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.

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... Add to 200 Add to 200 By selecting digits for an addition grid, what targets can you make? 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. Dicey Operations Dicey Operations Who said that adding and subtracting couldn't be fun? G	Subtraction Surprise Subtraction Surprise Try out some calculations. Are you surprised by the results? Forwards Add Backwards Forwards Add Backwards What happens when you add a three digit number to its reverse?	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? 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?
	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?	Farey Sequences Farey Sequences There are lots of ideas to explore in these sequences of ordered fractions.				Difference Sudoku Difference Sudoku Use the differences to find the solution to this Sudoku.
	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?









	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. A 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? Number Families Number Families How many different number families can you find? Statement Snap Statement Snap You'll need to know your number properties to win a game of Statement Snap... 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? G Dozens Dozens Can you select the missing digit(s) to find the largest multiple? Multiples Sudoku Multiples Sudoku Each clue in this Sudoku is the product of the two numbers in adjacent cells. Sieve of Eratosthenes Sieve of Eratosthenes Follow this recipe for sieving numbers and see what interesting patterns emerge.	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... Stars Stars Can you work out what step size to take to ensure you visit all the dots on the circle? 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? Multiple Surprises Multiple Surprises Sequences of multiples keep cropping up... Gabriel's Problem Gabriel's Problem Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was? 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.	Counting Factors Counting Factors Is there an efficient way to work out how many factors a large number has? 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? Alison's quilt Alison's quilt Nine squares are fitted together to form a rectangle. Can you find its dimensions? Product Sudoku Product Sudoku The clues for this Sudoku are the product of the numbers in adjacent squares. Xavi's T-shirt Xavi's T-shirt How much can you read into a t-shirt?	Latin Numbers Latin Numbers Can you create a Latin Square from multiples of a six digit number? Differences Differences Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number? Filling the gaps Filling the gaps Which numbers can we write as a sum of square numbers? 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?	Beelines Beelines Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses? LCM Sudoku LCM Sudoku Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.	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? Shopping Basket Shopping Basket The items in the shopping basket add and multiply to give the same amount. What could their prices be?
		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?	Diminishing Returns Diminishing Returns How much of the square is coloured blue? How will the pattern continue?











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?	Pocket money Pocket money Which of these pocket money systems would you rather have?	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?
		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.
		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?	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?










	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?	Tiny Nines Tiny Nines What do you notice about these families of recurring decimals?	Repetitiously Repetitiously Can you express every recurring decimal as a fraction?
		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?	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?	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...
		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.	Matching Fractions, Decimals and Percentages Matching Fractions, Decimals and Percentages Can you match pairs of fractions, decimals and percentages, and beat your previous scores?	Fractions and Percentages Card Game Fractions and Percentages Card Game Can you find the pairs that represent the same amount of money?










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? A 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? 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? Fractions jigsaw Fractions jigsaw A jigsaw where pieces only go together if the fractions are equivalent. Countdown fractions Countdown fractions Here is a chance to play a fractions version of the classic Countdown Game. Going round in circles Going round in circles Mathematicians are always looking for efficient methods for solving problems. How efficient can you be? 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? Cryptarithms Cryptarithms Can you crack these cryptarithms? Almost One Almost One Choose some fractions and add them together. Can you get close to 1? Strange Bank Account Strange Bank Account Imagine a very strange bank account where you are only allowed to do two things... 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. Consecutive Numbers Consecutive Numbers An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. Up, down, flying around Up, down, flying around Play this game to learn about adding and subtracting positive and negative numbers G 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? Have you got it? Have you got it? Can you explain the strategy for winning this game with any target?	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? 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? Consecutive negative numbers Consecutive negative numbers Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers? Weights Weights Different combinations of the weights available allow you to make different totals. Which totals can you make? Keep it simple Keep it simple Can all unit fractions be written as the sum of two unit fractions? 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. Making a difference Making a difference How many different differences can you make? Same Answer Same Answer Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples? 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.	Largest product Largest product Which set of numbers that add to 100 have the largest product? 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? 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. Overlaps Overlaps Can you find ways to put numbers in the overlaps so the rings have equal totals? Cinema Problem Cinema Problem A cinema has 100 seats. How can ticket sales make £100 for these different combinations of ticket prices? Impossibilities Impossibilities Just because a problem is impossible doesn't mean it's difficult...	More Twisting and Turning More Twisting and Turning It would be nice to have a strategy for disentangling any tangled ropes...
		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?
		Recognise and use relationships between operations including inverse operations
		Missing Multipliers Missing Multipliers What is the smallest number of answers you need to reveal in order to work out the missing headers? Countdown Countdown Here is a chance to play a version of the classic Countdown Game. G The Remainders Game The Remainders Game Play this game and see if you can figure out the computer's chosen number. G 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? 5 by 5 Mathdokus 5 by 5 Mathdokus Can you use the clues to complete these 5 by 5 Mathematical Sudokus?
		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.
		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...	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. Olympic Measures Olympic Measures These Olympic quantities have been jumbled up! Can you put them back together again?		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?
		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?
		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?		Mixing Paints Mixing Paints Can you work out how to produce different shades of pink paint? Mixing More Paints Mixing More Paints Can you find an efficent way to mix paints in any ratio?		Areas and Ratios Areas and Ratios Do you have enough information to work out the area of the shaded quadrilateral?
		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?
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? 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?
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? 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?
Ratio, Proportion & Rates of Change short problems

