At a glance. One lesson of about 60 minutes. It assumes factorizing (Lesson 2) and the vertex form. Students complete the square with a not 1, use h = −b ÷ (2a) and k = c − b² ÷ (4a), and find the least or greatest value of a model.
Calculator: GDC optional. Everything can be done by hand. The optional section shows students with a GDC how the minimum tool checks a vertex, and warns that a GDC reports rounded decimals.
What students will learn
Know
- Completing the square writes ax² + bx + c as a(x − h)² + k.
- The vertex is (−b ÷ (2a), c − b² ÷ (4a)).
- The least or greatest value of a quadratic is k.
Understand
- The three forms are one function; each form makes one feature visible.
- The vertex formula is completing the square done once in general.
- Every parabola is the basic parabola moved, and stretched or flipped.
Do
- Convert standard form to vertex form for any a.
- Derive or use the vertex formula.
- Find the extreme value of a quadratic model in 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 reading the vertex (questions 1 to 7).
- Criterion B. Investigating patterns: the what-if. Students vary a with b = 0 and with b not 0 and generalize where the vertex is.
- Criterion C. Communicating: the derivation of the vertex formula (worked example 3) and the equivalence in the exit ticket.
- Criterion D. Applying mathematics in real-life contexts: the cable and the football, where the vertex is the best or worst case (question 5).
The one thing to do with the figure
Ask for a prediction before every slide. With a = 1, b = 4, c = 1 on the screen, ask: “If I change b to 6, where does the axis go?” Then check with −b ÷ (2a).
Then switch a and ask why the vertex moves when b is not 0 but not when b is 0. Students should link it to the 2a in the denominator.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the cable. Collect predictions for the lowest point. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or D, return to the subtraction and the sign. |
| 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 + 26; (x − 5)(x + 3); −2. |
|---|---|
| Hook | At x = 4 m, with a lowest height of 3 m above the deck. |
| Worked examples | (x + 3)² − 14; greatest value 4; the vertex formula. |
| Faded example | 2(x − 5)² − 9, vertex (5, −9). |
| 1. Completing the square with a not 1 | D. A forgets to subtract 18; C halves 12 instead of 6; B flips the sign of h. |
| 2. Least value of a quadratic | 8. k = 20 − 144 ÷ 12. |
| 3. Reading the vertex from vertex form | 2. h = 5, k = −3. |
| 4. Completing the square, the constant | 1. 9 − 8. |
| 5. Greatest height of a model | 8.2. At t = 1.2. |
| 6. Finding a coefficient from the least value | 8. b² = 64. |
| 7. Standard form from the vertex | 34. y = x² − 12x + 34. |
| 8. Equivalent forms | C. It expands to 2x² + 4x − 16, a mirror image; the others equal 2x² − 4x − 16. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Forgot to subtract, flipped the sign, or halved the wrong number. | Expand each option and compare with 2x² − 12x + 7. |
| 20, 2 or 56 in question 2 | Gave the intercept or the x-coordinate, or used the wrong sign for x. | Find x = −b ÷ (2a), then substitute. |
| 8, −8 or −2 in question 3 | Read k or h with the wrong sign. | Write (x − 5) as (x − (+5)) and read h. |
| 9, 5 or 17 in question 4 | Did not subtract what the square added, forgot the 2, or added. | Expand 2(x + 2)² and compare. |
| 1.2, 1 or 9.4 in question 5 | Gave the time, the starting height, or did not square. | Ask: what does the question want, the time or the height? |
| 16, 4 or 64 in question 6 | Stopped early in the equation for b. | Write each line: b² ÷ 4 = 16, b² = 64, b = 8. |
| −2, 36 or 38 in question 7 | Gave k, h² or added instead of subtracting. | Expand (x − 6)² − 2 on one line. |
| A, B or D in question 8 | Expanded an equivalent form wrongly and thought it was a different function. | Expand each option fully and compare. |
Questions worth asking
- Why is the vertex formula only completing the square once? The same two steps, add and subtract half the coefficient squared, are done with letters instead of numbers.
- Which form would you use to find where a ball lands? Its highest point? Its starting height? Factorized for landing (zeros), vertex for the highest point, standard for the starting height.
- Does a change the x-coordinate of the vertex? Yes when b is not 0, because h = −b ÷ (2a); no when b = 0.
- Can every quadratic be put in vertex form? Yes: unlike factorizing, completing the square always works.
Differentiation
A way in for anyone stuck
- Give a four-column card: standard, vertex, factorized, y-intercept, with one worked example filled in.
- Start with a = 1 and only then introduce a not 1; write the x terms in a bracket with the a taken out.
- Provide x = −b ÷ (2a) on a card and have students write a, b and c first.
Stretch
- Show that the line x = −b ÷ (2a) is an axis of symmetry by showing f(h + t) = f(h − t).
- Find the values of k for which y = x² + kx + 25 has its least value at y = 0, and explain what that says about the graph.
Students learning in English as an additional language
“Vertex” is the turning point. “Completing the square” means writing a quadratic as a bracket squared plus a number. “Least value” and “greatest value” are the lowest and highest y. Say “opens upward” and “opens downward”.
| Word | Plain meaning | Example |
|---|---|---|
| vertex form | a(x − h)² + k, which shows the turning point (h, k) | y = (x − 11)² + 10 |
| completing the square | rewriting a quadratic as a squared bracket plus a number | x² + 10x = (x + 5)² − 25 |
| turning point | the vertex, where a parabola changes from falling to rising | the top of a thrown ball's path |
| least value | the lowest y a function takes | k when a > 0 |
| greatest value | the highest y a function takes | k when a < 0 |
| equivalent forms | different expressions that give the same function | x² − 49 and (x − 7)(x + 7) |
Sentence starters for explaining aloud or in writing:
- I took out … first because …
- The vertex is … so the least value is …
- These forms are equivalent because …
Two things not to say
“The number in the bracket is the x-coordinate.” It is the opposite sign: x minus h.
“Completing the square only works when a = 1.” It works for any a once a is taken out of the x terms.
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.
- Vertex formWrite the vertex of y = (x − 6)² + 9.
(6, 9). - Least valueFind the least value of y = x² + 10x + 40.
15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA fountain's jet in Granada has height y = −0.5x² + 3x + 1 metres at distance x metres. Find its greatest height.
x = 3, y = −4.5 + 9 + 1 = 5.5 metres. - Earlier work (Lesson 2)Factorize 6x² + x − 2.
(3x + 2)(2x − 1).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWrite y = 3x² + 12x + 20 in vertex form and say why it has no zeros.
3(x + 2)² + 8; the least value is 8, above the axis. - EquivalenceShow that y = 2(x − 2)² − 18 and y = 2(x − 5)(x + 1) are the same function.
Expand both: 2x² − 8x − 10.
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.