Practice Test

Why it is hard

Each step is one multiplication, division or addition; the risk is direction. The two rates point opposite ways, three zogs to a mink but four minks to a bart, and the fraction in the second rate invites dividing by 4. Asking of each unit whether one of it is worth more or less than a mink fixes the direction before any arithmetic, and it also shows why the 60 and the 24 can never simply be added.

1. Turning words into math moderate Understanding the question

   Both rates put the mink on the left, so “1 mink = 1/4 bart” has to be turned around into 1 bart = 4 minks, and the invented units give no sense of which is worth more. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Many steps minor Working it out

   Two conversions run in opposite directions, dividing the zogs by 3 and multiplying the barts by 4, and their results are added, so a slip in either step changes the total. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

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

   The total is typed into a box with no choices to compare, so a conversion done the wrong way still produces a believable whole number that nothing on screen questions. (Enright, Morley & Sheehan, 2002)

Strategy: For each kind of point, decide whether one of it is worth more or less than a mink, then multiply or divide to turn it into minks, and add the two results.

Traps

• 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).
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

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

  Reveals partial knowledge Expected pull: strong

  Gets 20 minks from the zogs but divides the barts by 4, 24 ÷ 4 = 6, as if a bart were worth a quarter of a mink, for 20 + 6 = 26.

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

  Reveals a misconception Expected pull: moderate

  Multiplies the 60 zogs by 3, as if each zog were worth 3 minks, then adds the 96 minks from the barts correctly: 180 + 96 = 276.

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

  Reveals a surface strategy Expected pull: weak

  Adds the 60 zogs and 24 barts as they stand, 60 + 24 = 84, without converting either kind of point into minks first.

How the questions are analyzed, and the research behind it

Why it is hard

The situation is familiar and the method direct, so this is one of the easier questions on the test. Points are lost in two places: turning 3 years into 36 payments, and remembering the money paid up front after the long multiplication. Each slip, like paying the down payment once a year, leads to a listed total, so checking that the answer holds exactly one $1,200 and 36 payments of $275 is worth the seconds.

1. Much to keep track of minor Working it out

   The $1,200 down payment is paid once while the $275 repeats every month, so it has to be set aside while 36 payments are totaled and then added at the end. (Sweller, 1988; Carpenter, Just & Shell, 1990)

2. Error-prone arithmetic minor Working it out

   The product 275 × 36 = 9,900 is a two-digit multiplication done by hand; splitting it as 275 × 4 = 1,100 and then 1,100 × 9 makes it quick and safe. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Turn 3 years into a number of months, multiply the monthly payment by it, and add the down payment once at the end.

Traps

• Stops one step early: A choice is a number from partway through the solution.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

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

  Reveals a slip Expected pull: moderate

  Multiplies the $275 by 3, the number of years, instead of by 36 months, then adds the $1,200 down payment: 825 + 1,200 = 2,025.

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

  Reveals partial knowledge Expected pull: strong

  Finds the 36 monthly payments, 36 × $275 = $9,900, and stops without adding the $1,200 down payment made at the start.

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

  Reveals partial knowledge Expected pull: weak

  Works out one year as $1,200 + 12 × $275 = $4,500 and multiplies by 3, so the down payment is paid three times instead of once.

How the questions are analyzed, and the research behind it

Why it is hard

Nothing needs computing, but each statement attaches words to two numbers, and three of them attach the wrong words. The point (1, 0.15) is the unit rate students are trained to look for, and the “bags” in the stem’s story make its statement sound right, though the graph measures pounds. Saying each point aloud as so many pounds for so many dollars exposes the swapped coordinates and the extra people as well.

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

   Each statement must be checked against the axis labels: x is the weight of ice in pounds and y the cost in dollars, on a grid ruled every 2 pounds and every $0.30. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Every choice must be checked minor Choosing the answer

   All four statements name a point that really lies on the line, (1, 0.15) included, so being on the graph rules nothing out; only the meaning given to each coordinate does. (Embretson & Wetzel, 1987)

Strategy: Read both axis labels and their units first, then for each choice say in words what its x-coordinate and its y-coordinate stand for, and compare that with the statement.

