Grade 6 B.E.S.T. benchmarks

Plotting starts with reading. The easiest questions show four lettered points on a number line and ask which one is at −3, or which one is the opposite of −4. Then the student plots a decimal or a fraction on an interactive number line marked in halves, fifths, fourths, tenths or eighths, so the first job is working out what one tick is worth. A few review questions give a direction and a distance (6 units left of 0, 3 units above 0 on a vertical line) and ask for the numbers.

Ordering and comparing climb from integers to decimals to fractions, sometimes with the values given only as points on a number line. Decimal comparisons always involve a negative, so the place-value trap turns around: −0.8 is less than −0.79. Comparison questions ask for the symbol, or for the one statement with both the right comparison and the right reason ("−3 > −7, because −3 is to the right of −7"). The hardest comparisons set a positive fraction beside a positive decimal (5/8 and 0.63), and the mixed-form questions order a fraction, a decimal and a percent, with fractions such as 3/8, 7/20 and 13/16, never 1/3 or 2/3.

Wrong answers come from real mistakes: picking the opposite of the target point, counting from the left end of the line instead of from 0, ordering negatives by their distance from 0, treating a longer decimal as a larger one, and reading 1/2 as 0.12. Error-analysis questions show a student's plotted point or ordered list and ask what went wrong. The coverage gate requires a number line on at least a quarter of the practice and quiz questions, and a lint rule fails any same-form item with no negative value and any mixed-form item with a negative value or a repeating fraction. FAST practice adds a two-part item that compares 3/8 with 0.4 and then plots 3/8, and ordering items followed by a count ("how many of the three numbers in the list are greater than 0.6?").

The easiest questions give a quantity, often only as a lettered point on a number line, and ask for its opposite and what that opposite means: a diver shown at −22 feet has the opposite +22, which is 22 feet above the surface. Other forms of the same question plot the opposite on an interactive number line, match quantities to their opposites, or complete a sentence ("the opposite value is −45, which means a loss of $45"). Many elevation and temperature questions use a vertical number line, which FDOE's clarification names.

The middle level tells a short story in words, says which direction is positive, and asks for three things: each quantity as a signed number, what 0 represents, and a comparison in everyday words. A pelican 9 feet above sea level is +9 and a fish 14 feet below it is −14, 0 is sea level, and the fish is lower than the pelican. The stories cover temperatures (warmer, colder), elevations and floors (higher, lower), money owed (more in debt, less in debt) and positions on an east-west trail. Wrong choices flip a sign, use the other quantity's number, call 0 the freezing point on the Fahrenheit scale, or compare 8 and 3 instead of −8 and −3.

The top level plots two values and asks whether they are opposites, with a reason: a submarine at −30 feet and a helicopter at +30 feet are both 30 feet from sea level, in opposite directions. The wrong choices include "a positive value and a negative value are always opposites" and "they are not opposites, because −30 is less than +30". The coverage gate requires a number line on at least a quarter of these questions, and a lint rule fails any item on this benchmark that asks for a sum or prints an arithmetic operation. FAST practice asks the same things, with its own stories.

The easiest questions ask for |n| directly, or ask how many units a number is from 0, as a typed answer, a choice of four, a point to plot on a number line, or a match between numbers and their absolute values. Select-all questions ask for every number exactly 4 units from zero, which has to include both 4 and −4. The next set moves to decimals, fractions and mixed numbers, such as |−3.5| and |−1 1/2|.

Comparison questions ask which of two numbers is farther from zero and why, first as bare numbers on a number line, then in context: which day's temperature, −18°F or 15°F, is the more extreme departure from 0°F. In the practice and quiz questions every choice gives a reason written with absolute values, and one choice is true. The wrong ones are the beliefs students bring: a negative or a positive number is always farther from zero, the greater number has the greater absolute value (4 > −7, so |4| > |−7|), and of two readings below zero the one closer to zero has the greater absolute value. When a story compares two readings, each name appears in two of the four choices, and the wrong choices are worded briefly in some questions and in full in others, so neither a name nor the length of a choice points to the answer.

A further set orders numbers from closest to farthest from 0, picks the city whose temperature is farthest from 0°F, and compares two absolute values, such as |−1/4| and |−3/4|. A lint rule fails any item on this benchmark that prints an arithmetic operation or asks no distance question, any reasoned set where only the key carries absolute-value bars, and any story whose two names do not each appear in two choices. FAST practice adds two-part items (find |−23|, then choose what it represents) and a reverse question: a number's absolute value is 27 and the number is less than zero, so what is the number?

The comparing skill starts with bare pairs: which of −7 and 5 has the greater absolute value, with the reason as part of the answer. Every choice reasons with absolute values, briefly or in full, so a wrong one reads "5, because 5 is positive, so |5| > |−7|" or "2, because positive numbers always have the greater absolute value, so |2| > |−9|". Then two readings appear on a vertical number line (a thermometer, or heights above and below sea level), and the student says which is colder, farther from sea level or closer to it, and by how much. The distance across zero is |a| + |b|, and a standard wrong answer subtracts the two sizes instead (7°F apart instead of 23°F). The top level asks for every reading within a limit, such as depths no more than 40 feet below sea level, where −28 is inside the limit and −42 is outside it.

The calculating skill evaluates expressions such as |−10| + |−3| and |−15| − |8| + 2, typed or chosen from four. Distractors come from subtracting the sizes, multiplying, and keeping the negative signs. Select-all and matching items ask which expressions equal a given value.

The hardest calculating questions put a sum inside the bars, as in |−12 + 5| − |−7|: simplify inside first, then take both absolute values, then subtract. A lint rule fails any item on this benchmark with more than two operations in one expression, or with a fraction or a decimal. FAST practice mirrors both skills: a select-all by absolute value followed by a typed value, a colder-by-how-many question on a vertical number line, within-the-limit readings for a cable, a crane load and a water tank alarm, and expressions such as |5 − 17| − |−7|.

Multiplication starts with a decimal times a whole number (4.5 × 6), then a decimal times a decimal (3.75 × 2.4, 0.125 × 5.6). Students type the product, choose it from four, match expressions to products, or fill in two dropdowns: the number of decimal places in the factors, then the placed product. Division runs the same way, from 14.4 ÷ 6 to 4.125 ÷ 0.75.

A lint rule checks both FDOE limits on every practice, quiz and FAST practice question, and no question breaks either one.

The top level puts the operation in a short story: ribbon at 12.50 dollars a yard, or how many complete 1.5-meter pieces can be cut from 8.4 meters of wire, where the student has to drop the remainder. FDOE lists this benchmark as a mathematical context only, so these stories go beyond the FAST's format.

