Practice Test

Why it is hard

Finding the price is one division, 12 ÷ 4 = 3, so the question is really about checking: each chosen dot must fit y = 3x exactly, and the item does not say how many do. The graph punishes judging by eye, since a straight row of dots climbs from (1, 3) at $2 a notebook and two dots sit a single gridline off the true line. Multiplying each x by 3 and reading each y from the axis, not from the pattern, settles all seven.

1. Every choice must be checked moderate Choosing the answer

   Seven dots must each be tested against $3 a notebook, and the item never says how many fit; one missed point or one extra loses the whole item, so no dot can be skipped. (Embretson & Wetzel, 1987)

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

   The graph stacks two dots above 1 notebook and two above 3, and each total cost has to be read off a grid ruled at every dollar up to 15. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

3. A very close wrong answer minor Choosing the answer

   (5, 14) sits one gridline below (5, 15), where 5 notebooks at $3 would be, and (2, 5) sits one below (2, 6), so judging by the line's direction cannot rule either out. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Work out the cost of one notebook from the total given in the question, then multiply each point's x-coordinate by that price and keep only the points whose y-coordinate matches exactly.

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.
• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.

The wrong answers, classified

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

  Reveals a surface strategy Expected pull: weak

  No working from 4 notebooks for $12 makes one notebook cost $1, so this lowest dot mostly catches guesses, such as taking the dot nearest the origin as a start.

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

  Reveals a misconception Expected pull: moderate

  Takes the straight row (1, 3), (2, 5), (3, 7) as the graph because its dots line up, though it climbs $2 a notebook and would meet the y-axis at $1, not the origin.

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

  Reveals a misconception Expected pull: weak

  Prefers the lower of the two dots above 3 notebooks because it continues the evenly spaced row from (1, 3) and (2, 5), instead of checking that 3 notebooks cost $9.

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

  Reveals a slip Expected pull: strong

  Finds that 5 notebooks cost 5 × $3 = $15, then reads the dot above 5 as sitting on the top gridline, 15, when it is one line lower, at 14.

How the questions are analyzed, and the research behind it

Why it is hard

Nothing needs solving here; the question is which pairings keep the usage rate fixed. Students who check only for buns over customers on both sides find exactly three such setups, two of them wrong, and pass over the two right ones written another way. The wrong pair gives 10.8 dozen, fewer buns for more customers, which a quick sense check would catch, and cross-multiplying every setup settles all five.

1. Every choice must be checked moderate Choosing the answer

   All five setups are built from 18, 150, 250 and x, so none can be dismissed on sight; each has to be cross-multiplied and compared with 150x = 18 × 250. (Embretson & Wetzel, 1987)

2. A very close wrong answer moderate Choosing the answer

   The first two setups differ only in where 150 and 250 sit, 18/150 = x/250 against 18/250 = x/150, and both show buns over customers on each side, so a units check passes both. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

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

   One right setup puts customers over dozens of buns, and another sets today’s x over yesterday’s 18 equal to 250 customers over 150, forms that look wrong to students who learned only rate equals rate. (Enright, Morley & Sheehan, 2002)

4. Much to keep track of minor Working it out

   Three of the five must be picked and all three must be right to score, so the verdict on each setup has to be kept until the last one is tested. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Write one proportion you trust by matching both units and days, cross-multiply it, then cross-multiply each choice and keep only those that give the same equation.

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.
• Answer in another form: The right answer is written differently from the way you worked it out.

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: moderate

  Checks only that both fractions show dozens of buns over customers, missing that 18 sits over today’s 250 customers; cross-multiplying gives 250x = 2,700, so x = 10.8.

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

  Reveals a misconception Expected pull: strong

  Left out by students who expect each fraction to compare buns with customers, not seeing that today over yesterday for buns and for customers also gives 150x = 4,500.

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

  Reveals a misconception Expected pull: moderate

  Accepts x over 150 because it shows buns over customers like the trusted setup, but today’s buns sit over yesterday’s customers, which yields only 10.8 dozen for more people.

How the questions are analyzed, and the research behind it

Why it is hard

The divisions are easy, so the question tests what proportional means. Three tables grow by the same amount for every 5 books, and the one with a $10 head start looks just as regular as the right one it copies; only its cost per book, drifting from $9 down to $8.40, gives it away. The flat $100 table and the one that jumps to $250 catch fewer students, and dividing the first and last column of each table is a quick, reliable check.

