Practice Test

Why it is hard

This is one formula and one square, so the question comes down to recall and to the form of the answer. The area and circumference formulas both pair π with the radius, and 2πr gives a tidy value with π in it that looks just as right. The other losses are about form: replacing π with 3.14, which the instructions rule out twice, or dropping π after squaring 5.

1. Something you must already know moderate Working it out

   The formula A = πr² is not printed with the question, and the circumference formula 2πr uses the same r and π, so recalling the wrong one gives a value that looks just as finished. (Anderson & Krathwohl, 2001)

2. An answer in an unexpected form minor Choosing the answer

   The answer is built on a keypad with a π key and must keep π as a symbol; a decimal worked out with 3.14 is not accepted, and no choices show the expected form. (Enright, Morley & Sheehan, 2002)

Strategy: Recall the area formula rather than the circumference formula, square the radius before multiplying, and leave π as a symbol instead of replacing it with 3.14.

Traps

• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

• 10π Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: strong

  Uses the circumference formula 2πr, or doubles the radius instead of squaring it, so 2 × 5 gives 10π, the distance around the circle.

• 78.5 Misreads the question (Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: moderate

  Finds the area correctly but replaces π with 3.14, 25 × 3.14 = 78.5, though the question asks for the area in terms of π.

• 25 Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: weak

  Squares the radius, 5 × 5 = 25, and stops before multiplying by π, so the entry leaves out the π the instructions ask for.

How the questions are analyzed, and the research behind it

Why it is hard

Once the angle fact is in place, 5x = 180 is a one-line equation, so the question tests recall more than computation. Students who use 90° for the pair land on half the right value, and the two smallest choices come only from combining the labels' 3 and 2. A sense check helps: x = 36 makes the angles 108° and 72°, one obtuse and one acute, just as the drawing shows.

1. Something you must already know moderate Working it out

   The figure gives no number of degrees, only 3x° and 2x°, so everything rests on recalling that the two angles at the ends of one side of a parallelogram add to 180°. (Anderson & Krathwohl, 2001)

2. Reading a figure, table or graph minor Understanding the question

   The marked angles sit at the upper and lower ends of the left side, consecutive rather than opposite, which is what makes them supplementary instead of equal. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

Strategy: Decide what two angles that share a side of a parallelogram add up to, set 3x + 2x equal to that total, solve for x, and check that the two angles fit the figure.

Traps

• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• Stops one step early: A choice is a number from partway through the solution.
• Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.

The wrong answers, classified

• B Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: strong

  Uses 90° as the total of the two angles, the complementary total, so 5x = 90 gives 18, though consecutive angles of a parallelogram add to 180°.

• C Numbers taken from the question (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: weak

  Multiplies the two coefficients from the labels, 3 × 2 = 6, without using any angle fact or writing an equation.

• D Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: weak

  Combines 3x + 2x into 5x and gives the coefficient 5, stopping before dividing the 180° total by it.

How the questions are analyzed, and the research behind it

Why it is hard

Each fact needs only a subtraction, but the student has to plan a route from the 73° angle at line p over to r at line n. The first number the route gives, the triangle's 17° angle, is listed, and it is the acute partner of r rather than r itself. The figure helps here: r is drawn plainly obtuse, and 163 is also 90 + 73, the triangle's exterior angle, which confirms it.

1. Reading a figure, table or graph moderate Understanding the question

   The 73° sits between line p and the vertical segment, not between p and m, and r is the obtuse angle above m on the left of line n, far from every marked angle. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Combines ideas moderate Working it out

   Three facts chain together: the triangle's angles sum to 180°, the angles along straight line m sum to 180°, and corresponding angles at the parallel lines n and p are equal. (Embretson, 1998; Daniel & Embretson, 2010)

3. Something you must already know minor Working it out

   The angle pairs formed by a line crossing two parallel lines are a grade 8 topic, and the question uses them without naming them. (Anderson & Krathwohl, 2001)

Strategy: Find the third angle of the right triangle, use the straight line m to get the angle beside it, then carry that angle over to line n using the parallel lines.

Traps

• Stops one step early: A choice is a number from partway through the solution.
• An arithmetic slip: The right method with a plausible arithmetic error: a digit, a place value or a times-table fact.