Wrong answers are slips students make. Many come from the decimal point: the point one place off, in one direction per question (102 for 10.2 in one, 3.01 for 30.1 in another); the digits with no point at all; or a divisor scaled without scaling the dividend. In a story, the larger point slip appears only when its total stays believable: never more than one unit costs for less than one unit, and at most 500 dollars for a purchase, 200 dollars for part of a day's pay or 50 miles for a runner or cyclist. Others add instead of multiplying, multiply instead of dividing, or use only the whole-number part of a number (3.65 × 8 for 3.65 × 8.4, or 9 + 0.4 = 9.4 for 3.4 × 3). Estimates round too far or cut the decimals off, and counts of complete pieces keep the leftover or round it up. Error-analysis questions show a student's answer, such as 3.4 × 6 = 2.04, and ask which statement describes the mistake. FAST practice adds two-step purchase problems (the total, then the change from a bill) and a price-per-pound comparison.

Each skill starts with bare computation: 2/3 × 3/25, 2 1/2 × 3, 1 1/3 × 9/28, 5/8 ÷ 3/4, 4 1/3 ÷ 2/3. Students type the answer, choose it from four, build it in numerator and denominator dropdowns, or match expressions to values. Typed answers are graded by value, so 15/2 and 7 1/2 both earn credit. The explanations convert mixed numbers first, flip the divisor and multiply, then simplify.

Word problems come next: a fraction of a quantity (8/21 of a recipe that uses 3/4 cup of sugar, or 14/27 of a 4 1/2-mile trail), the area of a small plate or tile measured to the 1/32 inch (2 2/3 by 1 5/32 inches), and "how many complete pieces fit", where a 5 1/4-foot board cut into 1 3/4-foot shelves gives a count, not a fraction. The size-reasoning questions sit alongside these.

Wrong answers are errors students make, such as adding instead of multiplying, multiplying the numerators but keeping the first denominator, cross-multiplying, flipping the first fraction instead of the second, multiplying without flipping, multiplying only the whole-number parts of mixed numbers, and giving the part left over instead of the part used. In a count of complete pieces they include keeping or rounding up the leftover and dividing only the whole-number parts. Error-analysis questions show one of these and ask for the true statement.

Both limits hold in practice, quiz and FAST practice questions, and a lint rule checks them: it fails any fraction-by-fraction product with no denominator above 20 and any unit fraction divided by a whole number or the reverse. FAST practice adds two-part items that ask for the expression first ("which expression gives the number of portions?"), then the value.

The decimal skill uses a dozen problem structures: the total of several counted amounts (2 backpacks at 3.25 kilograms plus a 1.5-kilogram bag), what is left after some is used, a decimal part of an amount added or taken away, the difference between two rate-times-time amounts (a car and a truck over 6 hours), a rate that changes after the first hours, and a total shared equally. Every problem gives a value to the hundredths and uses at least two different operations, one of them multiplication or division.

The fraction skill has three main structures: take a fraction of a whole and find what is left (16 cookies, 5/8 eaten); combine two mixed-number amounts and then multiply or divide by a whole number (6 1/2 feet minus 2 1/4 feet, cut into 2 equal shelves); and find a whole from a part, then adjust it (a tank 3/5 full holds 18 gallons, then 1/4 of its capacity is drained). Answers print as fractions or mixed numbers (10 1/2 gallons). A typed 10.5 still counts unless the question asks for a fraction or mixed number; then only that form scores.

Wrong answers are slips students make, such as stopping one step early, giving the part instead of the rest, taking the fraction of the current amount instead of the capacity, dividing where the story multiplies, and slipping the decimal point in a factor (0.08 read as 0.8). Error-analysis questions show a student's answer and ask what went wrong.

FAST practice follows the same limits. A lint rule checks every question for money, perimeter or area words, a decimal item with no hundredths, a problem that uses one operation or only addition and subtraction, a small-denominator fraction product, a printed numerical expression, and decimals and fractions mixed anywhere in one problem, answer choices and explanation included.

GCF questions are solved two ways in the explanations: listing factors, or prime factorization with the lowest power of each common prime. Tasks include typing the GCF, matching pairs to their GCFs, selecting every common factor of 24 and 40, and reverse questions ("the GCF of 12 and n is 4; which could n be?"). Stories cover identical bags or bouquets, the largest square that tiles a 48 cm by 36 cm rectangle, and kits that ask for both the number of groups and what goes in each. Every number a student works with in a GCF question stays under FDOE's 1,000; the largest numbers a student has to test are the candidates in reverse questions, up to 252.

LCM questions list multiples for small pairs and use prime factorization, the highest power of each prime, for pairs where listing runs long, such as 18 and 24 or 21 and 24. Stories are events that repeat on a schedule (buses, lights, runners on a track) and packages sold in different counts (hot dogs in 8s, buns in 6s).

Wrong answers match the real mix-ups: the LCM when the GCF was asked (and the reverse), a common factor that is not the greatest, a factor of only one number, one of the two numbers itself, the product of the two numbers offered as their LCM, and the lower power of a prime taken where the LCM needs the higher one. Method questions ask which to use and why, with the reason built into each choice. Lint rules flag any "identical groups, none left over" story that does not ask for the greatest number, and any LCM question with a third number or a number above 25. FAST practice adds two-part items: choose GCF or LCM with a reason, then solve.

Factoring questions ask for the rewrite of a sum such as 30 + 42 as the GCF times a sum, 6(5 + 7), as a choice of four, three dropdowns (the factor and the two addends), a select-all, or a matching set. At the top level the same sums come with a story, such as 30 square tiles and 42 round tiles set out in equal rows. Missing-number questions fill one blank: 16 + 40 = 8(2 + ?).

Expanding questions go the other way: which sum equals 4(5 + 9)? The hardest questions model a story with the distributive property: 12 plates each hold 2 tacos and 3 empanadas, so the total is 12(2 + 3).

Distractors are the classic slips: factoring out a common factor that is not the greatest, dividing only one addend by the factor (6(2 + 18) for 12 + 18) or subtracting the factor from an addend (15 + 35 = 5(10 + 7)), multiplying only the first term when expanding (6(4 + 7) = 24 + 7), adding the factor instead of multiplying (5(3 + 8) = 8 + 13), and writing digits side by side (45 + 49 for 4(5 + 9)). The menu for one addend never offers the other addend's value inside the parentheses, because 9(3 + 2) is as true as 9(2 + 3). Error-analysis questions show one of these and ask which statement is true. Lint rules check that every sum to be factored has two composite addends with a common factor, and that no common factor passes 1,000. FAST practice follows the same directions and adds two-part items that expand one expression and factor another.

Questions start with a single power (5³, 13², then 0.4² and (2/3)³), typed or chosen from four. The next level combines two powers with a sum or difference (3⁴ − 2⁵), compares two powers with <, > or =, matches powers to values, or selects every power equal to a given number. At the top, two fraction or decimal powers are combined ((1/2)³ + (1/4)², 1.5² − 0.5³), three powers are combined left to right (5² − 3² + 2⁴, 10² − 2⁵ − 0.4²), and fraction, decimal and mixed-number bases appear in area and volume: a square tile with 1 1/2-foot sides, a cube-shaped box with 0.4-meter edges.

