NRICH Secondary Curriculum Map

The problems linked below have detailed Teachers' Resources suggesting how they can be used in the classroom.

Please email any comments to secondary.nrich@maths.org

Looking for primary problems? See the NRICH Primary Curriculum Map.

Key

Games are indicated by ‘G’ and Articles by 'A'.

Tasks badged filled star are suitable for the whole class;
Tasks badged filled starfilled star are suitable for the majority;
Tasks badged filled starfilled starfilled star are for those who like a serious challenge.

Highlight ‘Thinking mathematically’ or ‘Mathematical mindset’ problems

NUMBER

Pre-Secondary Age 11 – 12 Age 15 - 16 Extension

Place Value, Integers, Ordering & Rounding

Number and Place Value

Understand and use place value for decimals, measures and integers of any size.
Nice or Nasty Nice or Nasty
There are nasty versions of this dice game but we'll start with the nice ones...
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Add to 200 Add to 200
By selecting digits for an addition grid, what targets can you make?
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More Less is More More Less is More
In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
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Dicey Operations Dicey Operations

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

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Subtraction Surprise Subtraction Surprise
Try out some calculations. Are you surprised by the results?
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Forwards Add Backwards Forwards Add Backwards
What happens when you add a three digit number to its reverse?
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Legs Eleven Legs Eleven
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
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How Many Miles To Go? How Many Miles To Go?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
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Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
Number Lines in Disguise Number Lines in Disguise
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
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Farey Sequences Farey Sequences
There are lots of ideas to explore in these sequences of ordered fractions.
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Difference Sudoku Difference Sudoku
Use the differences to find the solution to this Sudoku.
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Round numbers and measures to an appropriate degree of accuracy (for example, to a number of decimal places or significant figures)
Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b Apply and interpret limits of accuracy when rounding or truncating (including upper and lower bounds)
Place your orders Place your orders
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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Place Value, Integers, Ordering & Rounding short problems

Factors, Multiples & Primes

Properties of Numbers

Use concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
Divisibility Tests Divisibility Tests
This article explains various divisibility rules and why they work. An article to read with pencil and paper handy.
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Factors and Multiples Puzzle Factors and Multiples Puzzle
Using your knowledge of the properties of numbers, can you fill all the squares on the board?
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Multiples Sudoku Multiples Sudoku
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
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Sieve of Eratosthenes Sieve of Eratosthenes
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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Number Families Number Families
How many different number families can you find?
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Statement Snap Statement Snap
You'll need to know your number properties to win a game of Statement Snap...
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Dozens Dozens
Can you select the missing digit(s) to find the largest multiple?
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Factors and Multiples Game Factors and Multiples Game

A game in which players take it in turns to choose a number. Can you block your opponent?

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American Billions American Billions
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
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Stars Stars
Can you work out what step size to take to ensure you visit all the dots on the circle?
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How much can we spend? How much can we spend?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
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Multiple Surprises Multiple Surprises
Sequences of multiples keep cropping up...
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Gabriel's Problem Gabriel's Problem
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
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Power mad! Power mad!

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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Counting Factors Counting Factors
Is there an efficient way to work out how many factors a large number has?
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Funny Factorisation Funny Factorisation
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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Product Sudoku Product Sudoku
The clues for this Sudoku are the product of the numbers in adjacent squares.
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Alison's quilt Alison's quilt
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
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Xavi's T-shirt Xavi's T-shirt
How much can you read into a T-shirt?
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Latin Numbers Latin Numbers
Can you create a Latin Square from multiples of a six digit number?
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Differences Differences
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
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Filling the gaps Filling the gaps
Which numbers can we write as a sum of square numbers?
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Take Three From Five Take Three From Five

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

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Beelines Beelines
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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LCM Sudoku LCM Sudoku
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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Expenses Expenses
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
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Shopping Basket Shopping Basket
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
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Appreciate the infinite nature of the sets of integers, real and rational numbers
Route to infinity Route to infinity
Can you describe this route to infinity? Where will the arrows take you next?
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Diminishing Returns Diminishing Returns
How much of the square is coloured blue? How will the pattern continue?
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Factors, Multiples & Primes short problems

Powers and Roots

Properties of Numbers

Use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations Estimate powers and roots of any given positive number
Sticky Numbers Sticky Numbers
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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Pocket money Pocket money
Which of these pocket money systems would you rather have?
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Generating Triples Generating Triples
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
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Calculate with roots, and with integer (and fractional) indices
Power Countdown Power Countdown
In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.
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Interpret and compare numbers in standard form A x 10n 1≤A<10, where n is a positive or negative integer or zero
Standard Index Form Matching Standard Index Form Matching
Can you match these calculations in Standard Index Form with their answers?
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A question of scale A question of scale
Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?
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Powers and Roots short problems

Fractions, Decimals & Percentages

Fractions, Decimals and Percentages

Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and ⁷⁄₂ or 0.375 and ⅜) Change recurring decimals into their corresponding fractions and vice versa
Terminating or not Terminating or not
Is there a quick way to work out whether a fraction terminates or recurs when you write it as a decimal?
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Tiny Nines Tiny Nines
What do you notice about these families of recurring decimals?
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Repetitiously Repetitiously
Can you express every recurring decimal as a fraction?
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Interpret fractions and percentages as operators
Peaches today, Peaches tomorrow... Peaches today, Peaches tomorrow...
A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?
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Ben's Game Ben's Game
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
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A Chance to Win? A Chance to Win?
Imagine you were given the chance to win some money... and imagine you had nothing to lose...
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Define percentage as 'number of parts per hundred', interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%
Doughnut percents Doughnut percents
A task involving the equivalence between fractions, percentages and decimals which depends on members of the group noticing the needs of others and responding.
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Matching Fractions, Decimals and Percentages Matching Fractions, Decimals and Percentages
Can you match pairs of fractions, decimals and percentages, and beat your previous scores?
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Fractions and Percentages Card Game Fractions and Percentages Card Game
Can you find the pairs that represent the same amount of money?
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Fractions, Decimals & Percentages short problems

