Week 3 — Systems, Graphing, and Full Simulation

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Goal this week: convert the biggest point-loss area on the June 25 exam (Part III multi-step systems and Part IV graphing) into reliable points. End with a full-length simulation.

Time: ~60 min weekdays, ~3 hours Saturday.

Heuristics in heavy use: H1 (read prompt twice, circle every noun), H4 (simplest form), plus the graphing rubric checklist.

Monday — Linear-Quadratic Systems

Objective

Solve linear-quadratic systems all the way through to ordered pairs $(x, y)$ . The June 25 Q34 lost a point because $x$ -values were found but $y$ -values were missed or had wrong signs.

The completion checklist (use on EVERY problem)

Did I find every $x$ value the quadratic produces?
Did I back-substitute each $x$ into the linear equation (it’s easier)?
Did I keep the negative signs through back-substitution?
Did I write my final answer as ordered pairs $(x, y)$ , not just $x$ values?

Practice problems

Solve each system algebraically. Report all solutions as ordered pairs.

1. $\begin{cases} y = x^2 - 5x + 6 \\ y = x + 1 \end{cases}$

2. $\begin{cases} y = x^2 + 9x + 4 \\ y - 2x = -6 \end{cases}$ (Q34)

3. $\begin{cases} y = x^2 - 4 \\ y = 3x \end{cases}$

4. $\begin{cases} y = 2x^2 + 3x - 1 \\ y = x + 2 \end{cases}$

5. $\begin{cases} y = -x^2 + 4x \\ y = -x + 4 \end{cases}$

6. $\begin{cases} y = x^2 + 2x - 8 \\ y = 2x + 1 \end{cases}$

Answers

1. $(1, 2)$ and $(5, 6)$ . 2. $(-5, -16)$ and $(-2, -10)$ . 3. $(4, 12)$ and $(-1, -3)$ . 4. $(1, 3)$ and $(-\tfrac{3}{2}, \tfrac{1}{2})$ . 5. $(1, 3)$ and $(4, 0)$ . 6. $(3, 7)$ and $(-3, -5)$ .

Worked solution for #2 (the Q34 redo)

H1: prompt asks for $x$ and $y$ .

Linear equation: $y - 2x = -6 \implies y = 2x - 6$ .

Substitute into the quadratic: $2x - 6 = x^2 + 9x + 4$ .

$0 = x^2 + 7x + 10 \implies 0 = (x + 5)(x + 2)$ .

$x = -5$ or $x = -2$ .

Back-substitute into the linear equation (H3 — keep signs):

$x = -5 \implies y = 2(-5) - 6 = -10 - 6 = -16$
$x = -2 \implies y = 2(-2) - 6 = -4 - 6 = -10$

$\boxed{(-5,\, -16) \text{ and } (-2,\, -10)}$

Worked solution for #5 (downward parabola)

Set equal: $-x^2 + 4x = -x + 4 \implies 0 = x^2 - 5x + 4 \implies 0 = (x - 1)(x - 4)$ .

$x = 1$ or $x = 4$ .

Back-substitute into $y = -x + 4$ :

$x = 1 \implies y = 3$
$x = 4 \implies y = 0$

$\boxed{(1, 3) \text{ and } (4, 0)}$

Self-check question

After solving the quadratic, which equation should you back-substitute into and why? (The linear equation — arithmetic is simpler and less error-prone than re-squaring.)

Tuesday — Systems of Inequalities (Algebra + Graph)

Objective

Graph a system of inequalities with correct line types, correct shading, and labels — the three things graders specifically check.

Graph rubric (every problem must satisfy this)

Element	Required
Axes labeled	" $x$ " and " $y$ " or context labels
Scale	tick marks at consistent intervals
Line	solid for $\leq$ or $\geq$ ; dashed for $<$ or $>$
Plotted points	at least 2 per line, both intercepts when possible
Arrows	both ends of every infinite line
Shading	correct side of each boundary; overlap distinctly visible
Labels	each inequality named on its line (e.g., " $y \leq 2x + 1$ ")

Workflow

Convert each inequality to slope-intercept form.
Decide solid vs dashed from the inequality sign.
Plot the line.
Test the point $(0, 0)$ — if it satisfies, shade that side; if not, shade the other side.
Label the line.
Identify the overlap and outline it.

Practice problems

Graph each system. Show your work on each step.

#	System
1	$y \leq 2x + 1$ and $y > -x + 3$
2	$y \geq x - 2$ and $y \leq -2x + 8$
3	$x + y < 10$ and $y > 2x - 1$
4	$2x + y \leq 12$ and $x \geq 0$ and $y \geq 0$
5	$3x - y > 6$ and $y \geq -x + 2$

Worked solution for #2

Inequality A: $y \geq x - 2$ . Solid line, slope 1, $y$ -intercept $-2$ . Test $(0, 0)$ : $0 \geq -2$ ✓ — shade where $(0,0)$ lives (above the line).

