At a glance. One lesson of about 60 minutes. Students convert among the three forms on aligned scales, then compare and order mixed sets by changing them to percentages.
Calculator: no calculator. Nothing here needs one. The conversions use benchmarks and scaling to 100, and a calculator would hide why 3/8 = 37.5%.
What students will learn
Know
- A fraction is the top divided by the bottom, and a percentage is the decimal times 100.
- Benchmarks: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/10 = 10%.
- A percentage is a number over 100, so divide the top and bottom by their greatest common factor to simplify.
Understand
- The three forms name one amount, so changing form does not change the value.
- To compare different forms, change them all to the same one first.
- The digits of one form are not the digits of another: 1/4 is 25%.
Do
- Convert between fractions, decimals and percentages in either direction.
- Compare and order mixed sets using <, = and >.
- Choose the form that makes a comparison or a calculation simplest.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: conversions among the three forms and ordering (questions 1 to 6).
- Criterion B. Investigating patterns: the aligned-scales figure. Ask students to find the rule that links the number of parts to the decimal and percentage, and to say which denominators give a terminating decimal.
- Criterion C. Communicating: explaining why 1/4 is not 4% and why 0.35 = 35%, in words and in symbols (question 9).
- Criterion D. Applying mathematics in real-life contexts: sale signs, tank levels and ways of travelling to school (the hook, question 7 and the extra questions).
The one thing to do with the figure
Ask for a prediction before the first slide. “Is 3/8 more or less than 1/3? Roughly what percentage?” Take 38%, 40% or 33% from the room and let the three aligned scales settle it.
Then run through the benchmarks in the steps and ask which denominators give a decimal that stops. In the second figure, ask for the order before the line is read, and stop at set 5 to ask what “equal” looks like on the line.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea and conversions: scale to 100, divide, or use a benchmark. |
| 10 min | The second figure; worked example, then the faded ordering example in pairs. |
| 5 min | Question 1, the check. If many choose A or C, return to the first figure and read the decimal line. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 30%; 0.4; 2/3. |
|---|---|
| Worked example | 7/20 = 0.35 = 35%. |
| Faded example | 43% < 0.45 < 3/5. |
| 1. Decimal to percentage | B, 7%. A reads 0.07 as 0.7 before multiplying by 100; C is 0.007; D is 0.7. |
| 2. Fraction to percentage | 60. 3/5 = 60/100. |
| 3. Percentage to simplest fraction | B, 2/5. A is right but not simplest; C is 4%; D is 2.5%. |
| 4. Fraction to percentage by scaling | 36. 9/25 = 36/100. |
| 5. Percentage to decimal | 0.625. 62.5 ÷ 100 = 0.625. |
| 6. Ordering mixed forms | A. B swaps 2/5 and 41%; C is largest first; D puts 0.42 first. |
| 7. Difference between mixed forms | 1.5. 3 × 12.5 = 37.5, then 37.5 − 36 = 1.5. |
| 8. Missing numerator | 17. 85/100 = 17/20. |
| 9. Digits are not the percentage | B. A and D treat the 4 as the percentage; C joins the digits 1 and 4. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or C in question 1 | Read 0.07 as 0.7 before multiplying by 100 (A), or read it as 0.007 (C). | Use the aligned scales: 0.07 sits at 7 on the percentage line. |
| 35, 15 or 0.6 in question 2 | Joined the digits, multiplied top by bottom, or stopped at the decimal. | Ask: what does 3/5 mean as a division? Then scale to 100. |
| 4/10 or 4/100 in question 3 | Left it unsimplified, or wrote 4% as the fraction. | Say “40 out of 100” aloud, then ask for the greatest common factor of 40 and 100. |
| 9 or 0.36 in question 4 | Scaled the bottom and not the top, or stopped at the decimal. | Whatever you do to the bottom you do to the top: draw the arrow × 4 on both. |
| 6.25, 0.0625 or 62.5 in question 5 | Moved the digits one or three places, or left it as a percentage. | Write the percentage over 100, and ask how many places 100 moves a digit. |
| B, C or D in question 6 | Compared the digits, not the values, or read the order the wrong way. | Change all three to percentages and write them in a column. |
| 1, 2 or 0.015 in question 7 | Rounded 37.5% before subtracting, or left the difference as a decimal. | Ask whether 37% and 38% are equally close to 37.5%, and why that matters here. |
| 85 or 4.25 in question 8 | Gave the percentage as the numerator, or divided 85 by 20. | Write 85% as 85/100, then ask what 100 is divided by to reach 20. |
| A, C or D in question 9 | Treated the 4 as the percentage, or joined the digits 1 and 4. | Ask what a quarter of 100 is, then place 1/4 on the aligned scales. |
Questions worth asking
- Which is bigger, 1/3 or 33%? 1/3 is 33.3...%, so 1/3. Listen for “the decimal goes on for ever”.