The wrong answers, classified

• A Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: strong

  Finds the triangle's third angle, 180 − 90 − 73 = 17, and stops, though r is the obtuse angle next to the 17° angle, not that angle itself.

• B Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)

  Reveals a surface strategy Expected pull: weak

  No angle fact in the figure gives 117; even 180 − 73 is 107, so this choice mostly catches guesses at an obtuse angle.

• D An arithmetic slip (Junker & Sijtsma, 2001)

  Reveals a slip Expected pull: weak

  Finds the 17° angle and subtracts it from 180 but borrows without reducing the 8, so 180 − 17 comes out as 173.

How the questions are analyzed, and the research behind it

Why it is hard

Circle A is direct: its labeled radius of 5 gives an area of 25π. Circle C is where points are lost, since its size comes from a sentence rather than a label, and the segment through its center is a diameter, so its circumference is 5π while its area is only 6.25π. The row labels favor guessers, because the A and C in A = 25π and C = 5π match the circles' names, though they stand for area and circumference.

1. Reading a figure, table or graph moderate Understanding the question

   Circle C's size is shown only by segment DE, drawn through center C with no label, and only the stem's sentence that DE is congruent to AB makes it a diameter of 5. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Something you must already know moderate Working it out

   Neither A = πr² nor C = πd is given, and one circle is measured by a radius and the other by a diameter, so each formula needs the right length. (Anderson & Krathwohl, 2001)

3. Much to keep track of minor Working it out

   Four values are in play, areas of 25π and 6.25π and circumferences of 10π and 5π, and both rows must be matched correctly for the item to count. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Write down the radius and the diameter of each circle, compute both circles' areas and circumferences in terms of π, then match each value in the table.

Traps

• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.

The wrong answers, classified

• 1:B Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: weak

  Squares the diameter instead of the radius, so Circle C's area seems to be π × 5² = 25π while Circle A's becomes 100π.

• 1:C Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Uses 2πr for the area, so Circle A seems to have an area of 10π and Circle C one of 5π, and no circle seems to have 25π.

• 2:A Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Uses π times the radius for the circumference, so Circle A's radius of 5 seems to give a circumference of 5π.

• 2:C Misreads the question (Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: strong

  Overlooks that DE is congruent to AB, so Circle C has no known size, and with Circle A's circumference at 10π picks Neither.

How the questions are analyzed, and the research behind it

Why it is hard

This is a direct composite-volume problem with whole numbers: one cube, one hole, one subtraction. What it checks is that both volumes are real volumes, the hole as tall as the cube, and that the subtraction is done at the end. Students who stop at the hole’s volume, or who subtract only the square face, give amounts that are easy to type and hard to doubt without choices.

1. Something you must already know minor Working it out

   The volume of a rectangular prism, length times width times height, is not given, and it is needed twice, once for the 20 cm cube and once for the hole. (Anderson & Krathwohl, 2001)

2. Reading a figure, table or graph minor Understanding the question

   The figure labels only the 5 cm opening and the cube’s 20 cm edge; the hole’s 20 cm height comes from the words, while the drawing shows it reaching from top to bottom. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

3. An answer in an unexpected form minor Choosing the answer

   The volume is typed with no choices to compare, so the hole’s 500 cubic centimeters or the cube’s 8,000 can be entered without any warning. (Enright, Morley & Sheehan, 2002)

Strategy: Find the volume of the full cube, then the volume of the square hole that runs from the top face to the bottom, and subtract the hole from the cube.

Traps

• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.

The wrong answers, classified

• 500 Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: moderate

  Finds the volume of the hole, 5 × 5 × 20 = 500 cubic centimeters, and gives the clay removed instead of the clay that is left.

• 7,875 Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  Reveals a slip Expected pull: weak

  Treats the hole as a 5 cm cube, 5 × 5 × 5 = 125, though it runs the full 20 cm height, so 8,000 − 125 = 7,875.

• 7,975 Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: weak

  Subtracts only the area of the square opening, 5 × 5 = 25, from the cube’s 8,000, mixing a face’s area with the hole’s volume.

