ks3 problem solving questions

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KS3 maths question bank

Create KS3 maths practice papers and assessments tailored to your classes with this bank of over 250 maths questions. The ready-made exam paper template means you can create exam-style assessments with a full mark scheme of answers without any of the stress!

What’s included

- 250+ maths problems and questions for key stage 3

- Questions covering the following maths topics: Number, Algebra, Shape, space and measure and Data handling

- Mark scheme and answers for every question

How to use this bank of KS3 maths questions

Whether it’s for regular assessment or end of year tests, choose from over 250 maths questions in the bank to create exam-style test papers tailored to your classes.

Simply select the questions you’d like to use for your maths test and copy them into the ready-made template. Read the teacher's instructions to help make the process as simple as possible.

Please note: The password to save the bank is 'password'. Macros will need to be enabled – please see your school network administrator if you need help with this.

Looking for more like this?

Browse our collection of revision cards, workbooks and worksheets to support KS3 maths revision . If you’re looking for support for your GCSE classes, try our GCSE maths revision collection instead.

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• Free Teaching Resources For The Entire Ks3 Maths Curriculum

Key Stage 3 maths worksheets – Resources for whole curriculum

An entire breakdown of the KS3 maths curriculum in England with worksheets, activities, ideas, PowerPoints and more for every element…

Unlock a treasure trove of free Key Stage 3 maths worksheets tailored to the curriculum. These worksheets will save you time and boost students’ learning.

We’ve split this post up into six main areas. Underneath each area we’ve outlined the Key Stage 3 maths curriculum and highlighted worksheets and lesson plans that you can download right here on Teachwire – for free!

Key Stage 3 maths worksheets

Ratio, proportion and rates of change, geometry and measures, probability, free homework question packs.

Help students practise their maths skills at home with these free Key Stage 3 maths worksheets for Year 7 , Year 8 and Year 9.

100+ problem-solving questions

You can also download 100+ problem-solving questions in our KS3 maths worksheets pack from White Rose Maths.

Mega KS3 revision PowerPoint

We also have a mega KS3 maths questions PowerPoint featuring 100+ slides of questions on a range of topics, with answers included.

KS3 maths quiz

This comprehensive maths quiz PowerPoint offers 30 multiple-choice quizzes spread across over 300 slides. It’s perfect for KS3 and GCSE students and covers a wide array of topics.

Place value

• Understand and use place value for decimals, measures and integers of any size

Resource: Use this set of problem cards to get children thinking about place value with decimals.

Ordering and number lines

• Order positive and negative integers, decimals and fractions
• Use the number line as a model for ordering of the real numbers
• Use the symbols =, ≠, <, >, ≤, ≥

Resource : This Key Stage 3 maths worksheet covers negative numbers . Pupils will add positive values to negative values.

Or try this ordering fractions and decimals worksheet . It contains questions which require students to think about the relative sizes of fractions and decimals, and the order that they would be written on a number line.

Four operations

• Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

Resources: This adding fractions with the same denominator worksheet gives pupils the opportunity to practice fluency, reasoning and problem-solving.

This adding and subtracting fractions worksheet lets students pick the section on which they need more work.

These maths codebreakers cover a wide range of maths topics including multiplying and dividing fractions .

This simple PowerPoint for multiplying and dividing fractions shows how simplifying can be done before the multiplying.

Priority of operations and relationships between them

• Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
• Recognise and use relationships between operations including inverse operations

Resources: This order of operations worksheet encourages pupils to recognise that we need to reduce calculations to those involving sums and difference.

These four carefully thought-out worksheets will help your class take their first steps working with the order of operations.

Meanwhile, these PowerPoints cover the precedence of operations, making it clear that multiplication/division have the same precedence as each other, as do addition/subtraction.

Integer powers and real roots

• Use integer powers and associated real roots (square, cube and higher)
• Recognise powers of 2, 3, 4, 5
• Distinguish between exact representations of roots and their decimal approximations

Resource: This three-part lesson covers finding squares, cubes and square roots. There are also mini-plenary and plenary activities, with answers.

This activity helps to build an understanding of what powers are. Students need to copy and complete the table showing the first six powers of 2. They then need to explain in words how they would calculate 2 20 , before making similar tables showing the first six powers of 10, 3 and  a .

