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Conjecturing and Generalising is part of our Thinking Mathematically collection.
Play this game and see if you can figure out the computer's chosen number.
There are nasty versions of this dice game but we'll start with the nice ones...
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
Try out some calculations. Are you surprised by the results?
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
Got It game for an adult and child. How can you play so that you know you will always win?
A game in which players take it in turns to choose a number. Can you block your opponent?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
What happens when you add a three digit number to its reverse?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
How much of the square is coloured blue? How will the pattern continue?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.
Explore the effect of reflecting in two parallel mirror lines.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Imagine a very strange bank account where you are only allowed to do two things...
Can all unit fractions be written as the sum of two unit fractions?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Which set of numbers that add to 100 have the largest product?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Can you figure out how sequences of beach huts are generated?
Who said that adding, subtracting, multiplying and dividing couldn't be fun?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.
Can you explain the strategy for winning this game with any target?
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.
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Is there an efficient way to work out how many factors a large number has?
Can you work out what step size to take to ensure you visit all the dots on the circle?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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.
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
There are lots of ideas to explore in these sequences of ordered fractions.
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?
Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Is there a quick way to work out whether a fraction terminates or recurs when you write it as a decimal?
Explore the effect of reflecting in two intersecting mirror lines.
Play around with the Fibonacci sequence and discover some surprising results!
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
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?
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?
Can you describe this route to infinity? Where will the arrows take you next?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
Explore the effect of combining enlargements.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
A collection of short Stage 3 and 4 problems on Conjecturing and Generalising
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
Can you find the values at the vertices when you know the values on the edges?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
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?
It would be nice to have a strategy for disentangling any tangled ropes...
What's the largest volume of box you can make from a square of paper?
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?
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Which armies can be arranged in hollow square fighting formations?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
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?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Can you find the area of a parallelogram defined by two vectors?
What is special about the difference between squares of numbers adjacent to multiples of three?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
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?
Charlie has moved between countries and the average income of both has increased. How can this be so?
There are unexpected discoveries to be made about square numbers...
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.
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Can you find an efficent way to mix paints in any ratio?
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?