Week 1 — Pattern Recognition + Quick Wins

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Goal this week: lock in the easy point-recovery items — domain/range, factoring patterns, and the universal heuristics. Build the Mistake-Log habit.

Time: ~60 min weekdays, ~75 min Saturday. Sunday off.

Heuristics to apply daily: H1 (read prompt twice), H2 (name the structure), H4 (simplest form), H5 (re-do heavy arithmetic).

Monday — Domain & Range Rebuild

Objective

Burn into muscle memory: domain lives on the $x$ -axis, range lives on the $y$ -axis. This single error cost a point on Q13.

Instructions

Set up your binder with a fresh Mistake Log notebook.
For each of the 10 graph descriptions below, sketch the graph on graph paper. Use green pen for the $x$ -axis extent and red pen for the $y$ -axis extent.
Write the domain and range underneath, using interval notation.
After all 10, check against the answer key. For every miss, write a Mistake Log entry tagged “H2 — domain/range swap.”

Practice problems

For each function $f(x)$ , state the domain and range.

#	Description	Domain	Range
1	Line from $(-3, -2)$ to $(5, 6)$ , both endpoints closed	$-3 \leq x \leq 5$	$-2 \leq y \leq 6$
2	Line from open $(0, 1)$ to closed $(10, 21)$	$0 < x \leq 10$	$1 < y \leq 21$
3	Parabola $y = x^2$ , all real $x$	$\mathbb{R}$ (all reals)	$y \geq 0$
4	Parabola $y = -x^2 + 4$	$\mathbb{R}$	$y \leq 4$
5	Open dot at $(-4, 2)$ , closed dot at $(8, 11)$ , line between	$-4 < x \leq 8$	$2 < y \leq 11$
6	$y = \sqrt{x}$	$x \geq 0$	$y \geq 0$
7	$y = \\|x\\|$	$\mathbb{R}$	$y \geq 0$
8	Horizontal line $y = 5$ for $-2 \leq x \leq 2$	$-2 \leq x \leq 2$	$\{5\}$
9	$y = 1/x$ for $x > 0$	$x > 0$	$y > 0$
10	Piecewise: closed at $(-5, -3)$ rising to open at $(0, 7)$ , then closed at $(0, 4)$ rising to closed at $(6, 10)$	$-5 \leq x \leq 6$	$-3 \leq y \leq 10$ (combined)

Self-check question

If a graph has an open dot, is that endpoint included in the domain/range? (No — use $<$ or $>$ , not $\leq$ or $\geq$ .)

Tuesday — Factoring Patterns

Objective

Recognize the four core factoring patterns within 5 seconds of seeing them. Practice the H2 heuristic: name the structure aloud before you write anything.

Instructions

Part A — Classify (15 expressions, no solving): For each, say aloud which of the four patterns it matches: GCF, Difference of Squares (DOS), Perfect-Square Trinomial (PST), or Standard Trinomial (ST). Some require two patterns stacked.

Part B — Factor (10 expressions): Pick 10 from Part A and factor them completely.

Part A — Classify

#	Expression	Pattern(s)
1	$x^2 - 49$	DOS
2	$x^2 + 10x + 25$	PST
3	$x^2 + 7x + 12$	ST
4	$4x^3 + 12x^2$	GCF
5	$x^3 - 36x$	GCF + DOS
6	$9x^2 - 16$	DOS
7	$x^2 - 14x + 49$	PST
8	$x^2 + 5x - 14$	ST
9	$2x^4 - 32$	GCF + DOS (then DOS again on $x^4 - 16$ )
10	$50x^3 - 18x$	GCF + DOS
11	$x^2 - 6x + 9$	PST
12	$x^2 - 64$	DOS
13	$6x^2 + 9x$	GCF
14	$x^2 + x - 12$	ST
15	$4x^2 - 4x + 1$	PST (factors as $(2x-1)^2$ )

Part B — Factor (worked solutions)

Show worked solutions for #5, #9, #10, #15

#5 $x^3 - 36x$ GCF first: $= x(x^2 - 36)$ . $x^2 - 36$ is DOS: $= x(x - 6)(x + 6)$ .

#9 $2x^4 - 32$ GCF: $= 2(x^4 - 16)$ . $x^4 - 16 = (x^2)^2 - 4^2$ is DOS: $= 2(x^2 - 4)(x^2 + 4)$ . $x^2 - 4$ is also DOS: $= 2(x - 2)(x + 2)(x^2 + 4)$ .

#10 $50x^3 - 18x$ GCF: $= 2x(25x^2 - 9)$ . $25x^2 - 9 = (5x)^2 - 3^2$ is DOS: $= 2x(5x - 3)(5x + 3)$ .

#15 $4x^2 - 4x + 1$ Recognize PST: $4x^2 = (2x)^2$ , $1 = 1^2$ , and $-4x = -2 \cdot 2x \cdot 1$ . $= (2x - 1)^2$ .

