# Resources tagged with: Generalising

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Broad Topics > Thinking Mathematically > Generalising ### Semi-square

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

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? ### 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? ### 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? ### AMGM

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

Can you use the diagram to prove the AM-GM inequality? ### 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? ### 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? ### 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. ### 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? ### Equilateral Areas

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

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF. ### Tilted Squares

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

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

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

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

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

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

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

A game for 2 players ### 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? ### 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? ### 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. ### A Tilted Square

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

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

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

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. ### 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? ### 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? ### Pareq Calc

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

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . . ### 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? ### More Number Pyramids

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge... ### 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. ##### Age 11 to 14 Challenge Level:

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. . . . ### 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. ### Jam

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

To avoid losing think of another very well known game where the patterns of play are similar. ### 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? ### Route to Infinity

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

Can you describe this route to infinity? Where will the arrows take you next? ### 2001 Spatial Oddity

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

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done. ### Cuboid Challenge

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

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

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

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? ### Pinned Squares

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

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

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met? ### Multiplication Arithmagons

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

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons? ### 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? ### Magic Letters

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

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

##### Age 7 to 14 Challenge Level:

It starts quite simple but great opportunities for number discoveries and patterns! ### Got it for Two

##### Age 7 to 14 Challenge Level:

Got It game for an adult and child. How can you play so that you know you will always win? ### 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. ### 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? ### Winning Lines

##### Age 7 to 16

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games. ### Konigsberg Plus

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

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. ### 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?