SAT Foundations — Lesson 2: Heart of Algebra

How to run this lesson. Send the pre-work a day ahead. Open with goals (5 min), then work the four algebra moves together: linear equations (12), lines & slope (12), systems (16), inequalities (8), the Vocabulary Lab (7), and a wrap that assigns the next-two-days work. Question cards reveal the answer on click; math cards open a “Solve it another way” panel. The dark navy notes are for the instructor only.

The frame for today

One line to open: “Algebra is the backbone of SAT math — own these four moves and you own a third of the test.” Today is about fluency and a backup route: a main method plus a check on every problem. Watch for the one habit that quietly costs points — forgetting to flip the inequality sign on a negative. Drill it until it’s automatic.

00 Before you begin

Pre-work handout

Hand this to the student the day before. Five minutes of warm-up makes the lesson land faster.

Heart of Algebra — Pre-work

Complete on paper before our session · about 10 minutes

Warm-up — no calculator

1. Linear: Solve 2x + 9 = 21. x = ______

2. Slope: Find the slope of the line through (0, 1) and (2, 7). ______

3. System: If x + y = 9 and y = 3, what is x? ______

4. Inequality: Solve x − 4 < 2. ______

One line

Answer key (instructor)

1 · x = 6 2 · slope = 3 3 · x = 6 4 · x < 6

≈ 5 min

01 Orientation

The four moves of algebra

About a third of SAT math is “Heart of Algebra.” Almost all of it comes down to four moves — and each has a backup route.

Linear equations

Isolate the variable with inverse operations. Back-solve to check.

Lines & slope

Slope is rise over run; read y = mx + b without graphing.

Systems

Substitution or elimination — or test the answer choices.

Inequalities

Solve like an equation — but flip the sign on a negative.

≈ 12 min

02 Linear equations

Isolate, then check

The golden rule: whatever you do to one side, do to the other. Undo operations in reverse order until the variable stands alone.

Multi-stepQ1

Solve for x: 5x − 3 = 2x + 12

Why C

Get the x’s together: 5x − 2x = 12 + 3 → 3x = 15 → x = 5.

Solve it another way

Back-solve: test C (x = 5): 5(5)−3 = 22 and 2(5)+12 = 22. ✓ Match, so C.

True shortcut Move all the variables to one side and all the numbers to the other first; then one division finishes it.

Distribute firstQ2

Solve for x: 3(x + 4) = 2x + 18

Why C

Distribute: 3x + 12 = 2x + 18 → 3x − 2x = 18 − 12 → x = 6.

Solve it another way

Back-solve: test C (x = 6): 3(10) = 30 and 2(6)+18 = 30. ✓

True shortcut Distribute before you do anything else — a stray parenthesis is where most sign errors start.

≈ 12 min

03 Lines & slope

Read a line two ways

Slope is the rate of change: rise over run. In y = mx + b, m is the slope and b is where the line crosses the y-axis.

Slope from two pointsQ3

A line passes through (1, 5) and (3, 11). What is its slope?

Why B

Slope = Δy / Δx = (11 − 5) / (3 − 1) = 6 / 2 = 3.

Solve it another way

Picture it: from (1, 5) to (3, 11) you go up 6, right 2 → 6/2 = 3.

True shortcut Subtract the y’s and the x’s in the same order. Order discipline kills sign errors.

Slope-intercept formQ4

Where does the line y = −2x + 7 cross the y-axis?

Why A

In y = mx + b, the y-intercept is b. Here b = 7, so the line crosses at (0, 7). The −2 is the slope, not the intercept.

Solve it another way

True shortcut The y-intercept is just the value of y when x = 0. Set x = 0: y = −2(0) + 7 = 7.

≈ 16 min

04 Systems

Two equations, one point

A system asks where two lines meet. Use substitution when one variable is already alone, elimination when adding or subtracting cancels a variable — or just test the answers.

The two routes, side by side

Substitution

One variable alone? Put its expression into the other equation, then solve for one variable.

Elimination

Line the equations up. If a variable matches, add or subtract to cancel it.

Back-solve

Plug each answer choice into both equations until one fits. Fast when choices are given.

SubstitutionQ5

If 3x + y = 10 and y = x − 2, what is the value of x?

