Math from First Principles

MERIDIAN™ · © 2026 · original teaching + practice (not real SAT questions). Not affiliated with or endorsed by the College Board. SAT® is a trademark of the College Board.

The Reading side of this course has a from-first-principles guide because reading felt like a vibe and needed an algorithm. Math has the opposite problem: it is usually taught as a pile of algorithms with no reason attached, so the pile is forgotten under time pressure and there is nothing underneath it. This document is the underneath. When you want real, official practice, use the College Board Question Bank (Bluebook) — that is the only source guaranteed to match the live test.

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#Before you read: three questions

Answer these now, from memory, before reading anything below. Getting them wrong is the point — a question you have failed at is the question you will remember the answer to. Answers are at the end of Part 0.

1. A rectangle's sides are both tripled. What happens to its area, and why is the answer not "tripled"?
2. Why does the quadratic formula have a ± in it?
3. x⁰ = 1. Why? Not "because that's the rule" — why does it have to be 1?

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#How this guide is organised

• Part 0 — The one idea: three representations of the same object.
• Part 1 — Number and operation. Why exponent rules, fraction rules and percent rules are one rule.
• Part 2 — Linearity. What "constant rate of change" forces, and everything that falls out of it.
• Part 3 — Nonlinearity. Quadratics from completing the square; exponentials from repeated multiplication.
• Part 4 — Geometry from similarity. Why area scales by k², why trigonometry exists at all.
• Part 5 — Proportion and data. Percent as multiplication, and what a statistic is allowed to claim.
• Part 6 — The checklist, the study order, and what is still worth memorising.

Read it once slowly. Then use Parts 1–5 as the thing you reach for when a method in this course says "this is why" and you want the why.

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#PART 0 · THE ONE IDEA

#An equation, a table and a graph are the same object

Everything in SAT Math is one relationship shown three ways: as an equation, as a table of values, and as a picture. Every question is solvable in whichever of the three is cheapest, and the answer is the same in all three because they are not three things.

This is not a study tip. It is the structural fact that makes the section beatable in 70 seconds a question.

Take y = 2x + 1.

A question that hands you the graph and asks for the equation, a question that hands you two points and asks for the y-intercept, and a question that hands you the equation and asks where it crosses the axis are the same question in three costumes. Students who see three questions have to learn three methods. Students who see one relationship have to learn one.

The practical consequence, and it is the single largest time saving available to you: when a question is expensive in the form it arrives in, translate it. Algebra hard, picture easy? Graph it in Desmos. Picture hard, algebra easy? Write the equation. Both hard? Build the table — plug in numbers and look.

#Why "plug in numbers" is not cheating

A great deal of SAT Math is a claim about all values of a variable. "Which expression is equivalent to…" means: equivalent for every x. So if you pick x = 3 and the candidate expression disagrees with the original, that candidate is dead — one counterexample kills a universal claim, permanently. That is a proof, not a trick.

The reverse does not hold, and this is where people get burnt: agreeing at x = 3 does not prove equivalence. It only fails to disprove it. So plugging in is a fast elimination tool and not a fast confirmation tool. If two choices survive your number, pick a second, uglier number — most commonly the one that breaks a wrong answer is negative, or a fraction, or 0, or 1.

Avoid 0 and 1 as your first choice. They make too many different expressions agree, which is exactly what you are trying to avoid. Start at 2 or 3.

#Answers to the three opening questions

1. Area becomes nine times as large. Area is a product of two lengths, and both were tripled: (3a)(3b) = 9ab. Anything built from two lengths scales by the square of the scale factor. Part 4 does this properly.
2. Because squaring destroys sign information. x² = 9 is true for x = 3 and for x = −3, so undoing a square must offer both. The ± is not decoration; it is the record of what was lost. Part 3.
3. Because dividing by one copy must remove one copy. x³ ÷ x³ = 1 because anything over itself is 1. But subtracting exponents gives x⁰. For the exponent rule to hold without exception, x⁰ has to be 1. Part 1.

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#PART 1 · NUMBER AND OPERATION

#The rule underneath the exponent rules

There is one definition and everything else is bookkeeping.

xⁿ means: multiply n copies of x together.

Now watch every "rule" appear on its own.

xᵃ · xᵇ = xᵃ⁺ᵇ — Three copies next to two copies is five copies. x³ · x² = (x·x·x)(x·x) = x⁵. There is nothing to remember.

xᵃ ÷ xᵇ = xᵃ⁻ᵇ — Dividing cancels copies one for one. x⁵ ÷ x² = (x·x·x·x·x)/(x·x) = x³.

