A trial-session algebra deck: how the digital test is built and why module one accuracy matters most, a six-question timed diagnostic with one skill per question, then the named rules behind each miss — clearing fractions, distributing negatives, choosing between substitution, elimination and the graphing calculator, the 'for what value of k' systems question, flipping inequalities, and translating word problems — closing with an error log and a four-week plan.
Subject: SAT Prep · 61 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
SAT Prep · trial session
Find where the points are leaking, then fix them one named rule at a time
Objectives
You already know your own gaps, which puts you ahead of most people starting prep. Today is about turning 'algebra' into a short list of specific named rules, so that practice has a target.
Nothing today is a test of you. Every miss we find is a point we now know how to get back, which is the whole reason to look for them.
College Board — Digital SAT Suite Assessment Specifications: the Algebra domain is roughly 35 percent of the Math section — the Algebra domain is about 35 percent of the Math section
Section
Section 1
Concept
Reading and Writing comes first, in two modules of twenty-seven questions with thirty-two minutes each. Math follows, in two modules of twenty-two questions with thirty-five minutes each. Each section is scored from two hundred to eight hundred.
The graphing calculator is built into every math question, and there is a reference sheet of formulas you never have to memorise.
College Board — Digital SAT Suite, including the official test description — the official test description
Picture it
Four modules, and one feature that changes how you should think about the first one.
Figure (svg): Four rows showing the two Reading and Writing modules of twenty-seven questions in thirty-two minutes and the two Math modules of twenty-two questions in thirty-five minutes
That last line matters more than anything else about the format: accuracy in module one is worth more than speed, because it decides which module two you are given.
Prediction
Given that module two adapts to module one.
Predict first
If you had to be more careful in one module, which?
Correct: Module 1.
Why: Module one decides which version of module two you receive, and the higher-scoring version is the one that makes the top of the range reachable. So the first module is worth a little extra care, and it is also the one where the questions are more evenly mixed in difficulty.
Concept
This is the tool for the pressure you described. It is a habit rather than a tip, and it is the single change that moves scores fastest for students who know the content.
Before you start writing, decide how you are going to do the question: by hand, on the calculator, or not at all right now. Then, at about sixty seconds, ask whether you are on a path to an answer. If you are not, take your best guess, flag it, and move.
A student I worked with recently went from six hundred and ten to six hundred and seventy on math with this, and said it helped on Reading and Writing too.
College Board — Digital SAT Suite, including the official test description — the flag-and-review tool is built into the testing app
Picture it
Pre-decide, clock, mark, return.
Figure (svg): A four step flow: decide the method before writing, check at sixty seconds whether you are on a path, guess and flag if not, and return with saved time
The hardest part is the third box, because leaving a question feels like losing. It is the opposite: it is protecting three easy questions further on.
Section
Section 2
Concept
Each question below targets exactly one thing, so that whatever happens the result points at a specific rule rather than at a vague area.
Work them on paper, timed. We will sort the results together afterwards.
Khan Academy — Official Digital SAT Prep, Math — the same skill breakdown the official practice platform reports
Pattern
Four rules, so that whatever happens the result is useful.
Nothing here is graded, and there is no score at the end of it. We are looking for named rules, and a miss is how we find one.
Warm-up
One minute, no writing.
Discussion prompt
When you did your practice test, was there a question where you knew the method but still got it wrong? What happened on that one?
Hint: Think about the last few questions of a module rather than the hard ones.
Answer:
Almost always it is one of three things: a sign lost while distributing, the wrong quantity bubbled, or a rushed guess near the end of a module.
All three have specific fixes and none of them is 'be more careful'. That phrase is not actionable, which is why we are going to replace it today with named rules.
Khan Academy — Official Digital SAT Prep, Math — the official platform reports misses by skill for the same reason
Check
Two minutes. Show the first line of working.
Check your understanding
Solve for x: two-thirds of x, minus the quantity x minus 4 over 2, equals 3.
Answer: A
Why: Multiply every term by 6: 4x minus 3 times the quantity x minus 4, equals 18. Distribute carefully to get 4x minus 3x plus 12 equals 18, so x equals 6. Check: two-thirds of 6 is 4, and 6 minus 4 over 2 is 1, and 4 minus 1 is 3.