1. A very close wrong answer moderate Choosing the answer

   The table running $90, $130, $170, $210 is the $80-to-$200 table with $10 added to every cost; both climb $40 for each 5 books, and only the cost per book separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. Every choice must be checked minor Choosing the answer

   Five tables of four columns each have to be tested by dividing cost by books, and both right tables must be picked to score, so no table can be judged from one column. (Embretson & Wetzel, 1987)

3. Something you must already know minor Working it out

   Students must know that proportional means the same cost per book in every column, a line through zero, not merely a cost that grows by the same amount each time. (Anderson & Krathwohl, 2001)

Strategy: Divide the total cost by the number of books in every column of each table; a table is proportional only when that cost per book stays the same all the way across.

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.

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

  Sees the cost climb $40 for each 5 books across the first three columns and stops checking, though the cost per book goes $7, about $7.33, $7.50, then $10.

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

  Reveals a misconception Expected pull: strong

  Takes the steady rise of $40 for every 5 books as proportional, but 90 ÷ 10 = 9 while 210 ÷ 25 = 8.4, so the cost per book is not constant.

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

  Reveals a misconception Expected pull: weak

  Mistakes a cost that never changes for a constant ratio, though $100 for 10 books is $10 a book and $100 for 25 books is $4 a book.

How the questions are analyzed, and the research behind it

Why it is hard

Two equal decreases in a row feel like one decrease of twice the size, and that is the trap: 20% and 20% suggest 40% gone. The stem states the rule, each loss measured from that week’s start, in one quick phrase. Working week by week from 100 shows the second loss is 16, not 20, which separates the answer from the choice four points away, while stopping after one week or reporting the loss costs the point more simply.

1. A very close wrong answer major Choosing the answer

   The choices 60% and 64% sit four percentage points apart, so a rough estimate cannot choose between them; only seeing that the second 20% is taken from 80, not 100, does. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. Turning words into math moderate Understanding the question

   The phrase “the amount at the beginning of that same week” sets each week’s base, so the second week’s 20% is measured from what was left after the first week. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

3. Variables instead of numbers minor Understanding the question

   No starting amount is given, so students must pick one, such as 100 grams, or work with a percent of a percent, 0.8 × 0.8. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)

Strategy: Use a starting amount of 100 so that percents become units, find what remains after the first week, then apply the same 20% loss to that remainder.

Traps

• Percent of the wrong amount: Finding the whole from a part, or a percent change measured from the wrong starting amount.
• Stops one step early: A choice is a number from partway through the solution.
• 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

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

  Reveals a misconception Expected pull: moderate

  Adds the weekly losses, 20% + 20% = 40%, and gives that figure, which describes the amount lost rather than the percentage still left.

• B Percent of the wrong amount (Parker & Leinhardt, 1995)

  Reveals a misconception Expected pull: strong

  Subtracts 20% of the original amount twice, 100 − 20 − 20 = 60, though the second week’s 20% comes from the 80% left, not the starting amount.

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

  Reveals partial knowledge Expected pull: moderate

  Finds that 80% remains after the first week and stops, never taking the second week’s 20% from that remainder.

How the questions are analyzed, and the research behind it

Why it is hard

The calculation is long, and its key fact is missing from the page. Students who know the 128 fluid ounces still face a division by 33 that does not come out evenly, which tempts them to round the price per ounce to 5 cents and lose about 50 cents a gallon. Keeping the exact quotient to the end, or scaling 33 ounces up by about 3.9, keeps the difference near $2.95, which rounds to 3.

1. Something you must already know moderate Working it out

   The question never says that a gallon holds 128 fluid ounces, yet the whole comparison depends on it; the 6-pack holds only 6 × 5.5 = 33 ounces. (Anderson & Krathwohl, 2001)

2. Many steps moderate Working it out

   Five moves follow one another: the ounces in a pack, the scale-up to 128 ounces, the division by 33, the subtraction of $3.99 and the rounding to a whole dollar. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

3. Error-prone arithmetic minor Working it out

   1.79 × 128 = 229.12 and 229.12 ÷ 33 are long decimal calculations without a calculator, and a slip in either moves the price of a gallon. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

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

   The rounded difference is typed with no choices, so an unrounded $2.95, or a figure rounded too soon, looks just as finished as the whole-dollar answer. (Enright, Morley & Sheehan, 2002)

Strategy: Find the ounces in one six-pack, scale its price up to the 128 fluid ounces in a gallon, subtract the milk's price, and round only at the end.

