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.

Proving or disproving a statement

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

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

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

It is often easy to make a mistake when substituting values or measuring. 

It is often impossible to test all relevant cases.

Proofs are more convincing if they have diagrams or practical demonstrations.

Why is testing a conjecture often not a good way to prove it is true?

Laura wants to prove Pythagoras' theorem. Her first few steps are shown. Which of these is the correct next step of her proof?

If $$p$$ is a prime number and $$n$$ is a positive integer, which is a counterexample to the conjecture "$$p^n$$ is always odd"?

Which of these is a counterexample to the conjecture "The digit sum of any number of the form $$9n$$ where $$n$$ is a positive integer is always 9"?

Aisha writes the conjecture "For all integer values of $$x, (x-2)^2 > x-2$$". There is a counterexample to this when $$x=$${{}}.

An odd number subtract an odd number is an odd number.

An even number subtract an even number is an even number.

An even number multiplied by an integer is an even number. 

Which of these statements are always true?

Which of these is a general form for any odd number (where $$n$$ is an integer)?

The diagram shows a right-angled triangle. Which of these is an expression for the area?

The diagram shows a right-angled triangle. Which of these formula show the correct relationship between the lengths of the sides?

Which of these shows the first 4 numbers of the form $$9^n$$ where $$n$$ is a positive integer?

Numbers of this form end in either 1 or 9

Numbers of the form $$9^{n + 1}$$ always end in 1

Numbers of the form $$9^{2n}$$ always end in 9

Numbers of the form $$9^{2n-1}$$ always end in 9

Numbers of the form $$9^{5n}$$ always end in 9

Andeep has written out the first 4 numbers of the form $$9^n$$ where $$n$$ is a positive integer. Which of these conjectures hold true for these 4 values?

This lesson was made for home learning during the pandemic.

Proving or disproving a statement

Lesson outcome

This lesson was made for home learning during the pandemic.