Traps

• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• 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 Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  Reveals a slip Expected pull: moderate

  Reads the point (20, 3) backward, taking 20 as the price and 3 as the weight, though x counts pounds of ice and y counts dollars.

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

  Reveals a slip Expected pull: strong

  Confirms that (1, 0.15) is on the line, the cost of 1 pound, but accepts “1 bag” because the friends buy bags, though the x-axis measures pounds.

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

  Reveals a misconception Expected pull: weak

  Knows (0, 0) is on every proportional graph and accepts a true-sounding story about people, though the graph never counts people; the point means 0 pounds cost $0.

How the questions are analyzed, and the research behind it

Why it is hard

A single division answers the question, and its numbers are friendly: four lengths of 2.5 centimeters make 10. Every wrong choice is simply a number from the stem, so the only real risk is reaching for a given value instead of converting. Inches are longer than centimeters, so the rainfall in inches has to be less than 10 but more than the 1 inch that 2.5 centimeters make.

1. Turning words into math minor Understanding the question

   The fact “1 inch is equal to approximately 2.5 centimeters” has to be turned into a division, how many 2.5-centimeter lengths fit into 10, rather than a multiplication. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Ask how many groups of 2.5 centimeters fit into the 10 centimeters of rain, since each group of 2.5 centimeters makes one inch.

Traps

• 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.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

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

  Reveals a surface strategy Expected pull: weak

  Picks the 1 from the conversion fact, 1 inch for 2.5 centimeters, reading the given fact as the answer instead of applying it to 10 centimeters.

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

  Reveals a surface strategy Expected pull: weak

  Picks 2.5, the number of centimeters in one inch, straight from the conversion fact rather than finding how many inches 10 centimeters make.

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

  Reveals a surface strategy Expected pull: moderate

  Keeps the 10 from the rain gauge, as if centimeters and inches were the same size, so no conversion is done at all.

How the questions are analyzed, and the research behind it

Why it is hard

With both conversion facts printed, the question is mainly about keeping the units straight. Students who skip the word each convert 36 cups instead of 72, and those who stop at cups find a total that is right in the wrong unit. Working out that a gallon is 4 × 4 = 16 cups before dividing turns two conversions into one and avoids adding the 4s.

1. Many steps minor Working it out

   A multiplication, 36 × 2, comes before two divisions by 4, cups to quarts and quarts to gallons, and stopping after the first step leaves a believable 72. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

2. Turning words into math minor Understanding the question

   The word “each” turns 36 people into 72 cups, and the bold word gallons sets the unit of the answer, a change the note supplies in two steps. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Find the total number of cups first, then divide by 4 to get quarts and by 4 again to get gallons, or work out how many cups make a gallon and divide once.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

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

  Reveals a slip Expected pull: moderate

  Converts 36 cups to gallons, 36 ÷ 16 = 2 1/4, missing the word each, so every person’s second cup of juice is left out.

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

  Reveals a misconception Expected pull: weak

  Uses 8 cups to a gallon, adding the two 4s in the note instead of multiplying them, so 72 cups become 72 ÷ 8 = 9 gallons.

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

  Reveals partial knowledge Expected pull: strong

  Finds the 72 cups of juice and stops, giving the amount in cups though the bold word gallons asks for it in gallons.

What was corrected from the DOE handbook, and why

• Explanation: The DOE’s explanation ends with a line that reads only “lineExtend”, a formatting code left in the document by mistake. The site leaves it out.

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

Each step is ordinary, but the two changes are stated in different units, one a percent and one in meters, which tempts students to force them into the same form. Adding the two percents is the most natural shortcut and lands close to the right value, because it misses only the small extra area where both changes overlap. Computing the old and new areas directly is slower but leaves nothing to chance.

