The second most common SAT Math question type (9.2% of the bank): modelling and interpreting a constant rate of change. Covers naming the rate and the starting value in the story's own units, function notation as a point on a graph, reading a linear function from a table by constant first differences, solving for an input that hits a target, and comparing two linear plans — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
Open the interactive version of this deck
Title
SAT Math · Type 2 of 19
9.2% of the question bank — 154 of 1675 questions
Objectives
A linear function is one that changes by the same amount every step. Nearly every version of this question is a short story with two numbers hidden in it: something that happens repeatedly, and something that is true before anything happens. Find those two numbers in the story's own units and the question is finished.
Two questions answer this entire type: what is true at zero, and what happens each step? Everything else is arithmetic.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 154 tagged questions of this type in the site's bank
Section
Section 1
Concept
A linear function is one that changes by the same amount every step. Nearly every version of this question is a short story with two numbers hidden in it: something that happens repeatedly, and something that is true before anything happens. Find those two numbers in the story's own units and the question is finished.
You will see it phrased in these ways:
The giveaway is that these are almost always word problems. The algebra is trivial; the marks are in translating English into m and b without swapping them.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Linear functions: the tell, the move, and the trap
The trap panel is the whole difficulty of this type. English puts the flat fee first; the equation puts the rate first. That mismatch is worth several points a test.
Prediction
Recognition first: linear means the same amount each step, not the same proportion.
Predict first
Three of these are linear. Which one is not?
Correct: An investment grows 6 per cent each year
Why: Growing by a percentage means multiplying by the same factor each period, which compounds — the amount added grows as the balance grows. That is exponential. The other three add or subtract a fixed quantity per unit, which is exactly what constant rate of change means. The tell in a word problem: per cent of signals exponential, while a fixed number of dollars, litres or degrees per unit signals linear.
Concept
The same function arrives in four different wrappers. The routine does not change.
| how it arrives | where m is hiding | where b is hiding |
|---|---|---|
| A word problem | the amount attached to per, each, or every | the fee, deposit or starting amount |
| A table | the constant change in y per one-step change in x | the y-value when x is 0, or extrapolated back to it |
| A graph | rise over run between two clear lattice points | where the line crosses the vertical axis |
| An equation to interpret | the coefficient of x | the constant term |
Interpretation questions — what does this number represent? — are the most common single variant, and they are pure units.
Definition probe
Four stories. Sort each number into rate or starting value.
Sort into buckets
In each story, is the highlighted number m or b?
Discrimination
Which of these describe a linear function?
Sort into buckets
Linear, or not?
Warm-up
Do not write an equation yet. Just name them.
Discussion prompt
A repair company charges a 30 dollar callout fee plus 45 dollars for each hour of work. What is the rate of change, and what is the value at zero? Then write the function.
Hint: Which number would you pay even if the job took no time at all?
Answer:
b = 30 dollars — the callout fee is what you owe at zero hours. It is the value at zero.
m = 45 dollars per hour — the amount added for each additional hour.
So C(h) = 45h + 30.
Note the order swap: the sentence said 30 first, but 30 is the constant and 45 is the coefficient. Writing C(h) = 30h + 45 is the single most common error on this type, and it models a 45 dollar fee with a 30 dollar hourly rate.
Pattern
Three steps, and the first two are reading rather than algebra.
Find the value at zero and write it down with its units.
Why: Look for a fee, a deposit, an initial height, a starting balance — something true before the process begins. That is b.
Find the repeated change and write it down with its units and its sign.
Why: Look for per, each, every, a. Decreasing quantities take a negative rate, and the minus sign is part of the number.
Assemble f(x) = mx + b, then re-read the question to see what it actually wants.
Why: It may want the function, a value, an input that reaches a target, or the meaning of one of the numbers. Those are four different answers from the same two facts.
Notice that step 3 exists because writing the function feels like finishing. On this type it usually is not.
Section
Section 2
Concept
A function is linear exactly when equal changes in x always produce equal changes in y.
This one test settles every is-it-linear question, whether the data arrives as a table, a graph, or a story.
