Calculating

These materials will help you to assess pupils' progress in Calculation at level 8. The examples of what pupils should be able to do and the probing questions will help you to secure evidence of progress in relation to the assessment criteria.

Use fractions or percentages to solve problems involving repeated proportional changes or the calculation of the original quantity given the result of a proportional change

Examples of what pupils should know and be able to do

Solve problems involving, for example, compound interest and population growth using multiplicative methods.

Use a spreadsheet to solve problems such as:

• How long would it take to double your investment with an interest rate of 4 % per annum?
• A ball bounces to $\frac{3}{4}$ of its previous height each bounce. It is dropped from 8 m. How many bounces will there be before it bounces to approximately 1 m above the ground?

Solve problems in other contexts, such as:

• Each side of a square is increased by 10 %. By what percentage is the area increased?
• The length of a rectangle is increased by 15 %, the width is decreased by 5 %. By what percentage is the area changed?

Probing questions

Talk me through why this calculation will give the solution to this repeated proportional change problem.

How would the calculation be different if the proportional change was…?

What do you look for in a problem to decide the product that will give the correct answer?

How is compound interest different from simple interest?

Give pupils a set of problems involving repeated proportional changes and a set of calculations. Ask pupils to match the problems to the calculations.

Solve problems involving calculating with powers, roots and numbers expressed in standard form, checking for correct order of magnitude and using a calculator as appropriate.

Examples of what pupils should know and be able to do

Use laws of indices in multiplication and division, for example, to calculate:

$\frac{4^3\times 4^5}{{\left(4^2\right)}^3}$

What is the value of c in the following question?

$48\times 56=3\times 7\times 2^c$

Understand index notation with fractional powers, for example, knowing that $n^{\frac{1}{2}}=\sqrt{n}$ and $n^{\frac{1}{3}}=\sqrt[3]{n}$ for any positive number n.

Convert numbers between ordinary and standard form, for example:

$734.6=7.346\times {10}^2$

$0.0063=6.3\times {10}^{-3}$

Use standard form expressed in conventional notation and on a calculator display. Know how to enter numbers on a calculator in standard form.

Use standard form to make sensible estimates for calculations involving multiplication and division.

Solve problems involving standard form, such as:

Given the following dimensions

Diameter of the eye of a fly: $8\times {10}^{-4}\mathrm{m}$

Height of a tall skyscraper: $4\times {10}^2\mathrm{m}$

Height of a mountain: $8\times {10}^3\mathrm{m}$

How many times taller is the mountain than the skyscraper?

How high is the skyscraper in km?

Probing questions

Convince me that:

$3^7\times 3^2=3^9$

$3^7÷3^{-2}=3^9$

$3^7\times 3^{-2}=3^5$

When working on multiplications and divisions involving indices, ask:

Which of these are easy to do? Which are difficult? What makes them difficult?

How would you go about making up a different question that has the same answer?

What does the index of $\frac{1}{2}$ represent?

What are the key conventions when using standard form?

How do you go about expressing a very small number in standard form?

Are the following statements always, sometimes or never true?

• Cubing a number makes it bigger.
• The square of a number is always positive.
• You can find the square root of any number.
• You can find the cube root of any number.

If sometimes true, precisely when is the statement true and when is it false?

Which of the following statements are true?

${16}^{3/2}=8^2$

The length of an A4 piece of paper is $2.97\times {10}^{-5}\mathrm{km}$

$8^{-3}=\frac{1}{2^{-9}}$

${27}^2=3^6$

$3\sqrt{7}\times 2\sqrt{7}=5\sqrt{7}$