At a glance. One lesson of about 60 minutes. Students see equivalent fractions by cutting parts, write fractions with a given denominator, and simplify using common factors from Unit 1.
Calculator: no calculator. Nothing here needs one. Every number divides cleanly, and the skill is seeing why the amount stays the same.
What students will learn
Know
- n/d means n of d equal parts of one whole, and a point on the number line.
- Multiplying or dividing the top and bottom by the same number gives an equivalent fraction.
- Simplest form: only 1 divides both the top and the bottom.
Understand
- Equivalent fractions are one amount with different names, because smaller parts are matched by more of them.
- Adding the same number to top and bottom changes the amount, unless the fraction is 1 already.
- The greatest common factor reaches simplest form in one step; smaller common factors need repeating.
Do
- Write a fraction with a given denominator or numerator.
- Simplify a fraction fully, by one division or several.
- Explain why a fraction is or is not equal to another.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: equivalent fractions and simplest form (questions 1 to 8).
- Criterion B. Investigating patterns: the cutting figure. Ask students to find what always stays the same as k changes, then predict the top and bottom for a k they choose.
- Criterion C. Communicating: explaining in words why 4/9 is not 6/11 (question 5), using the words parts, whole, multiply and common factor.
- Criterion D. Applying mathematics in real-life contexts: sharing a bottle of water and a tray of fruit (the hook and question 4).
The one thing to do with the figure
Ask for a prediction before the first slide. “2/3 and 4/6: the same, or is 4/6 more because the numbers are bigger?” Take both views, then set the fraction to 2/3 and let the first figure settle it: slide the cut from 1 to 2.
Then change the fraction and the cut, and ask students to say the new name before it appears. In the second figure, ask for a number that does not divide both and read what the figure says, then find the largest one that works.
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 cut of 1 and a cut of 6. |
| 10 min | Key idea; the second figure. Students find the greatest common factor by trying numbers. |
| 10 min | Worked examples; students finish the faded example on their own. |
| 5 min | Question 1, the check. If many choose C, 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 | 6; 1 × 28, 2 × 14, 4 × 7; 75%. |
|---|---|
| Worked examples | 4/7 = 20/35; 42/56 = 3/4. |
| Faded example | 5/6 = 45/54. |
| 1. An equivalent fraction | B, 6/10. A doubles only the bottom; C adds 1 to both; D is the fraction inverted. |
| 2. Missing numerator | 35. 40 ÷ 8 = 5 and 7 × 5 = 35. |
| 3. Missing denominator | 9. 20 ÷ 4 = 5 and 36 ÷ 4 = 9. |
| 4. Equivalent fraction in context | 14. 7/12 = 14/24. |
| 5. Why adding does not work | B. A treats adding as scaling; C changes only the top; D wrongly says the bottom cannot change. |
| 6. Simplest form by repeated division | 5. 40/64 = 5/8. |
| 7. Equivalent fraction needing simplification first | 10. 6/9 = 2/3 = 10/15. |
| 8. Using equivalence to solve | 18. 11k = 66, k = 6, 18/48. |
| 9. Is it in simplest form? | B. A stops at smaller numbers; C says 2 is not a factor; D uses a number that divides neither. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| C in question 1; 39 in question 2 | Added the same number to top and bottom. | Draw the bar for 1/2 and for 3/4. Ask if they are the same, then ask what multiplying by 2 does to 1/2. |
| A in question 1; 7 in question 2 | Changed only the bottom, or only the top. | Ask what happens to the size of the parts when the bottom changes, and what the top counts. |
| 21 or 16 in question 3 | Subtracted the gap between top and bottom. | Write the division 20 ÷ 4 = 5 and ask what 36 ÷ 4 is. |
| 19 in question 4 | Added the gap between 12 and 24 to the top. | Use 24 pieces: ask how many pieces make one twelfth. |
| 20 or 10 in question 6 | Stopped dividing too early. | Ask: do 20 and 32 both still divide by something? Say “simplest form means only 1 divides both.” |
| 12 or 90 in question 7 | Did not simplify first, then added or multiplied by the new bottom. | Ask whether 9 multiplies to 15. Simplify to thirds, then go from 3 to 15. |
| 33 in question 8 | Halved the total. | Draw 3 parts and 8 parts of the same size and count the parts altogether. |
| A, C or D in question 9 | Stopped early, or divided by a number that does not divide both. | Ask what divides both 10 and 14, then both of the new numbers. |
Questions worth asking
- Does cutting the parts smaller change how much is shaded? No: there are more parts, each smaller. Listen for “same amount, smaller pieces”.