Check
Two minutes. Any method.
Check your understanding
If y equals 2x minus 1 and 3x plus y equals 14, what is the value of x?
Answer: A
Why: One variable is already alone, so substitute: 3x plus 2x minus 1 equals 14, giving 5x equals 15 and x equals 3. Then y is 5.
Check
One minute.
Check your understanding
Solve for x: negative 2x plus 5 is greater than 11.
Answer: A
Why: Subtract 5 to get negative 2x greater than 6, then divide by negative 2 and reverse the sign, giving x less than negative 3. Check with x equal to negative 4: 8 plus 5 is 13, which is greater than 11.
Section
Section 3
Concept
Two columns, and the wording is deliberate. Anything correct goes under already solid. Anything missed goes under points we just found.
| already solid | points we just found |
|---|---|
| what you got right, at speed | the specific rule behind each miss |
| keep, do not re-drill | this is today's work |
A found point is worth more than a solid one right now, because a found point is a score change and a solid one is not. That is not encouragement; it is arithmetic.
Worked example
The first diagnostic question, done slowly.
Find the smallest number both denominators divide
Why: Three and two both divide six, so six is the multiplier.
Multiply every term, including the right-hand side
Why: Missing the right-hand side is the second most common error here, after the sign.
\[ 6 \cdot \frac{2x}{3} - 6 \cdot \frac{x-4}{2} = 6 \cdot 3 \]
\[ 4x - 3(x-4) = 18 \]
Distribute the negative into BOTH terms
Why: Minus three times x is minus three x, and minus three times minus four is plus twelve. The plus is the whole difficulty of this line.
\[ 4x - 3x + 12 = 18 \]
\[ x = 6 \]
Verify: by substituting back into the original
Why: Two-thirds of six is four, and six minus four over two is one. Four minus one is three, which matches. The check takes ten seconds and it catches the sign error every time it happens.
Picture it
One multiplication, applied to every term.
Figure (svg): A fractional equation being multiplied through by six to become an equation with no fractions
Trap
The line everything depends on.
\[ -3(x - 4) = -3x - 12 \]
Multiply the negative three into x, then copy the rest
Why: The minus sign was applied once and then forgotten.
Get an answer of 2 instead of 6
Why: Both numbers are in the answer choices, so nothing about the result looks wrong.
Two multiplications, written out, every time.
\[ -3(x-4) = (-3)(x) + (-3)(-4) = -3x + 12 \]
Write the second product before simplifying
Why: Negative times negative is positive, and writing the intermediate line is what makes that visible.
Then substitute back
Why: The check is what makes this a rule you own rather than one you have been told.
Fill the middle
The setup and the answer are given.
Fill in the blanks
4x - 3(x-4) = 18 \;\Rightarrow\; 4x - 3x + 12 = 18 \;\Rightarrow\; x = 6
Why: Both terms inside the bracket get multiplied by negative three, so the minus four becomes plus twelve. Collecting the x terms leaves x plus twelve, and subtracting twelve gives six.
Concept
Substitution when a variable is already alone. Elimination when the coefficients line up or cancel. The graphing calculator when the numbers are ugly or the clock is short.
All three give the same answer. Choosing quickly is what saves the time, not being good at any one of them.
Picture it
Look at the equations before you touch them.
Figure (svg): A decision flow choosing substitution when a variable is isolated, elimination when coefficients cancel, and the graphing calculator when numbers are messy
Worked example
Two equations where the y terms cancel on sight.
\[ 2x + 3y = 12 \qquad \text{and} \qquad 4x - 3y = 6 \]
Notice the coefficients of y are opposites
Why: Plus three and minus three. Adding the equations removes y with no preparation at all.
\[ 6x = 18 \Rightarrow x = 3 \]
Substitute back into the simpler equation
Why: Six plus three y equals twelve, so three y is six and y is two.
\[ (3, 2) \]
Verify: in the equation you did not use
Why: Four times three minus three times two is twelve minus six, which is six. It checks, so both values are right rather than just consistent.
Picture it
The same system, done the other way.
