At a glance. One lesson of about 60 minutes. Students meet fraction equations with one fraction, then two with different denominators, then a bracket in the numerator and the unknown on both sides. The figure is the evidence that multiplying does not change the solution.
Calculator: no calculator. Nothing here needs one. Every denominator is 2 to 9, and every solution is a whole number.
What students will learn
Know
- Multiplying both sides of an equation by the same non-zero number keeps the solution.
- A common multiple of the denominators clears every fraction; the lowest one keeps numbers smallest.
- Every term is multiplied, including plain numbers and the whole of a numerator.
Understand
- Clearing fractions is the balance idea again: the same operation to both sides.
- The solution of an equation is a property of the equation, not of how it is written.
- A bracket on top is one quantity, so it needs its own brackets once the denominator is cleared.
Do
- Clear fractions by multiplying every term.
- Solve equations with fractions, brackets and the unknown on both sides.
- Write an equation for a situation involving shares of a whole, and solve it.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: clearing fractions and solving the resulting linear equation (questions 1 to 8).
- Criterion B. Investigating patterns: use the dot row on the figure to say which multipliers clear every fraction, then state a rule for it.
- Criterion C. Communicating: write a line of working that shows what each term became after multiplying.
- Criterion D. Applying mathematics in real-life contexts: the soap batch in the hook and the squad in question 5.
The one thing to do with the figure
Ask for the multiplier before sliding. “Which number clears both fractions in x/2 + x/3 = 5?” Take 2, 3, 5 and 6 and test each on the figure.
Then ask what the filled dots have in common, and what the first filled dot is for each equation. Finish by asking why the value of x at the bottom of the panel never changes.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions first. Include the other two equations. |
| 10 min | Key idea; worked examples on the soap batch and the bracket on top. |
| 10 min | Faded example in pairs; compare the first line. |
| 5 min | Question 1, the check. If many choose B, go back to the figure and test 15. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 8, on paper or on screen. |
The answers
| Warm-up | x = 6; 12; 3x. |
|---|---|
| Soap batch | x = 30. |
| Worked example | x = 7. |
| Faded example | x = 5. |
| 1. Lowest common multiple of the denominators | C, 18. A is a common factor; B adds the denominators; D works but is larger than needed. |
| 2. A single fraction | 15. x/3 = 5. |
| 3. A fraction with a bracket on top | 9. x + 5 = 14. |
| 4. Two fractions, one unknown | 20. 5x + 2x = 140; 7x = 140. |
| 5. Write and solve from a context | 30. 2x + 30 = 3x. |
| 6. Bracket on top, unknown on both sides | 14. 4x − 6 + 20 = 5x. |
| 7. Subtracting a fraction with a bracket | 13. 4x + 8 − 3x + 3 = 24. |
| 8. Multiply every term | B. A leaves the 3; C leaves the right side; D multiplies the numerator as well as cancelling the denominator. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Took a factor, the sum, or the product instead of the lowest common multiple. | List the multiples of 6 and of 9 until one appears in both. |
| 5, 39 or 1.67 in question 2 | Undid the +4 or the ÷3 in the wrong direction. | Say the operations in order, then undo them in reverse order. |
| 2, 12 or 19 in question 3 | Moved the 5 without first clearing the denominator, or the wrong way. | Multiply both sides by 2 first and write the new line. |
| 140, 98 or 2 in question 4 | Stopped early, added denominators, or left the right side unmultiplied. | Draw an arrow from the multiplier to every term, including 14. |
| 5, 10 or −30 in question 5 | Left the 5 unmultiplied or multiplied it by the wrong number. | Rewrite with the multiplier in a bracket: 6(x/3 + 5) = 6 × x/2. |
| 17, −4 or 26 in question 6 | Multiplied only part of the numerator, left the 2 alone, or slipped on a sign. | Put the numerator in a bracket the moment you clear the denominator. |
| 19, −9 or 35 in question 7 | Lost the minus sign across the second bracket, or left the right side alone. | Write −3(x − 1) with its bracket and expand carefully. |
| A, C or D in question 8 | Left one term unmultiplied, or multiplied the numerator as well. | Check every term: how many are there, and what did each become? |
Questions worth asking
- Why can we multiply both sides by 6 and not change the answer? Listen for “both sides stay equal” or “the balance is still level”.
- Is 12 always the best multiplier for x/4 + x/6? It is the lowest; 24 also works, with bigger numbers.
- What if there is no fraction on one side? Multiply it all the same: 3x becomes 18x if the multiplier is 6.
- Where do the brackets come from? A numerator is one quantity: the denominator cancels against the whole of it.
Differentiation
A way in for anyone stuck
- Give a list of multiples of 2 to 9 to find the lowest common multiple.
- Start with one fraction (x/3 + 4 = 9), then add a second fraction.
- Provide a template line: ___(x/__ + __) = ___ × ___ to fill in.
Stretch
- Question 7, then: solve x/2 + x/3 + x/6 = 12 and say what is special about the answer.
- Make up an equation with fractions whose solution is x = 10 and whose lowest common multiple is 12.
Students learning in English as an additional language
“Denominator” is the bottom of a fraction and “numerator” is the top. “Clear” here means “remove”. “Every term” is each part separated by + or −.
| Word | Plain meaning | Example |
|---|---|---|
| denominator | the number on the bottom of a fraction | in x/4 the denominator is 4 |
| numerator | the number or expression on top | in (x + 1)/3 the numerator is x + 1 |
| common multiple | a number that two numbers both divide into | 12 is a common multiple of 4 and 6 |
| term | one part of an expression, separated by + or − | 3x/4 − 5 has two terms |
| clear | remove the fractions from an equation | multiply every term by 6 |
| balance | both sides stay equal | multiply both sides by the same number |
Sentence starters for explaining aloud or in writing:
- The denominators are … and …, so I multiply every term by …
- After multiplying, the first term becomes …, and the right side becomes …
- I check by putting x = … into the original: the left side is …
Two things not to say
“Cross-multiply.” It works for one fraction equal to one fraction only, and hides why it works.
“Just get rid of the bottom.” It must be done to every term, and say how.
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.
- One fractionSolve x/4 = 6.
x = 24. - Add a numberSolve x/5 + 2 = 7.
x = 25.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextHalf of a bag of rice plus a quarter of it weighs 9 kg. How much does the whole bag weigh?
12 kg, because x/2 + x/4 = 9 gives 2x + x = 36, so 3x = 36. - Earlier workSolve 3(x + 2) = 2x + 11.
x = 5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWithout solving, which of x/3 + x/4 = 7 and x/6 + x/8 = 7 has the bigger solution?
The second: its fractions are smaller, so it needs a bigger x (x = 24 against x = 12). - Working backwardsWrite a fraction equation with denominators 3 and 5 whose solution is x = 15.
Any correct one, for example x/3 + x/5 = 8.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write multiples of the denominators in two rows and circle the first match.