The shape of the lesson
By the end of the lesson, learners can
- Factorise, then cancel: In the lab choose Cancel and press Next step.
- Multiply and divide: Multiply tops together and bottoms together, cancelling common factors on the way.
- Add and subtract: Find a common denominator, rewrite each fraction over it, then combine the numerators.
- Simplify: factorise top and bottom fully, then cancel common factors.
- Multiply: tops times tops, bottoms times bottoms, cancelling first.
- Divide: flip the second fraction and multiply.
- Add/subtract: common denominator; numerators in brackets.
- Finish: factorise the final numerator to see if anything else cancels.
One picture every idea returns to
Only cancel factors, never terms. Factorise the top and bottom completely first; then a bracket that appears on both can be cancelled. To add or subtract, write both fractions over a common denominator.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Fractions with letters.. Extended. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Factorise, then cancel | In the lab choose Cancel and press Next step. Project tab 6 of the lab (Algebra lab, algebraic fractions) and let a student drive it while the class predicts. Model the worked example Cancel a common bracket on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Multiply and divide | Multiply tops together and bottoms together, cancelling common factors on the way. Model the worked example Multiply on the board, one line at a time. Your turn: Simplify \(\dfrac{4x}{15} \div \dfrac{8x^{2}}{5}\). | Pairs |
| 21 to 30 min | Teach and check: Add and subtract | Find a common denominator, rewrite each fraction over it, then combine the numerators. Model the worked example Numerical denominators on the board, one line at a time. Your turn: Write \(\dfrac{2}{x + 3} + \dfrac{5}{x - 4}\) as a single fraction in its simplest form. | Numbered Heads Together |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 18 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Add and subtract, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Cancelling the \(x^{2}\) terms in \(\dfrac{x^{2} + 3x}{x^{2} - 9}\) to get \(\dfrac{3x}{-9}\) is wrong: \(x^{2}\) is a term, not a factor. | Factorise first: the answer is \(\dfrac{x}{x - 3}\). |
| Flip the SECOND fraction only, and only when dividing. | Flipping the first, or flipping when multiplying, gives the reciprocal of the right answer. |
| In \(\dfrac{x + 1}{4} - \dfrac{x - 3}{6}\) the minus applies to the whole of \(x - 3\): \(3(x + 1) - 2(x - 3) = 3x + 3 - 2x + 6\). | Writing \(-6\) instead of \(+6\) is the classic slip. |
Algebraic fractions: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
Simplify (a) \(\dfrac{15a^{2}b}{25ab^{3}}\) (b) \(\dfrac{4x + 12}{x^{2} + 3x}\).
Simplify \(\dfrac{x^{2} - x - 6}{x^{2} - 4}\).
Simplify \(\dfrac{4x}{15} \div \dfrac{8x^{2}}{5}\).
Write \(\dfrac{2x - 1}{3} - \dfrac{x - 4}{5}\) as a single fraction in its simplest form.
Write \(\dfrac{2}{x + 3} + \dfrac{5}{x - 4}\) as a single fraction in its simplest form.
(a) Show that \(\dfrac{1}{x} - \dfrac{1}{x + 1} = \dfrac{1}{x(x + 1)}\).
(b) Hence, without a calculator, find the exact value of \(\dfrac{1}{1 \times 2} + \dfrac{1}{2 \times 3} + \dfrac{1}{3 \times 4} + \dots + \dfrac{1}{9 \times 10}\).
Answer key and marking notes
- (a) \(\dfrac{3a}{5b^{2}}\)
- (b) \(\dfrac{4(x + 3)}{x(x + 3)} = \dfrac{4}{x}\)
- Top: \((x - 3)(x + 2)\). Bottom: \((x - 2)(x + 2)\).
- Cancel \((x + 2)\): \(\dfrac{x - 3}{x - 2}\)
- \(\dfrac{4x}{15} \times \dfrac{5}{8x^{2}} = \dfrac{20x}{120x^{2}}\)
- \(= \dfrac{1}{6x}\)
- \(\dfrac{5(2x - 1) - 3(x - 4)}{15}\)
- \(= \dfrac{10x - 5 - 3x + 12}{15} = \dfrac{7x + 7}{15}\)
- \(= \dfrac{7(x + 1)}{15}\)
- \(\dfrac{2(x - 4) + 5(x + 3)}{(x + 3)(x - 4)}\)
- \(= \dfrac{7x + 7}{(x + 3)(x - 4)} = \dfrac{7(x + 1)}{(x + 3)(x - 4)}\)
- (a) \(\dfrac{(x + 1) - x}{x(x + 1)} = \dfrac{1}{x(x + 1)}\)
- (b) Each term is \(\dfrac{1}{n} - \dfrac{1}{n + 1}\): \(\left(1 - \dfrac{1}{2}\right) + \left(\dfrac{1}{2} - \dfrac{1}{3}\right) + \dots + \left(\dfrac{1}{9} - \dfrac{1}{10}\right)\)
- Everything cancels except \(1 - \dfrac{1}{10}\)
- \(= \dfrac{9}{10}\)
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Simplify \(\dfrac{x^{2} - 5x}{x^{2} - 3x - 10}\).