Concept
In f(x) = mx + b, the coefficient of x is what happens per unit, and the constant is what is true before anything happens.
| in the story | in the function | units |
|---|---|---|
| 45 dollars per hour | m = 45 | dollars per hour |
| a 30 dollar callout fee | b = 30 | dollars |
| drains 4 litres a minute | m = negative 4 | litres per minute |
| the tank starts with 200 litres | b = 200 | litres |
Attaching units is not decoration — it is the fastest check that you have not swapped m and b.
Prediction
The order in the sentence is not the order in the equation.
Predict first
A technician charges a 60 dollar visit fee plus 35 dollars per hour. Which function is correct?
Correct: C(h) = 35h + 60
Why: The 35 is attached to per hour, so it multiplies h. The 60 is charged once regardless of duration, so it is the constant. Checking at h equals 0 gives 60, which is the visit fee alone — exactly right. The second option models a 60 dollar hourly rate and a 35 dollar fee, and the third adds them as though every hour cost 95.
Concept
f(3) = 7 says exactly one thing: when the input is 3, the output is 7 — so the graph passes through (3, 7).
Whenever a question hands you f(something) = something, immediately rewrite it as a coordinate pair. It converts notation into geometry you can use.
Prediction
Watch the step in x.
Predict first
A table shows x of 0, 3, 6, 9 with y of 5, 11, 17, 23. What is the rate of change?
Correct: 2 per unit
Why: The y-values rise by 6 each row, but x steps by 3 each row, so the rate is 6 divided by 3, which is 2 per unit of x. Answering 6 is the classic slip: it is the change per ROW, not per unit. The 3 is the step in x, and the 5 is the value at zero.
Concept
Subtract consecutive y-values and divide by the change in x.
| x | y | change in y |
|---|---|---|
| 0 | 12 | — |
| 2 | 19 | plus 7 |
| 4 | 26 | plus 7 |
| 6 | 33 | plus 7 |
Forgetting to divide by the step in x is the standard error here, and it produces a rate exactly double or triple the true one.
Concept
When asked what a number represents, answer with a quantity, a unit, and a per.
The wrong answers on interpretation questions are usually correct sentences about the wrong number, so decide which number you are describing before you read them.
Prediction
Notation into geometry.
Predict first
If g(4) = negative 2, which point lies on the graph of g?
Correct: (4, negative 2)
Why: The number inside the brackets is the input, which is the first coordinate, and the value the function returns is the output, which is the second. So g(4) = negative 2 places the point at (4, negative 2). Reversing them is the standard error and it is always offered as a choice.
Concept
Positive m rises left to right, negative m falls, and m equal to zero is a flat line.
If a story says a quantity decreases and your m came out positive, you have dropped a minus sign rather than made an arithmetic error.
Concept
Questions asking when or after how long become a one-variable equation.
This is the bridge to type 7. Setting a linear function equal to a target turns a modelling question into a one-variable solve.
Prediction
Units, not vocabulary.
Predict first
A pool's volume is V(t) = 5000 minus 60t litres, t in minutes. What does 5000 represent?
Correct: the volume in litres before draining began
Why: The constant is the value at t equals 0, which is the volume before any draining. The 60 is the rate, in litres per minute. The time to empty would be 5000 divided by 60, about 83 minutes, which is a different question — and note that the total drained by the time it empties is also 5000, which is why that distractor tempts.
Two truths and a lie
Three statements about linear functions are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: Steepness is about the ABSOLUTE VALUE of m, not its signed size. A slope of negative 8 is much steeper than a slope of positive 2, even though negative 8 is the smaller number. And a larger m only means a steeper upward line among positive slopes. The correct statement is that a larger absolute value of m means a steeper line, in whichever direction its sign points.
Check
One story, and the whole point is which number goes where.
Check your understanding
A caterer charges a 250 dollar setup fee plus 18 dollars per guest. Which function gives the total cost for g guests?
Answer: A
Why: The 18 is attached to per guest, so it multiplies g. The 250 is charged once no matter how many guests attend, so it is the constant. Checking at g equals 0 gives 250, the setup fee alone, which is exactly what the story describes.
