Fractions and decimals: equivalence, ordering and the 4 operations
Converting exactly between fractions and terminating decimals, ordering a mixed set, and adding, subtracting, multiplying and dividing fractions and mixed numbers
About four lessons of 45 to 60 minutes
Same amount, 2 different outfits
A recipe says 'use 3/4 of the milk'. A price tag says '£0.75 off'. A race result says the winner finished in 0.375 of an hour less than 2nd place. Fractions and decimals are 2 different ways of writing exactly the same amount, and real life switches between them constantly, so being able to convert instantly, in either direction, matters far more than picking a favourite.
Once fractions and decimals can both be trusted, the 4 operations, adding, subtracting, multiplying and dividing, work on fractions exactly as they do on whole numbers, just with a couple of extra steps: find a common denominator before adding or subtracting, and flip the second fraction before dividing.
- 3/4 of a pizza leftexactly the same amount as 0.75 of the pizza
- A 0.4 discount on a £20 jumperexactly the same as a 2/5 discount
- Adding 1/4 cup of sugar to 1/3 cup of flourneeds a common denominator (twelfths) before the amounts can be added
- Sharing 3/4 of a lap distance between 2 runners3/4 divided by 2, using the reciprocal method
What students will be able to do
Students will convert exactly between fractions and terminating decimals, order a mixed set of fractions and decimals from smallest to largest, and add, subtract, multiply and divide fractions and mixed numbers, always giving answers in their simplest form.
- I can convert a fraction to a decimal by dividing the numerator by the denominator.
- I can convert a terminating decimal to a fraction using place value, then simplify it.
- I can order a mixed set of fractions and decimals by converting every value to a decimal first.
- I can add and subtract fractions and mixed numbers with different denominators by finding a common denominator.
- I can multiply fractions directly, and divide fractions by multiplying by the reciprocal of the divisor.
Standards this unit teaches
- KS3 Maths: NumberUK National Curriculum (England)Number
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Number" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8)"; "order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers" (the fraction/decimal half; UK KS3 batch 2, lib/content_uks3math2.ts, already built the negative-integer half); "use the 4 operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative" (the fraction-arithmetic half; batch 2 already built the negative-integer half).
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 3 fractions teaching unitnaming fractions, equivalence and comparing, the foundation this unit builds on
- Grade 5 fractions: division and multiplication teaching unitthe reciprocal ('keep, change, flip') method for dividing fractions
- Place value in the glossaryneeded to read a decimal's digits as tenths, hundredths and so on
Words to teach and display
- Numerator
- the top number of a fraction, counting how many parts are taken
- Denominator
- the bottom number of a fraction, showing how many equal parts the whole is split into
- Terminating decimal
- a decimal that ends after a fixed number of digits (like 0.375), rather than repeating forever
- Mixed number
- a whole number and a fraction written together, such as 1 3/4
- Improper fraction
- a fraction where the numerator is greater than or equal to the denominator, such as 7/4
- Common denominator
- a shared denominator that 2 or more fractions can both be rewritten with, needed before adding or subtracting them
- Reciprocal
- a fraction flipped upside down (numerator and denominator swapped); multiplying by the reciprocal is how fraction division works
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Fractions and decimals: the same amount, 2 ways
ConcreteA fraction n/d means 'n divided by d'. Do that division and a decimal comes out, exactly, as long as the denominator's only prime factors are 2 and/or 5 (so the division eventually reaches a remainder of 0). Going the other way, a decimal's digits are already fraction digits in disguise: 0.75 means 75 hundredths, or 75/100.
A denominator like 8 (= 2 x 2 x 2) always terminates: 3/8 = 3 ÷ 8 = 0.375. A denominator like 3 never terminates (1/3 = 0.333...), because 3 is not a factor of any power of 10. This unit only uses denominators that terminate, so every decimal answer is exact, never a rounded approximation.
Write 3/8 as a decimal, and write 0.6 as a fraction in its simplest form.
- 3/8 means 3 ÷ 8. Dividing: 3.000 ÷ 8 = 0.375.
- 0.6 means 6 tenths, so 0.6 = 6/10.
- Simplify 6/10 by dividing top and bottom by their HCF, 2: 6/10 = 3/5.
Answer: 3/8 = 0.375. 0.6 = 3/5.
- Why does 1/4 terminate as a decimal but 1/3 does not?
- Write 0.25 as a fraction in its simplest form.
2. Ordering a mixed set of fractions and decimals
PictorialTo order fractions and decimals together, convert every value to the SAME form first. Decimals are usually easiest, since they can be compared digit by digit like whole numbers once the values line up on the number line.
