Let a(n) be the number of ways of expressing the integer n as an
ordered sum of 1's and 2's. Let b(n) be the number of ways of
expressing n as an ordered sum of integers greater than 1. (i)
Calculate. . . .
Show that all pentagonal numbers are one third of a triangular number.
Given the mean and standard deviation of a set of marks, what is
the greatest number of candidates who could have scored 100%?
Can you find a rule which connects consecutive triangular numbers?
A introduction to how patterns can be deceiving, and what is and is not a proof.
Can you find a rule which relates triangular numbers to square numbers?
Prove that 3 times the sum of 3 squares is the sum of 4 squares.
Rather easier, can you prove that twice the sum of two squares
always gives the sum of two squares?
Crack this code which depends on taking pairs of letters and using
two simultaneous relations and modulus arithmetic to encode the
Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.
Explore the two quadratic functions and find out how their graphs
Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11.
Then find formulas giving all the solutions to
7x + 11y = 100
where x and y are integers.
Here are some more quadratic functions to explore. How are their
A moveable screen slides along a mirrored corridor towards a
centrally placed light source. A ray of light from that source is
directed towards a wall of the corridor, which it strikes at 45
degrees. . . .
Rotate a copy of the trapezium about the centre of the longest side
of the blue triangle to make a square. Find the area of the square
and then derive a formula for the area of the trapezium.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Can you work out which processes are represented by the graphs?
Find a quadratic formula which generalises Pick's Theorem.
Which line graph, equations and physical processes go together?
The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?