GCSE Number Skills: Decimals, Fractions and Percentages
Mastering GCSE Number Skills
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Mastering GCSE Number Skills
This lesson brings together linked GCSE number topics: decimals, significant figures, fractions, recurring decimals, and percentage change.
Today’s learning goal
I can calculate confidently with decimals and fractions, round using significant figures, connect recurring decimals with fractions, and use percentage and repeated proportional change.
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One word or phrase you link with decimals, fractions or percentages
Use this to surface prior knowledge and misconceptions before the first teaching block. Expect terms such as place value, rounding, equivalent, recurring, multiplier, divide, exact and estimate.
Decimals in place value
Each digit has a place value
Digits to the right of the decimal point are tenths, hundredths, thousandths and beyond
Keeping place value clear helps with all four operations
Place value errors are a common cause of mistakes
Emphasise that decimals can be any size, including very small numbers such as 0.0048 and numbers greater than 100.
Adding and subtracting decimals
To add or subtract decimals, line up the decimal points first. Use place value columns so that tenths go under tenths and hundredths go under hundredths. Adding zeroes at the end can help, for example `4.6 = 4.60`.
Model examples such as 3.47 + 12.8 and 15.02 - 7.9. Stress that extra zeroes after the decimal do not change the value.
Multiplying and dividing decimals
For multiplication, ignore the decimal points first, multiply, then place the decimal point in the answer
Count the total number of decimal places in the factors
For division by a decimal, make the divisor a whole number by multiplying both numbers by the same power of 10
Estimation helps check whether the answer is sensible
Useful examples: 2.4 × 0.35 and 7.56 ÷ 0.12. Point out that students should estimate first to judge the size of the answer.
Checking decimal answers
Estimate before calculating
Compare with a whole-number version of the problem
Watch for misplaced decimal points
Use inverse operations where possible
A typical misconception is getting 2.3 × 4.1 as 94.3 or 9.43 without checking size. Estimation prevents this.
Decimal practice 1–20
Questions 11–20
`8.4 ÷ 2`
`7.2 ÷ 0.3`
`0.96 ÷ 0.12`
`15.75 ÷ 0.5`
`3.04 + 7.8 - 1.29`
`6 × 0.025`
`14.08 - 9.999`
`2.75 × 0.06`
`4.2 ÷ 0.07`
`18.5 + 0.37 - 2.9`
`4.7 + 8.35`
`12.06 + 0.9`
`15.4 - 7.28`
`20 - 3.675`
`0.48 + 2.307`
`9.004 - 1.89`
`3.6 × 4.2`
`0.7 × 0.08`
`12.5 × 0.4`
`0.35 × 1.2`
Questions 1–10
These are in-class practice questions only, as requested. Encourage students to show written methods for the harder items.
Why is estimation useful before multiplying decimals?
It checks the answer size
It removes decimal points
It changes place value
It finds an exact answer
Targets slide 5's key idea: estimation helps spot misplaced decimal points. A common misconception is that estimation is part of the formal method rather than a checking strategy.
Complete the decimal method summary
When adding or subtracting decimals, line up the decimal points first so place value stays correct. When multiplying decimals, count the total number of decimal places in the factors. When dividing by a decimal, make the divisor a whole number by multiplying both numbers by the same power of 10.
All three blanks echo the exact methods from slides 3 and 4.
What are significant figures?
Significant figures begin at the first non-zero digit
They show the precision of a value
Zeroes can be significant or not, depending on position
Example: `0.004560` has 4 significant figures
Clarify the difference between decimal places and significant figures. Decimal places count from the decimal point; significant figures count from the first non-zero digit.
How to round to significant figures
To round to a given number of significant figures, keep the required digits, look at the next digit, then round up or down. Replace later digits with zeroes if needed for whole numbers. For decimals, stop after the rounded digit.
Examples to model: 5382 to 2 significant figures gives 5400; 0.07846 to 3 significant figures gives 0.0785.
Choosing an appropriate degree of rounding
Use the context to decide what is appropriate
Measurements are often rounded to match their precision
Money is usually rounded to 2 decimal places
In science or statistics, significant figures may be more useful than decimal places
Link this to real-world use: measurements, costs, and reported data. Students often round too early in multi-step calculations, so remind them to round at the end unless instructed otherwise.
Common rounding pitfalls
Starting from the decimal point instead of the first non-zero digit
Forgetting place value in large numbers
Rounding halfway cases incorrectly
Rounding too early during calculations
Use quick examples such as 0.003049 to 2 significant figures and 1499 to 2 significant figures.
Significant figures practice 1–20
Questions 1–10
Round each number to the stated number of significant figures.
`4837` to 1 s.f.
`4837` to 2 s.f.
`0.06729` to 2 s.f.
`0.06729` to 3 s.f.
`915.48` to 3 s.f.
`1.995` to 3 s.f.
