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Maths problem solving questions – KS2 lesson plan

Zipped file containing 3 PDFs

The need to prepare children for their times tables test often means that we overlook building their reasoning and problem-solving skills. This is to allow time to develop number fluency and recall.

However, NRICH’s curriculum-linked activities enable your learners to become more fluent alongside developing their reasoning and problem-solving skills.

In this maths problem solving questions lesson you’ll work through NRICH’s Shape Times Shape worksheet to practise and develop some of these underused skills. 

Maths problem solving questions KS2 learning objectives

• Times tables facts  
• Reasoning skills  
• How to think about finding a way to a solution 
• The idea of a symbol (in this case a shape) representing a number 

Make sure all the children can see the Shape Times Shape worksheet. You could display it on the board or give out copies of the printable sheet.

Try to say very little as you introduce the task. Just review what the problem itself states, then give learners a few minutes to think on their own. 

NRICH is a maths outreach project, which is a collaboration between the Faculties of Education and Mathematics, based at the University of Cambridge. NRICH resources are free for teachers to use with their classes. 

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25 Fun Maths Problems For KS2 And KS3 (From Easy To Very Hard!)

Fun maths problems are one of the things mathematicians love about the subject; they provide an opportunity to apply mathematical knowledge, logic and problem solving skills all at once.  In this article, we’ve compiled 25 fun maths problems, each covering various topics and question types. They’re aimed at students in KS2 & KS3. We’ve categorised them as:

Maths word problems

Maths puzzles, fraction problems, multiplication and division problems, geometry problems, problem solving questions, maths puzzles are everywhere, how should teachers use these maths problems.

Teachers could make use of these maths problem solving questions in a number of ways, such as:

• embed into a relevant maths topic’s teaching.
• settling tasks at the beginning of lessons.
• break up or extend a maths worksheet.
• keep students thinking mathematically after the main lesson has finished.

Some are based on real life or historical maths problems, and some include ‘bonus’ maths questions to help to extend the problem solving fun! As you read through these problems, think about how you could adjust them to be relevant to your students or to practise different skills. 

These maths problems can also be used as introductory puzzles for maths games such as those introduced at the following links:

• KS2 maths games
• KS3 maths games

Need more support teaching reasoning, problem solving and planning for depth ? Read here for free CPD for you and your team of teachers.

1. Home on time – easy

Type: Time, Number, Addition

A cinema screening starts at 14:35. The movie lasts for 2 hours, 32 minutes after 23 minutes of adverts. It took 20 minutes to get to the cinema. What time should you tell your family that you’ll be home?

Answer: 17:50

2. A nugget of truth – mixed

Type: Times Tables, Multiplication, Multiples, Factors, Problem Solving 

Chicken nuggets come in boxes of 6, 9 or 20, so you can’t order 7 chicken nuggets. How many other impossible quantities can you find (not including fractions or decimals)?

Answer: 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, or 43

There is actually a theorem which can be used to prove that every integer quantity greater than 43 can be ordered.

3. A pet problem – mixed

Type: Number, Problem Solving, Forming and Solving Equations, Simultaneous Equations, Algebra

Eight of my pets aren’t dogs, five aren’t rabbits, and seven aren’t cats. How many pets do I have?

Answer: 10 pets (5 rabbits, 3 cats, 2 dogs)

4. The price of things – mixed

Type: lateral thinking problem

A mouse costs £10, a bee costs £15, and a spider costs £20. How much does a duck cost? Answer: £5 (£2.50 per leg)

Looking for more word problems, solutions and explanations? Read our article on word problems for primary school.

25 Fun Maths Problems - Printable

Download a printable version of these fun maths problems together with answers and mark scheme.

5. A dicey maths challenge – easy

Type: Place value, number, addition, problem solving

Roll three dice to generate three place value digits. What’s the biggest number you can make out of these digits? What’s the smallest number you can make?

Add these two numbers together. What do you get?

Answer: In most cases, 1,089.

Bonus: Who got a different result? Why?

