 Use the equivalence of fractions, decimals and percentages to compare proportions; calculate percentages and find the outcome of a given percentage increase or decrease

Examples of what pupils should know and be able to do
Convert fraction and decimal operators to percentage operators by multiplying by 100. For example:
 0.45 $0.45\times 100\%=45\%$
 $\frac{7}{12}\left(7\xf712\right)\times 100\%=58.3\%$ (1 d.p.)
Continue to use mental methods for finding percentages of quantities.
Use written methods:
Using an equivalent fraction
 13% of 48
$\frac{13}{100}\times 48=\frac{624}{100}=6.24$
Using an equivalent decimal
 13% of 48
$0.13\times 48=6.24$
Using a unitary method
 13% of 48
1% of $48=0.48$
13% of $48=0.48\times 13=6.24$
Find the outcome of a given percentage increase or decrease, for example:
An increase of 15% on an original cost of £12 gives a new price of
$\mathrm{\pounds}12\times 1.15=\mathrm{\pounds}13.80$
or
15% of $\mathrm{\pounds}12=\mathrm{\pounds}1.80$
$\mathrm{\pounds}12+\mathrm{\pounds}1.80=\mathrm{\pounds}13.80$ Secondary mathematics exemplification: Use the equivalence of fractions, decimals and percentages to compare simple proportions and solve problems
Probing questions
 Which sets of equivalent fractions, decimals and percentages do you know?
 From one set that you know (e.g. $\frac{1}{10}=0.1=10\%$), which others can you deduce?
 How would you go about finding the decimal and percentage equivalents of any fraction?
 How would you find out which of these is closest to $\frac{1}{3}$: $\frac{10}{31}$; $\frac{20}{61}$; $\frac{30}{91}$; $\frac{50}{151}$?
 Give me a fraction between $\frac{1}{3}$ and$\frac{1}{2}$. How did you do it? Is it closer to $\frac{1}{3}$ or $\frac{1}{2}$? How do you know?
 Talk me through how you would increase or decrease a price of £12 by, for example, 15%. Can you do it in a different way? How would you find the multiplier for different percentage increases/decreases?
 The answer to a percentage increase question is £10. Make up an easy question. Make up a difficult question.
What if pupils find this a barrier?
 Lesson idea: Making links with percentages
 Lesson idea: Stepping stones to percentages
 Lesson idea: Fractions, decimals and percentages 1
 Lesson idea: Calculating with fractions
 Order fractions by writing them with a common denominator or by converting them into decimals

Examples of what pupils should know and be able to do
Find the larger of $\frac{7}{8}$ and $\frac{4}{5}$:
 using common denominators:
$\frac{7}{8}$ is $\frac{35}{40}$, $\frac{4}{5}$ is $\frac{32}{40}$, so $\frac{7}{8}$ is larger.  using decimals:
$\frac{7}{8}$ is 0.875, $\frac{4}{5}$ is 0.8, so $\frac{7}{8}$ is larger.
Position fractions on a number line. Which is greater, 0.23 or $\frac{3}{16}$? Which fraction is exactly halfway between $\frac{3}{5}$ and $\frac{5}{7}$?
 Secondary mathematics exemplification: Converting decimals to fractions
Probing questions
 Which is larger, $\frac{4}{5}$ or $\frac{5}{6}$, and how do you know?
 Give me a fraction that is close to one half. How would you find one that is closer?
 How do you change a fraction into a decimal?
What if pupils find this a barrier?
If necessary revisit ideas of equivalence:
 between fractions
 between fractions and decimals.
Also, see Step 6.
 using common denominators:
 Divide a quantity into two or more parts in a given ratio; use the unitary method to solve simple word problems involving ratio and direct proportion

Examples of what pupils should know and be able to do
Solve problems such as the following.
 Potting compost is made from loam, peat and sand in the ratio 7 : 3 : 2 respectively. A gardener used 1.5 litres of peat to make compost. How much loam did she use? How much sand?
 The angles in a triangle are in the ratio 6 : 5 : 7. Find the sizes of the three angles.
 Secondary mathematics exemplification: Simplify ratios
 Assessment example: Ratio (adapting a recipe): Oscar (PDF379 KB) Attachments
Probing questions
 If the ratio of boys to girls in a class is 3 : 1, could there be exactly 30 children in the class? Why?
 Could there be 25 boys? Why?
 5 miles is about the same as 8 km. Can you make up some conversion questions that you could answer mentally?
 Can you make up some conversion questions for which you would have to use a more formal method?
 How would you work out the answers to these questions?
What if pupils find this a barrier?
 Lesson idea: Ratio and proportion
 Lesson idea: Proportion or not?
 Lesson idea: Thinking proportionally