Grade 7 B.E.S.T. benchmarks

Evaluating powers starts with a product or quotient of powers on a whole-number base, such as 2² · 2¹ or 3⁶/3², and moves to negative bases, fraction bases such as (2/3)¹ · (2/3)², and decimal bases, then to a law combined with a second power, such as (-3)⁶/(-3)² - 2³. Formats include typed values, a select-all for every expression equal to 25, and dropdowns that ask for a value along with the reason for its sign (negative because the single power's exponent is odd) or its number of decimal places.

The laws skill asks which expression equals 9² · 9³, 7⁶/7² or (5²)³, asks for the missing exponent, or asks students to name the rule in a dropdown. Harder items chain two laws, as in (3²)³ · 3², match four expressions built on one base to their single powers, or simplify first and then evaluate.

Wrong answers come from real errors: multiplying base by exponent (7³ read as 21), swapping the base and the exponent, adding exponents where they multiply and the reverse, multiplying the bases (8² · 8⁵ as 64⁷), raising only the numerator or only the denominator of a fraction, and losing the sign of a negative base. Both skills also show worked solutions with one faulty step, and the student finds the step where the work first goes wrong.

FAST practice uses the same two skills as bare expressions, and none of its expressions appears in the practice bank. One item has students plot the value of (-2)² · (-2)³ - (-3)³ on a number line.

The benchmark is split into two skills. The first moves between fractions and decimals: which decimal equals 2/3, write 1/3 as a decimal rounded to the nearest hundredth, turn 0.41666... into 5/12 by setting x equal to it, multiplying and subtracting, sort fractions into terminating and repeating, and compare 5/12 with 0.41. The second brings in percents: 25% as a decimal, 0.09 as a percent, 4/9 as a percent rounded to the nearest tenth, 33 1/3% as a fraction, select every value equal to 16 2/3%, and put 0.34, 1/3 and 33% in order.

A typed answer that names a form is graded in that form, so typing the given number back scores nothing, and a repeating value is typed only after the item says how to round it. Wrong answers are the classic slips: moving the decimal point the wrong way or only one place, attaching a percent sign without moving the point, dividing the denominator by the numerator (8/9 as 1.125), cutting the repeat off (0.88), drawing the bar over the wrong digits, and putting the numerator under the bar (1/3 as 0.1 repeating). The percent skill also shows worked solutions with one faulty step to find.

Practice and quiz questions are almost all bare numbers. FAST practice adds the contexts: two-part items compare a 13-out-of-15 quiz score with 80.5% on a 200-question test, three discounts on one jacket written as 7/12, 0.6 and 55%, and two funds that changed by -5/12 and -39%. It also plots -137.5% on a number line with ticks 0.125 apart. A lint rule fails any item that writes a cut-off repeating decimal, such as 1.333333, as if it were exact.

Each family of expressions adds one idea: nested brackets as in 2 · [(8 - 3) + 1], a power as in (4 - 1)² · 2 - 5, a negative base as in (-2)³ + 5 · 4, absolute value as in |-9 + 3| · 5 - 12, subtracting a negative group as in 4 - [(3 - 11) · 3 - 12], and fraction or decimal coefficients as in 0.2 · 10 + 3 · 6 + |-1|. Difficulty climbs through the numbers and the number of steps.

Most items ask for the value, typed or chosen. Others ask which partly simplified expressions still equal the starting value, which written steps are correct, or to match four expressions that differ by one term or one operation to their values. Worked solutions with one faulty step, such as adding 7 + 3 before multiplying, ask where the first error is.

Wrong answers come from real errors: working strictly left to right, multiplying base by exponent (3² as 6), keeping a negative inside absolute value bars (|-9 + 3| as -6), reading -3² as 9, treating 4 - (-36) as 4 - 36, and stopping one step early.

FAST practice uses the same expression families with new numbers, all as bare expressions, the way the FAST presents them.

Each operation runs from integers to decimals, fractions and mixed numbers. Adding starts at -13 + 5 and reaches three-term chains like -1/4 + 1/2 + 5/8. Subtracting reaches -3 1/6 - (-1 1/4). Multiplying ends with three and four factors, where the sign comes from counting the negatives. Dividing ends with mixed numbers and quotients such as (3/4 + 1/2) ÷ (-5/8).

Beyond computing a value, students plot a sum on a number line, select every expression equal to 0, pick every product that is positive, rewrite subtracting a negative as adding in a dropdown, walk through the sign rule for a sum in three blanks, and match expressions to values. Each operation also shows worked solutions with one faulty step to find.

Wrong answers are named slips: keeping the sign of the smaller addend, adding sizes when the signs differ, adding the denominators (4/5 - 3/5 as 1/10), subtracting a negative as if it were positive, multiplying the whole parts and fraction parts of mixed numbers separately, flipping the wrong fraction in a division, and placing the decimal point one place off (0.4 · 0.9 as 3.6).

FAST practice adds a missing-term item (what must be added to -5/8 to get -1/3), a number line marked every 0.2, and two-part items: correct a student's value for 3/4 - (-1/6) and name the error behind it, or compare 1 1/2 - 3 2/3 with 1 1/2 - (-3 2/3).

Adding and subtracting items track temperatures that cross zero, bank balances, the elevations of divers and hikers, football yardage and golf scores against par, plus fractional amounts such as 1/3 cup and 1/4 cup. Harder items keep a running total over three or four changes, group gains and losses, or ask which statements about a hiker's elevation are true. Some items choose the expression in a dropdown, match temperature stories to end temperatures, or plot the result on a number line.

