Algebra 1 (course 1200310) B.E.S.T. benchmarks

In the evaluating skill, difficulty climbs from a single root (27^1/3) to a root followed by a power (16^3/4), then to negative exponents that become reciprocals (8^-2/3 = 1/4), fractional bases such as (4/9)^-1/2, and two rational-exponent values added or multiplied. Reverse items ask which of four expressions equals 6. In the converting skill, students choose the radical form of 6^3/4 or the rational-exponent form of 1 over the fifth root of 13⁴, select every expression equivalent to 4^3/2, and match exponent forms to radical forms with one radical left over. Matching boards in the evaluating skill also carry one value that no row uses.

Wrong answers come from the errors students make with these exponents: dividing the base by the denominator (9 for 27^1/3), taking the wrong root, reading a negative exponent as a negative number (4^-1/2 = -2), flipping the fraction so the power becomes the root, and multiplying or adding the index and the power (5¹² or 5⁷ for the fourth root of 5³). Where options pair a rewrite with a value, the two slips are crossed, so the right exponent and the right value each appear twice; options that convert a form cross the swapped index with the reciprocal the same way. The correct value sits anywhere in the order of the options, not always in the middle.

FAST practice uses larger perfect powers (2197^1/3, 4096^3/4), asks for the single radical equal to 31^2/3 · 31^1/2, and includes a multiselect of expressions equal to 1/12. A lint rule fails any item on this benchmark with no fractional exponent in the stem, the options or the typed key, in both banks.

Most questions give a monomial expression and ask for its simplest form with positive exponents, as a choice or as a typed answer. Harder items combine a product and a quotient in one fraction with two variables, or ask for a missing exponent or coefficient, such as the value of a in (5x^a)(3x^3/4) = 15x^5/2. Multiselect and matching items ask which expressions equal a target such as 6x^3/2. Error-analysis items show a classmate's three-step work and ask for the first wrong step and what it should say.

The wrong answers are the slips students make with fractional exponents: multiplying exponents that should be added, adding numerators and denominators (3/4 + 1/2 becoming 4/6), dividing exponents that should be subtracted, subtracting in the wrong order, adding the coefficients instead of multiplying, forgetting to raise the coefficient to the power, and leaving a negative exponent in the answer. On product, quotient and power items whose answer has a coefficient and an exponent to get right, the options cross a coefficient slip with an exponent slip, so no wrong answer is the lone near miss.

Typed monomials always simplify to whole-number exponents, such as x⁹ or -12/x, so the answer box never has to read a fractional exponent; answers that keep a fractional exponent are chosen from options. Many stems state that every variable is positive, so each fractional power is defined. FAST practice adds a dropdown item that asks for the coefficient and each exponent separately, and a reasoned item on whether (625x⁸)^3/4 = 5x⁶ is correct. Lint rules fail any item here without a fractional exponent, and any that puts a variable under a radical sign.

In the multiplying and dividing skill, items climb from whole-number results (√50 ÷ √2 = 5) to ones that need simplifying (√112 ÷ √2 = 2√14), coefficients out front (-2√5 · 3√2 = -6√10) and cube roots (∛4 · ∛2 = 2). Reverse items ask for a missing radicand, as in √2 · √k = 6. The adding and subtracting skill starts with like radicals, then needs one or both radicals simplified first (√80 - √180 = -2√5), then cube roots (∛250 - ∛16 = 3∛2), and then sums that cannot combine.

Wrong answers come from real errors: adding or multiplying the radicands under one root, dropping the root (36 for √3 · √12), pulling out a perfect square whole, multiplying the coefficients when adding, leaving a radical unsimplified, and sign slips. Most single-term answers cross a coefficient slip with a radical slip, so the correct answer is never the one option a single change from all the others; it is sometimes the smallest, sometimes the largest. A negative answer never comes paired with its opposite as the only such pair. In the multiplying and dividing skill the build rejects any option or menu entry that repeats another's value or is half or double the correct value. Every matching board has one value left over.

FAST practice adds short contexts: the other side of a turf strip whose area is 6√96 and one side 2√2, and how much of a √605-kilometer trail is left after √180 kilometers. It also asks which cube-root products and quotients are integers, which sums cannot be written as one radical term, and a three-blank dropdown that walks 2√108 - 3√27 through each simplification, with its own menu for each blank. No figures appear. A lint rule fails more than two radicals, more than one operation, mixed square and cube roots, or a lone radical with no operation.

Questions climb from linear models, where the student names the starting value or the rate, to exponential models (the 800 in V = 800(1.05)^t, the base 2 in N = 200(2)^t), to quadratics and geometry formulas (the 4 in P = 4s, the 32 in a launch model), and then to single-entity items on compound interest and profit. Dropdown and matching items place two or three parts at once, so a student who swaps the rate and the start gets both wrong. Matching boards carry two spare descriptions and no row for a variable the stem defines.

The wrong answers are the swaps students actually make: the rate read as the starting value and the reverse, a coefficient called the total, the growth factor 1.04 read as "grows by 1.04 dollars" or "grows by 104%", and the number of units named when the question asks for a cost. Where the correct choice explains its reading ("multiplied by 1.04 each year, a 4% increase"), at least two wrong choices explain theirs too, and a lint rule fails any family whose correct option is nearly always the longest.

FAST practice adds formulas from geometry and finance: the base area inside a candle's V = 24h, the side panels 4sh inside a box's surface area, the factor (1 + rt) in simple interest, what (0.965)^t means for a fish population, and a time estimate with two variable terms where the student picks the one that depends on trees. No graphs appear: interpreting needs only the expression and its context. Lint rules fail any stem with no expression or equation, any ask for an equivalent or rewritten form, and any change over a period the model does not count in (two years, or a month on a yearly model).

Items climb from one-step formulas (d = rt for t, A = ℓw for w) to two-step ones (P = 2ℓ + 2w for w, the Fahrenheit formula for C), to linear equations in standard form solved for y, and to quadratic forms solved for a, k or x². Multiple-choice items offer four rearrangements of the same formula; typed items ask for the expression the variable equals. Error-analysis items show a classmate solving a standard-form equation for y in four steps and ask where the first error is. Every matching item has one more solved form than rows, and some solve four related formulas for the same letter (k in y = kx, y = k/x, y = x/k and y = k - x).

Wrong answers come from the moves students get wrong: undoing operations in the wrong order, dividing only one term by the coefficient, keeping the wrong sign when a term moves to the other side, and dividing where the formula calls for multiplying. Many option sets cross two of these slips (d/r, r/d, d - r and r - d for d = rt solved for t), so no wrong answer is the lone near miss of the correct one.

A typed answer accepts the spellings a student is likely to enter (with or without "y =", terms in either order, the fraction button's form), and a stem that asks for a fraction requires one. check-grading.mjs runs every key and alternate through the real grader, and check-alternates.mjs confirms each alternate is equivalent to the key.

FAST practice adds volume and surface-area formulas (a pyramid solved for h, a sphere solved for r³, a rectangular prism's surface area solved for h), a Fahrenheit-to-Kelvin conversion solved for F, the literal equation px - q = r - sx, and a reverse item that asks which formula a student started from.

Questions come at the skill from every direction. The student names the type of one written rule, names the type of a graphed curve, or decides which type could produce a table. Other questions give four rules or four lists of outputs and ask which are exponential, or match rules, tables and shape descriptions ("Two straight arms meeting at a sharp corner") to their types. The hardest ask for the type and the reason together, such as "Quadratic, because the second differences are all 4."