ALGEBRA

Pre-Secondary	Age 11 – 12	→	→	→	Age 15 - 16	Extension
Patterns and Sequences
Patterns Sequences	Generate terms of a sequence from either a term-to-term or a position-to-term rule			Deduce expressions to calculate the nth term of linear and quadratic sequences
	Beach Huts Beach Huts Can you figure out how sequences of beach huts are generated? Summing Consecutive Numbers Summing Consecutive Numbers 15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?	Frogs Frogs How many moves does it take to swap over some red and blue frogs? Do you have a method? Seven Squares Seven Squares Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100? Charlie's delightful machine Charlie's delightful machine Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?	1 Step 2 Step 1 Step 2 Step Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps? Triangle Numbers Triangle Numbers Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue? Tower of Hanoi Tower of Hanoi The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.	A little light thinking A little light thinking Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
	Recognise arithmetic sequences and find the nth term
	Train Spotters' Paradise Train Spotters' Paradise Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising. A Squares in rectangles Squares in rectangles A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all? Odds, Evens and More Evens Odds, Evens and More Evens Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions... Shifting Times Tables Shifting Times Tables Can you find a way to identify times tables after they have been shifted up or down?	spaces for exploration spaces for exploration Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms. A Coordinate Patterns Coordinate Patterns Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead? What numbers can we make? What numbers can we make? Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?	Days and Dates Days and Dates Investigate how you can work out what day of the week your birthday will be on next year, and the year after... Elevenses Elevenses How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results? Growing Surprises Growing Surprises Can you find the connections between linear and quadratic patterns? What numbers can we make now? What numbers can we make now? Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
	Recognise geometric sequences and appreciate other sequences that arise			Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rⁿ where n is an integer, and r is a positive rational number or a surd) and other sequences
	Picturing Square Numbers Picturing Square Numbers Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?		Picturing Triangular Numbers Picturing Triangular Numbers Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?	Attractive Tablecloths Attractive Tablecloths Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs? Painted Cube Painted Cube Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? Mystic Rose Mystic Rose Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes. Steel Cables Steel Cables Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?	Picture Story Picture Story Can you see how this picture illustrates the formula for the sum of the first six cube numbers? Partly Painted Cube Partly Painted Cube Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use? Double Trouble Double Trouble Simple additions can lead to intriguing results...	Summing geometric progressions Summing geometric progressions Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?