Multiplying and dividing items cover areas with fractional sides, equal shares, a rate over time (a steady drop of 2°F per hour for 5 hours is -10), and an average change from a total (-$11.25 over 5 days). Students also select the expressions that model a recipe, judge statements about servings, and sort stories into multiplication or division.

Wrong answers come from real slips: using the wrong operation or dividing the wrong way, dropping or flipping a sign, applying a change in the wrong direction, mixing up a.m. and p.m., moving the decimal point, rounding too early (a $78.50 bill rounded to $80 before it is shared), stopping before the last step, and fraction slips such as adding the denominators or forgetting to regroup. Both skills also show worked solutions with one faulty step to find.

FAST practice adds two-part items: a submarine's change in elevation and then its final elevation, a stock's daily change and then its change over 12 days, and the exact number of batches a bakery's flour makes (7 4/5) followed by how many full batches that is.

Items build toward the equivalence judgment. Students distribute (which expression equals 6(x + 2)), combine like terms (simplify (5x + 9) - x), factor (which expression equals 5x + 15), and then judge pairs such as 3(x + 4) and 3x + 7. Select-all items ask for every expression equal to 3(x + 2) - x, and matching items pair four expressions with their simplified forms.

Difficulty climbs through the coefficients: whole numbers first, then fractions such as (2/3)(9x + 12) - 5, then a leading negative, as in -(4x - 7) against -4x - 7.

The wrong statements carry the real errors: distributing to the first term only (4(x + 7) as 4x + 7), adding the factor to the constant (4x + 11), changing the sign of only one term after a minus sign, combining unlike terms into a single term, and factoring out of only one term (5(x + 15)).

FAST practice uses the same item kinds with new numbers and adds harder fraction cases, such as the expression equal to (2/3)(12x + 9) - 8x, where the x-terms cancel and leave 6. A lint rule holds both banks to the limits: every expression an item shows must be linear, none may combine more than two linear expressions, and a run of loose terms longer than a ± bx is written as its two expressions, as in (10x + 9) - (6x + 4).

Adding starts with whole-number coefficients, as in (5x + 8) + (3x + 4), then moves to fraction coefficients, sometimes with dropdowns for the new coefficient and constant, and to expressions that mix fractions and decimals. Subtracting runs from (7x + 8) - (3x + 3) and 10 - (4x + 3) to 2(3x + 5) - (4x + 2). Some expressions are written constant first, as the clarification allows, such as (3x + 4) - (5 - x) and ((1/3)x + 8) - (2 - (1/6)x). Both skills have select-all and matching items.

Every item adds or subtracts exactly two linear expressions, given in symbols. No item simplifies a long expression on its own, such as 7x - 4 - 2x + 9; that work appears as (7x - 4) + (9 - 2x), and the triangle perimeter item uses an isosceles triangle, 2(x + 4) + (2x - 1). A reverse item asks what was subtracted from 9x + 8 to leave 5x + 2. Error-analysis items show a student's work in a table with four statements, one per verdict (step 1 wrong, step 2 wrong, both wrong, correct), exactly one true.

Wrong answers come from real errors: changing the sign of only the first term after a minus sign, combining x-terms with constants (8x + 12 read as 20x), treating x as 0x, subtracting constants when adding, and adding the numerators and denominators of fractions. The right expression is never the choice closest to all the others. Typed answers accept any order, such as -4 + 4x for 4x - 4. Lint rules fail any item that combines three or more linear expressions, runs loose terms together beyond a ± bx, shows no sum or difference of two, or shows a nonlinear expression.

FAST practice adds two-part items: find a difference and evaluate it at x = 4, correct a student's subtraction of two decimal-coefficient expressions and name the error behind it, and check a classmate's rewrite of (10x + 3) - (3/4)(8x - 12).

The benchmark is split into three skills: adding and subtracting forms, multiplying and dividing forms (where the symbol may reverse), and writing and solving from a story. Students solve and choose the solution, type it, complete it in dropdowns, graph it with an open or closed circle and a ray, solve x - 2 > 1 and then select the listed values in the solution set, match inequalities to solutions, and pick the inequality behind a graphed solution. Some items put the variable on the right, such as 20 > x + 8.

Harder items check the reasoning: which statements about solving -2x ≥ -14 are true, which inequalities have the solution x ≤ -4 (including traps that match only if the reversal is skipped), which forms say the same thing (-3 ≥ x and x ≤ -3), and a student's work in a table where exactly one of four statements is true. Story items turn at least, more than and at most into one-step inequalities, such as n + 7 ≥ 11 for a roster that already has 7 players.

Wrong answers come from forgetting to reverse the symbol after dividing by a negative, reversing it after adding or subtracting a negative number, using a closed circle for < or an open one for ≤, shading the wrong way, and reading at least as more than. Lint rules fail any given or keyed inequality outside FDOE's four forms, any written inequality with no step, any check of listed values without solving, and any graph whose boundary misses a tick.

FAST practice adds fraction boundaries (x - 3/4 ≤ -5/6), a graph of -(3/5)x ≥ 15, and two-part stories: ferry passengers, party guests where the boundary is 18.5 so only whole numbers below it work, and a draining tank where the student explains why the symbol reverses.

The benchmark is split into three skills: equations like 5x + 8 = 33, equations like 6(x + 4) = 42, and writing and solving from a story. Coefficients run from whole numbers to decimals, negatives and fractions such as (2/5)x + 3 = 11, where the last step multiplies by the reciprocal. Some items put the variable on the right, as in 43 = 6x + 7.