`30501` to 2 s.f.
`30501` to 3 s.f.
`0.004806` to 2 s.f.
`76.499` to 4 s.f.
Questions 11–20
`9999` to 2 s.f.
`12.0501` to 4 s.f.
`508000` to 3 s.f.
`0.70049` to 3 s.f.
`3.141592` to 5 s.f.
`87049` to 1 s.f.
`87049` to 4 s.f.
`0.09995` to 3 s.f.
`650.05` to 2 s.f.
`2.5004` to 4 s.f.
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How many significant figures does 0.004560 have?
3
5
4
6
Checks the example given on slide 7. Distractors reflect counting from the decimal point or counting all digits.
Complete the rounding summary
Significant figures begin at the first non-zero digit. To round, keep the required digits and look at the next digit. In multi-step calculations, round at the end unless the question tells you otherwise.
This reinforces slides 7 to 10, especially the pitfall of starting from the decimal point and rounding too early.
Recurring decimals
A recurring decimal has one digit or a block of digits that repeats forever
We show this with a dot or bar notation, such as `0.3̇` or `0.1\overline{27}`
Recurring decimals are exact, not approximate
They can be written as fractions
Students often think a recurring decimal is only an approximation because it never ends. Stress that it represents an exact value.
Turning recurring decimals into fractions
A recurring decimal can be written as a fraction by using algebra. For example, if `x = 0.3̇`, then `10x = 3.3̇`. Subtracting gives `9x = 3`, so `x = 1/3`. The repeated part is removed by subtraction.
Also model a two-digit repeat such as x = 0.272727... so 100x = 27.272727..., giving 99x = 27 and x = 27/99 = 3/11.
Fractions that give recurring decimals
Some fractions end as terminating decimals, such as `1/4 = 0.25`
Others give recurring decimals, such as `1/3 = 0.3̇`
A fraction in simplest form terminates only if its denominator has prime factors of 2 and/or 5
Other prime factors lead to recurring decimals
This is a strong GCSE connection. Examples: 3/8 terminates, 2/15 recurs because 15 includes factor 3.
Exact values matter
`1/3` is exact
`0.333` is approximate
`0.3̇` is exact
In algebra and proof, exact forms avoid rounding errors
Good chance to reinforce mathematical language: exact versus approximate.
Recurring decimals practice 1–20
Write each recurring decimal as a fraction in simplest form.
`0.3̇`
`0.6̇`
`0.1̇`
`0.27̇`
`0.72̇`
`0.4̇5̇`
`0.81̇`
`0.09̇`
`0.12̇`
`0.5̇`
Questions 1–10
State whether each fraction gives a terminating or recurring decimal.
`1/2`
`3/5`
`7/8`
`1/3`
`2/9`
`5/6`
`7/20`
`11/25`
`4/15`
`7/12`
Questions 11–20
For question 6, the notation means 0.454545... If needed, explain bar notation clearly before students begin.
Which statement about 0.3̇ is correct?
It is approximate
It equals 3/10
It is exact
It terminates
Checks the exact-versus-approximate distinction from slides 12 and 15.
Complete the recurring decimal facts
A recurring decimal repeats forever and can be written as a fraction . A fraction in simplest form gives a terminating decimal only if its denominator has prime factors of 2 and 5 only.
Anchors the two key ideas in this block: exactness and the denominator test for terminating decimals.
Why is 0.333 not the same as 0.3̇?
Expected elements: recurring decimals are exact; examples such as 1/3 = 0.3̇ and 0.333 is only approximate; exact forms help avoid rounding errors in algebra and proof.
Adding and subtracting fractions
Fractions must have a common denominator before they can be added or subtracted
Equivalent fractions help create matching denominators
Add or subtract the numerators only
Simplify the final answer if possible
Model both easy and non-unit examples, including mixed numbers where appropriate.
Multiplying fractions and mixed numbers
To multiply fractions, multiply the numerators and multiply the denominators. Convert mixed numbers to improper fractions first. Simplify before multiplying when possible to make the arithmetic easier.
Demonstrate cancelling common factors before multiplication to reduce errors and working.
Dividing fractions
Keep the first fraction
Change division to multiplication
Flip the second fraction to its reciprocal
Convert mixed numbers first, then simplify the result
Students often remember this as 'keep, change, flip', but explain why the reciprocal is used.
Improper fractions and mixed numbers
A mixed number combines a whole number and a fraction
An improper fraction has a numerator greater than or equal to its denominator
Convert between the two forms when needed
Final answers may be left as improper fractions or mixed numbers, depending on the question
Useful reminder before practice, since operations with mixed numbers usually begin by converting them.