	Patterns and Sequences short problems
	Creating & Manipulating Linear & Quadratic Expressions
		Use and interpret algebraic notation For example: • ab in place of a x b • 3y in place of y + y+ y and 3 x y • a² in place of a x a; a³ in place of a x a x a; a²b in place of a x a x b • a/b in place of a ÷ b • coefficients written as fractions rather than as decimals • brackets
		Your number is... Your number is... Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know? The Number Jumbler The Number Jumbler The Number Jumbler can always work out your chosen symbol. Can you work out how?	Crossed Ends Crossed Ends Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends? Always a multiple? Always a multiple? Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...	Think of Two Numbers Think of Two Numbers Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How? Reversals Reversals Where should you start, if you want to finish back where you started? Special Numbers Special Numbers My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?	Puzzling Place Value Puzzling Place Value Can you explain what is going on in these puzzling number tricks?
		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			Simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by: • Factorising quadratic expressions of the form x² + bx + c, including the difference of two squares; factorising quadratic expressions of the form ax² + bx + c • Simplifying expressions involving sums, products and powers, including the laws of indices
		Perimeter Expressions Perimeter Expressions Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make? More Number Pyramids More Number Pyramids When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... Number Pyramids Number Pyramids Try entering different sets of numbers in the number pyramids. How does the total at the top change?	Fibonacci Surprises Fibonacci Surprises Play around with the Fibonacci sequence and discover some surprising results!		Plus Minus Plus Minus Can you explain the surprising results Jo found when she calculated the difference between square numbers? What's Possible? What's Possible? Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make? Multiplication square Multiplication square Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice? Harmonic Triangle Harmonic Triangle Can you see how to build a harmonic triangle? Can you work out the next two rows? Pythagoras Perimeters Pythagoras Perimeters If you know the perimeter of a right angled triangle, what can you say about the area? Hollow Squares Hollow Squares Which armies can be arranged in hollow square fighting formations? Pair Products Pair Products Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? Quadratic Patterns Quadratic Patterns Surprising numerical patterns can be explained using algebra and diagrams...	Finding factors Finding factors Can you find the hidden factors which multiply together to produce each quadratic expression? Factorising with Multilink Factorising with Multilink Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units? Difference of Two Squares Difference of Two Squares What is special about the difference between squares of numbers adjacent to multiples of three? Square Number Surprises Square Number Surprises There are unexpected discoveries to be made about square numbers...	2-Digit Square 2-Digit Square A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number? Why 24? Why 24? Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results. Always Perfect Always Perfect Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. Perfectly Square Perfectly Square The sums of the squares of three related numbers is also a perfect square - can you explain why?











Creating & Manipulating Linear & Quadratic Expressions short problems
Equations & Formulae
Equations Formulae		Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors			Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
		Shape Products Shape Products These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?					Iff Iff Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?
		Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement)
			Your number was... Your number was... Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?		Fair Shares? Fair Shares? A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
		Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs			Find approximate solutions to equations numerically using iteration
				Temperature Temperature Is there a temperature at which Celsius and Fahrenheit readings are the same?
		Solve two simultaneous equations in two variables (linear / linear or linear / quadratic) algebraically; find approximate solutions using a graph
				What's it worth? What's it worth? There are lots of different methods to find out what the shapes are worth - how many can you find? Fruity Totals Fruity Totals In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?	Warmsnug Double Glazing Warmsnug Double Glazing How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price? Arithmagons Arithmagons Can you find the values at the vertices when you know the values on the edges?	Matchless Matchless There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ? Multiplication arithmagons Multiplication arithmagons Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons? CD Heaven CD Heaven All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at each price?
		Solve linear inequalities in one or two variables, and quadratic inequalities in one variable; represent the solution set on a number line, using set notation, and on a graph
					Which is cheaper? Which is cheaper? When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?	Which is bigger? Which is bigger? Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?
		Solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square, and by using the quadratic formula; find approximate solutions using a graph
						How old am I? How old am I? In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?	Mega Quadratic Equations Mega Quadratic Equations What do you get when you raise a quadratic to the power of a quadratic?
		Substitute numerical values into formulae and expressions, including scientific formulae			Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution
					Terminology Terminology Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles? Pick's Theorem Pick's Theorem Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.	Training schedule Training schedule The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?
		Understand and use standard mathematical formulae; rearrange formulae to change the subject





