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30 Problem Solving Maths Questions And Answers For GCSE

Sophie Bessemer

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.

GCSE MATHS 2024: STAY UP TO DATE Join our email list to stay up to date with the latest news, revision lists and resources for GCSE maths 2024. We’re analysing each paper during the course of the 2024 GCSEs in order to identify the key topic areas to focus on for your revision. Thursday 16th May 2024: GCSE Maths Paper 1 2024 Analysis & Revision Topic List Monday 3rd June 2024: GCSE Maths Paper 2 2024 Analysis & Revision Topic List Monday 10th June 2024: GCSE Maths Paper 3 2024 Analysis GCSE 2024 dates GCSE 2024 results (when published) GCSE results 2023

How to teach problem solving

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.

If you’re a KS2 teacher needing more support and CPD around teaching reasoning, problem solving & planning for depth we have a whole series of word problems and strategies to teach them available for you.

30 Problem Solving Maths Questions, Solutions & Strategies

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

6 tips to tackling problem solving maths questions

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.

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 For children who need more support, our maths intervention programmes for KS3 achieve outstanding results through a personalised one to one tuition approach.

10 problem solving maths questions (Foundation tier)

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.

solution to finding perimeter of l-shape

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}

10 problem solving maths questions (Foundation & Higher tier crossover)

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

10 problem solving maths questions (Higher tier)

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.

prism with a trapezium cross-section image

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

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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.

$A teacher is teaching three students with a whiteboard happily.$

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?

Five middle school students sitting at a row of desks playing Prodigy Math on tablets.

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

A hand holding a pen is doing calculation on a pice of papper

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

A female teacher is instructing student math on a blackboard

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

A student is drawing on a notebook, holding a pencil.

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

Four students are sitting together and discussing math questions

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?

A tablet showing an example of Prodigy Math's battle gameplay.

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

$Two students are calculating on a whiteboard$

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

$A hand is calculating math problem on a blacboard$

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

Two teachers are discussing math with a pen and a notebook

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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Math Word Problems

Welcome to the math word problems worksheets page at Math-Drills.com! On this page, you will find Math word and story problems worksheets with single- and multi-step solutions on a variety of math topics including addition, multiplication, subtraction, division and other math topics. It is usually a good idea to ensure students already have a strategy or two in place to complete the math operations involved in a particular question. For example, students may need a way to figure out what 7 × 8 is or have previously memorized the answer before you give them a word problem that involves finding the answer to 7 × 8.

There are a number of strategies used in solving math word problems; if you don't have a favorite, try the Math-Drills.com problem-solving strategy:

Question : Understand what the question is asking. What operation or operations do you need to use to solve this question? Ask for help to understand the question if you can't do it on your own.
Estimate : Use an estimation strategy, so you can check your answer for reasonableness in the evaluate step. Try underestimating and overestimating, so you know what range the answer is supposed to be in. Be flexible in rounding numbers if it will make your estimate easier.
Strategize : Choose a strategy to solve the problem. Will you use mental math, manipulatives, or pencil and paper? Use a strategy that works for you. Save the calculator until the evaluate stage.
Calculate : Use your strategy to solve the problem.
Evaluate : Compare your answer to your estimate. If you under and overestimated, is the answer in the correct range. If you rounded up or down, does the answer make sense (e.g. is it a little less or a little more than the estimate). Also check with a calculator.

Most Popular Math Word Problems this Week

$Easy Multi-Step Word Problems$

Arithmetic Word Problems

$maths problem solving question$

Addition Word Problems One-Step Addition Word Problems Using Single-Digit Numbers One-Step Addition Word Problems Using Two-Digit Numbers
Subtraction Word Problems Subtraction Facts Word Problems With Differences from 5 to 12
Multiplication Word Problems One-Step Multiplication Word Problems up to 10 × 10
Division Word Problems Division Facts Word Problems with Quotients from 5 to 12
Multi-Step Word Problems Easy Multi-Step Word Problems

To get additional practice, check out the sample problems in each of the topic above. We provide full solutions with steps for all practice problems. There's no better way to find math help online than with Cymath, so also make sure you download our mobile app for and today! Learn more than what the answer is - with the math helper app, you'll learn the steps behind it too.

