Logical arguments | Maths | Quizalize & Oak National Academy

Starter quiz

The sum of any two integers is always __________.

even

odd

positive

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

$12n +m$

$2nm$ ✓

$3(n+2m)$

$2(m+n)+4$ ✓

$4(n^2 + n + m^2 + m)$ ✓
How could we generalise the difference between any two consecutive multiples of 10?

$10m-10n$ 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. ✓
A rational number can be written in the form __________.

$\sqrt a$ where $a$ in an integer

$ab$ 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 $a \ne 0$
Which of these numbers are rational?

$\sqrt{1\over 4}$ ✓

$\sqrt 1$ ✓

$\sqrt 2$

$\sqrt 5$

$\sqrt 6$
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 $n=$ ______?

'0' ✓

Exit quiz

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

All basketball players are short.

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 integers greater than -1 are positive"?

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 multiples of 10 are even"?

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.

No multiples of 10 are even.
Which is the correct opposing statement to "If $a$ and $b$ are even then $a+b$ is 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.
Arrange these steps for the proof that "For any integer $a$ if $a^2$ is even then $a$ is even" into the correct order.

1

⇔

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

2

⇔

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

3

⇔

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

4

⇔

$4m^2 + 4m + 1 = 2n$

5

⇔

$1=2n-4m^2-4m$

6

⇔

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

7

⇔

This is mathematically unsound so if $a^2$ is even $a$ cannot be odd.
Arrange these steps for the first half of the proof that "The number $\sqrt 2$ is irrational" into the correct order.

1

⇔

Assume $\sqrt2$ is rational.

2

⇔

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

3

⇔

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

4

⇔

$2=\frac{a^2}{b^2}$

5

⇔

$2b^2 = a^2$

6

⇔

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

7

⇔

$a=2n$ for some integer $n$ .

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Lesson Details

Key learning points

A logical argument is a series of statements that progress in a clear way.
Each subsequent statement is based on the previous statement being true.
If a contradiction is reached, it shows that the original assumption was wrong.

Common misconception

Pupils may struggle to write a correct opposing statement.

Encourage pupils to discuss the various proposed opposing statements presented in the first learning cycle. What does each statement actually mean?

Keywords

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