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 proof

Maths

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 are 3 consecutive even integers $$2n, 2n+2, 2n+4$$ ($$n$$ is an integer).

The middle number was $$2n +2$$ so this is 8 times that.

Therefore we have proven the original statement.

Arrange these statements into the correct order to prove: "For any three consecutive even integers, the difference between the squares of the first and the last number is 8 times the middle number".

$$n^2 + 6n + 6n + 36 - ( n^2 + 2n + 2n + 4)$$

The first step to the proof "If $$n$$ is an integer $$(n+6)^2-(n+2)^2$$ is a multiple of 8" is shown. Which is the correct next step?

Izzy wants to prove that $$(n+6)^2-(n+2)^2$$  is a multiple of 8 for any integer $$n$$. Which of these is equivalent to this expression?

If something is a multiple of 3 it is also a multiple of 9

Therefore the product of two multiples of 3 is a multiple of 9

The first step to the proof that "The product of any two multiples of 3 is a multiple of 9" is shown. What is the best next step?

Any two odd numbers have an odd product. 

An odd number squared is one more than a multiple of 4.

The product of consecutive odd numbers cannot be a multiple of 4.

Here are some steps to a proof Aisha is constructing. What do you think she is trying to prove?

The generalisation has not been explained. 

Too many steps have been skipped in the algebraic manipulation.

The expression has not been fully factorised. 

The brackets have been expanded incorrectly. 

Here are some steps to a proof Aisha is constructing. Why is this not a complete proof?

Which of these is equivalent to $$(n+5)^2-(n+1)^2$$?

If $$n$$ is an integer, which of these must be true for the expression $$12n + 60$$?

Which is the correct first step for the proof that "the square of any even number is a multiple of 4"?

$$n, n+1, n+2$$ 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

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

$$2n+1, 2n+3, 2n+5$$ where $$n$$ is an integer

Which of these could be a general form for any 3 consecutive integers?

Which is the correct first step for the proof that "the product of any two multiples of 3 is a multiple of 9"?

Take two odd integers $$2n + 1$$ and $$2m + 1$$.

Any integer multiplied by 2 is an even number.

The sum of the squares of two odd integers is even.

Starting with the first step, arrange these steps so that they form a complete proof for "the sums of the squares of two odd integers is even", (assume $$n$$ and $$m$$ are integers).

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