Besides solving, students solve an equation such as 2x - 6 = 8 or 7(x + 3) = 91 and then select every equation with the same solution, solve an equation and then pick the statements that describe its steps, match equations to their solutions, find a missing constant or coefficient (3x + c = 26 when x = 7), and choose valid first steps. Stories use a fee plus a rate (a bus that costs $100 plus $7 per student, $310 in all), per-item discounts in the p(x ± q) form, and elevations below zero; some give the model, such as 9p + 12 = 174, and ask for the solution and what the solution or the term 9p means.

Wrong answers come from real errors: adding the constant instead of subtracting it (x = 8.2 for 5x + 8 = 33), dividing before undoing the constant, stopping after the first step, distributing to the first term only (7x + 3 = 91), charging a one-time fee on every item, and swapping the fee and the rate (50p + 12 = 110). Error-analysis items show a student's work in a table with four statements about it, exactly one true.

FAST practice adds a solution plotted on a number line (-(2/3)x + 6 = 24), decimal coefficients, and two-part stories where Part B checks a claim, such as a submersible's descent that supposedly took 11 minutes. Lint rules fail any given or keyed equation outside the two forms, and any item that asks only what an equation's numbers represent, distributes one side, or checks a given solution by substitution, without solving.

Tax, tips and fees items find a total with one percent, a percent plus a flat fee, or tax and a tip together; reverse items recover the price before an 8% tax from a $54 total, or the fee rate from $40 and $46. Discounts and markups run from a sale price to two changes in a row (20% off, then a $10 coupon), back to the original price, and to a budget check: which purchases fit in $40. Simple interest items use I = Prt to find the interest, the total, or a missing rate, principal or time (some in months), and compare two accounts. Percent change items find increases and decreases, the original before a change, the overall change after a 20% rise and a 20% fall, which of two clubs grew by the larger percent, and which of six counts grew or fell by at least (or by more than) a given percent.

Select-all and dropdown items test structure: which expressions give the total (75 + 0.08 · 75 and 1.08 · 75), which multiplier a total uses, which values go into I = Prt and the interest they give, and which price changes are a 40% decrease. Ratio items run both ways: a 3:2 ratio of boys to girls becomes 60%, and 23% of juniors taking art becomes a ratio.

Wrong answers come from real errors: adding the percent as dollars, subtracting a markup or fee, taking the percent off the total in a reverse problem, writing the rate as the whole percent or as the multiplier 1 + r, using months as years, dividing a change by the new value, adding successive percents (a 20% rise and fall read as no change), and comparing raw gains instead of percents.

FAST practice adds two-part simple interest items: the interest and then the total, a 30-month loan followed by true statements about it, and one account's rate followed by a comparison of two rates.

Items range from one factor (a door 7 feet tall in meters, using 1 ft ≈ 0.3 m) to a chain through a middle unit (96 ounces to pounds to grams), a rate with two units to convert (38 miles per gallon in kilometers per liter, or $6 per pound in Canadian dollars per kilogram), and an area or volume found by converting each dimension before multiplying. Money items convert between US dollars, euros, Canadian dollars and British pounds at a stated rate.

Other formats ask students to select every measurement equal to 2 meters, match four conversions into one unit from two starting units whose results interleave (so neither the unit words nor the sizes finish the match), decide whether a 10-pound bag or a 5-kilogram bag is heavier and by how much, work out a shortfall in a two-part item (60 inches of ribbon against 2 meters needed), and judge a student's conversion.

Wrong answers come from multiplying when the question needs dividing (16 kilometers to miles as 16 × 1.6), adding the factor, stopping after the first step of a chain, running the factor for a rate's bottom unit the same way as its top unit, converting an area only once, and leaving the number unchanged.

FAST practice adds two-part items that compare a converted value with a posted limit or goal, such as a package's mass in pounds against a 17-pound shipping limit, or a delivery van's fuel economy against a goal of 25 miles per gallon.

Table items ask for a reasoned verdict on one table, which of several tables are proportional, which pair breaks the pattern, or the constant ratio of a table stated to be proportional. Graph items always draw the graph: a line through labeled points, plotted points that may or may not share one ratio, part of a line that has to be extended to see where it meets the y-axis, and a plane with four or five labeled graphs to sort into proportional and not proportional. Some grid items have students draw the proportional line through given points. Description items cover a steady rate with or without a starting fee, amounts that stay the same every hour or change from hour to hour, a first game that is free or a first lap that costs the same as the rest, two prices for two quantities (4 hats for $36, 7 hats for $56), and a cup fee that applies only to a first order.

The wrong options are beliefs students hold: it increases, so it is proportional; a constant rate is enough, even with a fee; passing through (0, 0) is enough; constant differences (9, 15, 21, 27) mean proportional; different totals mean different rates; the values are not evenly spaced.

FAST practice adds a graph whose points look lined up but are not, a grid item where the student plots the point that would make a relationship proportional, an e-book site that gives the first chapters free, and two-part items that find a ratio and then pick the pairs that could also appear in the table.

Table items find k from a bare x-y table or a story table, give k and then the value at a new x in dropdowns, pick the true statements about k (18/3, 6 and each y-value is 6 times its x-value are all right), pick the pairs that fit k = 8, match pairs to their constants, fill a missing cell, and decide which of two tables has the larger k.

Graph items always draw the graph: they read k from a line through the origin and a labeled point, plot the point (1, k), match four drawn lines to their constants, use k to find another value, decide which point is also on the line, and compare two drawn lines.