(xᵃ)ᵇ = xᵃᵇ — b groups of a copies each. (x³)² = (x·x·x)(x·x·x) = x⁶.

x⁰ = 1 — Forced, as above. Follow the division rule down: x³ ÷ x³ is 1 by arithmetic and x⁰ by the rule. Both are true, so they are equal.

x⁻ⁿ = 1/xⁿ — Keep going below zero. x² ÷ x³ = 1/x by arithmetic and x⁻¹ by the rule. Negative exponents are not "the opposite of" positive ones; they are what the same rule says when you run out of copies.

x^(1/n) = ⁿ√x — What number, multiplied by itself n times, gives x? By the power rule, (x^(1/2))² = x^1 = x. So x^(1/2) is the thing that squares to x, which is exactly what "square root" means. Fractional exponents are not a separate topic. x^(3/2) is (√x)³, and the SAT tests this every time it wants to look hard.

What you now do not have to memorise: seven rules. You memorise one sentence and re-derive in four seconds. Under time pressure, four seconds of derivation beats a half-remembered rule you are not sure about, every time.

#Fractions are division, and division is multiplication

a/b means a divided by b, and dividing by b is multiplying by 1/b. That single collapse explains the fraction rules that people get wrong most often:

• Dividing by a fraction flips it. Dividing by 2/3 is multiplying by 3/2, because 2/3 and 3/2 multiply to 1, and multiplying by something and then by its reciprocal must return you to where you started.
• You may cancel factors, never terms. (3x)/(3y) = x/y because the 3s are multiplied in and one undoes the other. (3 + x)/(3 + y) cancels nothing, because addition does not undo multiplication. This one mistake costs more points on the Math section than any other single algebra error, and it is a favourite trap on "equivalent expression" items.

#Percent is a multiplier, not a procedure

"p percent" means "multiply by p/100". That is the whole definition.

Every percent question on the test is now one line of arithmetic:

Two consequences the SAT tests deliberately:

Percent changes do not add. Up 20% then down 20% is × 1.2 × 0.8 = × 0.96 — a 4% loss, not a return to the start. The second percentage is taken of a different, larger number.

Percent change order does not matter. × 0.85 × 1.08 equals × 1.08 × 0.85, because multiplication commutes. A question that asks whether applying the discount before or after tax gives a different total is testing whether you know that; the answer is no.

#✅ Part 1 retrieval (answers at the end of Part 1)

1. Simplify (2x³)⁴.
2. Write x^(5/2) using a root sign.
3. A price rises 25%, then falls 20%. What single multiplier is that, and what happened?
4. Is (x + 5)/(x + 7) equal to 5/7? Say why in one sentence.

Answers. 1. 16x¹² — the 4 applies to the 2 as well, giving 2⁴ = 16, and (x³)⁴ = x¹². Forgetting to raise the coefficient is the standard error. 2. (√x)⁵, equivalently √(x⁵). 3. × 1.25 × 0.80 = × 1.00 — exactly back to the start, which is a coincidence of these two numbers and not a rule. 4. No. The 5 and 7 are added, not multiplied, so nothing cancels; test it with x = 3, which gives 8/10, not 5/7.

Self-explanation prompt. In one sentence, why does x⁻² mean 1/x² rather than −x²? Write it down before moving on. Nobody marks it — producing the sentence is the benefit.

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#PART 2 · LINEARITY

#One assumption, and the whole of Algebra follows

A relationship is linear when equal steps in x always produce equal steps in y. That is the definition. Everything else about lines is a consequence of it.

Suppose that is true, and call the change in y per one-unit change in x the number m. Then starting from the value at x = 0 — call it b — moving x units across adds m each time:

y = b + mx

That is slope-intercept form, and it was not memorised, it was derived from the definition in one line. m is the constant step and b is where you started. Nothing else can be true of a relationship with a constant rate of change.

Slope from two points. If (x₁, y₁) and (x₂, y₂) are both on the line, then going from the first to the second changes x by x₂ − x₁ and y by y₂ − y₁. Since each unit of x buys m units of y:

m = (y₂ − y₁)/(x₂ − x₁)

Again: derived, not memorised. And now the classic error is visible rather than arbitrary — subtract in the same order on top and bottom, because you are describing one journey, not two.

Parallel lines have equal slopes because same constant rate of change means never catching up.

Perpendicular slopes multiply to −1. This is the one linear fact whose reason is not obvious in one line, and it is worth the thirty seconds. Rotate a run-then-rise journey of (1, m) by a quarter turn and it becomes (−m, 1). The slope of the rotated journey is 1/(−m) = −1/m. So the perpendicular slope is the negative reciprocal, and m × (−1/m) = −1.