Comparing numbers

• Interpret and compare numbers in standard form A x 10n 1≤A<10, where n is a positive or negative integer or zero

Resources: This worksheet and PowerPoint explain why putting numbers in standard form is useful and how to use the x10^ button on your calculator

These maths codebreakers cover standard from, among other topics. Unjumble the letters to find out the answer to some (very bad) jokes.

Terminating decimals

• Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8)

Resource: Use this resource to think about the relative sizes of fractions and decimals, and the order that they would be written on a number line.

At the end of these fractions, decimals and percents worksheets students will look at fractions that produce terminating decimals. 

Percentages

• Define percentage as ‘number of parts per hundred’
• Interpret percentages and one quantity as a percentage of another
• Compare two quantities using percentages
• Work with percentages greater than 100%

Resource: This percentage calculations worksheet includes different types of percentage questions with answers.

Alternatively, use this PowerPoint and worksheets to introduce students to the method of calculating percentages without a calculator.

• Interpret fractions and percentages as operators

Resources: This fractions, decimals and percents worksheet includes recurring decimals and values greater than 1.

Meanwhile, this resource covers simplifying, converting recurring decimals to fractions and everything in between.

Units of measurement

• Use standard units of mass, length, time, money and other measures, including with decimal quantities

Resource: This units of measurement worksheet covers simple conversions, metric to imperial and area conversion.

Rounding up

• Round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures]

Resource: Use this rounding and approximation worksheet to practise rounding a number to a given number of decimal places or significant figures.

This free rounding numbers worksheet helps students avoid getting in a muddle with zeroes and columns when rounding numbers.

Approximation

• Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b

Resource: This resource takes you from basic rounding to whole numbers up through decimal places, significant figures and beyond.

Calculating

• Use a calculator and other technologies to calculate results accurately and then interpret them appropriately

Resources: This lesson plan runs through calculator basics and there’s a tarsia puzle and past papers to practise on.

This PowerPoint presentation explains every single button and mode on standard black Casio calculators.

Algebraic notation

Use and interpret algebraic notation, including:

• ab in place of a × b
• 3y in place of y + y + y and 3 × y
• a2 in place of a × a
• a3 in place of a × a × a
• a2b in place of a × a × b
• a/b in place of a ÷ b
• coefficients written as fractions rather than as decimals

Resource: This KS3 maths formula sheet first asks students to link each statement with a formula in the given table. They then move on to writing their own formula for each of the given statements.

This resource covers algebraic notation and simplifying by collecting like terms and multiplying/dividing expressions, as well as substituting values.

Substituting

• Substitute numerical values into formulae and expressions, including scientific formulae

Resource: This resource on the topic of algebra substitution gets pupils working in pairs and practising substituting values into expressions.

This algebra resource tasks students with playing a Connect 4-style game of connecting four squares to learn about how to substitute in simple and more complex algebraic expressions.

This nine-lesson simplifying expressions PowerPoint takes students from the most basic simplifying up through substitution and proof of formulae.

Expressions, equations, inequalities, terms and factors

• Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors

Resource: This algebraic expressions resource covers algebraic notation and simplifying, substituting values and forming algebraic expressions.

Maintaining equivalence

Simplify and manipulate algebraic expressions to maintain equivalence by:

• collecting like terms
• multiplying a single term over a bracket
• taking out common factors
• expanding products of two or more binomials

Resources: These Key Stage 3 maths worksheets feature funny codebreakers covering a wide range of maths topics, including simplifying expressions .

This algebraic expressions resource covers algebraic notation and simplifying, substituting values and forming algebraic expressions.

• Understand and use standard mathematical formulae
• Rearrange formulae to change the subject

Resources: This worksheet contains the full range of exam-type questions that require students to simplify algebraic fractions by factorising. 

This free algebraic fractions worksheet with answers asks students to factorise then simplify algebraic fractions.

Modelling and graphs

• Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs

Resource: This full lesson introduces solving simultaneous equations using graphical methods.

Linear equations

• Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement)

Resources: Use these matching activities to join equations that match the central cloud but also solve the ones that don’t match.

These four solving equations worksheets from Cazoom Maths range from levels 4-7.

This algebra worksheet contains four pages of questions on solving linear equations of increasing complexity. 

