New
New
Year 11
•
Higher
Multiple approaches to logical arguments
I can construct a logical argument.
New
New
Year 11
•
Higher
Multiple approaches to logical arguments
I can construct a logical argument.
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Lesson details
Key learning points
- A logical argument does not have to only be algebraic.
- Geometrical reasoning can be used to argue that something is true.
- Values can be used to demonstrate whether something is true.
Keywords
Apex - The apex is the point (vertex) which is the greatest perpendicular distance from the base.
Congruent - If one shape can fit exactly on top of another using rotation, reflection or translation, then the shapes are congruent.
Hypotenuse - The hypotenuse is the side of a right-angle triangle which is opposite the right angle.
Common misconception
Proofs have to be solely algebraic and do not involve diagrams.
Proofs involve showing that a conjecture holds for multiple cases (general case). A diagram can represent multiple cases.
You may wish do to one step of each proof at a time and get pupils to think about what they could do next. Depending on time, you could explore some of the pupils' suggestions to see where they lead.
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.
If one shape can fit exactly on top of another using rotation, reflection or translation they are __________.
compound
Correct answer: congruent
congruent
invariant
isosceles
regular
Q2.
The interior angles of any pentagon sum to °.
Correct Answer: 540
540
Q3.
The diagram shows two parallel lines and a transversal. Which angle is the alternate angle to $$a$$?
$$b$$
$$c$$
Correct answer: $$d$$
$$d$$
$$e$$
$$f$$
Q4.
The diagram shows two parallel lines and a transversal. Which angle is the corresponding angle to $$a$$?
$$b$$
$$c$$
$$d$$
$$e$$
Correct answer: $$f$$
$$f$$
Q5.
Use the diagram to match the lines to their equations.
Correct Answer:A (purple),$$y=x+2$$
A (purple) -
$$y=x+2$$
Correct Answer:B (blue),$$y=-x+2$$
B (blue) -
$$y=-x+2$$
Correct Answer:C (pink),$$y=2x$$
C (pink) -
$$y=2x$$
Correct Answer:D (green),$$y=x-2$$
D (green) -
$$y=x-2$$
Correct Answer:E (black),$$y={1\over2}x-2$$
E (black) -
$$y={1\over2}x-2$$
Q6.
Use the diagram to match the curves to their equations.
Correct Answer:A (blue),$$y = x^3$$
A (blue) -
$$y = x^3$$
Correct Answer:B (purple),$$y= x^2 - 2$$
B (purple) -
$$y= x^2 - 2$$
Correct Answer:C (pink),$$y = x^2$$
C (pink) -
$$y = x^2$$
Correct Answer:D (green),$$y = (x-2)^2$$
D (green) -
$$y = (x-2)^2$$
Exit quiz
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6 Questions
Q1.
What is the next step for the proof of the circle theorem "The angle in a semicircle is a right angle"?
Correct answer: The radius $$OB$$ can be drawn to split $$ABC$$ into 2 isosceles triangles.
The radius $$OB$$ can be drawn to split $$ABC$$ into 2 isosceles triangles.
The angle at $$B$$ is 90° so a right angle.
Interior angles of a triangle sum to 180°.
$$(AB)^2 + (BC)^2 = (AC)^2$$
Q2.
What is the next step for the proof that "The sum of the interior angles in any polygon is $$180(n − 2)$$ where $$n$$ is the number of sides" ?
The angles in a polygon sum to $$180(n-2)$$ where $$n$$ is the number of sides.
The number of triangles is two less than the number of sides.
The sum of the interior angles increases by 180 every time a side is added.
Correct answer: The sum of the interior angles of $$n$$ triangles is $$180n$$.
The sum of the interior angles of $$n$$ triangles is $$180n$$.
Q3.
Arrange the steps of this proof in order to show that "The tangents to a circle from an external point are equal in length".
1 - Take a circle with centre $$O$$ and tangents $$AB, BC$$ as shown in the diagram.
1
- Take a circle with centre $$O$$ and tangents $$AB, BC$$ as shown in the diagram.
2 - $$OA$$ and $$OC$$ are radii so are the same length.
2
- $$OA$$ and $$OC$$ are radii so are the same length.
3 - The tangents to a circle meet the radius at 90°.
3
- The tangents to a circle meet the radius at 90°.
4 - $$ABCO$$ can be split into two right-angled triangles.
4
- $$ABCO$$ can be split into two right-angled triangles.
5 - The triangles have the same hypotenuse, another side the same and a right-angle.
5
- The triangles have the same hypotenuse, another side the same and a right-angle.
6 - By the RHS law the triangles are congruent so length $$AB=$$ length $$BC$$.
6
- By the RHS law the triangles are congruent so length $$AB=$$ length $$BC$$.
7 - Therefore the tangents to a circle from an external point are equal in length.
7
- Therefore the tangents to a circle from an external point are equal in length.
Q4.
Which of these values of $$x$$ are counterexamples to the conjecture "For all values of $$x, (x-3)^2 > x-3$$"?
$$x=0$$
$$x=2.5$$
Correct answer: $$x=3.5$$
$$x=3.5$$
Correct answer: $$x=4$$
$$x=4$$
$$x=5$$
Q5.
Which of these best describes all the counterexamples to the conjecture "For all values of $$x, 2x > x-2$$"?
The counterexamples are when $$x$$ is positive.
The counterexamples are when $$x$$ is negative.
The counterexamples are when $$x \ge -2$$ .
Correct answer: The counterexamples are when $$x \le -2$$ .
The counterexamples are when $$x \le -2$$ .
The counterexamples are when $$x \le -4$$ .
Q6.
Use the graphs to decide which of these conjectures are true.
Correct answer: For all values of $$x, -x^2 \le 0$$
For all values of $$x, -x^2 \le 0$$
For all values of $$x, -x^2 < x$$
For all values of $$x, -x^2 < -x$$
Correct answer: For all negative values of $$x, -x^2 < -x$$
For all negative values of $$x, -x^2 < -x$$
For all integer values of $$x, -x^2 < x$$