Geometry (course 1206310) B.E.S.T. benchmarks

We practice the benchmark as two skills, solving with the relationships and completing proofs of them, and every item is built on a figure. In the first, easier items mark one measure where a transversal crosses two parallel lines and ask for a numbered angle, or have students sort angle pairs into corresponding, alternate interior, alternate exterior and consecutive interior. Middle items label two angles with expressions such as (4x - 7)° and ask for x or an angle. Harder items chain relationships across two pairs of parallel lines, two transversals or three parallel lines, follow a bent path between parallel roads (solved with a parallel through the bend), and use the perpendicular bisector in both directions. About one item in five has a setting, such as streets and rails.

Proof items usually show a Step, Statement and Reason table. The student picks the reason for a step, fills menus in the table, matches statements to reasons, orders the statements, finds the flawed steps, or finishes a numeric proof. A proof of a listed relationship names it in the stem, so citing it inside its own proof is only ever a wrong, circular choice.

Wrong answers come from real slips: taking the supplement of a congruent pair, treating consecutive interior angles as congruent, subtracting from 90 instead of 180, reporting x for the angle, stopping after the first link of a chain, and using half the segment as the equal distance on a perpendicular bisector. In proofs: calling corresponding angles alternate interior, justifying the right angles at a midpoint with vertical angles, or using a fact before it is proved.

The templates compute each figure from its givens and fail the build when a drawn angle disagrees with the item, and check-grading runs every typed answer through the real grader.

We practice the benchmark as two skills: naming the criterion and judging whether information is enough, and completing proofs. In the first, students name the criterion a marked pair fits, often after finding a pair the figure leaves unmarked; give a reasoned verdict ("No, because the congruent angles are not between the pairs of congruent sides"); pick the added fact that completes a named criterion, or select every fact that would; write the congruence or similarity statement in matching vertex order; and sort several pairs on a matching board with a "Not enough information" column. A few items are picture choices, such as which of four marked pairs SSS proves congruent. Difficulty climbs from two separate, fully marked triangles to shared sides, bowties and overlapping triangles, then to proportional sides.

Proof items climb the same way: one blank in a short proof, then two blanks or a hidden pair to supply, then chains through matching parts, proportional-side similarity, and checking a classmate's proof.

Wrong answers come from the usual confusions: calling Angle-Angle-Side marks Angle-Side-Angle, missing the shared side, calling two legs and the right angle Hypotenuse-Leg, closing a similarity proof with a congruence criterion, accepting Side-Side-Angle, pairing vertices by their position on the page, dividing the two pieces of a cut side instead of a piece by the whole side, and citing matching parts before the triangles are proved congruent.

When an item asks which added facts would be enough, the figure draws the parts not yet given visibly unequal, and every Side-Side-Angle pair is drawn as the two different triangles those parts allow, so no answer can be measured off the drawing. The templates check every drawing against the item's claims, and the proof skill re-derives each criterion from the congruent parts.

We split the benchmark into three skills: the angle facts, the segment facts, and proofs. Angle items start with a third angle, a base angle from the vertex angle, or an exterior angle from its neighbour, move to expressions and ratios such as 2 : 3 : 7, and end with quadrilaterals and pentagons split by diagonals and angle chases across two triangles. Segment items ask whether three lengths form a triangle, the range for a third side, a midsegment forward and in reverse, which segments are medians (from midpoint marks or picture choices), the midpoint a median reaches, and the 2 to 1 split. Triangle inequality items have no figure, so a drawing never shows one candidate length; most other items draw a filled triangle with its tick marks. About one item in five has a setting, such as a sail.

Proof items are Statement and Reason tables with menus, matching, ordering and flawed steps, plus picture choices such as which auxiliary line to add. They prove the 180° sum, the 360° exterior set, isosceles base angles, that an equilateral triangle is equiangular, the midsegment through similarity, and a polygon's angle sum by splitting it into triangles.

Wrong answers come from real slips: subtracting one known angle from 180, ignoring a right-angle box, forgetting to halve after subtracting the vertex angle, summing the exterior set to 180°, using the adjacent interior angle as a remote one, accepting 3, 3, 6 as a triangle, halving a midsegment when it should double, reversing the 2 to 1 split, and citing the fact being proved. The angle templates measure every labelled angle from the drawn figure, and a lint rule fails any figure with a curved edge.

We practice the benchmark as two skills: solving with the properties, and proving them or deciding whether a quadrilateral must be a parallelogram. Solving items draw the figure and never label or mark the measure asked. They start with an angle from one given angle, a side from its opposite side, or a diagonal from its half; move to expressions and to perimeters worked in reverse; and end with a rectangle's diagonals written with x, angles a diagonal cuts, and "Select all the statements that must be true."

Proof items are Statement and Reason tables in the formats above. Classification items ask whether given facts guarantee a parallelogram or a rectangle, described in words (a quilt patch whose diagonals cut each other in half), as picture choices of marked quadrilaterals, or as the value of x that makes a quadrilateral a parallelogram. SSS, SAS, ASA and AAS appear by name; HL and SSA appear only as wrong menu choices.

Wrong answers come from swapping opposite and consecutive angles, stopping at half a diagonal, giving every parallelogram congruent diagonals, giving a rectangle perpendicular diagonals or diagonals that bisect its angles, and in proofs citing the property being proved. On the classification side, students take one parallel pair plus one congruent pair (an isosceles trapezoid), congruent diagonals alone, a kite's two pairs of adjacent sides, or one diagonal bisecting the other as enough.

No item needs area or the Pythagorean Theorem, and the test "one pair of sides both parallel and congruent" never appears, even as a wrong option, since an option stating a true test would have to be keyed. Lint rules fail an item whose subject is a rhombus, square or kite, or that prints a theorem, postulate or property name outside the list above.

