS1 and KS2 Maths – Problem solving

• Author: Mike Askew
• Main Subject: CPD
• Subject: Maths
• Date Posted: 20 June 2012

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If children use well worn techniques to solve problems without understanding or modelling the context, their maths skills won't fully evolve

Children, from birth, are proficient problem solvers. By the age of two or three they have solved what are probably life’s two biggest problems - how to walk and how to talk. As they get older they solve practical problems, such as sharing a bag of sweets fairly with others, long before they’ve heard of division. In this article, I look at how we can build on this natural propensity to solve problems in teaching mathematics.

Routine and non-routine problems

Routine problems are problems children know how to solve based on their previous experiences. The sort of thinking required by routine problems can be described as reproductive: the child only needs to recall or reproduce a procedure or method they have previously learnt. A problem like ‘Apples’, for example (see below), is likely to be a routine problem for most children at the upper end of primary school; they know to multiply the two numbers together without having to think deeply about what operation to use.

• Apples on a supermarket shelf are in bags of eight. • If Jane buys six bags, how many apples is that?

In contrast, non-routine problems are where the learner does not immediately have a solution tucked under his belt. The problem solver has to put some effort into understanding the problem and creating, rather than recalling, a solution strategy. Non-routine problems engage learners in productive thinking.

We often think of non-routine problems as needing to be unusual or not having, to us as adults, an immediately obvious method of solution. ‘Stamps’ is typical of this type of non-routine problem.

• Clearing out a desk draw I found a collection of 5p and 6p stamps. • I have a parcel to post that needs 58p worth of stamps on it. • Can I create this exactly using the stamps I found? • If so, is there more than one way of doing this?

In choosing problems to work with, we need to decide whether or not we think a problem will be routine or non-routine for the particular children working on it. In the rest of this article, the problems chosen are being treated as though they are non-routine problems for the children working on them. That’s not to say that I don’t think routine problems have a place in the curriculum - they do. Here, however, I want to deal with some of the issues around teaching and learning non-routine problems.

The importance of context

• Four hungry girls share three pizzas equally. • Eight hungry boys share six pizzas equally. • Does each girl get more pizza than each boy, less or the same?

As a routine problem, the ‘story’ of pizzas and hungry children doesn’t serve any real purpose: children quickly learn to disregard the context, to strip out the mathematics and to work some procedure. The problem could just as easily have been put in the context of builders sharing bricks and many learners would happily say each builder would get 3/4 of a brick, without stopping to question the near impossibility of sharing out bricks.

We can, however, treat ‘Pizzas’ as a non-routine problem and use it to introduce children to thinking about fractions and equivalences. The context of hungry children and pizzas then becomes important. It is not chosen simply to be window-dressing for a fraction calculation. Nor are pizzas chosen because children are intrinsically motivated by food, making the unpalatable topic of fractions digestible. No, the context s chosen because children know about fair shares and slicing up pizzas - they can solve this problem without any formal knowledge of fractions. As the researcher Terezhina Nunes once pointed out, young children would not be able to solve the ‘bald’ calculation 3 divided by 4 but, “show me four young children who, given three bars of chocolate to share out fairly, hand the bars back saying ‘it can’t be done.”

Children have ‘action schemas’ for solving problem like ‘Pizzas’ - they can find ways to solve this with objects, pictures, diagrams and, eventually, symbols. Teaching can then build on the children’s informal solutions to draw out the formal mathematics of fractions. From being one of 20 ‘problems’ on a worksheet to complete in a lesson, ‘Pizzas’ can become a ‘rich task’ taking up the best part of a lesson, if children work on it in pairs and carefully selected solutions are then shared with the class.

Creating mathematical models

Part of the productive thinking in working on rich, non-routine problems requires children to create mathematical models, and we can teach to support this.

• At the supermarket Myprice, milk costs £1.08 per litre. • This is 7 pence less per litre than milk costs at Locost. • How much does 5 litres of milk cost at Locost?

What is missing from this approach is attention to setting up an appropriate model of the problem. Ultimately this could be a mental model of the problem context, but it helps initially to encourage children to put something on paper that can be shared and discussed. In problems involving quantities, like ‘Milk’, simple bar diagrams can help children create the appropriate model. These help children examine the relationships between the quantities (as opposed to simply fixing on specific numbers and keywords).

Setting up a diagrammatic model begins with creating a representation of what is known in the situation. In this example, we know milk at Myprice costs £1.08, so a diagram for this would look like:

MYPRICE £1.08

This provides the basis for talking about what the picture for the price of milk at Locost is going to be. Will the bar be longer or shorter? Where is the bar for the 7 pence to be drawn?

Two different models can be set up and children asked to describe the relationship between the prices at the two supermarkets, to see which diagram fits with the information in the problem. If the diagram for the price at Locost is shorter by 7, then two statements can be made:

MYPRICE £1.08p LOCOST     7p

• Myprice milk costs 7 pence more than milk at Locost.

• Locost milk costs 7 pence less than milk at Myprice.

In comparison, making the bar for milk at Locost longer by 7 gives different comparative statements:

MYPRICE £1.08p 7p LOCOST    

• Myprice milk costs 7 pence less than milk at Locost.

• Locost milk costs 7 pence more than milk at Myprice.

Children can then talk about which of these situations fits with the wording in the problem.

Having established that Locost milk must be £1.15 a litre, children can go on to produce the bar diagram model for this.

Supporting non-routine problem solving

Where the problems were played out as non-routine, three factors identified are worth noting. First, in choosing the tasks, the teachers made sure they would build on learners’ prior knowledge - as I suggest a problem like ‘Pizzas’ can. Second, in contrast to focusing on getting the answer, the researchers observed what they called ‘sustained pressure for explanation and meaning’. In other words, the teachers pressed for children to explain what and why they were doing what they were doing rather than simply focusing on whether or not they had got the correct answer. Third, the amount of time children were allowed to work on the problem was neither too long or too short: children need enough time to ‘get into’ a problem, but too much time can lead to a loss of engagement.

Share good practice

Gather together a collection of problems covering all the years of education in your school (or ask teachers to each contribute two or three problems).

Working together in small groups, teachers sort the problems into three groups:

1. Problems they think would be routine for the children they teach 2. Problems they think would be non-routine for their children 3. Problems they think would be much too difficult for their age group

Everyone agrees to try out a problem from group 2 with their class. Discuss how too much focus on getting the answer can reduce the challenge and stress the importance of pressing children to explain their working. At a subsequent meeting, people report back, focusing in particular on strategies they used to keep the problem solving non-routine.

About the author

Mike Askew is Professor of Primary Education at Monash University, Melbourne. Until recently, he was Professor of Mathematics Education at King’s College, University and Director of BEAM.

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Fluency, reasoning and problem solving in primary maths

Australia and new zealand, primary maths, tes resources team.

Develop fluency, reasoning and problem solving in Maths with the mastery approach

The skills of fluency, reasoning and problem solving are well-known to all primary Maths teachers, and in mastery teaching they play an essential role. To help you develop your mastery approach, we have hand-picked this selection of Maths resources, with everything from reasoning lessons and problem solving worksheets, to daily fluency questions and independent investigations. So, why not take a look?

To find out more about Maths mastery explore Teaching for mastery in primary maths .

Reasoning and Problem Solving Questions Collection - KS1 and KS2

Maths Mastery Times Tables Activities.

Length and Perimeter

Division Investigation

Compare and order negative integers and decimals - mastery worksheet

KS2: Time - Problem Solving

Measures Problem Solving Worksheet

Fluency: Bridging (Sample Set)

Daily Maths Fluency - Year 5 - NEW - 6 weeks of Questions and Answers

Decimals Reasoning and Problem Solving

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Some students may need support to learn effective problem-solving skills. This resource can assist students to think of and evaluate options to a problem or situation. 

You can encourage and support students to use this tool to:

- come up with two options

- write the pros and cons of each option, and

- implement the option they think is best. 

