Writing a generalised statement about specific number properties | Maths

Starter quiz

Match the number properties with their generalised forms.

Any integer

⇔

$n$ where $n$ is an integer. ✓

Any odd number

⇔

$2n + 1$ where $n$ is an integer. ✓

Any positive number

⇔

$n$ where $n > 0$ ✓

Any even number

⇔

$2n$ where $n$ is an integer. ✓
Which of these is true for the expression $4n + 7$ where $n$ is an integer?

It is always odd. ✓

It is sometimes odd.

It is never odd.
Which of these must be even for any integer value of $n$ ?

$2(n + 1)$ ✓

$3(n+1)$

$3(2n + 2)$ ✓

$12n + 9$

$(2n +1)(2n+1)$
Which of these must be odd for any integer value of $n$ ?

$6n + 8$

$3(n+1)$

$5(2n-1)$ ✓

$4(n+2) + 1$ ✓

$7n + 13$
Which of these represent any two consecutive integers?

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

$9n^2 -1$

$9n^2 +1$

$9n^2 -6n - 1$

$9n^2 -6n + 1$ ✓

$9n^2 -9n + 1$

Exit quiz

Which of these represent any 3 consecutive even numbers?

$n, n+ 1, n+2$ 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 could be the general form for any multiple of 5 greater than 10?

$n+ 5$ 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 would be a good first step to prove "The product of any two odd numbers is an odd number"?

None of these are good first steps. ✓
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?

It is always an even number

It is always a multiple of 3 ✓

It is always a muliple of 4

It is always 3 more than a multiple of 6 ✓

It is always 1 less than a multiple of 8
Starting with step 1 put these steps in order to form a complete proof for "The product of two odd numbers is always odd".

1

⇔

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

2

⇔

$(2n +1)(2m+1)$

3

⇔

$4nm + 2n + 2m + 1$

4

⇔

$2(2nm + n + m) + 1$

5

⇔

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

6

⇔

Therefore the product of any two odd numbers is odd.

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Lesson Details

Key learning points

A conjecture about a generalisation can be expressed algebraically.
The algebraic expression can be used to test the conjecture.
Substituting values can reveal whether the conjecture is wrong.
Substituting values cannot prove the conjecture unless all possibilities are exhausted.

Common misconception

Pupils may think it is acceptable to use the same letter for all expressions.

It may be acceptable for the letter to be the same but that depends on what the letter is representing.

Keywords

Conjecture - A conjecture is a (mathematical) statement that is thought to be true but has not been proved yet.
Generalise - To generalise is to formulate a statement or rule that applies correctly to all relevant cases.