Graphs sit on a fixed plane with no labeled points, so the student reads the shape itself. Tables are input-output tables, and the explanation works through first differences, then second differences or ratios.

Wrong answers come from real confusions: cubic and cube root swapped because both carry a 3, a squared rule read as a square root, a V read as a line, a constant difference read as a constant ratio, first and second differences mixed up, and the belief that a shift or a reflection changes the type. In the questions that pair a type with a difference or ratio claim ("Quadratic, because the second differences are all 4"), each type appears in two choices and every reason is a true or false fact about that table, so no choice refutes itself and neither half alone gives the answer away. The reasoned yes-or-no questions are keyed "No" as well as "Yes".

FAST practice uses fresh rules and tables in the same formats, including select-all sets with five statements, and tags every wrong answer with the misconception behind it. Those tags feed the class FAST practice report and the study plan.

Items climb from a linear rule at a whole number to quadratics at negative inputs, fractional inputs such as f(-1/3), and exponentials such as 5 · 2ˣ or 4 to the 3/2 power. Real-world models cover savings, temperatures outdoors, at a mountain station and inside a freezer, a lake level read against a mark, plant growth, a draining tank, projectile height and depreciation. Some questions ask for the value; others ask which sentence correctly states what f(a) = b means. A few give the function as a table or a graph and ask for a value read from it.

Numeric distractors come from specific slips: adding the constant instead of subtracting it, stopping after the multiplication, squaring a negative input and keeping the negative, multiplying the coefficient into the base of an exponential before raising it to the power, and multiplying by an exponent instead of raising to it. Interpretation choices pair a right or wrong value with a right or wrong reading (input and output swapped, or the value read as a change, a rate or a partial cost), so no single slip gives the key away. An error-analysis set shows a student's work, for example (-4)² written as -16, and asks which statement about the work is correct; in two of the eight the work has no error.

FAST practice adds new contexts, tables whose inputs are spaced 3 apart, and items that combine two function values, such as g(-2) - f(-1/2). That is arithmetic on two outputs, not composition, so it stays inside FDOE's limit.

The student starts from one of four places: tables of readings, often taken at uneven times, a quadratic or exponential model with a named interval, a graph with two labeled points on the curve, or a stated average rate that has to be run backwards to find an ending value. Harder questions interpret the rate, compare rates over two intervals or between two models, or pick every one-second interval on which the rate is positive. A lint rule fails any item whose table, graph or rule is linear, and a story with no rule, table or graph says why its change is not constant ("so its height does not change at a constant rate"); a lint rule checks that too.

Wrong answers come from forgetting to divide by the length of the interval, dividing by the ending input or by the number of points instead of the run, reading one step of the interval alone, adding or averaging the two outputs, subtracting the endpoints in the wrong order, comparing final or starting values instead of rates, and reading an average as a promise ("increased by exactly 1 meter during every minute"). Where the sign slip appears it comes with the undivided change in both signs, so the key is never one of a lone plus-or-minus pair. One interpretation item uses an interval in which the ball rises and then falls, so a positive average does not mean it climbed the whole time.

FAST practice adds a quadratic model compared with an exponential one over the same interval, a "which statement must be true" item about an average rate, a secant line drawn from a table, and a model whose starting amount has to be found from its average rate.

Each question sets a linear rule beside a nonlinear function, usually a graph. Most ask for one comparison: the greater y-intercept, more x-intercepts, different domains or ranges, where both increase, a horizontal asymptote, or which eventually has the greater values. A few matching questions pair each function with its own range, x-intercepts or end behavior. Typed questions ask how far apart two intercepts are, which constant gives f a stated y-intercept, where both functions increase, or, from a table, where the exponential first passes the line.

Graphs carry no labeled points that would give the answer away, and the exponential-versus-linear tables print both functions side by side. Comparing two outputs at one input is not on FDOE's list, so no item asks for it (the only single inputs named are where the exponential first passes the line), and a lint rule enforces that. Select-all statements stay on the list too: no growth factors.

Wrong answers come from claims students actually make: every parabola has two x-intercepts, a square root function's domain is all real numbers, a line "levels off" at its y-intercept, 2 to the power 0 is 0, and a function that starts ahead stays ahead. Each function is named in the same number of choices, so picking the odd name out never finds the key. In the y-intercept questions every choice is a y-intercept claim and the key is g as often as f; "Both" and "Neither" are keyed too.

FAST practice adds statements about an absolute value graph's minimum and sign, a square root graph compared on domain and range at once, and a story of two beetle colonies, one gaining 150 beetles a week and the other growing 40% a week.

Questions range from one sentence ("Each year, a tractor is worth 85% of its value the year before") to tables that have to be tested with first differences, second differences and ratios. Every table and every description is a real-world situation with named columns or quantities (never "a quantity"), such as a bouncing ball's heights or the cells in a lab culture, and a lint rule fails a bare x and y table. Other formats ask for the model and the reason together, match several situations to their models, select every situation that fits a named model, or name the model of a table in one blank and then pick its next value in another, one or two steps on, from a menu whose wrong values are what another model or a miscounted step gives.

Nothing is graphed, because FDOE's clarification limits the stimulus to descriptions and tables.

Wrong answers come from "it keeps increasing, so it is exponential", "it started with money in it, so it is linear", a constant amount read as exponential growth, and calling every percent situation exponential. In the model-and-reason questions written as sentences two models each appear in two choices. In the table questions the model named twice is the key's model in about a third of them, and the choice that shares nothing with the others is the key in about another third, so neither is a reliable tell.

FAST practice uses fresh contexts, such as a warehouse conveyor, a weekly pond treatment and a work crew that slows down once it grows too large, and includes a rebate of 4% of an original cost each year, which is linear even though it is stated as a percent. Every wrong answer there is tagged with the misconception behind it.

This benchmark has no practice and quiz questions of its own, because FDOE assesses it only together with MA.912.F.1.8. Its questions sit in that benchmark's pool and carry that benchmark's code, so a student's work on them is reported under MA.912.F.1.8.

About a dozen of those questions are about interest. Some describe one account ("An account earns 5% simple interest each year, always computed on the original deposit") and ask which type of function models the balance. Others put a simple-interest account and a compound-interest account in a matching set with a kicked ball and a doubling bacteria colony, or complete a sentence such as "Interest that is paid each year on the CURRENT balance grows the balance in a way that is ...". Reasoned versions ask for the model and why, for a 4,000-dollar loan at 7% simple interest or an account earning 3% compound interest.

Wrong answers treat any interest as exponential, or call a compound-interest account linear because it "earns interest every year". No interest question uses continuously compounded interest or asks the student to write an equation, and a lint rule fails any that does. The interest questions in MA.912.FL.3.2 come at the same idea from the computing side, with statements such as "the balance grows by the same amount each year" to judge for a simple-interest account.

In FAST practice the idea appears inside the MA.912.F.1.8 set: a club account earning interest on its current balance, and a rebate of 4% of an original cost each year.

Most questions describe the change: "How does the graph of g compare to the graph of f?" with shifts, stretches, compressions and reflections as the options. Others work with numbers: find k from a table of f and g, evaluate g(3) for g(x) = f(2x) from a table of f, or locate the new vertex of an absolute value graph after f(x - 3/2). Grid questions have the student draw g with a line, parabola or V tool over a drawn f, and the explanation shows the finished graph of g (in the horizontal set, with f dashed beside it). FDOE sets this benchmark in a mathematical context, so there are no stories: f is always a bare equation, graph or table.