Figure (svg): The Bluebook Desmos panel with two equations entered and their intersection at three comma five highlighted
Worth it when the numbers are ugly. Not worth it here, where elimination took fifteen seconds and typing would take thirty.
Concept
The test asks about systems with no solution or infinitely many solutions constantly, and always in the same disguise: a coefficient is replaced by a letter.
Two lines with the same slope never meet unless they are the same line. So compare the equations term by term and see whether one is a multiple of the other.
Khan Academy — Official Digital SAT Prep, Math — systems of linear equations
Picture it
Same slope both times. The intercept decides everything.
Figure (svg): Two coordinate planes, the first with parallel lines that never meet and the second with two identical overlapping lines
Worked example
For what value of k does the system have no solution?
\[ 3x + ky = 8 \qquad \text{and} \qquad 6x + 10y = 20 \]
Find the factor between the x coefficients
Why: Six is twice three, so if the lines are parallel the whole left side must scale by two.
Match the y coefficients under that factor
Why: Twice k must be ten, so k is five.
\[ 2k = 10 \Rightarrow k = 5 \]
Now check the constants, which is what decides which case it is
Why: Twice eight is sixteen, and the second equation says twenty. They disagree, so the lines are parallel rather than identical.
\[ k = 5 \Rightarrow \text{no solution} \]
Verify: by imagining the other outcome
Why: Had the second constant been sixteen instead of twenty, the same k would have given infinitely many solutions instead. The constant is the whole difference, and the test asks both versions.
Picture it
The left column is this question; the right column is the version with a different constant.
Figure (svg): Two bars comparing a system whose constants disagree giving no solution against one whose constants agree giving infinitely many
Elimination
A system where doubling the first equation matches the second on both variables but not on the constant.
Eliminate the wrong options
What can you say?
Survives elimination: A
Why: Two equations describe parallel lines exactly when the variable coefficients are proportional and the constants are not. That is the whole test, and it takes about ten seconds.
Section
Section 4
Worked example
A line passes through two, seven and five, sixteen. Find the equation, and say what the slope means.
Compute the slope as rise over run
Why: Sixteen minus seven is nine; five minus two is three.
\[ m = \frac{9}{3} = 3 \]
Find the starting value using either point
Why: Seven equals three times two plus b, so b is one.
\[ y = 3x + 1 \]
Say what the three means in context
Why: For each additional unit of x, y increases by three. That phrasing is the answer pattern the test wants.
Verify: with the second point
Why: Three times five plus one is sixteen, which matches. Always check with the point you did not use to find b.
Picture it
The dashed steps show the rise and the run.
Figure (svg): A line through two comma seven and five comma sixteen with dashed run of three and rise of nine and the intercept at one marked
When the test asks what a number represents in a context, this sentence pattern is the answer: for each additional unit of the input, the output changes by the slope.
Matching
A gym charges a thirty dollar signup fee plus fifteen dollars a month.
Match the pairs
Why: Every linear word problem on this test has this shape: a fixed amount that does not depend on the variable, plus a rate that multiplies it. Naming which is which before writing the equation is the whole translation step.
Worked example
The gym charges thirty dollars to join and fifteen dollars a month. After how many whole months does the total exceed one hundred and eighty dollars?
Write the total as a fixed part plus a rate part
Why: Thirty does not depend on the number of months; fifteen multiplies it.
\[ C = 30 + 15m \]
Translate 'exceeds' into an inequality
Why: Exceeds means strictly greater than.
\[ 30 + 15m > 180 \]
Solve, then read the context
Why: Fifteen m is greater than one hundred and fifty, so m is greater than ten.
\[ m > 10 \Rightarrow m = 11 \]
Verify: both sides of the boundary
Why: At ten months the total is one hundred and eighty exactly, which does not exceed it. At eleven it is one hundred and ninety-five, which does. So eleven is right, and this is why 'exceeds' had to be a strict inequality.
Trap
Solving the third diagnostic question.
\[ -2x > 6 \;\Rightarrow\; x > -3 \]
Divide both sides by negative two and keep the direction
Why: Every other operation preserves the direction, so this one is easy to do on autopilot.
Get exactly the wrong half of the number line
Why: And the wrong half is always one of the answer choices.