6. PIN problem solving – mixed

Type: Logic, problem solving, reasoning

I’ve forgotten my PIN. Six incorrect attempts locks my account: I’ve used five! Two digits are displayed after each unsuccessful attempt: “2, 0” means 2 digits from that guess are in the PIN, but 0 are in the right place.

What should my sixth attempt be?

Answer: 6347

7. So many birds – mixed

Type: Triangular Numbers, Sequences, Number, Problem Solving

On the first day of Christmas my true love gave me one gift. On the second day they gave me another pair of gifts plus a copy of what they gave me on day one. On day 3, they gave me three new gifts, plus another copy of everything they’d already given me. If they keep this up, how many gifts will I have after twelve days?

Answer: 364

Bonus: This could be calculated as 1 + (1 + 2) + (1 + 2 + 3) + … but is there an easier way? What percentage of my gifts do I receive on each day?

8. I 8 sum maths questions – mixed

Type: Number, Place Value, Addition, Problem Solving, Reasoning

Using only addition and the digit 8, can you make 1,000? You can put 8s together to make 88, for example.

Answer: 888 + 88 + 8 + 8 + 8 = 1,000 Bonus: Which other digits allow you to get 1,000 in this way?

9. Quizzical – easy

Type: Fractions, Adding Fractions, Equivalent Fractions, Fractions to Percentages

4 friends entered a maths quiz. One answered \frac{1}{5} of the maths questions, one answered \frac{1}{10} , one answered \frac{1}{4} , and the other answered \frac{4}{25} . What percentage of the questions did they answer altogether?

Answer: 71%

10. Ancient problem solving – mixed

Type: Fractions, Reasoning, Problem Solving

Ancient Egyptians only used unit fractions (like \frac{1}{2} , \frac{1}{3} or \frac{1}{4} ). For \frac{2}{3} , they’d write \frac{1}{3} + \frac{1}{3} . How might they write \frac{5}{8} ?

Answer: \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} is correct. So is \frac{1}{2} + \frac{1}{8} .

Bonus: Which solution is better? Why? Can you find any more? What if subtractions are allowed?

Learn more about unit fractions here

11. everybody wants a pizza the action – hard.

An infinite number of mathematicians buy pizza. The first wants \frac{1}{2} pizza. The second wants \frac{1}{4} pizza. The third & fourth want \frac{1}{8} and \frac{1}{16} each, and so on. How many pizzas should they order?

Answer: 1 Each successive mathematician wants a slice that is exactly half of what is left:

12. Shade it black – hard

Type: Fractions, Reasoning, Problem Solving What fraction of this image is shaded black?

Answer: \frac{1}{3}

Look at the L-shaped part made up of two white and one black squares: \frac{1}{3} of this part is shaded. Zoom in on the top-right quarter of the image, which looks exactly the same as the whole image, and use the same reasoning to find what fraction of its L-shaped portion is shaded. Imagine zooming in to do the same thing again and again…

13. Giving is receiving – easy

Type: Number, Reasoning, Problem Solving

5 people give each other a present. How many presents are given altogether?

14. Sharing is caring – mixed

I have 20 sweets. If I share them equally with my friends, there are 2 left over. If one more person joins us, there are 6 sweets left. How many friends am I with?

Answer: 6 people altogether (so 5 friends!)

15. Times tables secrets – mixed

Type: Area, 2D Shape, Rectangles

Here are 77 letters:

BYHRCGNGNEOEAAHGHGCURPUTSTSASHHSBOBOREOPEEMEMEELATPEPEFADPHLTLTUT IEEOHOHLENRYTITIIAGBMTNTNFCGEIIGIG

How many different rectangular grids could you arrange all 77 letters into?

Answer: Four: 1⨉77, 77⨉1, 11⨉7 & 7⨉11. If the letters are arranged into one of these, a message appears, reading down each column starting from the top left.

Bonus: Can you find any more integers with the same number of factors as 77? What do you notice about these factors (think about prime numbers)? Can you use this system to hide your own messages?