Coordinates

• Work with coordinates in all four quadrants

Resource: This 4 quadrants coordinates worksheet will get your KS3 students practising plotting and identifying coordinates in all four quadrants.

Recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane

Resource: This worksheet covers sketching quadratic graphs by finding out where the graph crosses each axis.

This worksheet about solving simultaneous equations graphically features questions on solving a quadratic that is equal to an integer and solving a quadratic and linear equation simultaneously using graphs.

Interpreting relationships

• Interpret mathematical relationships both algebraically and graphically

Resource: This solving simultaneous equations graphically worksheet covers solving a quadratic that is equal to an integer and solving a quadratic and linear equation simultaneously using graphs.

Reducing and interpreting

• Reduce a given linear equation in two variables to the standard form y = mx + c
• Calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically

Resource: This lesson introduces the general form of all straight line graphs (y=mx+c), explaining the key features.

Linear and quadratic graphs

• Use linear and quadratic graphs to estimate values of y for given values of x and vice versa
• Find approximate solutions of simultaneous linear equations

Resource: This lesson will show students how plotting two linear equations in x and y allows them to solve simultaneous equations.

Graphs of a variety of functions

• Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs

Resource: Cut the right coloured wires to save everyone in this ‘Defuse the bomb’ worksheet . This involves basic and more complicated linear graphs, quadratics and non-linear graphs (cubics, reciprocals and exponentials).

Terms of a sequence

• Generate terms of a sequence from either a term-to-term or a position-to-term rule

Resources: Use this resource to draw the next pattern and find the nth term of an arithmetic sequence.

This resource features a discussion of what sequences are and builds to the term-to-term rule.

Arithmetic sequences

• Recognise arithmetic sequences and find the nth term

Resources: These nth term worksheets contain material that is typically introduced at KS3. 

This arithmetic sequence worksheet helps students master arithmetic sequences through an array of thoughtfully curated questions.

Geometric sequences

• Recognise geometric sequences and appreciate other sequences that arise

Resource: This resource covers a recap of linear sequences, geometric sequences, quadratic sequences and sequence proofs.

Standard units

• Change freely between related standard units [for example time, length, area, volume/capacity, mass]

Resource: This ratio and proportion resource includes methods for solving problems involving direct and indirect proportion and problems involving ratio.

Scale factors, scale diagrams and maps

• Use scale factors, scale diagrams and maps

Resources: This Key Stage 3 maths worksheet covers enlargement with negative scale factors . It involves enlarging shapes to produce a picture. There’s also this scale factors worksheet .

Expressing quantities as a fraction of another

• Express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1

Resource: This worksheet involves finding fractions of amounts. The extension is an equivalent fraction investigation.

Ratio notation

• Use ratio notation, including reduction to simplest form

Resource: In this simplifying ratios worksheet , students need to fully simplify a ratio and write it as a unitary ratio.

• Divide a given quantity into two parts in a given part:part or part:whole ratio
• Express the division of a quantity into two parts as a ratio

Resources: Use this ratio worksheet to simplify ratios and share amounts by a given ratio.

Alternatively, this simplifying ratios treasure hunt gives students practice at converting ratios into their simplest form by dividing both parts.

Multiplicative relationships

• Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction

Resource: This resource features three lessons on ratio problems, but with an emphasis on using fractions in calculations and not relying on calculators or decimals.

Arithmetic of fractions

• Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions

Resource: These fraction, ratio and percentage reasoning tasks are based on an Edexcel GCSE question about fractions, ratio and percentages of an amount.

Percentage change

Solve problems involving percentage change, including:

• percentage increase, decrease and original value problems
• simple interest in financial mathematics

Resources: This activity involves working out percentage changes in footballers’ wages.

Alternatively, use this Homes Under the Hammer worksheet to calculate whether people have made a profit or loss. There’s also a Storage Hunters version !

In this percentage change KS3 lesson , students explore what happens to a certain amount of money when it undergoes a particular percentage increase or decrease.

Direct and inverse proportion

• Solve problems involving direct and inverse proportion, including graphical and algebraic representations

Resource: This ratio and proportion resource includes methods for solving problems involving direct and indirect proportion.

