1. Expressions
[ A1.2 Extension Plenary]
Use algebra to show that any three consecutive numbers added together are divisible by 3. Can you find a similar rule for four, five, ... consecutive numbers?
Take any three consecutive numbers. Multiply the first and third together. Subtract the square of the middle number from this total. Repeat with some other sets of three consecutive numbers. What do you notice? Use algebra to see if this is always the case.
Use algebra to solve this Ancient Chinese puzzle: Four less than my number multiplied by one more than my number gives nothing. What is my number?
Try to solve this old puzzle: The sum of two numbers is 3, the sum of each number squared is 5. What are the two numbers? Hint: let one of the numbers be x.
Make up some more puzzles like 3) and 4) for yourself. Try them on friends – but make sure you know the answers!
Preamble
These problems/puzzles give pupils practice using algebraic skills. In order to make
efficient use of time working in pairs or small groups would be most appropriate.
Question (5) asking children to make up and solve there own puzzles, would form the
basis of a useful homework.
Possible content
Setting up and solving equations, some involving simple quadratics, in a
problem/puzzle scenario, using brackets.
Resources
None.
Solution/Notes
x + (x + 1) + (x + 2) = 3x + 3 = 3(x + 1), which is divisible by 3. There is a similar rule for the sum of five consecutive numbers.
The difference is 1 Let x be the first number then the three numbers are x (x + 1) and (x + 2). So (x + 1)² = x² + 2x and x(x + 2) = x² + 2x Showing that the difference is always 1.
Let x be the number, so (x – 4)(x + 1) = 0, so the number is 4 or -1.
Let x be one of the numbers, so 3 – x is the other number. Also x² + (3 – x)² = 5, which gives (x - 1)(x - 2) = 0 so the numbers are 1 and 2. Trial and improvement may be more accessible to many pupils.
Children's own puzzles and solutions.
Original Material © Cambridge University Press 2010
Original Material © Cambridge University Press 2010