Traps

• Stops one step early: A choice is a number from partway through the solution.
• Rounds too early: Rounds a value partway through, or rounds the wrong way, so the final answer drifts from the exact one.
• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

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

  Reveals partial knowledge Expected pull: strong

  Finds the juice's cost per gallon, about $6.94, rounds it to $7, and enters that without subtracting the $3.99 price of the milk.

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

  Reveals partial knowledge Expected pull: moderate

  Rounds the price per ounce, 1.79 ÷ 33 ≈ 0.054, to $0.05, so a gallon costs $6.40 and the difference, $2.41, rounds to 2.

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

  Reveals a slip Expected pull: moderate

  Works out the difference of $2.95 correctly but enters it as it stands, though the question asks for it to the nearest dollar.

What was corrected from the DOE handbook, and why

• Explanation: The DOE’s explanation goes from 33x = $229.12 to “x = $6.94”. But 229.12 ÷ 33 = 6.9430…, so $6.94 is the cost per gallon rounded to the cent. The site writes x ≈ $6.94. Nothing else changes: $6.94 − $3.99 = $2.95, which rounds to $3, and the unrounded cost gives the same answer.

How the questions are analyzed, and the research behind it

Why it is hard

The scale factor comes from one division, 45 ÷ 5 = 9, so the risk lies in what to do with it. Two tempting results are listed: the photographs alone, and the whole show with the drawings added in. Multiplying the 3 + 4 = 7 parts for paintings and photographs by 9 gives the total in one step and avoids both.

1. Much to keep track of moderate Working it out

   The ratio has three parts, the known count belongs to the middle part, drawings, and the question wants the first and third parts combined, so three roles must be tracked. (Sweller, 1988; Carpenter, Just & Shell, 1990)

2. Turning words into math minor Understanding the question

   The word respectively assigns 3, 5 and 4 to paintings, drawings and photographs in that order, and the question then names two of them in the other order. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Find how many works each ratio part stands for from the drawings, multiply it by the parts for paintings and for photographs, and add only those two.

Traps

• Stops one step early: A choice is a number from partway through the solution.
• 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

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

  Reveals partial knowledge Expected pull: strong

  Finds the photographs, 4 × 9 = 36, and stops before adding the 27 paintings that the question also asks about.

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

  Reveals a surface strategy Expected pull: weak

  Multiplies the 7 parts for paintings and photographs by 12, the total of the ratio's parts, instead of by 9.

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

  Reveals a slip Expected pull: moderate

  Finds every group correctly but adds the 45 drawings too, giving the whole show, 108, rather than the paintings and photographs.

How the questions are analyzed, and the research behind it

Why it is hard

Two fractions and two units, feet and yards, leave students to decide whether to multiply, divide or add, and each wrong choice records one of those decisions. A sense check rules out the most tempting one before any arithmetic: a whole foot is more than 3/8 foot, so it needs more than 5/6 yard, yet multiplying gives only 5/16. Dividing the yards by the feet then picks out the answer among the rest.

1. Turning words into math moderate Understanding the question

   “For each 3/8 foot of curtain width, 5/6 yard of fabric” gives the rate as a pair of fractions, and “per foot” means dividing the yards by the feet. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Error-prone arithmetic minor Working it out

   Dividing by 3/8 means multiplying by 8/3, and the product 40/18 has to be reduced to 20/9, an improper fraction rather than a whole number. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Divide the yards of fabric by the feet of curtain width, since a rate per foot puts feet in the denominator, then simplify the fraction.

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).

The wrong answers, classified

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

  Reveals a misconception Expected pull: strong

  Multiplies 5/6 by 3/8 instead of dividing, getting 15/48 = 5/16, so a whole foot seems to need less fabric than 3/8 foot.