16. Laugh it up – hard

Type: Multiples, Lowest Common Multiple, Times Tables, Division, Time

One friend jumps every \frac{1}{3} of a minute. Another jumps every 31 seconds. When will they jump together? Answer: After 620 seconds

17. Pictures of matchstick triangles – easy

Type: 2D Shapes, Equilateral Triangles, Problem Solving, Reasoning

Look at the matchsticks arranged below. How many equilateral triangles are there?

Answer: 13 (9 small, 3 medium, 1 large)

Bonus: What if the biggest triangle only had two matchsticks on each side? What if it had four?

18. Dissecting squares – mixed

Type: Reasoning, Problem Solving

What’s the smallest number of straight lines you could draw on this grid such that each square has a line going through it?

19. Make it right – mixed

Type: Pythagoras’ theorem

This triangle does not agree with Pythagoras’ theorem. 

Adding, subtracting, multiplying or dividing each of the side lengths by the same integer can fix it. What is the integer?

Answer: 3 

8 – 3 = 5

The new side lengths are 3, 4 and 5 and  32 + 42 = 52.

20. A most regular maths question – hard

Type: Polygons, 2D Shapes, tessellation, reasoning, problem-solving, patterns

What is the regular polygon with the largest number of sides that will self-tessellate?

Answer: Hexagon.

Regular polygons tessellate if one interior angle is a factor of 360°. The interior angle of a hexagon is 120°. This is the largest factor less than 180°.

21. Pleased to meet you – easy

Type: Number Problem, Reasoning, Problem-Solving

5 people meet; each shakes everyone else’s hand once. How many handshakes take place?

Person A shakes 4 people’s hands. Person B has already shaken Person A’s hand, so only needs to shake 3 more, and so on.

Bonus: How many handshakes would there be if you did this with your class?

22. All relative – easy

Type: Number, Reasoning, Problem-Solving

When I was twelve my brother was half my age. I’m 40 now, so how old is he?

23. It’s about time – mixed

Type: Time, Reasoning, Problem-Solving

When is “8 + 10 = 6” true?

Answer: When you’re telling the time (8am + 10 hours = 6pm)

24. More than a match – mixed

Type: Reasoning, Problem-Solving, Roman Numerals, Numerical Notation

Here are three matches:

How can you add two more matches, but get eight? Answer: Put the extra two matches in a V shape to make 8 in Roman Numerals:

25. Leonhard’s graph – hard

Type: Reasoning, Problem-Solving, Logic

Leonhard’s town has seven bridges as shown below. Can you find a route around the town that crosses every bridge exactly once?

Answer: No!

This is a classic real life historical maths problem solved by mathematician Leonhard Euler (rhymes with “boiler”). The city was Konigsberg in Prussia (now Kaliningrad, Russia). Not being able to find a solution is different to proving that there aren’t any! Euler managed to do this in 1736, practically inventing graph theory in the process.

Many of these 25 maths problems are rooted in real life, from everyday occurrences to historical events. Others are just questions that might arise if you say “what if…?”. The point is that although there are many lists of such problem solving maths questions that you can make use of, with a little bit of experience and inspiration you could create your own on almost any topic – and so could your students. 

For a kick-starter on creating your own maths problems, read our article on KS3 maths problem solving .

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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Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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KS2 - Problem Solving and Reasoning Questions

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

10 March 2023

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Just the type of problems I was looking for.

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Watch CBS News

How two high school students solved a 2,000-year-old math puzzle

By Bill Whitaker , Aliza Chasan , Sara Kuzmarov, Mariah Campbell

May 5, 2024 / 7:00 PM EDT / CBS News

A high school math teacher at St. Mary's Academy in New Orleans, Michelle Blouin Williams, was looking for ingenuity when she and her colleagues set a school-wide math contest with a challenging bonus question. That bonus question asked students to create a new proof for the Pythagorean Theorem, a fundamental principle of geometry, using trigonometry. The teachers weren't necessarily expecting anyone to solve it, as proofs of the Pythagorean Theorem using trigonometry were believed to be impossible for nearly 2,000 years.

But then, in December 2022, Calcea Johnson and Ne'Kiya Jackson, seniors at St. Mary's Academy, stepped up to the challenge. The $500 prize money was a motivating factor.