Compound units

• Use compound units such as speed, unit pricing and density to solve problems

Resource: Use this resource to solve simple problems involving speed, distance, time, density, mass and volume.

Perimeter and area

Derive and apply formulae to calculate and solve problems involving:

• perimeter and area of triangles
• parallelograms
• volume of cuboids (including cubes)
• other prisms (including cylinders)

Resources: These maths codebreakers cover a wide range of maths topics including area and perimeter.

Or try this resource where students find the perimeter, circumference and area of a variety of shapes.

Perimeters and areas of 2D shapes

Calculate and solve problems involving:

• perimeters of 2D shapes (including circles)
• areas of circles
• composite shapes

Resource: These two Key Stage 3 maths worksheets cover the area of a circle . They focus on finding areas of circles and circular shapes.

Line segments and angles

• Draw and measure line segments and angles in geometric figures, including interpreting scale drawings

Resource: This simple sheet allows students to practise compass constructions.

Ruler and compass constructions

• Derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle)
• Recognise and use the perpendicular distance from a point to a line as the shortest distance to the line

Resource: Construct tool shapes and cut shapes in half using only a ruler and a protractor in this Walking Dead themed worksheet .

Describing, sketching and drawing

Describe, sketch and draw using conventional terms and notations:

• points, lines, parallel lines, perpendicular lines and right angles
• regular polygons and other polygons that are reflectively and rotationally symmetric

Resource: In this reflective symmetry worksheet themed around logos, students will explore the fundamental principles of symmetry through a practical exercise.

Labelling triangles

• Use the standard conventions for labelling the sides and angles of triangle ABC
• Know and use the criteria for congruence of triangles

Resource: This properties of 2D shapes KS3 worksheet asks students to link the shape to the properties. Students need to read each description carefully then draw a line to match up the correct shapes and properties.

Meanwhile, this right-angled triangles matching activity focuses on finding sides and angles.

Properties of shapes

• Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures using appropriate language and technologies

Resources: Use this PowerPoint listing the properties of quadrilaterals – including angles, equal/parallel sides, line and rotational symmetry and diagonals – as part of your teaching on geometry and measures.

Translations, rotations and reflections

• Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures

Resource: With this resource students will describe and carry out translations, reflections and rotations and identify a single transformation that has the same effect as certain combinations of multiple other transformations.

Constructing shapes

• Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

Properties of angles

• Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles

Resource: This KS3 maths worksheet looks at angles on a straight line . The objective is to be able to find missing angles on a straight line and around a point without using a protractor.

Parallel lines

• Understand and use the relationship between parallel lines and alternate and corresponding angles

Resources: Rather than finding particular unknown angles in each parallel lines question, this angles in parallel lines worksheet involves working out any angles possible and justifying your reasoning.

Alternatively, in this activity , students have to solve angle problems and specify each angle fact that they used in a grid.

Finally, this PowerPoint on angles of parallel lines shows alternate, corresponding and interior angles, followed by some questions to practise.

• Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons

Resource: This activity is perfect for lower secondary students. It allows them to practise calculating angles in isosceles and scalene triangles.

Angles and sides

• Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem
• Use known results to obtain simple proofs

Resources: Prevent confusion among your students when studying the intricacies of Pythagoras’ theorem with the aid of this worksheet .

Or try this Pythagoras PRET homework worksheet . In PRET homeworks, pupils practise, recall, extend and think.

Alternatively, this lesson uses Pythagoras’ Theorem to find the distance between coordinates and includes extended and reversed questions.

Right-angled triangles

• Use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles

Resources: This trigonometry lesson introduces sin, cos and tan, with a little quiz and questions at the the end.

Alternatively, this trigonometry resource includes a levelled activity with progressively harder questions.

Finally, this right-angled triangles activity takes students through six matching activities, three on finding sides and three on finding angles.

Solving problems in 3D

• Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3D

Resource: This advanced trigonometry resource covers 3D Trigonometry, 3D Pythagoras, angles between a line and a plane, angles between two planes and sine/cosine rules.

Mathematical relationships

• Interpret mathematical relationships both algebraically and geometrically.

Resource: This KS3 maths worksheets booklet from White Rose Maths contains over 100 problem-solving questions.

Probability experiments

• Record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale

Resources: This PowerPoint introduces probability and using the probability scale. It then moves on to show how to calculate simple probability using a tube of Smarties.