Self-check question

What’s the giveaway that distinguishes $x^2 - 36$ (DOS) from $x^2 - 12x + 36$ (PST)? (Two terms = DOS; three terms with the middle = $\pm 2ab$ pattern = PST.)

Wednesday — Polynomial Arithmetic + Linear Equations/Inequalities

Objective

Tighten up the basics: sign distribution in polynomial subtraction, the box method for products, and the inequality-flip rule.

Instructions

Do all 14 problems. Show full work. Use the box method for every multiplication, even simple ones — building the habit matters more than speed this week.

Part A — Polynomial arithmetic (6 problems)

Simplify each.

#	Problem	Answer
1	$(3x^2 - 5x + 7) - (x^2 + 2x - 4)$	$2x^2 - 7x + 11$
2	$(-2x^2 + 6) - (4x^2 - 3x + 1)$	$-6x^2 + 3x + 5$
3	$(2x + 3)(x - 5)$	$2x^2 - 7x - 15$
4	$(3x - 4)(2x + 1)$	$6x^2 - 5x - 4$
5	$(x + 6)^2$	$x^2 + 12x + 36$
6	$(5x - 2)(-3x + 4)$	$-15x^2 + 26x - 8$

Worked solution for #6 (box method)

$(5x - 2)(-3x + 4)$

	$-3x$	$+4$
$5x$	$-15x^2$	$+20x$
$-2$	$+6x$	$-8$

Sum: $-15x^2 + 20x + 6x - 8 = -15x^2 + 26x - 8$ .

Part B — Linear equations and inequalities (8 problems)

Solve for the variable. Pay attention to the inequality-flip in #5, #6, #8.

#	Problem	Answer
1	$3(x - 4) = 2x + 5$	$x = 17$
2	$5 - 2(y + 3) = 4y$	$y = -\tfrac{1}{6}$
3	$\dfrac{2x - 1}{3} = x - 4$	$x = 11$
4	$4(2x + 1) - 3 = 5x + 10$	$x = 3$
5	$-3x + 7 \geq 19$	$x \leq -4$
6	$-2(y - 5) < 8$	$y > 1$
7	$7 - 4x \leq 3x + 35$	$x \geq -4$
8	$\dfrac{1 - 2y}{4} > 3$	$y < -\tfrac{11}{2}$

Worked solution for #5 (sign flip)

$-3x + 7 \geq 19$ $-3x \geq 12$ Divide by $-3$ (negative — flip the inequality): $x \leq -4$ .

Self-check question

When do you flip the inequality sign? (Whenever you multiply or divide both sides by a negative number — never when adding or subtracting.)

Thursday — Mixed Part-I Timed Set

Objective

Simulate Part I conditions and stress-test the heuristics under time pressure.

Instructions

Set a timer for 18 minutes.
Answer all 12 questions below. Show your scratch work to the right.
After the timer ends, check answers. For every miss, write a Mistake Log entry tagged with the H-bucket that would have caught it.

Practice set

The expression $\dfrac{12}{\sqrt{3}}$ is equivalent to: (1) $\sqrt{3}$ (2) $4\sqrt{3}$ (3) $12\sqrt{3}$ (4) $36$
The graph of $f(x) = x^2 - 4$ has a minimum at: (1) $(0, -4)$ (2) $(-4, 0)$ (3) $(2, 0)$ (4) $(0, 4)$
Which scenario is exponential? (1) saves $10 each week (2) earns 8% interest each year (3) loses 5 lbs per month (4) drives 60 mph
Which polynomial has degree 4 and leading coefficient $-3$ ? (1) $4x^3 - 3x$ (2) $-3x^4 + 2x^2$ (3) $3x^4 - x$ (4) $-4x^3 + 3$
Simplify $(2x^2 + 3) - (x^2 - 5x + 1)$ . (1) $x^2 - 5x + 2$ (2) $x^2 + 5x + 2$ (3) $x^2 + 5x + 4$ (4) $3x^2 + 5x + 4$
The function $g(x)$ is shown with values: $g(1) = 3,\ g(3) = 11,\ g(5) = 27$ . The average rate of change from $x = 1$ to $x = 5$ is: (1) 6 (2) 7 (3) 8 (4) 24
Which is equivalent to $a^6 - b^4$ ? (1) $(a^3 - b^2)(a^3 + b^2)$ (2) $(a^2 - b^2)^3$ (3) $(a^3)^2 - b^4$ only (4) cannot be factored
Solve for $y$ : $4y - 7 = 2y + 11$ . (1) $y = 2$ (2) $y = 4$ (3) $y = 9$ (4) $y = -9$
Which represents $f(x) = |x|$ shifted 3 units left? (1) $|x| + 3$ (2) $|x| - 3$ (3) $|x + 3|$ (4) $|x - 3|$
The domain of the graph that is open at $(-2, 1)$ and closed at $(6, 9)$ is: (1) $[1, 9]$ (2) $(1, 9]$ (3) $[-2, 6]$ (4) $(-2, 6]$
Solve $\dfrac{3(x + 2)}{2} = x + 6$ . (1) $x = 6$ (2) $x = 4$ (3) $x = 2$ (4) $x = 0$
Which sequence is geometric with $a_1 = 8$ and common ratio $\tfrac{1}{2}$ ? (1) $8, 16, 32, 64, \ldots$ (2) $8, 4, 2, 1, \ldots$ (3) $8, 10, 12, 14, \ldots$ (4) $8, 0, -8, -16, \ldots$