The substitution check at zero separates A from B in about two seconds, and those are the only two choices a careless reader is deciding between.
Section
Section 3
Worked example
A moving company charges a 90 dollar base fee plus 40 dollars per hour. Write the cost function and find the cost of a 6-hour job.
Figure (svg): A line starting at 90 on the vertical axis rising 40 per hour, with the point (6, 330) marked
Value at zero: 90 dollars, because the base fee is owed for a job of no length.
Why: b is what is true before any hours are worked.
Rate: 40 dollars per hour, positive because the cost grows.
Why: m carries the per, and its units are dollars per hour.
Write C(h) = 40h + 90, then substitute h equals 6: 240 plus 90.
Why: Assemble first, then answer the specific question asked.
The verify step here is the whole safety net for this type. Substituting zero instantly exposes a swap.
Verify: check at h equals 0 the model gives 90, the base fee alone.
Why: A model that does not reproduce the starting value has m and b swapped.
Answer: C(h) = 40h + 90, and a 6-hour job costs 330 dollars
Worked example
A candle is 20 cm tall and burns down 3 cm per hour. Write its height function and find when it burns out.
Figure (svg): A falling line from 20 cm crossing zero at about 6.7 hours
Value at zero: 20 cm, the height before lighting.
Why: b is the initial height.
Rate: negative 3 cm per hour, because the candle shrinks.
Why: The minus sign is part of m whenever the quantity decreases.
Set H(t) = 20 minus 3t equal to 0 and solve: 3t equals 20, so t is 20 divided by 3.
Why: Burns out means height reaches zero, which is the x-intercept.
Burns out, empties, and reaches zero all mean the same thing: set the function equal to zero.
Verify: check t equals 6 gives 20 minus 18, which is 2 cm remaining.
Why: Still positive at 6 hours and negative after 7, so the burnout falls between — consistent with 6.67.
Answer: H(t) = 20 minus 3t, and it burns out after about 6.67 hours
Worked example
A table gives x of 3, 5, 7 and y of 14, 22, 30. Write the linear function.
Figure (svg): A line through (3, 14) and (7, 30) with its y-intercept at 2
First differences: 22 minus 14 is 8, and 30 minus 22 is 8. Constant, so it is linear.
Why: Equal changes in y over equal changes in x confirm a constant rate.
x steps by 2 each row, so the rate is 8 divided by 2, which is 4 per unit.
Why: The rate must be per one unit of x, not per row of the table.
Step back from (3, 14) to x equals 0: three steps of 4 down, so b is 14 minus 12, which is 2.
Why: The y-intercept is the value at x equals 0 even when the table omits it.
Answering 8 for the rate is the designed error. Always ask what one step of x is worth.
Verify: check x equals 7: 4 times 7 plus 2 is 30, matching the table.
Why: A model must reproduce every row it was built from, not just the one used.
Answer: f(x) = 4x + 2
Step zero
Only the opening line, not the solution.
Discussion prompt
A question reads: a scuba tank holds 80 cubic feet of air and a diver uses 1.6 cubic feet per minute. After how many minutes will 20 cubic feet remain? What do you write first?
Hint: Which of the two numbers is true before the dive starts?
Answer:
You write A(t) = 80 minus 1.6t. The 80 is the value at zero and the 1.6 is the rate, negative because air is being consumed.
Only then do you deal with the question, which is a target: set A(t) equal to 20, not to 0.
That gives 80 minus 1.6t equals 20, so 1.6t equals 60, and t equals 37.5 minutes.
The trap is solving for empty. The question asked when 20 cubic feet REMAIN, and answering 50 minutes — the time to empty — is the designed wrong answer.
Worked example
A biologist models a fish population as P(t) = 1200 minus 45t, where t is years since 2020. What do 1200 and negative 45 represent?
Figure (svg): A falling line from 1200 fish declining by 45 each year
1200 is the value at t equals 0, and t is measured from 2020.
Why: The constant is always the output when the input is zero, so the year matters.
So 1200 is the fish population in the year 2020.