In high school settings, some students may respond better to a short conversation. For these students, you can use the first page of the guide as a prompt sheet to facilitate talking through a problem. Short notes in a workbook of a student’s choosing as a reminder of decisions made may also be helpful.

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Primary teachers, SLSOs

This resource can be used to support students to think of and evaluate options to a problem or situation. It includes a template for students to consider and compare two potential solutions.

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• Primary 5 Math

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P5 Math Problem-solving Technique: Common Difference

Posted by Christine Tham | May 16, 2024 | Primary 5 Math | 0 |

Introduction

Welcome to another P5 Math blog post!

We previously talked about the Gap & Difference Method — the keywords to look out for and a step-by-step guide to solving problems using this method.

In this blog post, we will be discussing another P5 Math Problem-solving Technique — the Common Difference Method .

Common Difference is one of the 4 Change Strategies.

🔎   What Are The 4 Change Strategies? 🔎 1️⃣ One Item Kept The Same 2️⃣ Difference Kept The Same (or Common Difference) 3️⃣ Total Kept The Same 4️⃣ All Items Changed

Let’s Take A Look At This Question

P5 Math Problem-solving Technique: Unit X Value Method

• Beginner’s Guide To Cross Multiplication

Identifying Keywords

To help us decide which problem-solving strategy to use in this question, let us first highlight the keywords.

In the first sentence, we can highlight the keywords Karen is 10 years older than her younger brother .

Moving on to the second sentence, the keywords are the total age in 12 years time is 50 .

Are we talking about the present time here? No. “In 12 years’ time” means 12 years later. So this means that 12 years later, their total age will be 50.

However, this is not what the question is asking. It wants us to compute Karen’s age 2 years ago .

Solving P5 Age Problems Using Common Difference Method

When solving Age Problem questions, we are going to use the Common Difference Strategy.

💡 What Is Common Difference? 💡 This means that there is a constant amount of change in the numbers.

To explain it further, let’s say that Mummy and you have a 25-year age difference, which means that Mummy is 25 years older than you.

Next year, that difference will remain the same because both of you will grow older by one year.

👩‍👦   Remember This When Solving An Age Question 👩‍👦 The age difference will always stay the same.

Thought Process

Let us draw a model to illustrate the question.

This figure illustrates the age of Karen and her brother now . Karen’s block is longer because she is older.

In 12 years’ time, Karen will be 12 years older so let us add 12 for her.

How about her brother? Will her brother’s age stay the same after 12 years? No. He will also be 12 years older.

Let’s compute for the difference in their age after 12 years .

The difference remains 10 years.

But can you easily visualise the 10-year difference when you look at the illustration above? It can be tricky, right? Let us redraw our model so we can see the difference better.

We added 12 years at the beginning of the illustration so you won’t get confused about the 10-year age difference.

Next, we know that the sum of their ages is 50.

Let us fill in the missing information in our table. We know that these are going to have the same value so let us write 1 unit for each.

Let us now solve it.

We can now put 8 each for Karen and her brother’s current age.

Next, let us compute Karen’s age.

Karen is 18 years old now.

However, the question wants to know Karen’s age two years ago. So let us subtract 2 from her current age.

Therefore, Karen was 16 years old two years ago .

How Can This Question Be Tested Differently?

There are different types of Change Strategies questions.

This question is not an Age Problem, but you can see that each of them gave away the same number of stickers , so just like in the previous question, the difference will always stay the same.

I hope that after reading this blog post, you have gained a better understanding of how to solve Age Problems using the Common Difference Method.

Make sure to read the given information in the question thoroughly to avoid missing out on important keywords.

Stay tuned for more Math problem-solving tips!

Christine Tham

As an Accountancy graduate from Singapore Management University (SMU) with a diploma in Early Childhood Care & Education from Asian International College, Ms. Christine is passionate about fostering confidence & independence in learning by helping students navigate the thought processes and answering techniques required to excel in the PSLE Math examination. With over 13 years of teaching experience, she provides a fun, positive and supportive learning environment with her warm & approachable personality, as she believes that all children have the potential to learn and succeed in their own way. Going beyond rote memorisation and emphasising the development of higher-order thinking skills, she encourages her students to analyse, evaluate and apply their knowledge to real-world situations, fostering a deeper understanding of mathematical concepts and their practical applications.

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120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

• Teaching Tools
• Subtraction
• Multiplication
• Mixed operations
• Ordering and number sense
• Comparing and sequencing
• Physical measurement
• Ratios and percentages
• Probability and data relationships

You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

120 Math word problems, categorized by skill

Addition word problems.

Best for: 1st grade, 2nd grade

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

Subtraction word problems

Best for: 1st grade, second grade

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

Practice math word problems with Prodigy Math

Join millions of teachers using Prodigy to make learning fun and differentiate instruction as they answer in-game questions, including math word problems from 1st to 8th grade!

12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

Multiplication word problems

Best for: 2nd grade, 3rd grade

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?

Division word problems

Best for: 3rd grade, 4th grade, 5th grade

22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?

23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?

24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?

25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

Mixed operations word problems

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

Fractions word problems

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

Decimals word problems

Best for: 4th grade, 5th grade

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

Comparing and sequencing word problems

Best for: Kindergarten, 1st grade, 2nd grade

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?

63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.

68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

Probability and data relationships word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

96. Introducing Perimeter:  The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

Variables word problems

Best for: 6th grade, 7th grade, 8th grade

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

How to easily make your own math word problems & word problems worksheets

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

• Link to Student Interests:  By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
• Make Questions Topical:  Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
• Include Student Names:  Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
• Be Explicit:  Repeating keywords distills the question, helping students focus on the core problem.
• Test Reading Comprehension:  Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
• Focus on Similar Interests:  Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
• Feature Red Herrings:  Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

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Using Questioning to Stimulate Mathematical Thinking

Types of questions, levels of mathematical thinking, combining the categories.

10 Helpful Worksheet Ideas for Primary School Math Lessons

M athematics is a fundamental subject that shapes the way children think and analyze the world. At the primary school level, laying a strong foundation is crucial. While hands-on activities, digital tools, and interactive discussions play significant roles in learning, worksheets remain an essential tool for reinforcing concepts, practicing skills, and assessing understanding. Here’s a look at some helpful worksheets for primary school math lessons.

Comparison Chart Worksheets

Comparison charts provide a visual means for primary school students to grasp relationships between numbers or concepts. They are easy to make at www.storyboardthat.com/create/comparison-chart-template , and here is how they can be used:

• Quantity Comparison: Charts might display two sets, like apples vs. bananas, prompting students to determine which set is larger.
• Attribute Comparison: These compare attributes, such as different shapes detailing their number of sides and characteristics.
• Number Line Comparisons: These help students understand number magnitude by placing numbers on a line to visualize their relative sizes.
• Venn Diagrams: Introduced in later primary grades, these diagrams help students compare and contrast two sets of items or concepts.
• Weather Charts: By comparing weather on different days, students can learn about temperature fluctuations and patterns.

Number Recognition and Counting Worksheets

For young learners, recognizing numbers and counting is the first step into the world of mathematics. Worksheets can offer:

• Number Tracing: Allows students to familiarize themselves with how each number is formed.
• Count and Circle: Images are presented, and students have to count and circle the correct number.
• Missing Numbers: Sequences with missing numbers that students must fill in to practice counting forward and backward.

Basic Arithmetic Worksheets

Once students are familiar with numbers, they can start simple arithmetic. 

• Addition and Subtraction within 10 or 20: Using visual aids like number lines, counters, or pictures can be beneficial.
• Word Problems: Simple real-life scenarios can help students relate math to their daily lives.
• Skip Counting: Worksheets focused on counting by 2s, 5s, or 10s.

Geometry and Shape Worksheets

Geometry offers a wonderful opportunity to relate math to the tangible world.

• Shape Identification: Recognizing and naming basic shapes such as squares, circles, triangles, etc.
• Comparing Shapes: Worksheets that help students identify differences and similarities between shapes.
• Pattern Recognition: Repeating shapes in patterns and asking students to determine the next shape in the sequence.