Inequality B: $y \leq -2x + 8$ . Solid line, slope $-2$ , $y$ -intercept $8$ . Test $(0, 0)$ : $0 \leq 8$ ✓ — shade below the line.

Overlap: the region above the first line AND below the second. Outline that region in pen.

Label each line with its inequality near a prominent part of the line.

Self-check question

What test point should you use to determine which side to shade? ( $(0, 0)$ is almost always easiest — unless the line passes through the origin, in which case use $(1, 0)$ or $(0, 1)$ .)

Wednesday — Word-to-System Translation + Q35 Redo

Objective

Translate two-quantity word problems into a full system: inequalities + graph + valid combination + justification. Then re-do Q35 end-to-end.

Translation template

Every two-quantity word problem decomposes into:

Money / value inequality: $\text{rate}_x \cdot x + \text{rate}_y \cdot y\ [\geq,\ \leq,\ =]\ \text{total}$
Quantity / capacity inequality: $x + y\ [\leq,\ =]\ \text{cap}$
Domain limits if $x, y$ are real-world counts: $x \geq 0$ , $y \geq 0$

Solution template (4 deliverables)

System of inequalities written clearly.
Graph following the Tuesday rubric.
Combination — a point in the feasible region. Verify both inequalities numerically.
Justification — a sentence stating which inequalities are satisfied and why the combination is valid.

Practice problems

1. Marcus sells lemonade at $2/cup and cookies at $3/cookie. He wants to earn at least $60 at the fair. He can carry max 40 items total. Let $x$ = cups, $y$ = cookies. Write the system, graph it, state a valid combination, justify.

2. A bookstore sells novels for $15 and cookbooks for $25. To break even today they must earn at least $300. They have shelf space for at most 30 books. Let $x$ = novels, $y$ = cookbooks. System, graph, combination, justification.

3. Sarah’s babysitting + tutoring problem (Q35 redo): $6/hr babysitting ( $x$ ), $12/hr tutoring ( $y$ ), at least $120/week, max 14 hours total. System, graph, combination, justification.

4. A coffee shop has two drinks: lattes ($4) and teas ($3). The owner needs to sell at least $200 worth in an hour. The shop can make at most 60 drinks per hour. Let $x$ = lattes, $y$ = teas. System, graph, combination, justification.

Worked solution for #3 (Q35 redo)

Full worked solution

System: $\begin{cases} 6x + 12y \geq 120 & \text{(money: at least \$120)} \\ x + y \leq 14 & \text{(time: at most 14 hours)} \\ x \geq 0,\ y \geq 0 & \text{(non-negative hours)} \end{cases}$

Convert for graphing:

$6x + 12y \geq 120 \implies y \geq -\tfrac{1}{2}x + 10$ . Solid line. Test $(0, 0)$ : $0 \geq 10$ ? No — shade above (away from origin).
$x + y \leq 14 \implies y \leq -x + 14$ . Solid line. Test $(0, 0)$ : $0 \leq 14$ ? Yes — shade below (toward origin).

Graph the feasible region as the intersection of those two shaded regions, restricted to the first quadrant ( $x \geq 0$ , $y \geq 0$ ).

Combination: $x = 4$ babysitting hours, $y = 10$ tutoring hours.

Money: $6(4) + 12(10) = 24 + 120 = 144 \geq 120$ ✓
Hours: $4 + 10 = 14 \leq 14$ ✓

Justification: “Working 4 hours babysitting and 10 hours tutoring satisfies both constraints. Sarah earns $144 (which meets the $120 goal) and works exactly 14 hours (within the 14-hour limit).”

Self-check question

How do you decide whether to shade above or below a boundary line? (Pick a test point not on the line — usually $(0, 0)$ . If it satisfies the inequality, shade that side; otherwise, the other side.)

Thursday — Graphing Functions from Scratch

Objective

Drill graphing fidelity. Every graph must satisfy the rubric: labeled axes, scale, plotted points, arrows, and (for inequalities) shading.

Instructions

Graph all 8 functions on separate coordinate grids.
For each, plot at least 3 points and the key feature (intercept for lines, vertex for parabolas, etc.).
Apply arrows on both ends of every infinite curve.
Self-check against the rubric checklist after each one.

