At a glance. One lesson of about 60 minutes. It assumes factorizing and the quadratic formula from Year 10. Students solve a quadratic inequality using the roots and the direction of the parabola, handle a negative leading coefficient, count integer solutions, and use the discriminant to bound a coefficient.
Calculator: no calculator. No calculator: the roots are rational and the reasoning is the sign diagram. A graphing calculator is used in the next lesson to check graphically.
What students will learn
Know
- A quadratic changes sign only at its roots.
- For a positive leading coefficient the quadratic is below zero between the roots and above outside them.
- No real roots means the discriminant is negative.
Understand
- Why the sign can be read from the opening direction and the roots.
- Why dividing by x loses solutions.
- Why a strict inequality excludes the roots and a non-strict one includes them.
Do
- Solve a quadratic inequality exactly.
- Count integer solutions in an interval or a union.
- Use the discriminant to find the values of a parameter.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving and writing the solution set (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say how the strip changes when the direction flips.
- Criterion C. Communicating: writing the solution with the correct inequality signs and a sketch.
- Criterion D. Applying mathematics in real-life contexts: profit and garden area (hook and question 5).
The one thing to do with the figure
Ask for the prediction first. Then keep the roots fixed and flip the direction, and ask which parts of the strip change colour and why the roots themselves do not.
Ask: “What happens to the strip if the two roots move closer together?” The middle region narrows, and with a gap of 1 it holds no whole number.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tea stall; collect predictions for the range of cups. |
| 12 min | Figure, key idea and warning card; sketch each parabola next to its sign strip. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask where the parabola is above the axis. |
| 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 + 7)(x − 3); 4 or 5; x < −3. |
|---|---|
| Hook | P = −(x − 2)(x − 6), which is positive for 2 < x < 6, so between 200 and 600 cups. The most profit is half way, at x = 4 (400 cups), where P = 4. |
| Worked examples | −3 < x < 4; x ≤ −3 or x ≥ 0.5; −2 ≤ x ≤ 5. |
| Faded example | x < −5 or x > 3. |
| 1. Reading a quadratic inequality | B. A is the region below zero; C drops one side; D has the roots' signs wrong. |
| 2. Counting integer solutions | 7. 0 to 6. |
| 3. Largest integer in a solution | 4. −7 ≤ x ≤ 4. |
| 4. Inequality with a common factor | 6. Three negative and three above 7. |
| 5. Inequality in context | 3. 0 < x < 4. |
| 6. Discriminant condition | 11. −5 to 5. |
| 7. Two inequalities together | 5. −3 to 1. |
| 8. A negative leading coefficient | A. B is the upward pattern (or the sign not reversed); C keeps one root; D swaps the roots' signs. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Gave the region between the roots, one side only, or wrong signs on the roots. | Ask them to test x = 0 in the original inequality. |
| 9 or 8 in question 2 | Included the roots, or gave the gap. | List the integers between the roots and cross out the endpoints. |
| −7 or 7 in question 3 | Gave the smaller root, or the wrong sign. | Ask: in which direction is the interval, and which end is the largest? |
| 8 or 3 in question 4 | Included the roots, or divided by x. | Ask them to test x = −1 and x = 0 in x2 > 7x. |
| 4 or 9 in question 5 | Included the equality, or used the wrong root. | Substitute x = 4: is the area less than 36? |
| 13 or 5 in question 6 | Included the repeated root, or counted only positive values. | Ask what the discriminant is at k = 6. |
| 7 or 9 in question 7 | Used one inequality only, or included 4 and −4. | Draw both solutions on one number line and shade the overlap. |
| B, C or D in question 8 | Used the upward pattern or did not reverse the sign, kept one root, or swapped the roots' signs. | Test x = 0 and x = 7 in 24 + 2x − x2. |
Questions worth asking
- How does the sign of the leading coefficient change the answer? Listen for: the parabola opens down, so the regions swap.
- Why can we not divide by x in x2 > 3x? Listen for: x could be negative, which reverses the inequality, or zero.
- When do the roots belong to the solution? Listen for: when the inequality is non-strict.
- What does a negative discriminant tell us about the sign? Listen for: the parabola never crosses the axis, so it has one sign everywhere.
Differentiation
A way in for anyone stuck
- Give a pre-drawn number line with the two roots marked and three blank regions to label positive or negative.
- Provide a template: Roots …; opens up/down; above zero is outside/between; solution ….
- Give the frame: “I tested the value x = … and the expression was …, so this region is ….”
Stretch
- For which k does kx2 + 4x + k > 0 hold for every x?
- Solve x3 − 4x < 0 using a sign diagram with three roots.
Students learning in English as an additional language
“Inequality” means a statement with < or > rather than =. “Critical value” means a number where the sign can change. “Strict” means that the end values are not included. “Region” here means a stretch of the number line.
| Word | Plain meaning | Example |
|---|---|---|
| inequality | a statement that one expression is less than or greater than another | x2 < 10 |
| critical value | a number where an expression changes sign | a root of the quadratic |
| sign diagram | a number line showing where an expression is positive or negative | plus, minus, plus |
| strict inequality | one that uses < or > and excludes the end values | x > 5 |
| discriminant | b2 − 4ac, which tells the number of real roots | negative: no real roots |
Sentence starters for explaining aloud or in writing:
- The roots are … so the regions are …
- I tested … and the sign was …
- The parabola opens … so above zero is …
Two things not to say
“Positive means outside.” Say instead: for an upward parabola positive is outside the roots, and the rule flips for a downward one.
“Just take the square root.” Say instead: factorize and use the sign diagram, because both roots matter.
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.
- FindSolve x2 − 9 < 0.
−3 < x < 3. - FindSolve (x − 1)(x + 4) ≥ 0.
x ≤ −4 or x ≥ 1.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA ball's height is h = 20t − 5t2 metres. For which times is it above 15 m?
5t2 − 20t + 15 < 0, so 1 < t < 3. - Earlier work (Year 10)Find the roots of x2 − 6x − 16.
−2 and 8.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFor which values of m does x2 + mx + 4 > 0 hold for all x?
m2 < 16, so −4 < m < 4. - Working backwardsA quadratic is negative exactly when −1 < x < 6. Write one such quadratic.
(x + 1)(x − 6), which is x2 − 5x − 6.
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.