The apex is the point (vertex) which is the greatest perpendicular distance from the base.

If one shape can fit exactly on top of another using rotation, reflection or translation, then the shapes are congruent.

The hypotenuse is the side of a right-angle triangle which is opposite the right angle.

Multiple approaches to logical arguments

Maths

Checking and securing understanding of functions

Defining function notation

Finding the inverse of a function

Writing composite functions

Solving equations involving functions

Solving equations involving composite functions

General algebraic forms for specific number properties

Making conjectures about patterns and relationships

Proving or disproving a statement

Writing a generalised statement about specific number properties

Writing a proof

Logical arguments

Problem solving with functions and proof

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

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°.

What is the next step for the proof of the circle theorem "The angle in a semicircle is a right angle"?

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.

The sum of the interior angles of $$n$$ triangles is $$180n$$.

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" ?

Take a circle with centre $$O$$ and tangents $$AB, BC$$ as shown in the diagram.

$$OA$$ and $$OC$$ are radii so are the same length.

The tangents to a circle meet the radius at 90°.

$$ABCO$$ can be split into two right-angled triangles.

The triangles have the same hypotenuse, another side the same and a right-angle.

By the RHS law the triangles are congruent so length $$AB=$$ length $$BC$$.

Therefore the tangents to a circle from an external point are equal in length.

Arrange the steps of this proof in order to show that "The tangents to a circle from an external point are equal in length".

Which of these values of $$x$$ are counterexamples to the conjecture "For all values of $$x, (x-3)^2 > x-3$$"?

The counterexamples are when $$x$$ is positive.

The counterexamples are when $$x$$ is negative. 

The counterexamples are when $$x \ge -2$$ . 

The counterexamples are when $$x \le -2$$ . 

The counterexamples are when $$x \le -4$$ . 

Which of these best describes all the counterexamples to the conjecture "For all values of $$x, 2x > x-2$$"? 

For all negative values of $$x, -x^2 < -x$$

For all integer values of $$x, -x^2 < x$$

Use the graphs to decide which of these conjectures are true.

If one shape can fit exactly on top of another using rotation, reflection or translation they are __________.

The interior angles of any pentagon sum to {{}}°.

The diagram shows two parallel lines and a transversal. Which angle is the alternate angle to $$a$$?

The diagram shows two parallel lines and a transversal. Which angle is the corresponding angle to $$a$$?

Use the diagram to match the lines to their equations.

Use the diagram to match the curves to their equations.

Multiple approaches to logical arguments

Lesson outcome