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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

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Advent Calendar 2011 - Secondary

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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Tower of Hanoi

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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Cyclic Quadrilaterals

Stage: 3 Challenge Level: Challenge Level:1

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

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Pythagoras Proofs

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you make sense of these three proofs of Pythagoras' Theorem?

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Power Mad!

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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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Elevenses

Stage: 3 Challenge Level: Challenge Level:1

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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Iffy Logic

Stage: 4 Short Challenge Level: Challenge Level:1

Can you rearrange the cards to make a series of correct mathematical statements?

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Multiplication Square

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

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?

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Seven Squares - Group-worthy Task

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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Perfectly Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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More Number Pyramids

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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Always Perfect

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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Children at Large

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

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For What?

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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Leonardo's Problem

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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Take Three from Five

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Largest Product

Stage: 3 and 4 Challenge Level: Challenge Level:1

Which set of numbers that add to 10 have the largest product?

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Proof Sorter - Quadratic Equation

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

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1 Step 2 Step

Stage: 3 Challenge Level: Challenge Level:1

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?

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Number Rules - OK

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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Lens Angle

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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Why 24?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Angle Trisection

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

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Zig Zag

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

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Unit Interval

Stage: 4 and 5 Challenge Level: Challenge Level:1

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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Prime AP

Stage: 4 Challenge Level: Challenge Level:1

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

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Common Divisor

Stage: 4 Challenge Level: Challenge Level:1

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

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There's a Limit

Stage: 4 and 5 Challenge Level: Challenge Level:1

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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Square Mean

Stage: 4 Challenge Level: Challenge Level:1

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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Cosines Rule

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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Picture Story

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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Areas and Ratios

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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Growing Ls

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Can you fit Ls together to make larger versions of themselves?

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Kite in a Square

Stage: 4 Challenge Level: Challenge Level:1

Can you make sense of the three methods to work out the area of the kite in the square?

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What Numbers Can We Make Now?

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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Calculating with Cosines

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

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On the Importance of Pedantry

Stage: 3, 4 and 5

A introduction to how patterns can be deceiving, and what is and is not a proof.

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What Numbers Can We Make?

Stage: 3 Challenge Level: Challenge Level:1

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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L-triominoes

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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Geometric Parabola

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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Breaking the Equation ' Empirical Argument = Proof '

Stage: 2, 3, 4 and 5

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

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Sticky Numbers

Stage: 3 Challenge Level: Challenge Level:1

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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Folding Fractions

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

What fractions can you divide the diagonal of a square into by simple folding?

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Air Nets

Stage: 2, 3, 4 and 5 Challenge Level: Challenge Level:1

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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Truth Tables and Electronic Circuits

Stage: 2, 3 and 4

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

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Logic, Truth Tables and Switching Circuits Challenge

Stage: 3, 4 and 5

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

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Proof: A Brief Historical Survey

Stage: 4 and 5

If you think that mathematical proof is really clearcut and universal then you should read this article.

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A Long Time at the Till

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

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Road Maker

Stage: 4 and 5 Challenge Level: Challenge Level:1

Which of these roads will satisfy a Munchkin builder?