At a glance. One lesson of about 60 minutes. It assumes the formula (Lesson 5) and the graph of a quadratic (Lesson 3). Students calculate Δ, count real solutions, find a parameter for a given number of solutions, and choose a method.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students with a GDC how the graph and zero tool confirm the count of solutions, and notes that complex numbers from a solver are not real solutions.
What students will learn
Know
- The discriminant Δ = b² − 4ac.
- Δ > 0, Δ = 0 and Δ < 0 give two, one or no real solutions.
- If a, b and c are whole numbers and Δ is a perfect square, the quadratic factorizes.
Understand
- Δ is the part of the formula under the root, so its sign decides whether the root exists.
- The sign of Δ is what the parabola does to the x-axis: cross, touch or miss.
- No real solutions is a valid answer that can mean a target is never reached.
Do
- Calculate Δ with correct signs.
- Use Δ to count solutions and to find a parameter value.
- Choose the most efficient method and justify it.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: calculating and using the discriminant (questions 1 to 8).
- Criterion B. Investigating patterns: the what-if, and the figure. Students vary one parameter and describe when the number of solutions changes.
- Criterion C. Communicating: justifying a method choice by the value of Δ (the key card and the exit ticket).
- Criterion D. Applying mathematics in real-life contexts: the largest plot, where the discriminant gives the limit of a model (question 5).
The one thing to do with the figure
Ask for a prediction before every slide. With a = 1 and b = 4 on the screen, ask: “What is the largest c for which the parabola still reaches the x-axis?” before anyone moves c.
Then ask where the parabola is when Δ is exactly 0, and what that means for the number of solutions.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: before you solve. Collect predictions for x² + 4x + 7. |
| 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 A or D, return to the sign of −4ac. |
| 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 | Two solutions, 1 and −5; 36 and −36; 44. |
|---|---|
| Hook | No: Δ = 16 − 28 = −12, so it never reaches the x-axis. |
| Worked examples | One solution, x = 3; x = 1/3 or −3; greatest area 36 m². |
| Faded example | k = 16. |
| 1. Counting solutions from the discriminant | C. A drops the sign of −4ac; B confuses a negative Δ with zero; D does not know the test. |
| 2. Calculating the discriminant | 169. 25 + 144. |
| 3. A parameter for one solution | 6. k² = 36. |
| 4. A condition for two solutions | 9. 36 − 4k > 0. |
| 5. Using the discriminant in a model | 49. A square plot, 7 m by 7 m. |
| 6. Touching the axis | 32. 256 − 8c = 0. |
| 7. A range of parameters | 5. k² < 25. |
| 8. When a quadratic factorizes | B. Only B has a perfect square Δ = 81. A has 69, C has 48 and D has −119. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Dropped the sign of −4ac, confused negative with zero, or did not know the test. | Calculate Δ on the board with every bracket, then say what its sign means. |
| −119 or 119 in question 2 | Dropped the sign of c, or squared a negative as negative. | Write (−5)² and 4(3)(−12) in full. |
| 36 or 3 in question 3 | Gave k², or the square root of 9. | Write Δ = k² − 36 and say what k² must equal. |
| 36 or 3 in question 4 | Gave b², or half of b. | Write the inequality 36 − 4k > 0 and solve it. |
| 196, 98 or 14 in question 5 | Gave b², half of b², or b. | Ask: what must Δ be for a plot to exist? Write 196 − 4A. |
| 256 or 64 in question 6 | Forgot 4a, or forgot that a = 2. | Write 4ac with the numbers in place: 4(2)(c). |
| 25 or 10 in question 7 | Gave k² or the root of 100. | Divide 4k² < 100 by 4 before taking the square root. |
| A, C or D in question 8 | Did not check for a perfect square. | Compute Δ for each option and test it against the squares up to 100. |
Questions worth asking
- How does the graph show the sign of Δ? Listen for “crosses twice, touches once, misses”.
- Why is ‘no real solutions’ a good answer in a model? It means the target (a height, an area) is never reached.
- If Δ is a perfect square, why does it factorize? The square root in the formula is a whole number, so the solutions are rational.
- When would you use the formula rather than factorizing? When Δ is not a perfect square, or the pair search is slow.
Differentiation
A way in for anyone stuck
- Give a Δ template with brackets: Δ = (b)² − 4(a)(c), and have students fill in each value.
- Provide a table: Δ negative, zero, positive, with a sketch and a count for each.
- Use the figure to match each sketch to its sign before any calculation.
Stretch
- Find the values of k for which kx² + 6x + 3 = 0 has two real solutions (k < 3 and k ≠ 0).
- Show that x² + bx + c = 0 and x² + bx − c = 0 cannot both have no real solutions.
Students learning in English as an additional language
“Discriminant” comes from a word meaning “to tell apart”: it tells the cases apart. “Real” here means an ordinary number on the number line. “Repeated” means the same solution twice. Say “delta” for Δ.
| Word | Plain meaning | Example |
|---|---|---|
| discriminant | the value b² − 4ac that tells how many real solutions a quadratic has | Δ = 9 − 16 for x² + 3x + 4 |
| real solution | a solution that is an ordinary number on the number line | x = 2.5 |
| repeated solution | a solution counted twice because the factor is squared | x = 5 in (x − 5)² = 0 |
| perfect square | the square of a whole number | 121 is 11² |
| rational | a number that can be written as a fraction of integers | 0.75 = 3/4 |
| parameter | a letter in a formula whose value can be chosen | k in x² + kx + 1 |
Sentence starters for explaining aloud or in writing:
- The discriminant is …, which is positive, zero or negative, so …
- Because Δ is a perfect square, …
- I chose this method because …
Two things not to say
“A negative discriminant means you made a mistake.” It means there are no real solutions, which is a valid result.
“Just use the formula every time.” Factorizing is quicker when Δ is a perfect square; look first.
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.
- DiscriminantFind the discriminant of x² + 6x + 2.
36 − 8 = 28. - CountingHow many real solutions has 2x² + x + 1 = 0?
Δ = 1 − 8 = −7, so none.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA water spray in a garden in Busan has height h = −2t² + 8t + c metres. What is the smallest c for which the jet reaches a height of 10 m?
−2t² + 8t + c − 10 = 0 needs Δ = 64 + 8(c − 10) ≥ 0, so c ≥ 2. - Earlier work (Year 10, Lesson 5)Solve x² − 2x − 5 = 0 exactly.
x = 1 ± √6.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFor which values of k does x² + kx + 4 = 0 have two different real solutions?
k > 4 or k < −4, because k² − 16 > 0. - MethodWithout solving, say whether 6x² − 7x − 3 = 0 factorizes.
Δ = 49 + 72 = 121 = 11², so yes.
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.