- A\(\dfrac{x}{x + 2}\)
- B\(\dfrac{5x}{3x + 10}\)
- C\(\dfrac{x}{x - 2}\)
- D\(\dfrac{1}{x + 2}\)
Simplify \(\dfrac{2x^{2} - 7x + 3}{x^{2} - 9}\).
- A\(\dfrac{2x - 1}{x + 3}\)
- B\(\dfrac{2x + 1}{x + 3}\)
- C\(\dfrac{2x - 1}{x - 3}\)
- D\(\dfrac{2x + 1}{x - 3}\)
Simplify \(\dfrac{5x}{6} \div \dfrac{10x^{2}}{9}\).
- A\(\dfrac{3}{4x}\)
- B\(\dfrac{25x^{3}}{27}\)
- C\(\dfrac{4x}{3}\)
- D\(\dfrac{3x}{4}\)
Write \(\dfrac{x + 2}{3} + \dfrac{x - 1}{4}\) as a single fraction.
- A\(\dfrac{7x + 5}{12}\)
- B\(\dfrac{2x + 1}{7}\)
- C\(\dfrac{7x + 11}{12}\)
- D\(\dfrac{2x + 1}{12}\)
Write \(\dfrac{3}{x - 1} - \dfrac{2}{x + 2}\) as a single fraction in its simplest form.
- A\(\dfrac{x + 8}{(x - 1)(x + 2)}\)
- B\(\dfrac{x + 4}{(x - 1)(x + 2)}\)
- C\(\dfrac{1}{(x - 1)(x + 2)}\)
- D\(\dfrac{5x + 4}{(x - 1)(x + 2)}\)
Simplify \(\dfrac{x^{2} - 4}{3x} \times \dfrac{6x^{2}}{x + 2}\).
- A\(2x(x - 2)\)
- B\(2x(x + 2)\)
- C\(2(x - 2)\)
- D\(\dfrac{x - 2}{2x}\)
Quiz key
Top: \(x(x - 5)\). Bottom: \((x - 5)(x + 2)\). Cancel the common bracket \((x - 5)\) to get \(\dfrac{x}{x + 2}\). \(\dfrac{5x}{3x + 10}\) cancels the \(x^{2}\) terms, which are terms, not factors. \(\dfrac{1}{x + 2}\) also cancels the \(x\) on top, which has no partner below.
\(2x^{2} - 7x + 3 = (2x - 1)(x - 3)\) and \(x^{2} - 9 = (x - 3)(x + 3)\). Cancel \((x - 3)\): \(\dfrac{2x - 1}{x + 3}\). \(\dfrac{2x - 1}{x - 3}\) cancels the wrong bracket from the bottom. \((2x + 1)\) comes from a sign slip when factorising the top: \((2x + 1)(x - 3) = 2x^{2} - 5x - 3\).
\(\dfrac{5x}{6} \times \dfrac{9}{10x^{2}} = \dfrac{45x}{60x^{2}} = \dfrac{3}{4x}\). \(\dfrac{25x^{3}}{27}\) multiplies without flipping. \(\dfrac{4x}{3}\) flips the first fraction instead of the second, giving the reciprocal. \(\dfrac{3x}{4}\) cancels \(x^{2}\) with \(x\) the wrong way round.
\(\dfrac{4(x + 2)}{12} + \dfrac{3(x - 1)}{12} = \dfrac{4x + 8 + 3x - 3}{12} = \dfrac{7x + 5}{12}\). \(\dfrac{2x + 1}{7}\) adds the tops and adds the bottoms, which is never allowed. \(\dfrac{2x + 1}{12}\) changes the denominator but forgets to multiply the numerators.
\(\dfrac{3(x + 2) - 2(x - 1)}{(x - 1)(x + 2)} = \dfrac{3x + 6 - 2x + 2}{(x - 1)(x + 2)} = \dfrac{x + 8}{(x - 1)(x + 2)}\). \(\dfrac{x + 4}{(x - 1)(x + 2)}\) writes \(-2 \times -1\) as \(-2\). \(\dfrac{1}{(x - 1)(x + 2)}\) just subtracts the numerators 3 and 2 without cross-multiplying.
\(\dfrac{(x - 2)(x + 2)}{3x} \times \dfrac{6x^{2}}{x + 2}\). Cancel \((x + 2)\), and \(\dfrac{6x^{2}}{3x} = 2x\), leaving \(2x(x - 2)\). \(2x(x + 2)\) cancels the wrong bracket. \(2(x - 2)\) treats \(\dfrac{6x^{2}}{3x}\) as 2, losing the \(x\) that is left over. \(\dfrac{x - 2}{2x}\) flips the second fraction, which is only done when dividing.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.