MarkScanner

Worked example · Grade 11 Functions (MCR3U) style

Vertex form, zeros, domain, and range: four students, one rubric

Two students get full marks by different methods. One forgets the ± and loses a zero. One reverses the direction of the parabola and loses two marks for one reason. Here is how a rubric, the teacher’s rulings, and specific feedback turn the same question into four different next steps.

By Oscar ONeill ·

The question

What the students were asked

Quiz question

For f(x) = −2(x − 1)2 + 8:
(a) State the vertex and whether it is a maximum or a minimum.
(b) Find the zeros.
(c) State the domain and range.

[6 marks]

Teacher's answer key

  1. (a) Vertex (1, 8). a = −2 < 0, so it is a maximum.
  2. (b) −2(x − 1)2 + 8 = 0
  3.     (x − 1)2 = 4
  4.     x − 1 = ±2, so x = 3 or x = −1
  5. (c) Domain {x ∈ ℝ}. Range {y ∈ ℝ | y ≤ 8}

The rubric

The marking criteria the teacher approved

MarkScanner drafts a rubric from the teacher’s test and answer key. The teacher edits it and approves it, and the final marks follow it. This is one reasonable rubric for this question, not the only one.

Rubric items and marks
Item What earns the mark Marks
a Vertex (1, 8), identified as a maximum. 1
b1 Sets f(x) = 0 and isolates the squared term, or expands correctly to standard form. 1
b2 Takes the square root with ±, or factors correctly. 1
b3 Both zeros: x = −1 and x = 3. 1
c1 Domain: all real numbers. 1
c2 Range: y ≤ 8. 1

Decisions the teacher made before marking

  • Any valid method earns b1 and b2. Expanding and factoring is as good as taking square roots.
  • Set notation isn’t required on this quiz: “all real numbers” and “y ≤ 8” are accepted.
  • Follow-through does not cover a missing zero. Item b3 needs both values.

Another teacher could reasonably decide differently. What matters is deciding before marking, so every paper gets the same decision.

Four responses

The students' work, the draft marks, and the teacher's review

Each response is transcribed line by line. Under it: the marks for each rubric item, and what happened when the teacher reviewed the draft. In MarkScanner, nothing is final until the teacher approves it.

Lena

Square roots with ±

Final mark: 6/6

Student working, line by line:

  1. (a) (1, 8), max because a < 0
  2. (b) −2(x − 1)2 = −8
  3.     (x − 1)2 = 4
  4.     x − 1 = ±2
  5.     x = 3 or x = −1
  6. (c) D: {x ∈ ℝ}, R: {y ∈ ℝ | y ≤ 8}
Marks by rubric item for Lena
a Earned 1/1
b1 Earned 1/1
b2 Earned 1/1
b3 Earned 1/1
c1 Earned 1/1
c2 Earned 1/1
Teacher review

No flag. The teacher agreed with the draft.

Omar

Expands, then factors

Final mark: 6/6

Student working, line by line:

  1. (a) vertex (1, 8), maximum
  2. (b) f(x) = −2(x2 − 2x + 1) + 8
  3.     = −2x2 + 4x + 6
  4.     = −2(x − 3)(x + 1)
  5.     zeros: x = 3, x = −1
  6. (c) domain all real numbers, range y ≤ 8
Marks by rubric item for Omar
a Earned 1/1
b1 Valid method: expands correctly. 1/1
b2 Factors correctly. 1/1
b3 Earned 1/1
c1 Earned 1/1
c2 Earned 1/1
Teacher review

Flagged because the method differs from the answer key. The teacher confirmed the expansion and factoring are correct and kept every mark.

Tess

Square roots, missing ±

Final mark: 4/6

Student working, line by line:

  1. (a) (1, 8) maximum
  2. (b) (x − 1)2 = 4
  3.     x − 1 = 2 ±2
  4.     x = 3
  5. (c) D: all reals, R: y ≤ 8
Marks by rubric item for Tess
a Earned 1/1
b1 Earned 1/1
b2 Square root taken without ±. 0/1
b3 Only one zero. Changed by the teacher, under the ruling on missing zeros. 0/1
c1 Earned 1/1
c2 Earned 1/1
Teacher review Teacher changed the draft

In this example, the draft gave b3 by follow-through, because x = 3 follows from Tess’s work. The teacher’s ruling says follow-through doesn’t cover a missing zero, so the teacher changed b3 to 0.