Flip when you multiply or divide by a negative, and only then.
\[ -2x > 6 \;\Rightarrow\; x < -3 \]
Test one number from your answer
Why: Try negative four: two times four is eight, plus five is thirteen, which is greater than eleven. It works, so the direction is right.
Use the test point on graphing questions too
Why: For a shaded-region question, try the origin. If it satisfies the inequality, the origin is in the shaded region.
Picture it
The solution is everything to the left of negative three.
Figure (svg): A number line shaded to the left of negative three with a hollow circle at negative three
Translation
Word problems are a translation exercise, and the dictionary is short.
Match the pairs
Why: The third one is the trap: 'five less than x' reverses the order of what you read, giving x minus five rather than five minus x. Translate line by line rather than reading the whole problem and then writing an equation from memory.
Pattern
Six rules. Between them they cover most of what the Algebra domain asks.
| before you bubble | why |
|---|---|
| reread the last line of the question | x, 2x and x plus 3 are all in the choices |
| check units and whole numbers | eleven months, not ten point something |
| substitute your answer back | catches every sign error |
College Board — Digital SAT Suite Assessment Specifications: the Algebra domain is roughly 35 percent of the Math section — the Algebra domain
Check
Solve it on paper before you click.
Check your understanding
If 5x minus 3 equals 12, what is the value of 10x minus 6?
Answer: A
Why: The target is exactly double the given expression, so double both sides: 10x minus 6 is twice 12, which is 24. Solving for x also works and gives x equal to 3, then 30 minus 6 equals 24.
Check
Solve it on paper before you click.
Check your understanding
A plan costs C equals 45 plus 12h, where h is hours of service. What does the 12 represent?
Answer: A
Why: The number multiplying the variable is a rate: cost per hour. The answer pattern the test rewards is 'for each additional unit of the input, the output changes by this much'.
Sorting
Look at the equations only. Do not solve any of them.
Sort into buckets
Substitute, eliminate, or graph?
Making this decision in five seconds, before touching anything, is worth more than being fast at any one method.
Error analysis
A first attempt at clearing the denominators.
Annotate
On: \( \frac{x}{4} + \frac{x-1}{3} = 2 \;\Rightarrow\; 3x + 4(x-1) = 2 \)
Multiplying every term means every term, including the one with no fraction in it. Checking by substituting back catches this instantly.
Estimation
A gym charges thirty dollars to join plus fifteen a month.
Predict first
Roughly how many months until the total passes two hundred dollars?
Correct: About 11.
Why: Ignore the joining fee for a second: two hundred divided by fifteen is a bit over thirteen. The thirty dollar fee covers about two months of that, so the answer is around eleven. Estimating first means an answer of six or twenty gets rejected before you check anything.
Section
Section 5
Concept
Beyond linear algebra the test moves to quadratics, and the same expression gets written three ways depending on what the question is hiding.
\[ y = x^2 - 5x + 6 = (x-2)(x-3) = (x - 2.5)^2 - 0.25 \]
Standard form gives the y-intercept, factored form gives the roots, and vertex form gives the minimum. Recognising which form you have been handed is most of the question.
Picture it
The same parabola, three times.
Figure (svg): Three rows pairing standard, factored and vertex form with the fact each one reveals
Worked example
Solve the equation whose standard form has a leading coefficient of one, a linear coefficient of negative five and a constant of six.
Find two numbers multiplying to the constant and adding to the linear coefficient
Why: Negative two and negative three multiply to six and add to negative five.
\[ (x-2)(x-3) = 0 \]
Set each factor to zero
Why: A product is zero exactly when one of its factors is.
\[ x = 2 \quad \text{or} \quad x = 3 \]
Verify: by substituting the smaller root
Why: Four minus ten plus six is zero. And the two roots average to two point five, which is the vertex input from the vertex form, so all three forms agree.
Picture it
Roots, vertex and intercept, marked.
Figure (svg): A parabola crossing at two and three with the vertex at two point five and the y-intercept at six marked
That halfway shortcut is worth knowing: if a question gives you the roots and asks for the vertex, you can average them and skip the algebra.
Discrimination
The shape of the equation tells you which toolkit to open.
Sort into buckets
Which kind of equation is each?
