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It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

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

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Choose any three by three square of dates on a calendar page...

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These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

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Can all unit fractions be written as the sum of two unit fractions?

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Can you explain the strategy for winning this game with any target?

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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?

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Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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Can you find a way to identify times tables after they have been shifted up or down?

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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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Can you describe this route to infinity? Where will the arrows take you next?

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

A collection of short Stage 3 and 4 problems on Exploring and Noticing Structure.

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Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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Can you find the values at the vertices when you know the values on the edges?

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Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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

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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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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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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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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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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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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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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?

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Look for the common features in these graphs. Which graphs belong together?

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Which of these triangular jigsaws are impossible to finish?

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A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

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Can you make matrices which will fix one lucky vector and crush another to zero?