#What a system of equations actually is

Two linear equations are two constraints. A solution is a pair (x, y) satisfying both at once. Graphically, each equation is a line, and satisfying both means sitting on both lines — so the solution is the intersection point. Three cases, and they are the three things two lines can do:

Every "for what value of k does this system have no solution" question is asking one thing: make the slopes equal and the intercepts different. Every "infinitely many solutions" question is asking: make one equation an exact multiple of the other. You do not need a method per question type; you need to know what two lines can do.

#Inequalities: the same thing, with one asymmetry

An inequality behaves exactly like an equation under adding, subtracting, and multiplying by a positive number. There is one break:

Multiplying or dividing both sides by a negative number reverses the inequality sign.

Why: 3 < 5 is true. Multiply both by −1 and you get −3 and −5. On the number line −3 is to the right of −5, so the true statement is −3 > −5. Multiplying by a negative reflects the number line, and reflection swaps left and right. This is the single most-missed step in the Algebra domain, and knowing the reason is what makes you notice it at speed.

Graphically, y > mx + b is the region above the line and y < mx + b is the region below — because "greater y" means "higher up". Systems of inequalities shade to an overlap region, and "which point could be a solution" means "which of these four points is in the overlap". Plot the four; do not solve anything.

#✅ Part 2 retrieval

1. A line passes through (2, 7) and (5, 16). Find m, then b.
2. For what k does y = kx + 3 never meet y = 4x − 1?
3. Solve −2x + 1 < 9.
4. 2x + 3y = 12 and 4x + 6y = c. For what c are there infinitely many solutions?

Answers. 1. m = (16 − 7)/(5 − 2) = 3; then 7 = 3(2) + b gives b = 1, so y = 3x + 1. 2. k = 4 — equal slopes, and the intercepts already differ (3 ≠ −1), so they are parallel and never meet. 3. −2x < 8, then divide by −2 and flip: x > −4. 4. c = 24 — the second equation must be exactly twice the first, and 2 × 12 = 24.

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#PART 3 · NONLINEARITY

#Quadratics: three forms, one object

A quadratic is y = ax² + bx + c. Its graph is a parabola, symmetric about a vertical line, because x² treats +x and −x identically — that symmetry is the origin of every other fact about parabolas.

The three forms are not three topics. They are one curve written to make one particular thing easy to read:

Choosing the form is the method. "Where does it cross the x-axis" → factored. "What is the maximum value" → vertex. "What is the value when x = 0" → standard. An enormous fraction of Advanced Math questions are one conversion away from trivial, and knowing which conversion is most of the skill.

Why the vertex sits midway between the roots. The curve is symmetric, and the two x-intercepts are mirror images across the axis of symmetry. So the axis is at their average, x = (p + q)/2. This is often faster than any formula: factor, average the roots, done.

#Where the quadratic formula comes from

Nobody should memorise a formula they cannot produce. Completing the square produces it, and completing the square is itself just the observation that (x + d)² = x² + 2dx + d² — so to turn x² + 2dx into a perfect square you add d², which is half the middle coefficient, squared.

Start from ax² + bx + c = 0. Divide by a, move c across, add the square-completing term to both sides, and rewrite:

x² + (b/a)x = −c/a
x² + (b/a)x + (b/2a)² = (b/2a)² − c/a
(x + b/2a)²            = (b² − 4ac)/4a²
x + b/2a               = ±√(b² − 4ac) / 2a
x                      = (−b ± √(b² − 4ac)) / 2a

Two things worth taking from that derivation, both of which the SAT tests directly:

The ± is the record of lost sign information. Squaring maps 3 and −3 to the same place, so undoing it must return both. That is why quadratics have two roots and why "extraneous solutions" exist when you square both sides of an equation to clear a radical — squaring can create solutions that were never there, so every radical equation demands you check answers back in the original.

The discriminant b² − 4ac is the thing under the root, so it decides how many real roots there are. Positive → two, because ±√positive gives two different numbers. Zero → one, because ±0 is one number. Negative → none real, because no real number squares to a negative. Every "for what value of k does this equation have exactly one solution" question is b² − 4ac = 0 and nothing else.

#Exponentials: what changes is the multiplier, not the step

Linear growth adds a constant. Exponential growth multiplies by a constant.

y = a · bˣ means: start at a, and multiply by b once per unit of x.

Read the parts straight off. a is the starting amount, because at x = 0, b⁰ = 1. b is the per-period multiplier: b = 1.05 is 5% growth per period, b = 0.9 is 10% decay per period. Growth when b > 1, decay when 0 < b < 1, and the curve never reaches zero because multiplying a positive number by a positive number never yields zero.

