Evaluating iterative formulas

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Lesson details

Learning outcome

I can evaluate and interpret iterative formula for various real-world situations.

Key learning points

Iterative formulae use the previous iteration's output for this iteration's input
A population can be estimated using an iterative formula
Efficient use of the calculator makes this process much faster

Keywords

Iteration - Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.

Common misconception

Some pupils want to clear their calculator display and type every single level of iteration in each time.

Encourage the pupils to trust their 'Ans' button. Get them to work on an example whereby they physically type the input value in and type the whole formula again. Then get them to repeat the same question using the 'Ans' button. Time this if you can!

Teacher tip

Effective use of a calculator is very helpful for this topic. You may wish to use the Classpad feature from the Casio website to display the Classwiz on the screen to help pupils see what buttons to press.

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

(Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
Iteration is the __________ application of a function or process in which the output of each iteration is used as the input for the next iteration.

Correct answer: repeated

mathematical

algebraic

calculative

Q2.
If $$\text{f}(x)=4x-7$$ evaluate $$\text{f}(1)$$

Correct Answer: -3

Q3.
If $$\text{f}(x)=4x-7$$ evaluate $$\text{fff}(1)$$

Correct Answer: -83

Q4.
$$£30\;000$$ is invested at a rate of $$3.25$$% interest for $$4$$ years. Which calculation will work out the value of the investment after $$4$$ years?

Correct answer: $$30\;000\times1.0325^4$$

$$30\;000\times1.04^{3.25}$$

$$30\;000\times103.25^4$$

$$30\;000\times1.0325$$

Q5.
Match the functions to the correct iterative formula we would use if we were to perform multiple repetitions of the same function.

Correct Answer:$$\text{f}(x)=3x-7$$,$$x_{n+1}=3x_{n}-7$$

$$x_{n+1}=3x_{n}-7$$

Correct Answer:$$\text{f}(x)=3x^2-7$$,$$x_{n+1}=3(x_{n})^2-7$$

$$x_{n+1}=3(x_{n})^2-7$$

Correct Answer:$$\text{f}(x)=3x^2-7x$$,$$x_{n+1}=3(x_{n})^2-7(x_{n})$$

$$x_{n+1}=3(x_{n})^2-7(x_{n})$$

Correct Answer:$$\text{f}(x)=x^3-7$$,$$x_{n+1}=(x_{n})^3-7$$

$$x_{n+1}=(x_{n})^3-7$$

Correct Answer:$$\text{f}(x)=x^3-7x^2$$,$$x_{n+1}=(x_{n})^3-7(x_{n})^2$$

$$x_{n+1}=(x_{n})^3-7(x_{n})^2$$

Q6.
$$£150$$ is invested at a rate of $$5.5$$% compound interest p.a. After how many years will the savings first be worth $$£500$$? years.

Correct Answer: 23, 23 years

Assessment exit quiz

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6 Questions

Q1.
Iteration is the repeated application of a function or process in which the of each iteration is used as the input for the next iteration.

Correct Answer: output

Q2.
Using the iterative formula $$x_{t+1}=8x_t-17$$ and $$x_0=5$$ find the value of $$x_1$$

Correct Answer: 23

Q3.
Using the iterative formula $$x_{t+1}=8x_t-17$$ and $$x_0=5$$ find the value of $$x_5$$

Correct Answer: 84263, 84 263

Q4.
Using the iterative formula $$x_{t+1}={3(x_t)^2-12(x_t)+15}$$ and $$x_0=1$$ find the value of $$x_2$$

Correct Answer: 51

Q5.
Using the iterative formula $$x_{t+1}={3(x_t)^2-12(x_t)+15}$$ and $$x_0=1$$ find the value of $$x_4$$

$$7206$$

$$7602$$

$$155\;692$$

$$692\;851$$

Correct answer: $$155\;692\;851$$

Q6.
There are $$20\;000$$ fish in a lake. Every year $$4000$$ fish are fished and removed. The remaining population reproduces and grows by $$12$$%. After how many years will the stock of fish reach zero?

$$5$$ years

$$6$$ years

Correct answer: $$7$$ years

$$8$$ years

$$9$$ years

To help you plan your 11 maths lesson on: Evaluating iterative formulas, download all teaching resources for free and adapt to suit your pupils' needs...

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Evaluating iterative formulas

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.Iteration is the __________ application of a function or process in which the output of each iteration is used as the input for the next iteration.

Q2.If $$\text{f}(x)=4x-7$$ evaluate $$\text{f}(1)$$

Q3.If $$\text{f}(x)=4x-7$$ evaluate $$\text{fff}(1)$$

Q4.$$£30\;000$$ is invested at a rate of $$3.25$$% interest for $$4$$ years. Which calculation will work out the value of the investment after $$4$$ years?

Q5.Match the functions to the correct iterative formula we would use if we were to perform multiple repetitions of the same function.

Q6.$$£150$$ is invested at a rate of $$5.5$$% compound interest p.a. After how many years will the savings first be worth $$£500$$? years.

Assessment exit quiz

6 Questions

Q1.Iteration is the repeated application of a function or process in which the of each iteration is used as the input for the next iteration.

Q2.Using the iterative formula $$x_{t+1}=8x_t-17$$ and $$x_0=5$$ find the value of $$x_1$$

Q3.Using the iterative formula $$x_{t+1}=8x_t-17$$ and $$x_0=5$$ find the value of $$x_5$$

Q4.Using the iterative formula $$x_{t+1}={3(x_t)^2-12(x_t)+15}$$ and $$x_0=1$$ find the value of $$x_2$$

Q5.Using the iterative formula $$x_{t+1}={3(x_t)^2-12(x_t)+15}$$ and $$x_0=1$$ find the value of $$x_4$$

Q6.There are $$20\;000$$ fish in a lake. Every year $$4000$$ fish are fished and removed. The remaining population reproduces and grows by $$12$$%. After how many years will the stock of fish reach zero?

How to plan a lesson using our resources

Q1.
Iteration is the __________ application of a function or process in which the output of each iteration is used as the input for the next iteration.

Q2.
If $$\text{f}(x)=4x-7$$ evaluate $$\text{f}(1)$$

Q3.
If $$\text{f}(x)=4x-7$$ evaluate $$\text{fff}(1)$$

Q4.
$$£30\;000$$ is invested at a rate of $$3.25$$% interest for $$4$$ years. Which calculation will work out the value of the investment after $$4$$ years?

Q5.
Match the functions to the correct iterative formula we would use if we were to perform multiple repetitions of the same function.

Q6.
$$£150$$ is invested at a rate of $$5.5$$% compound interest p.a. After how many years will the savings first be worth $$£500$$? years.

Q1.
Iteration is the repeated application of a function or process in which the of each iteration is used as the input for the next iteration.

Q2.
Using the iterative formula $$x_{t+1}=8x_t-17$$ and $$x_0=5$$ find the value of $$x_1$$

Q3.
Using the iterative formula $$x_{t+1}=8x_t-17$$ and $$x_0=5$$ find the value of $$x_5$$

Q4.
Using the iterative formula $$x_{t+1}={3(x_t)^2-12(x_t)+15}$$ and $$x_0=1$$ find the value of $$x_2$$

Q5.
Using the iterative formula $$x_{t+1}={3(x_t)^2-12(x_t)+15}$$ and $$x_0=1$$ find the value of $$x_4$$

Q6.
There are $$20\;000$$ fish in a lake. Every year $$4000$$ fish are fished and removed. The remaining population reproduces and grows by $$12$$%. After how many years will the stock of fish reach zero?