Description items read a rate stated per one unit, divide a rate given for several units (540 gallons in 9 minutes), handle signed rates, and select every correct description of k (21 rolls per tray). In the practice and quiz questions, a multiple-choice or dropdown item that reads k straight from a stated rate also gives a sample (when h = 7, e = 63), so its wrong options are numbers the student can see.

Wrong answers come from inverting the ratio (1/18 for $18 per sweater), taking a coordinate or table value as k (2 for a line through (2, 26)), reporting a sample total as k (72 for 4 sweaters), dropping the sign, reading the relationship backwards (the number of chapters is 8 times the number of pages), and using the difference between values instead of the ratio. FAST practice adds grid items that plot (1, k) from a story, two-part items that find k and then use it, and a candle item where k comes from the difference between two batches.

Drawing items ask students to graph y = 4x or a line through the origin and (3, 2) with the line tool, or to plot the point on the graph of y = (5/2)x where x = 4. Other items test points against an equation (select every point that is part of y = (3/4)x), find y or x on a line (the point (x, 22) on y = 4x), name the equation of a drawn line, match points to the lines they lie on, turn a stated rate into an equation of the form y = kx, and use a rate or two drawn lines to decide which point belongs.

Grid windows follow the grid rules that lint checks: both axes inside the window, no more than about 24 ticks per axis, and every answer point on a grid intersection. Negative constants get windows that extend below the x-axis.

Wrong answers come from inverting k (dividing x by y), taking one coordinate of the given point as k, subtracting the coordinates, swapping the coordinates of a point (a student who puts the quantities on the wrong axes picks (60, 3) for 20 miles per hour), adding the rate instead of multiplying (5 hours and 25 miles at 20 miles per hour), and choosing points that sit just off the line. In the practice and quiz rate stories that ask for a point, those two slips are crossed (the swap, the added rate, and both), so the right point is not the one closest to all the others.

FAST practice adds graphing from a table with the line tool, plotting on lines with negative constants (the point where y = -9 on a line through (-4, 6)), and two-part items that find a value and then decide whether a claimed point such as (16, 25) lies on the line. Its story items say which quantity goes on which axis.

Writing an equation starts from a rate per one unit (25 pages per hour), two amounts (27 marbles in 9 jars), a decreasing quantity (a diver descending 4 feet per second, so y = -4t), a table, or a graph through the origin and one labeled point. Formats include typed equations, dropdowns for k and the equation, select-all for equivalent forms (d = 7w and d = 7 · w), and matching tables to equations.

Interpreting an equation asks what the constant means (in c = 14n, the 14 is the cost of one shirt), which statements about y = 10x are true, how many passengers fit on 7 buses from p = 30b, how much more one pay-per-minute plan costs than another at 150 minutes, and which description goes with each equation (8 items cost $2 for y = 0.25x).

Translating into a table: items match x-values to y-values from y = 5x, match each equation to the list of ordered pairs that fits it (the lists share values, so size alone cannot match them), and pick the pairs that satisfy an equation.

In the multiple-choice and dropdown equation menus, the wrong equations cross two real slips, so no option is one change away from all the others. The wrong number is a real misreading on either side of k: a line through (3, 18) offers y = 18x (the y-coordinate read as k), a line through (2, 14) offers y = 2x (the x-coordinate), and a table can offer the ratio upside down. So the right coefficient is the larger number in some items and the smaller in others. The other slip swaps the variables (x = 6y) or adds instead of multiplying (y = x + 6). Other wrong answers drop the sign of a decrease. FAST practice adds two-part items, including a claim that 4.5 meters per second means 4.5 seconds per meter to check.

Items find a total from a rate (24 granola bars per box, 6 boxes), find the input from a total (180 miles at 24 miles per hour), find a negative change (a temperature that drops 6 degrees every 2 hours, after 9 hours), convert and scale (24 crates every 5 minutes, crates in 1 hour), or count whole groups by rounding up (bucket loads of 9 liters to fill a 150-liter tub). Reasoning items select true statements about a filling machine, select the distance and time pairs that fit 40 miles per hour, match rates to equations, and compare two brands to name the better buy and the difference per ounce.

Every wrong answer is a real mistake, offered only at a size a student could believe for the story; a mistake that lands far from any sensible amount (millions of hours for a water plant) is left out. For a car that goes 28 miles in 2 hours, asked about 6 hours, the choices include 56 miles (the 4 extra hours only), 34 miles (28 and 6 added) and 168 miles (28 used as the rate). Other items build up by adding (47 pages in 2 days when 48 pages take 3 days), round the scale factor to a whole number, write a remainder as decimal digits (270 miles at 60 miles per hour read as 4.3 hours), count an hour as 100 minutes, use one part of a two-part total, lose the sign of a decrease, or compare total prices instead of unit prices.

FAST practice adds number-line items (a candle's change in height after 7 hours), steps on top of the rate (cable left to lay after 7.5 hours), FDOE's fill-the-tank question (360 miles on a full tank, 4 fills for 1,400 miles), and two-part items, including a student's claim about a pool's change in water to check.

Most items find a missing value. When the unit rate is clean, the explanation finds it and scales; otherwise it sets up a proportion and cross-multiplies, as in 9 ounces of sauce for 4 servings, scaled to 14 servings. Other items ask only for the unit rate (84 miles on 3.5 gallons), carry a signed rate (a diver descends 60 feet in 4 minutes, and the stem asks for a negative number for a descent), convert a time unit first (5 cans every 2 seconds, for 4 minutes), or scale a blueprint and then find an area.

