SAT Preparation: Advanced Punctuation Grammar and Polynomial Algebra II-Algebraic Polynomials Polynomials

Administrative Updates and Diagnostic Logistics

• Date Context: The lecture took place on Wednesday, July 8.

• Typo Hunt: Students are encouraged to find typos in assignments, homework, and classwork. The first person to catch a typo earns a point toward a reward (e.g., bubble tea).

• Testing Site Cancellations: August and September SAT testing sites, particularly in New York, are prone to cancellation due to lack of proctors or other logistical issues. Students must monitor their testing sites daily.

• Manhasset Test Site: A site in Manhasset (Long Island) briefly opened up. It is approximately $20$ to $25$ minutes away from the current location. Commutes up to $30$ to $45$ minutes are considered worth it to secure a seat.

• First Diagnostic Exam: A real, full-length SAT diagnostic will occur this Friday.

  • Setup: In-person and media room students arrive by $08:30$ or $08:45$ for setup on personal devices.

  • Timing: Release is at $09:00$. Conclusion is typically between $11:15$ and $11:20$.

  • Review: The test will be reviewed the following Monday.

Writing and Grammar: Parentheses and Dashes

• Parentheses in Formal Writing: Used to set off information that interrupts the flow of a sentence.

  • Content: Can contain a single word, a compound sentence, or even two sentences linked by a semicolon.

  • Grammar Rule: The information inside is not grammatically essential; removing it will not break the sentence.

  • Requirement: Must always be used in pairs. An opening parenthesis must have a closing one.

  • Example: "Savvy insiders at hedge funds (unbeknownst to the general public)."

• Hyphens vs. Dashes: The primary difference is length.

  • Hyphens: Short lines used for compound numbers (e.g., $30-5$), fractions ($\frac{3}{5}$), specific prefixes (ex-, all-, self-, semi-), or double vowels (anti-inflammatory). Hyphens are generally not tested on the SAT.

  • Dashes (Em Dashes): Longer lines ($-$) that are heavily tested. Used for abrupt changes of thought or to add emphasis.

  • Dash Placement Rules:

    • End of Sentence: Use only one dash if the interjection follows the main clause. The period acts as the second boundary. Example: "A driver must use turn signals — a concept too difficult for some."

    • Middle of Sentence: Must use a pair of dashes if the information is interjected in the middle. Example: "Teachers — not parents or students — receive the blame."

Writing and Grammar: Apostrophes and Ownership

• Possessive Nouns:

  • Singular: Add $'s$ (e.g., "the child's cat").

  • Plural: Add $s'$ if the noun ends in $s$ (e.g., "the players' salaries").

• Contractions and Omissions: Apostrophes replace missing letters (e.g., "can't", "could've"). In formal college essays, full words (e.g., "cannot") are preferred.

• When NOT to use Apostrophes:

  • Possessive Pronouns: "Its", "hers", "his", and "whose" never take apostrophes. "It's" means "it is"; "Who's" means "who is".

  • Pluralizing Nouns: Simply add an "s". Never use an apostrophe to make a noun plural.

• Shared vs. Separate Possession (Hard Level Rule):

  • Shared Possession: If two or more people own the same item together, only the last person gets the apostrophe. Example: "Aaron and Samuel's high school" (they both attend the same one).

  • Separate Possession: If two people own separate items of the same category, both get apostrophes. Example: "Aaron's and Samuel's homework scores" (different scores for different students).

Writing and Grammar: Colons and Semicolons

• Colons ($:$): Used to link two statements where the second supports, explains, clarifies, or defines the first.

  • The Full Sentence Rule: The clause before the colon must be an independent clause (a full sentence). If you can put a period there, you can likely use a colon.

  • Lists: Used to introduce a series of items, but only if preceded by a full sentence. Example: "The surgeon declared four majors: biology, economics, and engineering."

• Semicolons ($;$):

  • Independent Clauses: Connect two closely related full sentences without a conjunction.

  • The "Super Comma" (Hard Level Rule): Used to separate items in a list when the items themselves already contain internal commas. This prevents confusion. Example: "Item one, with details; item two, with more details; and item three."

