MATH 130-002 · Monday, August 31, 2026 · 50 minutes
Split the middle term.
Factor by grouping.
Monday’s core is factoring quadratics with leading coefficient not 1. Teach the ac method explicitly: find a pair with product ac and sum b, split the middle term, factor each group, and factor out the common binomial. Do not skip from the quadratic to its answer. Slides 1–20 are the core; slides 21–24 are optional follow-up. Keep this answer guide off the projector.
What changed
The earlier plan allocated too little time to this prerequisite and moved to common denominators too soon. The revised lesson includes two full models, a GCF-first model, guided practice, partner explanation, an independent attempt, and a written exit. Rational-expression simplification, LCDs, and addition are now optional follow-up. This is a change in lecture emphasis; the published Quiz 1 scope and Canvas dates have not been changed.
Prepare once; save paper
- Replace the old E1, E2, and H1 Top Hat questions with the three below. The IDs are reused but the content and keys have changed. Create single-select questions, preserve option order, and hide results until voting closes. Follow the course’s existing participation policy. These questions are prepared here, not uploaded into Top Hat.
- No course printing is required if students use their own notebooks or annotate/fill the PDF. Every problem is also on the slides. If you want a collected sheet, print the two-page PDF double-sided, long-edge flip: one sheet per student, including the exit check. For 40 students, that is 40 sheets plus any spares. Do not print a separate exit ticket.
- The PDF has 12 fillable fields. Students download, fill, and save a copy; typing
x^2and(3x+1)(2x-3)is fine. The PDF does not submit automatically. Use an existing submission channel if one exists; none has been created. If collecting paper, allow a photo of notes first and return the sheet next class. - Launch the HTML deck or PowerPoint. PowerPoint Presenter View keeps speaker notes separate; the HTML Notes button shows notes on the shared screen. Keep this guide on a second screen. No new account setup or in-class ALEKS switch is planned.
Controls: arrows/Space move; R reveals a key where available; N opens notes; T starts/pauses the timer; F enters full screen; End goes to slide 20. The timers are work prompts, not a synchronized 50-minute class clock. Top Hat responses are not collected through the deck.
Why this focus fits the evidence
Your explicit teaching priority is the deciding factor. The August 31 ALEKS snapshot also shows a need for grouping and nonmonic quadratic factoring:
| ALEKS topic | Progress | Instructional response |
|---|---|---|
| Factoring a univariate polynomial by grouping: Problem type 2 | 33% | Show the common binomial and the outside coefficients, including 1 and −1. |
| Factoring a quadratic with leading coefficient greater than 1: Problem type 1 | 46% | Use product ac and sum b; require the split and grouping lines. |
| Factoring out a constant before factoring a quadratic | 31% | Remove the GCF first and retain it through every line. |
Source: user-provided itemsReport.xlsx, class_progress!B42:C42,B44:C45, generated August 31, 2026 at 2:15 PM. These are ALEKS progress values, not percent correct on a common assessment. They do not establish that Friday caused a gain. R.1–R.3 aggregate progress is 78.05%; R.4–R.5 is 51.62% (C5,C31).
The supplied DetailedProgress (1).xlsx has 42 history records for 40 unique students; history records are not additional students, and missing entries are not zeros. No student identities are included here. Friday’s instructor-reported entrance results were 33/37 on interval notation, 27/37 on factoring, and 30/37 on quotient exponents. Those results do not establish mastery of this specific factoring method.