Midsegment items start with the midsegment from the two bases, then a missing base worked in reverse, then bases and midsegment written with x, midsegments chained down an A-frame shelf, and areas found as the midsegment times the height, including one part of a trapezoid cut by its midsegment. Isosceles items find base angles and opposite angles, often from expressions, and set the two diagonals equal. Every item draws a trapezoid with exactly one pair of parallel sides. Midsegment figures draw the legs visibly unequal; isosceles figures tick the legs, or the stem says "isosceles". A small share has a setting, such as a patio or a tile.

Property items ask which statements must be true, match each statement to the property behind it (with a spare "Diagonals bisect each other" column), or give a reasoned verdict. Some show a trapezoid not marked isosceles, where the right answer refuses all three isosceles properties. The hard end adds proofs as tables with menus, matching and a first error: why the base angles, opposite angles or diagonals behave as they do (through SAS, HL or a parallelogram drawn inside), and the midsegment formula itself.

Wrong answers come from adding the bases without halving, taking half the longer base, subtracting the base from the midsegment in reverse, setting the midsegment equal to the sum of the bases, calling a supplementary pair congruent, carrying parallelogram facts over (congruent opposite angles, diagonals that bisect each other), applying an isosceles property to an unmarked trapezoid, using the whole height for one part's area, and citing a property inside its own proof.

Lint rules fail an item whose subject is a kite or rhombus, or that prints a theorem, postulate or property name other than the Trapezoid Midsegment Theorem and the course names listed for MA.912.GR.1.4.

Items cover two chords, two secants, a secant and a tangent, congruent tangent segments (including a circle inscribed in a triangle, where the tangent pieces give the perimeter), the right angle at a point of tangency, two relationships chained, and naming which relationship a figure uses. Easier items give every number but one. Middle items ask for the inside piece of a secant (whole minus outside), put an expression such as 5x - 4 on a piece, or ask "Which equation is true?" as released item 32 does. Harder items make the outside piece the unknown, chain relationships as item 36 does and ask for the perimeter, or find a radius from a tangent and a distance. Situations run through every level: paths across a circular park, a fence touching a flower bed, a satellite's line of sight to the horizon.

Circles are drawn to scale with the given pieces on the figure and repeated in bullets, a right-angle box at each point of tangency, secants that stop at the outside point, and a "?" or nothing on the piece asked for. The circle-diagram layout test replays the chord, secant and tangent figures and fails one whose drawn lengths are out of proportion to their labels.

Wrong answers multiply the outside piece by the inside piece, forget to square the tangent, stop at the whole secant when an inside piece is asked, pair the chord pieces wrongly or add them, add or subtract lengths where the Pythagorean Theorem squares them, or keep the negative root. The equation choices include angle-style sums such as (3x + 4) + (5x - 12) = 180, which treat tangent lengths as angles. The FLDOE coverage gate requires a figure in at least a quarter of the items.

In the first skill, easier items read one relationship off the figure: a central angle equals its arc, an inscribed angle is half its arc, a major arc is 360° minus the minor arc. Middle items add arcs around the circle or across a diameter, compare inscribed angles on one arc, solve expressions such as (x + 22)° and (3x - 8)°, or write the relationship of released item 8. Harder items chain arcs into an angle, use an obtuse inscribed angle on a major arc, or work backward. Situations use equally spaced cars on a Ferris wheel and hours on a clock face.

The second skill starts from the right angle at a point of tangency: the angle at P beside a line through the center, the angle between two tangents (180° minus the minor arc), and a perpendicular bisector cutting a chord's arcs in half. Harder items combine two tangents with the line through the center, put an inscribed angle on the bisector, or use expressions on both arcs. Situations include a pond, a satellite's view of Earth and a rope pulled around a post.

Every circle is drawn to scale, and the asked angle or arc never carries its number. A layout test fails any of these circles whose drawn arc or angle is more than a degree off its label.

Wrong answers set an inscribed angle equal to its arc or to twice it, halve a central angle, use the arc that contains the vertex, add arcs to 180° instead of 360°, swap minor and major arcs, forget the 90° at a point of tangency, or report x for the measure. check-alternates proves each listed form of a typed relationship equivalent, and the FLDOE coverage gate requires a figure in at least a quarter of the items.

For the diameter case, easier items give the 90° opposite a diameter, find the other acute angle, or find the radius of a circle around a rectangle. Middle items use the diameter as the hypotenuse to find a side or the radius (to the tenth, or exactly), also for squares and right kites whose diagonal is a diameter. Harder items find a circumference from the two legs and complete a short proof that the angle opposite a diameter is 90°, choosing each reason from a menu. The converse, a right inscribed angle standing on a diameter, appears throughout.

For quadrilaterals, easier items give one angle and ask for the opposite one, sometimes in the form of released item 31 (the relationship word from a menu, then the measure). Harder items solve expressions on opposite angles and find angles from arcs and central angles. Other items sort which quadrilaterals can be inscribed always, sometimes or never, from their names, from four drawn figures, or from their angles listed in order around them.

Figures are circles with the triangle or quadrilateral drawn by its chords, built from the given angles so that each drawn angle matches its label, with a right-angle box only where a right angle is given. The asked angle is never labelled.

Wrong answers make opposite angles congruent or complementary, pair adjacent angles, put the right angle at an end of the diameter, give 180° or 45° for the angle opposite a diameter, mix up the radius and the diameter, or report x instead of the angle. Error-analysis items test a named student's claim, such as "m∠C = 74° because opposite angles are congruent". The FLDOE coverage gate requires a figure in at least a quarter of the items.

Easier items use friendly angles (90°, 60°, 120°, 180°) with the radius given and ask for an arc length or a sector area in terms of π. Middle items give a diameter or an angle such as 72° or 150°, ask for a rounded answer, tell an arc's measure apart from its length, and work back from a length or an area to the central angle. Harder items work back to the radius (a sector area gives r², then r), find a sector's perimeter, the area a wiper blade cleans between two radii, the region between a chord and its arc, or the cost of fencing around a bed.

