Algebra I — 3-Week Study Plan

Designed from the June 18, 2025 Regents results. A compressed plan targeting the five weakness patterns from the evaluation (conceptual mis-classification, failing to finish the last 20%, off-by-one indexing, graphing fidelity, simplest-form discipline) and the five priority skill areas (multi-step systems, factoring patterns, quadratic formula end-to-end, domain vs. range, arithmetic-sequence indexing).

Assumes ~60 minutes per weekday plus a longer Saturday timed-practice session. Sunday is rest. Three weeks = 21 study sessions in total.

Weekly files (detailed daily instructions + practice problems)

Week 1 — Pattern Recognition + Quick Wins — domain/range, factoring patterns, polynomial arithmetic, Part I timed practice
Week 2 — Quadratics + Sequences End-to-End — discriminants, quadratic formula in simplest radical form, factor-vs-formula method choice, arithmetic & geometric sequences
Week 3 — Systems, Graphing, and Full Simulation — linear-quadratic systems, systems of inequalities, word-to-system translation, full-length Regents simulation

Each weekly file contains the day-by-day plan with concrete practice problems and worked solutions.

How to Use This Plan

Work in pen, not pencil (except graphs). Mirrors test conditions and exposes sloppy steps you’d otherwise erase.
Keep a single “Mistake Log” notebook. Every wrong answer gets one page: the original problem, the wrong step, the correct step, and a sentence in plain English explaining the category of error.
The five heuristics below are the spine of the plan. Apply them to every problem before declaring it done.

The Five Universal Self-Check Heuristics

Tape these to the inside of your binder.

#	Heuristic	What it catches
H1	”Read the prompt twice; circle every noun it asks for.”	Stops Q34-style misses where you find $x$ but forget $y$ .
H2	”Name the structure.” Before computing, say out loud: “This is a difference of squares” / “This is a system” / “This is a domain question.”	Stops Q13 (domain vs range) and Q30 ( $a^2 - b^2$ vs $(a-b)^2$ ) misclassifications.
H3	”Count the gaps, not the dots.” When sequences or intervals appear, draw the dots and count between them.	Stops Q28-style off-by-one on common difference.
H4	”Is this simplest form?” Radical reduced? Fraction reduced? Trinomial in standard order?	Stops Q32-style stop-too-early.
H5	”Re-do the most arithmetic-heavy line.” Pick the line with the most digits and recompute from scratch on the side.	Stops the slip that turned $\sqrt{28}$ into $\sqrt{30}$ .

Week-at-a-Glance Summary

Week 1 →

Day	Focus
Mon	Domain & range rebuild — 10 graph problems
Tue	Factoring patterns — 15 classify + 10 factor
Wed	Polynomial arithmetic + linear equations/inequalities — 14 problems
Thu	Mixed Part-I timed set — 12 MC in 18 min
Fri	Re-do Q13, Q30 + 4 siblings each
Sat	Full Part I + Part II timed (~70 min)

Week 1 success bar: Part I 45+/48, Part II 11+/12, every Mistake Log entry tagged with an H-bucket.

Week 2 →

Day	Focus
Mon	Discriminants + radical simplification — 8 problems + memorize the radical table
Tue	Quadratic formula end-to-end — 6 full problems
Wed	Factor vs. formula method choice — 6 mixed problems
Thu	Arithmetic & geometric sequences with dot-and-gap — 10 problems
Fri	Mixed Part II/III timed — 6 problems in 30 min
Sat	Full Part II + Part III timed (~75 min)

Week 2 success bar: Part II 11–12/12, Part III 10+/16, zero unsimplified quadratic answers, zero off-by-one sequence errors.

Week 3 →

Day	Focus
Mon	Linear-quadratic systems — 6 problems with the completion checklist
Tue	Systems of inequalities (algebra + graph) — 5 problems
Wed	Word-to-system translation + Q35 redo — 4 problems
Thu	Graphing functions from scratch — 8 functions
Fri	Timed Part III + Part IV (~60 min)
Sat	Full-length Regents simulation (3 hours)

Week 3 success bar: Raw score $\geq$ 70 on the full simulation (vs. 62 baseline) — that’s a scale score of ~85+, clearing the Mastery threshold.