1. Combines ideas moderate Working it out

   Percent change has to be combined with area: the length grows by a percent and the width by meters, and the question asks for the percent change of their product. (Embretson, 1998; Daniel & Embretson, 2010)

2. Many steps moderate Working it out

   Five computations are chained: the new length 92, the new width 60, the areas 4,000 and 5,520, and the increase 1,520 as a share of 4,000. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

3. A very close wrong answer minor Choosing the answer

   The 35% from adding 15% and 20% sits only 3 points below 38%, so a rough estimate of the new area cannot rule it out. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Compute the new length and the new width, find the old and new areas, and divide the increase in area by the original area.

Traps

• 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.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

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

  Reveals a slip Expected pull: moderate

  Treats the 10-meter widening as a 10% increase, so multiplies 1.15 × 1.10 = 1.265 and reports a 26.5% rise.

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

  Reveals a misconception Expected pull: strong

  Adds the length's 15% to the width's 20% increase, as if percent increases in two dimensions simply add, missing the 3% where they overlap.

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

  Reveals a slip Expected pull: weak

  Treats the 15% as 15 meters, making the field 95 by 60 meters, an area of 5,700 square meters and a 42.5% increase.

How the questions are analyzed, and the research behind it

Why it is hard

The rate itself is one division; what makes the question work is that the distance and the time both arrive in the wrong units. Converting only the time gives a speed in kilometers per minute, and converting only the distance and stopping gives the whole trip in meters. Writing the units beside every number, meters over minutes, shows when the answer is ready.

1. Something you must already know minor Working it out

   The conversions are not given: 1 kilometer is 1,000 meters and 1 hour is 60 minutes, so 8 kilometers becomes 8,000 meters and 1 hour 40 minutes becomes 100 minutes. (Anderson & Krathwohl, 2001)

2. Many steps minor Working it out

   Two conversions, one for distance and one for time, come before the single division, and skipping the distance conversion leaves an answer a thousand times too small. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Rewrite the time entirely in minutes and the distance entirely in meters, then divide distance by time; check that the answer's units are meters per minute.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• 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 Units or scale (King et al., 2004)

  Reveals partial knowledge Expected pull: strong

  Divides the 8 kilometers by the 100 minutes but never changes kilometers to meters, giving a speed of 0.08 kilometer per minute.

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

  Reveals a slip Expected pull: weak

  Converts both units correctly but cancels only one zero when dividing 8,000 by 100, getting a rate ten times too large.

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

  Reveals partial knowledge Expected pull: weak

  Converts 8 kilometers to 8,000 meters and stops, giving the whole distance before dividing by the 100 minutes it took.

How the questions are analyzed, and the research behind it

Why it is hard

This is a one-proportion map question, and the only judgments are the direction of the scale and when to round. The phrase to the nearest kilometer tempts some students to round the scale itself before multiplying, which lands on a listed answer two kilometers too far. Rounding only the final 6.75 keeps the result right.

1. Turning words into math minor Understanding the question

   “2 centimeters represents 3 kilometers” gives 1.5 kilometers per centimeter; set up the other way round, the scale shrinks the 4.5 centimeters instead of enlarging them. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Error-prone arithmetic minor Working it out

   The map distance is a decimal, 4.5 × 3 ÷ 2 = 6.75, and the result must then be rounded to the nearest kilometer, two places where a quick estimate drifts. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Work out how many kilometers 1 centimeter on the map stands for, multiply by 4.5, and round only at the very end.

Traps

• 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).
• Rounds too early: Rounds a value partway through, or rounds the wrong way, so the final answer drifts from the exact one.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

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

  Reveals a misconception Expected pull: moderate

  Turns the scale upside down, multiplying 4.5 by 2/3 instead of 3/2, so the real distance comes out smaller than the map distance.

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

  Reveals a surface strategy Expected pull: weak

  Takes the 4.5 centimeters on the map as kilometers and rounds to 5, never applying the scale that turns each centimeter into 1.5 kilometers.