	Equations & Formulae short problems
	Functions and Graphs
	Position, Direction and Movement	Where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the 'inverse functions'; interpret the succession of two functions as a 'composite function'
		Work with coordinates in all four quadrant			Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts
		Dotty grids in the classroom Dotty grids in the classroom This article suggests ways in which a dotty grid can be used in the classroom as an environment for rich exploration A Treasure Hunt Treasure Hunt Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?	Route to infinity Route to infinity Can you describe this route to infinity? Where will the arrows take you next?
		Recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and in the Cartesian plane			Plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration
		Interpret mathematical relationships both algebraically and graphically			Recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point
				Parallel lines Parallel lines How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?
		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			Use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points, or through one point with a given gradient
			How steep is the slope? How steep is the slope? On the grid provided, we can draw lines with different gradients. How many different gradients can you find? Can you arrange them in order of steepness?	Reflecting Lines Reflecting Lines Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings. Translating Lines Translating Lines Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings. Diamond Collector Diamond Collector Collect as many diamonds as you can by drawing three straight lines. G	Perpendicular lines Perpendicular lines Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines? At right angles At right angles Can you decide whether two lines are perpendicular or not? Can you do this without drawing them? Surprising Transformations Surprising Transformations I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?	Doesn't add up Doesn't add up In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
Use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous equations			Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1/x with x ≠ 0, the exponential function y = kˣ for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size
				Negatively Triangular Negatively Triangular How many intersections do you expect from four straight lines ? Which three lines enclose a triangle with negative co-ordinates for every point ?		Back fitter Back fitter 10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes? What's that graph? What's that graph? Can you work out which processes are represented by the graphs?
Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs			Sketch translations and reflections of the graph of a given function
			Fill Me Up Fill Me Up Can you sketch graphs to show how the height of water changes in different containers as they are filled? Speeding up, slowing down Speeding up, slowing down Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its speed at each stage. How far does it move? How far does it move? Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects the distance it travels at each stage. Up and across Up and across Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its vertical and horizontal movement at each stage.			Parabolic Patterns Parabolic Patterns The illustration shows the graphs of fifteen functions. Two of them have equations $y=x^2$ and $y=-(x-4)^2$. Find the equations of all the other graphs. Tangled Trig Graphs Tangled Trig Graphs Can you work out the equations of the trig graphs I used to make my pattern? Exploring cubic functions Exploring cubic functions Quadratic graphs are very familiar, but what patterns can you explore with cubics?
Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically, and turning points by completing the square
						Curve fitter Curve fitter This problem challenges you to find cubic equations which satisfy different conditions.





Functions and Graphs short problems

GEOMETRY

Pre-Secondary	Age 11 – 12	→	→	→	Age 15 - 16	Extension
Angles, Polygons and Geometrical Proof
Identifying Shapes and Their Properties Comparing and Classifying Angles	Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures (for example, equal lengths and angles) using appropriate language and technologies			Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
	Quadrilaterals game Quadrilaterals game A game for 2 or more people, based on the traditional card game Rummy. G An Equilateral Triangular Problem An Equilateral Triangular Problem Take an equilateral triangle and cut it into smaller pieces. What can you do with them? Guess my Quad Guess my Quad How many questions do you need to identify my quadrilateral?	Completing Quadrilaterals Completing Quadrilaterals We started drawing some quadrilaterals - can you complete them? Property chart Property chart A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?	Square coordinates Square coordinates A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?	Circles in quadrilaterals Circles in quadrilaterals Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
	Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric
	Hidden Squares Hidden Squares Can you find the squares hidden on these coordinate grids? Square It Square It Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square. Parallelogram It Parallelogram It Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a parallelogram.	Shapely pairs Shapely pairs A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow... Rhombus It Rhombus It Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.	Opposite vertices Opposite vertices Can you recreate squares and rhombuses if you are only given a side or a diagonal?
	Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles
	Draw and measure line segments and angles in geometric figures, including interpreting scale drawings
	Estimating angles Estimating angles How good are you at estimating angles? G
	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
		Triangles in circles Triangles in circles Can you find triangles on a 9-point circle? Can you work out their angles? Polygon Rings Polygon Rings Join pentagons together edge to edge. Will they form a ring? Polygon Pictures Polygon Pictures Can you work out how these polygon pictures were drawn, and use that to figure out their angles?	Which solids can we make? Which solids can we make? Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids? Star Polygons Star Polygons Draw some stars and measure the angles at their points. Can you find and prove a result about their sum? Semi-regular Tessellations Semi-regular Tessellations Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?	Isosceles Seven Isosceles Seven Is it possible to find the angles in this rather special isosceles triangle?
	Understand and use the relationship between parallel lines and alternate and corresponding angles
	Use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles
	Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids			Apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures
				Quad in Quad Quad in Quad Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it? Triangle in a Triangle Triangle in a Triangle Can you work out the fraction of the original triangle that is covered by the inner triangle? Nicely Similar Nicely Similar If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle? Making sixty Making sixty Why does this fold create an angle of sixty degrees? Same length Same length Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?	Napkin Napkin A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed . Kite in a Square Kite in a Square Can you make sense of the three methods to work out what fraction of the total area is shaded? The square under the hypotenuse The square under the hypotenuse Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle? Triangle midpoints Triangle midpoints You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle? Two Ladders Two Ladders Two ladders are propped up against facing walls. At what height do the ladders cross?	Angle Trisection Angle Trisection It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square. Squirty Squirty Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle. Trapezium Four Trapezium Four The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
	Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras' Theorem, and use known results to obtain simple proofs			Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
		Right angles Right angles Can you make a right-angled triangle on this peg-board by joining up three points round the edge? Triangles in circles Triangles in circles Can you find triangles on a 9-point circle? Can you work out their angles?	Subtended angles Subtended angles What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it? Tilted Squares Tilted Squares It's easy to work out the areas of most squares that we meet, but what if they were tilted? Cyclic Quadrilaterals Cyclic Quadrilaterals Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?			Sitting Pretty Sitting Pretty A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r? Partly Circles Partly Circles What is the same and what is different about these circle questions? What connections can you make?
	Interpret mathematical relationships both algebraically and geometrically
			Quadrilaterals in a Square Quadrilaterals in a Square What's special about the area of quadrilaterals drawn in a square?	Of all the areas Of all the areas Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?