Cross-Week Reference Material

The four factoring patterns

Pattern	Looks like	Factors as
GCF	every term shares a factor	pull it out first, always
Difference of squares	$a^2 - b^2$ (two terms, both perfect squares, minus sign)	$(a - b)(a + b)$
Perfect-square trinomial	$a^2 \pm 2ab + b^2$	$(a \pm b)^2$
Standard trinomial	$x^2 + bx + c$	$(x + p)(x + q)$ where $p + q = b,\ pq = c$

The quadratic-formula 6-step checklist

Write $a$ , $b$ , $c$ on their own lines.
Substitute into $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with parentheses around negatives.
Compute discriminant on a separate line. Recompute (H5).
Simplify the radical (largest perfect-square factor).
Reduce the fraction (factor a GCF out of numerator and denominator).
Write both answers (or use $\pm$ ).

Radical-simplification table to memorize

$n$	$\sqrt{n}$	$n$	$\sqrt{n}$
8	$2\sqrt{2}$	32	$4\sqrt{2}$
12	$2\sqrt{3}$	45	$3\sqrt{5}$
18	$3\sqrt{2}$	48	$4\sqrt{3}$
20	$2\sqrt{5}$	50	$5\sqrt{2}$
27	$3\sqrt{3}$	72	$6\sqrt{2}$
28	$2\sqrt{7}$	75	$5\sqrt{3}$

The sequence dot-and-gap fix

$a_1 \xrightarrow{d} a_2 \xrightarrow{d} a_3 \xrightarrow{d} a_4 \xrightarrow{d} a_5 \xrightarrow{d} a_6 \xrightarrow{d} a_7 \xrightarrow{d} a_8$

There are 7 gaps between $a_1$ and $a_8$ , not 8. The formula $a_n = a_1 + (n - 1)\,d$ uses that gap count.

Linear-quadratic system completion checklist

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?

Graphing rubric — what graders actually check

Element	Required for full credit
Axes labeled	" $x$ " and " $y$ " (or context labels like “Hours Tutoring”)
Scale shown	Tick marks at consistent intervals
Line/curve drawn correctly	Vertex, intercepts, and at least 2 plotted points
Arrows on rays	Both ends of every line, or both ends of every parabola branch
For inequalities: shading	Correct side of each boundary, with the overlap (feasible region) visually distinct
For inequalities: solid vs dashed	Solid for $\leq$ or $\geq$ , dashed for $<$ or $>$
Labels on each graphed object	" $f(x)$ " or " $y = -3x$ " near the line

Final Self-Assessment Rubric

Score yourself at the end of each week. Move forward only when every row is “Comfortable” or higher.

Skill	Shaky	OK	Comfortable	Automatic
I name the structure of a factoring problem on sight
I always reduce radicals and fractions to simplest form
I never confuse domain ( $x$ ) with range ( $y$ )
I count gaps, not dots, in sequences
I back-substitute to find every requested variable
My graphs are labeled, scaled, with correct line types and shading
I re-do the most arithmetic-heavy line of every problem
I read the prompt twice and circle every noun it asks for

Realistic Projection After 3 Weeks

If the plan is followed faithfully:

Part I: 22/24 → 23/24 (one Q13/Q21-type slip removed via H2)
Part II: 10/12 → 12/12 (Q28 and Q30-type slips removed via H3 and factoring drills)
Part III: ~5/16 → 10–12/16 (the biggest swing — end-to-end discipline and the completion checklist)
Part IV: ~3/6 → 4–5/6 (graphing rubric)

Projected raw score: $62 \to \sim 72$ . Projected scale score: $80 \to \sim 87$ , clearing the Mastery threshold (85).

The largest gains aren’t from learning new content — they’re from finishing problems you already mostly know how to start.

Companion Documents

Original exam evaluation — per-question rationale for every June 25 item, with the heuristic mapping that produced this plan. (Not yet published on this site.)
Week 1 | Week 2 | Week 3