Why: Interpretation must name the real-world quantity, not just say intercept.
Negative 45 is the rate: the population falls by 45 fish per year.
Why: The units of m are output units per input unit — fish per year.
Notice that t is years SINCE 2020. Interpretation questions frequently hide the reference point in that phrase.
Verify: check t equals 2 gives 1200 minus 90, which is 1110 fish in 2022.
Why: A drop of 90 over two years is consistent with 45 per year.
Answer: 1200 is the population in 2020; negative 45 is the annual decline of 45 fish per year
Worked example
Plan A costs 50 dollars plus 8 dollars a class. Plan B costs 20 dollars plus 14 dollars a class. After how many classes do they cost the same?
Figure (svg): Two lines crossing at five classes and ninety dollars
Write both: A(n) = 8n + 50 and B(n) = 14n + 20.
Why: Each plan is its own linear function with its own rate and fee.
Set them equal: 8n plus 50 equals 14n plus 20.
Why: Cost the same means the two outputs are equal at the same input.
Gather: 30 equals 6n, so n equals 5.
Why: Subtracting 8n and 20 from both sides isolates n.
Beyond 5 classes, the plan with the smaller rate wins. The steeper line overtakes despite starting lower — worth stating when a question asks which is cheaper.
Verify: check both at n equals 5: A gives 40 plus 50 and B gives 70 plus 20, both 90.
Why: Equal outputs at n equals 5 confirms the intersection.
Answer: after 5 classes, both cost 90 dollars
Faded example
From memory. These four lines convert English into a linear function.
Fill in the blanks
The number attached to a per or an each is the rate. The fee, deposit or initial amount is the value at zero. A quantity that decreases has a negative rate. And f(3) = 7 means the graph passes through the point (3, 7).
Why: Those four conversions handle every linear-function question on the test. The one that costs the most marks is the second, because English puts the fee first in the sentence while algebra puts it last in the equation.
Worked example
A line passes through (2, 7) and (6, 19). What is its rate of change, and what is its value at zero?
Figure (svg): A line through (2, 7) and (6, 19) crossing the vertical axis at 1
Rise: 19 minus 7 is 12. Run: 6 minus 2 is 4.
Why: Subtract in the same order on both coordinates or the sign will flip.
Rate: 12 divided by 4, which is 3 per unit.
Why: Slope is rise over run, the change in output per one unit of input.
Step back from (2, 7) by two units of x: 7 minus 6, so b is 1.
Why: Two steps left at 3 per step lowers the value by 6.
Always verify with the point you did NOT use to find b. Verifying with the same point proves nothing.
Verify: check (6, 19): 3 times 6 plus 1 is 19.
Why: The point that was not used to find b confirms the whole function.
Answer: the rate is 3 per unit and the value at zero is 1, so f(x) = 3x + 1
Fill the middle
Units are the check that you picked the right number.
Fill in the blanks
In C(h) = 45h + 30 for a repair job, the units of the 45 are dollars per hour and the units of the 30 are dollars. In V(t) = 5000 minus 60t for a draining pool, the units of the 60 are litres per minute.
Why: The rate always carries a per and the constant never does. If the number you called m does not have per in its units, you have selected the constant by mistake — which is exactly the swap this type is built to catch.
Estimation
A rough answer protects you from a decimal-point error.
Predict first
A tank holding 4,800 litres drains at 32 litres per minute. Roughly how long until it is empty?
Correct: about 150 minutes
Why: Round 32 up to a friendly number: at 30 litres a minute, 4,800 divided by 30 is 160 minutes, so the true answer is a little under that. The exact value is 150. Estimating first catches the two errors this question invites — dividing by 320 or by 3.2 — because both give answers off by a factor of ten from the estimate.
Check
Watch the step in x before you compute anything.
| x | 0 | 4 | 8 | 12 |
|---|---|---|---|---|
| f(x) | 7 | 19 | 31 | 43 |
Check your understanding
The table shows values of a linear function f. What is f(6)?