	Angles, Polygons and Geometrical Proof short problems
	Construction
	Drawing and Constructing	Derive and use 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
			Constructing Triangles Constructing Triangles Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?	The Farmers' Field Boundary The Farmers' Field Boundary The farmers want to redraw their field boundary but keep the area the same. Can you advise them?












Construction short problems
Perimeter, Area and Volume
Comparing and Estimating Measuring and Calculating		Derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)			Calculate surface areas and volumes of spheres, pyramids, cones and composite solids
		Shear Magic Shear Magic Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas? On the Edge On the Edge If you move the tiles around, can you make squares with different coloured edges? Changing areas, changing perimeters Changing areas, changing perimeters How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same? Perimeter Possibilities Perimeter Possibilities I'm thinking of a rectangle with an area of 24. What could its perimeter be? Colourful Cube Colourful Cube A colourful cube is made from little red and yellow cubes. But can you work out how many of each?	Fence it Fence it If you have only 40 metres of fencing available, what is the maximum area of land you can fence off? Isosceles Triangles Isosceles Triangles Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? Can they be equal? Can they be equal? Can you find rectangles where the value of the area is the same as the value of the perimeter? Changing areas, changing volumes Changing areas, changing volumes How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same? Isometric Areas Isometric Areas We usually use squares to measure area, but what if we use triangles instead? More Isometric Areas More Isometric Areas Isometric Areas explored areas of parallelograms in triangular units. Here we explore areas of triangles... Cuboid Challenge Cuboid Challenge What's the largest volume of box you can make from a square of paper?	Cuboids Cuboids Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? Sending a Parcel Sending a Parcel What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres? Triangle in a Trapezium Triangle in a Trapezium Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?		Immersion Immersion Various solids are lowered into a beaker of water. How does the water level rise in each case?	Funnel Funnel A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ? Fill Me Up Too Fill Me Up Too In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?
		Calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes			Calculate arc lengths, angles and area of sectors of circles
		Perimeter Challenge Perimeter Challenge Can you deduce the perimeters of the shapes from the information given?	Triangles in a Square Triangles in a Square What are the possible areas of triangles drawn in a square?	Blue and White Blue and White Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest? An Unusual Shape An Unusual Shape Can you maximise the area available to a grazing goat? Efficient cutting Efficient cutting Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end. Cola Can Cola Can An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?	Arclets Arclets Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes". Gutter Gutter Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter? Curvy areas Curvy areas Have a go at creating these images based on circles. What do you notice about the areas of the different sections? Track design Track design Where should runners start the 200m race so that they have all run the same distance by the finish?	Triangles and petals Triangles and petals An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers? Salinon Salinon This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?











		Perimeter, Area and Volume short problems
	3D Shapes
	Identifying Shapes and Their Properties	Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D			Construct and interpret plans and elevations of 3-D shapes
				Tet-Trouble Tet-Trouble Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units? Nine Colours Nine Colours Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour? Marbles in a box Marbles in a box How many winning lines can you make in a three-dimensional version of noughts and crosses? Triangles to Tetrahedra Triangles to Tetrahedra Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?