Wrong answers come from treating the exponent as a factor (6² = 12, (2/3)³ = 6/3), switching the base and the exponent, using the base one time too many or too few, raising only the numerator of a fraction ((3/4)² = 9/4), putting the decimal point in the wrong place (0.2³ = 0.8), adding fractions across, and taking 3² + 2⁴ away all at once in 5² − 3² + 2⁴. Error-analysis questions show a student's answer and ask which statement is true.

Powers come first and then the terms combine left to right, so the explanations evaluate every power on its own line before combining. FDOE lists this benchmark as a mathematical context only, so the tile and box stories go beyond the FAST's format. FAST practice keeps to bare expressions, with whole-number bases at the first two levels and fraction and decimal bases at the top, such as (1/2)³ + (3/4)² and 0.5² + 1.2² − 0.1³. Lint rules fail any item with an exponent above 5 or a negative base.

The core task is writing the factorization, 52 = 2² × 13, either typed or matched from a list of numbers to their factorizations. Other questions work the same idea from different sides: choose which product of primes equals 60, find the number whose factorization is 2³ × 3 × 7, or say how many times 2 and 3 appear as factors of 72. The explanations divide out each prime in turn (72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9 gives three 2s) and then write the result with exponents.

Wrong answers are near misses a student could really write: a factor that is not prime (2² × 15 for 60), a missing prime (2 × 3² for 90), and an exponent one too high or too low. The how-many-times menus list four counts in a row, from 0 to 3 up to 4 to 7, so the right count does not always sit in the same place, and no menu shows both half and double the right answer. Error-analysis questions show a student's factorization and ask which statement is true.

The GCF and LCM questions use the factorizations of two numbers, take the lowest or highest power of each prime, and multiply; one dropdown form asks for both from the same pair, such as 18 and 24. A lint rule fails any LCM question with a number above 25, and any factorization asked of a number that is not composite. FAST practice is all factorization, with numbers up to about 2,000 at the top level and multiple-choice foils such as 2² × 10² for 400.

Fraction-to-decimal questions start with fractions like 3/4 and 3/8, then mixed numbers such as 2 5/8. Students type the decimal, choose it, match fractions to decimals, or select every fraction equal to 0.8. Comparison and ordering questions convert first: which comparison of 7/8 and 0.85 is true, or which of four boards is longest when one length is 2 3/8 feet.

Decimal-to-fraction questions write the decimal over its place value and simplify (0.15 = 15/100 = 3/20). The percent skill converts fractions to percents (5/8 = 62.5%), decimals to percents and back, and uses matching grids where each value has one or two equivalent forms to find. A typed question that asks for a fraction or a decimal accepts only that form: 0.6 typed back into "Write 0.6 as a fraction" does not score.

Wrong answers are the conversion errors students make: writing the digits after a decimal point (1/8 = 0.18, 1/2 = 12%), dividing the denominator by the numerator, reading hundredths as tenths, dropping the whole number of a mixed number, stopping or rounding the division too soon (2 3/8 = 2.37 or 2.38), reading a remainder as tenths (7/5 = 1.2), mixing up neighbouring eighths (7/8 = 0.625), stopping short of simplest form (6/10), and moving the point one place, three places or the wrong way for a percent (7% or 0.007% for 0.7, 300% for 0.3), never one place and three places in the same list of choices. Error-analysis questions show each of these. Lint rules fail any decimal past the thousandths, any fraction that does not terminate, and any typed conversion that does not check the form. FAST practice adds conversions inside short measurement stories, some of which add a fraction and a decimal (a 5/8-meter ribbon plus 0.5 meter), and percent claims to check (a survey finds 0.6 of students ride the bus; a newsletter reports 6%). FDOE lists this benchmark as a mathematical context only, so these stories, like the measurement contexts in the harder practice questions, go beyond the FAST's format.

Addition moves from two integers (−4 + (−5), −15 + 8) to four-term chains added left to right. Some addition questions have the student find the sum on an interactive number line and plot it, and missing-addend questions ask what must be added to −7 to get 5.

Subtraction starts with a larger positive taken from a smaller one (3 − 10), then subtracting a negative (−5 − (−12)), then four-term chains. Rewrite questions ask which addition expression equals −23 − (−16). The story skill covers temperatures, elevations, bank balances, elevators and football yardage: students write the expression, pick + or − for each change, compute the result, or work backward to the starting value. FDOE lists this benchmark as a mathematical context only, so the stories go beyond the FAST's format.

Wrong answers come from the sign rules: adding the absolute values when the signs differ, subtracting them when the signs match, the right size with the wrong sign, treating every number as positive, subtracting in the wrong order, and subtracting 7 where subtracting −7 means adding 7; longer chains and stories add leaving out one number or one change and taking a change the wrong way. A second ± pair is one of these slips with its sign lost too, and no answer choice in a story repeats its starting number. When a story fixes the sign, as for a diver who never comes back up to the surface, every choice keeps that sign. Error-analysis questions show a student's work and name the slip. A lint rule fails any item whose stem and answer show no negative integer and no larger subtrahend, running each story forward to see whether it goes below zero, and any select-all option or matching row that is whole-number work. FAST practice adds a number-line item for (−6) + 5 and two-part stories that ask for the expression first, then the value.

Multiplication starts with one negative factor ((−4) × 6), then two ((−7) × (−8)), then three factors multiplied left to right, then missing factors: (−12) × ? = 84. Division follows the same sign cases, from −12 ÷ 3 to −110 ÷ (−10), then a division followed by a multiplication, (−81 ÷ 9) × (−3), and missing dividends or divisors. Other forms choose the sign and then the value in two dropdowns, compare two quotients with <, > or =, or match expressions to values. Every quotient comes out to a whole integer.

The story skill has three kinds of question: a repeated change (a submarine dives 9 feet each minute for 3 minutes, a change of −27 feet), a negative total split into equal parts (−96 feet over 6 equal dives), and two phases combined (a drop of 4°F an hour for 7 hours, then a rise of 2°F an hour for 5 hours). Some ask only for the expression that models the story, which tests multiply-or-divide as well as the sign.

Wrong answers come from the sign rules (two negative factors called negative, three called positive), from adding or subtracting instead of multiplying or dividing, from a fact slip, from one repeat or one share too many or too few, from multiplying or copying the product where a missing factor needs dividing, from stopping after two factors, and from leaving out one part of a two-part story. A positive answer never has its sign-flipped twin as the only negative choice. In a story whose changes all go one way (a drop repeated each hour, a loss shared equally), every numeric choice keeps that sign and no share is larger than the whole. Error-analysis questions ask which statement about a student's work is true. A lint rule fails any item, select-all option or matching row with no negative integer (6 × 6 became (−6) × (−6)). FAST practice writes some divisions as fractions (−63 over 7, times −5) and pairs stories with a choice of expression.