Answer: A
Why: The y-values rise by 12 for every 4 units of x, so the rate is 12 divided by 4, which is 3 per unit. The value at zero is 7, so f(x) = 3x + 7 and f(6) is 18 plus 7, which is 25. Alternatively, 6 sits midway between 4 and 8, so f(6) is midway between 19 and 31 — also 25.
The midpoint shortcut is worth noticing: on a linear function, the value halfway between two inputs is halfway between their outputs.
Section
Section 4
Trap
The trap. A gym charges 40 dollars to join plus 25 dollars a month, and you write C(m) = 40m + 25.
The sentence gave 40 first, and equations are read left to right, so the 40 landed in the coefficient slot. The model now charges 40 dollars a month with a 25 dollar joining fee.
The two are impossible to tell apart by looking at the equation, which is why this error survives all the way to the answer sheet.
The fix. Substitute zero before you do anything else.
C(0) should be the joining fee alone. With the wrong model, C(0) is 25 — which is not the 40 the story stated, so the swap is exposed instantly.
Trap
The trap. A tank starts with 200 litres and drains 4 litres a minute, and you write V(t) = 4t + 200.
The model now says the tank FILLS. After 10 minutes it predicts 240 litres, and the tank never empties at all.
Every subsequent step is executed correctly on a model that describes the opposite process.
The fix. Decide whether the quantity grows or shrinks before writing m, and attach the sign at that moment.
Draining, depreciating, cooling, burning down, using up and paying off all mean a negative rate.
Error analysis
A student's work on a phone-plan question. One line is wrong. Find it before reading the notes.
Annotate
On: \( \text{20 dollars monthly, 5 cents a minute} \;\Rightarrow\; C(m) = 20m + 0.05 \)
Substituting zero costs three seconds and catches the single most expensive error on this type. Make it automatic.
Trap
The trap. A table steps x by 3 and y rises by 12 each row, and you report the rate as 12.
The rate of change is per ONE unit of x. Here y rises 12 for every 3 units, so the rate is 4.
Reporting 12 overstates the rate by exactly the size of the x-step, and it is only visible if you look at the x column.
The fix. Compute rise over run, not rise alone. The run is the change in x, and it is only 1 when the table happens to step by 1.
Elimination
A car starts with a full 60-litre tank and uses 7 litres per 100 km.
Eliminate the wrong options
Which function gives the litres remaining after d hundred kilometres? Three can be ruled out on structure alone.
Survives elimination: b
Why: The tank starts full, so the constant is 60, and fuel is consumed, so the rate is negative 7. Only one choice has both. Each wrong answer is a distinct predictable failure: the wrong sign, the swap, and the missing constant. All three are visible without any arithmetic — checking the value at d equals 0 rules out two of them on its own.
Trap
The trap. A question asks when 20 litres remain, and you set the function equal to 0 because that is the familiar move.
Empty and reaches 20 are different targets, and they give different times. The habit of solving for the x-intercept overrides what the stem asked.
This is a reading failure, not an algebra one — and the time-to-empty is always offered as a choice.
The fix. Underline the target value in the stem before you write the equation.
The equation to solve is function equals target, and the target is whatever number the question names.
Counterexample
A claim that sounds reasonable and is not.
Discussion prompt
A student says: if a line has a bigger slope, it always has bigger values. Give a counterexample.
Hint: Think about where the two lines start.
Answer:
Counterexample: f(x) = 10x and g(x) = x + 1000. The slope of f is far bigger, but at x equals 1, f gives 10 while g gives 1001.
Slope controls how fast a line grows, not how high it currently is. The starting value decides where it begins.
A line with a bigger slope will eventually overtake, but eventually can be a long way out — here not until x equals about 111.
This is exactly the structure of a which-plan-is-cheaper question: the plan with the higher rate but lower fee wins only after the crossover point, and the question is usually about which side of that point you are on.
Edge cases
Linear models are convenient and they are not always true.
Discussion prompt
A model says a plant grows 2 cm per week: H(w) = 5 + 2w. For what values of w is this model sensible, and where does it break?
Hint: What does the model predict for very large w, and for negative w?
Answer:
Negative w is meaningless here — it would describe the plant before it was planted, and the model predicts a height of 5 cm at w equals 0, not before.