Even simple math problems become easier to solve when broken down into steps. From basic additions to calculus, the process of problem solving usually takes a lot of practice before answers could come easily. As problems become more complex, it becomes even more important to understand the step-by-step process by which we solve them. At Cymath, our goal is to take your understanding of math to a new level.

If you find Cymath useful, try today! It offers an ad-free experience and more detailed explanations. In short, goes into more depth than the standard version, giving students more resources to learn the step-by-step process of solving math problems.

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Solving Word Questions

With LOTS of examples!

In Algebra we often have word questions like:

Example: Sam and Alex play tennis.

On the weekend Sam played 4 more games than Alex did, and together they played 12 games.

How many games did Alex play?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

Read the whole thing first
Do a sketch if possible
Assign letters for the values
Find or work out formulas

You should also write down what is actually being asked for , so you know where you are going and when you have arrived!

Also look for key words:

When you see		Think
add, total, sum, increase, more, combined, together, plus, more than		+
minus, less, difference, fewer, decreased, reduced		−
multiplied, times, of, product, factor		×
divided, quotient, per, out of, ratio, percent, rate		÷
maximize or minimize		geometry formulas
rate, speed		distance formulas
how long, days, hours, minutes, seconds		time

Thinking Clearly

Some wording can be tricky, making it hard to think "the right way around", such as:

Example: Sam has 2 dollars less than Alex. How do we write this as an equation?

Let S = dollars Sam has
Let A = dollars Alex has

Now ... is that: S − 2 = A

or should it be: S = A − 2

or should it be: S = 2 − A

The correct answer is S = A − 2

( S − 2 = A is a common mistake, as the question is written "Sam ... 2 less ... Alex")

Example: on our street there are twice as many dogs as cats. How do we write this as an equation?

Let D = number of dogs
Let C = number of cats

Now ... is that: 2D = C

or should it be: D = 2C

Think carefully now!

The correct answer is D = 2C

( 2D = C is a common mistake, as the question is written "twice ... dogs ... cats")

Let's start with a really simple example so we see how it's done:

Example: A rectangular garden is 12m by 5m, what is its area ?

Turn the English into Algebra:

Use w for width of rectangle: w = 12m
Use h for height of rectangle: h = 5m

Formula for Area of a Rectangle : A = w × h

We are being asked for the Area.

A = w × h = 12 × 5 = 60 m 2

The area is 60 square meters .

Now let's try the example from the top of the page:

Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

Use S for how many games Sam played
Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4

And we know that together they played 12 games: S + A = 12

We are being asked for how many games Alex played: A

Which means that Alex played 4 games of tennis.

Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games. Yes!

A slightly harder example:

Example: Alex and Sam also build tables. Together they make 10 tables in 12 days. Alex working alone can make 10 in 30 days. How long would it take Sam working alone to make 10 tables?

Use a for Alex's work rate
Use s for Sam's work rate

12 days of Alex and Sam is 10 tables, so: 12a + 12s = 10

30 days of Alex alone is also 10 tables: 30a = 10

We are being asked how long it would take Sam to make 10 tables.

30a = 10 , so Alex's rate (tables per day) is: a = 10/30 = 1/3

Which means that Sam's rate is half a table a day (faster than Alex!)

So 10 tables would take Sam just 20 days.

Should Sam be paid more I wonder?

And another "substitution" example:

Example: Jenna is training hard to qualify for the National Games. She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days. She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?

The number of "5 hour" days: d
The number of "3 hour" days: e

We know there are seven days in the week, so: d + e = 7

And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d + 3e = 27

We are being asked for how many days she trains for 5 hours: d

The number of "5 hour" days is 3

Check : She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours

Some examples from Geometry:

Example: A circle has an area of 12 mm 2 , what is its radius?

Use A for Area: A = 12 mm 2
Use r for radius

And the formula for Area is: A = π r 2

We are being asked for the radius.