On the words-to-expression side, the easiest questions translate one operation ("the difference of f and 11"). Middle questions stack two ("12 less than twice a number e"), and the hardest are cost and profit stories: a $25 monthly fee plus $0.10 per text, or $24 earned per dog minus $45 a day for supplies. Students pick one of four expressions or type their own, and typed answers also accept common rearrangements, such as 20 + h for h + 20.

On the expression-to-words side, students choose the phrase that matches an expression such as k/3 − 4, match four expressions to four phrases, or select every phrase or expression that fits. In the hardest items a story defines s as the number of smoothies sold, and two drop-down menus ask what 6s and 75 represent in 6s − 75 (the money from the smoothies sold, the rent paid for the day). Every menu choice is a short phrase of the same shape (the rate, the variable, the whole total), so the answer never stands out by length.

Wrong answers are the slips students make: reading "less than" in the wrong order (18 − x for "18 less than x"), grouping too much (2(e − 12) for "12 less than twice e"), swapping the coefficient and the constant, or adding the rate to the daily cost. Error-analysis items show a student's translation, such as 3 − k for "k decreased by 3", and ask which slip produced it. FAST practice adds two-part items: write the expression, then say what a number in it represents.

Both FDOE limits are machine-checked: a lint rule fails any item with more than two unknowns or none, and any item that asks for a value. Every practice, quiz and FAST practice item passes it.

There are three kinds of question. The first gives a sentence ("A water slide requires riders to be at least 42 inches tall") and asks for the inequality: pick it, pick the symbol from a drop-down, type it, or match four descriptions to four inequalities. The second has the student graph an inequality on an interactive number line from −10 to 10 by choosing an open or closed circle and the direction of the shading. A smaller set runs the other way: a graphed inequality is shown and the student names it.

The hardest questions take two steps. A container holds at most 250 milliliters and already has 145, so the student finds m ≤ 105 and then decides whether m = 100 works. Each option pairs an inequality with a yes or no; the wrong ones use the full 250 instead of what remains, add or drop the "or equal to", or reach the wrong verdict.

Elsewhere the wrong answers mix up "at least" with "more than", draw a closed circle for a strict inequality, or shade the wrong way. FAST practice adds bare inequalities to graph and more of the two-step boundary questions, told with the same measured quantities.

Lint rules fail any item that tells a counted, money or points story, any graphed story that shades below zero for a quantity that cannot be negative, and any answer choice that is not a single variable against a number. Another fails any graphing item whose explanation does not end with the finished graph.

The easiest questions have one variable and two operations, such as 4h − 5 when h = 8. Middle questions bring two variables and negative values (5a − 6b + 1 when a = −3 and b = 2), or a formula whose answer is negative, such as a lemonade stand's profit 2c − 15 after 6 cups. The hardest include like terms that can be combined first, three terms with a square, as in 5m² + 3n − 2m when m = 2 and n = −4, or a formula with a grouping bar, such as the average (a + b + c)/3 of three low temperatures. Students type the value or pick it from four.

Distractors come from order-of-operations and sign slips: adding before multiplying, reading 3x at x = 4 as 34, computing (3k)² instead of 3k², writing (−3)² as −9, mishandling a subtracted negative, and skipping the division under a fraction bar. Error-analysis items show a student's wrong value ("Jaden evaluated n² + 5 when n = 6 and got 17") and ask which slip produced it.

FAST practice adds two-part items that evaluate one expression and then a changed version with the same values. For every typed key, check-grading runs the key and each accepted alternate through the real grader, so an entry such as 17.0 for 17 is known to score. A lint rule fails any item with more than three variables, and any item with a decimal or a fraction in its values, choices, answer or worked steps; a division such as 30/6 or −16/4 that comes out to an integer passes.

Distributive-property questions start with a positive number outside the parentheses (6(a + 7)), move to negative factors and subtraction inside (−4(3n + 2)), and end with select-all items over five candidate forms of −5(2y − 3), such as 5(−2y + 3). Combining-like-terms questions grow from two terms (15k − 9k) to six (8b − 6 + 4b + 3 − 9b − 2) and include perimeter and garden stories whose parts are expressions. Equivalence questions combine both moves, as in 4(3k + 1) − 7k.

Answers are chosen, typed, assembled from drop-downs, or matched. Wrong options come from named errors: multiplying only the first term (8k + 5 for 8(k + 5)), losing a sign when distributing a negative, squaring the variable (8x² for 5x + 3x), multiplying the coefficients (42k for 7k + 6k), and folding the constants into the variable terms. In a multiple-choice rewrite the wrong options cross two errors (each alone, then both), so no choice sits at the center of the others: for 3(x + 4) they are 3x + 12, 4x + 12 (added 3 and 1), 3x + 4 (the 4 never multiplied) and 4x + 4 (both). A stem with no subtraction never gets a choice with a minus sign. Error-analysis items show one of these answers and ask which slip made it.

Typed answers accept rearranged forms, so 18y − 12 and −12 + 18y both score, and a check confirms every accepted alternate equals the key. FAST practice adds two-part items: pick the equivalent form, then select every other expression that also matches, such as 4(2d + 10) for 8(d + 5).

A lint rule enforces FDOE's limit (at most two variables; integer coefficients, factors and terms, x/2 excluded; a rewrite, never a value), and every item passes.

Easy practice and quiz questions are one-step equations with four candidate values (which value is a solution of 3x = 27?) or a set in braces to check. Most middle questions are one-sided inequalities such as 5x − 4 < 21 with five or six candidates, and the student selects every value that works. The hardest put the variable on both sides (5x − 3 ≤ 2x + 9), and their select-all lists include zero or a negative number. A few questions show the candidates in a table.

About a quarter of the easy and middle practice and quiz questions put the variable alone on the right (15 = x + 9, 17 ≥ 3x + 2), and only those use > or ≥. Six practice and quiz questions tell a short story (a backpack that holds at most 15 pounds); a story's candidates are counts, never negative.

The candidates often include the boundary, where students slip: counting it for a strict inequality, dropping it for "or equal to", or comparing the candidate with a number in the inequality instead of substituting. Reasoned items give a claim ("Jada says x = 4 is a solution of 3x − 2 ≤ 13") and four yes-or-no answers with reasons, two of each. The wrong ones cross a slip in the work (reading 3x as 3 + 4, or not substituting) with a slip in the comparison (reading the symbol backwards, or as an equals sign).

FAST practice follows the same pattern: easy items pick one value (one of them for the inequality 3x < 12), the rest are select-all lists, and some use > or ≥ with the variable on the left (3x − 2 > 4). A lint rule fails any item whose equation or inequality has a decimal or fraction coefficient, a second variable or an exponent, uses the not-equal sign, or offers a value to check that is not an integer.