Proportional-reasoning items change one quantity and ask what happens to the arc or the sector, often as a matching item, and a pair of concentric circles cut by one central angle asks for the outer arc from the inner one. Mathematical items draw the circle with its radii, the central angle and the arc, or shade the sector; the asked value is never labelled. Situations (pizza slices, sprinklers, cakes, clock hands, wipers, a curve in a road) are mostly in words.

Wrong answers use the diameter as the radius, 2πr for an area or πr² for a length, θ/180 instead of θ/360, the whole circle, the arc's degree measure as its length, or the circle's area in place of the sector's. The circle lint fails any item, explanations included, that uses radians or s = rθ. check-grading runs every exact and rounded key through the real grader, and on an exact key a decimal and a doubled π value must both fail. The FLDOE coverage gate expects a figure in at least a tenth of the items.

Every item is text only, as on the EOC. Easier items name one form of a short statement, rewrite an "all" statement as an if-then, name the hypothesis or the conclusion, or fill a three-row table. Middle items add a row that is none of the three, decide which forms are true once the original is known to be true, and build or split a biconditional. The hardest refute a biconditional with a counterexample, work with statements that already contain "not", negate an "all" claim, select every statement with the same meaning, and read "only if".

Truth is always settled by geometry: a theorem, or a named figure that breaks the claim (a 3 by 12 rectangle and a 6 by 6 square have equal areas but are not congruent). The theorems include the true converses a Geometry course proves, such as lines being parallel when alternate interior angles are congruent, and a triangle with two congruent angles having two congruent sides. True or false judgments are reasoned choices or matching boards over written statements, never tables of T and F values.

Wrong answers come from swapping the converse and the inverse, giving the swap without the negations (or the negations without the swap) as the contrapositive, negating only one part, assuming the converse of a true statement is true, assuming an inverse is always false, reading "not all" as "none", reading "all A are B" as "all B are A", and reading "only if" backward. A verdict always comes with its reason.

Lint rules fail any item that mentions or shows a truth table (explanations included), and any statement with no GR.1 geometry in it or with content from outside GR.1, such as chords, slopes or transformations.

Items come in two families. Counterexample items give a claim and have the student choose from candidates described in words or measures, choose from four or five drawn figures, name an angle pair on lines cut by a transversal, judge one candidate with its reason, or type the one measure that works (90° angles break "any two supplementary angles include an obtuse angle"). Matching boards sort candidates into "Counterexample", "fails the hypothesis" and "meets the conclusion".

Validity items set out premises and a conclusion and ask for a verdict with its reason, which conclusions follow from a chain of statements, or which of several arguments are valid. The valid forms are the ones proofs use: a conditional with its hypothesis, a chain of conditionals, and the contrapositive. The errors are assuming the converse or the inverse, running a chain backward, taking examples as proof, and trusting a conclusion because its premises are true. Difficulty climbs from one claim and one candidate to select-alls that need two kinds of counterexample and arguments with three premises.

Each wrong counterexample fails in one way: it meets the conclusion, it fails the hypothesis, or it is not a figure at all (a triangle with sides 3, 3 and 7). Every converse or inverse error uses a statement whose converse is false, so a student who knows a true converse theorem is never marked wrong for using it, and no distractor reason is one that is also sound.

A lint rule fails any statement with no GR.1 geometry in it or with content from outside GR.1, such as chords, slopes or transformations.

We practice congruent figures and similar figures as two skills. Congruence items name the part that corresponds in a statement such as ABCDE ≅ PQRST, carry a side or angle across, solve an equation such as 4x - 21 = 2x + 49 for corresponding angles, find the partner's perimeter or area, find a missing angle through the angle sum and the correspondence, decide congruence from given measures, and choose the congruence statement that fits marked figures. Triangles appear only here, as congruent pairs.

Similarity items find a side of a similar rectangle described in words (a photo 5 inches wide and 7 tall, enlarged to 12.5 inches wide), a side of similar trapezoids, pentagons or L-shaped floor plans drawn with their sides marked, a perimeter, a side from two perimeters, and an area by finding both dimensions first, never by squaring a ratio. Others decide whether two figures are similar or choose the proportion that finds a side. Most similarity items have a real-world setting; most congruence items are abstract.

Wrong answers come from shifting the vertex order by one letter, matching parts by their place on the page, reporting x for the length, taking equal areas or perimeters as proof of congruence, adding the difference instead of scaling, pairing the new side with the wrong old side, running the proportion backward, scaling a perimeter like an area or an area like a length, converting one figure's unit, and taking proportional sides as enough without the angles.

The FLDOE coverage gate requires a figure on at least a quarter of the items, so drawn pairs and picture choices are a steady minority. Lint rules fail an item that states or asks for a scale factor, calls two triangles similar, or calls open figures congruent or similar.

The benchmark is practiced as two skills: describing translations and reflections, and describing rotations and dilations. Every question gives a figure and its image, as vertex lists or on a coordinate grid. The student picks the rule, types one number of it, builds it from menus, matches figure pairs to moves, or selects the true statements. Difficulty climbs from one translation, axis reflection or origin turn through reflections across x = a, y = a, y = x and y = −x and turns or dilations about a vertex, to two-move sequences written as one rule, circles dilated and translated, and reverse questions that recover the preimage.

About three questions in ten draw both figures on a coordinate plane. On the interactive grid the student draws the mirror line with the line tool or plots a center with the point tool.

Wrong answers are real slips: subtracting the image from the preimage, swapping the two shifts, negating the wrong coordinate, mixing up y = x and y = −x, writing (a − x, y) where (2a − x, y) belongs, turning the wrong way, using the origin rule when the center is a vertex, inverting a scale factor, taking for the center a vertex that an image vertex lands on, and doing a sequence's moves in the other order.