Wrong answers come from the classic confusions: reading f(x + 4) as a shift right, treating f(2x) as a vertical change, using k where 1/k belongs, reflecting over the wrong axis, reading f(x) + 5 as a stretch or a horizontal shift, and subtracting instead of dividing when finding k from a table. In the questions that describe the change, the choices cross two real slips, such as the sign and the reciprocal factor, or the direction and the size of a shift, so no choice sits one step from all the others, and a build guard rejects any such option set whose key is the one choice closest to all the rest.

FAST practice adds a vertical scale factor found from a single point on g, a V graphed after both a flip and a compression, and f(-x/2), which combines a horizontal stretch with a reflection across the y-axis.

Most classification questions show a real display, such as a scatter plot of engine size and fuel efficiency or a two-way table of language by grade, and ask how the data should be classified; a few describe several data sets in words and sort them. Reading questions use box plots, histograms, dot plots, stem-and-leaf plots, circle graphs, bar charts and frequency tables: find the interquartile range, count the students with at least 2 pets, or find how many of 240 concert tickets were balcony seats. Comparison questions set two box plots or two histograms side by side and ask which statements they support. Every question that interprets a distribution shows the display, and a lint rule fails any that does not.

Histogram bins share clean boundaries (0-10, 10-20) and equal widths, so any threshold a question names falls on a printed tick; the build rejects bins that do not. Every matching offers one more choice than it has rows, so the last match is never free.

Wrong answers come from counting "how many" as a second variable, treating number labels such as grade levels as numerical data, using the whisker ends for the IQR, reading a box edge as the median, counting only the boundary bar or the other side of the threshold on an "at least" question, reading a circle graph's percent as a count, comparing percents across groups of different sizes, and calling a skewed histogram symmetric.

FAST practice spreads the displays so no type repeats within a difficulty level, and adds two line plots compared on center and spread.

This benchmark has no practice and quiz questions of its own, because FDOE assesses it only together with MA.912.DP.1.2. Its questions sit in that benchmark's pool and carry that code, so a student's work on them is reported under MA.912.DP.1.2.

Inside that pool, a set of questions describes a data set and a purpose and asks for the best display, such as "A coach records the number of free throws made out of 10 by each of 40 players and needs every individual value to stay visible." Others match three or four data sets to displays, with one extra display that fits none of them, so the last match is never free. The hardest ask for the display and the reason together. These questions describe the data rather than show it.

Wrong answers pick a display of the right data type but the wrong purpose (a circle graph when counts must be compared), a one-variable display for paired data, a scatter plot for a single variable, or a box plot when every value must stay visible. Every choice of display offers at least one wrong display that holds the same kind of data as the right one, so knowing numerical from categorical never settles it alone, as FDOE's limit requires. A lint rule fails any multiple-choice item where it would. The classification questions in MA.912.DP.1.2 practice the other half of the choice: deciding whether data is numerical or categorical, univariate or bivariate.

FAST practice includes select-a-display items in the same set, such as monthly bus ridership, which calls for a line graph, and charging time against run time for phone batteries, which calls for a scatter plot.

Questions start with scaling: 40% of a 200-student sample ride the bus, so about 600 of the school's 1,500 students do. They build to intervals: the plausible range for a percentage, the same range turned into counts of people, and a mean plus or minus its margin turned into a population total. The hardest interval questions give two polls, each with its result and margin, and ask which percentage both make plausible; some wrong answers also sit between the two results, so the key is never simply the value between them. Some run backwards, such as finding the school's size from the estimate and the percentage. The last group judges claims, such as a reporter's "Support for the bond is exactly 45%."

Wrong answers come from the mistakes students make with surveys: applying the percent to the sample size instead of the population, counting the people who do not, taking the margin as a percent of the percent or of the estimate, applying it in only one direction or twice, applying it to the count after scaling, multiplying a mean by the sample size, multiplying where a backwards question needs a division, and treating the sample result as exact. In matchings of polls to ranges, two polls share a result or a decoy range holds the same value, so no row can be matched by finding the one range that contains its result.

Every question, in practice, quiz and FAST practice alike, states a sample of at least 100 and gives both the sample result and the margin of error; a lint rule checks both.

FAST practice runs the same moves in new settings, such as a claim of "at least 65% of all students" tested against a sample result of 64% plus or minus 6, and adds one that works backwards from the lowest plausible count to the size of the population.

The table skill starts with reading one joint or marginal count from a finished table, then filling a missing cell by subtraction, then building the table from a description (of 80 students, 45 play a sport, 36 are in band and 21 do both: how many do neither?). Every table has a 2 by 2 body with a Total row and a Total column, and no completion menu offers a count larger than the printed total it belongs to. All the arithmetic is whole-number addition and subtraction. Other items sort values into joint and marginal frequencies or find the error in a student's answer, such as using the "both" count where "owns a bike but not a skateboard" was asked.

The association skill compares groups. With equal-size groups the counts can be compared directly; with unequal groups the student compares each group's share. Each kind of comparison keys "association" in some questions and "no association" in others. Reasoned items ask for the judgment and a valid reason, with two "association" and two "no association" options; every option cites numbers from the table, and a wrong option talks about a share too, so neither a reason nor the word "share" marks the right answer.

Wrong answers come from comparing raw counts across groups of different sizes, reading one column of the table (the share of all the people who do something that each group makes up), deciding that groups of different sizes cannot be compared at all, taking equal group sizes as evidence of no association, reading the column totals as a sign of association, and, in the table skill, subtracting from the wrong total or mistaking a cell for a total.

FAST practice uses fresh contexts, such as a science fair and a regional airport, including a table with three missing entries to complete at once, and a retailer's survey where the smaller group has the larger share.

The easiest items give a starting amount and a rate (a joining fee and a monthly charge, a booking fee and an hourly rate, a tank draining from a full level) and ask which equation finds the count. The wrong choices cross two real slips, such as the fee and the rate swapped and the fee subtracted or put inside parentheses (12(m + 30) = 114), so the right one never stands out as the option that shares the most. Next come write-and-solve items with decimal rates, variables on both sides (two phone plans), perimeter, consecutive-number and age problems, and percent problems with a second step (a shipping charge, a coupon, a fee or a second percent), so none solves in one step.

Other wrong answers name the usual slips: the percent off used for the percent paid (0.20p for 0.80p), a 15% discount undone by adding 15% to the sale price, consecutive numbers stepped by 2, one length or one width of a perimeter left out, and, for two plans set equal, one plan's fee divided by the gap in rates or the two fees added. For a count, the choices come from ignoring the fee, adding it, forgetting to divide or dividing by the fee, and the right answer moves among them from item to item. Dropdown items build the equation piece by piece; in the fee-and-rate matching each pair of numbers appears in two equations, so a match takes knowing which number is the fee.

No figures appear, since the skill is translating words into an equation. FAST practice adds a markup followed by a delivery surcharge, a job billed at one hourly rate for the first four hours and a lower rate after that, two funds that meet (one shrinking, one growing), and three consecutive odd locker numbers.

In the graph skill, the student reads two lattice points from an unmarked line and picks or types its equation in slope-intercept, point-slope or standard form. Other items ask only for the slope (typed as a fraction, which is how it is graded), for the slope and intercept through dropdowns, or for every equation that represents the line. In the table and description skill, a four-row table or a situation such as a kayak rental (a launch fee plus 6.50 dollars an hour, 27 dollars for 3 hours) leads to the equation or to b. Both skills include converting between forms and error-analysis items where a classmate's four steps contain one mistake.

