At a glance. One lesson of about 60 minutes. Students recut fractions to a common denominator, use one half as a benchmark, and order sets of fractions on a number line.
Calculator: no calculator. Nothing here needs one. The lowest common multiple is a Unit 1 skill, and the benchmark test needs only doubling.
What students will learn
Know
- To compare fractions, give them a common denominator.
- n/d is more than 1/2 exactly when 2 × n is more than d.
- With equal tops, the fraction with the bigger bottom is smaller.
Understand
- Counts of parts only compare when the parts are the same size.
- The lowest common multiple of the bottoms gives the smallest numbers.
- Position on a number line is the order.
Do
- Compare two fractions using a common denominator.
- Order three or more fractions.
- Use one half as a quick test.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: common denominators, benchmarks and ordering (questions 1 to 8).
- Criterion B. Investigating patterns: the what-if question, where adding one to top and bottom moves a fraction towards 1. Ask students to test more pairs and state the rule.
- Criterion C. Communicating: explaining why a bigger bottom does not mean a bigger fraction (question 9), using the words part, whole and size.
- Criterion D. Applying mathematics in real-life contexts: rows of lamps and results of games (the hook and question 6).
The one thing to do with the figure
Ask for a prediction before the first slide. “3 of 5 lamps lit, or 5 of 8?” Collect the votes, then set the sliders to 3/5 and 5/8 and recut the rows. Ask which vote was right and why the numbers alone could not tell you.
Then set the two sliders to two fractions with the same top, such as 7/10 and 7/12, and ask which bar has bigger parts. In the second figure, ask for the order before pressing, and have a student explain a position.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, prediction and the first figure. Include a pair with equal tops. |
| 10 min | Key idea; the second figure. Students order each set before the order appears. |
| 10 min | Worked examples; students finish the faded example on their own. |
| 5 min | Question 1, the check. If many choose A or D, go back to the warn card. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 21/35; 5 2/3; 24. |
|---|---|
| Worked examples | 5/6 is greater (15/18 against 14/18); 11/24 is less than 1/2. |
| Faded example | 7/8 is greater (49/56 against 44/56). |
| 1. Which fraction is greater | B, 4/5. A and D compare one of the two numbers; C is a near miss. |
| 2. Common denominator | 12. 3/4 = 9/12 and 5/6 = 10/12. |
| 3. Comparing four fractions | 18. 18, 15, 16, 14: Lin ate the most. |
| 4. Ordering three fractions | 20. 18/30 < 20/30 < 21/30. |
| 5. Using one half as a benchmark | 2. 5/9 and 6/11. |
| 6. Comparing in context | 4. 49 − 45 = 4. |
| 7. A fraction between two others | 13. 12/20 < 13/20 < 14/20. |
| 8. Comparing with an unknown | 4. 3/4 is greater than 5/8; 3/5 is not. |
| 9. Why a bigger bottom is a smaller part | B. A gets the answer wrong; C accepts a bad reason; D adds numbers that do not belong together. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or D in question 1 | Compared one of the numbers, a bottom or a top. | Ask what the parts look like when the bottom is 9 and when it is 5, then recut both. |
| 24 or 10 in question 2 | Used a common multiple that is not the lowest, or added the bottoms. | List multiples of each until the lists meet. |
| 7 in question 3 | Chose the biggest top number. | Over 24, show 7/12 as 14/24. Ask what 7 is a count of. |
| 18 or 21 in question 4 | Gave the smallest or the greatest instead of the middle. | Underline the word middle and order the three tops on the board. |
| 3 or 5 in question 5 | Counted an equal fraction, or did not test them all. | Say aloud: “more than”, not “at least”. Test 4/8 on its own. |
| 2 in question 6 | Subtracted the tops of fractions with different bottoms. | Ask whether 7 ninths and 5 sevenths count the same size of part. |
| 12 or 14 in question 7 | Gave one of the ends. | Write the ends on a number line over 20 and ask what is strictly between. |
| 5 or 8 in question 8 | Thought a bigger n gives a bigger fraction, or stopped at the first case. | Try 3/n for n = 3, 4, 5 and compare each with 5/8 over 40. |
| A, C or D in question 9 | Used the size of a bottom number as the reason, or added tops and bottoms. | Ask for 1/8 and 1/3: which is greater, and why does 8 > 3 not help? |
Questions worth asking
- Why can you not compare 3/5 and 5/8 by looking at 3 and 5? They count different sizes of part. Listen for “fifths and eighths are different sizes”.
