Problem solving with functions and proof

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Lesson details

Learning outcome

I can use my knowledge of functions and proof to solve problems.

Key learning points

In order to prove something, you may need to use any of your maths skills and knowledge.
It is important to define any algebraic expressions at the start of a proof.
This is so it is clear what cases you are proving.

Keywords

Function - A function is a mathematical relationship that uniquely maps values of one set to the values of another set.
Expression - An expression contains one or more terms, where each term is separated by an operator.

Common misconception

Pupils may struggle with showing why the false proofs are false.

Checking what each variable represents will help pupils see why each of the 'proofs' do not work.

Teacher tip

If pupils are interested by the triangle problem, ask them to research the dissection fallacy to see more examples.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0

except where otherwise stated. See Oak's terms & conditions

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Prior knowledge starter quiz

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6 Questions

Q1.
$${\text{fg}}(x)$$, $${\text{gh}}(x)$$, and $${\text{fgh}}(x)$$ are all examples of __________.

equations

quadratic equations

multiple functions

Correct answer: composite functions

Q2.
If $${\text{f}}(x)=5x−1$$ and $${\text{g}}(x)=2x+8$$ which is an expression for $${\text{fg}}(x)$$?

$$7x+7$$

$$10x+40$$

Correct answer: $$10x+39$$

$$10x+8$$

$$10x+6$$

Q3.
If $${\text{f}}(x)=5x−1$$ then $$7{\text{f}}(8)=$$ .

Correct Answer: 273, 7f(8)=273

Q4.
If $${\text{f}}(x)=x+5$$ and $${\text{g}}(x)=x^2+4x-7$$ which is an expression for $${\text{gf}}(x)$$?

$$5(x+5)-7$$

$$(x+5)^2+4x-7$$

Correct answer: $$x^2+14x+38$$

$$x^2+4x+3$$

$$x^2+10x+25$$

Q5.
$$x^2=100$$ has how many solutions?

$$0$$

$$1$$

Correct answer: $$2$$

$$3$$

Q6.
Which of the below are accurate manipulations of the equation $$2x=y+3$$?

$$3=y-2x$$

Correct answer: $$2xy=y^2+3y$$

$$2xy=y^2+3$$

Correct answer: $$2x(y-3)=(y+3)(y-3)$$

Correct answer: $$2xy-6x=y^2-9$$

Assessment exit quiz

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6 Questions

Q1.
If $${\text{f}}(x)=5x$$ and $${\text{g}}(x)=7-x$$ then $$5(7-x)$$ is __________ for $${\text{fg}}(x)$$.

Correct answer: an expression

an equation

a formula

Q2.
If $${\text{f}}(x)=5x-18$$ which is an expression for $${\text{f}}(x+3)$$?

$$5(x+3)$$

$$5x-15$$

Correct answer: $$5x-3$$

$$5x+3$$

Q3.
If $${\text{f}}(x)=5x-18$$ and $${\text{g}}(x)=-2x$$ which is an expression for $$2{\text{gf}}(5x)$$?

$$72-20x$$

Correct answer: $$72-100x$$

$$100x+72$$

$$100x-72$$

$$-2(5x-18)$$

Q4.
$${\text{f}}(x)=k(x-3)$$ If $${\text{f}}(-1)=-24$$ find $$k$$ where $$k$$ is a constant.

$$k=-3$$

$$k=1$$

$$k=3$$

Correct answer: $$k=6$$

Q5.
Where is the error in this proof?

$$a^2-b^2=ab-b^2$$ as you can’t subtract a square from both sides.

$$(a+b)(a-b)=b(a-b)$$ because $$a-b=0$$ and you can’t multiply by zero.

Correct answer: From $$(a+b)(a-b)=b(a-b)$$ to $$a+b=b$$ because dividing by zero is undefined.

$$a^2=a\times{a}$$, not $$a^2=a\times{b}$$

Q6.
Where is the error in this proof?

$$(c+a)(c-a)$$ does not expand to $$c^2-a^2$$

$$c^2-2bc=a^2-2ab$$ is not an accurate rearrangement of $$c^2-a^2=2bc-2ab$$

$$c^2-2bc+b^2$$ does not factorise to $$(c-b)^2$$

Correct answer: From $$(c-b)^2=(a-b)^2$$ to $$c-b=a-b$$ because it assumes the positive root.

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Problem solving with functions and proof

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.$${\text{fg}}(x)$$, $${\text{gh}}(x)$$, and $${\text{fgh}}(x)$$ are all examples of __________.

Q2.If $${\text{f}}(x)=5x−1$$ and $${\text{g}}(x)=2x+8$$ which is an expression for $${\text{fg}}(x)$$?

Q3.If $${\text{f}}(x)=5x−1$$ then $$7{\text{f}}(8)=$$ .

Q4.If $${\text{f}}(x)=x+5$$ and $${\text{g}}(x)=x^2+4x-7$$ which is an expression for $${\text{gf}}(x)$$?

Q5.$$x^2=100$$ has how many solutions?

Q6.Which of the below are accurate manipulations of the equation $$2x=y+3$$?

Assessment exit quiz

6 Questions

Q1.If $${\text{f}}(x)=5x$$ and $${\text{g}}(x)=7-x$$ then $$5(7-x)$$ is __________ for $${\text{fg}}(x)$$.

Q2.If $${\text{f}}(x)=5x-18$$ which is an expression for $${\text{f}}(x+3)$$?

Q3.If $${\text{f}}(x)=5x-18$$ and $${\text{g}}(x)=-2x$$ which is an expression for $$2{\text{gf}}(5x)$$?

Q4.$${\text{f}}(x)=k(x-3)$$ If $${\text{f}}(-1)=-24$$ find $$k$$ where $$k$$ is a constant.

Q5.Where is the error in this proof?

Q6.Where is the error in this proof?

How to plan a lesson using our resources

Q1.
$${\text{fg}}(x)$$, $${\text{gh}}(x)$$, and $${\text{fgh}}(x)$$ are all examples of __________.

Q2.
If $${\text{f}}(x)=5x−1$$ and $${\text{g}}(x)=2x+8$$ which is an expression for $${\text{fg}}(x)$$?

Q3.
If $${\text{f}}(x)=5x−1$$ then $$7{\text{f}}(8)=$$ .

Q4.
If $${\text{f}}(x)=x+5$$ and $${\text{g}}(x)=x^2+4x-7$$ which is an expression for $${\text{gf}}(x)$$?

Q5.
$$x^2=100$$ has how many solutions?

Q6.
Which of the below are accurate manipulations of the equation $$2x=y+3$$?

Q1.
If $${\text{f}}(x)=5x$$ and $${\text{g}}(x)=7-x$$ then $$5(7-x)$$ is __________ for $${\text{fg}}(x)$$.

Q2.
If $${\text{f}}(x)=5x-18$$ which is an expression for $${\text{f}}(x+3)$$?

Q3.
If $${\text{f}}(x)=5x-18$$ and $${\text{g}}(x)=-2x$$ which is an expression for $$2{\text{gf}}(5x)$$?

Q4.
$${\text{f}}(x)=k(x-3)$$ If $${\text{f}}(-1)=-24$$ find $$k$$ where $$k$$ is a constant.

Q5.
Where is the error in this proof?

Q6.
Where is the error in this proof?