3D Shapes short problems
Transformations
Position, Direction and Movement		Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures			Describe the changes and invariance achieved by combinations of rotations, reflections and translations
		Shady Symmetry Shady Symmetry How many different symmetrical shapes can you make by shading triangles or squares? Mirror, mirror... Mirror, mirror... Explore the effect of reflecting in two parallel mirror lines. ...on the wall ...on the wall Explore the effect of reflecting in two intersecting mirror lines. Attractive rotations Attractive rotations Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or... Reflecting Squarely Reflecting Squarely In how many ways can you fit all three pieces together to make shapes with line symmetry?	Transformation Game Transformation Game Why not challenge a friend to play this transformation game? G Robotic Rotations Robotic Rotations How did the the rotation robot make these patterns?
		Interpret and use fractional and negative scale factors for enlargements
					Who is the fairest of them all ? Who is the fairest of them all ? Explore the effect of combining enlargements.	Growing Rectangles Growing Rectangles What happens to the area and volume of 2D and 3D shapes when you enlarge them? Fit for photocopying Fit for photocopying Explore the relationships between different paper sizes.











		Transformations short problems
	Vectors
		Describe translations as 2D vectors
					Vector Racer Vector Racer The classic vector racing game. G Vector Gem Collector Vector Gem Collector Use vectors to collect as many gems as you can and bring them safely home! G	Vector journeys Vector journeys Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?	Vector walk Vector walk Starting with two basic vector steps, which destinations can you reach on a vector walk?
		Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs
						Spotting the loophole Spotting the loophole A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?	Areas of parallelograms Areas of parallelograms Can you find the area of a parallelogram defined by two vectors?











Vectors short problems
Pythagoras' Theorem & Trigonometry
		Use Pythagoras' Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles			Apply Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles (and, where possible, general triangles) in two and three dimensional figures; interpret and use bearings
				Garden Shed Garden Shed Can you minimise the amount of wood needed to build the roof of my garden shed?	Compare Areas Compare Areas Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle? Inscribed in a Circle Inscribed in a Circle The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?	Semi-detached Semi-detached A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius. Far Horizon Far Horizon An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see? The Spider and the Fly The Spider and the Fly A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly? Where to Land Where to Land Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?	Ladder and Cube Ladder and Cube A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?
		Know the exact values of sin θ and cos θ for θ = 0˚, 30˚, 45˚, 60˚ and 90˚; know the exact value of tan θ for θ = 0˚, 30˚, 45˚ and 60˚
					Where is the dot? Where is the dot? A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?	Three cubes Three cubes Can you work out the dimensions of the three cubes?
		Know and apply the sine rule and cosine rule to find unknown lengths and angles
							Bendy Quad Bendy Quad Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load. Hexy-Metry Hexy-Metry A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle? Three by One Three by One There are many different methods to solve this geometrical problem - how many can you find? Cubestick Cubestick Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.
		Know and apply Area = ½ ab sin C to calculate the area, sides or angles of any triangle









