Resources tagged with: Generalising

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Broad Topics > Thinking Mathematically > Generalising

Polycircles

Age 14 to 16
Challenge Level

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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.

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. . . .

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?

Squaring the Circle and Circling the Square

Age 14 to 16
Challenge Level

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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?

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.

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?

Jam

Age 14 to 16
Challenge Level

A game for 2 players

Train Spotters' Paradise

Age 11 to 16

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

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?

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.

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?

Loopy

Age 14 to 16
Challenge Level

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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

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?

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?

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?

More Number Pyramids

Age 11 to 14
Challenge Level

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

Arithmagons

Age 11 to 16
Challenge Level

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

More Twisting and Turning

Age 11 to 16
Challenge Level

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

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. . . .

Make 37

Age 7 to 14
Challenge Level

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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?

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?

Chocolate 2010

Age 14 to 16
Challenge Level

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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.

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?

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?

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.

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?

Enclosing Squares

Age 11 to 14
Challenge Level

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

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.

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?

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?

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?

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?

Spaces for Exploration

Age 11 to 14

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

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?

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. . . .

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?

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?

Thank Your Lucky Stars

Age 14 to 16
Challenge Level

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

Route to Infinity

Age 11 to 14
Challenge Level

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

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.

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. . . .

More Magic Potting Sheds

Age 11 to 14
Challenge Level

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?

Beach Huts

Age 11 to 14
Challenge Level

Can you figure out how sequences of beach huts are generated?