Resources tagged with: Generalising
There are 229 NRICH Mathematical resources connected to Generalising, you may find related items under Thinking mathematically.
Broad Topics > Thinking mathematically > Generalising
More Magic Potting Sheds
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?
Areas of Parallelograms
Can you find the area of a parallelogram defined by two vectors?
Litov's Mean Value Theorem
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Squares in Rectangles
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?
Harmonic Triangle
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Cubes Within Cubes Revisited
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
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
Multiplication Square
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Number Differences
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
More Numbers in the Ring
If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
Arithmagons
Can you find the values at the vertices when you know the values on the edges?
Lots of Lollies
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
Painted Cube
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?
Tilted Squares
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Coordinate Patterns
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Seven Squares - Group-worthy Task
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?
Shear Magic
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
Break it Up!
In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?
Up and Down Staircases
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
More Number Pyramids
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Number Pyramids
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Odd Squares
Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?
Pair Products
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Picturing Square Numbers
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
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Mind Reading
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 know?
Make 37
Four bags contain a large number of 1s, 3s, 5s and 7s. Can you pick any ten numbers from the bags so that their total is 37?
Pick's Theorem
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.
Take Three from Five
Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
Got It
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.
Frogs
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Nim-7
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Egyptian Fractions
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.
Attractive Tablecloths
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?
Enclosing Squares
Can you find sets of sloping lines that enclose a square?
Where Can We Visit?
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?
What's Possible?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Beelines
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Cut it Out
Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
Of All the Areas
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Plus Minus
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Odd Differences
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.
Summing Consecutive Numbers
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Is There a Theorem?
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?
Counting Factors
Is there an efficient way to work out how many factors a large number has?
Have You Got It?
Can you explain the strategy for winning this game with any target?
Shape and Territory
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Sticky Triangles
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Round and Round the Circle
What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
Doplication
We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?