# Resources tagged with: Generalising

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Broad Topics > Mathematical Thinking > Generalising ### Attractive Tablecloths

##### Age 14 to 16 Challenge Level:

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? ### Mirror, Mirror...

##### Age 11 to 14 Challenge Level:

Explore the effect of reflecting in two parallel mirror lines. ### ...on the Wall

##### Age 11 to 14 Challenge Level:

Explore the effect of reflecting in two intersecting mirror lines. ### Painted Cube

##### Age 14 to 16 Challenge Level:

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? ### Cubes Within Cubes Revisited

##### Age 11 to 14 Challenge Level:

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? ### Cuboid Challenge

##### Age 11 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN? ### Frogs

##### Age 11 to 14 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes. ### Jam

##### Age 14 to 16 Challenge Level:

A game for 2 players ### Hidden Rectangles

##### Age 11 to 14 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard? ### Christmas Chocolates

##### Age 11 to 14 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

To avoid losing think of another very well known game where the patterns of play are similar. ### Partly Painted Cube

##### Age 14 to 16 Challenge Level:

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? ### AMGM

##### Age 14 to 16 Challenge Level:

Can you use the diagram to prove the AM-GM inequality? ### Overlap

##### Age 11 to 14 Challenge Level:

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . . ### Sliding Puzzle

##### Age 11 to 16 Challenge Level:

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves. ### Route to Infinity

##### Age 11 to 14 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next? ### Who Is the Fairest of Them All ?

##### Age 11 to 14 Challenge Level:

Explore the effect of combining enlargements. ### Squares in Rectangles

##### Age 11 to 14 Challenge Level:

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? ### Is There a Theorem?

##### Age 11 to 14 Challenge Level:

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? ### Chess

##### Age 11 to 14 Challenge Level:

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board? ### Steel Cables

##### Age 14 to 16 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible. ### In a Spin

##### Age 14 to 16 Challenge Level:

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse? ### Multiplication Square

##### Age 14 to 16 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles? ### Number Pyramids

##### Age 11 to 14 Challenge Level:

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

##### Age 7 to 16

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games. ### One, Three, Five, Seven

##### Age 11 to 16 Challenge Level:

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses. ### Pair Products

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . . ### Games Related to Nim

##### Age 5 to 16

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning. ### Special Sums and Products

##### Age 11 to 14 Challenge Level:

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. ### Partitioning Revisited

##### Age 11 to 14 Challenge Level:

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 ### Tourism

##### Age 11 to 14 Challenge Level:

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. ### Squares, Squares and More Squares

##### Age 11 to 14 Challenge Level:

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares? ### Dotty Triangles

##### Age 11 to 14 Challenge Level:

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw? ### Tower of Hanoi

##### Age 11 to 14 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Can you find the values at the vertices when you know the values on the edges? ### Consecutive Negative Numbers

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important. ### More Number Pyramids

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

Make some loops out of regular hexagons. What rules can you discover? ### What's Possible?

##### Age 14 to 16 Challenge Level:

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

##### Age 5 to 14 Challenge Level:

This task encourages you to investigate the number of edging pieces and panes in different sized windows. ### Nim-7 for Two

##### Age 5 to 14 Challenge Level:

Nim-7 game for an adult and child. Who will be the one to take the last counter? ### Square Pizza

##### Age 14 to 16 Challenge Level:

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?