We need to rearrange the formula to find the area

Example: A cube has a volume of 125 mm 3 , what is its surface area?

Make a quick sketch:

Use V for Volume
Use A for Area
Use s for side length of cube
Volume of a cube: V = s 3
Surface area of a cube: A = 6s 2

We are being asked for the surface area.

First work out s using the volume formula:

Now we can calculate surface area:

An example about Money:

Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate. One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?

Joel's normal rate of pay: $N per hour
Joel works for 40 hours at $N per hour = $40N
When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour
Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N
And together he earned $660, so:

$40N + $(12 × 1¼N) = $660

We are being asked for Joel's normal rate of pay $N.

So Joel’s normal rate of pay is $12 per hour

Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660

More about Money, with these two examples involving Compound Interest

Example: Alex puts $2000 in the bank at an annual compound interest of 11%. How much will it be worth in 3 years?

This is the compound interest formula:

So we will use these letters:

Present Value PV = $2,000
Interest Rate (as a decimal): r = 0.11
Number of Periods: n = 3
Future Value (the value we want): FV

We are being asked for the Future Value: FV

Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33 What was the annual rate of interest for the savings account?

The compound interest formula:

Present Value PV = $1,000
Interest Rate (the value we want): r
Number of Periods: n = 9
Future Value: FV = $1,551.33

We are being asked for the Interest Rate: r

So the annual rate of interest is 5%

Check : $1,000 × (1.05) 9 = $1,000 × 1.55133 = $1,551.33

And an example of a Ratio question:

Example: At the start of the year the ratio of boys to girls in a class is 2 : 1 But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3 How many students are there altogether now?

Number of boys now: b
Number of girls now: g

The current ratio is 4 : 3

Which can be rearranged to 3b = 4g

At the start of the year there was (b + 4) boys and (g − 2) girls, and the ratio was 2 : 1

b + 4 g − 2 = 2 1

Which can be rearranged to b + 4 = 2(g − 2)

We are being asked for how many students there are altogether now: b + g

There are 12 girls !

And 3b = 4g , so b = 4g/3 = 4 × 12 / 3 = 16 , so there are 16 boys

So there are now 12 girls and 16 boys in the class, making 28 students altogether .

There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3 At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1

And now for some Quadratic Equations :

Example: The product of two consecutive even integers is 168. What are the integers?

Consecutive means one after the other. And they are even , so they could be 2 and 4, or 4 and 6, etc.

We will call the smaller integer n , and so the larger integer must be n+2

And we are told the product (what we get after multiplying) is 168, so we know:

n(n + 2) = 168

We are being asked for the integers

That is a Quadratic Equation , and there are many ways to solve it. Using the Quadratic Equation Solver we get −14 and 12.

Check −14: −14(−14 + 2) = (−14)×(−12) = 168 YES

Check 12: 12(12 + 2) = 12×14 = 168 YES

So there are two solutions: −14 and −12 is one, 12 and 14 is the other.

Note: we could have also tried "guess and check":

We could try, say, n=10: 10(12) = 120 NO (too small)
Next we could try n=12: 12(14) = 168 YES

But unless we remember that multiplying two negatives make a positive we might overlook the other solution of (−14)×(−12).

Example: You are an Architect. Your client wants a room twice as long as it is wide. They also want a 3m wide veranda along the long side. Your client has 56 square meters of beautiful marble tiles to cover the whole area. What should the length of the room be?

Let's first make a sketch so we get things right!:

the length of the room: L
the width of the room: W
the total Area including veranda: A
the width of the room is half its length: W = ½L
the total area is the (room width + 3) times the length: A = (W+3) × L = 56

We are being asked for the length of the room: L

This is a quadratic equation , there are many ways to solve it, this time let's use factoring :

And so L = 8 or −14

There are two solutions to the quadratic equation, but only one of them is possible since the length of the room cannot be negative!

So the length of the room is 8 m

L = 8, so W = ½L = 4

So the area of the rectangle = (W+3) × L = 7 × 8 = 56

There we are ...

... I hope these examples will help you get the idea of how to handle word questions. Now how about some practice?