50-minute run of show
| Time | Slides | Action |
|---|---|---|
| 0–4 | 2–3 | E1 and E2: write, then vote. Product/sum and reverse distribution. Keys C/A. |
| 4–6 | 4 | Debrief; introduce GCF → ac → split → group → factor → check. |
| 6–13 | 5–7 | W1: model 6x² + 11x + 3, showing every line. |
| 13–18 | 8–9 | P1: guided attempt on 6x² + 7x + 2, then check. |
| 18–20 | 10 | H1: choose a valid split for 6x² + x − 2. Key B. |
| 20–26 | 11–12 | W2: group with a negative factor; distribute −1 to check signs. |
| 26–30 | 13 | W3: GCF first, then split and group the remaining quadratic. |
| 30–35 | 14–15 | P2: partner practice combining the GCF and negative signs. |
| 35–41 | 16–17 | P3: independent full-method attempt; inspect work and debrief. |
| 41–46 | 18 | X1: independent factoring with every step and an expansion check. |
| 46–49 | 19 | X2: correct a sign error, explain it, and finish factoring. |
| 49–50 | 20 | Collect work if desired; remind students of Wednesday’s unchanged dates. |
If time slips: shorten partner discussion by at most two minutes. Protect the independent attempt and the exit. Do not add an optional rational-expression slide on top of the 50 minutes.
What to say at the difficult steps
- “The two numbers must multiply to ac, not just c, and add to b.”
- “We have not changed the polynomial: 9x + 2x is still 11x.”
- “Each group must contain the exact same binomial, including its signs.”
- “A binomial by itself has a coefficient of 1.”
- “Factoring out −1 changes both signs. Distribute it back to check.”
- “After taking out a GCF, compute ac using the quadratic that remains.”
Accept either order of the split when the work is correct. For example, 6x² + 7x + 2 can split as 3x + 4x or 4x + 3x. A different common binomial during grouping can still produce the same final factors. If no integer pair exists after a systematic search, do not force one; all core examples today do factor over the integers.
Top Hat: exactly three questions
Use single-select multiple choice. Keep the labels and option order below. Students write a short attempt before voting. Do not reveal the key on the projected slide until the vote closes.
E1 · Product and sum · Slide 2
Prompt: Which pair of numbers has product 18 and sum 11?
- 6 and 3
- 12 and −1
- 9 and 2
- −9 and −2
Correct: C. 9 × 2 = 18 and 9 + 2 = 11.
Timing: 60 seconds of work, then vote; two minutes total. A checks only the product. B checks only the sum. D misses the required sum/sign. If students struggle, list factor pairs of 18 before introducing the split.
E2 · Reverse distribution · Slide 3
Prompt: Factor out the common binomial: 3x(2x + 3) + (2x + 3).
- (3x + 1)(2x + 3)
- 3x(2x + 3)
- 4x(2x + 3)
- (3x)(1)(2x + 3)
Correct: A. The outside coefficients are 3x and 1; add them.
Timing: 60 seconds of work, then vote; two minutes total. B loses the last term. C combines unlike terms. D multiplies the outside coefficients instead of adding them. If needed, substitute a box: 3x□ + 1□ = (3x + 1)□.
H1 · Valid middle-term split · Slide 10
Prompt: To factor 6x² + x − 2 by grouping, which split uses two numbers whose product is ac = −12 and whose sum is b = 1?
- 6x² + 3x − 2x − 2
- 6x² + 4x − 3x − 2
- 6x² − 4x + 3x − 2
- 6x² + 6x − 5x − 2
Correct: B. 4 × (−3) = −12 and 4 + (−3) = 1.
Timing: 60 seconds of work/voting, then one minute to explain. A and D are equivalent rewrites of the original polynomial but their split coefficients fail the ac condition. C has the required product but the wrong middle coefficient.
Decision: at 85% or more correct, ask for one justification and continue. At 70–84%, brief partner explanation and revote if time permits. Below 70%, list signed factor pairs during W2 and shorten the later partner discussion if necessary. Use the number who actually responded, not a fixed denominator. These are teaching thresholds, not a new grading policy.
Written exit: keep the working visible
Use the bottom of side two, a saved digital copy, or students’ own paper. Cover earlier work. No choices and no revealed answers until collection. No additional printed exit ticket is needed.
X1 · Full method · Slide 18 · Five minutes
Prompt: Factor 6x² − 7x − 3 by splitting the middle term and then grouping. Show every step. Expand to check.
ac = −18; sum = −7; pair = −9 and 2 6x² − 7x − 3 = 6x² − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (3x + 1)(2x − 3) Check: 6x² − 9x + 2x − 3 = 6x² − 7x − 3
Inspect: correct product/sum pair; an equivalent four-term line; valid GCFs for both groups; the common binomial factored out; expansion back to the original. Accept the reverse split order. A final answer alone does not establish that the required method was learned.
