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Year 11
Higher

Problem solving with functions and proof

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

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New
New
Year 11
Higher

Problem solving with functions and proof

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

Download all resources
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Share activities with pupils
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Lesson details

Key learning points

  1. In order to prove something, you may need to use any of your maths skills and knowledge.
  2. It is important to define any algebraic expressions at the start of a proof.
  3. 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.

If pupils are interested by the triangle problem, ask them to research the dissection fallacy to see more examples.
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.
$${\text{fg}}(x)$$, $${\text{gh}}(x)$$, 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: $$2xy-6x=y^2-9$$

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.
$$k=-3$$
$$k=1$$
$$k=1$$
$$k=3$$
Correct answer: $$k=6$$
Q5.
Where is the error in this proof?
An image in a quiz
$$a^2-b^2=ab-b^2$$ 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?
An image in a quiz
$$(c+a)(c-a)$$ 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.