- Which is greater, 99/100 or 100/101? How do you know without a calculator? 100/101: the missing piece is 1/101, which is less than 1/100. Accept common denominators.
- How could you compare 5/8 and 2/3 using 1/2? Both are more than 1/2, so it does not decide. Over 24 they are 15/24 and 16/24.
- Does the lowest common multiple always work as a common denominator? Yes. Any common multiple works; the lowest keeps numbers small.
Differentiation
A way in for anyone stuck
- Give each student two paper strips of the same length. They fold one into 5 and one into 8 and compare lit lengths.
- Start with bottoms where one divides the other: 1/2 and 3/8, 2/3 and 5/6.
- Write the lowest common multiple on its own line before any conversion.
Stretch
- Find a fraction with a bottom number below 50 that lies strictly between 5/8 and 2/3. How do you know it is between? (31/48: 5/8 = 30/48 and 2/3 = 32/48. 9/14 also works: over 336, 5/8 = 210/336, 9/14 = 216/336 and 2/3 = 224/336.)
- Order 5/6, 7/9 and 11/12 by thinking about what each is missing from a whole. (Missing 1/6, 2/9 and 1/12; 1/12 is least and 2/9 is most, so 7/9 < 5/6 < 11/12.)
Students learning in English as an additional language
“Greater” and “less” are used for numbers, and “bigger” and “smaller” for size; both are fine here. Point to the parts when you say “same size”.
| Word | Plain meaning | Example |
|---|---|---|
| common denominator | a bottom number that two fractions can both be written over | 12 is a common denominator for 3/4 and 5/6 |
| lowest common multiple | the smallest number that is a multiple of two numbers | the lowest common multiple of 4 and 6 is 12 |
| benchmark | a friendly number to compare against | 1/2 is a benchmark |
| order | to arrange from least to greatest | order 1/3, 1/4, 1/2 |
| greater than | further along the number line, more | 3/4 is greater than 2/3 |
Sentence starters for explaining aloud or in writing:
- Over … the fractions are … and …, so … is greater.
- … is more than 1/2 because double the top is …, which is more than the bottom.
- I cannot compare the tops because the parts are not the same size.
Two things not to say
“Cross multiply.” It works, but hides that you are making a common denominator. Show the common denominator first.
“The bigger the bottom, the smaller the fraction.” Only for equal tops. 5/8 is greater than 1/3.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Which is greater?3/8 or 2/5?
2/5. 15/40 and 16/40. - OrderPut 1/2, 3/5 and 2/5 in order, smallest first.
2/5, 1/2, 3/5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo ropes: one is 5/8 of a metre and one is 2/3 of a metre. Which is longer, and by how much?
2/3 m, by 1/24 m: 16/24 − 15/24. - Earlier work (Lesson 2 of this unit)Which is greater, 11/4 or 2 3/5?
11/4 = 2 3/4, and 3/4 is more than 3/5, so 11/4.
3Challenge
Reasoning, generalising and working backwards.
- BetweenWrite a fraction with a bottom number of 48 that lies between 5/8 and 2/3.
31/48. 5/8 = 30/48 and 2/3 = 32/48. - ReasoningWhich is greater, 99/100 or 100/101? Explain using what is missing from a whole.
100/101. 1/101 is missing, which is less than 1/100.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use strips of squared paper 24 squares long, which divide neatly into 2, 3, 4, 6 and 8.