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Geometry and Measure - Short Problems

A collection of short Stage 3 and 4 problems on geometry and measure.

Handling Data - Short Problems

A collection of short Stage 3 and 4 problems on handling data.

Thinking Mathematically - Short Problems

A collection of short problems which require students to think mathematically.

The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

These problem solving starter packs are great to support students with problem solving skills. I've used them this year for two out of four lessons each week, then used Numeracy Ninjas as starters for the other two lessons. When I first introduced the booklets, I encouraged my students to use scaffolds like those mentioned here , then gradually weaned them off the scaffolds. I give students some time to work independently, then time to discuss with their peers, then we go through it as a class. The levels correspond very roughly to the new GCSE grades.

Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

@TessMaths has written some great stuff for BBC Bitesize.

There are some intersting though-provoking problems at Open Middle.

I'm sure you've seen it before, but if not, check it out now! Nrich is where it's at if your want to provide enrichment and problem solving in your lessons.

MathsBot by @StudyMaths has everything, and if you scroll to the bottom of the homepage you'll find puzzles and problem solving too.

I may be a little biased because I love Edexcel, but these question packs are really useful.

The UKMT has a mentoring scheme that provides fantastic problem solving resources , all complete with answers.

I have only recently been shown Maths Problem Solving and it is awesome - there are links to problem solving resources for all areas of maths, as well as plenty of general problem solving too. Definitely worth exploring!

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A different question each day, from all subjects. Every question has a worked solution.

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Unit 1: Algebra foundations

Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.

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Student Question Bank: Math Questions

Because each question on the Math section deals with different numbers and mathematical scenarios, it's not as simple as the Reading and Writing section to identify exactly what each question stem will look like. You can still use the descriptions in this section to determine which math domains and skills you want to focus on in the Student Question Bank.

Math Questions

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Abstract purple lines funnelling towards the right with white dotted light sources becoming smaller towards the right.

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

The zeta function as a function of all prime numbers

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

The coordinates of z = x + iy illustrate a complex number

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

A visual mapping of the zeta function looks like a mountainscape with peaks and troughs

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

The dark blue area demarcates a stretch along the x axis where the Riemann zeta function contains nontrivial zeros

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

The Hechinger Report

Covering Innovation & Inequality in Education

Why schools are teaching math word problems all wrong

Teacher takeaways

Don’t instruct students to focus mainly on “key words” in word problems such as “and” or “more”
Mix question types in any lesson so that students don’t assume they just apply the same operation (addition, subtraction) again and again
Teach students the underlying structure — or schema — of the word problem

Yet most experts and many educators agree that too many schools are doing it wrong, particularly in the elementary grades. And in a small but growing number of classrooms, teachers like Molina are working to change that. “With word problems, there are more struggling learners than non-struggling learners” because they are taught so poorly, said Nicole Bucka, who works with teachers throughout Rhode Island to provide strategies for struggling learners.

Too many teachers, particularly in the early grades, rely on key words to introduce math problems. Posters displaying the terms — sum, minus, fewer, etc. — tied to operations including addition and subtraction are a staple in elementary school classrooms across the country.

Key words can be a convenient crutch for both students and teachers, but they become virtually meaningless as the problems become harder, according to researchers. Key words can help first graders figure out whether to add or subtract more than half of the time, but the strategy rarely works for the multi-step problems students encounter starting in second and third grade. “With multi-step problems, key words don’t work 90 percent of the time,” said Sarah Powell, a professor at the University of Texas in Austin who studies word problems and whose research has highlighted the inefficacy of key words . “But the average kindergarten teacher is not thinking about that; they are teaching 5-year-olds, not 9-year-olds.”

Many teachers in the youngest grades hand out worksheets featuring the same type of word problem repeated over and over again. That’s what Molina’s colleague, Cassandra Santiago, did sometimes last year when leading a classroom on her own for the first time. “It was a mistake,” the first grade teacher said. “It’s really important to mix them up. It makes them think more critically about the parts they have to solve.”

