Resources tagged with: Generalising

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Broad Topics > Thinking Mathematically > Generalising

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. . . .

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?

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?

Magic Squares II

Age 14 to 18

An article which gives an account of some properties of magic squares.

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?

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.

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?

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.

All Tangled Up

Age 14 to 18 Challenge Level:

Can you tangle yourself up and reach any fraction?

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.

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?

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?

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?

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?

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?

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?

Areas of Parallelograms

Age 14 to 16 Challenge Level:

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

Mind Reading

Age 11 to 14 Challenge Level:

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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?

Where Can We Visit?

Age 11 to 14 Challenge Level:

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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?

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?

How Much Can We Spend?

Age 11 to 14 Challenge Level:

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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. . . .

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?

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.

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?

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?

More Number Pyramids

Age 11 to 14 Challenge Level:

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

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?

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?

Enclosing Squares

Age 11 to 14 Challenge Level:

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

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...

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?

Problem Solving, Using and Applying and Functional Mathematics

Age 5 to 18 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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. . . .

Magic Squares

Age 14 to 18

An account of some magic squares and their properties and and how to construct them for yourself.

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?

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.

Neighbourly Addition

Age 7 to 14 Challenge Level:

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

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?

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?

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. . . .

Cunning Card Trick

Age 11 to 14 Challenge Level:

Delight your friends with this cunning trick! Can you explain how it works?

Searching for Mean(ing)

Age 11 to 16 Challenge Level:

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

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?