Measurement Worksheets

Measurement is another area where real-life application and math converge.

• Length and Height: Comparing two or more objects and determining which is longer or shorter.
• Weight: Lighter vs. heavier worksheets using balancing scales as visuals.
• Time: Reading clocks, days of the week, and understanding the calendar.

Data Handling Worksheets

Even at a primary level, students can start to understand basic data representation.

• Tally Marks: Using tally marks to represent data and counting them.
• Simple Bar Graphs: Interpreting and drawing bar graphs based on given data.
• Pictographs: Using pictures to represent data, which can be both fun and informative.

Place Value Worksheets

Understanding the value of each digit in a number is fundamental in primary math.

• Identifying Place Values: Recognizing units, tens, hundreds, etc., in a given number.
• Expanding Numbers: Breaking down numbers into their place value components, such as understanding 243 as 200 + 40 + 3.
• Comparing Numbers: Using greater than, less than, or equal to symbols to compare two numbers based on their place values.

Fraction Worksheets

Simple fraction concepts can be introduced at the primary level.

• Identifying Fractions: Recognizing half, quarter, third, etc., of shapes or sets.
• Comparing Fractions: Using visual aids like pie charts or shaded drawings to compare fractions.
• Simple Fraction Addition: Adding fractions with the same denominator using visual aids.

Money and Real-Life Application Worksheets

Understanding money is both practical and a great way to apply arithmetic.

• Identifying Coins and Notes: Recognizing different denominations.
• Simple Transactions: Calculating change, adding up costs, or determining if there’s enough money to buy certain items.
• Word Problems with Money: Real-life scenarios involving buying, selling, and saving.

Logic and Problem-Solving Worksheets

Even young students can hone their problem-solving skills with appropriate challenges.

• Sequences and Patterns: Predicting the next item in a sequence or recognizing a pattern.
• Logical Reasoning: Simple puzzles or riddles that require students to think critically.
• Story Problems: Reading a short story and solving a math-related problem based on the context.

Worksheets allow students to practice at their own pace, offer teachers a tool for assessment, and provide parents with a glimpse into their child’s learning progression. While digital tools and interactive activities are gaining prominence in education, the significance of worksheets remains undiminished. They are versatile and accessible and, when designed creatively, can make math engaging and fun for young learners.

The post 10 Helpful Worksheet Ideas for Primary School Math Lessons appeared first on Mom and More .

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Maths Word Problems

Want to help your child with their maths?  These activity sheets are a great way to practise problem solving with your child.  Some questions in the Mid Primary section could be used with younger students as an oral problem solving activity to model how to solve the problems. 

Mid Primary Problem-solving

Worksheet 1

Worksheet 2

Worksheet 3

Worksheet 4

Worksheet 5

Worksheet 6

Worksheet 7

Worksheet 8

Worksheet 9

Worksheet 10

Worksheet 11

Worksheet 12

Worksheet 13

Worksheet 14

Worksheet 15

Worksheet 16

Worksheet 17

Worksheet 18

Worksheet 19

Worksheet 20

Worksheet 21

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Worksheet 23

Worksheet 24

Worksheet 25

Worksheet 26

Worksheet 27

Worksheet 28

Worksheet 29

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Worksheet 31

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Worksheet 34

Worksheet 35

Upper Primary Problem-solving

Worksheet 18  

Extension Problem-solving

© 2024 Castletown Primary School

MindMap Gallery Problem solving, transfer and creativity among primary school students

Problem solving, transfer and creativity among primary school students

This is a mind map about primary school students' problem solving, transfer and creativity. The main content includes: creativity and its measurement, problem solving, transfer and its main types.

• Recommended to you

Migration and main types

Transfer meaning: refers to the impact of one type of learning on another type of learning, or the impact of acquired experience on the completion of other activities

Type of migration

Migration nature: positive migration, negative migration

Different levels of abstraction and generalization of migrating content: horizontal migration and vertical migration

Time sequence of migration: forward migration and reverse migration

Differences in migration content: general migration and specific migration

Migration scope: near migration and far migration

Basic theory of migration

Form training theory: Learning content is not important, form and training are important

Common elements theory (Thorndike and Woodworth)

The Theory of Generalization of Experience (Judd) - Target Shooting Experiment in Water

Relationship Transformation Theory (Gestalt Psychologist-Koehler)

Cognitive structure theory (Ausubel)

Promote the migration of primary school students

Main factors affecting migration

Similarity of learning materials (Thorndike-Common Elements Theory)

Similarity in structural properties and similarity in surface properties (Jick and Holyoke)

Original cognitive structure (Ausubel)

Learning set (set: a state of mental readiness or tendency to an activity caused by previous activities)

Teaching Strategies to Promote Primary School Students’ Migration

Selected teaching material content (the most basic and typical factual materials; variations and counterexamples)

Reasonably arrange teaching content (from general to individual, from overall to details; comprehensive and thorough)

Reasonable handling of teaching procedures (macro - the overall arrangement of the learning sequence; micro - the specific arrangement of each lesson)

teach learning methods

Raise students’ awareness of migration

problem solved

Meaning: A problem usually refers to a difficult situation in which an individual has no ready-made solution to the current task.

Type of question

Well-defined problems (well-structured problems): usually means that the starting state, goal state and solution of the problem are determined

Poorly defined problems (poorly structured problems): refer to the problem where the starting state, goal state and solution are uncertain

Problem-solving characteristics: purposefulness, cognition, sequence

problem solving model

Dewey: Problems, Analysis, Hypothesis, Testing and Evaluation, Conclusion

Wallace: Preparation, Incubation, Clarity, Verification

Dunkel: Scope reduction aggregation model

The first stage: general scope

Phase 2: Functional solution

Stage Three: Special Solutions

Newell and Simon's information processing model, four steps

Glass: The modern cognitive model

Develop an initial representation of the problem

Specify a plan to resolve the problem

Reconstruct problem representation

Execution plan and inspection results

Improve primary school students’ problem-solving skills

Main factors affecting primary school students’ problem solving

objective external environmental factors

How the problem is presented

The intuitiveness and specificity of the problem

The amount of information provided in the question

personal reason

Function fixation (Dunkel): When a person sees an item, his thinking is often bound by its usual use, and it is difficult to think of other functions of the item.

Thinking set: "Inertial thinking" is a special state of mental preparation for activities caused by previous activities, or the tendency of activities

brewing effect

reserve of knowledge

cognitive strategies

Cultivation of primary school students’ problem-solving abilities

Improve students’ knowledge reserve level

Combining the teaching content of various subjects to train students’ problem-solving abilities

Cultivate students’ habit of thinking about problems

Adopt scientific problem-solving training methods

Creativity and its measurement

Concept: Creativity refers to an individual’s personality traits or tendencies to produce unique, socially valuable products

Main factors affecting primary school students' creativity

Intelligence (most creative people have IQs between 110 and 130; people with IQs above 130 have very scattered levels of creativity, ranging from high to low; people with low IQs are also less creative)

knowledge experience

personality and motivational factors

envirnmental factor

Cultivation of creativity among primary school students

Training methods for primary school students’ creative thinking

Brain stimulation method (brainstorming method), social stimulation

metaphor training method

Multiple Strategies (Guilford)

Teaching suggestions for cultivating creativity among primary school students

Create an independent and relaxed learning atmosphere

Pay attention to the development of children's intuitive thinking and divergent thinking

Encourage hands-on experience and learn from creative role models

Map Options

PISA Scores by Country 2024

Every three years in the worldwide education systems, OECD conducts PISA to test the critical thinking of 15-year-olds in math, science, and reading in over 65 countries that make up 90% of the world’s economies.

What Is PISA? Why Does It Matter?

Programme for International Student Assessment (PISA) is a global program whereby around 600,000 15 year students from different countries are put under a two-hour test to gauge their skills and knowledge, mainly in science, reading, and mathematics. PISA test questions don't gauge memorization of facts but demand that students draw on real-world problem-solving skills and knowledge.