$maths problem solving question$

Another flaw with word problem instruction is that the overwhelming majority of questions are divorced from the actual problem-solving a child might have to do outside the classroom in their daily life — or ever, really. “I’ve seen questions about two trains going on the same track,” said William Schmidt, a University Distinguished Professor at Michigan State University. “First, why would they be going on the same track and, second, who cares?”

Schmidt worked on an analysis of about 8,000 word problems used in 23 textbooks in 19 countries. He found that less than one percent had “real world applications” and involved “higher order math applications .”

“That is one of the reasons why children have problems with mathematics,” he said. “They don’t see the connection to the real world … We’re at this point in math right now where we are just teaching students how to manipulate numbers.”

He said a question, aimed at middle schoolers, that does have real world connections and involves more than manipulating numbers, might be: “Shopping at the new store in town includes a 43% discount on all items which are priced the same at $2. The state you live in has a 7% sales tax. You want to buy many things but only have a total of $52 to spend. Describe in words how many things you could buy.”

Schmidt added that relevancy of word problems is one area where few, if any, countries excel. “No one was a shining star leading the way,” he said.

In her brightly decorated classroom one Tuesday afternoon, Santiago, the first grade teacher, gave each student a set of animal-shaped objects and a sheet of blue paper (the water) and green (the grass). “We’re going to work on a number story,” she told them. “I want you to use your animals to tell me the story.”

“ Once upon a time,” the story began. In this tale, three animals played in the water, and two animals played in the grass. Santiago allowed some time for the ducks, pigs and bears to frolic in the wilds of each student’s desk before she asked the children to write a number sentence that would tell them how many animals they have altogether.

Some of the students relied more on pictorial representations (three dots on one side of a line and two dots on the other) and others on the number sentence (3+2 = 5) but all of them eventually got to five. And Santiago made sure that her next question mixed up the order of operations (so students didn’t incorrectly assume that all they ever have to do is add): “Some more animals came and now there are seven. So how many more came?”

One approach to early elementary word problems that is taking off in some schools, including Segue Institute, has its origins in a special education intervention for struggling math students. Teachers avoid emphasizing key words and ask students instead to identify first the conceptual type of word problem (or schema, as many practitioners and researchers refer to it) they are dealing with: “Total problems,” for instance, involve combining two parts to find a new amount; “change problems” involve increasing or decreasing the amount of something. Total problems do not necessarily involve adding, however.

$maths problem solving question$

“The schemas that students learn in kindergarten will continue with them throughout their whole career,” said Powell, the word-problem researcher, who regularly works with districts across the country to help implement the approach.

In Olathe, Kansas — a district inspired by Powell’s work — teachers had struggled for years with word problems, said Kelly Ulmer, a math support specialist whose goal is to assist in closing academic gaps that resulted from lost instruction time during the pandemic. “We’ve all tried these traditional approaches that weren’t working,” she said. “Sometimes you get pushback on new initiatives from veteran teachers and one of the things that showed us how badly this was needed is that the veteran teachers were the most excited and engaged — they have tried so many things” that haven’t worked.

In Rhode Island, many elementary schools initially used the strategy with students who required extra help, including those in special education, but expanded this use to make it part of the core instruction for all, said Bucka. In some respects, it’s similar to the recent, well publicized evolution of reading instruction in which some special education interventions for struggling readers — most notably, a greater reliance on phonics in the early grades — have gone mainstream.

There is an extensive research bas e showing that focusing on the different conceptual types of word problems is an effective way of teaching math, although much of the research focuses specifically on students experiencing difficulties in the subject.

Molina has found asking students to identify word problems by type to be a useful tool with nearly all of her second graders; next school year she hopes to introduce the strategy much earlier.

$maths problem solving question$

One recent afternoon, a lesson on word problems started with everyone standing up and chanting in unison: “Part plus part equals total” (they brought two hands together). “Total minus part equals part ” (they took one hand away) .

It’s a way to help students remember different conceptual frameworks for word problems. And it’s especially effective for the students who learn well through listening and repeating. For visual learners, the different types of word problems were mapped out on individual dry erase mats.

