A prime number is an integer greater than one with exactly 2 factors.

A triangular number is a number that can be represented by a pattern of dots arranged into an equilateral triangle. The term number is the number of dots in a side of the triangle.

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

Making conjectures about patterns and relationships

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

Proving or disproving a statement

Writing a generalised statement about specific number properties

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

What sort of numbers always have an odd number of factors?

Give an example of a number less than 100 which is both square and triangular.

A conjecture is a (mathematical) statement which __________.

Lucas writes the conjecture "All numbers of the form $$2^{n}-1$$ (where $$n$$ is an integer greater than 1) are prime". Which of the following are true?

Which of these shows the first 4 numbers of the form $$7^n$$ where $$n$$ is a positive integer?

Numbers of this form always end in either 1, 3, or 9

Numbers of this form cannot be multiples of 10

Numbers of the form $$7^{2n}$$ where $$n$$ is a positive integer end in 9

Numbers of the form $$7^{4n}$$ where $$n$$ is a positive integer end in 1

Jacob has written out the first 4 numbers of the form $$7^n$$ where $$n$$ is a positive integer. Which of these conjectures hold true for these 4 values?

The number 100 has exactly {{}} factors. 

Which of these expressions must represent a number 'one more than a multiple of 5' when $$n$$ is an integer?

What is the value of $$3^n$$ when $$n=4$$?

Making conjectures about patterns and relationships

Lesson outcome