How the questions are analyzed, and the research behind it

Why it is hard

One formula and one multiplication finish the problem once the right formula is chosen. The cement in the story pulls toward area, and the area, 78.5, is listed, as is half the circumference, from multiplying 3.14 by the radius alone. Remembering that the circumference is π times the diameter, here 10 feet, and that the answer is in feet, not square feet, settles it.

1. Something you must already know moderate Working it out

   The circumference formula is not printed, and the area formula uses the same radius and π, so recalling the wrong one gives a number that looks just as finished. (Anderson & Krathwohl, 2001)

2. Turning words into math minor Understanding the question

   Pouring cement suggests covering a surface, but the question asks for “the distance, in feet, around the edge”, which is the circumference, not the area. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Decide which circle measurement the distance around the edge is, recall its formula, and substitute the radius and 3.14, doubling the radius wherever the formula calls for the diameter.

Traps

• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.
• A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

• A Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Multiplies π by the radius, 3.14 × 5 = 15.7, using the radius where C = πd needs the diameter, so it finds half the distance around.

• C A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)

  Reveals a misconception Expected pull: weak

  Starts from the area formula and squares π instead of the radius, 3.14 × 3.14 × 5 ≈ 49.3, mixing the wrong formula with a slip in squaring.

• D Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: strong

  Finds the patio’s area, 3.14 × 5² = 78.5 square feet, using the area formula instead of the circumference for the distance around.

How the questions are analyzed, and the research behind it

Why it is hard

Every number is small, so the work is choosing the right pieces: the area of the square, the area of the hole, and the difference. A radius of 1 makes the circle's area simply π, which is easy to confuse with the circumference, 2π, when the formulas are half remembered. The region asked for is the gray metal, not the hole, and the final rounding goes up, since the third decimal place is 8.

1. Something you must already know moderate Working it out

   Neither area formula is given, and the hole is labeled by its 2-inch diameter, so students must recall A = πr² and halve the diameter to get the radius 1 first. (Anderson & Krathwohl, 2001)

2. Combines ideas minor Working it out

   The shaded metal is a square with a circle removed, so two area formulas are combined by subtraction rather than either one used alone. (Embretson, 1998; Daniel & Embretson, 2010)

3. An answer in an unexpected form minor Choosing the answer

   The result is typed to the nearest hundredth with no choices to compare, so the long decimal 5.858… must be rounded up, not cut off after two places. (Enright, Morley & Sheehan, 2002)

Strategy: Halve the diameter to get the radius, find the area of the square and of the circle, subtract the circle from the square, and round only at the end.

Traps

• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.

The wrong answers, classified

• 2.72 Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: strong

  Uses the circumference, 2π × 1 ≈ 6.28, in place of the circle's area and subtracts it from the square's 9 square inches, getting about 2.72.

• 3.14 Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: moderate

  Finds the area of the circular hole, π × 1² ≈ 3.14 square inches, and enters it instead of the area of the shaded metal left around it.

• 5.85 Rounds too early (King et al., 2004)

  Reveals a slip Expected pull: weak

  Works the area out correctly as about 5.858 but cuts off the extra digits instead of rounding up to the nearest hundredth.

What was corrected from the DOE handbook, and why

• Explanation: The DOE’s explanation gives the area of the hole as “π(1 in.)² ≅ 3.14 sq in.” The symbol ≅ means “is congruent to”, which an area and a number cannot be; the line means “is approximately equal to”, written ≈. The site now says “π(1 in.)² ≈ 3.14 sq in.”

How the questions are analyzed, and the research behind it

Why it is hard

A scale drawing multiplies every length by one factor, and the perimeter shares that factor, so the work is short once the two perimeters are paired. The common loss is reasoning with differences: the perimeter grows by 5, so a side seems to grow by 5 as well. Converting to miles per inch first, then checking that the three real sides add back to 25 miles, guards against that error and against scaling the wrong side.

1. Turning words into math minor Understanding the question

   The 25 miles is the park's perimeter, not a side, so it must be matched with the drawing's perimeter, 5 + 7 + 8 = 20 inches, before any side can be scaled. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Many steps minor Working it out

   Three operations in a row, adding the sides, dividing 25 by 20 for 1.25 miles per inch, and multiplying the 5-inch side, each produce a number the next step needs. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Add the three drawn sides to get the drawing's perimeter, divide the real perimeter by it to find the miles per inch, then multiply the shortest drawn side by that scale.