Solving questions begin with equations that cross zero once (x + 5 = 2), move to negative answers (x + 14 = −6), and end with the variable on the right (−3 = x − 15). Students type the value, pick it, match equations to solutions, or plot the answer on a number line, and a few stories ask them to write and solve (a game score that drops 8 points to −3).

Writing questions describe a start, a change and an end (a temperature that drops 8 degrees to −3°F). The student picks the equation, completes it from drop-downs, matches situations to equations, or selects every equation that fits. Each situation-matching item uses one pair of numbers four ways, so every row needs the plus-or-minus decision. Select-all items accept reordered terms or swapped sides (−4 = w + 15), never a solved form. Nearly every top-level writing item also asks for the starting value.

Wrong answers are real slips: the same operation instead of the inverse, the inverse step taken as the answer (−11 for x + 11 = −4), a sign flipped or dropped (on the answer, the start or the end), a number copied from the equation, the wrong operation for the change (adding a drop, subtracting a rise or a change), the start and the end swapped, or a subtraction in the wrong order (18 − x = −7). The drop-down that picks the step also offers multiplying and dividing, and a few writing items offer a multiplication or division equation (8x = −3). Error-analysis items show one of these and ask which slip it was.

FAST practice adds two-part items (choose the equation, then solve it), and its hardest bare equations put the variable on the right (−30 = x + 13). Lint rules fail any item that neither shows a negative nor asks for one, any keyed equation outside x + p = q or p + x = q, any inequality, and any decimal or fraction.

Solving questions start with positive equations (4x = 28, x/8 = 6) and then bring in negative coefficients and divisors (−8x = 32, x ÷ (−8) = −4). The hardest ones also put the variable on the right side (−52 = −4x, 7 = x ÷ (−8)). Students type the answer, pick it, choose it from a drop-down, match equations to solutions, or select every equation with a given solution.

Writing questions come in three families: equal groups (54 cookies on 3 trays, so 3t = 54), constant rates of loss (a balloon descends for 4 minutes and changes by −80 feet, so 4r = −80), and equal shares of a total (five neighbors each pay $26, so d ÷ 5 = 26). Some of the hardest attach a sign to the share (a shared debt). Wrong equations add where they should multiply or divide, reverse the product (54t = 3), divide or subtract where they should multiply, multiply or subtract where they should divide, divide the count by the total (5 ÷ d = 24), or drop the negative sign. When a rate or share item offers four equations, they cross two slips (the operation with the sign, the order or the value), so no wrong choice stands alone.

Solving distractors are real slips of a believable size: a flipped sign, a number copied from the equation, adding or subtracting in either order (3 − 27 = −24 for 3x = 27), and on x/a = b one group too many or too few. No answer choice for a solution multiplies where it should divide, since 252 for 6x = 42 fails a quick size check; a solution found that way appears only in error-analysis items ("Noah solved 2x = 38 and got x = 76").

FAST practice adds two-part items: choose the equation, then solve it. Lint rules fail any given or keyed equation outside px = q or x/p = q, any inequality, and any decimal or non-whole fraction a student sees, choices and explanations included.

Decimal questions run from x + 1.5 = 4.3 through 1.2 · x = 13.2 to x ÷ 0.5 = 9.4, and students type, pick, match, or select every equation with a given solution. Fraction questions run from like denominators (x + 1/9 = 5/9) through unlike ones (x − 3/4 = 1/8) to a fraction coefficient ((2/3)x = 5/12), and are almost all typed. A typed fraction answer also scores as its decimal when the decimal ends (0.875 for 7/8), and an improper fraction also scores as a mixed number (1 1/6 for 7/6).

Wrong answers come from the move each form invites: doing the same operation instead of the inverse (15.84 for 1.2 · x = 13.2), subtracting instead of dividing (12), copying a number from the equation, a + b for a − x = b, a × b for a ÷ x = b, multiplying the two fractions (5/36 for 5/6 = x + 1/6), the reciprocal of the answer, fractions added or subtracted straight across, a decimal point one place off (90 for 0.6 · x = 5.4), and a sum one whole too big (6.4 for x − 1.6 = 3.8). Error-analysis items show one of these and ask which slip it was.

FAST practice is typed throughout and puts the unknown on the right or in the divisor more often (11.25 = 2.5 · x, 11.7 ÷ x = 6.5). Every FAST item states its equation; the one word problem that did not (a price per pound) is now 6.5 ÷ x = 1.25.

Lint rules check the benchmark's limits on every item: no decimal and fraction mixed anywhere a student sees them, choices and explanations included (a fraction key still accepts its ending decimal); no negative number or zero in an equation or an answer choice; and every equation relates exactly three numbers.

Easy questions give two counts (9 daisies and 4 sunflowers) and ask for a part-to-part ratio. Middle questions ask for one part to the total, in simplest form. Hard questions use three kinds of objects and ask for one part to the total. Interpretation questions give a ratio and ask which sentence describes it ("For every 7 cats there are 4 dogs"), or which statements about it are true, or which ratios are equivalent to the one in the story.

The wrong answers are the ones students write: the reversed ratio, part to whole when part to part was asked (6 to 15 instead of 6 to 9) or the other way round, the other part, a total that leaves out or double-counts a group, and the unsimplified ratio. Error-analysis items show a student's ratio ("daisies to the total, in simplest form, is 6 to 20") and ask which error it is.

There are no figures on this benchmark; every question is in words. Most answers use "a to b" for two quantities and "a:b" for three, and a few are typed as fractions (goals allowed to goals scored, 18 to 24, is 3/4). A lint rule fails any item on this benchmark that asks for a decimal or a percent, names a ratio of bare numbers with nothing counted ("write the ratio 14:21"), or prints a value that is not a whole number. FAST practice adds a drop-down item that writes one ratio with a colon, with "to" and as a fraction, a typed fraction answer, and three-part collections in simplest form.

Finding questions give a total and a count: 240 miles in 4 hours, $23.20 for 8 mugs, 7.2 cups of batter for 9 muffins. At the top level both numbers are decimals (8.28 miles in 3.6 hours). Each takes one division, and the wrong options include multiplying, subtracting, dividing the wrong way (0.02 miles per hour) and misplacing the decimal point.

Comparing questions put two stores, cars, pumps or printers side by side. Each option names a winner and a rate ("Store B at $0.45 per apple"), so a student who computes a rate correctly but picks the wrong store is caught. Harder items use decimal quantities such as 1.5 hours, include ties, or ask how much less per item the cheaper store charges. Every multiple-choice comparison in the practice and quiz questions (and the hardest FAST practice ones) also offers "Neither: both at" a rate, which is the answer only in a tie, so the wording of a tie is not a giveaway. Explanations show each rate as one division line and say which quantity goes on top and why.

