Resources tagged with: Generalising

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Broad Topics > Thinking Mathematically > Generalising

Areas of Parallelograms

Age 14 to 16
Challenge Level

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

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?

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.

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?

Jam

Age 14 to 16
Challenge Level

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

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?

Maths Trails

Age 7 to 14

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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?

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?

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?

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?

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.

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?

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.

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?

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?

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?

More Twisting and Turning

Age 11 to 16
Challenge Level

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

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?

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

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?

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?

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. . . .

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.

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?

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.

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?

Coordinate Patterns

Age 11 to 14
Challenge Level

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Route to Infinity

Age 11 to 14
Challenge Level

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

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?

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.

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?

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.

Jam

Age 14 to 16
Challenge Level

A game for 2 players

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.

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. . . .

Enclosing Squares

Age 11 to 14
Challenge Level

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

More Number Pyramids

Age 11 to 14
Challenge Level

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

Take Three from Five

Age 11 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?”

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.

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?

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.

Arithmagons

Age 11 to 16
Challenge Level

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

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.

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?