New
New
Year 11
•
Higher
Logical arguments
I can use algebra and algebraic manipulation to construct logical arguments.
New
New
Year 11
•
Higher
Logical arguments
I can use algebra and algebraic manipulation to construct logical arguments.
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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.
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
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?
This lesson looks at the proof that root 2 is irrational. You may wish to skip this section (and the last question of the task) if you feel this is too challenging.
Teacher tip
Licence
This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).
Lesson video
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Starter quiz
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6 Questions
Q1.
The sum of any two integers is always __________.
even
odd
positive
Correct answer: an integer
an integer
Q2.
If $$n$$ and $$m$$ are integers which of these are always even?
$$12n +m$$
Correct answer: $$2nm$$
$$2nm$$
$$3(n+2m)$$
Correct answer: $$2(m+n)+4$$
$$2(m+n)+4$$
Correct answer: $$4(n^2 + n + m^2 + m)$$
$$4(n^2 + n + m^2 + m)$$
Q3.
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.
Correct answer: $$10n + 10 - 10n$$ where $$n$$ is an integer.
$$10n + 10 - 10n$$ where $$n$$ is an integer.
Q4.
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 $$b\ne 0$$
Correct answer: $$\frac{a}{b}$$ where $$a$$ and $$b$$ are integers 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 $$a \ne 0$$
Q5.
Which of these numbers are rational?
Correct answer: $$\sqrt{1\over 4}$$
$$\sqrt{1\over 4}$$
Correct answer: $$\sqrt 1$$
$$\sqrt 1$$
$$\sqrt 2$$
$$\sqrt 5$$
$$\sqrt 6$$
Q6.
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 $$n=$$ ?
Correct Answer: 0, zero
0, zero
Exit quiz
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6 Questions
Q1.
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.
Correct answer: There exists a basketball player who is not tall.
There exists a basketball player who is not tall.
There exists a basketball player who is tall.
Q2.
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.
Correct answer: There is an integer greater than -1 which is not positive.
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.
Q3.
Which is the correct opposing statement to "All multiples of 10 are even"?
Correct answer: There exists a multiple of 10 which is not even.
There exists a multiple of 10 which is not even.
Correct answer: There exists a multiple of 10 which is odd.
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.
Q4.
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.
Correct answer: There exists values for $$a$$ and $$b$$ which are even but $$a+b$$ is odd.
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.
Q5.
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.
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.
2
- $$a=2m + 1$$ and $$a^2=2n$$ where $$m$$ and $$n$$ are integers.
3 - $$(2m+1)^2=2n$$
3
- $$(2m+1)^2=2n$$
4 - $$4m^2 + 4m + 1 = 2n$$
4
- $$4m^2 + 4m + 1 = 2n$$
5 - $$1=2n-4m^2-4m$$
5
- $$1=2n-4m^2-4m$$
6 - $$1=2(n-2m^2-2m)$$ this states 1 is an even number.
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.
7
- This is mathematically unsound so if $$a^2$$ is even $$a$$ cannot be odd.
Q6.
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.
1
- Assume $$\sqrt2$$ is rational.
2 - $$\sqrt{2}= \frac{a}{b}$$ where $$a$$ and $$b$$ are integers and $$b\ne 0$$.
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
3
- Assume any common factors have been cancelled so the HCF of $$a$$ and $$b$$ is 1
4 - $$2=\frac{a^2}{b^2}$$
4
- $$2=\frac{a^2}{b^2}$$
5 - $$2b^2 = a^2$$
5
- $$2b^2 = a^2$$
6 - Therefore $$a^2$$ is even so $$a$$ is even.
6
- Therefore $$a^2$$ is even so $$a$$ is even.
7 - $$a=2n$$ for some integer $$n$$.
7
- $$a=2n$$ for some integer $$n$$.