Very large w breaks it too: at w equals 500 the model predicts a plant over 10 metres tall. Real growth slows and stops.
The technical name for this is the domain of the model — the set of inputs for which it actually describes the situation.
The SAT tests this directly by asking which value of x makes sense in context, and the answer is usually the one that is non-negative and within a stated range.
It also drives the extrapolation warning on scatterplot questions: a linear fit is trustworthy inside the data and speculative outside it.
Check
The target is not zero. Read carefully.
Check your understanding
A printer starts with 900 sheets and uses 24 sheets per job. After how many jobs will 300 sheets remain?
Answer: A
Why: The model is S(j) = 900 minus 24j. Setting it equal to 300 gives 24j equals 600, so j equals 25. Checking: 25 jobs use 600 sheets, and 900 minus 600 is 300 remaining, exactly as required.
Choice B is the designed trap, and it is the answer to a question the stem did not ask. Underline the target before writing the equation.
Section
Section 5
Matching
Six pieces of English, six pieces of algebra. No computing.
Match the pairs
Why: Every one of these is a translation rather than a calculation, and translation is where this question type is won and lost. Notice that the last two rows both produce equations to solve — the difference is whether the right-hand side is a number or another function.
Sorting
Six phrases from real stems. Which slot does each fill?
Sort into buckets
Rate of change, or value at zero?
Item (e) is worth pausing on. When a story sets a reference time such as 6 a.m., that moment is t equals 0 — and interpretation questions love to hide the reference point in a phrase like years since 2020.
Comparison
Fill the blanks from memory. Same function, four presentations.
Comparison matrix
| arrives as | how to get m | how to get b |
|---|---|---|
| A word problem | the number attached to per or each | the fee or initial amount |
| A table | change in y divided by change in x | the y-value at x equals 0, or step back to it |
| A graph | rise over run between two lattice points | where it crosses the vertical axis |
| An equation | the coefficient of x | the constant term |
Only the middle two rows involve any calculation. The other two are pure reading, which is why this type rewards care rather than speed.
Trade off
Once you have the rate, there is more than one route to b. Fill in what each costs.
Comparison matrix
| method | when it is fastest | what can go wrong |
|---|---|---|
| Read it off the table at x equals 0 | when the table includes a zero row | the table often omits x equals 0 |
| Step backwards from a known point | when x equals 0 is only a step or two away | easy to step the wrong direction |
| Substitute a point into y = mx + b | always works, any point | one more line of algebra than stepping |
| Read it off the graph | when the axes are clearly scaled | axes that do not start at zero |
The substitution method never fails, so make it your fallback. The stepping method is faster but inverts the sign if you step the wrong way, which is the error worth guarding against.
Real world
One minute on why this type is worth more than its 9.2 per cent.
Discussion prompt
Phone plans, taxi fares, salary plus commission, and utility bills are all the same function. What makes them linear, and why does the SAT keep using them?
Answer:
They are linear because they combine a fixed part that does not depend on usage with a variable part that is charged per unit. That is precisely mx + b.
The fixed part is the line rental, the flag drop, the base salary, the standing charge. The variable part is the per-minute, per-mile, per-sale, per-kilowatt rate.
The SAT uses them because the translation step is genuinely hard while the algebra is easy, so the question discriminates on reading rather than on computation.
It is also the most transferable thing on the Math section: comparing two plans by finding the crossover point is a calculation most adults actually perform.
Ranking
All four are linear. Order them from cheapest to most expensive at exactly 10 units.
Put in order
Why: At 10 units: (b) is 20, (a) is 5 plus 20 which is 25, (c) is 40, and (d) is 10 plus 30 which is 40. So (c) and (d) tie at 40 — order them either way and the reasoning is the same. The point of the exercise is that the ranking depends entirely on the input: at 1 unit the order is completely different, with (c) cheapest at 4 and (b) most expensive at 20. A plan is never cheaper in the abstract, only at a given usage.