Wrong answers are built from the usual errors: run over rise, the raw change in y used as the slope when x steps by 2 or more, the first table value or a rounded crossing read as b, b computed as y + mx instead of y - mx, sign errors in point-slope form, the anchor point's coordinates swapped, and a standard form that keeps a negative x-coefficient. Most equation sets cross two of these slips, so the right one is never the option that shares the most with the rest. In select-all items the wrong equations are one wrong line written in two forms, just as the right ones are, so spotting the equivalent pair does not find the answer. Matching rows reuse the same numbers in different roles, so no row can be matched by spotting its numbers in a column.

The FDOE coverage gate checks that this benchmark keeps enough graph and table items. FAST practice adds a line whose y-intercept the student brackets between two integers from the graph, a melting block of ice graphed with titled axes, a phone battery draining between two decimal readings, and a kennel model matched to a description that never states the fee outright.

Items climb from parallel lines through a point, to perpendicular lines (where the slope flips and changes sign), to given lines in standard form or as point pairs or graphs, to horizontal and vertical cases. Classification items ask whether two lines are parallel, perpendicular or neither. Some typed items ask only for b once the new line is in slope-intercept form. On grid items the student draws the new line with the line tool through a marked point on a -10 to 10 grid, and the explanation shows the finished graph.

Wrong answers come from the rules students mix up: using the same slope for a perpendicular line or the perpendicular slope for a parallel one, taking the reciprocal without changing the sign, changing the sign without taking the reciprocal, keeping the given line's intercept, and reading a standard-form coefficient as the slope. In the which-equation items with a slanted line, every choice passes through the given point and only the slopes differ, so substituting the point rules nothing out and the parallel or perpendicular rule decides. In the horizontal and vertical items two choices pass through the point, one of each orientation, so the orientation decides. Error-analysis items show a classmate's four steps with one mistake, such as a sign error when solving for b.

The FDOE coverage gate checks that enough items show the given line as a graph. FAST practice adds a context (a loading lane perpendicular to a conveyor belt on a floor plan), items with a parameter such as y = (2a + 1)x - 3 parallel to y = -9x + 4, a point with an unknown coordinate on a line perpendicular to a given one, and a grid item.

In the graphing skill, grid items ask the student to draw the line with the line tool from f(x) = 2x - 3, 2x + y = 4 or y - 2 = (3/2)(x + 2), and horizontal lines such as f(x) = -4 are included. From a table, two points or a description, the student plots the x-intercept and the y-intercept instead. Other items ask which point continues a table, whether the graph rises or falls, or what the slope is. The explanation shows the completed graph.

In the key-features skill, students find intercepts from any form and write them as ordered pairs, read intercepts off an unmarked graphed line and the rate of change off a graph, plot both intercepts with the point tool, give the domain and range of a horizontal line or of a segment with closed endpoints, match features to values, and solve for a missing m or b. In the feature-matching items (x-intercept, y-intercept, rate of change) each feature has a decoy of the same form, such as the y-intercept's value placed on the x-axis, so no feature can be matched by its shape alone. Error-analysis items show a classmate finding intercepts with one mistake.

Wrong answers come from the errors students make: swapping the coordinates of an intercept ((4, 0) for (0, 4)), giving the other intercept, sign errors when dividing by a negative, the reciprocal slope, reading the domain as the range, and calling a restricted range "all real numbers". Ordered-pair, equation and move choices cross two real slips, so the right one is never the option one change from all the others, and no value menu leaves the answer as one half of its only plus-or-minus pair. The FDOE coverage gate holds the share of items with a coordinate plane. FAST practice adds a point-slope line to graph, a "which move lands on another point" item that reads slope as rise and run, and the range of f(x) = -3x + 5 over a restricted domain.

Items climb from what a number in the model means (the 120 in V(t) = 120 - 8t, or the point (0, 100) on a battery graph), to when the quantity reaches zero, to evaluating or solving (the cost of 40 pages, the hours until a candle burns out), to the domain and range that fit the situation, to graphing over the practical domain and comparing two plans. Context graphs have titled axes, and most mark a few lattice points, from which the student reads the rate before interpreting it.

Wrong answers come from the swaps students make: reading the rate as the starting value or the reverse, writing a domain that uses the output limit (0 ≤ t ≤ 600 for a 600-liter tank that fills in 40 minutes), using the rate as the end of the domain or multiplying where they should divide, and leaving the domain with no upper end (t ≥ 0). When a practice or quiz question asks when a quantity runs out, the choices come from reading the rate or the starting amount as the time, multiplying instead of dividing, or a dropped zero, and the starting amounts are small enough that every choice is a believable time. Matching and dropdown items ask for two parts of a model at once, so a swap costs both.

The FDOE coverage gate checks the share of items with a graph. FAST practice adds two catering plans graphed until they cross, the range of a fuel model read from a graph that runs past the stated domain, reverse items that find a drain rate from the time a tank is empty or a starting balance from a later one, and a comparison of two draining rain barrels.

The multi-step skill solves inequalities with parentheses (4(x - 1) + 6 > 18), a negative coefficient that flips the symbol, or the variable on both sides, and shows the solution in set-builder form or on a number line. Context items write and solve an inequality, then round the right way (a cab fare of at most 19 dollars allows at most 8 miles). The compound skill covers sandwich inequalities, "or" and "and" pairs, compound inequalities written from words (at least 42 inches and less than 78), and solving one, then saying which values are in its solution set or judging a classmate's number-line graph of it.

On number-line grids the student sets each boundary, open or closed, and the direction or, for a compound inequality, whether the solution lies between the two boundaries or outside them; all of it is graded. Two of the six compound grids are "or" pairs. Every grid explanation shows the finished number line.

Wrong answers come from not flipping the symbol when dividing by a negative, flipping it when it should stay, distributing to the first term only, changing only the middle of a sandwich, reading an endpoint's inclusion backwards, and rounding the wrong way. Most solve-and-choose items cross two of these slips, so the answer is not the choice closest to the rest. A lint rule fails any given inequality that is not compound and takes two steps or fewer (counting the shorter path, with a fraction cleared in one step), and a second fails any item with no inequality to write or solve. FAST practice adds an inequality whose graph is given and whose constant k the student finds, a compound inequality matched to a drawn solution set, and an "and" pair with no solution.

From a graph, students choose or build the inequality in slope-intercept form (with dropdowns for the symbol and the expression), in standard form, or as y > 2 or x ≤ -1 for horizontal and vertical boundaries, and match described graphs to inequalities. From words, they write the inequality for a budget or a fundraising goal in standard form, or translate a sentence such as "y is less than 5 more than 3 times x" into symbols. Multiselect items ask which ordered pairs are solutions, and reasoning items ask which inequality includes one point but not another.

Wrong answers come from the four parts of the answer: the wrong symbol for the line style, the wrong direction for the shading, a slope with the wrong sign or flipped, and coefficients attached to the wrong variable (5x + 8y for 8x + 5y). Most multiple-choice items cross one symbol slip with one expression slip, so each symbol and each expression appears in two of the four choices and the answer is not the choice closest to all the others. Diagnosis items show a drawn half-plane and a classmate's inequality and always offer the same four claims (line style, direction, slope, intercept), so the student has to find which part is wrong. Phrase-matching items pair "at most", "more than", "at least" and "fewer than" with their symbols.