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

  Reveals partial knowledge Expected pull: moderate

  Rounds the scale of 1.5 kilometers per centimeter up to 2 before multiplying, so 4.5 × 2 = 9 instead of 6.75.

How the questions are analyzed, and the research behind it

Why it is hard

The only real trap is the mixed units: 20 and 14 are yards, 6 is feet, and the answer is wanted in cubic yards. Multiplying the three numbers as printed gives a listed choice three times too large. Underlining the unit of each measurement before multiplying, and converting the odd one out, takes seconds.

1. Combines ideas minor Working it out

   The length and width are in yards but the depth is in feet, so a conversion, 6 feet = 2 yards, must come before the volume formula is used. (Embretson, 1998; Daniel & Embretson, 2010)

2. Something you must already know minor Working it out

   Neither 3 feet = 1 yard nor volume = length × width × depth is given; both are assumed in a stem that asks for cubic yards. (Anderson & Krathwohl, 2001)

Strategy: Change the depth from feet to yards first, so that all three measurements are in yards, then multiply the length, width and depth.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Wrong formula: Uses a related but wrong formula: perimeter for area, diameter for radius, area for volume, or a misremembered one.
• 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 Numbers taken from the question (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: weak

  Adds the three measurements, 20 + 14 + 6 = 40, with no conversion and no volume formula, combining the stem’s numbers.

• B Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: weak

  Finds the area of the water’s surface, 20 × 14 = 280, then adds the depth of 6 instead of multiplying by it.

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

  Reveals partial knowledge Expected pull: strong

  Multiplies 20 × 14 × 6 without changing 6 feet to 2 yards, mixing yards with feet and getting a volume three times too large.

How the questions are analyzed, and the research behind it

Why it is hard

Every wrong choice is a conversion handled badly: multiplying by 0.3 when the fact 1 foot = 0.3 meter calls for dividing, dividing by 3 instead of 0.3, or stopping in meters per hour. Writing the units as fractions, so that seconds and meters cancel, shows which way each factor goes. A size check also helps: a foot is shorter than a meter, so the number of feet must be larger than the 36,000 meters.

1. Many steps moderate Working it out

   Both parts of the rate change: seconds become hours by multiplying by 3,600, and meters become feet by dividing by 0.3, two conversions that each have a wrong direction. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

2. Error-prone arithmetic moderate Working it out

   Dividing 36,000 by 0.3 means moving the decimal point in the divisor, and a slip of one place gives 12,000 instead of 120,000. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

3. Something you must already know minor Working it out

   The stem gives 1 foot = 0.3 meter but not the 3,600 seconds in an hour, which students must build from 60 × 60. (Anderson & Krathwohl, 2001)

Strategy: Write the speed as a fraction, multiply by 3,600 seconds per hour, then by 1 foot over 0.3 meter, checking that seconds and meters cancel.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• 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).
• 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 Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)

  Reveals a misconception Expected pull: strong

  Multiplies 36,000 meters per hour by 0.3 instead of dividing, turning the fact 1 foot = 0.3 meter upside down and getting 10,800.

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

  Reveals a slip Expected pull: weak

  Divides 36,000 by 3 instead of 0.3, putting the decimal point in the wrong place and getting an answer ten times too small.

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

  Reveals partial knowledge Expected pull: moderate

  Converts seconds to hours, 10 × 3,600 = 36,000, and stops, leaving the speed in meters per hour rather than feet per hour.

How the questions are analyzed, and the research behind it

Why it is hard

The arithmetic is light, $135 − $45 = $90 and then $90 ÷ $60, so the question is decided by the setup and by the form of the entry. The trip charge is paid once, not every hour, and has to come off the budget before dividing. The time works out to an hour and a half, and students who think in hours and minutes may write it the way a clock shows it rather than as a decimal.