- How do you know 0.35 and 7/20 are equal without a calculator? 7/20 = 35/100 by scaling by 5. Listen for the scale factor.
- Is 0.5% the same as 0.5? No: 0.5% is 0.005. The % sign divides by 100.
- Which form is best for comparing, and which for sharing? Percentages for comparing shares of different wholes; fractions for sharing equally.
Differentiation
A way in for anyone stuck
- Give each student a conversion strip: fraction, decimal and percentage for 1/2, 1/4, 3/4, 1/5, 1/10 and 1/8, to build from.
- Keep to denominators that divide 100: 2, 4, 5, 10, 20, 25, 50. Add 8 and 40 only when the scaling is secure.
- Use the aligned scales as a rule: a number is converted when it sits at the same place on all three.
Stretch
- Question 7, then: which fractions in simplest form, with a bottom number up to 12, give a decimal that stops? (Bottom numbers 1, 2, 4, 5, 8 and 10. A decimal that stops is a fraction over 10, 100, 1000 and so on, and those have no prime factors except 2 and 5.)
- Put 5/9, 0.55 and 56% in order. (5/9 = 55.5...%, so 0.55 < 5/9 < 56%.)
Students learning in English as an additional language
The three words look unrelated. Pair them on one card: “fraction: out of any number; decimal: out of tens; percentage: out of 100”.
| Word | Plain meaning | Example |
|---|---|---|
| fraction | a number written as one number over another | 3/8 |
| decimal | a number written with a decimal point | 0.375 |
| percentage | a number out of 100 | 37.5% |
| equivalent | equal in value, written differently | 1/2, 0.5 and 50% are equivalent |
| simplest form | a fraction where the top and bottom have no common factor but 1 | 45/100 in simplest form is 9/20 |
| benchmark | a value worth knowing by heart, to build others from | 1/8 = 12.5% |
Sentence starters for explaining aloud or in writing:
- … as a percentage is … because …
- To compare these I change all of them to …
- … and … are equal because they are both … out of 100.
Two things not to say
“Just divide the top by the bottom and times by 100.” True, but give the reason first: a percentage is a fraction scaled to a bottom of 100.
“The longer decimal is bigger.” 0.5 is bigger than 0.45. Compare by changing both to hundredths.
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.
- Fraction to percentageWrite 4/5 as a decimal and as a percentage.
0.8, 80%. - Past one wholeWrite 1.4 as a percentage and as a fraction in simplest form.
140% and 7/5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextIn a class of 40, 3/8 walk to school, 0.4 cycle and 20% come by bus. Which way do most students travel?
Cycling: 0.4 is 40%, against 37.5% and 20%. - Earlier work (Lesson 1 of this unit)A bar is cut into 25 equal parts and 6 are shaded. Give the share as a fraction, a decimal and a percentage.
6/25, 0.24 and 24%.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningFind a fraction that is greater than 0.45 and less than 46%. Show that it is.
For example 91/200 = 0.455 = 45.5%, which lies between 45% and 46%. - Spot the errorZoe says 5/8 = 5.8%. What did she do, and what is the percentage?
She put the digits side by side. 5/8 = 0.625 = 62.5%.
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? Draw three parallel number lines on squared paper, marked as fractions, decimals and percentages, and place cards on them.