Why B

Substitute y = x − 2: 3x + (x − 2) = 10 → 4x − 2 = 10 → 4x = 12 → x = 3.

Solve it another way

Back-solve: test B (x = 3): then y = 1, and 3(3) + 1 = 10. ✓

EliminationQ6

If 2x + 3y = 12 and 2x − y = 4, what is the value of y?

Why B

Both have 2x, so subtract: (2x + 3y) − (2x − y) = 12 − 4 → 4y = 8 → y = 2.

Solve it another way

True shortcut When a variable’s coefficient already matches in both equations, subtraction cancels it instantly — no rearranging needed.

≈ 8 min

05 Inequalities

One rule changes everything

Solve an inequality exactly like an equation, with a single exception: flip the direction of the sign whenever you multiply or divide by a negative number.

The sign flipQ7

Solve: −2x + 5 > 1

Why A

−2x > −4. Divide by −2 and flip the sign: x < 2. (Forgetting to flip gives the trap answer B.)

Solve it another way

True shortcut Circle the negative coefficient before you divide — it’s your reminder to flip. That one habit saves the most common inequality mistake on the test.

Greatest valueQ8

What is the greatest integer value of x for which 3x − 4 ≤ 11?

Why B

3x − 4 ≤ 11 → 3x ≤ 15 → x ≤ 5. No flip needed (we divided by +3). The greatest integer that works is 5.

Build the two-route habit

For each problem, have the student name their main method, then show one alternative. By August they should have a primary route and a backup (back-solve, elimination) so no question stalls them. And rehearse the inequality flip aloud — saying “negative, so flip” makes it stick.

≈ 7 min

06 Vocabulary Lab

Week 2 words — think in synonyms and antonyms

Tap a card to flip it. Learn each word next to its opposite — that’s how the test frames them.

Quick check

SynonymV1

Which word is closest in meaning to concise?

Why B

Concise = clear and brief, like succinct. “Verbose” is its opposite — the trap.

AntonymV2

Which word is most nearly opposite to objective?

Why B

Objective means unbiased; its opposite is biased. Impartial and neutral are synonyms placed to bait a fast read.

Word in contextV3

Without documents to back her up, the lawyer could not ______ the claim.

Why A

To substantiate is to support with evidence — exactly what she can’t do without documents.

≈ 5 min

07 Wrap-up

Three things to carry out the door

Say them out loud — that’s how they stick.

Isolate, then check

Get the variable alone, then back-solve to confirm in seconds.

Slope is Δy/Δx

Same order top and bottom; in y = mx + b, b is the intercept.

Flip on a negative

Multiply or divide an inequality by a negative → flip the sign.

Close the loop

Have the student set one specific target for the mid-week mini-lesson. Confirm the two lesson days, point them to the workbook, and preview that Lesson 3 is Advanced Math — quadratics, functions, and exponentials, which build directly on today’s linear work.

08 Keep the fire lit

Next-two-days handout

Short, daily, cumulative. Reinforces today and loads Lesson 3.

Heart of Algebra — Work for the Next Two Days

About 30 minutes a day · name your method on every problem

Day 1 · Reps

Run the four moves

1. Solve 6x − 5 = 3x + 7. x = ______

2. Slope through (0, 2) and (4, 10). ______

3. x + 2y = 11 and x = y + 2. x = ______

4. Solve −4x + 9 ≥ 1. ______

5. A printer costs $80 plus $0.05 per page. Cost for p pages? ______

Day 2 · Timed + words

Beat the clock

1. Solve 2(x − 3) = x + 1. x = ______

2. 3x − y = 5 and y = x + 1. x = ______

3. Solve −2x ≤ 6. ______

4. Tickets cost $12 each plus a $4 fee. Total for n tickets? ______

Then: write a synonym and an antonym for all 10 words from memory, and bring one problem you want to review.

Answer key (instructor)

Day 1: 1 · x = 4 2 · slope = 2 3 · x = 5 4 · x ≤ 2 (flip) 5 · 0.05p + 80
Day 2: 1 · x = 7 2 · x = 3 3 · x ≥ −3 (flip) 4 · 12n + 4

This sets up Lesson 3. Advanced Math (quadratics and functions) is built on the linear moves you ran today — factoring is just the reverse of distributing.