At a glance. One lesson of about 60 minutes. It builds on lessons 1 and 2. Students find a common denominator from factorized bottoms, combine tops with brackets, and simplify; a rate context ends the practice.
Calculator: no calculator. No calculator: the structure of the algebra is the lesson, and the numbers are small.
What students will learn
Know
- The common denominator contains each factor once.
- Each top is multiplied by the factors its bottom lacks.
- A subtraction needs brackets round the second top.
Understand
- Adding tops and bottoms does not work for numbers, so it cannot work for letters.
- Factorizing first can shorten the common denominator.
- A rate sum is a sum of fractions with letters when the times are unknown.
Do
- Write a sum or difference of algebraic fractions as a single fraction.
- Choose a common denominator after factorizing.
- Use the single fraction to answer a rate question.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: common denominators and brackets (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe how the coefficients depend on p and q.
- Criterion C. Communicating: writing each step with the factors shown.
- Criterion D. Applying mathematics in real-life contexts: the filling taps (question 7).
The one thing to do with the figure
Ask for the prediction first. Set p = 3 and q = 2, then ask students to work out both fractions at x = 2 on paper. Ask whether Kwame's 5/7 matches.
Ask: “What would the bars look like if adding tops and bottoms worked?” The total would be a fraction of the form 5 over 7.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: Kwame's fraction; evaluate both at x = 2. |
| 12 min | Figure, key idea and warning card; write the three lines of working for another p and q. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, expand the subtraction one term at a time. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 5/12; 2x − 14; (x − 4)(x + 4). |
|---|---|
| Hook | No: at x = 2 the fractions add to 18/12 = 1.5, but 5/7 is about 0.71. |
| Worked examples | (7x + 13) ÷ ((x + 3)(x − 1)); (x + 16) ÷ ((x − 2)(x + 4)); (x + 7) ÷ ((x − 3)(x + 3)). |
| Faded example | (5x − 13) ÷ ((x + 2)(x − 2)). |
| 1. Subtracting with brackets | A. B changes only the first sign; C subtracts the tops; D adds. |
| 2. Combining two fractions | 9. 7x + 2. |
| 3. Subtracting fractions | 17. 4x + 17. |
| 4. Factorizing to find the denominator | 12. 2x + 12. |
| 5. A constant numerator | 8. 4(x + 1) − 4(x − 1). |
| 6. Finding an unknown top | 6. (3 + a)x + 6 − a. |
| 7. A rate sum in x | 3.75. 16/60 per hour, so 60 ÷ 16. |
| 8. Choosing the common denominator | B. A multiplies the bottoms; C misses the factor x; D repeats x + 7. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Changed only the first sign, subtracted the tops, or added. | Expand (x − 3)(x + 2) inside a bracket first, then subtract the whole bracket. |
| 12 or 7 in question 2 | Multiplied by the wrong bottom, or added the tops. | Underline the bottom each fraction is missing. |
| 13 or −13 in question 3 | Dropped a sign, or swapped the multipliers. | Expand the bracket line by line. |
| 6 or 0 in question 4 | Left out the second top, or subtracted it. | Check the sign between the fractions. |
| 0 or −8 in question 5 | Subtracted the tops, or swapped the multipliers. | Write 4(x + 1) first, then 4(x − 1). |
| −6 or 3 in question 6 | Expanded a(x − 1) as ax + a, or copied a top. | Expand a bracket and write the constant separately. |
| 8 or 16 or 0.27 in question 7 | Averaged or added the times, or gave the fraction per hour. | Ask: should two taps take more or less time than one? |
| A, C or D in question 8 | Multiplied the bottoms, missed a factor, or repeated a factor. | Factorize each bottom first, then list each factor once. |
Questions worth asking
- Why can't tops and bottoms be added? Listen for: 1/2 + 1/3 is not 2/5.
- Why use brackets in a subtraction? Listen for: the minus sign multiplies every term of the second top.
- How do you choose the common denominator? Listen for: each factor once, after factorizing.
- How can you check your single fraction? Listen for: put in a value of x for both sides.
Differentiation
A way in for anyone stuck
- Give a three-line frame: bottoms factorized, tops multiplied, tops combined.
- Provide the common denominator and ask students to find only the top.
- Give the frame: “The first fraction is missing …. The second is missing …. The top is ….”
Stretch
- Show that 1/x − 1/(x + 1) = 1/(x(x + 1)), and use it to add 1/(1 × 2) + 1/(2 × 3) + 1/(3 × 4).
- Find p and q so that p/(x + 1) + q/(x + 2) has top 4x + 7.
Students learning in English as an additional language
“Common denominator” means a bottom that both fractions can share. “Combine” means write as one. “Collect” means add the like terms. “Rate” means how much is done in one unit of time.
| Word | Plain meaning | Example |
|---|---|---|
| common denominator | a bottom that both fractions can be written over | 15 for 1/3 and 1/5 |
| combine | write two fractions as one | p/3 and q/4 together |
| collect like terms | add the terms with the same letter part | 2x + 3x = 5x |
| rate | how much is done in one unit of time | 30 pages printed each minute |
| bracket | a pair of curved lines that group terms | (x − 2) |
Sentence starters for explaining aloud or in writing:
- The bottoms factorize as …, so the common denominator is ….
- The first fraction is missing ….
- After the brackets are expanded the top is ….
Two things not to say
“Add the tops, add the bottoms.” Say instead: write both fractions over a common bottom, then add the tops.
“The minus only touches the first term.” Say instead: it multiplies the whole bracket.
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.
- AddWrite 1/(x + 1) + 1/(x + 2) as one fraction.
(2x + 3) ÷ ((x + 1)(x + 2)). - SubtractWrite 3/x − 1/(x + 1) as one fraction.
(2x + 3) ÷ (x(x + 1)).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Factorize firstWrite 2/(x2 − 1) + 1/(x + 1) as one fraction.
(x + 1) ÷ (x2 − 1), which is 1/(x − 1). - Earlier work (lesson 2)Simplify (x2 − 9) ÷ (x + 2) × (x + 2) ÷ (x − 3).
x + 3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs 1/(x − 3) − 1/(x + 3) ever equal to 0? Explain.
No: the single fraction has top 6, which is never 0. - Working backwardsa/(x + 1) + 2/(x − 1) has top 5x − 1 over x2 − 1. Find a.
a(x − 1) + 2(x + 1) = (a + 2)x + 2 − a, so a = 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.