Alternatively, use these probability worksheets and PowerPoint to cover describing probability, the probability scale, how we calculate probability and more.

In this probability KS3 lesson plan with worksheet, students will carry out an experiment to estimate the probability that when they throw three ordinary dice, they will show consecutive numbers.

Probabilities of all possible outcomes

• Understand that the probabilities of all possible outcomes sum to 1

Resource: This KS3 probability resource contains a coin probability activity, dice probability demo, PowerPoint, worksheet pack and quiz.

• Enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams

Resource: Try this probability lesson from Dr Frost Maths .

Theoretical probabilities

• Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes
• Use these to calculate theoretical probabilities

Resources: Use this Year 8 resource to help pupils identify the sample space for both a single event and two combined events (eg adding two dice) and use this to calculate probabilities.

Or try this sample spaces and calculating probabilities worksheet to show how a sample space can help when finding combinations of events. 

Distributions of a single variable

Describe, interpret and compare observed distributions of a single variable through:

• appropriate graphical representation involving discrete, continuous and grouped data
• appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)

Resources: This worksheet involves finding the median and mean from sets of data, plus calculating the mean from a frequency table and estimating the mean from grouped data.

These funny maths codebreakers cover a wide range of maths topics including average and range.

Tables, charts and diagrams

• Construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data

Resources: This worksheet covers interpreting statistical graphs . Alternatively, these tutorials cover everything from bar charts to histograms and simple averages to estimating their mean.

In this lesson , students learn how to calculate the angles of the sectors in a pie chart for various data presented in frequency tables.

Scatter graphs

Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs.

Resources: In this lesson students draw and interpret scatter diagrams, identify correlation, draw a line of best fit and more.

Or try this PowerPoint and worksheet on plotting scatter graphs, identifying correlation and using lines of best fit.

Finally, help out the famous Weatherfield residents of  Coronation Street  with their data issues by drawing and using scatter graphs.

Browse more KS3 maths games and lesson ideas and GCSE maths games .

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Every Question Helps You Learn

Key Stage 3 Maths

When you answer 8 or more questions correctly your red streak will increase in length. The green streak shows the best player so far today. See our Hall of Fame for previous daily winners.

Each time you answer 10 questions correctly in a quiz your streak will lengthen.

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Go to the Quizzes

The problem with Maths is you only need to miss a few key topics and you’ll be more confused than a donkey in a spelling bee. Keeping on top of everything you learn in class is a must. That way, when you progress to new topics, you’ll have a firm understanding of previous topics to fall back on.

How can you keep on top of the entire Maths syllabus? It’s quite a task, but luckily we have Maths quizzes for topics covered in Year 7, Year 8 and Year 9. That means you’re sorted for the whole of KS3! Problem solved.

Listening ears at the ready, future Maths geniuses. Here’s what you’ll be learning...

Subjects You Will Learn About

We’ll give you a comprehensive overview of Algebra, including sequences, powers, factors, expressions and more. Then we’ll look at Data Handling (you know, collecting/recording data and those pesky frequency diagrams). Next up is a thorough look at Numbers - decimals, percentages, rounding, place value, the lot! We’ll also teach you all about shapes, including Pythagoras’ Theorem and other juicy stuff.

We cover all levels. You can progress through our quizzes at your own pace - even if that means going over old topics to refresh them in your memory, or taking a peek at new topics to familiarise yourself before you cover it in class!

Thankfully, we teach you all this stuff without making you sleepy. That’s right, we feed all this Mathematics knowledge into your brain in a way that you’ll really enjoy!

We’ve taken all the core bits from the Mathematics National Curriculum and whizzed them up into engaging and enjoyable quizzes.

Ready to get started? Numbers here we come! 3, 2, 1...

Maths and Science Go Hand in Hand

It really is true that you can't be a good scientist without a thorough understanding of mathematics. So, if ever you feel that maths is a little "dry" just remember that it will do wonders for your work in science and then treat yourself to a few scientific quizzes in our KS3 Science section!

KS3 Maths Need not be a Chore!

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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!

Problem solving with trigonometry

I can use my knowledge of trigonometry to solve problems.

Lesson details

Key learning points.