Kai

Correct zeros, direction of opening reversed

Final mark: 4/6

Student working, line by line:

  1. (a) (1, 8) minimum a < 0
  2. (b) (x − 1)2 = 4
  3.     x − 1 = ±2
  4.     x = 3, −1
  5. (c) D: {x ∈ ℝ}, R: {y ∈ ℝ | y ≥ 8}
Marks by rubric item for Kai
a Coordinates correct, but it’s a maximum. 0/1
b1 Earned 1/1
b2 Earned 1/1
b3 Earned 1/1
c1 Earned 1/1
c2 Range should be y ≤ 8. 0/1
Teacher review

No flag. The teacher agreed with the draft, and wrote one comment for both lost marks because they share a cause.

One idea, two lost marks

Kai lost marks in (a) and (c), but both come from one misunderstanding about what a negative a does. Two separate comments would read like two problems. One comment naming the shared cause gives Kai one thing to fix, and his own correct zeros give him a way to check it.

A ruling the draft got wrong

Follow-through is usually generous to students, and the draft applied it to Tess. But the teacher had decided before marking that a missing zero isn’t covered. The rubric and the rulings are the teacher’s, so the teacher’s decision wins, and it applies to every paper the same way.

Feedback

What each student reads

A mark says how much. Feedback should say what worked, where to look again, and one thing to try next. MarkScanner drafts comments in this pattern; the teacher edits them before anything is printed. These are written for this example.

Lena

6/6

What worked
Complete and clearly set out, and you said why the vertex is a maximum.
Where to look again
Nothing to fix on this question.
What to try next
Find the zeros a second way, by expanding and factoring, and confirm you get the same values.

Omar

6/6

What worked
Expanding and factoring is a valid route, and every line is correct.
Where to look again
Nothing to fix on this question.
What to try next
When a function is already in vertex form, try isolating the square and taking ± square roots. It’s often quicker.

Tess

4/6

What worked
Your vertex, domain, and range are all correct, and you isolated the square cleanly.
Where to look again
(b): (x − 1)2 = 4 has two solutions, because 22 and (−2)2 both equal 4. You found x = 3 but not the other zero.
What to try next
Write ± every time you take a square root in an equation. Then check each zero by substituting it into f(x).

Kai

4/6

What worked
Your zeros are correct, and your vertex coordinates are right.
Where to look again
(a) and (c) come from one idea: the −2 in front means the parabola opens down. So (1, 8) is the highest point, a maximum, and the range is y ≤ 8.
What to try next
Use your own zeros as a check. They’re at y = 0, below the vertex at y = 8, which can only happen if the vertex is a maximum. A quick sketch shows it.

After the quiz

A follow-up question for fresh evidence

A corrected answer shows a student can fix this question. A new question with different numbers shows whether the idea has landed. The teacher gives this one in class after the papers go back.

For g(x) = 3(x + 2)2 − 12, state the vertex and whether it is a maximum or minimum, find the zeros, and state the range.

Show the answer

Vertex (−2, −12), a minimum. (x + 2)2 = 4, so x + 2 = ±2: x = 0 or x = −4. Range y ≥ −12.

This one opens up, so Kai has to reason about the sign of a instead of remembering the last answer. Tess has to write ± again to find both zeros.

More on this routine: math test corrections that give you new evidence.

What this doesn't show

One question is one piece of evidence

A marked question shows what a student wrote on one day. It doesn’t show everything they understand, and two students with the same mark can need very different help. The conversation in class and the next piece of work still matter.

Real student work is also messier than these transcriptions. When MarkScanner can’t read something clearly, or its draft looks doubtful, it flags the answer for you. An answer without a flag can still be wrong, so the teacher’s review is part of the workflow, not an extra.

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