Explain it
Two sentences, out loud.
Discussion prompt
Why is guessing and moving on at sixty seconds better than pushing through a hard question?
Hint: Compare the value of the hard question against the value of the two questions at the end you never reached.
Answer:
Because every question is worth the same, and a minute spent on a hard one is a minute taken from two easier ones later in the module.
And you can come back. Flagging costs nothing, and returning with a fresh look often solves in twenty seconds what was stuck for two minutes.
College Board — Digital SAT Suite, including the official test description — the flag and review tool
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the useful combination to find.
Predict first
If 4x + 8 = 20, what is the value of x + 2?
Correct: 5.
Why: Divide the whole equation by 4: x plus 2 equals 5. No solving for x is needed at all. Solving anyway gives x equal to 3, which is the distractor sitting in the choices for exactly that reason.
Two truths and a lie
Three statements about the digital test.
Eliminate the wrong options
Which is true?
Survives elimination: A
Why: The adaptive structure is the one genuinely new thing about the digital format, and it changes the priority: get module one right rather than fast, and never leave a blank anywhere.
Real world
A phone plan costs twenty-five dollars a month plus eight cents a minute.
Discussion prompt
Write the cost as an equation, and say what each number means.
Hint: Which number does not depend on how much you use it?
Answer:
Cost equals twenty-five plus zero point zero eight times the number of minutes.
Twenty-five is the fixed part, paid whether or not you make a call. Eight cents is the rate, the amount the total changes for each extra minute.
It is the same structure as the gym problem, and as almost every linear word problem on the test: a fixed part plus a rate part.
Section
Section 6
Concept
Every missed question gets one row, written the same day, with four columns: the question, what you did, the rule, and whether you would get it now.
The third column is the important one. A miss recorded as a rule is a thing you can fix. A miss recorded as 'careless' is not.
Khan Academy — Official Digital SAT Prep, Math — reviewing misses by skill is what the official platform reports too
Picture it
Three real rows from a first week.
Figure (svg): An error log table with columns for the question, what the student did, the rule, and whether they would get it now
After two weeks the third column starts repeating, and the repeats are the study plan. It writes itself.
Concept
A mid-September test gives roughly four weeks. Two sessions a week plus about twenty minutes a day of practice is enough time if it is aimed.
| week | math | reading and writing |
|---|---|---|
| 1 | linear equations, systems, functions | punctuation between clauses |
| 2 | quadratics and exponents, first timed module | subject-verb agreement, pronouns |
| 3 | problem solving and data, geometry basics | transitions and vocabulary in context |
| 4 | two full timed tests, pacing drills, taper | error-log review across all four weeks |
Scores move in steps, not in a line. A flat week is normal and expected, which is why we track module accuracy and the error log rather than only the headline number.
College Board — Digital SAT Suite, including the official test description — official full-length practice tests for weeks three and four
Exit ticket
One honest answer. There is no wrong one, and it decides what we open with next time.
Predict first
Of everything today, which felt least automatic?
Correct: Whichever you picked is where we start next session.
Why: Every item on that list is a named rule with a fifteen-minute fix, which is exactly why we spent today turning 'algebra' into this list. Naming the weak one is worth more than another pass over the strong ones.
Connect it up
Ten minutes, and then stop.
Draw it
Take the practice test you already did. Pick the five math questions you missed and write one error-log row for each: what you did, the rule, and whether you would get it now.
Bring the five rows next session. They are a better plan than anything either of us could invent in advance.
Recap
The point of the diagnostic was not the score. It was turning one word, algebra, into a short list of named rules.
| question | answer | the rule it tested |
|---|---|---|
| fractions equation | x = 6 | multiply every term, distribute the negative |
| substitution system | x = 3, y = 5 | substitute when a variable is alone |
| elimination system | (3, 2) | add when coefficients are opposites |
| no solution | k = 5 | constants decide the case |
| inequality | x less than -3 | flip for a negative divide |
| gym word problem | 11 months | fixed part plus rate part |
College Board — Digital SAT Suite, including the official test description — official practice, and the site's own SAT page for daily drilling
Want this taught 1-on-1? Alexander tutors SAT Prep — $55/session, free consultation.