There are no figures: the numbers are in the story. No item makes the student convert units, and a lint rule fails any item on this benchmark that puts two units of one kind (hours and minutes, feet and inches) in the same situation, or that asks for anything other than a rate (the summer review questions used to scale ratio tables). FAST practice adds two-part items where the student computes one unit rate and then picks the statement with the right answer and the right reason, a pace question in minutes per mile, and a repeating rate typed exactly in simplest form.

Easy questions show a complete two-column table (4 and 6 cups of flour and sugar, up to 16 and 24) and ask for the ratio in simplest form. Middle questions blank one cell with a question mark, in either column of a later row; the student finds the multiplier and types the value. Hard questions ask for a value past the end of the table: how many cups of melon go with 21 cups of grapes when the table stops at 12.

Three-column questions give two parts and a total (boys, girls, total) and ask for a part or the total. Every question on this benchmark reads or completes a table, including the summer review questions: a scale model's conversion chart (1 inch on the model for 24 on the car), a teacher-to-student ratio held for a bigger wing, and a table of equivalent rows scaled down to simplest form.

Wrong answers come from additive thinking (adding the same amount down a column), copying the known value, using the neighboring row's multiplier, reading the wrong column, and reversing the ratio (3:2 for 2:3). Our coverage gate checks that this benchmark's items carry the table FDOE's text calls for, and a lint rule fails any item without a table, with more than three columns, or with three columns and no total. FAST practice adds tables whose rows are not consecutive multiples and, at the top level, three-column part, part and total tables where the student scales the total past the end of the table.

Finding-the-percent questions give a part out of a whole and ask for the percent, or for the percent NOT in the group. Easy ones scale to a denominator of 100; harder ones divide to a decimal such as 68.75%, or round (4 out of 9 is about 44%). Finding-the-part questions ask for 80% of 65 or a percent of a count, then money (the 8% tax on a $72 sweater). Finding-the-whole questions give the percent and the part (12% of a zoo's animals are reptiles, and there are 6 reptiles) and ask for the total.

Wrong answers are traceable slips. They include the part or its complement reported as the percent, a decimal point one place off, dividing the whole by the percent, taking the percent of the part, and dividing by 12 instead of 0.12. The complement and the answer moved one decimal place (1.3 or 130 beside 13) each appear in only some sets, and no set moves the point both ways. A question asking for a percent NOT in a group may offer the group's own percent, the given count as a percent with that count taken from 100 (30% and 70%), or the NOT count read as a percent (51% when 51 of 60 cookies are not chocolate chip). Answers are chosen, typed, matched or built from drop-downs. Summer review questions compare two groups by percent (18 of 24 peaches ripe against 26 of 40); the group with more ripe peaches has the smaller percent.

There are no figures on this benchmark. A lint rule fails any item that asks for a sale price, asks about the part NOT in a group (or the amount left) in more than one step from the numbers it gives, asks for a unit rate, converts units, or offers a decimal choice for a count. FAST practice adds two-part sale items (a discount, then the price a discount came from), percents such as 62.5%, and NOT questions (75 of 375 pastries are defective).

Comparison questions give two or three win records, recipes or rates (Team A won 3 of 5 games, Team B won 4 of 7) and ask which is greater. The student turns each ratio into a decimal or finds each unit rate. Most options pair the winner with a reason or a rate. Wrong options include comparing numerators or totals, flipping the comparison, giving the winning rate to the other name, and claiming a tie that is not there. Mixture questions give a ratio and a total (juice and soda in a 3:1 ratio, 20 cups in all) and ask for one part, at the middle level from a tape diagram.

Conversion questions combine a rate with a count or a total in customary units (length, weight, capacity) or metric units (kilometers, grams, milliliters): 3 pints of soup in each of 5 jars is how many cups, 6 laps of 1,320 feet is how many miles, a 3,600-gallon pool filled at 8 quarts per minute. Wrong answers include skipping the conversion, converting one item only, using a neighboring factor (12 ounces in a pound), converting the wrong way, stopping a step early, and moving a decimal point too far.

Ratio-of-length questions work with a model train at 1:48, a map at 1 inch to 5 miles, two ribbons in the ratio 3:4 with one length in yards and the answer in inches, and flags or gardens compared by length-to-width ratio, where the longer one is the trap. Lint rules fail any item on this benchmark that only converts, states no ratio or rate, mixes customary and metric units, or is a percent problem. FAST practice adds two-part mixtures whose second part converts the answer (cups of cranberry juice to fluid ounces), and a ratio of two ribbon lengths with one length in yards and the answer in inches.

For plotting, students click the point on an interactive coordinate plane, name the quadrant a point lies in, and read the coordinates of a point already drawn. The easiest plotting questions use halves in the first quadrant and whole numbers in the other three and on the y-axis; the next level adds halves in every quadrant and on both axes. The hardest are short stories: a kite at (-4, -5) drifts 7 units right and 12 units up, and the student finds where it lands. Whenever the student plots, the worked solution draws the answer point.

For reflections, students reflect a point across a named axis (choosing the image, picking each coordinate from a list, or matching four points to their images), identify the axis from a point and its image drawn on the plane, select every pair that shows a reflection across a given axis, and at the top level apply two reflections in a row.

Wrong answers come from the usual slips: swapping x and y, moving the wrong way on one axis, changing the sign of the wrong coordinate or of both, a swap and a lost sign together, and, in two-reflection questions, stopping after the first reflection, doing only the second, or assuming the two reflections cancel. Some questions show a named student's plotted point and ask what went wrong.

A lint rule fails any item that asks a student to plot a first-quadrant point with a whole-number coordinate, and any coordinate plane that draws or keys a point between its grid lines. The usual grid rules hold too: the answer lands on a grid line, both axes are in view, and the grid is not too dense to read. FAST practice adds its own quadrant, plotting, translation and reflection items; its two plotting items use (3.5, 1.5) and (-2 1/2, -4 1/2) on a half-unit grid.

Questions that give all four vertices and ask for the area or perimeter draw the rectangle. They start in the first quadrant, then span the axes, then move into contexts: a garden plot, a dog run that needs fencing, a patio, each with a stated scale such as "each unit = 1 yard". A few add a cost step, such as $0.50 per square foot.

Vertex questions give three corners and ask for the fourth: students plot it, choose it, or pick its x and y from two lists. Harder ones give an area, such as 24 square units, and ask for the vertex or the missing side that produces it, sometimes for a rectangle that does not start at the origin.

Wrong answers come from counting grid points instead of units (one side or each side one too long, or one side one too short), adding the width and height, giving the perimeter for the area or the area for the perimeter, doubling only one side, and subtracting instead of adding across an axis. For the fourth vertex they come from measuring a side the wrong way from the right corner, swapped coordinates, the side lengths written as a point, a side miscounted by one, a dropped minus sign, and a coordinate copied from a given vertex. Error-analysis items ask which of these a named student made.