The real work began when Molina passed out questions, and the students— organized into the Penguin, Flower Bloom, Red Panda and Marshmallow teams — had to figure out which framework they were dealing with on their own and then work toward an answer. A few months ago, many of them would have automatically shut down when they saw the text on the page, Molina said.

For the Red Pandas, the question under scrutiny was: “The clothing store had 47 shirts. They sold 21, how many do they have now?”

“It’s a total problem,” one student said.

“No, it’s not total,” responded another.

“I think it’s about change,” said a third.

None of the students seemed worried about their lack of consensus, however. And neither was Molina. A correct answer is always nice but those come more often now that most of the students have made a crucial leap. “I notice them thinking more and more,” she said, “about what the question is actually asking.”

This story about word problems was produced by The Hechinger Report , a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for the Hechinger newsletter .

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In the last example of the word problem involving the shirts would the teacher have been fine with either explanation. One thought would be that there is a total number of shirts (41) with one part being sold (21) leaving the other part on hand (26). Another way of thinking about the problem would be a change happening with the beginning number of (47) shirts being reduced or changed by the number being taken away (21), leaving the rest (26).

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Math is tripping up community college students. Some schools are trying something new

ALBANY, Ore. – It’s 7:15 on a Monday morning in May at Linn-Benton Community College in northwestern Oregon. Math professor Michael Lopez, a tape measure on his belt, paces in front of the 14 students in his “math for welders” class. “I’m your OSHA inspector,” he says. “Three-sixteenths of an inch difference, you’re in violation. You’re going to get a fine.”

He has just given them a project they might have to do on the job: figuring out the rung spacing on a steel ladder that attaches to a wall. Thousands of dollars are at stake in such builds, and they’re complicated: Some clients want the fewest possible rungs to save money; others want a specific distance between steps. To pass inspection, rungs must be evenly spaced to within one-sixteenth of an inch.

Math is a giant hurdle for most community college students pursuing welding and other career and technical degrees. About a dozen years ago, Linn-Benton’s administrators looked at their data and found that many students in career and technical education, or CTE, were getting most of the way toward a degree but were stopped by a math course, said the college’s president, Lisa Avery. That’s not unusual: Up to 60% of students entering community college are unprepared for college-level work, and the subject they most often need help with is math .

The college asked the math department to design courses tailored to those students, starting with its welding, culinary arts and criminal justice programs. The first of those, math for welders, rolled out in 2013.

More than a decade later, welding department instructors say math for welders has had a huge effect on student performance. Since 2017, 93% of students taking it have passed, and 83% have achieved all the course’s learning goals, including the ability to use arithmetic, geometry, algebra and trigonometry to solve welding problems, school data shows. Two years ago, Linn-Benton asked Lopez to design a similar course for its automotive technology program; the college began to offer that course last fall.

Math for welders changed student Zane Azmane’s view of what he could do. “I absolutely hated math in high school. It didn’t apply to anything I needed at the moment,” said Azmane, 20, who failed several semesters of math early in high school but got a B in the Linn-Benton course last year. “We actually learned equations I’m going to use, like setting ladder rungs.”

Linn-Benton’s aim is to change how students pursuing technical degrees learn math by making it directly applicable to their technical specialties.

Some researchers say these small-scale efforts to teach math in context could transform how it’s taught more broadly.

Among the strategies to help college students who struggle with math, giving them contextual curriculums seems to have "the strongest theoretical base and perhaps the strongest empirical support,” according to a 2011 paper by Dolores Perin, now professor emerita at Columbia University Teachers College. (The Hechinger Report is an independent unit of Teachers College.)

Perin’s paper echoed the results of a 2006 study of math in high school CTE involving almost 3,000 students. Students in the study who were taught math through an applied approach performed significantly better on two of three standardized tests than those taught math in a more traditional way. (The applied math students also performed better on the third test, though the results were not statistically significant.)

There haven’t been systematic studies of math in CTE at the college level, according to James Stone, director of the National Research Center for Career and Technical Education at the Southern Regional Education Board, who ran the 2006 study.