• Sometimes there can be more than one valid approach to a problem.
• You should consider all the skills you have.
• Consider multiple approaches and take time to evaluate each.
• Keep the goal in mind throughout, remember what you are trying to find out.

Common misconception

Pupils may struggle to identify a right-angled triangle in the problem.

Encourage pupils to sketch the question out and add any known right angles or perpendicular lines, as this may lead to a right-angled triangle.

Trigonometric ratios - The trigonometric ratios are ratios between each pair of lengths in a right-angled triangle.

This content is © Oak National Academy Limited ( 2024 ), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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6 questions.

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FREE secondary maths resources

Here you can find all of our free secondary resources; explore the collections or navigate to each section to filter resources by topic

15 Algebra Questions And Practice Problems (KS3 & KS4): Harder GCSE Exam Style Questions Included

Beki Christian

Algebra questions involve using letters or symbols to represent unknown values or values that can change. Here you will find 15 algebra questions to test your knowledge and show you the different ways that algebra can be used to solve a problem to find an unknown value or to make generalisations.

Algebra in KS3 and KS4

There are many topics and techniques within algebra . In KS3 we learn to write and manipulate basic algebraic expressions and linear equations. In KS4 we develop these techniques to allow us to deal with more complicated algebra problems such as ones that involve quadratic equations or a system of equations.

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How to solve algebraic questions

When you are presented with an algebraic problem it is important to establish what you are being asked to do. Here are some of the key terms along with what they mean:

• Solve the equation – find out the value of the unknown
• Substitute – put the values you have been given into the algebraic expression
• Simplify – collect together like terms to make the expression or equation look simpler
• Expand brackets – multiply out the brackets
• Factorise – put into brackets
• Make x the subject – rewrite the equation in the form x =…..

Remember, when working with algebra, we must still apply BODMAS / BIDMAS. i.e. B rackets, I ndices (powers, exponents, square roots), D ivision, M ultiplication, A ddition, S ubtraction.

When working with algebraic expressions and equations we must consider carefully which operations to deal with first.

Download this 15 Algebra Questions And Practice Problems (KS3 & KS4) Worksheet

Help your students prepare for their Maths GCSE with this free Algebra worksheet of 15 multiple choice questions and answers.

Algebra in KS2

The ideas of writing and simplifying expressions, solving equations and substitution are introduced in KS2. Here are some example KS2 algebra questions:

Algebra questions KS2

1. A chocolate bar costs c pence and a drink costs d pence. Write down an expression for the cost of 2 chocolate bars and 2 drinks.

2 chocolate bars would cost 2 lots of c, or 2c, and 2 drinks would cost 2 lots of d, or 2d.

2. Simplify the expression 4m+5+2m-1

We need to collect together like terms here so 4m + 2m = 6m and 5 – 1 = 4 (watch out for the negative).

Algebra questions KS3

In KS3 we learn a variety of different algebra techniques to answer algebra questions and to practise problem solving with algebra. These include:

• Simplifying algebraic expressions
• Expanding brackets and factoring
• Forming algebraic equations from word problems
• Solving algebraic equations and inequalities
• Substituting into expressions
• Changing the subject of an equation
• Working with real life graphs and straight line graphs

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Algebra questions KS3: basic algebra

1. In this pyramid, you add two adjacent blocks to find the value of the block above.

What expression will be in the top box?

2.  Brian is a window cleaner. He uses the following formula to calculate the amount to charge his customers:

Charge = £20 + 4n

Where n is the number of windows a house has.

If a house has 7 windows, how much would Brian charge?

In this question, n is 7 so we can substitute 7 into the formula.

Charge = £20 + 4 × 7

Charge = £48

3. The area of a rectangle is 4x-6.

Which of the following pairs could be the length and width of the rectangle?

2x and 2x-3

There are two ways of attempting this question. We know that area \;of \;a \;rectangle = length × width so we could multiply each pair together to see which pair makes 4x − 6.

Alternatively, if we factorise 4x − 6 we get 2(2x − 3) meaning the sides could be 2 and 2x − 3.

4.  The formula for changing degrees Celsius to degrees Fahrenheit is

F=\frac{9C}{5}+32

Rearrange this formula to make C the subject.