The FDOE coverage gate checks the share of items with a coordinate plane. FAST practice adds typed inequalities written from a graph and solved for y, a sentence solved for x rather than y, a context where x and y are named in the opposite order from the story, and a vertical boundary described by two points on it.

Items build the graph in parts before asking for all of it. Students rewrite 3x + y ≤ 7 with y alone, find the boundary's slope and intercepts, choose solid or dashed and above or below through dropdowns, sort inequalities by boundary style or shaded side, and test which points lie in a drawn solution set. The graphing items then ask for the whole half-plane, and diagnosis items give the inequality with a classmate's graph and ask whether the style, the side, both, or neither is wrong, from the same four statements every time.

Wrong answers come from the errors that matter on this skill: a solid line for a strict inequality or the reverse, shading the wrong side, forgetting to reverse the symbol after dividing by a negative y-coefficient, and a boundary with the reciprocal slope. Real-world items put a goal such as 4x + 8y ≥ 48 on a graph and ask where a combination's point lies and whether it meets the goal. Every item gives the inequality: a lint rule fails any item that shows a graph and asks which inequality it represents, which is the previous benchmark's skill.

The FDOE coverage gate checks the share of items with a coordinate plane. FAST practice adds a reasoned item on whether 4 small and 3 large yards meet a 36-dollar goal, a which-inequalities-shade-above multiselect that mixes point-slope form and a negative y-coefficient, and two half-plane graphing items, one given in standard form and one in point-slope form.

Solving items give a system and ask for the ordered pair, one coordinate, or which statements about the solution are true. The work grows from one equation already solved for y, or opposite x terms, to a distributed minus sign, fractional slopes or both equations scaled; the hardest items run backward (the solution has x = 2, so what is the missing constant?).

Graphing items give an empty plane; the student draws both lines with a two-line tool and the crossing is graded, and some elimination items ask for the solution plotted as a point. Every reasoned solution-count item offers the same four choices: one solution because the slopes differ, none because the lines are parallel, infinitely many because they are the same line, and the trap "one solution because the y-intercepts differ". The stem asks for the number AND the reason, since the trap states a true fact on many one-solution systems, and the explanation says why the y-intercepts decide nothing. Each count is the answer somewhere.

Story items (ticket sales, coin jars, two-price purchases) ask for the count and value equations, or for one quantity. Wrong answers are real slips: coordinates reversed, a point checked in one equation only, totals swapped, a price on the wrong variable. Where the choices are ordered pairs and the solution's coordinates differ, they cross the order swap with one slip, so no choice sits one step from all the others; in matchings a solution also fits one equation of another row. Find-the-error items ask which step of a classmate's work holds the first mistake.

FAST practice adds its own items in the EOC's voice, including a solution read off two graphed lines, a basketball score split into two-point and three-point baskets, and a two-price hardware purchase. A build check rejects any FAST item that repeats a practice item's numbers and answers.

The most common task gives a system and asks which point is a solution, which points from a list are, or whether one point is a solution with the correct reason. Harder items put the point on a dashed boundary, ask for the corner where the boundary lines meet (typed, or plotted on the grid), or ask how many integer values of y make (2, y) a solution.

Graph items give the system: the student graphs one inequality while the other is drawn, or judges a classmate's graph of the whole system from the same four statements (correct, a boundary with the wrong style, a half-plane shaded on the wrong side, a boundary in the wrong place). Style and shading items name one boundary of a given system, and matching items sort whole systems by their pair of boundary styles or by where the solution region lies. At the top end students decide which systems have no solution, which is where parallel boundaries shaded away from each other come in.

Wrong answers come from accepting a point that satisfies only one inequality, testing only the first inequality, swapping solid and dashed, shading the wrong side, taking every point in either region instead of the overlap, and counting a point on a dashed line. Error-analysis items show a classmate's graph or claim with one of these mistakes. For every graphed inequality in the practice and quiz bank, the grading check also submits the answer with the other side shaded, and it must score zero.

FAST practice adds a vertical dashed boundary read against a slanted one, two parallel boundaries shaded the same way (the solution is the smaller half-plane, not a strip), and an item where the first inequality is drawn and the student graphs the second with the half-plane tool.

Writing items give a short story, a parking lot that holds at most 40 vehicles or a car wash with a vehicle limit and an earnings goal, and ask which inequality or system represents it. Wrong choices reverse "at most" and "at least", confuse a strict symbol with an inclusive one, or attach each price to the wrong variable, and in the system choices the reversed symbols are crossed with the swapped prices, so each choice shares a part with two others and the answer is not the one closest to the rest. A few items give the story alone and ask the student to write the system and solve it.

Judging items give a candidate pair and ask whether it is viable, which pairs from a list are viable, or which verdict goes with each pair. Difficulty climbs from pairs that fail on sight to whole-number pairs that fail a named constraint, where the student has to substitute to find out. Every list of candidate pairs includes a whole, non-negative pair that breaks a constraint, so the format alone never finds the viable ones. Other items fix one quantity and ask for the greatest or fewest whole number of the other.

Error-analysis items show a classmate who calls (9, 5.5) viable because it satisfies both equations, or who writes "at most 24 jobs" with ≥ instead of ≤. Graph items give the story without its equations: the student writes each constraint, then graphs it with the half-plane tool, or draws both constraint lines and marks where both hold.

FAST practice adds its own stories, including a phone repair kiosk where the pair is whole and under the job limit but misses the points goal, so only substituting reveals the verdict.

Most questions set Function A beside Function B and ask which has the greater slope, the greater y-intercept or x-intercept, or the greater maximum over a stated domain. Options pair a verdict with the comparison that justifies it ("Function B, because 4 > 3"), and most items name each function in two options, so the student has to check the reason, not just the name. Typed questions ask for the greater or lesser of two values (two slopes, two intercepts or two maximums) or the difference between them, so they compare too. Some items compare four functions in four forms, and harder ones ask what happens as x increases without bound.

Tables list a few ordered pairs, graphs show one line, and descriptions read like "a plant that is 4 cm tall and grows 3 cm each week." A few grid questions ask the student to graph whichever function has the greater growth rate or the greater y-intercept.

Wrong answers come from comparing intercepts when the question asks about slopes, taking a table's output or its change in y as the slope, reading 3y = 12x - 6 as a slope of 12 without dividing by 3, assuming the function that starts higher stays higher, and believing that two lines that meet stay equal from then on.

FAST practice leans harder on real-world descriptions (a food pantry, a solar array, a cable car) and on rearranged equations, and asks for intercepts of functions given only as ordered pairs.

Many questions pair an interpretation with a prediction: "Which statement correctly interprets the slope of the line of fit AND gives its predicted sales on a day with a high of 80°F?" In the multiple-choice versions the options cross one interpretation slip with one prediction slip, so each half of the key appears in two options. Others give the line and an input, give a target output and ask when the model reaches it, or show a scatter plot with its line of fit through two named points and ask for a prediction, the rate and a prediction, or the equation and a prediction. Some ask whether a prediction is an interpolation or an extrapolation and how far to trust it.

Grid questions have the student plot the point the line of fit predicts. No question has the student draw a line of fit by eye, because the grader can only check an exact line; the choose-the-best-line question covers that skill instead.