Reasoning items ask students to select every proportion that models a rate, where 90/5 = x/12 and 5/90 = 12/x are both right, to select equivalent rates or true statements, to compare unit prices (where a dropdown asks for the better buy, each choice names its reason: the price per unit, the sticker total or the package size), and to match situations to their unit rates or to the amounts they scale to.

Wrong answers come from additive thinking (adding the difference instead of scaling), inverting the rate, cross-multiplying the wrong pair, skipping the unit conversion, reading 2 hours 45 minutes as 245 minutes, scaling an area by the scale factor for only one dimension, and judging the better buy by the sticker price.

FAST practice adds a number-line item for the total change in a reservoir's level, a map scale with a decimal distance, a four-brand unit-price comparison, and a planting job timed in hours and minutes.

About four questions in ten show the figure and ask for its area; the rest give the measurements in words. Numbers move from whole numbers to decimals, and settings include patios, flags, tiles and drainage channels. Reverse questions give the area and one measurement and ask for the missing base, height, parallel side or rhombus side. The hardest items write a side as (16 − x) and ask for x, compare or add two areas, carry an area into a cost, or ask what happens when one dimension doubles (the area doubles) or when both the side and the height of a rhombus double (the area is four times as large).

Every drawn figure labels only its bases and its perpendicular height, as FDOE's limit requires. When a question also gives a slanted side, it gives it in words, so the student still has to choose the perpendicular height. Rhombus questions use a side as the base together with the perpendicular height, the A = bh form on the reference sheet, so no rhombus question needs its diagonals.

Wrong answers are real mistakes: multiplying the base by the slanted side, finding the perimeter, taking half as if the figure were a triangle, forgetting the one-half in the trapezoid formula, multiplying the two bases, adding the measurements, and multiplying a rhombus's side by itself as if it were a square. About one question in ten shows a named student's work and asks which mistake they made. Select-all items ask which expressions give the area, such as (8 + 6) × 5 and 8 × 5 + 6 × 5. FAST practice adds two-part trapezoid items: Part A finds the area, and Part B prices it, diagnoses a classmate's wrong answer, or tests a claim about doubling every dimension.

Nearly every question shows the figure with its measurements labeled: an L-shape, a rectangle with a rectangular notch cut from one edge (a wooden sign, a boat slip in a plot of land, a concrete slab with a planter), a T-shape, a pentagon made of a rectangle with a triangle on top (a barn end or a house gable), and a hexagon whose dashed lines split it into a rectangle and two triangles (a tabletop, a garden bed, a stage floor). Difficulty climbs from whole numbers to decimals and from bare figures to settings such as concrete pads and garden beds.

Select-all items ask which calculations belong in the solution, such as 9 × 3, 4 × 5 and 27 + 20 for a T-shape, and reject tempting ones such as adding side lengths; others ask which two shapes a figure is made of. Matching items describe four composite figures in words and pair each with its area; every figure in one item uses the same unit, so the units cannot give the answer away.

Wrong answers come from real slips: never subtracting the cutout, adding it instead, using the whole bounding rectangle, forgetting the one-half on a triangle, adding the side lengths, and subtracting side lengths instead of areas. Error-analysis items show a student's work and ask which of these happened.

Forward questions give a radius or diameter, often on a drawn circle, and ask for the circumference in terms of π or with a named approximation. Contexts include bicycle wheels, running tracks, tabletops and ponds. Harder questions multiply by a number of laps or revolutions, subtract an inner circumference from an outer one, find the curved edge of a semicircle, compare a circle given by its radius with one given by its diameter, or ask how the circumference changes when the radius is multiplied by 3.

Backward questions give a circumference (12π m, 50.24 cm with π ≈ 3.14, or 220 m with π ≈ 22/7) and ask for the diameter or the radius. Some ask the student to plot the radius on a number line; others write the circumference as (5x − 6)π and ask for x, or ask which equations could be used to find r.

Wrong answers are built from the classic mix-ups: πr in place of 2πr, the area formula πr², doubling twice (2πd), dropping π, and on the backward side reporting the diameter when the radius was asked, dividing by π a second time, or running the area formula backward. Each skill has error-analysis items in which a named student makes one of these slips. FAST practice adds two-part items that carry the circumference into a second step, such as the distance a wheel rolls in four turns or the cost of edging around a rink.

Whole-circle questions give a radius or a diameter, usually on a drawn circle, and ask for the area in terms of π or with π ≈ 3.14. Fractional-part questions, often with the slice shaded on a drawn circle, ask for the area of one slice, of a half-circle plus a quarter-circle, or of a sector such as 315° of a circle. Harder items compare two circles, ask what tripling or quadrupling the radius does to the area (it multiplies by 9 or by 16), write the radius as (x − 2), or ask the student to plot k on a number line when the area is kπ.

The reverse skill starts from an area such as 25π cm² and asks for the radius or the diameter, sometimes through a two-blank dropdown (divide by π to get r², then take the square root) or by matching four areas to their radii.

Wrong answers come from the formulas students mix up: the circumference 2πr given as the area, πr without squaring, squaring the diameter instead of the radius, the whole circle when a slice was asked, and scaling the area by the radius's own factor (area × 3 when the radius triples). On the reverse side the traps are stopping at r², halving instead of taking a root, keeping the 2 from the circumference formula (solving A = 2πr²), dividing by 3.14 again, and giving the radius when the diameter was asked. FAST practice adds two-part items: the area of a patio and then the cost of paving it, the pizza left after 3 of 8 slices are taken, and the part of a square plaza not covered by a round fountain.