X2 · Negative-factor error analysis · Slide 19 · Three minutes
Prompt: A student writes 6x² − x − 2 = 3x(2x + 1) − 2(2x − 1). Correct the grouping line, then factor. Explain why the sign must change.
Correct grouping: 3x(2x + 1) − 2(2x + 1) Final product: (3x − 2)(2x + 1) Reason: −2(2x + 1) = −4x − 2. The incorrect −2(2x − 1) produces −4x + 2.
Inspect: correct inner sign, distributive explanation, and identical binomial factored out. These observations guide follow-up; they are not a new points policy. A strong X1 with a weak X2 needs sign repair; a weak pair or split needs more ac practice; a correct split with mismatched groups needs reverse-distribution practice. Two exit items do not establish mastery of all R.1–R.5 topics.
Worked-example and practice key
| Item / slides | Pair after GCF | Grouping and final form |
|---|---|---|
| W1 / 5–7 6x² + 11x + 3 | ac = 18; 9, 2 | 3x(2x + 3) + 1(2x + 3) = (3x + 1)(2x + 3) |
| P1 / 8–9 6x² + 7x + 2 | ac = 12; 3, 4 | 3x(2x + 1) + 2(2x + 1) = (3x + 2)(2x + 1) |
| W2 / 11–12 6x² + x − 2 | ac = −12; 4, −3 | 2x(3x + 2) − 1(3x + 2) = (2x − 1)(3x + 2) |
| W3 / 13 6x² + 15x + 9 | GCF 3; ac = 6; 2, 3 | 3[2x(x + 1) + 3(x + 1)] = 3(2x + 3)(x + 1) |
| P2 / 14–15 6x² − 15x + 9 | GCF 3; ac = 6; −2, −3 | 3[2x(x − 1) − 3(x − 1)] = 3(2x − 3)(x − 1) |
| P3 / 16–17 4x² − 4x − 3 | ac = −12; −6, 2 | 2x(2x − 3) + 1(2x − 3) = (2x + 1)(2x − 3) |
After class and assessment alignment
Canvas remains the master: ALEKS R.1–R.5 homework and Quiz 1 are Wednesday, September 2, at 2:10 PM; the quiz is in class on paper. No due dates have changed. Homework is completed in ALEKS.
Direct students to these available ALEKS topics in order: Factoring a univariate polynomial by grouping: Problem type 2; Factoring a quadratic with leading coefficient greater than 1: Problem type 1; and Factoring out a constant before factoring a quadratic. Topic availability can differ by readiness, so use the corresponding Unit 1A practice when a target is not yet available. Then use the full R.1–R.5 review/checkpoint; do not imply today’s factoring sheet replaces the full assignment.
Coverage check before Quiz 1: today’s core does not teach LCDs or rational-expression addition. Compare the planned quiz with the instruction and practice students have actually had. Adjust item coverage or provide the missing preparation if needed, while retaining the Canvas date. The existing published R.1–R.5 coverage has not been silently narrowed by this revision.
Optional follow-up slides and keys
- Simplify (6x² + 11x + 3)/(2x² + 5x + 3): (3x + 1)/(x + 1), with x ≠ −3/2, −1. Keep the restriction from the canceled factor.
- For denominators 2x² + 5x + 3 and 2x² + x − 3, the LCD is (2x + 3)(x + 1)(x − 1). Their common factor is needed once.
- Add 1/(2x² + 5x + 3) + 1/(2x² + x − 3): 2x/[(2x + 3)(x + 1)(x − 1)], with x ≠ −3/2, −1, 1.
- Factor 12x² − 2x − 4: 2(3x − 2)(2x + 1). Take out 2 first; then use the pair 3, −4.
Teaching examples are original. Data sources are the instructor-provided reports named above. No identifiable student data are embedded. Filenames retain “CommonDenominators” so existing links keep working; titles and content now reflect the factoring lesson.