Wrong answers come from swapping the slope and the intercept, reading the intercept as a rate per unit, dropping the intercept from a prediction, reversing the sign of a falling rate, reading the first plotted point as the y-intercept, and calling an extrapolation reliable. When an option pairs an equation with a prediction, the wrong predictions come from real slips, most of them the value a wrong equation gives, so the key is not the only option that agrees with itself. Prediction options include an honest overshoot whenever one is a believable size, and matchings that apply one line to several inputs carry one extra choice, set outside the range of the real answers, so neither "pick the largest" nor "pair them in order" works.

FAST practice adds a rate-and-prediction question read off a graph with two marked points, a student's claim about the intercept to judge, a trip-length prediction that has to be checked against the range of the plotted data, and a choose-the-best-line question on swim times.

Strength questions, and direction questions outside the causation set, show one scatter plot with the line of fit and the residuals drawn as dashed segments. They ask for the direction, the strength, or both ("strong negative"), or which sentence describes the relationship in the situation's own words. In most label and interpretation items the options cross two real slips, a wrong strength and a flipped direction, so the key is never simply the label that appears most often. Reasoned items ask for the label and the reason that supports it, and select-all items mix true statements with the common false ones.

Wrong answers come from treating strength and direction as the same thing ("strong correlation" offered as a direction), reading a steep line or a large point count as strong, deciding direction from how many residuals are negative, and claiming the line gives each point's value exactly.

The correlation-versus-causation questions show a scatter plot with a line of fit for two things a third factor may drive, such as ice cream sales and pool rescues, or compare two survey groups, and ask which statement the data justify. Every keyed answer refuses the causal claim without denying the pattern. Wrong answers claim causation, sometimes because the points follow the line so closely, reverse it, propose acting on it, or dismiss the pattern. In most items a wrong answer also holds back from causation, for a false reason: it names the factor that drives both and then denies the variables are related, or calls a strong correlation too weak to mean anything.

FAST practice follows the same rules. It also asks whether plotting more points would make a correlation stronger, and shows a six-point plot whose strength a student doubts because of the point count.

This benchmark has no practice and quiz questions of its own, because FDOE assesses it only together with MA.912.DP.2.6. Its questions sit in that benchmark's pool and carry that code, so a student's work on them is reported under MA.912.DP.2.6.

Those questions show a scatter plot with a line of fit for two quantities that move together because of something else: umbrella sales and car accidents, jackets and colds, shoe size and reading level, storks and births. The student picks the justified statement, completes "From these data alone, is it true that ...?" with "cannot be determined", or sorts claims ("The two variables are related", "One variable causes the other", "A third factor may explain the pattern") into justified and not justified. A categorical version compares two survey groups, such as students who eat breakfast most mornings and students who rarely do.

Wrong answers claim that one variable causes the other, often because the points follow the line so closely, reverse the direction of cause, recommend acting on the correlation, or deny that the pattern means anything. In most items one wrong answer also refuses the causal claim, but for a false reason: it names the factor that drives both and then denies the variables are related, or calls a strong correlation too weak. Several wrong answers open the way the correct one does ("The two variables are correlated, so the data show ..."), so the student has to read the conclusion, not just the first clause.

FAST practice includes one scatter-plot item (ice rink visits and flu cases) and one survey item (a study group and exam passing) of this kind.

In the adding and subtracting skill, students combine two or three polynomials, subtract with the negative distributed across every term, find one coefficient of the result, supply the missing polynomial in (2x² + 3x + 1) + ? = 5x² + 7x + 4, and total fencing or find profit (revenue minus cost) in context. The multiplying skills distribute a monomial, multiply binomials with and without a leading coefficient, model a rug's area as (x + 4)(x + 6), square binomials such as (3x - 1)² and (x + 2y)², and multiply (x - 1)(x² + 2x + 5).

Wrong answers come from the mistakes these problems produce: adding exponents when combining like terms (8x² - 3x² as 5x⁴), subtracting only the first term of the second polynomial, dropping the middle term of a square, writing (2x)² as 2x², writing (-4)² as -16, and adding the last terms of two binomials instead of multiplying them. Error-analysis items show a classmate's four-step work and ask where the first error is; select-all items ask which statements about closure and degree are always true.

The adding-and-subtracting template refuses to build an item with a polynomial longer than three terms or without a term of degree two, and the monomial-times-polynomial generator refuses an item with no degree-two expression, so a row that breaks an FDOE limit stops the build, and a lint rule checks the same three limits on every stem. FAST practice adds a subtraction with fractional coefficients, a missing polynomial in a difference, a stage's area as a product of binomials, 3(4x + 1)(x - 3), and a binomial product with an unknown constant k.

Items climb from two-term dividends with a positive divisor, to three-term dividends where one term divided by itself is 1 ((8x³ + 12x² + 4x) ÷ 4x), to negative divisors that flip every sign, fractional coefficients, and two-variable divisions such as (20a³b - 15a²b) ÷ 5a²b. Reverse items give the quotient and ask for the missing monomial divisor. Error-analysis items show a classmate splitting the fraction term by term and ask where the first mistake is.

The wrong answers are the classic slips: dividing the coefficients but copying every exponent, leaving a coefficient undivided, dropping a term divided by itself instead of writing 1, inverting or rounding down a fraction, and losing a sign. Most option sets with a positive divisor cross two of these, the copied exponents and one other slip, so each wrong choice carries one mistake or both and a teacher can name it. Most option sets with a negative divisor cross the negative ignored with one other slip (copied exponents, a term divided by itself dropped, or a fraction rounded down).

The template refuses to build a dividend that is not two or three terms. Each term is divided in one line of the explanation; nothing is set up as long division. FAST practice adds a rectangle whose width is its area divided by its length, a which-expression-simplifies-to-3x + 2 item, a three-term dividend over 6ab, and a reasoned item on whether (20x² + 5x) ÷ 5x = 4x is correct.

The common-factor skill asks students to factor out the greatest common monomial factor, name that factor, fill in the other factor in 21m³n - 14m²n² = 7m²n(?), and pick every expression whose greatest common factor is 3x². The trinomial skill runs through the sign cases of x² + bx + c, then differences of squares, perfect squares, quartics such as x⁴ + 5x² + 6, the value of c that makes a perfect square, and matching polynomials to factored forms. The grouping skill factors 2x² + 7x + 3 by the ac method, four-term polynomials such as x³ + 3x² + 4x + 12, and 6x² + 10x + 4 as 2(3x + 2)(x + 1).

Wrong answers are the near misses students write: a common factor that is not the greatest, a shared variable left inside, factor pairs with the right product and the wrong sum, sign errors, (x + 5)² offered for x² - 25, and a factorization that is not complete. Error-analysis items walk through a classmate's grouping work, and "choose the split" items ask which rewriting of the middle term, such as 3x² + x + 9x + 3, makes grouping work.

Typed answers accept the factors in any order and the terms inside each factor in any order, since (x + 3)(x - 4) and (-4 + x)(3 + x) are the same answer. The common-factor template checks FDOE's term rule as it builds its factor-it-out items, a lint rule checks it on every expression a select-all or matching item asks about, and the grouping template expands every factorization back to the original before it builds a row. FAST practice adds a four-term common factor, a which-are-not-completely-factored multiselect, x⁴ - 25x² + 144 into four linear factors, and 15x³ + 39x² - 18x factored completely.

Factoring items climb from a common factor (x² - 7x = 0) to trinomials with a leading coefficient of 1, then to trinomials such as 2x² + 7x - 15 = 0 with fraction roots, then to equations that must first be moved to standard form. Square-root items climb from 2x² = 32 to shifted squares and to radicals that must be simplified. Completing-the-square items start with an equation whose left side is already a perfect square, such as x² - 6x + 9 = 16, and end with irrational and no-real cases; quadratic formula items include reading the discriminant for the number of real solutions.

