Resources tagged with: Generalising

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Broad Topics > Thinking Mathematically > Generalising

Chocolate 2010

Age 14 to 16Challenge Level

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

Litov's Mean Value Theorem

Age 11 to 14Challenge Level

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Odd Differences

Age 14 to 16Challenge 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.

Sum Equals Product

Age 11 to 14Challenge 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. . . .

Regular Hexagon Loops

Age 11 to 14Challenge Level

Make some loops out of regular hexagons. What rules can you discover?

For Richer for Poorer

Age 14 to 16Challenge Level

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

Multiplication Square

Age 14 to 16Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Lower Bound

Age 14 to 16Challenge 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 =

Card Trick 2

Age 11 to 14Challenge Level

Can you explain how this card trick works?

Loopy

Age 14 to 16Challenge 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?

Sums of Pairs

Age 11 to 16Challenge 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?”

Age 11 to 14Challenge 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. . . .

Nim-like Games

Age 7 to 16Challenge Level

A collection of games on the NIM theme

What's Possible?

Age 14 to 16Challenge Level

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Searching for Mean(ing)

Age 11 to 16Challenge 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?

Nim

Age 14 to 16Challenge 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.

Konigsberg Plus

Age 11 to 14Challenge 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.

Number Pyramids

Age 11 to 14Challenge Level

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Pair Products

Age 14 to 16Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Age 11 to 14Challenge 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?

Special Sums and Products

Age 11 to 14Challenge 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.

Converging Means

Age 14 to 16Challenge 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. . . .

Harmonic Triangle

Age 14 to 16Challenge Level

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Janine's Conjecture

Age 14 to 16Challenge 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. . . .

Age 11 to 14Challenge 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. . . .

Beach Huts

Age 11 to 14Challenge Level

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

How Much Can We Spend?

Age 11 to 14Challenge 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?

Chocolate Maths

Age 11 to 14Challenge Level

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Keep it Simple

Age 11 to 14Challenge Level

Can all unit fractions be written as the sum of two unit fractions?

Frogs

Age 11 to 14Challenge Level

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Route to Infinity

Age 11 to 14Challenge Level

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

What Numbers Can We Make Now?

Age 11 to 14Challenge Level

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

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 14Challenge 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.

Shear Magic

Age 11 to 14Challenge Level

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

Painted Cube

Age 14 to 16Challenge 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?

Got It

Age 7 to 14Challenge 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.

Coordinate Patterns

Age 11 to 14Challenge Level

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

Picturing Square Numbers

Age 11 to 14Challenge 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 14Challenge Level

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Got it for Two

Age 7 to 14Challenge Level

Got It game for an adult and child. How can you play so that you know you will always win?

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.

Nim-7 for Two

Age 5 to 14Challenge Level

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Counting Factors

Age 11 to 14Challenge Level

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

Jam

Age 14 to 16Challenge Level

A game for 2 players

Steel Cables

Age 14 to 16Challenge Level

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Steps to the Podium

Age 7 to 14Challenge Level

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

Arithmagons

Age 11 to 16Challenge Level

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

Cuboid Challenge

Age 11 to 16Challenge Level

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