Since a high ranking on PISA corresponds to economic success, researchers have concluded that PISA is one of the indicators of whether school systems are preparing students for the 21st-century global knowledge economy.

U.S. critics of PISA state that the state has a higher percentage of disadvantaged children and therefore cannot fair as well as other OECD countries on the test. Data from the OECD shows that the U.S actually has around the OECD average of disadvantaged students.

However, the U.S has more students from socio-economically advantaged backgrounds. Meaning students in the U.S are better off than the average OECD country.

Typically, PISA is on a three-yearly cycle, but due to the COVID-19 pandemic, PISA 2021 was postponed to 2022.

85 countries are taking part in PISA 2022, intended to take place from 26 July to 16 September 2022. Three key learning subject areas are tested: reading literacy, mathematics, and science. These are accompanied by an innovative domain that varies from cycle to cycle. Mathematics is the key focus, while creative thinking is the innovative domain.

Around 6,300 students aged between 15-years-3-months and 16-years-2-months at the time of assessment will participate in the PISA 2022 Main Survey, conducted under the auspices of the OECD.

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Which country scores highest in PISA?

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• PISA 2022 Results - OECD

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The Possible Collapse of the U.S. Home Insurance System

A times investigation found climate change may now be a concern for every homeowner in the country..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Sabrina Tavernise. And this is “The Daily.”

[MUSIC PLAYING]

Today, my colleague, Christopher Flavelle, on a “Times” investigation into one of the least known and most consequential effects of climate change — insurance — and why it may now be a concern for every homeowner in the country.

It’s Wednesday, May 15.

So, Chris, you and I talked a while ago about how climate change was really wreaking havoc in the insurance market in Florida. You’ve just done an investigation that takes a look into the insurance markets more broadly and more deeply. Tell us about it.

Yeah, so I cover climate change, in particular the way climate shocks affect different parts of American life. And insurance has become a really big part of that coverage. And Florida is a great example. As hurricanes have gotten worse and more frequent, insurers are paying out more and more money to rebuild people’s homes. And that’s driving up insurance costs and ultimately driving up the cost of owning a home in Florida.

So we’re already seeing that climate impact on the housing market in Florida. My colleagues and I started to think, well, could it be that that kind of disruption is also happening in other states, not just in the obvious coastal states but maybe even through the middle of the US? So we set out to find out just how much it is happening, how much that Florida turmoil has, in fact, become really a contagion that is spreading across the country.

So how did you go about reporting this? I mean, where did you start?

All we knew at the start of this was that there was reason to think this might be a problem. If you just look at how the federal government tracks disasters around the country, there’s been a big increase almost every year in the number and severity of all kinds of disasters around the country. So we thought, OK, it’s worth trying to find out, what does that mean for insurers?

The problem is getting data on the insurance industry is actually really hard. There’s no federal regulation. There’s no government agency you can go to that holds this data. If you talk to the insurers directly, they tend to be a little reluctant to share information about what they’re going through. So we weren’t sure where to go until, finally, we realized the best people to ask are the people whose job it is to gauge the financial health of insurance companies.

Those are rating agencies. In particular, there’s one rating company called AM Best, whose whole purpose is to tell investors how healthy an insurance company is.

Whoa. So this is way down in the nuts and bolts of the US insurance industry.

Right. This is a part of the broader economy that most people would never experience. But we asked them to do something special for us. We said, hey, can you help us find the one number that would tell us reporters just how healthy or unhealthy this insurance market is state by state over time? And it turns out, there is just such a number. It’s called a combined ratio.

OK, plain English?

Plain English, it is the ratio of revenue to costs, how much money these guys take in for homeowner’s insurance and how much they pay out in costs and losses. You want your revenue to be higher than your costs. If not, you’re in trouble.

So what did you find out?

Well, we got that number for every state, going back more than a decade. And what it showed us was our suspicions were right. This market turmoil that we were seeing in Florida and California has indeed been spreading across the country. And in fact, it turns out that in 18 states, last year, the homeowner’s insurance market lost money. And that’s a big jump from 5 or 10 years ago and spells real trouble for insurance and for homeowners and for almost every part of the economy.

So the contagion was real.

Right. This is our first window showing us just how far that contagion had spread. And one of the really striking things about this data was it showed the contagion had spread to places that I wouldn’t have thought of as especially prone to climate shocks — for example, a lot of the Midwest, a lot of the Southeast. In fact, if you think of a map of the country, there was no state between Pennsylvania and the Dakotas that didn’t lose money on homeowner’s insurance last year.

So just huge parts of the middle of the US have become unprofitable for homeowner’s insurance. This market is starting to buckle under the cost of climate change.

And this is all happening really fast. When we did the Florida episode two years ago, it was a completely new phenomenon and really only in Florida. And now it’s everywhere.

Yeah. And that’s exactly what’s so striking here. The rate at which this is becoming, again, a contagion and spreading across the country is just demolishing the expectations of anyone I’ve spoken to. No one thought that this problem would affect so much of the US so quickly.

So in these states, these new places that the contagion has spread to, what exactly is happening that’s causing the insurance companies to fold up shop?

Yeah. Something really particular is happening in a lot of these states. And it’s worth noting how it’s surprised everyone. And what that is, is formally unimportant weather events, like hailstorms or windstorms, those didn’t used to be the kind of thing that would scare insurance companies. Obviously, a big problem if it destroys your home or damages your home. But for insurers, it wasn’t going to wipe them out financially.

Right. It wasn’t just a complete and utter wipeout that the company would then have to pony up a lot of money for.

Exactly. And insurers call them secondary perils, sort of a belittling term, something other than a big deal, like a hurricane.

These minor league weather events.

Right. But those are becoming so frequent and so much more intense that they can cause existential threats for insurance companies. And insurers are now fleeing states not because of hurricanes but because those former things that were small are now big. Hailstorms, wildfires in some places, previous annoyances are becoming real threats to insurers.

Chris, what’s the big picture on what insurers are actually facing? What’s happening out there numbers-wise?

This is a huge threat. In terms of the number of states where this industry is losing money, it’s more than doubled from 10 years ago to basically a third of the country. The amount they’re losing is enormous. In some states, insurers are paying out $1.25 or even $1.50 for every dollar they bring in, in revenue, which is totally unsustainable.

And the result is insurers are making changes. They are pulling back from these markets. They’re hiking premiums. And often, they’re just dropping customers. And that’s where this becomes real, not just for people who surf balance sheets and trade in the stock market. This is becoming real for homeowners around the country, who all of a sudden increasingly can’t get insurance.

So, Chris, what’s the actual implication? I mean, what happens when people in a state can’t get insurance for their homes?

Getting insurance for a home is crucial if you want to sell or buy a home. Most people can’t buy a home without a mortgage. And banks won’t issue a mortgage without home insurance. So if you’ve got a home that insurance company doesn’t want to cover, you got a real problem. You need to find insurance, or that home becomes very close to unsellable.

And as you get fewer buyers, the price goes down. So this doesn’t just hurt people who are paying for these insurance premiums. It hurts people who want to sell their homes. It even could hurt, at some point, whole local economies. If home values fall, governments take in less tax revenue. That means less money for schools and police. It also means people who get hit by disasters and have to rebuild their homes all of a sudden can’t, because their insurance isn’t available anymore. It’s hard to overstate just how big a deal this is.

And is that actually happening, Chris? I mean, are housing markets being dragged down because of this problem with the insurance markets right now?

Anecdotally, we’ve got reports that in places like Florida and Louisiana and maybe in parts of California, the difficulty of getting insurance, the crazy high cost of insurance is starting to depress demand because not everyone can afford to pay these really high costs, even if they have insurance. But what we wanted to focus on with this story was also, OK, we know where this goes eventually. But where is it beginning? What are the places that are just starting to feel these shocks from the insurance market?

