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#### Resources tagged with Generalising similar to What's Possible?:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising ### 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? ### 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? ### Lower Bound

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 = ### Sums of Pairs

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

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?” ### Odd Differences

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares. ### 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. ### 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? ### 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? ### 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? ### Arithmagons

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

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

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

Can you explain the surprising results Jo found when she calculated the difference between square numbers? ### 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? ### Janine's Conjecture

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . . ### 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? ### 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? ### 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? ### AMGM

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

Can you use the diagram to prove the AM-GM inequality? ### Magic Squares II

##### Age 14 to 18

An article which gives an account of some properties of magic squares. ### Regular Hexagon Loops

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

Make some loops out of regular hexagons. What rules can you discover? ### Take Three from Five

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him? ### 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? ### 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? ### Triangle Numbers

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

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

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

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

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses? ### Searching for Mean(ing)

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

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have? ### 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? ### Winning Lines

##### Age 7 to 16

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

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

It starts quite simple but great opportunities for number discoveries and patterns! ### Route to Infinity

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

Can you describe this route to infinity? Where will the arrows take you next? ### 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. ### 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? ### What Numbers Can We Make Now?

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now? ### 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. ### 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? ### Sum Equals Product

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

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . . ### 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? ### What Numbers Can We Make?

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

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

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

A game for 2 players ### More Number Pyramids

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

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

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

Can you see how to build a harmonic triangle? Can you work out the next two rows? ### For Richer for Poorer

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

Charlie has moved between countries and the average income of both has increased. How can this be so?