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Resources tagged with Generalising similar to Magic Squares for Special Occasions:

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Masterclass Ideas: Generalising

Stage: 2 and 3 Challenge Level: Challenge Level:1

A package contains a set of resources designed to develop pupils’ mathematical thinking. This package places a particular emphasis on “generalising” and is designed to meet the. . . .

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Partitioning Revisited

Stage: 3 Challenge Level: Challenge Level:1

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

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Number Pyramids

Stage: 3 Challenge Level: Challenge Level:1

Using the same starter numbers 2, 1, 4 and 6 can you get a larger total at the top of the pyramid? What is the largest total you can get?

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Chocolate Maths

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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Sum Equals Product

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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 for. . . .

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Strike it Out

Stage: 1, 2 and 3 Challenge Level: Challenge Level:1

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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Make 37

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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GOT IT Now

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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Christmas Chocolates

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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Mind Reading

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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GOT IT

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Maths Trails

Stage: 2 and 3

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.

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Nim

Stage: 4 Challenge Level: Challenge Level:1

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.

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Janine's Conjecture

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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Pair Products

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Consecutive Sums

Stage: 3 Challenge Level: Challenge Level:1

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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Card Trick 1

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you explain how this card trick works?

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Handshakes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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AMGM

Stage: 4 Challenge Level: Challenge Level:1

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

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Cunning Card Trick

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Triangle Numbers

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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Frogs

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

You can solve frogs on the computer, using counters, or acting it out. Start with frogs in a line on one side, and toads on the other, with a space in between. They need to change places.

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What's Possible?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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More Number Pyramids

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Are there any patterns within the pyramid? Can you explain why you only get multiples of 4 at the top when you start with an integer in the bottom left hand corner?

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Nim-like Games

Stage: 2, 3 and 4 Challenge Level: Challenge Level:1

A collection of games on the NIM theme

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Cubes Within Cubes Revisited

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Consecutive Negative Numbers

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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Odd Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Harmonic Triangle

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Magic Squares

Stage: 4 and 5

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

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Mindreader

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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Card Trick 2

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you explain how this card trick works?

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All Tangled Up

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you tangle yourself up and reach any fraction?

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Enclosing Squares

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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More Twisting and Turning

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Arithmagons

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find the value of the circles when you know what's in the squares?

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Games Related to Nim

Stage: 1, 2, 3 and 4

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.

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Squares, Squares and More Squares

Stage: 3 Challenge Level: Challenge Level:1

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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Three Times Seven

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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Lower Bound

Stage: 3 Challenge Level: Challenge Level:1

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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Adding in Rows

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Mini-max

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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Winning Lines

Stage: 2, 3 and 4 Challenge Level: Challenge Level:1

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

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For Richer for Poorer

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Multiplication Square

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Searching for Mean(ing)

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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Sliding Puzzle

Stage: 1, 2, 3 and 4 Challenge Level: Challenge Level:1

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.

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Egyptian Fractions

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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Keep it Simple

Stage: 3 Challenge Level: Challenge Level:1

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

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How Much Can We Spend?

Stage: 3 Challenge Level: Challenge Level:1

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?