It is a common mistake to compare fractions by their numerators or denominators alone. 3/4 is bigger than 1/2, but neither the 3 nor the 4 alone tells you that at a glance; converting both to decimals (0.75 and 0.5) makes the comparison immediate and safe.
Order these values from smallest to largest: 1/4, 0.3, 0.45, 1/2.
- Convert every value to a decimal: 1/4 = 0.25, 0.3 stays 0.3, 0.45 stays 0.45, 1/2 = 0.5.
- Order the decimals: 0.25, 0.3, 0.45, 0.5.
Answer: 1/4, 0.3, 0.45, 1/2 (smallest to largest).
- Why is converting everything to decimals a safe way to order fractions and decimals together?
- Which is bigger, 0.4 or 3/8? Show your working.
3. Adding and subtracting fractions with unlike denominators
Abstract1/4 and 1/3 are cut into different-sized pieces, a quarter and a third, so their numerators cannot simply be added. Both fractions must first be rewritten using a COMMON denominator, a shared number of equal-sized pieces, before the numerators can be combined.
The smallest common denominator is the lowest common multiple (LCM) of the 2 original denominators. For mixed numbers, convert each one to an improper fraction first (multiply the whole number by the denominator and add the numerator), then follow the same common-denominator method.
Find 1/4 + 1/3, and find 2 1/2 - 1 1/3.
- 1/4 + 1/3: the LCM of 4 and 3 is 12. 1/4 = 3/12 and 1/3 = 4/12. 3/12 + 4/12 = 7/12.
- 2 1/2 - 1 1/3: convert to improper fractions first: 2 1/2 = 5/2, 1 1/3 = 4/3. The LCM of 2 and 3 is 6. 5/2 = 15/6 and 4/3 = 8/6. 15/6 - 8/6 = 7/6.
- 7/6 is improper, so convert back to a mixed number: 7/6 = 1 1/6.
Answer: 1/4 + 1/3 = 7/12. 2 1/2 - 1 1/3 = 1 1/6.
- Why can't 1/4 + 1/3 be found by just adding the numerators and adding the denominators?
- What is the LCM of 5 and 6, and why is that the right common denominator to use for 1/5 + 1/6?
4. Multiplying and dividing fractions
AbstractMultiplying fractions needs no common denominator at all: multiply the 2 numerators together, multiply the 2 denominators together, then simplify. Dividing by a fraction uses 'keep, change, flip': keep the first fraction, change divide to multiply, and flip (find the reciprocal of) the second fraction.
Dividing by a fraction can feel backward at first (dividing by 1/2 makes a number BIGGER, not smaller), but it makes sense once you think about what division means: 3/4 ÷ 1/2 asks 'how many halves fit into 3/4?', and the answer, 1 1/2, is correctly bigger than 3/4.
Find 2/3 x 3/5, and find 3/4 ÷ 1/2.
- 2/3 x 3/5 = (2 x 3)/(3 x 5) = 6/15.
- Simplify 6/15 by dividing top and bottom by their HCF, 3: 6/15 = 2/5.
- 3/4 ÷ 1/2 = 3/4 x 2/1 (keep, change, flip) = (3 x 2)/(4 x 1) = 6/4.
- Simplify 6/4 by dividing top and bottom by their HCF, 2: 6/4 = 3/2 = 1 1/2.
Answer: 2/3 x 3/5 = 2/5. 3/4 ÷ 1/2 = 1 1/2.
- Why does dividing 3/4 by 1/2 give an answer bigger than 3/4?
- Find 1/2 x 2/3, giving your answer in its simplest form.
Common misconceptions and how to address them
MisconceptionTo add 2 fractions, add the numerators together and add the denominators together.
Why it happens: It mirrors how whole-number addition works, and it is the simplest possible rule a student can invent without being taught the common-denominator method.
How to address it: Show a concrete counterexample with the fraction bars: 1/2 + 1/2 should equal 1 whole, but 'add tops and bottoms' gives 2/4 (a half), which is clearly wrong. Only a shared, equal-sized piece (a common denominator) lets numerators be added.
MisconceptionA decimal with more digits after the point is always the bigger number (e.g. thinking 0.45 is bigger than 0.5, because 45 is bigger than 5).
Why it happens: Students compare the digits as if they were whole numbers, ignoring that each digit's place value (tenths, hundredths) changes what it is worth.
How to address it: Line the decimals up by place value, padding with a trailing 0 if needed (0.50 vs 0.45), and compare tenths first, then hundredths only if the tenths are tied.
MisconceptionWhen dividing fractions, you flip the FIRST fraction, not the second.