Length questions give a scale such as 1 cm = 4 m or 2 cm = 5 km and ask for an actual length or a drawing length, or show a small triangle or rectangle and ask for the matching side on a similar, larger figure. Some ask the student to plot a length on a number line: the actual length, a drawing length on a ruler, or a side redrawn at a new scale. Harder items find the scale factor first and then apply it to a second side, or ask which statements about a scale factor of 3 are true.

Area questions multiply an area by k², work backward from two areas to the scale factor (4 cm² and 36 cm² give 3), find a smaller figure's area from a larger one, or start from a drawn rectangle or triangle whose area the student must find before scaling. One family compares two enlargements and asks which is larger and by how much.

Wrong answers follow the usual errors: adding the scale number instead of multiplying, multiplying when the question goes from real life to the drawing, using the reciprocal of the scale factor, multiplying an area by k instead of k², and forgetting the square root when going from areas back to k. Each skill has error-analysis items built on these. FAST practice adds two-part items, for example reading a garage wall off a floor plan and then counting how many 3-foot storage cubes fit along it, or finding a map distance and then a driving time.

Most questions show a labeled net, with the radius or the diameter on the circles and the height on the rectangle, and ask for the total surface area in terms of π, as the whole number N in Nπ, or with π ≈ 3.14. Other questions give the cylinder and ask which of four drawn nets matches it, or show the cylinder and ask for the rectangle's two sides and the total surface area. Harder items give the diameter instead of the radius, compare two or three cylinders, or ask what happens when the radius doubles (the surface area does not simply double: 42π becomes 120π).

Net-anatomy dropdowns ask what the rectangle's width equals (the circumference), its formula (2πr), and what its height equals (the cylinder's height). Select-all items ask which expressions give the exact surface area, and matching items pair several cylinders with their areas. A typed answer found with π ≈ 3.14 also accepts the results for 22/7 and the calculator's π.

Wrong answers include what students do with a net: using 2πr instead of πr² for the circles, counting only one circle or counting the circles twice, leaving out the rectangle, using πr (half the circumference) as the rectangle's width, swapping the rectangle's two sides, and using the diameter as the radius. Error-analysis items show a student's work with one of these slips. FAST practice draws on the same item families, the drawn net choice included, and its build checks that no FAST item reuses the numbers of a practice or quiz item.

Every question is about a real object (coffee canisters, soup-can labels, pencil holders, water heaters, grain silos), and its measurements are given in words or on a drawn cylinder. Students find the total area of a closed cylinder, the curved side alone, or the curved side plus one base. Harder items compare two containers, carry the area into a cost per square foot or the number of paint cans needed, or work backward from a label's area to the can's height.

Wrong answers include the ways students miscount faces: one end on a closed can, both ends on an open cup, the ends or the curved side left out, πr in place of 2πr for the curved side, the diameter used as the radius, and the volume πr²h offered as a surface area. Error-analysis items show a student's work and ask which of these mistakes it contains. A typed answer found with π ≈ 3.14 also accepts the results for 22/7 and the calculator's π.

FAST practice adds two-part items: a container's area and then which expressions still apply once the lid is removed, a tank's area checked against what one container of sealant covers, and a label's height carried into a second drum's total.

Volume questions give the radius (or a diameter) and the height, in words, on a drawn cylinder, or in a context such as tanks, cisterns and juice cans, and ask for the volume in terms of π, with π ≈ 3.14, or with π ≈ 22/7 when the radius is a multiple of 7. Harder items add the volumes of two tanks, compare two cylinders, list several (r, h) pairs and ask which give 36π, or ask what doubling or tripling the radius does (the volume multiplies by four or by nine).

Height questions give a volume such as 60π cm³ and a radius or diameter and ask for the height. Some ask the student to plot the height on a number line, compare the heights of two cylinders, or find the new height when the volume triples.

Wrong answers include forgetting to square the radius, squaring the diameter, swapping the radius and the height, stopping at the area of the base, confusing volume with surface area, adding the scale factor to the radius instead of multiplying by it, and on the height side dividing by πr instead of πr². Error-analysis items show these as a named student's work. FAST practice adds two-part items: a volume and then the effect of doubling the height, a planter's volume checked against the soil it must hold, and two cylinders where one has twice the radius and half the height.

Most questions describe a real data set (quiz scores, home prices, daily cupcake sales, puppy weights) as a list or as a dot plot, which is how we draw a line plot; the items that sort several data sets at once show bare lists of numbers. Most multiple-choice items ask for the measure and the reason together, so the right measure with the wrong reason is a wrong answer. Dropdown items build a sentence in two blanks: the measure and its reason, the outliers the data has and the measure that follows, or a computed range or interquartile range and the measure it supports. No menu offers the mode, which FDOE's limit leaves out.

Harder items compare two groups (the mean for one group, the median for the other, with the outlier in either one), sort several data sets into mean and median, or ask what happens to the median when an outlier grows or every value drops by 2. Some compute the range and the interquartile range from a dot plot before choosing. Error-analysis items show a student's claim, such as "the mean is best because it uses every value", and ask whether the measure and the reason are right.

Wrong answers are reasons students actually give: the mean is always best, an odd count calls for the median, the interquartile range is always smaller so it is always better, or the mean fits because the whole class took the same quiz. FAST practice adds two-part items that compute the range or the interquartile range first and then judge it, for example a farmers market's daily visitors with one festival day among them.

