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

A rational number is one that can be written in the form a/b where a and b are integers and b is not equal to 0

An irrational number is one that cannot be written in the form a/b where a and b are integers and b is not equal to 0

Logical arguments

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 generalised statement about specific number properties

Writing a proof

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

There exists a person who is not a basketball player but is tall.

There exists a basketball player who is not tall. 

There exists a basketball player who is tall.

Which is the correct opposing statement to "All basketball players are tall"?

There is an integer greater than -1 which is positive.

There is an integer greater than -1 which is negative.

There is an integer greater than -1 which is not positive.

There is an integer less than -1 which is positive.

There is an integer less than -1 which is negative.

Which is the correct opposing statement to "All integers greater than -1 are positive"?

There exists a multiple of 10 which is not even.

There exists a multiple of 10 which is odd. 

There exists a multiple of 10 which is even. 

There exists an even number which is not a multiple of 10.

Which is the correct opposing statement to "All multiples of 10 are even"?

If $$a$$ and $$b$$ are odd then $$a + b$$ must be even.

If $$a$$ and $$b$$ are even then $$a + b$$ must be odd.

There exists integer values for $$a$$ and $$b$$ where $$a+b$$ is even.

There exists values for $$a$$ and $$b$$ which are even but $$a+b$$ is odd.

There exists values for $$a$$ and $$b$$ which are odd and $$a+b$$ is odd.

Which is the correct opposing statement to "If $$a$$ and $$b$$ are even then $$a+b$$ is even"?

Assume there is an integer value for $$a$$ where $$a^2$$ is even but $$a$$ odd. 

$$a=2m + 1$$ and $$a^2=2n$$ where $$m$$ and $$n$$ are integers.

$$1=2(n-2m^2-2m)$$ this states 1 is an even number.

This is mathematically unsound so if $$a^2$$ is even $$a$$ cannot be odd.

Arrange these steps for the proof that "For any integer $$a$$ if $$a^2$$ is even then $$a$$ is even" into the correct order. 

$$\sqrt{2}= \frac{a}{b}$$ where $$a$$ and $$b$$ are integers and $$b\ne 0$$.

Assume any common factors have been cancelled so the HCF of $$a$$ and $$b$$ is 1

Therefore $$a^2$$ is even so $$a$$ is even.

Arrange these steps for the first half of the proof that "The number $$\sqrt 2$$ is irrational" into the correct order. 

The sum of any two integers is always __________.

If $$n$$ and $$m$$ are integers which of these are always even?

$$10m-10n$$ where $$n$$ and $$m$$ are integers.

$$10n + 1 - 10n$$ where $$n$$ is an integer. 

$$10(m+1)-10n$$ where $$n$$ and $$m$$ are integers.

$$10n + 10 - 10n$$ where $$n$$ is an integer. 

How could we generalise the difference between any two consecutive multiples of 10?

$$ab$$ where $$a$$ and $$b$$ are integers, $$a \ne 0$$ and $$b\ne 0$$

$$\frac{a}{b}$$ where $$a$$ and $$b$$ are integers and $$b\ne 0$$

$$a^b$$ where $$a$$ and $$b$$ are integers and $$a \ne 0$$ 

A rational number can be written in the form __________.

Alex writes the conjecture "All numbers of the form $$n^2$$ where $$n$$ is an integer are positive". There is a counterexample to this when $$n=$${{}}?

Logical arguments

Lesson outcome