1. Turning words into math moderate Understanding the question

   The stem never says “at most”; the limit comes from “a budget of $135”, and the one-time $45 trip charge is added once while the $60 multiplies the hours, giving 45 + 60x ≤ 135. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

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

   The grid-in demands a decimal, so an hour and a half must be entered as 1.5, not as 1:30 or 90 minutes, with no choices to confirm the form. (Enright, Morley & Sheehan, 2002)

Strategy: Take the one-time trip charge off the budget first, divide what is left by the hourly charge, and give the time in hours as a decimal rather than in hours and minutes.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
• Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

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

  Reveals partial knowledge Expected pull: strong

  Divides the whole $135 budget by $60 an hour, 135 ÷ 60 = 2.25, leaving out the one-time $45 trip charge that must come off first.

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

  Reveals a misconception Expected pull: moderate

  Finds that the $90 left pays for 1 hour 30 minutes and writes that as 1.3, as if minutes were tenths of an hour rather than sixtieths.

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

  Reveals partial knowledge Expected pull: weak

  Subtracts the trip charge and enters the $90 left for labor, stopping before dividing by the $60 hourly charge to turn dollars into hours.

How the questions are analyzed, and the research behind it

Why it is hard

The figure gives four of the six sides and the scale gives feet per half inch, so two conversions hide behind a simple-looking perimeter question. Students who add only the labeled sides, or who read the scale as 25 feet per inch, reach totals that look reasonable for a pool and are listed. Tracing the outline to count six sides, and doubling 25 before multiplying, avoids both.

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

   Two sides of the notch carry no labels, and the 5 in. arrow spans the full right side, so the missing lengths come from 5 − 2 = 3 and 6 − 3 = 3. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

2. Combines ideas moderate Working it out

   The perimeter of an L-shaped drawing has to be joined to a scale in which half an inch, not a whole inch, stands for 25 feet. (Embretson, 1998; Daniel & Embretson, 2010)

3. Many steps minor Working it out

   Four stages follow in order, the two missing sides, the 22-inch perimeter, the feet per inch and the product, and a slip at most of them reaches a listed choice. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Use opposite sides to fill in the two missing lengths, add all six sides of the drawing, then double the 25 feet to get the feet one inch stands for, and multiply.

Traps

• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• 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

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

  Reveals partial knowledge Expected pull: strong

  Finds the drawing's 22-inch perimeter but multiplies by 25, using the scale as if 1 inch, not 1/2 inch, stood for 25 feet.

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

  Reveals partial knowledge Expected pull: moderate

  Adds only the four labeled sides, 6 + 2 + 3 + 5 = 16 inches, and scales that to 800 feet, missing the two unlabeled sides of the notch.

• D Wrong formula (King et al., 2004)

  Reveals a misconception Expected pull: weak

  Multiplies the 6-inch and 5-inch sides to get 30, the outer rectangle's area, and scales that by 50 feet per inch as if it were a length.

How the questions are analyzed, and the research behind it

Why it is hard

This is a unit conversion dressed as a measuring story: four yardsticks and two rulers are laid end to end, so the total is simply a sum of lengths. Points are lost by forgetting that the rulers count too, or by giving the rulers the yardsticks' length. Writing every piece in feet before adding, 3 + 3 + 3 + 3 + 1 + 1, shows the total at a glance.

1. Something you must already know minor Working it out

   Neither conversion is given: a yardstick is 1 yard, or 3 feet, and each 12-inch ruler is 1 foot, so 4 yards and 24 inches become 12 + 2 feet. (Anderson & Krathwohl, 2001)

Strategy: Change every length to feet before adding: a yard is 3 feet and 12 inches is 1 foot. Then total the six pieces laid end to end.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• Stops one step early: A choice is a number from partway through the solution.

The wrong answers, classified

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

  Reveals partial knowledge Expected pull: strong

  Converts the four yardsticks to 12 feet and stops, forgetting to add the 2 feet of the two 12-inch rulers laid at the end.

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

  Reveals a surface strategy Expected pull: weak

  No common working gives 16 feet; it would need each 12-inch ruler to be 2 feet long, so it mostly draws guesses between nearby totals.