Each template recomputes every key from the moves and stops the build on a mismatch. Lint rules fail a turn other than 90°, 180° or 270°, a 90° or 270° turn without a direction, a center off the figure, a reflection line outside FDOE's four forms, a stretch even as a wrong answer, more than four moves, a stem that does not name both figures, and function or vector notation, explanations included. check-grading runs every key through the real grader.

Every question is text, as on the released items: nothing is drawn. The easiest name one move in words and ask whether it keeps distance. From the middle level up the student reads a coordinate rule first: (x, y) → (x, −y − 10) is a reflection across y = −5, (x, y) → (3x + 3, 3y + 3) is a dilation followed by a translation, and (x, y) → (2y, −2x) is a dilation with a quarter turn. Other questions sort moves on a congruent or noncongruent board, judge two moves in one sentence (each, neither, only the first, only the second), pick the rules that keep angle measure but not distance, find a length or angle after a stated move, and work backward to the rule that made every side half as long.

Wrong answers come from beliefs students hold: a reflection flips the figure, so it cannot be congruent; a dilation keeps the shape, so it keeps the size; the center of a dilation stays put, so lengths survive; a 41° turn must change lengths; a constant after a multiplier makes a plain translation; a negative sign shrinks the figure; angles grow with the scale factor.

We avoid stretches and shears even as wrong answers (two early anchor questions offered (x + 1, 2y) and (2x, 5y); both options were replaced), a sequence whose scale factors multiply back to 1, and a mirror line at a half unit, which is explained as an axis reflection plus a translation instead. Lint rules fail any rule that is not a translation, reflection, quarter or half turn, dilation or a sequence of them, any reflection line outside y = a, x = a, y = x and y = −x, and any function or vector notation, explanations included.

In the symmetry skill a drawn kite, rectangle, rhombus, parallelogram, isosceles trapezoid or triangle carries its candidate lines and center, with tick marks for the given facts. The student picks or counts the lines of symmetry, names the turns about the center that work, or chooses from four picture cards the figure with exactly two lines of symmetry. Harder questions move to the coordinate plane: plot the vertex that completes a symmetric figure, find a vertex set that meets two conditions at once, or find the second step that returns a figure to its place.

In the mapping skill the student chooses or completes a sequence, matches each vertex to its image and writes the congruence or similarity statement in corresponding order, justifies congruence or similarity from the moves, finds a scale factor, and moves circles rigidly.

Wrong answers: a rectangle's diagonals as lines of symmetry, a rectangle with a 90° turn, a turn about a vertex instead of the center, an isosceles trapezoid with 180° symmetry; and the moves in the other order, the wrong axis or direction, y = x for y = −x, "a reflection reverses orientation, so not congruent", a rigid sequence called similar, equal perimeters taken as congruence, an inverted scale factor, and image vertices written in the order given.

The FLDOE coverage gate requires a drawn figure on a quarter of the questions, and a coordinate plane does not count toward it. Lint rules fail any turn other than 90°, 180° or 270°, a 90° or 270° turn without a direction, a figure of five or more vertices, a dilated circle, a stretch, more than four moves, function notation, or a reflection line outside FDOE's four forms. Each template re-derives every key from the coordinates and stops the build on a mismatch.

This benchmark has no practice and quiz questions of its own, because FDOE assesses it only together with MA.912.GR.2.3. Its questions sit in that benchmark's skill on mapping one figure onto another and carry that benchmark's code, so a student's work on them is reported under MA.912.GR.2.3.

About two dozen of those questions are about congruence. A sequence is stated in words, or two figures are given by their vertices, in the text or plotted on a coordinate grid. The student picks or completes the conclusion and its reason, matches each sequence or coordinate rule to "Congruent" or "Similar, not congruent", selects the measures of the image that must equal given measures of the figure, types a side length of the image, or chooses the rigid sequence that shows two given triangles are congruent. Eight more questions move a circle onto another circle of the same radius, where only the center has to be tracked through the moves.

Wrong answers come from real mistakes: "a reflection reverses the order of the vertices, so the triangles are not congruent", a rigid sequence called similar but not congruent, equal perimeters taken as proof of congruence, a side matched with one that does not correspond, and a scaled rule such as (x, y) → (2x, 2y − 1) read as a rigid move.

Each template re-derives every key from the coordinates and stops the build on a mismatch, and check-grading runs every key through the real grader.

This benchmark has no practice and quiz questions of its own, because FDOE assesses it only together with MA.912.GR.2.3. Its questions sit in that benchmark's skill on mapping one figure onto another and carry that code, so a student's work on them is reported under MA.912.GR.2.3.

About twenty of those questions are about similarity. Two triangles or rectangles are given by their vertices, in the text or plotted on a coordinate grid, or a sequence is stated in words. The student picks the dilation and rigid move that map one figure onto the other, finds the scale factor from a pair of corresponding sides, types the scale factor or the shift in a pair of coordinate rules, selects every statement that must be true (MN = 2JK, m∠M = m∠J), or puts the steps of a classmate's similarity argument in order. Several questions turn on the order of the moves: a translation done before a dilation centered at the origin gets scaled along with the figure.

Wrong answers come from an inverted scale factor (1/2 where the image is twice as large), a translation placed before the dilation without scaling it, additive scaling ("each side is 3 units longer"), angles scaled with the sides ("m∠N = 2m∠K"), and a congruence statement written after a dilation.

The GR.2.3 lint rules run on these questions as well: no dilated circle, no stretch or other move outside FDOE's four, no more than four moves, and no function notation. Each template re-derives every key from the coordinates, and check-grading runs every key through the real grader.

About four questions in ten are drawings. The rest ask where one vertex or a circle's center lands, choose the one rule for the whole sequence, pick the final image from four graph cards drawn on the same window, compare two students who do the same moves in different orders, or work backward from the final image to the preimage or to a move's missing number.