Fraction practice 1–20
Questions 1–10
`1/4 + 3/8`
`5/6 - 1/4`
`2/3 + 5/9`
`7/10 - 3/5`
`1 1/2 + 2 3/4`
`4 2/3 - 1 5/6`
`3/5 × 4/7`
`5/8 × 12/15`
`1 2/3 × 3/10`
`2 1/4 × 1 1/3`
`3/4 ÷ 2/5`
`7/9 ÷ 14/27`
`1 1/2 ÷ 3/8`
`2 2/5 ÷ 1 1/5`
`5/12 + 7/18`
`11/15 - 2/9`
`4/9 × 3/16`
`5/6 ÷ 1/3`
`3 3/4 + 2/5`
`6 1/2 - 3 2/3`
Questions 11–20
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What must you do before adding 2/3 and 5/9?
Flip one fraction
Find a common denominator
Multiply denominators only
Convert to percentages
Checks the core prerequisite from slide 17.
Complete the fraction methods
To add or subtract fractions, first find a common denominator . To multiply mixed numbers, convert them to improper fractions first. To divide fractions, keep the first fraction, change to multiplication, and use the reciprocal of the second fraction.
Directly checks the three procedures from slides 17 to 20.
Percentages as operators
A percentage means 'out of 100'
It can describe part of a whole, an increase, or a decrease
`10% of` means multiply by `0.10`
A percentage change can be handled with a multiplier
Bridge from prior knowledge: percentages, decimals and fractions all represent parts of a whole.
Percentage increase and decrease
A percentage increase uses a multiplier greater than 1, and a percentage decrease uses a multiplier less than 1. For example, increasing by 8% means multiplying by `1.08`, while decreasing by 8% means multiplying by `0.92`.
Students often add or subtract the percentage numerically without linking it to a multiplier. Show both interpretations.
Repeated proportional change
When a percentage change happens again and again, the multiplier is repeated.
Original value → multiply by scale factor → new value
After `n` changes: `original × (multiplier)^n`
This is why repeated change is not found by simple addition of percentages.
Examples include compound interest, depreciation and population change. Stress that 10% increase twice is not 20% overall; it is ×1.1×1.1.
Where repeated change appears
Savings and loans
Depreciation of cars or technology
Population growth
Repeated price rises or discounts
Real-life context helps students see why multipliers matter. Connect to graphs if time allows.
Percentage practice 1–20
Find `15%` of `80`
Find `32%` of `250`
Increase `120` by `10%`
Decrease `450` by `12%`
Write the multiplier for a `7%` increase
Write the multiplier for a `7%` decrease
A jacket costs `£60` and rises by `15%`. Find the new price.
A bike costs `£240` and is reduced by `25%`. Find the sale price.
Increase `350` by `4%`
Decrease `96` by `30%`
A value increases by `5%` each year for `2` years from `200`
A value decreases by `10%` each year for `3` years from `500`
`£800` grows by `3%` for `4` years
A car worth `£12000` depreciates by `15%` for `2` years
A population of `6000` grows by `2%` for `5` years
Write the formula for repeated growth of `P` by `r%` for `n` periods
Write the formula for repeated decrease of `P` by `r%` for `n` periods
A price is increased by `20%` then decreased by `20%`. Is it back to the original price?
`£500` increases by `8%` then by `8%` again
`900` decreases by `5%` for `4` periods
Questions 1–10
Questions 11–20
Encourage students to identify the multiplier first before calculating. Question 18 is a useful conceptual check.
A 12% decrease uses which multiplier?
1.12
0.12
0.88
1.88
Targets the multiplier idea from slides 22 and 23. Students often confuse increase and decrease multipliers.
Complete the percentage change summary
A percentage increase uses a multiplier greater than 1 . A percentage decrease uses a multiplier less than 1 . For repeated proportional change, use the original value multiplied by the multiplier raised to the number of periods.
Reinforces the conceptual structure behind slides 23 and 24.
Which topic do you most need to practise again?
Decimals
Significant figures
Fractions
Percentages
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What is one thing you will remember from today?
Reflection prompt. Invite students to name one method, example or misconception that became clearer during the lesson.
Which method helped you most, and why?
Reflection prompt. Encourage students to mention a specific strategy such as lining up decimal points, using a common denominator, using a multiplier, or delaying rounding.
Glossary
decimal - A number written using place value and a decimal point
place value - The value of a digit because of its position in a number
significant figures - Digits used to show the precision of a value, starting from the first non-zero digit
recurring decimal - A decimal in which a digit or block of digits repeats forever
exact - A value written without rounding or approximation
approximate - A value that has been rounded or estimated
common denominator - A shared denominator used to add or subtract fractions
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Glossary
equivalent fractions - Fractions that have the same value
mixed number - A number made from a whole number and a fraction
improper fraction - A fraction with numerator greater than or equal to denominator
reciprocal - The result of flipping a fraction, used in fraction division
percentage - A proportion out of 100
multiplier - The decimal used to increase or decrease a value by a percentage
repeated proportional change - A change that uses the same multiplier again and again over time
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