At a glance. One lesson of about 60 minutes. It assumes surds (Unit 1), solving by factorizing (Lesson 4) and the vertex form of a quadratic (Lesson 3). Students complete the square, use the quadratic formula, and give exact or rounded answers.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students with a GDC how a solver gives decimal roots to check against exact answers, and that the exact form still comes from hand working.
What students will learn
Know
- x² + bx + (b ÷ 2)² = (x + b ÷ 2)².
- The quadratic formula x = (−b ± √(b² − 4ac)) ÷ (2a).
- An exact answer keeps the surd; a rounded answer states its decimal places.
Understand
- Every quadratic equation can be solved by completing the square, even when factorizing fails.
- The formula is completing the square with letters, so the methods are equivalent.
- A negative number under the root means no real solution.
Do
- Solve by completing the square, with a ± and an exact answer.
- Use the quadratic formula with correct brackets.
- Reject a solution that does 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: completing the square and the formula (questions 1 to 8).
- Criterion B. Investigating patterns: the what-if. Students complete the square on x² + bx + c and generalize the expression under the root.
- Criterion C. Communicating: showing each step of completing the square and writing the formula with brackets (the worked examples).
- Criterion D. Applying mathematics in real-life contexts: the water rocket, where the negative time is rejected (question 5).
The one thing to do with the figure
Ask for a prediction before every slide. With m = 3 on the screen, ask: “What c makes the equation have exactly one solution?” before anyone moves c. Then ask what happens when c goes past that value.
Then ask students to point to the dashed corner and say what is missing from the square, and what has to be done to both sides.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: when the pairs run out. Collect predictions. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and both worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to the missing corner. |
| 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 | x² + 10x + 25; 5√2; 10. |
|---|---|
| Hook | There are solutions; factorizing is the wrong tool. x = −2 ± √7, about 0.65 and −4.65, so the seed bed is about 0.65 m wide (the negative value is rejected). |
| Worked examples | x = −4 ± √19 (0.36 and −8.36); 2.18 and 0.15. |
| Faded example | 4.56 and 0.44. |
| 1. Completing the square | C. A adds 25 instead of subtracting; B does not halve; D has the wrong sign. |
| 2. Completing the square on an equation | 64. 15 + 49. |
| 3. Using the quadratic formula | 0.85. (−3 + √41) ÷ 4. |
| 4. Solving by completing the square | 5.16. 2 + √10. |
| 5. A quadratic model | 4.07. Positive solution. |
| 6. Working backwards from a solution | 11. 16 − c = 5. |
| 7. Completing the square with a leading coefficient | −13. 5 − 18. |
| 8. Exact solutions by completing the square | A. B adds 2 instead of subtracting; C has the wrong sign; D does not halve. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Added instead of subtracted, did not halve, or flipped the sign. | Expand each option and compare with x² − 10x + 3. |
| 15 or 34 in question 2 | Forgot to add 49, or moved the 15 with the wrong sign. | Write x² + 14x = 15 first, then add 49 to both sides. |
| 1.7, 2.35 or −1.4 in question 3 | Divided by a, used +b, or divided only the root. | Write the formula with brackets and 2a = 4 before substituting. |
| 3.16, −1.16 or 12 in question 4 | Stopped at the root, gave the negative solution, or did not root. | Underline what the question asks for: positive solution. |
| −0.07 or 8.15 in question 5 | Gave the negative time, or divided by a. | Ask: what does a negative time mean for the rocket? |
| 21, −11 or 5 in question 6 | Added instead of subtracted, a sign slip, or gave the root number. | Square both sides of x − 4 = √5 and compare. |
| −4 or 23 in question 7 | Forgot the factor 2, or added the 18. | Expand 2(x + 3)² in full and compare it with the given quadratic. |
| B, C or D in question 8 | Added 2, a sign slip, or did not halve. | Move the 2 across and add 9 to both sides; say each step. |
Questions worth asking
- Why does the strip diagram need a corner? Without it x² + bx is not a square; the corner m² makes (x + m)².
- Where does the ± come from? Both a number and its negative square to the same value.
- Which method would you try first for x² + 9x + 20 = 0? For x² + 9x + 3 = 0? Factorizing for the first; the formula or completing the square for the second.
- How do the formula and completing the square connect? The formula is completing the square on ax² + bx + c, once and for all.
Differentiation
A way in for anyone stuck
- Give a completing-the-square template: x² + bx = …, add (b ÷ 2)², (x + …)² = ….
- Use algebra tiles or the figure to see the missing corner.
- Provide the formula with brackets drawn, and have students write a, b and c on the line above.
Stretch
- Derive the quadratic formula by completing the square on ax² + bx + c = 0.
- Show that the sum of the two solutions of ax² + bx + c = 0 is −b ÷ a.
Students learning in English as an additional language
“Complete” means to finish a pattern: here, to finish a square. “Exact” means leaving the surd in the answer. “Formula” is a rule written with letters. Say “plus or minus” for ±.
| Word | Plain meaning | Example |
|---|---|---|
| completing the square | adding the missing corner so an expression becomes a perfect square | turning x² + 2x into (x + 1)² − 1 |
| perfect square | an expression that is something squared | (x + 9)² |
| quadratic formula | a formula that gives the solutions of ax² + bx + c = 0 | uses a, b and c |
| exact answer | an answer that keeps surds and fractions | 2 + √3 |
| coefficient | the number in front of a letter | the 6 in 6x² |
| plus or minus | the symbol ± stands for two values at once | 5 ± 2 is 3 and 7 |
Sentence starters for explaining aloud or in writing:
- I moved the constant across, then added … to both sides.
- The solutions are exactly … and about … to 2 decimal places.
- I used the formula because …
Two things not to say
“Just plug into the formula.” Have students write a, b and c first, and use brackets for negative values.
“The root of a sum is the sum of the roots.” √(9 + 16) is not 3 + 4; add under the root 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.
- SquareComplete the square: write x² + 10x as (x + p)² + q.
(x + 5)² − 25. - FormulaSolve x² + 2x − 1 = 0 exactly.
x = −1 ± √2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA sprinkler in a park in Sydney has height h = −5t² + 10t + 1 metres after t seconds. When does it reach the ground? Give the positive answer to 2 decimal places.
t = 2.10 s: (−10 − √120) ÷ (−10) = 2.095. - Earlier work (Year 10, Lesson 4)Solve x² − 7x + 10 = 0.
x = 2 or x = 5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFor which values of k does x² + kx + 9 = 0 have no real solutions?
−6 < k < 6, because k² − 36 < 0. - VertexWrite y = x² − 6x + 14 in vertex form and state the vertex.
y = (x − 3)² + 5; vertex (3, 5).
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.