Traps

• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

• B Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals a slip Expected pull: weak

  Finds the scale of 1.25 miles per inch but applies it to the 7-inch side, the middle one, giving 8.75 miles instead of the shortest side.

• C Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: strong

  Keeps a constant difference instead of a constant ratio: the perimeter grows by 5, from 20 to 25, so it adds 5 to the 5-inch side.

• D Misreads the question (Hegarty, Mayer & Monk, 1995)

  Reveals a slip Expected pull: weak

  Pairs the 25 miles with the longest side, 8 inches, instead of the 20-inch perimeter, so each inch seems to be 3.125 miles and 5 inches 15.625.

How the questions are analyzed, and the research behind it

Why it is hard

The figure gives one measurement on a slanted radius and two labels on the diameter, and the right angle at the foot of the vertical segment is a decoy. Once RP and PT are seen as radii, RT is 24, and x is that diameter less the 5. Students who never connect the slanted 12 to the horizontal line fall back on the labeled numbers, adding 12 and 5, or take x to end at the center.

1. Reading a figure, table or graph moderate Understanding the question

   The 12 labels the slanted segment SP, not the horizontal one, and the arrow for x runs from R past the center P to the foot of the vertical segment, so x is neither a radius nor a whole diameter. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. No obvious method minor Working it out

   The right-angle mark and the triangle under S suggest the Pythagorean theorem, which is no help; the way in is seeing that RP and PT are radii as well. (Schoenfeld, 1985)

Strategy: Mark every segment from P to the circle as 12, notice that R and T are the ends of a diameter, then write an equation for that diameter using x and the 5.

Traps

• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Information you don't need: The question gives numbers or facts that play no part in the answer.
• Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.

The wrong answers, classified

• A Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  Reveals a slip Expected pull: moderate

  Reads x as the radius RP, as if its arrow ended at the center P, though it runs past P to the foot of the vertical segment.

• B Trusting the picture (Lesh, Post & Behr, 1987)

  Reveals a surface strategy Expected pull: weak

  Judges x from the drawing, where it looks about three times as long as the piece labeled 5, instead of working from the radius of 12.

• C Numbers taken from the question (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: strong

  Adds the two labeled numbers, 12 + 5 = 17, without placing the 12 on the diameter, as if x were a radius plus the 5.

How the questions are analyzed, and the research behind it

Why it is hard

Setting the sum of the two labeled angles to 90° is the decision; the algebra after it is routine. The trap waits at the end: x = 19 is not an angle, and the expression at the top vertex, 2x + 9, gives the larger acute angle, 47°. Evaluating both expressions and checking that they add to 90° takes seconds and shows which one is smaller.

1. Combines ideas moderate Working it out

   The angle facts and the algebra meet in one equation: the right angle means the two labeled angles add to 90°, and only that sum makes (2x + 9) + (3x − 14) solvable. (Embretson, 1998; Daniel & Embretson, 2010)

2. Many steps minor Working it out

   After solving 5x − 5 = 90, both angle expressions must be evaluated and compared, because the question wants an angle, not x, and not the first angle found. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

3. Something you must already know minor Working it out

   The 180° angle sum of a triangle is not printed, and only the small square at the bottom left shows that one angle is 90°. (Anderson & Krathwohl, 2001)

Strategy: Use the right angle to see that the other two angles add to 90°, solve that equation for x, then substitute into both expressions and compare them.

Traps

• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.

The wrong answers, classified

• B Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)

  Reveals a surface strategy Expected pull: weak

  No common working gives 44°; it sits one degree from the right measure and fits no consistent value of x, so it mostly draws estimates and guesses.

• C Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)

  Reveals a surface strategy Expected pull: weak

  No common working gives 46°, which would need x = 20 and two angles adding to 95°, so it mostly catches guesses near 45°.

• D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: strong

  Solves x = 19 correctly but answers with 2x + 9 = 47°, the larger acute angle, instead of evaluating both expressions to find the smaller.

How the questions are analyzed, and the research behind it

Why it is hard

A horizontal cut runs parallel to the ground, and this prism stands on a square base, so the slice copies the base. The main danger is picturing the cut along the tall front face, which gives a 2-by-6 rectangle, the shape a vertical slice makes. The two triangles need a slanted cut through a corner and catch only students guessing.