The easiest questions reflect a triangle across an axis and translate it, with its coordinates also listed. The middle level adds turns about the origin, reflections across x = a, y = a, y = x and y = −x, quadrilaterals, circles, and vertices read off the grid. The hardest use three or four moves, dilations and turns about a vertex, pentagons, and reverse problems. From the middle level up, most sequences land somewhere else when their order is reversed, and that other-order image is offered as a wrong answer.

Other wrong answers: skipping a move, plotting the in-between image, reflecting across the other axis, turning the wrong way, reading (x, y) → (y, −x) as the counterclockwise turn, centering a vertex dilation at the origin, translating before dilating, and keeping a circle's radius under a dilation.

Lint rules fail a stem that states one move or more than four, any turn other than 90°, 180° or 270°, a 90° or 270° turn without a direction, a center that is not the origin or on the figure, a reflection line outside FDOE's four forms, a stretch, and function notation. Grid windows use square cells and put every keyed vertex on a grid point; a keyed or given circle has its center on a grid point, a whole-number radius and room to spare inside the window. check-grading runs every key through the real grader.

The ladder starts with the multipliers themselves: a scale factor of 3 makes an area 9 times as large and a volume 27 times. Next the student applies them to a stated measure (a prism of 96 cubic inches dilated by 1/2) or to a drawn square, rectangle, triangle, trapezoid or circle and the area of its image. The middle level runs the work backward, from two areas or two volumes to the scale factor by a square root or a cube root, and on to a side, a surface area or a volume. The top level adds percent change (edges 20% longer make a cube's volume 72.8% larger), two dilations in a row or one undone, and real situations: an enlarged photo, two pizzas compared, paint for a larger box, the mass of a clay sculpture from its model.

Wrong answers name the classic slips: an area multiplied by k instead of k², a volume by k² or a surface area by k³, a scale factor below 1 read as growth, an area ratio taken as the scale factor with no root, a root replaced by halving, the ratio flipped, a volume ratio applied to surface area, the same percent applied to every measure, only one of two dilations used, and pizzas compared by their diameters.

Typed answers are whole numbers or terminating decimals, and π appears only in answer choices. No solid is drawn, and nothing sits on a coordinate plane, since the center of the dilation never matters here. A lint rule fails any item that asks for a dilated solid's volume or surface area without stating the original's, and the FLDOE coverage gate expects a drawn figure in at least a tenth of the items.

Easier items name the construction from its marks or narrated steps, or say what the compass opening holds ("the length JK"). Middle items pick or complete the next step, select every length equal to BX, or match each arc to the width it uses and each step to its tool. Harder items put four or five steps in order, explain why the copy is congruent (SSS, then corresponding parts, sometimes in a short proof table), find a classmate's mistake and its effect, or build from two copies: a segment of length AB + CD or AB - CD, or an angle twice as large.

Figures are drawn to scale: arcs swung from their true centers, rays with arrowheads, primes on the copy's points and a dot at every crossing a step names. A figure shows exactly the steps narrated so far, so a next-step item never shows the next arc, and no figure carries a measure. The angle being copied is never 60°, where the chord would equal the first compass opening and the two widths could not be told apart. A lint rule fails a set of picture choices whose key card is the only largest or smallest, or the only one with labels.

Wrong answers are compass steps done wrong: opening the compass wider between the two arcs that must match, swinging the third arc from P instead of from X', setting the width to PX' instead of XY, drawing the ray through X' instead of Y'. The wrong reasons for why the copy works are SAS, similarity by AA and "the compass measures the angle in degrees". The construction lint also fails any item that neither describes nor draws a construction, and the FLDOE coverage gate requires a figure in at least a quarter of the items.

Telling the two arc patterns apart is the skill, so items work both bisectors side by side. Easier items name the construction from its marks or narrated steps, sort steps between the two constructions, or give the last step. Middle items pick or complete the next step, read the finished marks (M is the midpoint, PA = PB for any point P on the line, which arcs must share a radius) and say why the compass must open wider than half the segment (more than 4 cm for an 8 cm segment). Harder items order four or five steps, give the reason the construction works, sometimes inside a proof table, find a classmate's error and its effect, and chain bisectors: a 45° angle, a quarter of an angle, the point one fourth of the way along a segment.

Figures show arcs of one width crossing above and below a segment, or one arc from a vertex and two arcs crossing inside the angle, turned and slanted as well as level. They carry no right-angle box, tick marks or angle arcs, so the arc pattern alone decides. Some items follow FDOE's released item: four small construction figures as the choices, and the student picks the one whose steps lead to a perpendicular bisector.

Wrong answers come from real slips: a compass width at or under half the segment (the arcs never meet), unequal widths (the line misses the midpoint), arcs on one side only (one crossing, so no line yet), the final ray drawn from a side point instead of the vertex, and segment PQ between the side points taken as the bisector. The lint rules for copying apply here too: no ruler, protractor, folding or software, and every item describes or draws a construction.

Easier items ask which lines locate each center, with FDOE's four line types as the choices (altitudes, angle bisectors, medians, perpendicular bisectors), or which circle a construction in progress is heading for. Middle items ask for the step after the center is found (open the compass from O to a vertex; from the incenter, first construct the perpendicular to a side), which segment is a radius, where each center lies in an acute, right or obtuse triangle, and why the circle through A also passes through B and C. Harder items order the steps of the inscribed-circle construction, find a right triangle's circumradius as half its hypotenuse, and diagnose a classmate who used medians, opened the compass to a midpoint, or swung unequal arcs.

Figures show the triangle with two constructed bisectors drawn dashed with their arcs (right-angle boxes on the perpendicular bisectors), the center dotted, and the circle once it is finished. Obtuse triangles leave room for a circumcenter outside them. Some items show four constructions on one triangle as the choices, with no letter at a crossing or a center that would give the answer away. FDOE's sample also has the student drag the center into place; we ask for the center by name or by picture instead, because our graphing tool cannot draw construction arcs.