The first skill works from box plots: which city has the higher median, with the reason, which set is centered higher and which has the wider interquartile range, how much greater one interquartile range is than the other, what an extreme value does to the range but not to the interquartile range, which measure, the range or the interquartile range, fits a comparison when one value is extreme, whether each plot is symmetric or skewed, and where two middle halves overlap. A few give only summary numbers, such as the maximum that would give set B the same range as set A.

The second skill works from paired dot plots, histograms and lists: how much greater one group's range or median is, which interval holds each group's median, whether Monday's and Friday's median waits are under 10 minutes, which group has the greater range and by how much, and which statement gets both the median and the range right. Harder items ask which of two classes with the same median most likely has the greater mean, or check a list of statements against two lists.

Wrong answers come from comparing the wrong statistic (Q3 when the median was asked, the range when the interquartile range was asked), reading the tallest bar as the median, naming the other group, giving one group's value instead of the difference, and moving the mean away from a tail instead of toward it. Every statistic a question compares is a measure of center or of variation. A quartile or an extreme value is the answer only on the way to one, as in the maximum that gives set B the same range as set A; otherwise it appears in wrong answers and in lists of statements to check. FAST practice adds two-part items: compute one class's interquartile range and then decide which class was more consistent, or find the difference of two interquartile ranges and then judge a claim that the higher maximum means the better team.

Most questions give a sample count out of a sample size, plus a population total, and ask for the predicted count as a typed number, a multiple-choice answer, or a two-blank dropdown (the sample proportion, then the prediction). Others give the sample result as a percent or a decimal, such as 35% of 400 voters. Harder items mix count, percent and decimal inputs in one matching grid or ask for a population percent.

Select-all items ask which statements about one survey are true: the sample proportion, the prediction, the number who did not choose the option, and a plausible but wrong method such as 24 × 900. Error-analysis items show a student's prediction and ask which mistake produced it: predicting the complement, using the fraction upside down, reading the sample count as a percent, or adding the growth from sample to population. Every wrong option is computed from the item's own numbers, and the build checks that the four mistakes give four different answers.

Wrong answers in the prediction questions are the same kinds of mistake: adding the growth from sample to population, reading the sample count as a percent, predicting only for the people who were not sampled (which leaves the sample's own count out of the total), adding the sample's count on top of the prediction, applying the rate to the sample instead of the population, giving the sample percent as a count, or comparing the part with the rest of the sample instead of the whole sample. The complement, the number who did NOT choose the option, is among them in at most one question of each kind, so the key rarely sits beside the one option that adds with it to the population.

Reading questions show a labeled circle graph (the favorite fruits of 200 students, say) and ask how many chose one category, two together, how many more chose one than another, how many did not choose a category, or which categories passed a threshold. Harder items run backward from a slice's count to the total, apply one graph's percents to a larger group, or check a list of statements.

Angle questions convert a share to a central angle: 40% of a class walks (144°), 1/6 of students eat eggs (60°), 18 of 60 chose pizza (108°). Others show a partly built graph with one grey slice marked "?", or judge whether angles such as 120°, 90°, 80° and 60° can form a circle graph (they add to only 350°).

Construct questions give the data in a table or a sentence, and the student builds the whole graph. In percents, each count becomes its share of the whole (95 of 250 students ride the bus: 38%), and the totals always give whole percents. In degrees, percents, fractions or counts become angles (10 of 24 students: 150°). Harder ones add a category that is "the rest", combine two groups, or round.

Wrong answers are real slips: a different slice or everyone outside it, the percent read as the count, the percent difference instead of the count difference, a percent read as degrees (20% as 20°), 180 in place of 360, 4° per percent, a slice compared with the rest of the circle instead of the whole, and one part of a fraction such as 5/8 (45° instead of 225°). The rest of the circle is a wrong answer in at most one question of each kind. Matching items mix kinds of rows, so size alone cannot pair them. Every circle graph, read or built, has at most six slices, each at least 6% of the circle. FAST practice builds one graph in percents and one in degrees, and adds two-part items that pair a count with a judgment about a claim.

Most questions describe a scenario (30 students' homework minutes in 15-minute intervals, which of 4 main dishes each student picks, the percent of seats sold in each zone) and ask for the most appropriate graph, for the kind of data and the display in two blanks, or for every display that could work. The kind-of-data blank has three choices, each the key somewhere: categorical, numerical shown value by value, and numerical grouped into intervals. Matching items pair three or four scenarios with displays.

Harder items turn the question around: which scenarios fit a box plot, or which questions a shown histogram or box plot can answer (the median's interval, yes; the exact median, no). Others ask which displays could show one variable but not the other. Critique items show a choice someone made, such as a stem-and-leaf plot for 240 test scores, and ask for the best critique; two show a good choice, where the wrong critiques give a false reason or invent a flaw.

Create questions give the data in a table, a sorted list or a raw list to tally: 18 books lent by kind become a bar chart, 16 butterfly wingspans a histogram over 10-millimeter intervals, 16 students' numbers of pets a line plot. At the hardest level some values sit on an interval boundary (the question says which interval takes them), line plots use half sizes, and bar-chart labels look like numbers, such as zip codes. FAST practice builds a food drive's cans and 18 hives' honey yields.

Wrong answers come from the common confusions: a bar chart for numerical measurements, a histogram or line plot for categories, a box plot to show individual values or the mode, and a stem-and-leaf plot for hundreds of values. Critique items also offer the right flaw with the wrong fix, such as a box plot to replace a histogram that hid the individual values, since a box plot hides them too. FAST practice includes a transit example where route numbers look numerical but only name the routes.

