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

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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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Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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

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

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

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

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

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

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

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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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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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Can you find a rule which connects consecutive triangular numbers?

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

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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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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?

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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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Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

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Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

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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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If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

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Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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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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Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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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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Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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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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A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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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?

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There are unexpected discoveries to be made about square numbers...

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Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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Can you explain what is going on in these puzzling number tricks?

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The Number Jumbler can always work out your chosen symbol. Can you work out how?

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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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

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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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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 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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What is special about the difference between squares of numbers adjacent to multiples of three?

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Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?