Real-world items are areas (rectangles, gardens, square patios and tiles, a triangle), consecutive integers, prices that bring in a given revenue, and the height of a launched or falling object, such as a diver's, h(t) = -16t² + 48t + 64. The student writes or uses the equation and rejects the root that cannot be a length or a time. Some items ask students to plot both solutions on the x-axis.

Wrong answers come from named errors: dividing by x and losing the zero solution, solving a factor with the wrong sign, dropping the negative square root, using b instead of half of b when completing the square, and reporting the length when the width was asked. Find-the-error items show a classmate's factoring or square-root work and ask which step went wrong first.

FAST practice adds a walkway of uniform width around a patio, a round-robin league that plays n(n - 1)/2 games, and (x + 8)(x - 2) = 11, where using the zero product property before expanding is the trap.

Items give the graph, the table or the sentence and ask which equation represents the function, ask the student to enter a, or ask for the whole function. Tables list the vertex with its neighbors on each side, so the student has to find the turning row first. Harder items ask for standard or factored form, which means finding vertex form and then expanding, and some ask the student to plot new points of the function, such as the points where x = -1 and x = -5.

Matching items give a vertex and two symmetric points, as a description or as a row of three points, and ask for the equation, the standard form or the value of a that fits each, and select-several items ask which statements, equivalent forms or further points belong to a function.

Wrong answers come from the sign errors students make with vertex form (y = (x + 3)² - 4 for a vertex of (3, -4)), from assuming a = 1, from dividing by the horizontal distance instead of its square, from swapping the vertex coordinates, and from reading the first table row as the vertex.

FAST practice adds contexts, a skate ramp's lowest point and a delivery drone's highest point, and a table whose rows are not one unit apart, so a student who assumes one-unit steps finds the wrong a.

Most items state the intercepts and a point and ask for the equation or for a. The easiest items often use the y-intercept as the extra point and a = 1; harder items use a point elsewhere on the curve and a negative or fractional a. Other items use the function once it is found: the y-intercept, the value at another x, or a new point plotted on the grid.

Some items ask for the pieces: the two factors and the value of a from menus, or the two equations, factored and standard form, that name the function. A matching item sends four functions, two intercept pairs with two points each, to their equations, so the factors narrow each row to two choices and only the point decides. Real paths appear too: a ball kicked from the ground lands 12 feet away and is 20 feet high when x = 2.

Each set of equation choices crosses two real mistakes, most often the intercept signs copied into the factors and one slip with a (its sign lost, a left at 1, the point's y-value used as a, or the division done upside down). The choices are the key, each mistake alone, and both together, so the choice most like the others is never a shortcut. Error-analysis items show a student who wrote x + 3 for an intercept at 3, or divided -12 by -12 and got -1, and about one in four shows a student who is right and asks whether the student is correct and why. A lint rule fails any item that does not give both intercepts and another point, or gives one that is not an integer.

FAST practice adds a signal flare fired from the ground, a bouncing ball whose zeros are both away from the origin, and an item that asks for g(-6) once the function is built.

Zero items ask when a ball kicked from the ground lands, which prices make profit zero, or what the zero t = 0 means (the ball is on the ground at the instant it is kicked). Harder ones launch from a platform, so one root is negative and has to be rejected, or run backward: a rocket lands at t = 5, so what is b? The break-even and landing matchings give three models in standard form, so each row has to be solved rather than read.

Vertex items ask when the peak happens and how high it is, the maximum or minimum value itself, how far the peak sits above the launch point, or whether a cable's model has a minimum or a maximum and why. Some ask the student to plot the highest point of a fountain's arc on the grid.

Wrong answers read the coefficient of t as a time, report the input when the value was asked, leave the 2 out of -b/(2a), leave the starting height out of the maximum, drop the squared term or lose its sign, call a zero the maximum, take the halfway point between the zeros as the landing time, or decide maximum or minimum from the constant term instead of the sign of a. When a question asks when the peak happens and how high it is, its choices cross two of these mistakes. No choice for a landing time, a height or a price is negative. Error-analysis items put these mistakes in a classmate's words, and about one in four shows a classmate who is right.

FAST practice adds launches from cliffs and towers, a second break-even price found from the first, and select-several items where the profit is positive between the zeros and zero only at them.

Graphing items start from factored form, where the zeros are visible, move to standard form, where the student factors and uses x = -b/(2a), and to vertex form, where the zeros come from square roots. Table items give five points and ask for the vertex by symmetry; the vertex is not always the middle row. Description items give features and ask for the matching equation or statement.

Key-feature items show an unlabeled graph or give an equation and ask for the vertex, the axis of symmetry, the y-intercept, the range, the x-values where f is increasing or negative, and end behavior in the form "As x → ∞, f(x) → ?". Matching and select-several items ask for several features at once. In the key-feature matchings every value has a decoy in the same form (the vertex with the sign of h flipped, (0, k) beside the y-intercept, x = -h beside the axis), so no row can be matched by the look of its answer. The FLDOE coverage gate requires a coordinate plane in at least a quarter of the items, in both banks.

Wrong answers read the zeros with the sign shown in the factor, call the y-intercept the vertex, flip the range inequality, split the positive region at the axis instead of the zeros, and write the axis of symmetry as a horizontal line or at an x-intercept. Most choice sets cross two of these mistakes (for example, the zero signs copied and the y-intercept taken as the vertex height), so the choice most like the others is not a shortcut. When the equation shows the sign of a, every choice opens the same way. Error-analysis items give a classmate's claim, such as "f is increasing for x > 3" when a is negative, and ask what is wrong, or whether the classmate is right and why.

FAST practice adds a graph where a has to be found from a zero using the squared distance, and a positive-interval question asked from the graph alone.

Projectile items give a model such as h(t) = -16t² + 64t + 16, or a metric -5t² model, often with its graph, and ask for the greatest height, the time it happens, the height at a given time, the landing time, or the values of t that make sense. Area items fence a garden against a barn wall; revenue items depend on the number of tickets sold or the price charged. Harder items find where a profit or an area is largest, both the price and the revenue at the top, the two prices that bring in a given revenue, or the smaller of two break-even quantities. Matching items pair a feature with its meaning, and every meaning has a decoy in the same words (the peak read as the start, a landing time with the launch height left out).

Graph items have the student draw the model with the parabola tool (the vertex, then one more point) or plot the start, the vertex and the landing. The FLDOE coverage gate requires a coordinate plane in at least a quarter of the items.

Wrong answers read the coefficient of t as a height, take the starting height as the maximum or count it twice, drop the squared term or add it instead of subtracting it, swap the time and the height of the peak, and, for the domain, keep the negative zero, stop at the time of the peak, or both. No choice for a maximum height sits below the launch height, and no choice that names a single height or time is negative. A lint rule fails any item that gives neither an equation nor a graph.

FAST practice adds comparisons across representations, such as one water balloon given by a formula and another by a graph, revenue that reaches a target at two different prices, and profit that is positive only between the break-even prices.

Items climb from an absolute value already alone (|x - 4| = 9), to a coefficient or constant outside the bars (2|x - 1| = 12, |x + 2| + 3 = 10), to a coefficient inside the bars (|3x + 3| - 2 = 4). The student picks the solution list, enters one named solution, or plots both solutions on a number line.