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

  Reveals a slip Expected pull: weak

  Converts the rulers with the yard factor too, counting all six sticks as 3 feet long, 6 × 3 = 18, though a 12-inch ruler is 1 foot.

What was corrected from the DOE handbook, and why

• Explanation: The handbook says “Two 12-inch rulers is 2 feet in length.” (p. 142, question 86), repeating the verb of the line before it, “A 12-inch ruler is 1 foot in length.” The site now reads “Two 12-inch rulers are 2 feet in length.” The working and the answer, 14 ft, are unchanged.

How the questions are analyzed, and the research behind it

Why it is hard

Pricing the letter ounce by ounce, $0.55, $0.70, $0.85, $1.00, reaches the total at the fourth ounce without any algebra. The equation route works too, but the first ounce is easy to lose: dividing the $0.45 left by $0.15 counts only the ounces after the first. The two decimal choices come from merging the two prices or from treating dollars as ounces.

1. Turning words into math moderate Understanding the question

   “$0.55 for the first ounce of weight plus $0.15 for each additional ounce” makes the cost 0.55 + 0.15(x − 1), so the $0.15 covers every ounce except the first. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Error-prone arithmetic minor Working it out

   The money is in decimals, $1.00 − $0.55 = $0.45 and then $0.45 ÷ $0.15 = 3, and dividing by a decimal invites a misplaced point. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Write the cost of a letter weighing x ounces, $0.55 plus $0.15 for each ounce after the first, set it equal to $1.00 and solve, then check by pricing the letter ounce by ounce.

Traps

• 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.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

The wrong answers, classified

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

  Reveals a surface strategy Expected pull: weak

  Merges the two prices into $0.70 for every ounce and divides $1.00 by $0.70, about 1.43, which rounds to 1.4.

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

  Reveals a surface strategy Expected pull: weak

  Counts the first ounce, then treats the $0.45 left over as 0.45 ounce, getting 1.45 ounces, which rounds to 1.5.

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

  Reveals partial knowledge Expected pull: strong

  Finds the 3 additional ounces that the $0.45 left over pays for, 0.45 ÷ 0.15 = 3, and stops before adding the first ounce.

How the questions are analyzed, and the research behind it

Why it is hard

The numbers are friendly, but the rate is given per 6 seconds and the target per minute, so the units have to meet first. A minute holds ten 6-second stretches, so the robot takes 90 steps in it, and 270 feet over 90 steps settles the length. Students who divide 270 by the 9 steps forget that those steps take only 6 seconds, and those who scale 270 down to 6 seconds stop at 27 feet, a distance rather than a step length.

1. Much to keep track of moderate Working it out

   Three rates are in play, 9 steps per 6 seconds, x feet per step and 270 feet per minute, and their units must be lined up before any two are combined. (Sweller, 1988; Carpenter, Just & Shell, 1990)

2. Many steps minor Working it out

   The rate is first scaled from 6 seconds to 60, giving 90 steps a minute, and only then compared with the 270 feet covered in that minute. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Change the rate to steps per minute first, since 6 seconds goes into a minute evenly, then ask what step length makes that many steps cover 270 feet.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• 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 Numbers taken from the question (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: weak

  Multiplies the 9 steps by the 6 seconds, 54, and divides 270 by it, combining the stem’s numbers without a model of the rate.

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

  Reveals partial knowledge Expected pull: moderate

  Scales 270 feet per minute down to 27 feet per 6 seconds but stops before dividing by the 9 steps taken in that time.

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

  Reveals partial knowledge Expected pull: strong

  Divides 270 feet by 9 steps, as if the robot took 9 steps a minute, never converting the 6-second rate to a minute.

How the questions are analyzed, and the research behind it

Why it is hard

The main risk is thinking in differences: 2 more bicycles than 3 seems to call for 2 more hours than 4, a tidy wrong answer of 6 hours. Students who scale correctly still have to turn a fraction of an hour into minutes, and a careless 45 for the 2/3 hour lands on the key's neighbor. Working in minutes, 80 per bicycle, keeps every number whole and leaves one simple conversion at the end.