Changing the period is changing the exponent, not the base. Something growing 5% per year, examined monthly, is 1.05^(t/12) where t is in months — one twelfth of a year's worth of multiplication per month. Every "the population doubles every 7 years, write a model in months" question is this and only this.

How to tell linear from exponential in a table in five seconds. Take differences between consecutive y values. Constant differences → linear. Not constant → take ratios instead. Constant ratios → exponential. This works because those are precisely the two definitions.

#✅ Part 3 retrieval

1. y = 2(x − 3)(x + 5). Give the x-intercepts and the x-coordinate of the vertex.
2. For what k does x² + kx + 9 = 0 have exactly one real solution?
3. A culture starts at 400 and triples every 6 hours. Write the model in hours.
4. √(x + 6) = x. Solve, and say why one candidate must be thrown out.

Answers. 1. Intercepts at x = 3 and x = −5; vertex at their average, x = −1. 2. k² − 4(1)(9) = 0, so k² = 36 and k = ±6. Both work — a question asking "one solution" about k usually has two answers for k, which is a favourite trap. 3. y = 400 · 3^(t/6). 4. Squaring gives x + 6 = x², so x² − x − 6 = 0, so (x − 3)(x + 2) = 0 and x = 3 or x = −2. Check both in the original: √9 = 3 ✓, but √4 = 2 ≠ −2 ✗. The square root symbol denotes the non-negative root, so −2 is extraneous — it was manufactured by the squaring step.

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#PART 4 · GEOMETRY FROM SIMILARITY

#Similar figures, and the scaling law that follows

Two figures are similar when one is a scaled copy of the other. If every length is multiplied by k, then every area is multiplied by k², and every volume by k³.

This is not three facts. Area is built from two lengths, so both get multiplied by k. Volume is built from three. That is the entire derivation, and it disposes of a whole category of question: "the radius is doubled, what happens to the volume of the sphere" is 2³ = 8 times, with no formula needed and no arithmetic done.

The same idea handles the shadow-and-pole problems, the nested-triangle problems, and every "these two triangles are similar, find the missing side" item: set up the ratio of corresponding sides and solve. The only genuine difficulty is identifying which sides correspond, and the answer is always: the ones opposite equal angles.

#Why the trigonometric ratios exist at all

Here is the fact that makes trigonometry possible, and almost nobody is told it:

All right triangles with the same acute angle are similar to one another.

They must be: one angle is 90°, one is your chosen angle θ, and angles sum to 180°, so the third is determined too. Same three angles means similar.

And similar figures have equal ratios of corresponding sides. So for a given θ, the ratio opposite/hypotenuse is the same number for every right triangle containing that angle — regardless of size. That number depends only on θ, so it deserves a name. It is called sin θ. Cosine and tangent are the other two ratios, named for the same reason.

This is why SOH-CAH-TOA works, and why it can only be applied to right triangles. It also explains the fact the SAT likes most: sin θ = cos(90° − θ). The side opposite one acute angle is the side adjacent to the other, so the two ratios are literally the same fraction read from the other angle's point of view.

#The circle equation is the Pythagorean theorem

A circle is the set of points at distance r from a centre (h, k). Distance between two points comes from Pythagoras on the right triangle formed by their horizontal and vertical separation:

(x − h)² + (y − k)² = r²

So the circle equation is not a formula to memorise; it is "distance equals r", squared to avoid the root. Which is why "complete the square to find the centre and radius" works: you are converting an expanded distance statement back into the form that shows the centre.

#What is on the reference sheet, and what that tells you

The digital SAT gives you a reference sheet with basic area, volume and special-right-triangle facts. That is a strong signal about what is actually being tested: not recall of those formulas, but knowing which one applies and setting it up. Time spent memorising what is printed in front of you is time stolen from the parts that are not — the Formula Vault in this course lists precisely those.

#✅ Part 4 retrieval

1. A cone's radius and height are both halved. What fraction of the original volume remains?
2. In a right triangle, sin θ = 3/5. What is cos(90° − θ)?
3. x² + y² − 6x + 4y = 12. Find the centre and radius.

Answers. 1. (1/2)³ = 1/8. No volume formula required. 2. 3/5 — it is the same ratio seen from the other acute angle. 3. Complete both squares: (x − 3)² − 9 + (y + 2)² − 4 = 12, so (x − 3)² + (y + 2)² = 25. Centre (3, −2), radius 5.