Other items ask which equation has no solution, how many solutions each equation has, which value of k gives exactly one solution, or the product of the two solutions. In the matching boards the number printed on the right side can mislead: |x - 1| + 8 = 3 has no solution even though the 3 is positive, so the count comes from isolating first.

For |x - 4| = 9 the wrong solution lists cross two real slips: one in isolating (skipping the division, or carrying the ± onto the constant inside the bars), one in the cases (only the positive case, or the first answer and its opposite). The student sees x = 13 or x = -5, x = 5 or x = -13, x = 13 only and x = 5 only, so no wrong list is the only one close to the key. When the printed right side is negative but the equation isolates to a positive number, as in |x + 1| - 6 = -1, "No solution" (reading the sign before isolating) replaces one case slip, so it is not offered only as the answer; the no-solution items offer "One solution" at the vertex. Find-the-error items show a classmate who split 2|x - 3| + 4 = 10 before isolating.

FAST practice adds a greenhouse humidity tolerance, a centrifuge where the student picks the equation that gives both boundary speeds, and a unit-fraction coefficient, (1/4)|x + 5| - 2 = 1. A lint rule fails any equation outside FDOE's two forms.

Items start from an equation, a table, a written transformation (the graph of y = |x| shifted 3 units right and 2 units up), or an unlabeled graph. Easier items ask for the vertex, the axis of symmetry or the y-intercept. Harder ones ask for the range, the x-values where f is increasing or below the x-axis, end behavior, whether the graph has any x-intercepts, and whether it is wider or narrower than another. Those last two are reasoned choices: the student picks the answer and the reason together.

The FLDOE coverage gate requires a coordinate plane in at least a quarter of the items.

In the vertex, range, axis and interval items the options cross two real slips: the sign of h against the sign of k (or the coordinates swapped) for a vertex, the inequality sign against the value for a range, x = against y = for the axis. Vertex and range boards share numbers across rows, such as vertices (4, 2), (-4, 2) and (2, 4). Key-feature boards carry a fifth, real-slip value, and the y-intercept never sits between the x-intercepts, so no board is solved by sorting. Wrong answers also build the range from h, rise on the wrong side of the vertex, and read a larger |a| as wider. Error-analysis items give a classmate's claim, such as "increasing for x > -3" for f(x) = |x - 3| - 2, and ask what is wrong.

FAST practice adds the vertex from a table, and positive and negative intervals read from an unlabeled graph. A lint rule fails any item that asks which function matches a graph, table or description, puts a coefficient inside the bars, or writes a domain or range in interval notation.

Items climb from a whole-number or unit-fraction base, such as 5(3)^x or 8(1/5)^x, to the rate form, where the student adds or subtracts r first, to percent stories and tables. In a table the student finds the ratio of successive outputs, and a wrong reason on offer is that the outputs change by a constant amount. At the top, the student builds the model from a percent and classifies it in one step, or picks every decay function out of a list. Both growth and decay are keyed often in every kind of item.

Some items ask only for the base, typed. When the stem names the form, the base is accepted as its value or as written (1.07 or 1 + 0.07). A unit-fraction base must be typed as a fraction, and the grader marks a decimal typed back wrong.

Wrong answers use the rate as the base (0.12 instead of 0.88), decide from the coefficient, judge a unit fraction by its large denominator, subtract the kept percent instead of the lost one, and look at a single output instead of the ratio. The build-the-model items cross the base (right, or built the wrong way) with the verdict, so every option shares a part with two others.

FAST practice adds five-option lists of situations and functions, and a base found from two values two seasons apart, which takes a square root.

Table and graph items ask which equation defines f, or for a or b alone. Graphs label points on the curve. Some items show two tables, one of them linear or quadratic; the exponential one is f in some items and g in others, and the options name both tables (the other table's options use its first ratio), so the student has to check for a constant ratio, and some ask the student to plot a point the table does not show, such as the value at x = 4.

Description items give a starting amount and a percent change, a doubling or halving, or a fraction ("loses three-eighths of its mass"), and ask for the function, the starting value or the base. Reverse items give the function and ask for the percent rate, and one matching board mixes "loses 20%", "keeps 20%" and "gains 20%".

Answer choices cross two real slips. For "which equation defines f" from a table or graph: a read from the next output instead of the starting one, crossed with a base slip (a and b swapped, or the reciprocal base for decay), so every option shares its a or its b with another option. For a percent: growth against decay, crossed with the decimal moved one place (1 + 0.02 for 20%) or with the value after one period used as the start. One error-analysis item shows a student who wrote 900(1 - 0.45)^x for 45% growth.

FAST practice adds a table with a missing output, two tables where only one is exponential, and a point to plot at x = -1, one step before the table starts.

Feature items give an equation, a table or an unlabeled graph and ask for the y-intercept, the asymptote, the range, whether f increases or decreases, and what happens to f(x) as x → ∞ or x → -∞. Select-several items give four or five statements about one function, and matching items sort functions by growth or decay, direction or end behavior.

Rate items ask by what percent f changes each time x increases by 1: reading r from 350(1 + 0.06)^x, finding the ratio from a table such as 256, 192, 144, or turning a base of 1/4 into a 75% decrease and a base of 3 into a 200% increase. No given equation prints a decimal base, and a lint rule enforces it. The FLDOE coverage gate requires a coordinate plane in at least a quarter of the items.

In the multiple-choice asymptote, range and intercept items, the options cross two real slips, such as y = against x = and 0 against the intercept height for an asymptote, or > against ≥ and 0 against the intercept height for a range. Wrong answers put the asymptote at the coefficient or on the y-axis, include 0 in the range, start the range at the y-intercept, swap the intercept's coordinates, decide the direction from a instead of b, report the base itself as the percent (75% for a base of 0.75, 300% for a base of 3), and read a growth rate with the decay rule (94% for 6% growth).

FAST practice adds comparing two functions' percent rates, 500(3)^x against 500(2)^x, domain and range in set-builder notation read from a graph, and points at negative inputs, such as (-3, y) on f(x) = 20(1/2)^x.

The simple-interest skill starts with the interest or the total balance on a deposit, then moves to solving I = Prt for the rate, the time or the principal, loan repayments where the interest is the payment minus the amount borrowed, and select-all statements about how a simple-interest account behaves. The compound-interest skill uses A = P(1 + r/n)^nt with annual, semiannual, quarterly and monthly compounding: find the balance, find the interest earned, compare two accounts (by rate, by compounding frequency, or compound against simple), and identify r/n and nt for an account. Even the short definition questions name an account's rate, time and compounding, and a lint rule enforces the FDOE limits.

Almost every item is a money story: savings accounts, certificates of deposit, bonds and loans. Error-analysis items show a student's steps and ask for the first mistake, such as a rate left as 4 instead of 0.04, a per-period rate not divided by n, an exponent of t instead of nt, or a balance reported as the interest.

Wrong answers are real slips of a believable size: one year of interest instead of the whole term, one year too many or too few, compound interest where the account pays simple interest and the reverse, the wrong number of compounding periods, and a rate never converted to a percent (0.03%). A time is never offered as less than a year. In the compute items no answer choice is an impossible size, such as a balance below the deposit or interest larger than the deposit, and the key sits at different ranks among the options. Balance matching boards use close deposits whose rates and times cross, so sorting by the deposit does not give the balances.

FAST practice keeps only items with a money context, so the bank's short definition questions ("which value of n belongs in the formula?") are left out there. A harder FAST item asks whether the interest added in a quarterly account's second quarter is greater than in its first.