1. Reading a figure, table or graph minor Understanding the question

   The prism is drawn standing on end, 6 inches tall on a 2-by-2 base, so the cut has to be pictured in a flat drawing where the tall front face is the most visible shape. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Something you must already know minor Working it out

   Students must know that a slice parallel to the base of a prism has the same shape and size as the base, which here is the square at the bottom. (Anderson & Krathwohl, 2001)

Strategy: Picture a flat cut across the prism parallel to the ground, ask which face of the prism that cut runs parallel to, and read that face's edge lengths from the labels.

Traps

• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
• Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.

The wrong answers, classified

• B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: weak

  Expects a slice to cut off a corner and make a triangle, as a cut through a pyramid might, not a cut parallel to this square base.

• C Misreads the question (Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: moderate

  Describes a vertical slice, which runs down the 6-inch height and makes a 2-by-6 rectangle, mixing up horizontal and vertical.

• D Numbers taken from the question (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: weak

  Builds a triangle from the labels in the figure, one 2-inch and two 6-inch sides, a shape no horizontal cut of this prism can make.

How the questions are analyzed, and the research behind it

Why it is hard

Once supplementary is read as a 180° total, the problem is a one-variable equation with whole numbers. Students lose the point in two ways: by stopping at the smaller angle, which the equation produces first, or by using 90° for the total. Checking that the gridded angle is the larger of the pair, and that the two angles add to 180°, catches both.

1. Something you must already know moderate Working it out

   The stem never says what supplementary means; the 180° total has to be recalled, and a student who confuses it with complementary's 90° halves every angle. (Anderson & Krathwohl, 2001)

2. Turning words into math minor Understanding the question

   The sentence "The measure of angle R is 5 times the measure of angle M" has to become R = 5M, so the 180° splits into six equal parts, not five. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

3. An answer in an unexpected form minor Choosing the answer

   As a grid-in there is nothing to compare against, so gridding angle M's 30°, the first value the equation gives, looks just as finished as angle R. (Enright, Morley & Sheehan, 2002)

Strategy: Call angle M one part and angle R five parts; together the six parts make the supplementary total, so find one part and then multiply for angle R.

Traps

• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.

The wrong answers, classified

• 30 Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: strong

  Solves 6m = 180 correctly for angle M, 30°, and grids it instead of multiplying by 5 to get angle R, the angle the question asks for.

• 75 Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Uses 90° for supplementary, the complementary total, so 6m = 90 gives m = 15 and angle R = 75, half its real size.

• 36 Numbers taken from the question (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: weak

  Divides 180 by the 5 in the stem, combining the two numbers without an equation that gives angle M its own share of the 180°.

How the questions are analyzed, and the research behind it

Why it is hard

The question chains three short facts: from the square's area to its side, from the side to the radius, and from the whole circle to a quarter of it. None is hard, but each wrong choice marks a broken link, most often the whole circle's area, which looks like a finished answer. Pointing to S as the center before computing anything settles the radius.

1. Reading a figure, table or graph moderate Understanding the question

   The radius is never labeled: students must see that S is both a corner of the square and the circle's center, so sides PS and RS are radii. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Something you must already know minor Working it out

   Neither the circle area formula, π times the radius squared, nor the step from a square's area to its side length is given. (Anderson & Krathwohl, 2001)

3. Combines ideas minor Working it out

   Square and circle are linked in one chain: the area 4 gives a side of 2, the side is the radius, and the shaded part is a quarter of π × 2². (Embretson, 1998; Daniel & Embretson, 2010)

Strategy: Take the square root of the square's area to get its side, which is also the radius, then take one quarter of pi times the radius squared.

Traps

• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• Stops one step early: A choice is a number from partway through the solution.
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.

The wrong answers, classified

• A Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Halves the square's side as if it were a diameter, so it uses a radius of 1, and one quarter of π × 1² gives π/4.

• C Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: weak

  Misremembers the circle's area as 2πr², blending it with the circumference 2πr, so one quarter of 2π × 2² gives 2π.

• D Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: strong

  Finds the whole circle's area, π × 2² = 4π, and stops before taking the quarter that is shaded.