Wrong answers swap angle bisectors and perpendicular bisectors between the two circles, use the medians and the centroid, take the inscribed radius to a vertex or to a side's midpoint, put the circumcenter inside every triangle, or insist that all three bisectors are needed. The tool and construction lint rules of the other two benchmarks apply, and the FLDOE coverage gate requires a figure in at least a quarter of the items.

Questions classify a triangle by its sides or angles, decide whether an angle is right, give a quadrilateral its most precise name or every name that fits, decide whether a point is inside, on or outside a circle given by its center and radius, and plot a missing vertex with the point tool (the fourth corner of a square or rectangle). Justifications are dropdowns that pick two slopes or two lengths and then the conclusion, or ask which true computation proves a property, or whose reasoning is right. Real-world questions put a delivery radius or a cell tower on a town map.

Easier questions need one formula: two side lengths or two slopes. Harder ones chain them: a parallelogram first and then a right angle for a rectangle, diagonals that bisect each other and are perpendicular for a rhombus, or a square built on a given side, with a stated quadrant choosing between the two possible squares.

Wrong answers: run plus rise used as a length, reciprocal slopes without the sign change (or any opposite signs) taken as perpendicular, an obtuse triangle called acute because the squares were not compared with the longest side, a square called a rectangle when the most precise name is asked, congruent diagonals taken as a rectangle and perpendicular diagonals as a rhombus without the other conditions, d² compared with r, and the wrong fourth corner (B + C - A in place of A + C - B).

Every trapezoid has exactly one pair of parallel sides, and no question asks whether a parallelogram is a trapezoid. FDOE sets no limits here, so the lint rule is the benchmark text: it fails any question about a pentagon, hexagon or larger polygon. check-grading runs every key and accepted alternate through the real grader.

The averaging skill asks for a midpoint and the length from an endpoint to it, the other endpoint from a midpoint, a partition point from a ratio or a fraction of the way, the reverse (the ratio, or the far endpoint), the centroid, a median's endpoint and length, and weighted averages on a number line. The slope skill asks which sides of a quadrilateral are parallel, the slope a side needs to be perpendicular, a missing coordinate that makes sides parallel, the fourth vertex of a parallelogram or rectangle, a median's slope or equation, a tangent line's slope, equation or point of tangency, and a point on a circle on a stated side of its center.

Most coordinate questions are text, as on the released EOC items. The point tool appears where the answer is a lattice point, and a few easy questions draw a plane. A typed answer is one coordinate; a fractional one is asked as a fraction and graded in that form.

Wrong answers: adding the coordinates without halving, reading 2 : 1 as 2/1, measuring from the wrong endpoint, a side's midpoint or the middle of a median taken as the centroid, dividing the vertex sums by 2; and in the slope skill, the reciprocal without the sign change, the radius slope used as the tangent slope, the tangent drawn through the center, run over rise, the midpoint of the wrong side, and D = B + C - A for a parallelogram's fourth vertex.

Lint rules fail any question that names or describes the orthocenter (explanations included), mentions an altitude, circumcenter or incenter, or breaks the weighted-average limits. Grid answers sit on lattice points, and check-grading runs every key through the real grader.

There is no separate bank. The weighted-average questions sit in the MA.912.GR.3.3 pool, inside its skill on locating points by averaging, where they make up about a quarter of the questions. Each has exactly two points on a number line: drawn on about half of them, an interactive number line where the student plots the average on a few, and named in words ("Points A and B are at 6 and 26 on a number line") on the rest.

Easier questions give the two points and weights such as 1/4 and 3/4 or 70% and 30% and ask for the average, chosen, typed or plotted. The middle level asks which point the average is closer to and where it lands, and how it moves when a weight changes, which is the clarification's own instruction. The hardest give the two points and the average and ask for the weight.

Wrong answers come from swapping the weights so the average sits nearer the lighter point, taking the midpoint whatever the weights, dropping a negative sign, giving the other point's weight, and moving the average toward the point whose weight fell.

No question is filed under this benchmark's own code, so the FLDOE coverage gate cannot check it separately, and three lint rules carry FDOE's limit instead: a question that uses the word weight must show or name a number line, average exactly two points, and state its weights as fractions or percentages that total 1, never a decimal, a whole number or a ratio. A typed weight is graded only in the form the stem names, and check-grading runs every key through the real grader.

Questions ask for a polygon's perimeter from its vertices; an area found from slanted sides (a tilted square or rectangle, or a right triangle with slanted legs whose right angle is stated and marked); the perimeter and area of one figure together, or what a computation measures; real-world boundaries (fencing a dog run at a price per foot, edging a garden, a park drawn to a scale, a field walked off as "40 feet west of the third corner"); population density, people from area times density or the reverse; and a vertex or coordinate found from a given area or perimeter. The easiest have one slanted side; harder ones add pentagons and hexagons, scales and densities.

Wrong answers: run plus rise as a slanted side's length, a slanted side counted as its rise or run, radicands added under one root (√29 + √29 written √58), the area reported as the perimeter, the box around a figure taken as its area, the half forgotten for a right triangle, a square's diagonal squared as its area, a map scale applied once to an area, the area divided by the density, and a root outside the stated condition kept.

Two lint rules fail any polygon whose every side is horizontal or vertical and any area-only question on a figure other than a tilted rectangle or square or a right triangle with slanted legs. The walked-off questions, which the lint cannot read, are built with a slanted closing side. The template checks that every rounded answer comes out the same when each side is rounded first and sits away from a rounding boundary. Grid answers are lattice points made unique by a stated quadrant. check-grading runs every key and alternate through the real grader, and check-alternates proves each alternate equals its key.

The easiest questions draw one right solid with its cutting plane and ask for the cross section's most specific name: a cylinder, prism, pyramid or cone cut parallel or perpendicular to its base. The middle level adds pyramids cut away from the apex and a triangular prism lying on a rectangular face. The top level brings composites (a prism with a pyramid roof, a grain bin), labeled solids whose dimensions decide between a rectangle and a square or give the size of a scaled section, and typed questions asking for the cross section's area. Picture choices run through every level: the drawn shape that matches a described cut, or every drawn solid whose slice is a circle, a triangle or a trapezoid.