Concept
This type is 9.2 per cent of the section and is almost pure translation, which makes it one of the most improvable. Four sessions.
| session | what you do | why |
|---|---|---|
| 1 | Twenty word problems, writing only the function — do not solve any of them. | Isolates the translation step, which is where the marks actually are. |
| 2 | Twenty interpretation questions: what does this number represent? Answer aloud with units. | Interpretation is the most common variant and it is graded on units. |
| 3 | Tables and graphs: ten of each, always stating the x-step before computing the rate. | Builds the habit that prevents the per-row error. |
| 4 | Mixed set under time, including target questions and two-plan comparisons. | Target and comparison questions are where the type overlaps with types 5 and 7. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 154 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state where m and b hide in a word problem, in a table, and in a graph — and give the check that catches a swap.
Hint: The check involves a single substitution.
Answer:
Word problem: m is attached to per, each or every; b is the fee, deposit or initial amount.
Table: m is the change in y divided by the change in x; b is the value at x equals 0, or stepped back to it.
Graph: m is rise over run between two lattice points; b is where the line crosses the vertical axis.
The check: substitute zero. f(0) must equal the starting value the story gave. If it does not, m and b are swapped.
If you produced the check as well as the three locations, you have the whole type.
Explain it
Two minutes, out loud.
Discussion prompt
A friend keeps writing C(h) = 30h + 45 for a 30 dollar fee plus 45 an hour. How do you explain it so it does not come back?
Answer:
Ask them what the job costs if it takes zero hours. The answer from the story is 30 dollars — just the callout.
Now substitute zero into their equation: 30 times 0 plus 45, which is 45. That is not the fee the story gave, so the model is wrong.
Give them the rule as a sentence: the number that gets multiplied is the one with per in its units. Dollars per hour multiplies hours; plain dollars does not multiply anything.
Then have them re-derive it: 45 has a per, so it takes the h. 30 has no per, so it stands alone. C(h) = 45h + 30.
Commit first
Commit before you check.
Predict first
For W(t) = 152 minus 1.5t, a person's weight in kg after t weeks of training, what does 1.5 represent?
Correct: the weight lost in kilograms each week
Why: The 1.5 is the coefficient of t, so its units are output units per input unit — kilograms per week. The minus sign in front means the weight is decreasing, so it is 1.5 kg lost each week. The 152 is the starting weight, and the total lost depends on how many weeks pass, so it is not a fixed number at all.
Connect it up
Blank paper. Make the revision artefact.
Draw it
Draw a map of linear functions. Put f(x) = mx + b in the middle. From m, draw four arrows to where it hides: the per in a word problem, rise over run on a graph, change in y over change in x in a table, and the coefficient in an equation. Do the same for b: the fee, the vertical intercept, the value at x equals 0, and the constant term. Underneath, write the three question shapes — write the function, interpret a number, hit a target — and next to each write the first line you would put down. Finally, box the swap check: substitute zero and compare to the starting value.
Exit ticket
One question before you close the deck.
Predict first
You have just written C(g) = 250g + 18 for a caterer charging a 250 dollar setup fee plus 18 dollars per guest. What is the fastest way to discover you are wrong?
Correct: Substitute g equals 0 and compare with the setup fee
Why: At zero guests the cost should be the 250 dollar setup fee alone. The written model gives 18, which contradicts the story immediately. Graphing shows a straight line either way and proves nothing, expanding does nothing to a model already in simplest form, and testing three values takes three times as long as testing the one value whose answer you already know.
Recap
One type, one translation: find what is true at zero and what happens each step.
| never do this | do this instead |
|---|---|
| Put the fee in the coefficient slot because it came first | Give the coefficient to the number with per in its units |
| Write a positive rate for a draining tank | Attach the minus sign as you identify the rate |
| Report the change per table row as the rate | Divide by the step in x, every time |
| Answer slope when asked what a number represents | Answer with the quantity and its units |
| Solve for zero out of habit | Solve for the target the question named |
| Verify with the point you used to find b | Verify with the other point |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — and the site's SAT pages to drill this type in isolation, then mixed
Want this taught 1-on-1? Alexander tutors SAT Prep — $55/session, free consultation.