Why it happens: 'Keep, change, flip' is easy to garble under exam pressure into flipping the wrong one.
How to address it: Anchor it to a concrete question: 3/4 ÷ 1/2 asks 'how many halves fit into 3/4?', which only makes sense if the DIVISOR (the second fraction, 1/2) is the one that gets flipped and multiplied.
MisconceptionThe fraction with the bigger denominator is always the smaller fraction.
Why it happens: Overgeneralising from the true, but narrower, rule that among UNIT fractions (numerator 1), a bigger denominator does mean a smaller fraction (1/8 < 1/4).
How to address it: Compare with a genuine counterexample: 7/8 has a bigger denominator than 3/4, but 7/8 (0.875) is bigger than 3/4 (0.75). The numerator matters just as much as the denominator.
Guided practice (with answers)
1. Write 1/8 as a decimal.
Answer: 0.125, because 1 divided by 8 = 0.125.
2. Write 0.35 as a fraction in its simplest form.
Answer: 7/20, because 0.35 = 35/100, and dividing top and bottom by their HCF, 5, gives 7/20.
3. Order these from smallest to largest: 0.6, 1/2, 0.55, 3/5.
Answer: 1/2, 0.55, 0.6, 3/5, because as decimals these are 0.5, 0.55, 0.6, 0.6 (3/5 = 0.6, tied with 0.6, listed after it here).
4. Find 1/3 + 1/6.
Answer: 1/2, because the LCM of 3 and 6 is 6, so 1/3 = 2/6, and 2/6 + 1/6 = 3/6 = 1/2.
5. Find 3/5 x 2/3.
Answer: 2/5, because 3/5 x 2/3 = 6/15, and dividing top and bottom by their HCF, 3, gives 2/5.
6. Find 2/3 ÷ 1/6.
Answer: 4, because 2/3 ÷ 1/6 = 2/3 x 6/1 = 12/3 = 4.
Independent practice worksheets
Practise fraction and decimal equivalence, ordering, and all 4 operations with computed, never-wrong answer keys.
Differentiation
- Keep a printed fraction-bar reference strip (halves, thirds, quarters, fifths, sixths, eighths) visible until denominators can be compared by eye.
- For ordering, insist on writing every value as a decimal in a single row before comparing, rather than comparing fractions and decimals directly.
- Practise finding the LCM of 2 small denominators as its own short drill before combining it with fraction addition.
- For 'keep, change, flip', have students say the 3 words aloud every time until the sequence is automatic.
- Add or subtract 3 fractions with 3 different denominators in a single problem, requiring the LCM of all 3.
- Investigate which denominators from 2 to 20 give a terminating decimal and which do not, and explain the pattern using prime factors.
- Solve a word problem requiring 2 different operations, for example: 'a tank is 2/3 full; 1/4 of the full tank is removed; what fraction remains?'.
- Prove why dividing by a fraction less than 1 always gives a bigger result, using a bar diagram.
Assessment: exit ticket
A three-question exit ticket sampling equivalence, ordering and a fraction operation.
1. Write 5/8 as a decimal.
Answer: 0.625, because 5 divided by 8 = 0.625.
2. Order these from smallest to largest: 0.7, 3/4, 0.65.
Answer: 0.65, 0.7, 3/4, because 3/4 = 0.75, so the decimals 0.65, 0.7, 0.75 are already in that order.
3. Find 1 1/4 + 2/3.
Answer: 1 11/12, because 1 1/4 = 5/4, the LCM of 4 and 3 is 12, 5/4 = 15/12 and 2/3 = 8/12, and 15/12 + 8/12 = 23/12 = 1 11/12.
Teacher notes and timings
- Rough timing across 4 lessons: Lesson 1 equivalence (section 1), Lesson 2 ordering (section 2), Lesson 3 adding/subtracting (section 3), Lesson 4 multiplying/dividing (section 4) plus the exit ticket.
- Every fraction-decimal conversion in this unit and its matching worksheets uses a denominator whose only prime factors are 2 and/or 5 (2, 4, 5, 10, 20, 25, 50, 100), so every decimal is an EXACT terminating value, never a rounded approximation, matching this site's 'never wrong' answer-key policy.
- Batch 2 (PR #379, lib/content_uks3math2.ts) already built ordering and the 4 operations for NEGATIVE INTEGERS; this unit deliberately covers the fraction/decimal half of the same 2 curriculum bullets, not the integer half, to avoid duplicating that work.
- The number-line and bar/circle figures in this unit reuse the site's existing fraction-diagram engine (components/MathFigures.tsx) exactly as earlier fraction units do; no new figure type was added.