Number Operations and Calculation Methods

Addition and Subtraction

Multiplication, Division and Ratio

Calculating with Fractions and Decimals

Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
Adding and Subtracting Positive and Negative Numbers Adding and Subtracting Positive and Negative Numbers
How can we help students make sense of addition and subtraction of negative numbers?
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Consecutive Numbers Consecutive Numbers
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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Have you got it? Have you got it?
Can you explain the strategy for winning this game with any target?
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Number Daisy Number Daisy
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
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Consecutive Seven Consecutive Seven
Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?
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Fractions jigsaw Fractions jigsaw
A jigsaw where pieces only go together if the fractions are equivalent.
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Connect Three Connect Three
In this game the winner is the first to complete a row of three. Are some squares easier to land on than others?
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Countdown fractions Countdown fractions
Here is a chance to play a fractions version of the classic Countdown Game.
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Going round in circles Going round in circles
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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Magic Letters Magic Letters
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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Cryptarithms Cryptarithms
Can you crack these cryptarithms?
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Almost One Almost One
Choose some fractions and add them together. Can you get close to 1?
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Strange Bank Account Strange Bank Account

Imagine a very strange bank account where you are only allowed to do two things...

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Up, down, flying around Up, down, flying around

Play this game to learn about adding and subtracting positive and negative numbers

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Two and Two Two and Two

How many solutions can you find to this sum? Each of the different letters stands for a different number.

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Where can we visit? Where can we visit?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
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Egyptian Fractions Egyptian Fractions
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
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Method in multiplying madness? Method in multiplying madness?
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
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Consecutive negative numbers Consecutive negative numbers
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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Weights Weights
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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Keep it simple Keep it simple
Can all unit fractions be written as the sum of two unit fractions?
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the greedy algorithm the greedy algorithm
The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.
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Making a difference Making a difference
How many different differences can you make?
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Same Answer Same Answer
Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?
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Largest product Largest product
Which set of numbers that add to 100 have the largest product?
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Round and round and round Round and round and round
Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?
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Twisting and Turning Twisting and Turning
Take a look at the video and try to find a sequence of moves that will untangle the ropes.
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Impossibilities Impossibilities
Just because a problem is impossible doesn't mean it's difficult...
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Overlaps Overlaps
Can you find ways to put numbers in the overlaps so the rings have equal totals?
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Cinema Problem Cinema Problem

A cinema has 100 seats. How can ticket sales make £100 for these different combinations of ticket prices?

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More Twisting and Turning More Twisting and Turning
It would be nice to have a strategy for disentangling any tangled ropes...
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Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals
Can you Make 100? Can you Make 100?
How many ways can you find to put in operation signs (+, −, ×, ÷) to make 100?
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Recognise and use relationships between operations including inverse operations
5 by 5 Mathdokus 5 by 5 Mathdokus
Can you use the clues to complete these 5 by 5 Mathematical Sudokus?
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Remainders Remainders
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
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Missing Multipliers Missing Multipliers
What is the smallest number of answers you need to reveal in order to work out the missing headers?
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Countdown Countdown

Here is a chance to play a version of the classic Countdown Game.

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Calculate exactly with fractions, surds, and multiples of π; simplify surd expressions involving squares and rationalise denominators
The Root of the Problem The Root of the Problem
Find the sum of this series of surds.
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Use a calculator and other technologies to calculate results accurately and then interpret them appropriately
Number Operations and Calculation Methods short problems

Ratio, Proportion & Rates of Change

Ratio and Proportion

Change freely between related standard units (for example time, length, area, volume / capacity, mass)
Thousands and Millions Thousands and Millions
Here's a chance to work with large numbers...
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All in a jumble All in a jumble
My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.
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Olympic Measures Olympic Measures
These Olympic quantities have been jumbled up! Can you put them back together again?
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Nutrition and Cycling Nutrition and Cycling
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
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Use scale factors, scale diagrams and maps
Express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
Fractions Rectangle Fractions Rectangle
The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?
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Use ratio notation, including reduction to simplest form Compare lengths, areas and volumes using ratio notation and / or scale factors; make links to similarity (including trigonometric ratios)
Mixing Lemonade Mixing Lemonade

Can you work out which drink has the stronger flavour?

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Mixing Paints Mixing Paints
Can you work out how to produce different shades of pink paint?
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Mixing More Paints Mixing More Paints
Can you find an efficent way to mix paints in any ratio?
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Areas and Ratios Areas and Ratios
Do you have enough information to work out the area of the shaded quadrilateral?
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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
Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions
Solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics
Solve problems involving direct and inverse proportion, including graphical and algebraic representations Understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; construct and interpret equations that describe direct and inverse proportion
Triathlon and Fitness Triathlon and Fitness
The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?
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Interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
Interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts
Set up, solve and interpret the answers in growth and decay problems, including compound interest, and work with general iterative processes
Dating made Easier Dating made Easier
If a sum invested gains 10% each year how long before it has doubled its value?
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The Legacy The Legacy
Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?
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Use compound units such as speed, unit pricing and density to solve problems Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts
Speeding boats Speeding boats
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
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Speed-time problems at the Olympics Speed-time problems at the Olympics
Have you ever wondered what it would be like to race against Usain Bolt?
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Ratio, Proportion & Rates of Change short problems