Oregon appears to be one of the few places where this approach is spreading, if slowly.

Three hours south of Linn-Benton, Doug Gardner, an instructor in the Rogue Community College math department, had long struggled with a persistent question from students: “Why do we need to know this?”

“It became my life’s work to have an answer to that question,” said Gardner, now the department chair.

Meanwhile, at the college, about a third of students taking Algebra I or a lower-level math course failed or withdrew. For many who stayed, the lack of math knowledge hurt their job prospects, preventing them from gaining necessary skills in fields like pipefitting.

So, in 2010, Gardner applied for and won a National Science Foundation grant to create two new applied algebra courses. Instead of abstract formulas, students would learn practical ones: how to calculate the volume of a wheelbarrow of gravel and the number of wheelbarrows needed to cover an area, or how much a beam of a certain size and type would bend under a certain load.

Since then, the pass rate in the applied algebra class has averaged 73%, while the rate for the traditional course has continued to hover around 59%, according to Gardner.

One day in May, math professor Kathleen Foster was teaching applied algebra in a sun-drenched classroom on Rogue’s campus. She launched into a lesson about the Pythagorean theorem and why it’s an essential tool for building home interiors and steel structures.

James Butler-Kyniston, 30, who is pursuing a degree as a machinist, said the exercises covered in Foster’s class are directly applicable to his future career. One exercise had students calculate how large a metal sheet you would need to manufacture a certain number of parts at a time. “Algebraic formulas apply to a lot of things, but since you don’t have any examples to tie them to, you end up thinking they’re useless,” he said.

In 2021, Oregon state legislators passed a law requiring all four-year colleges to accept an applied math community college course called Math in Society as satisfying the math requirement for a four-year degree. In that course, instead of studying theoretical algebra, students learn how to use probability and statistics to interpret the results in scientific papers and how political rules like apportionment and gerrymandering affect elections, said Kathy Smith, a math professor at Central Oregon Community College.

“If I had my way, this is how algebra would be taught to every student: the applied version,” Gardner said. “And then if a student says ‘This is great, but I want to go further,’ then you sign up for the theoretical version.”

But at individual schools, lack of money and time constrain the spread of applied math. Stone’s team works with high schools around the country to design contextual math courses for career and technical students. They tried to work with a few community colleges, but their CTE faculty, many of whom were part-timers on contract, didn’t have time to partner with their math departments to develop a new curriculum, a yearlong process, Stone said.

Linn-Benton was able to invest the time and money because its math department was big enough to take on the task, said Avery. Both Linn-Benton and Rogue may be outliers because they have math faculty with technical backgrounds: Lopez worked as a carpenter and sheriff’s deputy and served three tours as a machine gunner in Iraq, and Gardner was a construction contractor.

Back in Lopez’s class, students are done calculating where their ladder rungs should go and now must mark them on the wall.

As teams finish up, Lopez inspects their work. “That’s one-thirty-second shy. But I wouldn’t worry too much about it,” he tells one group. “OSHA’s not going to knock you down for that.”

Three teams pass, two fail – but this is the place to make mistakes, not out on the job, Lopez tells them.

“This stuff is hard,” said Keith Perkins, 40, who’s going for a welding degree and wants to get into the local pipe fitters union. “I hated math in school. Still hate it. But we use it every day.”

This story was produced by The Hechinger Report , a nonprofit, independent news organization focused on inequality and innovation in education.

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The most rigorous math program you've never heard of.

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Math-M-Addicts students eagerly dive into complex math problems during class.

In the building of the Speyer Legacy School in New York City, a revolutionary math program is quietly producing some of the city's most gifted young problem solvers and logical thinkers. Founded in 2005 by two former math prodigies, Math-M-Addicts has grown into an elite academy developing the skills and mindset that traditional schooling often lacks.

"We wanted to establish the most advanced math program in New York," explains Ruvim Breydo, co-founder of Math-M-Addicts. "The curriculum focuses not just on mathematical knowledge, but on developing a mastery of problem-solving through a proof-based approach aligned with prestigious competitions like the International Mathematical Olympiad."