Writing and Grammar: Commas and Nonessential Clauses

• Common Usages: Comma-FANBOYS for compound sentences, items in a series (Oxford comma is required for clarity), and coordinate adjectives.

• Introductory Elements: Use a comma after a dependent clause or modifier starting a sentence. Example: "Having been raised in a traditional household, she…"

• Quoted Material: Use a comma to introduce dialogue unless the quote blends perfectly into the sentence syntax (often signaled by the word "that").

• Nonessential (Parenthetical) Clauses: Use a pair of commas to surround extra information that is not required for the sentence's meaning or grammar. Example: "The newspaper, which has boring editorials, won awards."

• Comma Exception: Labels and Titles (Hard Level Rule): Do not use commas if the noun is a label or specific title that cannot be removed without changing the meaning.

  • Incorrect: "Chemistry student, Teresa Dankovich, of McGill…"

  • Correct: "Chemistry student Teresa Dankovich of McGill…" (Chemistry student is a label for her specifically).

Advanced Math: Polynomial Operations

• Combining Like Terms: The fundamental process for adding and subtracting polynomials. Ensure signs are distributed correctly when subtracting.

• Multiplication:

  • Monomial $\times$ Polynomial: Use the distributive property.

  • Binomial $\times$ Binomial: Use the FOIL method ($\text{First, Outer, Inner, Last}$).

• Special Polynomial Patterns:

  • $(x+y)^2 = x^2 + 2xy + y^2$ (Common trap: $(x+y)^2 \neq x^2 + y^2$).

  • $(x-y)^2 = x^2 - 2xy + y^2$.

  • $x^2 - y^2 = (x+y)(x-y)$ (Difference of Squares).

• Remainder Theorem: If a polynomial $P(x)$ is divided by $(x - a)$, the remainder is $P(a)$. If $(x - a)$ is a factor, then $P(a) = 0$.

Advanced Math: Factoring Techniques

• Gratest Common Factor (GCF): Always check for a GCF first. On Desmos, you can use the function $\text{GCF}(n_1, n_2)$ to find the common factor of constant coefficients.

• Factoring by Grouping: Used for four-term polynomials. Group into two pairs, factor out GCFs from each, and then factor out the common binomial.

  • Metaphor: If you have $x^2$ apples and subtract $9$ apples, you have $(x^2 - 9)$ apples.

• Factoring Cubic Expressions (SOAP):

  • Formula: $x^3 \pm y^3 = (x ± y)(x^2 ∓ xy + y^2)$.

  • S.O.A.P. Mnemonic: Sign placement follows $\text{Same, Opposite, Always Positive}$.

Questions & Discussion

• Punctuation Drill Review:

  • Question 2 (Urns): Error occurred where students used apostrophes for simple plurals. "Urns" and "Archaeologists" specify multiple items/people, needing no apostrophes unless showing possession.

  • Question 9 (Cyclones): A dash was incorrect because removing the text between dashes broke the sentence (the remaining part had no subject-verb logic).

  • Question 13 (Gideon Lack): No comma before the quote because the quote was introduced by "that" and blended into the sentence.

  • Question 14 (Rock/Rap): Separate possession required two apostrophes because rock and rap have separate ratios.

  • Question 20 (OECD Nations): "Poland and Switzerland" was a label for the "OECD nations," meaning no comma should separate the label from the specific names.

• Math Drill Review:

  • Question 4 (Perfect Squares): Recognition of $x^4 - y^4$ as $(x^2+y^2)(x^2-y^2)$ allowed for a quick solution of $15 \times 10 = 150$.

  • Question 5 (Nested Logic): Grouping $x^2 - 2xy + y^2$ into $(x-y)^2$ allowed the substitution of $50$ to solve as $50^2 + 2(50) = 2600$.

  • System of Equations in Desmos: For complex factoring (e.g., Question 22), students should plot system equations like $2x + 3y = 17$ and $xy = 5$ to find non-integer solutions ($7.5$ and $0.667$).