At a glance. One lesson of about 60 minutes. It builds on lessons 1 to 5. Students clear fractions, solve the resulting linear or quadratic equation, and reject values that make a denominator zero; a rate context ends the practice.
Calculator: no calculator. No calculator: each quadratic factorizes, and the point is the structure and the check.
What students will learn
Know
- Multiply every term by the common denominator.
- Solve the linear or quadratic equation that results.
- Reject any value that makes a denominator of the original equation zero.
Understand
- Multiplying by an expression that can be zero can introduce a value that is not a solution.
- A graph shows a gap where a value is excluded.
- A negative time or length is not a solution of a context problem.
Do
- Solve equations with fractions that give linear or quadratic equations.
- State which values are rejected and why.
- Set up and solve a rate equation.
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 (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe at which height the line meets nothing.
- Criterion C. Communicating: writing the rejection with its reason.
- Criterion D. Applying mathematics in real-life contexts: filling taps (question 7).
The one thing to do with the figure
Ask for the prediction first. Set c = 4 and ask what happens when Naledi substitutes. Then move c one step either side and ask what changes.
Ask: “Why does the algebra give an answer that the graph does not have?” The cancelled factor was zero at that value.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: Naledi's check; collect predictions. |
| 12 min | Figure, key idea and warning card; substitute x = 2 into the original equation. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, substitute x = 3 into the original bottom, and ask how many roots a linear equation has. If many choose C, ask which terms were multiplied by x − 3. |
| 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 | x = 3 or x = −2; 1.2; (2x + 1) ÷ (x(x + 1)). |
|---|---|
| Hook | She finds 0 ÷ 0 on the left, which has no value: x = 2 is rejected and there is no solution. |
| Worked examples | x = 8; x = 4 or x = −1; x = −4 (x = 3 rejected). |
| Faded example | x = 10. |
| 1. Rejecting a value | B. A does not reject; C does not multiply the 2; D keeps x = 3 and expects a second root. |
| 2. Solving a proportion of fractions | 7. 5x − 15 = 2x + 6. |
| 3. A fraction with a constant | 2. 12 = 6x. |
| 4. A quadratic with two solutions | 6. x = 6 or −1. |
| 5. Rejecting a root | −4. x = 4 rejected. |
| 6. A sum of two fractions | 3. The other root is −1/3. |
| 7. Taps and a quadratic | 4. −8/3 rejected. |
| 8. Solving and rejecting | A. B keeps x = 3; C rejects the negative root; D uses the wrong multipliers. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Kept a value that makes the bottom zero, did not multiply the 2, or expected a second root. | Substitute each value into the original equation. |
| −7 in question 2 | Crossed over the wrong tops. | Write the multipliers beside each fraction first. |
| 3 in question 3 | Multiplied the fraction but not the constant. | Multiply every term, including numbers, by the common denominator. |
| 1 in question 4 | Sign slip in the quadratic. | Expand each side separately before collecting. |
| 4 in question 5 | Did not reject. | Ask which root makes the bottom zero. |
| 1.33 in question 6 | Added the bottoms. | Write the sum as one fraction with the common denominator. |
| −2.67 in question 7 | Kept a negative time. | Ask whether a tap can take a negative number of hours. |
| B, C or D in question 8 | Kept a value that makes a bottom zero, rejected a negative, or used the wrong multipliers. | Substitute each value into every original bottom. |
Questions worth asking
- Why check in the original equation? Listen for: a value may make a denominator zero there.
- Does a quadratic always give two answers here? Listen for: no, a root may be rejected.
- What does the gap in the graph represent? Listen for: the excluded value.
- How would you check a root in a context? Listen for: put it back in and ask if it makes sense.
Differentiation
A way in for anyone stuck
- Give the solution in boxes: clear the fractions, expand, collect, factorize, check.
- Provide the cleared equation and ask only for solving and rejecting.
- Give the frame: “The excluded values are …. The roots are …. I reject … because ….”
Stretch
- Find a value of k so that (x2 + kx)/(x − 3) = 10/(x − 3) has only one solution.
- Show that 1/x + 1/(x + 1) = k has a real solution for every value of k, and say how many solutions there are when k = 0.
Students learning in English as an additional language
“Clear” means remove the fractions by multiplying. “Reject” means decide an answer is not allowed. “Root” is a solution of an equation. “Check” means put the answer back in.
| Word | Plain meaning | Example |
|---|---|---|
| clear the fractions | multiply every term so that no bottoms are left | multiply by x(x + 2) |
| reject | decide a value is not allowed | a negative length is rejected |
| root | a value of x that solves an equation | the roots of x2 = 9 are 3 and −3 |
| extraneous solution | a value that the algebra gives but the original does not allow | a value that makes a bottom 0 |
| rate | how much is done in one unit of time | 30 pages printed each minute |
Sentence starters for explaining aloud or in writing:
- The values excluded by the denominators are ….
- After clearing the fractions I have ….
- I reject … because it makes … zero.
Two things not to say
“Cancel the x on both sides of the equation.” Say instead: multiply by the common denominator, then solve.
“The answer is the root of the quadratic.” Say instead: it is the root that survives the check.
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.
- SolveSolve 6/(x + 1) = 3.
x = 1. - SolveSolve x/(x − 2) = 3.
x = 3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- RejectSolve (x2 − 2x)/(x − 2) = 2.
x = 2 from the cancelled form, but it is rejected, so there is no solution. - Earlier work (lesson 5)State the asymptotes of y = 3/(x + 4) − 2.
x = −4 and y = −2.
3Challenge
Reasoning, generalizing and working backwards.
- QuadraticSolve 1/x + 1/(x + 1) = 5/6 for x > 0.
6(2x + 1) = 5x(x + 1) gives x = 2. - Working backwardsx/(x − 2) = k has no solution for one value of k. Find it.
k = 1.
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 the slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.