Answer key

(2) $4\sqrt{3}$ — rationalize: $\tfrac{12}{\sqrt{3}} \cdot \tfrac{\sqrt{3}}{\sqrt{3}} = \tfrac{12\sqrt{3}}{3}$ .
(1) $(0, -4)$ .
(2) Constant percent each period.
(2) Degree 4 with leading coefficient $-3$ .
(2) Distribute the minus across all three terms.
(1) $\tfrac{27 - 3}{5 - 1} = \tfrac{24}{4} = 6$ .
(1) $a^6 = (a^3)^2$ , $b^4 = (b^2)^2$ — difference of squares.
(3) $2y = 18$ .
(3) Shift left = inside $+$ .
(4) Open on left → $($ , closed on right → $]$ .
(1) Multiply by 2: $3(x+2) = 2x + 12 \implies 3x + 6 = 2x + 12 \implies x = 6$ .
(2) Each term is half the previous.

Self-grade

11–12 correct: on target. 8–10: review the misses and the H-bucket. <8: redo Mon and Tue before continuing.

Friday — Re-do Q13 and Q30, plus Sibling Problems

Objective

Convert the actual June 25 misses into wins, with extra siblings to prove the fix is durable.

Instructions

For each problem, write the H2 rationale next to your work (“This is a domain question because…” / “This is a difference of squares because…”). Then solve.

Q13 (original)

A graph runs from open $(0, 15)$ to closed $(12, 45)$ . State the domain.

Answer: $0 < x < 12$ written as $0 < x \leq 12$ in interval form. Wrong choice on June 25: $15 < x \leq 45$ — that’s the range, not domain.

Q13 siblings

#	Problem	Domain	Range
13a	Graph open at $(-7, 1)$ , closed at $(3, 21)$	$-7 < x \leq 3$	$1 < y \leq 21$
13b	Graph closed at $(-1, -1)$ , open at $(5, 11)$	$-1 \leq x < 5$	$-1 \leq y < 11$
13c	$f(x) = \sqrt{x - 4}$	$x \geq 4$	$y \geq 0$
13d	Horizontal segment $y = 8$ from $x = -3$ to $x = 7$ , both endpoints closed	$-3 \leq x \leq 7$	$\{8\}$

Q30 (original)

Factor $x^3 - 36x$ completely. Answer: $x(x - 6)(x + 6)$ . Mistake on June 25: wrote $x(x - 6)^2$ .

Q30 siblings

#	Problem	Answer
30a	$x^3 - 49x$	$x(x - 7)(x + 7)$
30b	$4x^3 - 100x$	$4x(x - 5)(x + 5)$
30c	$2y^4 - 50y^2$	$2y^2(y - 5)(y + 5)$
30d	$x^5 - x$	$x(x - 1)(x + 1)(x^2 + 1)$

Worked solution for 30d

$x^5 - x$ GCF: $= x(x^4 - 1)$ . $x^4 - 1 = (x^2)^2 - 1^2$ is DOS: $= x(x^2 - 1)(x^2 + 1)$ . $x^2 - 1$ is also DOS: $= x(x - 1)(x + 1)(x^2 + 1)$ . $x^2 + 1$ does not factor over the reals — leave it.

Self-check question

What two patterns are needed to factor $x^5 - x$ completely? (GCF, then DOS applied twice.)

Saturday — Full Part I + Part II Timed

Objective

Measure the week’s progress under real test conditions.

Instructions

Use a prior Regents Algebra I administration (any year; June 2024 or January 2025 are good). NYS releases them at https://www.nysedregents.org.
Print out Part I (Q1–Q24) and Part II (Q25–Q30) only.
Set timer for 70 minutes. Work in pen.
Score yourself strictly against the official answer key.
Compare scores to baseline (June 25 was Part I = 44/48, Part II = 10/12).

Success bar for Week 1

Part I score: 45+/48.
Part II score: 11+/12.
Every miss in the Mistake Log has an H-bucket tag.
Can name the structure of 9/10 factoring problems within 5 seconds.

Mistake Log review

At the end of the day, flip through every entry from the week. Group them by H-bucket. The H-bucket with the most entries is your top priority going into Week 2.

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