Questions ask how many outcomes an experiment has, which values are in its sample space, which list is the complete sample space, or which outcome is missing from a student's list. The experiments use FDOE's range of objects: dice with 8, 10, 12 or 20 sides, cards numbered 14 through 29, cards numbered by 4s from 4 to 40, letter tiles spelling COMMITTEE or MISSISSIPPI, spinners with colored or numbered sections, and a deck where only the suit is recorded.

The counting behind the harder items is inclusive counting (cards 14 through 29 are 29 − 14 + 1 = 16 outcomes), skip counting, and noticing what is recorded (a card's suit has 4 outcomes; the card itself has 52). Dropdown items ask for the counting method and the total, and statement items ask what changes the sample space: adding blue marbles does not, adding one white marble does.

Wrong answers come from counting objects instead of outcomes, forgetting the 1 in an inclusive count, including impossible values such as 0 or 7 on a number cube, and listing every marble separately. Error-analysis items show a student's count and ask which mistake produced it; every choice names only what the student did, and the reason it is wrong is in the explanation. Each list of counts mixes these slips so the answer is not always in the middle, and no list offers both neighbors of the answer (5, 6 and 7 around a 6), where picking the middle number would work. The same holds for lists of outcomes: no item offers both a list missing one outcome and a list with one extra outcome, which would put the answer in the middle of three nested lists.

Interpretation questions give a probability such as P(win) = 7/10 or P(rain) = 0.05 and ask for the likelihood word, where the value sits (between 1/2 and 1), or the reverse: the probability of an impossible or a certain event. Others sort probabilities into bands, pick the most or least likely of four events given in mixed forms, judge whether a stated probability is reasonable (a 0.95 chance of snow on a winter day in Miami is not), or find the smallest number of winning marbles out of 20 that makes a win likely.

Comparison questions ask which of two events is more likely, which probabilities are greater than 2/5, how to order three or four events, which bag gives the best chance of a winner when each holds a different number of tickets, and which value lies strictly between two others. Typed answers are graded in the form the question names: an answer asked for as a decimal does not score when it is typed as a fraction, and the reverse. A typed question that asks for any probability between two others accepts every value strictly between them, as a fraction or a decimal.

Wrong answers come from comparing numerators across forms (2/3 against 0.65 because 2 < 65), treating anything between 0 and 1 as equally likely, judging which value is closer to 1 backwards, reading 0.06 as larger than 0.6 because it has more digits, reading a count as hundredths (12 of 30 students as 0.12), and giving the probability of the other outcome. FAST practice also plots a probability as a percent on a number line, asks a game designer's question with one setting per likelihood band (enter the unlikely one as a decimal), and adds two-part items, such as testing a claim that 11 winners in a jar of 22 tokens already makes a win likely.

The basic question gives the contents of a bag, a spinner, a die or a club roster and asks for the probability of one outcome, typed in the form the question names or chosen from four. The favorable count then gets harder to find: a rule on numbered outcomes (a multiple of 4 that is greater than 8), repeated letters on tiles (BASKETBALL has 10 tiles but only 7 different letters), or a standard 52-card deck. When the count comes from a rule, the build computes it from the rule, so the answer, the options and the explanation always agree.

Other items ask the student to select the favorable outcomes, choose every event on one die with probability exactly 1/2, match events to probabilities, or select every form equal to one probability (3/4, 9/12, 0.75, 75%). The edge cases appear on purpose: a green bead from a jar with no green beads has probability 0, and a number less than 6 on a spinner numbered 1 to 5 has probability 1.

Wrong answers come from favorable over unfavorable instead of over the total, the complement, a favorable count one too many or too few, an outcome added to or dropped from the sample space, and treating each color as one equally likely outcome when the groups have different sizes. Most lists leave the complement out, or show it beside a second pair that also adds to 1 (a miscount and its complement), so looking for the two choices that add to 1 does not reliably find the answer. The answer's place in the list varies, and a lint rule fails any item whose answer is the largest choice beside its own half or the smallest beside its own double. Error-analysis items show a named student making one of these mistakes, and the build checks that the four mistakes give four different values. FAST practice also plots a probability as a percent on a number line marked in eighths.

Questions start from results given in the stem or in a frequency table: a spinner spun 200 times with Red 70 times, or a coin flipped 200 times. Students compute the experimental probability, the theoretical probability of the device, or the expected count (1/3 of 150 draws is 50), including how many more blue draws than red to expect and how many golds to expect in 240 more spins after 60 golds in 160. Matching items give each simulation its own number of trials (Ana counts red over 100 draws, Ben counts blue over 400), so a larger probability does not always mean a larger count and the rows cannot be matched by order alone.

Comparison items ask which of two students' results is closer to theory, which estimate is more reliable (100 tosses or 2,000), whether a spinner looks fair from its counts, and which statements about a set of flips are true. In the closeness items the run with fewer trials is sometimes the closer one, because closeness and reliability are different questions, and a choice that argues from the number of trials is always wrong there. A dropdown asks for the definition itself: favorable outcomes observed divided by the total number of trials.

Wrong answers come from favorable over unfavorable, the complement, the theoretical value where the experimental one was asked, a count read as hundredths, and the belief that the theoretical probability should change to fit the data. Apart from the one item that asks which values could be probabilities, no choice states a probability above 1: the reversed ratio (60/12) became dividing by the other results (24/96). A lint rule fails any item here that states a run of fewer than 100 flips, rolls, spins or draws. FAST practice adds two-part items, such as testing a raffle app's fairness from 300 draws, or a spinner section that came up 250 times in 1,200 spins.