FAST practice joins the steps in two-part items: find the fourth vertex, then the area; or find a missing side from the area, then the perimeter. A lint rule fails any item with an ordered pair that is not an integer, or with a rectangle whose sides are not parallel to the axes.

The easiest questions mostly keep both points on one side of an axis, such as (2, 3) and (2, 8). The next level brings in negative coordinates, and most pairs cross an axis, such as (-10, 2) to (9, 2), where the two distances from the axis add. At the top level most questions are maps, with places named by their coordinates and a stated scale ("each unit represents 2 miles").

Other questions run the idea the other way. Point B is 12 units to the left of (7, 1): the student gives its coordinates or plots it. Some compare two segments and ask which is longer, and by how much.

Wrong answers match real mistakes: adding the coordinates when the points are on the same side, subtracting when they are on opposite sides, reporting a negative distance, counting the starting grid line as 1, giving one point's distance from 0 or the shared coordinate, and on a map, leaving the answer in grid units or dividing by the scale. Error-analysis items show a named student's work and ask which of these happened.

FAST practice has its own distance items at every level: typed distances with the points drawn, multiple-choice distances across an axis, and five map questions with a scale. A lint rule fails any item with an ordered pair that is not an integer, or with a distance between two points that share neither coordinate.

The main task is the area of a drawn triangle, first with whole numbers, then with decimal lengths, then as named objects such as a triangular sign or garden. The same idea also appears as matching four triangles to their areas, completing "Area = ___ = ___" from two menus, and choosing which expression gives the area.

The derivation has its own questions. A rectangular sheet 12 inches by 9 inches is cut along its diagonal: students find each triangle's area, type the whole sheet's area when they are given one triangle's area, or pick the statement that explains why the 1/2 is there.

Wrong answers are forgetting the 1/2 (offered in about 6 of every 10 area items), doubling instead of halving, halving twice, using the slanted side as the height or pairing the height with it, a misread decimal or mixed-number base (8.5 read as 8), and one adding slip: the base plus the height, half of that sum, twice that sum, or all three sides. No item shows more than one adding slip, and no choice is more than twice base × height. For a rectangle cut along its diagonal, the wrong answers are the whole rectangle, three quarters of it (the triangle counted as 3 of the rectangle's 4 corners), a quarter of it, and one edge sum. Where the correct answer falls among the choices changes from item to item, and when it is the smallest choice its double is never beside it. Error-analysis items show a named student's work and ask which of these happened; on some of them the triangle is obtuse and the student used the leaning side.

FAST practice uses the same mix: right triangles, mixed-number bases such as 3 1/2 feet, acute and obtuse triangles with the height drawn inside or outside, and an obtuse triangle with its slanted side labeled. A lint rule fails any triangle item whose figure has no height, whose height is a question mark, or that asks for the height instead of giving it.

Quadrilateral questions climb from rhombi (four equal sides and a dashed height) to parallelograms (slanted side and height both labeled) to trapezoids. Composite questions move from L-shapes to rectangles with a hole (the outer rectangle minus the missing piece) to a rectangle with a triangle on top (found by adding). Questions name the part that counts, such as the figure with the hole removed, never "the shaded region", because the shading on screen is pale. In a missing-dimension figure the unknown is marked "?": the height of a parallelogram or triangle, or the depth of the notch in an L-shape.

Wrong answers come from named mistakes. Rhombus: side × side, twice side × height. Parallelogram: the base or the height times the slanted side, halving, adding lengths, the perimeter. Trapezoid: one base only, half of one base, the bases multiplied, every length added. Missing piece: the outer rectangle, the cutout alone, one piece only, subtracting the cutout's side lengths, the cutout twice or a full strip (fill-in items also offer adding the cutout). Missing dimensions: adding or subtracting instead of dividing, halving the area, dividing by twice the base, half the area minus the base, the labeled length; for triangles also twice the area and forgetting to double. No choice is bigger than the figure allows (base × slanted side, the rectangle around a trapezoid), and at most one missing length exceeds the printed area. Error-analysis items build on these, including a trapezoid missing its 1/2.

FAST practice uses parallelograms and trapezoids and adds two-part items: the area of one piece, then the whole figure. A lint rule fails any item whose given quadrilateral is a rectangle or a square, any L-shape or rectangle with a hole whose lengths are all whole numbers, and any triangle drawn without its height.

Volume questions give a labeled prism. Every item includes a fraction or a decimal edge: a half-unit edge at the first level, then decimals (a fish tank with a volume of 33.75 cubic feet), then fractions such as 5/6, 1/2 and 1/3. Missing-edge questions give the volume and two edges, and the student divides by the base area. Two-step same-volume questions give one prism's edges and a second prism's base and ask for the second prism's height. At the first level, Box B holds the same volume as Box A; later levels also move the contents of one container into another, such as sand from a sandbox into a storage bin, soil from a dump truck bed into a garden bed, or water from a loaf pan, a tray or a trough into a vase or a tank, and ask how high the sand reaches or how deep the soil or water is.

Wrong answers are adding the three edges, multiplying only two of them (the area of one face), finding the surface area, adding one face's area to the volume, doubling or halving the answer, dividing by one base edge instead of the base area, stopping before the division (giving the volume or the base area), giving a labeled edge, subtracting instead of dividing, and multiplying fractions without multiplying the denominators. Error-analysis items ask which of these a named student made.

FAST practice uses two-part items: find the volume, then the height of a second prism with the same base and a new volume. A lint rule fails any item whose given edges, volume and labels are all whole numbers.

Prism questions add face pairs (top and bottom, front and back, the two ends), and the worked solution follows that order; pyramid questions add the base and the triangles. Other versions match each pair of faces to its total area (shuffled, with one value left over), select every expression equal to the surface area (the two wrong ones make the same mistake, so their values match too), or ask how many cans of paint cover a chest.

Choosing the net: the student picks which of four drawn nets, labeled A to D, folds into a given box or square pyramid. The wrong nets: for a box, the top and bottom with the wrong edges (always), a box with one edge used twice, a missing or seventh face, or both flaps on one side; for a pyramid, the base edge and slant height swapped (always), a second square, or three triangles. Two nets always have wrong-size faces, so neither the familiar shape nor a face count finds the answer. Some are two-part items: the net, then its surface area. Top-level pyramids use decimals, such as a 4.5 cm base.

Wrong answers for prisms: one face of each pair, a pair left out or counted twice, faces with the wrong edges, covering every face when one is left bare or leaving out the wrong one, and the volume; for cubes, one, three, four or five faces, all six counted twice, the edge times 6, and the twelve edges added. For pyramids: only the base, no base, the base twice, one to three triangles instead of four, a pair of triangles counted twice, no 1/2 in each triangle, and one slant height for both pairs on a rectangular pyramid.