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#PART 5 · PROPORTION AND DATA

#A ratio is a multiplier waiting for a unit

"3 miles per hour" is not a pair of numbers; it is the instruction multiply hours by 3 to get miles. Once you read every rate that way, unit conversion stops being a procedure and becomes bookkeeping: chain the multipliers and cancel the units.

90 km/h → m/s :   90 × (1000 m / 1 km) × (1 h / 3600 s) = 25 m/s

The units cancel like factors because that is exactly what they are. If your units do not cancel to the units you want, your setup is wrong — and this is the fastest self-check available on any rate question. Check the units before you check the arithmetic.

#Averages, and the only useful way to think about them

The mean of n values is their total divided by n. Rearranged, and this is the version that solves SAT problems:

total = mean × count

Almost every average question on the test is a hidden total question. "The mean of 5 numbers is 12; a sixth number is added and the mean becomes 13. What was it?" — the first total is 60, the second is 78, so the number is 18. No algebra.

Median is the middle value; it ignores magnitude. That is why an outlier drags the mean and leaves the median alone, and why the SAT asks about which measure changes when a large value is added. Skewed right → mean above median, because the mean feels the size of the tail and the median only counts positions.

#What a statistic is allowed to claim

The SAT tests inference from studies, and it tests it strictly. Two rules cover nearly all of it:

• Random sampling lets you generalise to the population it was drawn from — and no further. A random sample of one school's students supports a claim about that school, never about the country.
• Random assignment is what licenses a causal claim. Without it, you may report association only. An observational study can never support "caused", however large.

And on margins of error: a margin of error describes the precision of an estimate. A wider interval means less precision, which comes from a smaller sample. The correct reading of "42% ± 3%" is "plausibly between 39% and 45%" — not "42% is wrong", and not "somewhere between 39 and 45 with certainty".

#✅ Part 5 retrieval

1. Convert 72 km/h to metres per second.
2. Seven test scores average 84. One score of 60 is removed. What is the new average?
3. A study randomly assigns volunteers to two exercise programmes and finds group A improves more. What may be concluded, and about whom?

Answers. 1. 72 × 1000 / 3600 = 20 m/s. 2. Total was 84 × 7 = 588; removing 60 gives 528 over 6 values, so 88. 3. Random assignment was used, so a causal claim about the programme is licensed — but the volunteers were not randomly sampled, so it applies only to people like those volunteers, not to the general population.

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#PART 6 · THE CHECKLIST AND STUDY ORDER

#The universal five steps

1. Name the object. Line, parabola, exponential, triangle, rate, distribution. This decides everything downstream.
2. Choose the cheapest representation. Equation, table, or picture. Translate if the one you were given is expensive.
3. Write what you know before you solve. Most Math errors are transcription, not reasoning — see Careless-Error Elimination in this course.
4. Solve, then answer the question that was actually asked. The single most common wrong answer on the Math section is a correct value of the wrong variable. If you solved for x and the question wanted 2x + 1, the test has a choice prepared for x.
5. Sanity-check the size and the units. A negative length, a probability above 1, or metres where you wanted seconds means stop and re-read.

#What is still worth memorising after all this

Deriving is fast but not free. These are the items where recall genuinely beats derivation under time pressure, and they are the reason the Formula Vault and the review deck exist in this course:

• The Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and the two special right triangles.
• The quadratic formula itself — you should be able to derive it, and should not be doing so on test day.
• Vertex x = −b/2a, and the discriminant.
• The exponential-model shape a · bˣ and how to convert the period.
• Slope from two points.

Everything else in this document is meant to be re-derivable in under ten seconds, which is what makes it survive a bad night's sleep.

#The order to learn this in

Week 1 — Parts 0, 1 and 2. Linearity is the largest and most automatable block on the section, and Part 0's translate-the-representation habit pays off immediately. Week 2 — Part 3. Advanced Math is where the hard second module lives; quadratics-in-three-forms is the highest-yield idea in this document. Week 3 — Parts 4 and 5. Geometry and data are smaller and more formula-driven, and the scaling law does a disproportionate share of the work. Throughout — official practice. Nothing here substitutes for real items from the Question Bank. This document exists so that when you get one wrong, the error log has somewhere to point.

#Where to go next in this course

• Algebra System, Advanced Math System, Problem-Solving & Data System, Geometry & Trig System — the drill methods, one per domain.
• Math Formula Vault — everything not on the reference sheet.
• Math Question-Type Playbook — the stem-to-method lookup.
• Desmos Mastery — how to make the picture route as fast as it should be.

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#REFERENCE CARD — MATH FROM FIRST PRINCIPLES