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Conjecturing and generalising is part of our Working Mathematically collection.
Got It game for an adult and child. How can you play so that you know you will always win?
Can you figure out how sequences of beach huts are generated?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
Explore the effect of reflecting in two parallel mirror lines.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Can you explain the strategy for winning this game with any target?
Can all unit fractions be written as the sum of two unit fractions?
This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
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.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
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?
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?
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?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Can you describe this route to infinity? Where will the arrows take you next?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
What's the largest volume of box you can make from a square of paper?
It would be nice to have a strategy for disentangling any tangled ropes...
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 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?
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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?
Can you find the area of a parallelogram defined by two vectors?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Charlie has moved between countries and the average income of both has increased. How can this be so?
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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 the values at the vertices when you know the values on the edges of these multiplication arithmagons?