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We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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

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How good are you at finding the formula for a number pattern ?

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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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Kyle and his teacher disagree about his test score - who is right?

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The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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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?

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A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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Show that all pentagonal numbers are one third of a triangular number.

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Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

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Can you find a rule which connects consecutive triangular numbers?

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Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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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?

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Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

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

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How many winning lines can you make in a three-dimensional version of noughts and crosses?

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Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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Make some loops out of regular hexagons. What rules can you discover?

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Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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Can you see how to build a harmonic triangle? Can you work out the next two rows?

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Can you find the area of a parallelogram defined by two vectors?

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First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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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.

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

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Can you find a rule which relates triangular numbers to square numbers?

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Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

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If a sum invested gains 10% each year how long before it has doubled its value?

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Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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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?

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Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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What is the total number of squares that can be made on a 5 by 5 geoboard?

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Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be. . . .

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However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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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?

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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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The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

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Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?