A conjecture is a (mathematical) statement that is thought to be true but has not been proved yet.

To generalise is to formulate a statement or rule that applies correctly to all relevant cases.

Writing a generalised statement about specific number properties

Maths

Checking and securing understanding of functions

Defining function notation

Finding the inverse of a function

Writing composite functions

Solving equations involving functions

Solving equations involving composite functions

General algebraic forms for specific number properties

Making conjectures about patterns and relationships

Proving or disproving a statement

Writing a proof

Logical arguments

Multiple approaches to logical arguments

Problem solving with functions and proof

In this future unit, pupils continue to explore composite functions, thereby building on their work from this unit. Pupils need to understand function notation from this unit as this is used in iteration.

The functions unit requires pupils to write composite functions and pupils will need to draw on their understanding of how to manipulate various algebraic expressions and equations, including algebraic fractions.

In this unit pupils will further develop their understanding of algebraic notation and graphs by learning about functions, their inverses and composite functions.

Functions and proof

$$n, n+ 1, n+2$$ where $$n$$ is an integer

$$n, n+ 2, n+4$$ where $$n$$ is an integer

$$2n, 2n+ 1, 2n+2$$ where $$n$$ is an integer

$$2n, 2n+ 2, 2n+4$$ where $$n$$ is an integer

$$2(n - 1), 2(n + 1), 2(n + 2)$$ where $$n$$ is an integer

Which of these represent any 3 consecutive even numbers?

$$n+ 5$$ where $$n>5$$ (and $$n$$ is an integer)

$$5n$$ where $$n>2$$ (and $$n$$ is an integer)

$$5n$$ where $$n>10$$ (and $$n$$ is an integer)

$$5n + 5$$ where $$n>0$$ (and $$n$$ is an integer)

$$5n + 10$$ where $$n>0$$ (and $$n$$ is an integer)

Which of these could be the general form for any multiple of 5 greater than 10?

Which of these would be a good first step to prove "The product of any two odd numbers is an odd number"?

Which of these would be the best first step to prove "The difference between the squares of consecutive odd numbers is always a multiple of 8"?

Which of these statements are true for the expression $$12n + 15$$ where $$n$$ is an integer?

Take two odd numbers $$2n + 1$$ and $$2m +1$$ where $$n$$ and $$m$$ are integers

Any number one more than a multiple of 2 is odd. 

Therefore the product of any two odd numbers is odd. 

Starting with step 1 put these steps in order to form a complete proof for "The product of two odd numbers is always odd".

Match the number properties with their generalised forms.

Which of these is true for the expression $$4n + 7$$ where $$n$$ is an integer?

Which of these must be even for any integer value of $$n$$?

Which of these must be odd for any integer value of $$n$$?

$$n-1$$ and $$n+1$$ where $$n$$ is an integer.

$$n$$ and $$2n$$ where $$n$$ is an integer.

$$n+3$$ and $$n+4$$ where $$n$$ is an integer.

$$n$$ and $$m$$ where $$n$$ and $$m$$ are integers.

$$2n+1$$ and $$2n + 3$$ where $$n$$ is an integer.

Which of these represent any two consecutive integers?

What is the correct expansion of $$(3n-1)^2$$?

Writing a generalised statement about specific number properties

Lesson outcome