- How do you know 15/40 can be simplified without trying every number? Both end in 0 or 5, so both divide by 5. Listen for a divisibility rule.
- Is 4/6 in simplest form? No: 2 divides both, and 2/3 is the simplest form.
- Could a fraction have two different simplest forms? No: dividing by the greatest common factor always lands on the same one. Listen for the point being unique.
Differentiation
A way in for anyone stuck
- Give each student two fraction strips of the same length, one cut in 3 and one cut in 6, so they can lay them side by side.
- Start with doubling and tripling only: 1/2 to 2/4 to 3/6.
- Write the multiplier above an arrow between the two fractions, so it is the same arrow on top and bottom.
Stretch
- How many different fractions with a bottom of 60 or less are equal to 1/3? (20: 1/3, 2/6, ..., 20/60.)
- Write the rule for every fraction equal to 3/5. Then say how many of them have a bottom number below 40. (k × 3 over k × 5, with k from 1 to 7, so 7 of them.)
Students learning in English as an additional language
“Numerator” and “denominator” are long, so say “top number” and “bottom number” as well. “Equivalent” comes from Latin for “equal value” and many languages share the root.
| Word | Plain meaning | Example |
|---|---|---|
| fraction | a number of equal parts out of a whole cut into equal parts | 3 parts out of 4 is 3/4 |
| numerator | the top number: how many parts are counted | in 3/5 the numerator is 3 |
| denominator | the bottom number: how many equal parts the whole is cut into | in 3/5 the denominator is 5 |
| equivalent fractions | different fractions that name the same amount | 1/2 and 3/6 |
| simplest form | a fraction where only 1 divides both top and bottom | 3/4 is the simplest form of 42/56 |
| common factor | a number that divides two numbers exactly | 7 is a common factor of 21 and 28 |
Sentence starters for explaining aloud or in writing:
- Cutting every part into … makes … times as many parts, so …
- …/… is not equal to …/… because …
- I can simplify this because … divides both the top and the bottom.
Two things not to say
“Whatever you do to the top, do to the bottom.” That includes adding, which changes the amount (unless the fraction is 1 already). Say “multiply or divide both by the same number”.
“Cancel the numbers.” Cancelling hides what is happening until students can say what it does, and then it is fine as a short name. Start with “divide the top and the bottom by the same number”.
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.
- Fill the gap3/4 = ☐/20.
15. - SimplifyWrite 24/60 in simplest form.
2/5. Both divide by 12.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA class of 30 students: 18 take the bus. Write 18/30 in simplest form, and say how many students out of 10 that is.
3/5; 6 out of 10. - Earlier work (Unit 2, lesson 1)A bar has 25 equal parts and 10 are shaded. Write the shaded part as a percentage and as a fraction in simplest form.
40% and 2/5. Each part is 4%, and 10 × 4 = 40.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningWhich of these is not equal to 4/6: 2/3, 8/12, 6/8, 20/30?
6/8, which is 3/4. The others are 2/3 scaled. - Working backA fraction equals 5/7 and its bottom number is 18 more than its top number. What is the fraction?
45/63. 7 − 5 = 2 parts are 18, so each part is 9, and 5 × 9 = 45, 7 × 9 = 63.
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? Fold or cut paper strips of the same length into 2, 3, 4 and 6 equal parts and lay them side by side.