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Conjecturing and generalising is part of our Developing Mathematical Thinking collection.
Play this game and see if you can figure out the computer's chosen number.
In each of these games, you will need a little bit of luck, and your knowledge of place value to develop a winning strategy.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Try out some calculations. Are you surprised by the results?
There are nasty versions of this dice game but we'll start with the nice ones...
Got It game for an adult and child. How can you play so that you know you will always win?
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
A game in which players take it in turns to choose a number. Can you block your opponent?
How much of the square is coloured blue? How will the pattern continue?
Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can you figure out how sequences of beach huts are generated?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Can all unit fractions be written as the sum of two unit fractions?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you explain the strategy for winning this game with any target?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
What happens when you add a three digit number to its reverse?
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?
Who said that adding, subtracting, multiplying and dividing couldn't be fun?
This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.
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?
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Imagine a very strange bank account where you are only allowed to do two things...
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.
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
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...
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?
Can you work out what step size to take to ensure you visit all the dots on the circle?
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.
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?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Explore the effect of reflecting in two intersecting mirror lines.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Is there an efficient way to work out how many factors a large number has?
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?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Aisha's division and subtraction calculations both gave the same answer! Can you find some more examples?
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?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Is there a quick way to work out whether a fraction terminates or recurs when you write it as a decimal?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Explore the effect of combining enlargements.
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?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?
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?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
Can you find the area of a parallelogram defined by two vectors?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
There are unexpected discoveries to be made about square numbers...
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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?
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Charlie has moved between countries and the average income of both has increased. How can this be so?
What is special about the difference between squares of numbers adjacent to multiples of three?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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 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 find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.