Can you find the area of a parallelogram defined by two vectors?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
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?
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
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.
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.
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.
Explore the effect of reflecting in two parallel mirror lines.
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two intersecting mirror lines.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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. . . .
It starts quite simple but great opportunities for number discoveries and patterns!
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Can you find sets of sloping lines that enclose a square?
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?
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. . . .
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
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. . . .
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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.
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.
Imagine you have a large supply of 3kg and 8kg weights. How many of
each weight would you need for the average (mean) of the weights to
be 6kg? What other averages could you have?
Can you see how to build a harmonic triangle? Can you work out the
next two rows?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative 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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Can all unit fractions be written as the sum of two unit fractions?
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Charlie has moved between countries and the average income of both
has increased. How can this be so?
It would be nice to have a strategy for disentangling any tangled