From its inception, Math-M-Addicts took an unconventional path. What began as an attempt to attract only the highest caliber high school students soon expanded to offer multiple curriculum levels. "We realized we couldn't find enough kids at the most advanced levels," says Breydo. "So we decided to develop that talent from an earlier age."

The program's approach centers on rigor. At each of the 7 levels, the coursework comprises just a handful of fiendishly difficult proof-based math problems every week. "On average, we expect them to get about 50% of the solutions right," explains instructor Natalia Lukina. "The problems take hours and require grappling with sophisticated mathematical concepts."

But it's about more than just the content. Class sizes are small, with two instructors for every 15-20 students. One instructor leads the session, while the other teacher coordinates the presentation of the homework solutions by students. The teachers also provide customized feedback by meticulously reviewing each student's solutions. "I spend as much time analyzing their thought processes as I do teaching new material," admits instructor Bobby Lee.

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Lee and the Math-M-Addicts faculty embrace an unconventional pedagogy focused on developing logic, creativity, and a tenacious problem-solving mindset over procedures. "We don't dumb it down for them," says Breydo. "We use technical math language and allow students to struggle through the challenges because that's where real learning happens."

Impressive results of Math-M-addicts students in selective math competitions highlight their ... [+] preparation and dedication.

For the Math-M-Addicts team, finding the right teachers is as essential as shaping brilliant students. Prospective instructors go through a rigorous multi-stage vetting process. "We seek passionate mathematical problem solvers first," says program director Sonali Jasuja. "Teaching experience is great, but first and foremost, we need people who deeply understand and enjoy the reasoning behind mathematics."

Even exceptional instructors undergo extensive training by co-teaching for at least a year alongside veteran Math-M-Addicts faculty before taking the lead role. "Our approach is different from how most US teachers learned mathematics," explains instructor Tanya Gross, the director of Girls Adventures in Math (GAIM) competition. "We immerse them in our unique math culture, which focuses on the 'why' instead of the 'how,' empowering a paradigm shift."

That culture extends to the students as well. In addition to the tools and strategies imparted in class, Math-M-Addicts alumni speak of an unshakable confidence and camaraderie that comes from up to several thousands of hours grappling with mathematics at the highest levels alongside peers facing the same challenges.

As Math-M-Addicts ramps up efforts to expand access through online classes and global partnerships, the founders remain devoted to their core mission. "Math education should not obsess with speed and memorization of math concepts," argues Breydo. "This is not what mathematics is about. To unlock human potential, we must refocus on cognitive reasoning and problem-solving skills. We are seeking to raise young people unafraid to tackle any complex challenge they face"

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30 Problem Solving Maths Questions And Answers For GCSE

How to teach problem solving

6 tips to tackling problem solving maths questions

10 problem solving maths questions (Foundation tier)

1) L-shape perimeter

2) Find the missing point

3) That rating was a bit mean!

Strategies

4) Changing angles

5) Base and power

6) Just an average problem

7) Square and rectangle

9) Unequal share

10) Somewhere between

Solution

10 problem solving maths questions (Foundation & Higher tier crossover)

11) What’s the difference?

13) What a waste

14) Tri-polygonometry

15) That’s a lot of Pi

16) Is it right?

17) Pie chart ratio

18) DIY Simultaneously

19) Triathlon pace

10 problem solving maths questions (Higher tier)

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22) Prism and force (Non-calculator)

23) Geometric sequences (Non-calculator)

24) Printing factory

25) Circles

26) Circle theorems

27) Quadratic function

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29) Ice cream tub volume

30) Translating graphs

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120 Math Word Problems To Challenge Students Grades 1 to 8

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120 Math word problems, categorized by skill

Subtraction word problems

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Mixed operations word problems

Ordering and number sense word problems

Fractions word problems

Decimals word problems

Comparing and sequencing word problems

Time word problems

Money word problems

Physical measurement word problems

Ratios and percentages word problems

Probability and data relationships word problems

Geometry word problems

Variables word problems

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