	Pythagoras' Theorem & Trigonometry short problems

STATISTICS AND PROBABILITY

Pre-Secondary	Age 11 – 12	→	→	→	Age 15 - 16	Extension
Statistics
Statistics	Describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendancy (mean, mode, median) and spread (range, consideration of outliers)
	Litov's Mean Value Theorem Litov's Mean Value Theorem Start with two numbers and generate a sequence where the next number is the mean of the last two numbers... Searching for mean(ing) Searching for mean(ing) If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg? Statistical shorts Statistical shorts Can you decide whether these short statistical statements are always, sometimes or never true? About Average About Average Can you find sets of numbers which satisfy each of our mean, median, mode and range conditions? M, M and M M, M and M If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?	How would you score it? How would you score it? Invent a scoring system for a 'guess the weight' competition. Half a minute Half a minute Anna, Ben and Charlie have been estimating 30 seconds. Who is the best? Unequal Averages Unequal Averages Play around with sets of five numbers and see what you can discover about different types of average...	Wipeout Wipeout Can you do a little mathematical detective work to figure out which number has been wiped out?	For richer for poorer For richer for poorer Charlie has moved between countries and the average income of both has increased. How can this be so?
	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			Interpret and construct tables and line graphs for time series data
	Substitution Cipher Substitution Cipher Find the frequency distribution for ordinary English, and use it to help you crack the code.	Who's the best? Who's the best? Which countries have the most naturally athletic populations? Olympic Records Olympic Records Can you deduce which Olympic athletics events are represented by the graphs?	Tree tops Tree tops Can you make sense of information about trees in order to maximise the profits of a forestry company? What's the weather like? What's the weather like? With access to weather station data, what interesting questions can you investigate?
	Construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use
	Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs			Use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predications; interpolate and extrapolate apparent trends whilst knowing the dangers of doing so
		Reaction timer Reaction timer This problem offers you two ways to test reactions - use them to investigate your ideas about speeds of reaction. Estimating time Estimating time How well can you estimate 10 seconds? Investigate with our timing tool.		Olympic Triathlon Olympic Triathlon Is it the fastest swimmer, the fastest runner or the fastest cyclist who wins the Olympic Triathlon?
	Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling
			In the bag In the bag Can you guess the colours of the 10 marbles in the bag? Can you develop an effective strategy for reaching 1000 points in the least number of rounds?	Picturing the world Picturing the world How can we make sense of national and global statistics involving very large numbers?	Perception versus reality Perception versus reality Infographics are a powerful way of communicating statistical information. Can you come up with your own?
	Interpret, analyse and compare the distributions of data sets from univariate empirical distributions through: • Appropriate graphical representation involving discrete, continuous and grouped data, (including box plots) • Appropriate measures of central tendency (including modal class) and spread (including quartiles and inter-quartile range)
				Box plot match Box plot match Match the cumulative frequency curves with their corresponding box plots.	Which list is which? Which list is which? Six samples were taken from two distributions but they got muddled up. Can you work out which list is which?
	Apply statistics to describe a population






	Statistics short problems
	Probability
		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			Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
		Statistical shorts Statistical shorts Can you decide whether these short statistical statements are always, sometimes or never true?	Sociable Cards Sociable Cards Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?	Do you feel lucky? Do you feel lucky? Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not? What does random look like? What does random look like? Engage in a little mathematical detective work to see if you can spot the fakes.	Which spinners? Which spinners? Can you work out which spinners were used to generate the frequency charts? Who's the winner? Who's the winner? When two closely matched teams play each other, what is the most likely result?	The Better Choice The Better Choice Here are two games you can play. Which offers the better chance of winning?
		Understand that the probabilities of all possible outcomes sum to 1			Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one
				At least one... At least one... Imagine flipping a coin a number of times. Can you work out the probability you will get a head on at least one of the flips?		Same Number! Same Number! If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?
		Enumerate sets and unions/ intersections of sets systematically, using tables, grids and Venn diagrams
		Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities			Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
		Odds and Evens Odds and Evens Are these games fair? How can you tell?	Flippin' discs Flippin' discs Discs are flipped in the air. You win if all the faces show the same colour. What is the probability of winning? Interactive Spinners Interactive Spinners This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events. Non-Transitive Dice Non-Transitive Dice Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?	Two's company Two's company Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning? Cosy corner Cosy corner Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?	Last one standing Last one standing Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?	In a box In a box Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair? Chances are Chances are Which of these games would you play to give yourself the best possible chance of winning a prize? Odds and Evens made fair Odds and Evens made fair In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.	Mathsland National Lottery Mathsland National Lottery Can you work out the probability of winning the Mathsland National Lottery?
		Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams








Probability short problems

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For younger learners

NRICH Secondary Curriculum Map

NUMBER

Place Value, Integers, Ordering & Rounding

Factors, Multiples & Primes

Powers and Roots

Fractions, Decimals & Percentages

Number Operations and Calculation Methods

Ratio, Proportion & Rates of Change

ALGEBRA

Patterns and Sequences

Creating & Manipulating Linear & Quadratic Expressions

Equations & Formulae

Functions and Graphs

GEOMETRY

Angles, Polygons and Geometrical Proof

Construction

Perimeter, Area and Volume

3D Shapes

Transformations

Vectors

Pythagoras' Theorem & Trigonometry

STATISTICS AND PROBABILITY

Statistics

Probability