And so I called around and asked insurance agents, who are the front lines of this. They’re the ones who are struggling to find insurance for homeowners. And I said, hey, is there one place that I should go if I want to understand what it looks like to homeowners when all of a sudden insurance becomes really expensive or you can’t even find it? And those insurance agents told me, if you want to see what this looks like in real life, go to a little town called Marshalltown in the middle of Iowa.

We’ll be right back.

So, Chris, you went to Marshalltown, Iowa. What did you find?

Even before I got to Marshalltown, I had some idea I was in the right spot. When I landed in Des Moines and went to rent a car, the nice woman at the desk who rented me a car, she said, what are you doing here? I said, I’m here to write a story about people in Iowa who can’t get insurance because of storms. She said, oh, yeah, I know all about that. That’s a big problem here.

Even the rental car lady.

Even the rental car lady knew something was going on. And so I got into my rental car and drove about an hour northeast of Des Moines, through some rolling hills, to this lovely little town of Marshalltown. Marshalltown is a really cute, little Midwestern town with old homes and a beautiful courthouse in the town square. And when I drove through, I couldn’t help noticing all the roofs looked new.

What does that tell you?

Turns out Marshalltown, despite being a pastoral image of Midwestern easy living, was hit by two really bad disasters in recent years — first, a devastating tornado in 2018 and then, in 2020, what’s called a derecho, a straight-line wind event that’s also just enormously damaging. And the result was lots of homes in this small town got severely damaged in a short period of time. And so when you drive down, you see all these new roofs that give you the sense that something’s going on.

So climate had come to Marshalltown?

Exactly. A place that had previously seemed maybe safe from climate change, if there is such a thing, all of a sudden was not. So I found an insurance agent in Marshalltown —

We talked to other agents but haven’t talked to many homeowners.

— named Bobby Shomo. And he invited me to his office early one morning and said, come meet some people. And so I parked on a quiet street outside of his office, across the street from the courthouse, which also had a new roof, and went into his conference room and met a procession of clients who all had versions of the same horror story.

It was more — well more of double.

A huge reduction in coverage with a huge price increase.

Some people had faced big premium hikes.

I’m just a little, small business owner. So every little bit I do feel.

They had so much trouble with their insurance company.

I was with IMT Insurance forever. And then when I moved in 2020, Bobby said they won’t insure a pool.

Some people had gotten dropped.

Where we used to see carriers canceling someone for frequency of three or four or five claims, it’s one or two now.

Some people couldn’t get the coverage they needed. But it was versions of the same tale, which is all of a sudden, having homeowner’s insurance in Marshalltown was really difficult. But I wanted to see if it was bigger than just Marshalltown. So the next day, I got back in my car and drove east to Cedar Rapids, where I met another person having a version of the same problem, a guy named Dave Langston.

Tell me about Dave.

Dave lives in a handsome, modest, little townhouse on a quiet cul-de-sac on a hill at the edge of Cedar Rapids. He’s the president of his homeowners association. There’s 17 homes on this little street. And this is just as far as you could get from a danger zone. It looks as safe as could be. But in January, they got a letter from the company that insures him and his neighbors, saying his policy was being canceled, even though it wasn’t as though they’d just been hit by some giant storm.

So then what was the reason they gave?

They didn’t give a reason. And I think people might not realize, insurers don’t have to give a reason. Insurance policies are year to year. And if your insurance company decides that you’re too much of a risk or your neighborhood is too much of a risk or your state is too much of a risk, they can just leave. They can send you a letter saying, forget it. We’re canceling your insurance. There’s almost no protection people have.

And in this case, the reason was that this insurance company was losing too much money in Iowa and didn’t want to keep on writing homeowner’s insurance in the state. That was the situation that Dave shared with tens of thousands of people across the state that were all getting similar letters.

What made Dave’s situation a little more challenging was that he couldn’t get new insurance. He tried for months through agent after agent after agent. And every company told him the same thing. We won’t cover you. Even though these homes are perfectly safe in a safe part of the state, nobody would say yes. And it took them until basically two days before their insurance policy was going to run out until they finally found new coverage that was far more expensive and far more bare-bones than what they’d had.

But at least it was something.

It was something. But the problem was it wasn’t that good. Under this new policy, if Dave’s street got hit by another big windstorm, the damage from that storm and fixing that damage would wipe out all the savings set aside by these homeowners. The deductible would be crushingly high — $120,000 — to replace those roofs if the worst happened because the insurance money just wouldn’t cover anywhere close to the cost of rebuilding.

He said to me, we didn’t do anything wrong. This is just what insurance looks like today. And today, it’s us in Cedar Rapids. Everyone, though, is going to face a situation like this eventually. And Dave is right. I talked to insurance agents around the country. And they confirmed for me that this kind of a shift towards a new type of insurance, insurance that’s more expensive and doesn’t cover as much and makes it harder to rebuild after a big disaster, it’s becoming more and more common around the country.

So, Chris, if Dave and the people you spoke to in Iowa were really evidence that your hunch was right, that the problem is spreading and rapidly, what are the possible fixes here?

The fix that people seem most hopeful about is this idea that, what if you could reduce the risk and cause there to be less damage in the first place? So what some states are doing is they’re trying to encourage homeowners to spend more money on hardening their home or adding a new roof or, if it’s a wildfire zone, cut back the vegetation, things that can reduce your risk of having really serious losses. And to help pay for that, they’re telling insurers, you’ve got to offer a discount to people who do that.

And everyone who works in this field says, in theory, that’s the right approach. The problem is, number one, hardening a home costs a fantastic amount of money. So doing this at scale is hugely expensive. Number two, it takes a long time to actually get enough homes hardened in this way that you can make a real dent for insurance companies. We’re talking about years or probably decades before that has a real effect, if it ever works.

OK. So that sounds not particularly realistic, given the urgency and the timeline we’re on here. So what else are people looking at?

Option number two is the government gets involved. And instead of most Americans buying home insurance from a private company, they start buying it from government programs that are designed to make sure that people, even in risky places, can still buy insurance. That would be just a gargantuan undertaking. The idea of the government providing homeowner’s insurance because private companies can’t or won’t would lead to one of the biggest government programs that exists, if we could even do it.

So huge change, like the federal government actually trying to write these markets by itself by providing homeowner’s insurance. But is that really feasible?

Well, in some areas, we’re actually already doing it. The government already provides flood insurance because for decades, most private insurers have not wanted to cover flood. It’s too risky. It’s too expensive. But that change, with governments taking over that role, creates a new problem of its own because the government providing flood insurance that you otherwise couldn’t get means people have been building and building in flood-prone areas because they know they can get that guaranteed flood insurance.

Interesting. So that’s a huge new downside. The government would be incentivizing people to move to places that they shouldn’t be.

That’s right. But there’s even one more problem with that approach of using the government to try to solve this problem, which is these costs keep growing. The number of billion-dollar disasters the US experiences every year keeps going up. And at some point, even if the government pays the cost through some sort of subsidized insurance, what happens when that cost is so great that we can no longer afford to pay it? That’s the really hard question that no official can answer.

So that’s pretty doomsday, Chris. Are we looking at the end of insurance?

I think it’s fair to say that we’re looking at the end of insurance as we know it, the end of insurance that means most Americans can rest assured that if they get hit by a disaster, their insurance company will provide enough money they can rebuild. That idea might be going away. And what it shows is maybe the threat of climate change isn’t quite what we thought.

Maybe instead of climate change wrecking communities in the form of a big storm or a wildfire or a flood, maybe even before those things happen, climate change can wreck communities by something as seemingly mundane and even boring as insurance. Maybe the harbinger of doom is not a giant storm but an anodyne letter from your insurance company, saying, we’re sorry to inform you we can no longer cover your home.

Maybe the future of climate change is best seen not by poring over weather data from NOAA but by poring over spreadsheets from rating firms, showing the profitability from insurance companies, and how bit by bit, that money that they’re losing around the country tells its own story. And the story is these shocks are actually already here.

Chris, as always, terrifying to talk to you.