Wrong answers come from real slips: an ellipse for a circle (the drawn section looks oval), a triangle for a pyramid's parallel cut, a section kept the size of the base inside a pyramid or cone, the face a lying prism rests on taken as its base, a triangle for a pyramid cut that misses the apex, a prism expected to taper, a rectangle when the given sides make a square, and only one part of a composite named.

We never offer Square for an unlabeled prism or cylinder cut perpendicular to its base, since the drawing cannot guarantee it, and a stem that offers Trapezoid or Ellipse beside a rectangle or circle always asks for the most specific name, so the key holds under either definition of a trapezoid. The FLDOE coverage gate requires a drawn figure in at least a quarter of the items, and lint rules fail any item with a solid outside FDOE's list, a slanted cut, or a prism or pyramid base that is not a rectangle or a regular polygon.

Every question has a drawing: usually the figure beside its axis, or, when working backward, the solid or candidate figures. The easiest name what a rectangle, a right triangle or a semicircle makes when turned about a side, or ask which part becomes which: the segment that becomes the radius, the leg that becomes the height, the side that sweeps the flat circular face. The middle level reads the solid's radius and height off the figure, turns a quarter circle into a hemisphere and a right triangle about its hypotenuse into two cones, and turns composite outlines: a model rocket, an ice cream cone, a pill.

Holes wait for the top level, where the figure sits away from the axis (a pipe, a spool, a flower pot with a drainage hole). Some items at every level run backward: which figure and line make a given solid, chosen from descriptions or drawn cards, or, from the middle level up, drawn on the grid with the polygon tool. Typed answers are lengths, such as an outer diameter or the radius two cones share.

Wrong answers come from the slips students make: a rectangular prism for a cylinder, a hemisphere for a sphere, the side on the axis or the hypotenuse taken as the radius, the radius and the height swapped, the diameter used as the radius, one cone instead of two, a rocket's total height used for its cone, and, for a figure set away from the axis, a solid with no hole or the figure's width taken as the outer radius.

The FLDOE coverage gate requires a drawn figure in at least a quarter of the items, and check-grading runs every key, grid answers included, through the real grader and confirms that a blank grid never scores.

Single triangles and quadrilaterals test the height that meets the base at a right angle: a parallelogram's slanted side, an obtuse triangle's outside height, a kite's diagonals, an isosceles trapezoid's legs. Circles appear whole, halved, quartered or given by a circumference. Shaded and composite regions include an arched window, a picture frame, a circle inside a square and a ring, and covering problems price carpet by the square yard or count tiles and pallets of sod. Easier items take one figure and one step; harder ones add a Pythagorean step, a quadratic, a composite region or a unit change.

Figures are drawn to scale from their givens: a shaded region is filled with its holes cut out, and a regular polygon shows its center with the radius and one other measure labeled. Exact answers (49π, 96√3, 100 - 25π) are chosen from a list or typed with the π and √ keys when the stem asks for the exact area. A rounded answer names its place, and the key accepts the result of the calculator's π, of 3.14 and of 22/7.

Wrong answers come from named slips: the slanted side used as the height, the 1/2 dropped, the diagonals left whole, the diameter used as the radius, the circumference instead of the area, the radius used as the apothem, a ring computed as (R - r)²π, the negative root kept, x reported when a side was asked, area divided by population, and square feet divided by 3 instead of 9.

Sectors, the coordinate plane and dilations stay with their own benchmarks. The FLDOE coverage gate expects a drawn figure in at least a tenth of the items, and check-grading confirms that an exact key scores however it is typed while its rounded decimal does not.

The prisms and cylinders skill starts with a labeled right prism or cylinder, then gives a diameter instead of a radius, a triangular or hexagonal base, an oblique cylinder or box, or a volume to work back to a missing height or radius. Density runs through every level: a block's mass, a bar's density, a cylinder's metal picked from a table of densities, a stone's density from the water it raises. The top items add hollow pipes, unit changes (27 cubic feet in a cubic yard) and Cavalieri's Principle in words: a straight stack of coins against a leaning one, or which solid has the same volume.

The pyramids, cones and spheres skill moves from a labeled cone or square pyramid to slant-height labels that hide the height, spheres and hemispheres (the hemisphere is not on the reference sheet), a radius found by a cube root, water rising when a ball is dropped into a glass, and composites: a silo, an ice cream cone, a capsule, a cylinder with a cone removed.

Wrong answers come from real slips: the base perimeter used as the base area, a triangular base's 1/2 dropped, the diameter used as the radius, the slanted side used as the height, a lean that seems to add volume, the one-third dropped or applied to a prism, a whole sphere for a hemisphere, a silo's total height used for its cylinder, density inverted, and grams never converted to kilograms.

No item prints a formula or says "Use 3.14". Solids are drawn to scale from their labels. Where 22/7 would stretch a rounded answer's accepted range past one unit, we changed the numbers, since check-grading proves that an answer one unit off never scores. A lint rule fails any item that names a solid outside the five.

Easier questions give a labeled prism, cylinder or sphere and ask for its total or lateral area. The middle level labels a pyramid or cone with its slant height, adds hemispheres and regular pyramids given by side and apothem, and sets the solid in a context that leaves faces out: a lidless aquarium, an open can, a paper label, a party hat, a roof with no floor. The top level labels only the height, so the slant height needs the Pythagorean Theorem; builds composites (a silo, a model rocket, a capsule, a birdhouse, a pipe coated inside and out); and works backward from a surface area to an edge, a height, a slant height or a diameter. Matching boards pair solids with their areas, or a job with the surface it covers.