How the questions are analyzed, and the research behind it

Why it is hard

Scale questions usually convert one length; this one needs a round trip, from the drawing to the real room and back. The real width of 24 feet appears partway through and is a choice, so students who lose track of units stop there. The last conversion, 24/20, then has to be written as a mixed number, and its fraction part is where a hurried student slips.

1. Combines ideas moderate Working it out

   The 960 square feet is a real-world area, so the scale cannot be applied to it directly; the length must go to feet, the width come from the area, and then return to inches. (Embretson, 1998; Daniel & Embretson, 2010)

2. Much to keep track of minor Working it out

   Four lengths in two units are in play, 2 inches, 40 feet, 24 feet and the width on the drawing, and each must stay labeled with its unit. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Convert the 2-inch length to real feet, divide the area by it to get the real width, then use the scale to change that width back to inches.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

The wrong answers, classified

• B An arithmetic slip (Junker & Sijtsma, 2001)

  Reveals a slip Expected pull: weak

  Gets the real width of 24 feet and divides by 20 correctly, but writes the remainder 4/20 as 1/4 instead of 1/5.

• C Units or scale (King et al., 2004)

  Reveals partial knowledge Expected pull: strong

  Finds the room's real width, 960 ÷ 40 = 24 feet, and gives it as the drawing's width without converting back at 1 inch per 20 feet.

• D Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)

  Reveals partial knowledge Expected pull: weak

  Converts the 2-inch length to the real 40 feet and gives that number, answering for the length instead of the width and in the wrong units.

How the questions are analyzed, and the research behind it

Why it is hard

Students have to build the pyramid in their heads from one flat triangle, then choose the right measurement from three labels. The 13-inch sides look like the natural height, and using them gives a listed total; picturing a pyramid with three sloping faces gives another. Sketching the square base with a triangle on each edge, and marking 12 as the height, settles both the face count and the formula.

1. Reading a figure, table or graph moderate Understanding the question

   The figure gives one triangle with three length labels, 13, 13 and 10, and puts its height, 12, on a separate arrow beside it, where it is easy to overlook. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Combines ideas moderate Working it out

   The solid is never drawn: students must see that the square's side equals the triangle's 10-inch base, count four triangles, and add triangle and square areas. (Embretson, 1998; Daniel & Embretson, 2010)

3. Something you must already know minor Working it out

   A triangle's area uses its perpendicular height, not a slanted side, and a pyramid on a square base has four triangular faces; neither fact is given. (Anderson & Krathwohl, 2001)

Strategy: Sketch the pyramid: a square base whose side is the triangle's base, and one triangular face on each side of the square. Find each area with the matching formula, then add them all.

Traps

• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Information you don't need: The question gives numbers or facts that play no part in the answer.
• Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.

The wrong answers, classified

• A Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: moderate

  Counts only three triangular faces, as if the pyramid had a triangular base, so 3 × 60 + 100 = 280.

• B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)

  Reveals a misconception Expected pull: weak

  Counts three faces and also uses the 13-inch side as each triangle's height: 3 × 65 + 100 = 295.

• D Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  Reveals partial knowledge Expected pull: strong

  Takes the 13-inch slanted side as each triangle's height instead of the 12-inch arrow beside it, so 4 × 65 + 100 = 360.

What was corrected from the DOE handbook, and why

• Explanation: The handbook ends “So the total surface area is 340 square feet.” The triangle’s dimensions are in inches and the question asks for square inches, so the site now reads “340 square inches.”

How the questions are analyzed, and the research behind it

Why it is hard

The figure hides the work: it labels four of the six sides, and the missing two are short pieces of the notch that are easy to overlook. Once all six are found, the scale step is a single multiplication, but it must go the right way, since the real garden is larger than the drawing. Tracing the outline and counting six sides guards against the most common loss.

1. Reading a figure, table or graph moderate Understanding the question

   Two sides of the notch carry no labels, so students must find them from the opposite sides: 13 − 8 = 5 cm across and 6 − 3 = 3 cm down. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Combines ideas minor Working it out

   The perimeter of an L-shaped drawing has to be joined to a scale of 2.5 meters per centimeter, two separate skills in one answer. (Embretson, 1998; Daniel & Embretson, 2010)

3. An answer in an unexpected form minor Choosing the answer

   With no choices, nothing flags a perimeter that is missing two sides or still in centimeters; the student must check the six-sided total alone. (Enright, Morley & Sheehan, 2002)

Strategy: Use the opposite sides to work out the two unlabeled lengths, add all six sides of the drawing, then multiply the total in centimeters by 2.5 meters per centimeter.