Always a pleasure, Sabrina.

Here’s what else you should know today. On Tuesday, the United Nations has reclassified the number of women and children killed in Gaza, saying that it does not have enough identifying information to know exactly how many of the total dead are women and children. The UN now estimates that about 5,000 women and about 8,000 children have been killed, figures that are about half of what it was previously citing. The UN says the numbers dropped because it is using a more conservative estimate while waiting for information on about 10,000 other dead Gazans who have not yet been identified.

And Mike Johnson, the Speaker of the House, gave a press conference outside the court in Lower Manhattan, where Michael Cohen, the former fixer for Donald Trump, was testifying for a second day, answering questions from Trump’s lawyers. Trump is bound by a gag order. So Johnson joined other stand-ins for the former president to discredit the proceedings. Johnson, one of the most important Republicans in the country, attacked Cohen but also the trial itself, calling it a sham and political theater.

Today’s episode was produced by Nina Feldman, Shannon Lin, and Jessica Cheung. It was edited by MJ Davis Lin, with help from Michael Benoist, contains original music by Dan Powell, Marion Lozano, and Rowan Niemisto, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Sabrina Tavernise. See you tomorrow.

• May 23, 2024   •   34:24 I.C.C. Prosecutor Requests Warrants for Israeli and Hamas Leaders
• May 22, 2024   •   23:20 Biden’s Open War on Hidden Fees
• May 21, 2024   •   24:14 The Crypto Comeback
• May 20, 2024   •   31:51 Was the 401(k) a Mistake?
• May 19, 2024   •   33:23 The Sunday Read: ‘Why Did This Guy Put a Song About Me on Spotify?’
• May 17, 2024   •   51:10 The Campus Protesters Explain Themselves
• May 16, 2024   •   30:47 The Make-or-Break Testimony of Michael Cohen
• May 15, 2024   •   27:03 The Possible Collapse of the U.S. Home Insurance System
• May 14, 2024   •   35:20 Voters Want Change. In Our Poll, They See It in Trump.
• May 13, 2024   •   27:46 How Biden Adopted Trump’s Trade War With China
• May 10, 2024   •   27:42 Stormy Daniels Takes the Stand
• May 9, 2024   •   34:42 One Strongman, One Billion Voters, and the Future of India

Hosted by Sabrina Tavernise

Featuring Christopher Flavelle

Produced by Nina Feldman ,  Shannon M. Lin and Jessica Cheung

Edited by MJ Davis Lin

With Michael Benoist

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Across the United States, more frequent extreme weather is starting to cause the home insurance market to buckle, even for those who have paid their premiums dutifully year after year.

Christopher Flavelle, a climate reporter, discusses a Times investigation into one of the most consequential effects of the changes.

On today’s episode

Christopher Flavelle , a climate change reporter for The New York Times.

Background reading

As American insurers bleed cash from climate shocks , homeowners lose.

See how the home insurance crunch affects the market in each state .

Here are four takeaways from The Times’s investigation.

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We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

Christopher Flavelle contributed reporting.

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30 Problem Solving Maths Questions And Answers For GCSE

Martin Noon

Problem solving maths questions can be challenging for GCSE students as there is no ‘one size fits all’ approach. In this article, we’ve compiled tips for problem solving, example questions, solutions and problem solving strategies for GCSE students. 

Since the current GCSE specification began, there have been many maths problem solving exam questions which take elements of different areas of maths and combine them to form new maths problems which haven’t been seen before. 

While learners can be taught to approach simply structured problems by following a process, questions often require students to make sense of lots of new information before they even move on to trying to solve the problem. This is where many learners get stuck.

How to teach problem solving

6 tips to tackling problem solving maths questions, 10 problem solving maths questions (foundation tier), 10 problem solving maths questions (foundation & higher tier crossover), 10 problem solving maths questions (higher tier).

In the Ofsted maths review , published in May 2021, Ofsted set out their findings from the research literature regarding the sort of curriculum and teaching that best supports all pupils to make good progress in maths throughout their time in school.

Regarding the teaching of problem solving skills, these were their recommendations:

• Teachers could use a curricular approach that better engineers success in problem-solving by teaching the useful combinations of facts and methods, how to recognise the problem types and the deep structures that these strategies pair to.
• Strategies for problem-solving should be topic specific and can therefore be planned into the sequence of lessons as part of the wider curriculum. Pupils who are already confident with the foundational skills may benefit from a more generalised process involving identifying relationships and weighing up features of the problem to process the information. 
• Worked examples, careful questioning and constructing visual representations can help pupils to convert information embedded in a problem into mathematical notation.
• Open-ended problem solving tasks do not necessarily mean that the activity is the ‘ideal means of acquiring proficiency’. While enjoyable, open ended problem-solving activities may not necessarily lead to improved results.  

Are you a KS2 teacher needing more support teaching reasoning, problem solving & planning for depth ? See this article for FREE downloadable CPD

There is no ‘one size fits all’ approach to successfully tackling problem solving maths questions however, here are 6 general tips for students facing a problem solving question:

• Read the whole question, underline important mathematical words, phrases or values.
• Annotate any diagrams, graphs or charts with any missing information that is easy to fill in.
• Think of what a sensible answer may look like. E.g. Will the angle be acute or obtuse? Is £30,000 likely to be the price of a coat?
• Tick off information as you use it.
• Draw extra diagrams if needed.
• Look at the final sentence of the question. Make sure you refer back to that at the end to ensure you have answered the question fully.

There are many online sources of mathematical puzzles and questions that can help learners improve their problem-solving skills. Websites such as NRICH and our blog on SSDD problems have some great examples of KS2, KS3 and KS4 mathematical problems.

Read more: KS2 problem solving and KS3 maths problem solving

In this article, we’ve focussed on GCSE questions and compiled 30 problem solving maths questions and solutions suitable for Foundation and Higher tier students. Additionally, we have provided problem solving strategies to support your students for some questions to encourage critical mathematical thinking . For the full set of questions, solutions and strategies in a printable format, please download our 30 Problem Solving Maths Questions, Solutions & Strategies.

30 Problem Solving Maths Questions, Solutions & Strategies

Help your students prepare for their maths GCSE with these free problem solving maths questions, solutions and strategies

These first 10 questions and solutions are similar to Foundation questions. For the first three, we’ve provided some additional strategies.

In our downloadable resource, you can find strategies for all 10 Foundation questions .

1) L-shape perimeter 

Here is a shape:

Sarah says, “There is not enough information to find the perimeter.”

Is she correct? What about finding the area?

• Try adding more information – giving some missing sides measurements that are valid. 
• Change these measurements to see if the answer changes.
• Imagine walking around the shape if the edges were paths. Could any of those paths be moved to another position but still give the same total distance?

The perimeter of the shape does not depend on the lengths of the unlabelled edges.

Edge A and edge B can be moved to form a rectangle, meaning the perimeter will be 22 cm. Therefore, Sarah is wrong.

The area, however, will depend on those missing side length measurements, so we would need more information to be able to calculate it.

2) Find the missing point

Here is a coordinate grid with three points plotted. A fourth point is to be plotted to form a parallelogram. Find all possible coordinates of the fourth point.

• What are the properties of a parallelogram?
• Can we count squares to see how we can get from one vertex of the parallelogram to another? Can we use this to find the fourth vertex?

There are 3 possible positions.

3) That rating was a bit mean!

The vertical line graph shows the ratings a product received on an online shopping website. The vertical line for 4 stars is missing.

If the mean rating is 2.65, use the information to complete the vertical line graph.

Strategies 

• Can the information be put into a different format, either a list or a table?
• Would it help to give the missing frequency an algebraic label, x ?
• If we had the data in a frequency table, how would we calculate the mean?
• Is there an equation we could form?

Letting the frequency of 4 star ratings be x , we can form the equation \frac{45+4x}{18+x} =2.65

Giving x=2 

4) Changing angles

The diagram shows two angles around a point. The sum of the two angles around a point is 360°.