FAST practice has similar prism and pyramid items, some split into two parts: the base area, then the total, or the net, then the surface area. A lint rule fails any pyramid item without its slant heights and any item that gives a surface area and asks for a length.

The first level sorts four questions into statistical and not statistical. The second asks students to select every statistical question from a list of five. The third gives a real purpose and asks for the one question that would collect the needed data, or asks students to respond to a classmate's wrong claim about a question.

Every statistical answer in the bank is a number that varies: minutes of exercise, heights, shoe sizes, bedtimes. The questions that are not statistical have one fixed answer: the number of sides of a hexagon, the minutes in an hour, the year the school was founded.

FAST practice adds items that ask which question is not statistical, why a question is statistical (because different students give different answers, not because the answer is a number), and which question could have led to a stated conclusion. No item on this benchmark uses a figure.

Mean questions go from adding and dividing, to finding the missing value that gives a target mean, to finding the new mean after a value is replaced, removed or added. Median questions go from a short sorted set to an unsorted set with an even count, where the two middle values are averaged. Mode and range questions include data sets with two or three modes.

Every item shows its data inside a real situation: a set in braces, such as the minutes students waited for the bus, a dot plot, or a table. Wrong answers are sizes a student could believe: a mean choice always lies inside the data's range, and when a change pushes the mean up or down, a wrong answer moves that way too. They include dividing by the wrong count, leaving a value out of the sum, averaging the middle values without sorting first, the point halfway between the least and greatest values, and, on dot plots, the height of the tallest stack given as the mode.

Choosing questions show a set with one value far from the rest, where the median is the better choice, and sets with no outlier, where the mean is. Harder ones give two groups' data sets with the same mean; students find each median and decide which group has extreme values pulling its mean. Mode questions always use numbers, such as the number of scoops in each order, never categories like favorite flavors, because the benchmark calls for a numerical data set. FAST practice adds two-part items: compute a mean, then choose the measure that fits.

A lint rule fails any item that writes its data as a bare list, gives only a summary with no data, or uses a negative value.

In practice and quiz questions, the first level reads a single box plot in a real-world context and asks for one of its five values, the range or the interquartile range, such as the first quartile of daily park visitors. The second uses the summary: the interquartile range, the percent or number of values in part of the plot, which quarter is most spread out, whether a claim about the plot is true, the shape of the data, and whether a value could be an outlier. On every single-plot question each value sits on a tick mark, so it reads exactly.

The top level compares two box plots on center, on the spread of the middle half, or on range and IQR together. On the multiple-choice versions, two choices start with each group's name, so the name gives nothing away. Spread answers read like "Class A, because its IQR of 20 is greater than Class B's IQR of 10", and each wrong reason cites a true fact about another measure, such as a higher maximum, or gets backwards what a larger IQR means. Center answers state the comparison and what it means ("so a typical fish length for Lake A is lower"); the wrong ones claim every value in one group beats every value in the other, read the wrong line of a box, or draw a conclusion the measure cannot support.

Shape questions ask for the shape (symmetric, skewed left or skewed right) and the reason in two menus. The wrong shapes include naming the skew for the side where the data bunch up, and the wrong reasons include a fact true of every box plot (each part holds about 25% of the data).

One gap remains: every plot is horizontal, because our display has no vertical mode yet, although FDOE allows vertical number lines. FAST practice also shows a box plot on every item; it adds two-part items that read one group's Q3 or IQR and then compare two groups, plus a shape question and an outlier question.

The first level names the shape of a histogram. The next adds dot plots with gaps, clusters and outliers, and asks which interval holds the most data. The top level compares two plots, draws a conclusion from where most of the data sit (which ages a movie ad should aim at), and explains what a long tail means.

Every display comes with a real-world story, such as test scores for two classes or wait times at two food trucks. Wrong answers come from real misreadings: taking the height of the tallest bar as a data value, calling the interval farthest to the right the most common, deciding that the group with more data in one bar or stack is the one with higher values, and claiming "most" for a peak that holds less than half of the data.

The shape word is computed from the data, so the answer cannot disagree with the picture. When two plots are compared they share one scale, and the generator refuses to build a pair that does not, so a wider spread really looks wider. FAST practice adds questions about where a gap falls, a histogram of ages at a family swim night with two clusters, and how the interval width of a histogram can hide or reveal gaps and clusters.

A lint rule fails any item that shows a display other than a histogram or dot plot, labels its axis only "Value", or names a mean, median or other statistic.

Box-plot questions start with 7 sorted values from a real-world setting, where each quartile is one middle value, then 8, where a quartile averages two. At the top level, 9 unsorted values hold one far value: students sort them, then find a quartile or the median, pick the five-number summary, or build the plot with a whisker out to the far value. Every level has box plots to build. A practice or quiz question that starts from a list alone ends its worked solution with the finished box plot, and every drawn box plot has a title and a value axis labeled with its unit, such as Time (minutes).

Wrong plots to pick from come from real construction errors: the range cut into four equal parts, the median halfway between the extremes, the median counted in both halves, or whiskers stopped one value short. The practice and quiz generator checks that the four plots look different. Wrong five-number summaries count the median in both halves, drop the far value, skip sorting, or stop the whiskers short.

Histogram questions read the count in one interval, choose an interval width (multiple-choice options state the bar count; dropdowns ask for it), pick true statements about shape and peak, or build the histogram over given intervals: from a sorted list first, then from unsorted lists with values on interval edges, with some decimals at the top level. Data sets have 20 values or fewer, and the generator calls a histogram skewed only when its bars visibly lean. FAST practice adds two-part items (a quartile, then the matching box plot or summary) and one box plot and one histogram to build.

A lint rule fails any item that asks for the IQR, a fence or a mean, shows more than 20 data points, writes an interval in interval notation, opens with a bare data list and no context, or has a plot to build with no title or axis labels.

Every question is set in a real-world scenario, such as pages read, customers served or test scores. The center skill starts with menus: after 18 is replaced by 100, the mean will increase significantly and the median will stay the same. Next, students decide which measure changes more and by how much, then find the score needed on the next test to reach a target mean.

The variation skill starts with the new range after an endpoint changes, then asks what adding an outlier or a middle value does to the range and the IQR. Every answer statement is checked against the recomputed range and IQR before the item is built.

The hardest questions ask which measure best describes a typical value or a typical spread after a change. The practice and quiz generator recomputes the measures and checks that the change moves the range far more than the IQR, or the mean far more than the median, before it keys the answer. Wrong answers include thinking a far value is simply ignored, that both measures change equally, that the IQR grows more than the range, and that the range describes the middle half. FAST practice adds two-part items: the new mean, then which measure of spread moved more.

A lint rule fails any choosing question that does not describe a change, any item that names the mode, any negative data value and any change to zero.