Algebra questions KS3: forming and solving equations

5. Work out the size of the smallest angle.

The angles in a triangle add up to 180^{\circ} therefore we can write

Now we have an equation we can solve.

The angles are :

The smallest angle is 34^{\circ} .

6.  Jamie’s dad is 4 times older than Jamie. In 14 years time, Jamie’s dad will be twice the age of Jamie.

What is the sum of Jamie’s age now and Jamie’s dad’s age now?

To solve this we need to write an equation.

Let Jamie’s age now be x . Then Jamie’s dad’s age is 4x .

In 14 years time Jamie’s age will be x + 14 and Jamie’s dad’s age will be 4x + 14 .

Since we know Jamie’s dad’s age will be two times Jamie’s age, we can write

4x+14=2(x+14)

Jamie is currently 7 years old meaning his dad is 28 years old. The sum of their ages is 35 .

Algebra questions KS3: graphs

7.  Which of the following lines passes through the point (2, 5)?

At the point (2, 5), x is 2 and y is 5. We can check which equation works when we substitute in these values:

Algebra questions KS4

Algebra is studied extensively in the GCSE and IGCSE curriculum.

In KS4 we build on the techniques learnt in KS3. Topics include:

• Expanding and factorising polynomials
• Solving quadratic equations
• Solving simultaneous equations
• Inequalities
• Algebraic fractions
• Further work on graphs

Algebra questions KS4: algebraic manipulation

8. Which of the following expressions has the smallest value when a=5 and b=-3?

9. Find an expression in terms of x for the volume of this cuboid

Volume = (5x+1)(2x-3)(3x-1)

Volume = (10x^2+2x-15x-3)(3x-1)

Volume = (10x^2-13x-3)(3x-1)

Volume = 30x^3-39x^2-9x-10x^2+13x+3

Volume = 30x^3-49x^2+4x+3

Algebra questions KS4: forming and solving equations

10. The area of this triangle is 24cm^2 .

Work out the perimeter of the triangle.

The area of a triangle is area = \frac{1}{2} × b × h.

If we fill in what we know we get:

Since x = 3 , the side lengths are 6m, 8cm and 10cm .

The perimeter is 6 + 8 + 10 = 24cm .

11.  Solve the equation x+2-\frac{15}{x}=0

x=-2 or x=15

x=-3 or x=5

x=-5 or x=3

x=-15 or x=2

We can make this a bit easier by getting rid of the fraction involving x. We do this by multiplying each term by x.

12. At a theme park the Jones family purchased 2 adult tickets and 3 child tickets for £48 .  The Evans family purchased 3 adult tickets and 1 child ticket for £44 . 

Calculate the cost of one child ticket.

We can write simultaneous equations to solve this.

2a+3c=48 (Equation 1) 3a+c=44 (Equation 2)

Multiply equation 2 by 3 to make the coefficients of c equal: 9a+3c=132 (Equation 3)

Subtract equation 1 from equation 3: 7a= 84 a=12

Substitute a into equation 3: 3×12+c=44 36+c=44 c=8

The cost of an adult ticket is £12 and a child ticket is £8 .

Algebra questions KS4: graphs

13.  Which of these lines is parallel to the line 2y = x + 7

For two lines to be parallel, their gradient must be equal.

If we rearrange 2y=x+7 to make y the subject we get y=\frac{1}{2}x+\frac{7}{2}.

The gradient is \frac{1}{2}

14. Find the minimum value of the function f(x) = x^2+4x+5 .

To find the minimum value we need to complete the square.

The minimum value is 1. This occurs when (x+2) is 0.

15. The diagram shows the circle x^2+y^2=25 . The line is a tangent to the circle at the point (3,4) . Work out the equation of the line.

To work out the gradient of the line we need to work out the gradient of the normal.

We know that the normal goes through the points (0, 0) and (3, 4) so we can calculate the gradient: \frac{4-0}{3-0}=\frac{4}{3}.

The gradient of the tangent will be \frac{-3}{4}.

We can now use y=mx+c . We know the tangent goes through the point (3, 4) and that it’s gradient is \frac{-3}{4} .

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GCSE Problem Solving Questions of the Day - Compilation

Subject: Mathematics

Age range: 11-14

Resource type: Worksheet/Activity

Last updated

10 March 2023

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