At a glance. One lesson of about 60 minutes. It assumes factorizing (Lesson 2) and the graph of a quadratic (Lesson 3). Students use the null factor law, rearrange to zero, keep the solution x = 0, and reject a solution that does not fit the context.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students with a GDC how the zero tool checks both solutions and warns that a solver may report only one.
What students will learn
Know
- If ab = 0 then a = 0 or b = 0.
- A quadratic equation must equal 0 before it is factorized.
- The solutions are the x-intercepts of the graph.
Understand
- The law is about zero only: no other product splits.
- Dividing by x can lose a solution.
- A solution of an equation may not be a solution of the situation it models.
Do
- Solve a quadratic equation by factorizing, including x² = kx.
- Rearrange an equation to equal 0 first.
- Reject solutions that do not fit the context.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving by factorizing (questions 1 to 8).
- Criterion B. Investigating patterns: the what-if. Students compare a repeated factor with two different factors, and generalize the number of solutions.
- Criterion C. Communicating: showing the rearrangement, the factorization and the check in a clear sequence (the worked examples).
- Criterion D. Applying mathematics in real-life contexts: the plot, where one solution is rejected, and the consecutive numbers (questions 5 and 7).
The one thing to do with the figure
Ask for a prediction before every slide. With p = −2 and q = 3, ask: “At which input will the product be zero?” before anyone moves x. Then ask for an input where neither bracket is zero and check the product.
Then set p = 0 and ask students to read the equation. Name x as a factor, and say why dividing by x loses a solution.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: a product of zero. Collect what students notice, then predictions for x. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and the worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, C or D, return to the figure. |
| 18 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 + 5)(x − 3); x = 4; 0. |
|---|---|
| Hook | At least one of the two numbers is 0; two numbers that are both not zero never multiply to 0. For (x − 6)(x + 9) = 0: x = 6 or x = −9. |
| Worked examples | x = 5 or −2; x = 0 or 8; x = 1.5 or −2. |
| Faded example | x = −7 or x = 3. |
| 1. Applying the null factor law | A. B reverses both signs; C misses the sign of the second; D adds the numbers. |
| 2. The larger solution | 7. (x − 2)(x − 7). |
| 3. Not dividing by x | 0. x(x − 12) = 0 gives 0 and 12. |
| 4. Solving with a leading coefficient | 0.5. (2x − 1)(x − 3). |
| 5. A solution that has to fit the context | 6. Reject x = −9. |
| 6. Rearranging before factorizing | 4. (x + 7)(x − 4) = 0. |
| 7. Consecutive numbers | 13. 12 × 13 = 156. |
| 8. Rearranging and solving | C. A reverses the signs; B forgets the 15; D treats the numbers as solutions. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Reversed the signs, missed one sign, or added the numbers. | Substitute each candidate back: which values make a bracket 0? |
| 2, 14 or −2 in question 2 | Gave the smaller solution, the constant, or reversed the sign. | Ask for both solutions first, then pick the larger. |
| 12, 144 or −12 in question 3 | Divided both sides by x and lost x = 0, squared 12, or changed the sign. | Substitute x = 0 into x² = 12x. Is it true? |
| 3 or 1 in question 4 | Gave the larger solution, or did not undo the 2x. | Ask: what is 2x − 1 = 0 as an equation for x? |
| 9 or −9 in question 5 | Gave the length, or kept a negative width. | Ask what x stands for and whether a width can be negative. |
| 7 or 28 in question 6 | Misread the sign of a bracket, or used the constant. | Rearrange to zero before factorizing: say it aloud. |
| 12 or −13 in question 7 | Gave the smaller number or the negative solution. | Read the question again: which of the two numbers is asked for? |
| A, B or D in question 8 | Reversed the signs, forgot the 15, or used the numbers as solutions. | Move 15 across so the right side is 0, then factorize. |
Questions worth asking
- Why does the law only work for zero? Listen for “many pairs multiply to 6, but only 0 forces a factor to be 0”.
- Why is dividing by x risky? Because x might be 0, and dividing by 0 is not allowed; the solution x = 0 vanishes.
- How many solutions can a quadratic equation have? Two, one repeated, or none; the graph shows which.
- Where do the solutions appear on the graph? At the x-intercepts of y = the quadratic expression.
Differentiation
A way in for anyone stuck
- Give a five-step solving checklist: rearrange, factorize, set each factor to 0, solve, check.
- Use the figure's three readouts to have students predict the product before they check.
- Provide factorized forms for the first examples, then remove them.
Stretch
- Solve (x − 2)(x + 3) = 14 by expanding and rearranging, and explain why x − 2 = 14 or x + 3 = 14 fails.
- Find a quadratic equation with integer coefficients whose solutions are 2.5 and −4.
Students learning in English as an additional language
“Solution” and “root” both mean a value of x that makes the equation true. “Factor” is a noun and “factorize” a verb. “Null” means zero. Say “either … or …” for a = 0 or b = 0.
| Word | Plain meaning | Example |
|---|---|---|
| null factor law | if a product is zero, one of its factors is zero | if mn = 0 then m = 0 or n = 0 |
| solution | a value of the unknown that makes the equation true | x = 2 for x + 1 = 3 |
| root | another name for a solution, or a zero of a function | the roots of x² − 1 = 0 |
| rearrange | move terms so one side is zero | x² = 5x becomes x² − 5x = 0 |
| repeated root | the same solution twice, from a squared factor | x = 1 in (x − 1)² = 0 |
| reject | decide a solution does not fit the situation | a negative length |
Sentence starters for explaining aloud or in writing:
- I rearranged to get 0 on one side, then …
- Each factor equals 0, so x = … or x = …
- I reject x = … because …
Two things not to say
“Divide both sides by x.” That loses x = 0; move everything to one side and factorize.
“Set each bracket equal to the right-hand side.” The law needs the product to be exactly 0.
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.
- SolvingSolve (x + 1)(x − 8) = 0.
x = −1 or x = 8. - Zero solutionSolve x² = 5x.
x = 0 or x = 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA pond in a park in Seoul is a rectangle 4 m longer than wide with area 45 m². Find its width.
w(w + 4) = 45 gives (w + 9)(w − 5) = 0, so w = 5 m. - Earlier work (Year 10, Lesson 3)Find the zeros of y = x² − 2x − 8.
(x − 4)(x + 2): 4 and −2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSolve 6x² − 5x − 4 = 0.
(3x − 4)(2x + 1) = 0: x = 4/3 or x = −0.5. - Repeated rootFor what value of k does x² + 6x + k = 0 have one solution?
k = 9, because (x + 3)² = 0.
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.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.