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What's the largest volume of box you can make from a square of paper?

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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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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?

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Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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It would be nice to have a strategy for disentangling any tangled ropes...

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Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

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Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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

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Delight your friends with this cunning trick! Can you explain how it works?

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Explore the effect of combining enlargements.

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Explore the effect of reflecting in two intersecting mirror lines.

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Explore the effect of reflecting in two parallel mirror lines.

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The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

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Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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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?

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

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

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Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

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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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Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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

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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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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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Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I know?

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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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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?

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A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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How many moves does it take to swap over some red and blue frogs? Do you have a method?

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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.

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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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Can you find sets of sloping lines that enclose a square?

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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?

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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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Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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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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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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Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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Is there an efficient way to work out how many factors a large number has?

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