Resources tagged with: Generalising

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There are 128 results

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?

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?

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?

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?

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.

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. . . .

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.

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?

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?

AMGM

Age 14 to 16 Challenge Level:

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

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?

Mindreader

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. . . .

Winning Lines

Age 7 to 16

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

Seven Squares - Group-worthy Task

Age 11 to 14 Challenge Level:

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?

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?

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?”

Nim-like Games

Age 7 to 16 Challenge Level:

A collection of games on the NIM theme

Nim

Age 14 to 16 Challenge Level:

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

Got It

Age 7 to 14 Challenge Level:

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.

Jam

Age 14 to 16 Challenge Level:

A game for 2 players

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?

Neighbourly Addition

Age 7 to 14 Challenge Level:

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

More Twisting and Turning

Age 11 to 16 Challenge Level:

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

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?

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:

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

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?

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.

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?

Mini-max

Age 11 to 14 Challenge Level:

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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 =

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. . . .

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?

Arithmagons

Age 11 to 16 Challenge Level:

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

Adding in Rows

Age 11 to 14 Challenge Level:

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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?

Counting Factors

Age 11 to 14 Challenge Level:

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

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?

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?

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?

Steps to the Podium

Age 7 to 14 Challenge Level:

It starts quite simple but great opportunities for number discoveries and patterns!

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?

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 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?

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?

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?

Plus Minus

Age 14 to 16 Challenge Level:

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

More Number Pyramids

Age 11 to 14 Challenge Level:

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