Traps

• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Stops one step early: A choice is a number from partway through the solution.
• Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).

The wrong answers, classified

• 75 Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  Reveals partial knowledge Expected pull: strong

  Adds only the four labeled sides, 13 + 3 + 8 + 6 = 30 cm, and scales that to 75 m, missing the two unlabeled sides of the notch.

• 38 Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: moderate

  Finds the drawing's perimeter, 38 cm, correctly and grids it without applying the scale of 2.5 meters per centimeter.

• 15.2 Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)

  Reveals a misconception Expected pull: weak

  Divides the 38 cm perimeter by 2.5 instead of multiplying, shrinking the garden though each centimeter of the drawing stands for 2.5 meters.

How the questions are analyzed, and the research behind it

Why it is hard

The question rewards knowing that a prism's volume is its base area times its height, then matching each length to its job. The drawing lies on its side, with a short 2-inch height and the triangle flat on top, which makes the base harder to spot. Most lost points come from the one half: leaving it out doubles the volume, and stopping at the triangle's area gives half of it.

1. Something you must already know moderate Working it out

   Neither formula is given: the volume is the area of the triangular base times the prism's height, and that area needs the one half in 1/2 × base × height. (Anderson & Krathwohl, 2001)

2. Reading a figure, table or graph minor Understanding the question

   The prism lies with its triangles on top and bottom, so the 2-inch vertical edge is its height, while the dashed 8-inch line is the triangle's height to the 4-inch side. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

3. An answer in an unexpected form minor Choosing the answer

   There are no choices, so a product that forgets the one half, or one that stops at the area of the triangle, gives no warning. (Enright, Morley & Sheehan, 2002)

Strategy: Find the area of the triangular face as one half times base times height, using the 4-inch side and the dashed 8-inch height, then multiply by the 2-inch edge of the prism.

Traps

• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• Stops one step early: A choice is a number from partway through the solution.
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.

The wrong answers, classified

• 64 Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: strong

  Multiplies 4 × 8 × 2 = 64 as for a rectangular box, leaving out the one half in the area of the triangular base.

• 16 Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: moderate

  Finds the triangle's area, 1/2 × 4 × 8 = 16 square inches, and stops before multiplying by the prism's 2-inch height.

How the questions are analyzed, and the research behind it

Why it is hard

Two area calculations and a subtraction, kept in terms of π, make this routine for students who know the area formula. Points are lost by using the labeled diameters as radii, by subtracting circumferences, or by subtracting the radii before squaring, and each of these has a listed answer. Writing both areas, 81π and 36π, before subtracting avoids all three.

1. Something you must already know moderate Working it out

   The area formula A = πr² is not given, and the figure labels each plate by its diameter, so 18 and 12 must be halved to radii of 9 and 6 first. (Anderson & Krathwohl, 2001)

2. Reading a figure, table or graph minor Understanding the question

   The dashed lines run all the way across each circle, which marks them as diameters; reading the 18 in. and 12 in. as radii leads to a listed choice. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

Strategy: Halve each diameter to get the radius, find each plate's area as pi times the radius squared, and subtract the smaller area from the larger.

Traps

• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
• A formula you must know: Needs a formula the test does not give: area, circumference, volume, probability rules.

The wrong answers, classified

• A Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: moderate

  Subtracts the circumferences, 18π − 12π = 6π, using πd, the distance around a circle, in place of the area formula.

• B A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)

  Reveals a misconception Expected pull: moderate

  Subtracts the radii first, 9 − 6 = 3, and then squares, π(3)² = 9π, as if the difference of two squares were the square of the difference.

• D Wrong formula (King et al., 2004)

  Reveals a slip Expected pull: strong

  Uses the labeled diameters as radii, π(18)² − π(12)² = 324π − 144π = 180π, forgetting to halve them first.

How the questions are analyzed, and the research behind it