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Resources tagged with Generalising similar to More Magic Potting Sheds:

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More Magic Potting Sheds

Stage: 3 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?

Picturing Triangular Numbers

Stage: 3 Challenge Level:

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

One O Five

Stage: 3 Challenge Level:

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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.

Nim-7

Stage: 1, 2 and 3 Challenge Level:

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Christmas Chocolates

Stage: 3 Challenge Level:

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

Make 37

Stage: 2 and 3 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.

Repeaters

Stage: 3 Challenge Level:

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Stage: 3 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. . . .

Triangle Numbers

Stage: 3 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?

Cunning Card Trick

Stage: 3 Challenge Level:

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

Hidden Rectangles

Stage: 3 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?

Card Trick 2

Stage: 3 Challenge Level:

Can you explain how this card trick works?

Special Sums and Products

Stage: 3 Challenge 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.

Cubes Within Cubes Revisited

Stage: 3 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?

Partitioning Revisited

Stage: 3 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

Handshakes

Stage: 3 Challenge Level:

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

Steps to the Podium

Stage: 2 and 3 Challenge Level:

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

More Number Pyramids

Stage: 3 Challenge Level:

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

Konigsberg Plus

Stage: 3 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.

Have You Got It?

Stage: 3 Challenge Level:

Can you explain the strategy for winning this game with any target?

Shear Magic

Stage: 3 Challenge Level:

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Tourism

Stage: 3 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.

Number Pyramids

Stage: 3 Challenge Level:

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

Enclosing Squares

Stage: 3 Challenge Level:

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

Three Times Seven

Stage: 3 Challenge Level:

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

Chess

Stage: 3 Challenge Level:

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Painted Cube

Stage: 3 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?

Stage: 3 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. . . .

Go Forth and Generalise

Stage: 3

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.

Intersecting Circles

Stage: 3 Challenge Level:

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Winning Lines

Stage: 2, 3 and 4

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

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.

Sum Equals Product

Stage: 3 Challenge 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 for. . . .

Stage: 3 Challenge 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?

2001 Spatial Oddity

Stage: 3 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.

Converging Means

Stage: 3 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. . . .

Lower Bound

Stage: 3 Challenge 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 =

Mini-max

Stage: 3 Challenge Level:

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. . . .

Jam

Stage: 4 Challenge Level:

A game for 2 players

Got It

Stage: 2 and 3 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.

Steel Cables

Stage: 4 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?

Egyptian Fractions

Stage: 3 Challenge Level:

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.

Magic Letters

Stage: 3 Challenge Level:

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

...on the Wall

Stage: 3 Challenge Level:

Explore the effect of reflecting in two intersecting mirror lines.

Route to Infinity

Stage: 3 Challenge Level:

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

Consecutive Negative Numbers

Stage: 3 Challenge Level:

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

Squares in Rectangles

Stage: 3 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?

Tilted Squares

Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?