Peter says “If we increase the small angle by 10% and decrease the reflex angle by 10%, they will still add to 360°.”

Explain why Peter might be wrong.

Are there two angles where he would be correct?

Peter is wrong, for example, if the two angles are 40° and 320°, increasing 40° by 10% gives 44°, decreasing 320° by 10% gives 288°. These sum to 332°.

10% of the larger angle will be more than 10% of the smaller angle so the sum will only ever be 360° if the two original angles are the same, therefore, 180°.

5) Base and power

The integers 1, 2, 3, 4, 5, 6, 7, 8 and 9 can be used to fill in the boxes. 

How many different solutions can be found so that no digit is used more than once?

There are 8 solutions.

6) Just an average problem 

Place six single digit numbers into the boxes to satisfy the rules.

The mean in maths is 5  \frac{1}{3}

The median is 5

The mode is 3.

How many different solutions are possible?

There are 4 solutions.

2, 3, 3, 7, 8, 9

3, 3, 4, 6, 7, 9

3, 3, 3, 7, 7, 9

3, 3, 3, 7, 8, 8

7) Square and rectangle  

The square has an area of 81 cm 2 . The rectangle has the same perimeter as the square.

Its length and width are in the ratio 2:1.

Find the area of the rectangle.

The sides of the square are 9 cm giving a perimeter of 36 cm. 

We can then either form an equation using a length 2x and width x . 

Or, we could use the fact that the length and width add to half of the perimeter and share 18 in the ratio 2:1. 

The length is 12 cm and the width is 6 cm, giving an area of 72 cm 2 .

8) It’s all prime

The sum of three prime numbers is equal to another prime number.

If the sum is less than 30, how many different solutions are possible?

There are 6 solutions. 

2 can never be used as it would force two more odd primes into the sum to make the total even.

9) Unequal share

Bob and Jane have £10 altogether. Jane has £1.60 more than Bob. Bob spends one third of his money. How much money have Bob and Jane now got in total?

Initially Bob has £4.20 and Jane has £5.80. Bob spends £1.40, meaning the total £10 has been reduced by £1.40, leaving £8.60 after the subtraction.

10) Somewhere between

Fred says, “An easy way to find any fraction which is between two other fractions is to just add the numerators and add the denominators.” Is Fred correct?

Solution 

Fred is correct. His method does work and can be shown algebraically which could be a good problem for higher tier learners to try.

If we use these two fractions \frac{3}{8} and \frac{5}{12} , Fred’s method gives us \frac{8}{20} = \frac{2}{5}

\frac{3}{8} = \frac{45}{120} , \frac{2}{5} = \frac{48}{120} , \frac{5}{12} = \frac{50}{120} . So \frac{3}{8} < \frac{2}{5} < \frac{5}{12}

The next 10 questions are crossover questions which could appear on both Foundation and Higher tier exam papers. We have provided solutions for each and, for the first three questions, problem solving strategies to support learners.

11) What’s the difference?

An arithmetic sequence has an nth term in the form an+b .

4 is in the sequence.

16 is in the sequence.

8 is not in the sequence.

-2 is the first term of the sequence.

What are the possible values of a and b ?

• We know that the first number in the sequence is -2 and 4 is in the sequence. Can we try making a sequence to fit? Would using a number line help?
• Try looking at the difference between the numbers we know are in the sequence.

If we try forming a sequence from the information, we get this:

We can now try to fill in the missing numbers, making sure 8 is not in the sequence. Going up by 2 would give us 8, so that won’t work.

The only solutions are 6 n -8 and 3 n -5.

12) Equation of the hypotenuse

The diagram shows a straight line passing through the axes at point P and Q .

Q has coordinate (8, 0). M is the midpoint of PQ and MQ has a length of 5 units.

Find the equation of the line PQ .

• We know MQ is 5 units, what is PQ and OQ ?
• What type of triangle is OPQ ?
• Can we find OP if we know PQ and OQ ?
• A line has an equation in the form y=mx+c . How can we find m ? Do we already know c ?

PQ is 10 units. Using Pythagoras’ Theorem OP = 6

The gradient of the line will be \frac{-6}{8} = -\frac{3}{4} and P gives the intercept as 6.

13) What a waste

Harry wants to cut a sector of radius 30 cm from a piece of paper measuring 30 cm by 20 cm. 

What percentage of the paper will be wasted?

• What information do we need to calculate the area of a sector? Do we have it all?
• Would drawing another line on the diagram help find the angle of the sector?

The angle of the sector can be found using right angle triangle trigonometry.

The angle is 41.81°.

This gives us the area of the sector as 328.37 cm 2 .

The area of the paper is 600 cm 2 .

The area of paper wasted would be 600 – 328.37 = 271.62 cm 2 .

The wasted area is 45.27% of the paper.

14) Tri-polygonometry

The diagram shows part of a regular polygon and a right angled triangle. ABC is a straight line. Find the sum of the interior angles of the polygon.

Finding the angle in the triangle at point B gives 30°. This is the exterior angle of the polygon. Dividing 360° by 30° tells us the polygon has 12 sides. Therefore, the sum of the interior angles is 1800°.

15) That’s a lot of Pi

A block of ready made pastry is a cuboid measuring 3 cm by 10 cm by 15 cm. 

Anne is making 12 pies for a charity event. For each pie, she needs to cut a circle of pastry with a diameter of 18 cm from a sheet of pastry 0.5 cm thick.

How many blocks of pastry will Anne need to buy?

The volume of one block of pastry is 450 cm 3 . 

The volume of one cylinder of pastry is 127.23 cm 3 .

12 pies will require 1526.81 cm 3 .

Dividing the volume needed by 450 gives 3.39(…). 

Rounding this up tells us that 4 pastry blocks will be needed.

16) Is it right?

A triangle has sides of (x+4) cm, (2x+6) cm and (3x-2) cm. Its perimeter is 80 cm.

Show that the triangle is right angled and find its area.

Forming an equation gives 6x+8=80

This gives us x=12 and side lengths of 16 cm, 30 cm and 34 cm.

Using Pythagoras’ Theorem

16 2 +30 2 =1156 

Therefore, the triangle is right angled.

The area of the triangle is (16 x 30) ÷ 2 = 240 cm 2 .

17) Pie chart ratio

The pie chart shows sectors for red, blue and green. 

The ratio of the angles of the red sector to the blue sector is 2:7. 

The ratio of the angles of the red sector to the green sector is 1:3. 

Find the angles of each sector of the pie chart.

Multiplying the ratio of red : green by 2, it can be written as 2:6. 

Now the colour each ratio has in common, red, has equal parts in each ratio.

The ratio of red:blue is 2:7, this means red:blue:green = 2:7:6.

Sharing 360° in this ratio gives red:blue:green = 48°:168°:144°.

18) DIY Simultaneously

Mr Jones buys 5 tins of paint and 4 rolls of decorating tape. The total cost was £167.

The next day he returns 1 unused tin of paint and 1 unused roll of tape. The refund amount is exactly the amount needed to buy a fan heater that has been reduced by 10% in a sale. The fan heater normally costs £37.50.

Find the cost of 1 tin of paint.    

The sale price of the fan heater is £33.75. This gives the simultaneous equations

p+t = 33.75 and 5 p +4 t = 167.

We only need the price of a tin of paint so multiplying the first equation by 4 and then subtracting from the second equation gives p =32. Therefore, 1 tin of paint costs £32. 

19) Triathlon pace

Jodie is competing in a Triathlon. 

A triathlon consists of a 5 km swim, a 40 km cycle and a 10 km run. 

Jodie wants to complete the triathlon in 5 hours. 

She knows she can swim at an average speed of 2.5 km/h and cycle at an average speed of 25 km/h. There are also two transition stages, in between events, which normally take 4 minutes each.

What speed must Jodie average on the final run to finish the triathlon in 5 hours?