Wrong answers come from miscounted faces and mixed-up lengths: a pair of faces missed, the volume given instead, the lateral area alone, one base on a closed cylinder or two on an open can, πrh without the 2, the diameter used as the radius, the height used as the slant height, the 1/2 left off the triangles, a floor the roof does not have, πr² or 2πr² for a sphere, a hidden face where two solids meet, and a dome counted as a whole sphere.

Every solid is a right solid: a lint rule fails the word oblique, a slanted or leaning solid, or an oblique drawing, and another fails any solid outside the five. Solids are drawn to scale from their labels, the asked measure is never labeled, radicals are simplified (45 + 9√3), and an apothem that is not a whole number is given to the tenth. The FLDOE coverage gate expects a drawn figure in at least a tenth of the items.

The first skill writes the equation from features: the center with a radius or diameter, a graph whose center and one labeled point sit on gridlines, the center and one point, the endpoints of a diameter, and, hardest, a condition that fixes the radius (a circle tangent to an axis, a concentric circle with a radius 3 units longer, a sprinkler moved with the same reach). Error-analysis questions show a classmate's equation and ask what went wrong. Graphs are static planes with square cells; the student never draws, since the product is an equation.

The second skill starts from an equation such as x² + y² - 6x + 4y - 12 = 0, text only. Students complete the square to reach standard form, read the center and radius, find the number added to complete a square, find the first wrong step in a worked solution, compare circles, and work back to a missing constant from a given radius. The layout varies: = 0, the constant on the right, y-terms first, one variable already complete.

Wrong answers: the center's signs kept as printed, the radius not squared, the diameter used as the radius, run plus rise as the radius, the labeled point read as the center, an endpoint taken as the center of a diameter; and in completing the square, the half added instead of its square, the numbers added to one side only, the constant moved with the wrong sign, and r² reported as the radius.

Lint rules fail a question that gives no coordinate, graph or equation, and any mention of a focus, directrix, eccentricity, domain or range. The template checks that every general-form equation is a real circle. check-grading runs each typed equation's accepted forms, squared terms in either order, through the real grader.

Easier questions read the center, radius, domain and range from standard form, graph the circle, or pick its graph from four cards. The middle level adds whether a point is inside, on or outside. The hardest find where the circle crosses an axis or a line, put the circle in a setting (which houses a cell tower covers, how far west of the harbor a lighthouse beam reaches, how many miles of highway lie inside a radar boundary, the area where music from a stage can be heard), or start from general form, so completing the square comes first.

Static planes draw the circle and its center, often a named tower or station, but never the points a question asks about, and a line the question names runs across the whole frame.

Wrong answers: the binomials' signs kept for the center, r² read as the radius, r² halved instead of square-rooted, h and k swapped, the domain built from k, a strict inequality, the distance compared with r², a point on the circle counted as inside, the other variable's term dropped when finding a crossing, only one root kept, half a chord reported, and the circumference found for the area.

Lint rules fail any question whose givens hold no circle equation, any mention of eccentricity, radians, a focus, a directrix or an axis of symmetry, and interval notation anywhere, explanations included. Grid rules check every circle-tool question: equal steps, the center on a grid point, a whole-number radius and a step to spare on every side. A rounded answer, such as an area to the nearest tenth, also accepts the exact form with π, and check-grading runs every key through the real grader.

In the ratio skill, easier items choose the equation that finds x; middle items find a side with one ratio or an acute angle with an inverse ratio; harder items put the unknown in the denominator or need the third side first. The applied skill climbs from one special right triangle worked forward and one angle of elevation, through special triangles worked backward, angles of depression, ladders and ramps, to special triangles inside squares, rhombi and equilateral triangles with exact perimeters and areas, a ladder that slides (a ratio, then the Pythagorean Theorem), and two triangles at once: two sightings of one tower, a window looking up and down at a building, two boats seen from a lighthouse.

Every triangle is drawn to scale from its givens, with a right-angle box, the given angle marked and labelled, and the right angle turned to different corners. Situations are plain triangles: level ground, a vertical tower, and an angle of depression marked at the observer between a dashed horizontal and the line of sight. One item shows four figures and asks which marks the angle of depression.

Every answer box has π and √ keys. A rounded key's band also covers ratios rounded to four places and lengths rounded along the way, so it widens in two-triangle chains. Explanations show each calculator value to four places, in degree mode, and round last.

Wrong answers use cosine for sine, tangent where the hypotenuse is given, divide instead of multiply, give the other acute angle, take sin⁻¹ where tan⁻¹ belongs, swap the legs of a 30°-60°-90° triangle, multiply by √2 where dividing is needed, or measure a depression from the vertical. check-grading runs every key and alternate through the real grader, with an off-by-one probe on each typed number.

The FLDOE coverage gate records that no items are filed under this benchmark and that its content rides on the MA.912.T.1.2 bank, where it makes up much of the ratio skill. Easier items name sin, cos or tan of either acute angle from a triangle whose sides are a Pythagorean triple (20-21-29, 12-35-37, 33-56-65 and others), match each ratio to its value as released item 37 does, select every ratio equal to 63/65, or type a ratio as a fraction in lowest terms. The released-form items give two bullets in words (m∠H = 90°, cos G = 7/25) and ask for sin I.

Harder items build on the definitions: sin A = cos(90° - A) with numbers and with expressions such as (5x - 2)° and (2x + 15)°, tan A from sin A = 2/7 through the missing side (2√5/15), and two right triangles with a congruent acute angle, where the student finds a side from the shared ratio or judges a claim that tripling the sides triples sin D. Statements to check include sin A + cos A = 1 and "sin A can be greater than 1".

Wrong answers name the opposite and adjacent sides from the wrong angle, use the hypotenuse as a leg, flip the ratio, read sin J as sin K, or call the two acute angles supplementary. The T.1.2 lint rules hold here too: no ratio of an angle that is not acute, no secant, cosecant or cotangent, and no unit circle.