1. An answer in an unexpected form moderate Choosing the answer

   The rate gives 5 × 4/3 = 20/3, or 6 2/3 hours, but every choice is in hours and minutes, so the 2/3 of an hour must become 40 minutes before the answer can be recognized. (Enright, Morley & Sheehan, 2002)

2. Error-prone arithmetic minor Working it out

   Each bicycle takes 4/3 of an hour, not a whole number; changing 4 hours to 240 minutes first gives a whole 80 minutes per bicycle and avoids the fraction. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

3. A very close wrong answer minor Choosing the answer

   The choices 6 hr 40 min and 6 hr 45 min are five minutes apart, so an estimate of a little under 7 hours fits both and only an exact conversion decides. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Change the 4 hours to minutes first so the time per bicycle comes out whole, multiply by 5, and convert the total minutes back into hours and minutes.

Traps

• 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.
• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.

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: strong

  Reasons additively: 2 more bicycles than 3 must take 2 more hours than 4, giving 6 hours, which keeps the difference instead of the rate of 4/3 hour per bicycle.

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

  Reveals a slip Expected pull: moderate

  Reaches 6 2/3 hours but converts the 2/3 hour to 45 minutes, the minutes in 3/4 of an hour, instead of 2/3 × 60 = 40.

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

  Reveals a surface strategy Expected pull: weak

  No common working gives 7 hr 15 min, which would mean 87 minutes per bicycle; it lies farthest from the answer and mostly draws guesses.

How the questions are analyzed, and the research behind it

Why it is hard

The heights mix two units, and the inch parts run the wrong way, 4.5 against 9, so subtracting part by part goes wrong unless a foot is regrouped as 12 inches. Converting both heights to inches first avoids the borrow entirely. The choices sit a single inch apart, so a rough estimate cannot stand in for the exact working, and a student who treats feet as tens of inches lands squarely on a wrong choice.

1. Error-prone arithmetic moderate Working it out

   Subtracting 5 feet 9 inches from 6 feet 4.5 inches needs a borrow, since 4.5 inches is less than 9, and the borrowed foot holds 12 inches, not the 10 that decimal habits suggest. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

2. Something you must already know minor Working it out

   The stem never states that a foot is 12 inches, yet both heights must be turned into inches, 76.5 and 69, before the difference can be taken. (Anderson & Krathwohl, 2001)

Strategy: Change both heights to inches by multiplying the feet by 12 and adding the extra inches, then subtract the shorter height from the taller one.

Traps

• Units or scale: The answer is wanted in other units, or a rate or scale must be converted first.
• 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 Numbers taken from the question (Hegarty, Mayer & Monk, 1995)

  Reveals a surface strategy Expected pull: moderate

  Subtracts the two inch figures alone, 9 − 4.5 = 4.5, taking the smaller from the larger and leaving the feet out of the comparison entirely.

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

  Reveals a misconception Expected pull: strong

  Treats feet as tens of inches, reading the heights as 64.5 and 59, or borrowing a foot as 10 inches, so 64.5 − 59 = 5.5.

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

  Reveals a surface strategy Expected pull: weak

  No conversion or borrow gives 6.5, one inch short of the true gap; only a miscount such as taking Joseph as 70 inches reaches it, so it mostly catches guesses.

What was corrected from the DOE handbook, and why

• Explanation: The handbook’s explanation ends “Then, subtract Joseph’s height from Robert’s height: 76.5 − 69 = 7.5” (p. 133, question 75). The taller boy is Roberto, as the question and the explanation’s own “Roberto:” line call him, so the site now reads “Roberto’s height.” Nothing else changes: Roberto is 6(12) + 4.5 = 76.5 inches, Joseph is 5(12) + 9 = 69 inches, and the difference is 7.5 inches.

How the questions are analyzed, and the research behind it