Dividing the distances by the average speeds for each section gives times of 2 hours for the swim and 1.6 hours for the cycle, 216 minutes in total. Adding 8 minutes for the transition stages gives 224 minutes. To complete the triathlon in 5 hours, that would be 300 minutes. 300 – 224 = 76 minutes. Jodie needs to complete her 10 km run in 76 minutes, or \frac{19}{15} hours. This gives an average speed of 7.89 km/h.

20) Indices

a 2x × a y =a 3

(a 3 ) x ÷ a 4y =a 32

Find x and y .

Forming the simultaneous equations

Solving these gives

This final set of 10 questions would appear on the Higher tier only. Here we have just provided the solutions. Try asking your learners to discuss their strategies for each question.  

21) Angles in a polygon

The diagram shows part of a regular polygon.

A , B and C are vertices of the polygon. 

The size of the reflex angle ABC is 360° minus the interior angle.

Show that the sum of all of these reflex angles of the polygon will be 720° more than the sum of its interior angles.

Each of the reflex angles is 180 degrees more than the exterior angle: 180 + \frac{360}{n}

The sum of all of these angles is n (180 + \frac{360}{n} ). 

This simplifies to 180 n + 360

The sum of the interior angles is 180( n – 2) = 180 n – 360

The difference is 180 n + 360 – (180 n -360) = 720°

22) Prism and force (Non-calculator)

The diagram shows a prism with an equilateral triangle cross-section.

When the prism is placed so that its triangular face touches the surface, the prism applies a force of 12 Newtons resulting in a pressure of \frac{ \sqrt{3} }{4} N/m^{2}

Given that the prism has a volume of 384 m 3 , find the length of the prism.

Pressure = \frac{Force}{Area}

Area = 12÷ \frac{ \sqrt{3} }{4} = 16\sqrt{3} m 2

Therefore, the length of the prism is 384 ÷ 16\sqrt{3} = 8\sqrt{3} m

23) Geometric sequences (Non-calculator)

A geometric sequence has a third term of 6 and a sixth term of 14 \frac{2}{9}

Find the first term of the sequence.

The third term is ar 2 = 6

The sixth term is ar 5 = \frac{128}{9}

Diving these terms gives r 3 = \frac{64}{27}

Giving r = \frac{4}{3}

Dividing the third term twice by \frac{4}{3} gives the first term a = \frac{27}{8}

24) Printing factory

A printing factory is producing exam papers. When all 10 of its printers are working, it can produce all of the exam papers in 12 days.

For the first two days of printing, 3 of the printers are broken.

At the beginning of the third day it is discovered that 2 more printers have broken down, so the factory continues to print with the reduced amount of printers for 3 days. The broken printers are repaired and now all printers are available to print the remaining exams.

How many days in total does it take the factory to produce all of the exam papers?

If we assume one printer prints 1 exam paper per day, 10 printers would print 120 exam papers in 12 days. Listing the number printed each day for the first 5 days gives:

Day 5: 5 

This is a total of 29 exam papers.

91 exam papers are remaining with 10 printers now able to produce a total of 10 exam papers each day. 10 more days would be required to complete the job.

Therefore, 15 days in total are required.

25) Circles

The diagram shows a circle with equation x^{2}+{y}^{2}=13 .

A tangent touches the circle at point P when x=3 and y is negative.

The tangent intercepts the coordinate axes at A and B .

Find the length AB .

Using the equation  x^{2}+y^{2}=13 to find the y value for P gives y=-2 .

The gradient of the radius at this point is - \frac{2}{3} , giving a tangent gradient of \frac{3}{2} .

Using the point (3,-2) in y = \frac {3}{2} x+c gives the equation of the tangent as y = \frac {3}{2} x – \frac{13}{2}

Substituting x=0 and y=0 gives A and B as (0 , -\frac {13}{2}) and ( \frac{13}{3} , 0)

Using Pythagoras’ Theorem gives the length of AB as ( \frac{ 13\sqrt{13} }{6} ) = 7.812.

26) Circle theorems

The diagram shows a circle with centre O . Points A, B, C and D are on the circumference of the circle. 

EF is a tangent to the circle at A . 

Angle EAD = 46°

Angle FAB = 48°

Angle ADC = 78°

Find the area of ABCD to the nearest integer.

The Alternate Segment Theorem gives angle ACD as 46° and angle ACB as 48°.

Opposite angles in a cyclic quadrilateral summing to 180° gives angle ABC as 102°.

Using the sine rule to find AC will give a length of 5.899. Using the sine rule again to find BC will give a length of 3.016cm.

We can now use the area of a triangle formula to find the area of both triangles.

0.5 × 5 × 5.899 × sin (46) + 0.5 × 3.016 × 5.899 × sin (48) = 17 units 2 (to the nearest integer).

27) Quadratic function

The quadratic function f(x) = -2x^{2} + 8x +11 has a turning point at P .

Find the coordinate of the turning point after the transformation -f(x-3) .

There are two methods that could be used. We could apply the transformation to the function and then complete the square, or, we could complete the square and then apply the transformation.

Here we will do the latter.

This gives a turning point for f(x) as (2,19).

Applying -f(x-3) gives the new turning point as (5,-19).

28) Probability with fruit

A fruit bowl contains only 5 grapes and n strawberries.

A fruit is taken, eaten and then another is selected.

The probability of taking two strawberries is \frac{7}{22} .

Find the probability of taking one of each fruit. 

There are n+5 fruits altogether.

P(Strawberry then strawberry)= \frac{n}{n+5} × \frac{n-1}{n+4} = \frac{7}{22}

This gives the quadratic equation 15n^{2} - 85n - 140 = 0

This can be divided through by 5 to give 3n^{2} - 17n- 28 = 0

This factorises to (n-7)(3n + 4) = 0

n must be positive so n = 7.

The probability of taking one of each fruit is therefore, \frac{5}{12} × \frac{7}{11} + \frac {7}{12} × \frac {5}{11} = \frac {70}{132}

29) Ice cream tub volume

An ice cream tub in the shape of a prism with a trapezium cross-section has the dimensions shown. These measurements are accurate to the nearest cm.

An ice cream scoop has a diameter of 4.5 cm to the nearest millimetre and will be used to scoop out spheres of ice cream from the tub.

Using bounds find a suitable approximation to the number of ice cream scoops that can be removed from a tub that is full.

We need to find the upper and lower bounds of the two volumes. 

Upper bound tub volume = 5665.625 cm 3

Lower bound tub volume = 4729.375 cm 3

Upper bound scoop volume = 49.32 cm 3  

Lower bound scoop volume = 46.14 cm 3  

We can divide the upper bound of the ice cream tub by the lower bound of the scoop to get the maximum possible number of scoops. 

Maximum number of scoops = 122.79

Then divide the lower bound of the ice cream tub by the upper bound of the scoop to get the minimum possible number of scoops.

Minimum number of scoops  = 95.89

These both round to 100 to 1 significant figure, Therefore, 100 scoops is a suitable approximation the the number of scoops.

30) Translating graphs

 The diagram shows the graph of y = a+tan(x-b ).

The graph goes through the points (75, 3) and Q (60, q).

Find exact values of a , b and q .

The asymptote has been translated to the right by 30°. 

Therefore, b=30

So the point (45,1) has been translated to the point (75,3). 

Therefore, a=2

We hope these problem solving maths questions will support your GCSE teaching. To get all the solutions and strategies in a printable form, please download the complete resource .

Looking for additional support and resources at KS3? You are welcome to download any of the secondary maths resources from Third Space Learning’s resource library for free. There is a section devoted to GCSE maths revision with plenty of maths worksheets and GCSE maths questions . There are also maths tests for KS3, including a Year 7 maths test , a Year 8 maths test and a Year 9 maths test Other valuable maths practice and ideas particularly around reasoning and problem solving at secondary can be found in our KS3 and KS4 maths blog articles. Try these fun maths problems for KS2 and KS3, SSDD problems , KS3 maths games and 30 problem solving maths questions . For children who need more support, our maths intervention programmes for KS3 achieve outstanding results through a personalised one to one tuition approach.

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