After months of work, they submitted their innovative proofs to their teachers. With the contest behind them, their teachers encouraged the students to present at a mathematics conference, and then to seek to publish their work. And even today, they're not done. Now in college, they've been working on further proofs of the Pythagorean Theorem and believe they have found five more proofs. Amazingly, despite their impressive achievements, they insist they're not math geniuses.

"I think that's a stretch," Calcea said.

The St. Mary's math contest

When the pair started working on the math contest they were familiar with the Pythagorean Theorem's equation: A² + B² = C², which explains that by knowing the length of two sides of a right triangle, it's possible to figure out the length of the third side.

When Calcea and Ne'Kiya set out to create a new Pythagorean Theorem proof, they didn't know that for thousands of years, one using trigonometry was thought to be impossible.  In 2009, mathematician Jason Zimba submitted one, and now Calcea and Ne'Kiya are adding to the canon.

Calcea and Ne'Kiya had studied geometry and some trigonometry when they started working on their proofs, but said they didn't feel math was easy. As the contest went on, they spent almost all their free time developing their ideas.

"The garbage can was full of papers, which she would, you know, work out the problems and if that didn't work, she would ball it up, throw it in the trash," Cal Johnson, Calcea's dad, said.

Neliska Jackson, Ne'Kiya's mother, says lightheartedly, that most of the time, her daughter's work was beyond her. 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

"Well, our teacher approached us and was like, 'Hey, you might be able to actually present this,'" Ne'Kiya said. "I was like, 'Are you joking?' But she wasn't. So we went. I got up there. We presented and it went well, and it blew up."

Why Calcea' and Ne'kiya's work "blew up"

The reaction was insane and unexpected, Calcea said. News of their accomplishment spread around the world. The pair got a write-up in South Korea and a shoutout from former first lady Michelle Obama. They got a commendation from the governor and keys to the city of New Orleans. 

Calcea and Ne'Kiya said they think there's several reasons why people found their work so impressive. 

"Probably because we're African American, one," Ne'Kiya said. "And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part."

Ne'Kiya said she'd like their accomplishment to be celebrated for what it is: "a great mathematical achievement."

In spite of the community's celebration of the students' work, St. Mary's Academy president and interim principal Pamela Rogers said that with recognition came racist calls and comments. 

"[People said] 'they could not have done it. African Americans don't have the brains to do it.' Of course, we sheltered our girls from that," Rogers said. "But we absolutely did not expect it to come in the volume that it came."

Rogers said too often society has a vision of who can be successful.

"To some people, it is not always an African American female," Rogers said. "And to us, it's always an African American female."

Success at St. Marys 

St. Mary's, a private Catholic elementary and high school, was started for young Black women just after the Civil War. Ne'Kiya and Calcea follow a long line of barrier-breaking graduates. Leah Chase , the late queen of Creole cuisine, was an alum. So was Michelle Woodfork, the first African American female New Orleans police chief, and Dana Douglas, a judge for the Fifth Circuit Court of Appeals. 

Math teacher Michelle Blouin Williams, who initiated the math contest, said Calcea and Ne'Kiya are typical St. Mary's students. She said if they're "unicorns," then every student who's matriculated through the school is a "beautiful, Black unicorn."

Students hear that message from the moment they walk in the door, Rogers said. 

"We believe all students can succeed, all students can learn," the principal said. "It does not matter the environment that you live in."

About half the students at St. Mary's get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. There's no test to get in, but expectations are high and rules are strict: cellphones are not allowed and modest skirts and hair in its natural color are required. 

Students said they appreciate the rules and rigor.

"Especially the standards that they set for us," junior Rayah Siddiq said. "They're very high. And I don't think that's ever going to change." 

What's next for Ne'Kiya and Calcea

Last year when Ne'Kiya and Calcea graduated, all their classmates were accepted into college and received scholarship offers. The school has had a 100% graduation rate and a 100% college acceptance rate for 17 years, according to Rogers.

Ne'Kiya got a full ride in the pharmacy department at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University. Neither one is pursuing a career in math, though Calcea said she may minor in math.

"People might expect too much out of me if I become a mathematician," Ne'Kiya said wryly. 

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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