Practice — graph these

#	Function	Key features
1	$y = -3x + 1$	$y$ -int $(0, 1)$ ; slope $-3$ ; passes through $(1, -2)$
2	$f(x) = 2x - 4$	$y$ -int $(0, -4)$ ; $x$ -int $(2, 0)$
3	$y = x^2 - 4$	vertex $(0, -4)$ ; $x$ -ints $(\pm 2, 0)$ ; opens up
4	$g(x) = -(x - 1)^2 + 4$	vertex $(1, 4)$ ; opens down; $x$ -ints $(-1, 0)$ and $(3, 0)$
5	$y = \\|x - 2\\|$	V-shape, vertex $(2, 0)$ , opens up
6	$y = \\|x\\| + 3$	V-shape, vertex $(0, 3)$ , opens up
7	$f(x) = -3x$ and $g(x) = x^2 + 2$ on same axes (Q31!) — find intersection visually	intersections at $(-1, 3)$ and $(-2, 6)$
8	$y = (x + 2)(x - 4)$	$x$ -ints $(-2, 0)$ and $(4, 0)$ ; vertex at $x = 1$ , $y = -9$

Worked plotting plan for #7 (Q31 redo)

Line $f(x) = -3x$ :

Passes through $(0, 0)$ .
Slope $-3$ : from $(0, 0)$ go right 1, down 3 to $(1, -3)$ .
Other points: $(-1, 3)$ , $(-2, 6)$ .
Draw as a straight line with arrows on both ends.

Parabola $g(x) = x^2 + 2$ :

Vertex at $(0, 2)$ .
Symmetric points: $(\pm 1, 3)$ , $(\pm 2, 6)$ , $(\pm 3, 11)$ .
Smooth U-shape opening up. Arrows on both branches.

Intersection points (visually and algebraically):

$(-1, 3)$ and $(-2, 6)$ — both curves pass through these.
Algebraically: $-3x = x^2 + 2 \implies x^2 + 3x + 2 = 0 \implies (x+1)(x+2) = 0$ .

State: $x = -1$ and $x = -2$ .

Self-check rubric

After each graph, score yourself:

☐ Both axes labeled
☐ Scale visible (tick marks)
☐ Key feature plotted (vertex / intercept)
☐ At least 2 other points plotted
☐ Arrows on infinite ends
☐ Line/curve labeled with its equation

A graph that misses any of these would lose points on the Regents.

Friday — Timed Part III + Part IV

Objective

Final dress rehearsal before Saturday’s full simulation. Apply every checklist under timed pressure.

Instructions

Pull Part III (Q31–Q34) and Part IV (Q35) from a prior Regents administration.
Timer: 60 minutes.
Work in pen. Use the linear-quadratic completion checklist on every system problem.
Use the graphing rubric on every graph.
Score strictly against the Model Response Set.

Self-score targets

Part III: 12+ / 16
Part IV: 5+ / 6

What to do with misses

For every lost point, identify which checklist item failed:

Did you skip the back-substitution step? (Completion checklist #2)
Did you forget a negative sign? (Checklist #3)
Did you give $x$ -values only? (Checklist #4)
Did you forget to shade or label a graph? (Graphing rubric)
Did you forget the justification on Q35-type problems? (Solution template #4)

Each of these is a process failure, not a knowledge failure. Re-do the problem on Saturday morning before the full simulation.

Saturday — Full-Length Regents Simulation (3 hours)

Objective

Realistic, end-to-end measurement of progress.

Instructions

Setup: print a complete prior Regents Algebra I (any released year). Sit at a clear table. No phone, no distractions.
Timer: 3 hours, no break. Bring a graphing calculator, ruler, pen, and pencil (graphs only).
Work the test in order. Apply every heuristic and checklist from the past three weeks.
Score immediately afterward against the Model Response Set.

Score breakdown to record

Part	Possible	Earned	vs. June 25 baseline
I (MC)	48	__	44
II (2-pt)	12	__	10
III (4-pt)	16	__	~5
IV (6-pt)	6	__	~3
Raw total	82	__	62

Success bar for Week 3

Raw score $\geq$ 70 (up from 62) — that’s a scale score of ~85+, the Mastery threshold.
Zero “didn’t finish” errors on multi-step problems.
Zero graphing rubric misses.

Post-test reflection

Write 1–2 paragraphs in the Mistake Log:

What worked? Which heuristics fired automatically?
What still needs work? Which patterns showed up multiple times?
What’s the next focus? Based on remaining gaps.

If raw score is below 70, repeat the day(s) of the corresponding weak area before retaking the test.

After Week 3

You’ve completed 21 sessions of focused work. The next 2–3 weeks before the actual Regents should be:

One full-length simulation per week under timed conditions.
Daily 20-minute mistake-log review — flip through the entries and re-do any problem you no longer instantly see the solution to.
Targeted drills on whatever pattern shows up most in your latest simulation.

The pattern is durable: every Regents mistake fits into one of the H1–H5 buckets. If you keep tagging and reviewing, the failure modes get harder and harder to repeat.

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