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

  Reveals a surface strategy Expected pull: moderate

  Adds 5/6 and 3/8 over a common denominator, 20/24 + 9/24 = 29/24, combining yards with feet instead of forming a rate.

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

  Reveals partial knowledge Expected pull: weak

  Finds that a foot is 8/3 times 3/8 foot, then adds 8/3 and 5/6 instead of multiplying them: 16/6 + 5/6 = 21/6 = 7/2.

How the questions are analyzed, and the research behind it

Why it is hard

Once the origin rule is recalled, the graph answers the question at a glance: only the line from O through T and R starts at (0, 0). Students who think of proportional as any steady pattern have three other candidates, from the falling line through M and S to the flat line at height 3. The steep line through V and P is the closest call, because its lower end disappears near the corner where the axes meet.

1. Something you must already know minor Working it out

   The rule that a proportional graph is a straight line through the origin is not printed, and all four lines are straight, so straightness alone rules none of them out. (Anderson & Krathwohl, 2001)

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

   Line VP runs off the grid just right of O, crossing the x-axis at (1, 0), so whether it passes through the origin has to be read from the gridlines, not from its general direction. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

Strategy: Recall what every proportional graph must look like, then test each line: see where it would be when x is 0, and whether y divided by x stays the same at both labeled points.

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.
• Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.

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: weak

  Takes MS as proportional because it is straight and meets both axes, though it starts at (0, 8) and its y-values fall as x rises.

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

  Reveals a slip Expected pull: moderate

  Sees the steep line’s lower end vanish near O and assumes it passes through the origin, though VP crosses the x-axis at (1, 0).

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

  Reveals a misconception Expected pull: weak

  Mistakes the steady y-value of 3 along VQ for a constant of proportionality, though y ÷ x falls from 3/2 at V to 1/2 at Q.

What was corrected from the DOE handbook, and why

• Explanation: The DOE’s explanation says “Both point T (1, 4) and point R (2, 8) have a y-value that is 4 times the x-value, and every point on the line fits the proportional equation y = 4x”, and then “When x = 0, y = 4(0).” On the graph, T is at (4, 1) and R at (8, 2): each y-value is 1/4 of its x-value, and the line through the origin, T and R is y = (1/4)x. (The line y = 4x would pass through (1, 4) and (2, 8), which are not marked.) The 2022–23 guide, which first printed this question with the same graph, has the same slip. The site now gives T (4, 1) and R (8, 2), “a y-value that is 1/4 of the x-value”, y = (1/4)x, and y = (1/4)(0) at x = 0. The answer is unchanged: TR is still the only line through the origin.

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

Any row gives the answer with one division, and the values divide evenly, so the points are lost on direction. Fruit fills the first column and comes first in the stem’s own ratio, which invites dividing it by the yogurt. The table also climbs by a steady half cup of yogurt, a number that looks like a constant but ignores the 2 cups of fruit added each time. Checking that 0.25 times each fruit value gives its yogurt value settles it.

1. Turning words into math moderate Understanding the question

   The opening sentence speaks of a ratio of fruit to yogurt, but the question asks for the constant of yogurt to fruit, so the division is y ÷ x, against the table’s column order. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

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

   The constant is typed into a box with no choices, so 4, the ratio taken the other way round, is a believable entry that nothing on screen questions. (Enright, Morley & Sheehan, 2002)

Strategy: Check which quantity the question names first, divide that column by the other in any row, and confirm the quotient is the same in a second row.

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).
• 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.
• 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

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

  Reveals partial knowledge Expected pull: strong

  Divides cups of fruit by cups of yogurt, 6 ÷ 1.5 = 4, following the stem’s “ratio of fruit to yogurt” and turning the requested constant upside down.

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

  Reveals a misconception Expected pull: moderate

  Takes the steady rise of 0.5 cup of yogurt from row to row as the constant, ignoring that the fruit rises by 2 cups each time.

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

  Reveals a surface strategy Expected pull: weak

  Gives 1.5, the first yogurt value in the table, as if the constant were an amount in one batch rather than a ratio of two amounts.

How the questions are analyzed, and the research behind it

Why it is hard

The proportion itself is routine; the question is built around two conditions that each change the numbers. The stem reports the damaged boxes, but the question asks about undamaged ones, and its bold not removes the 20 sampled boxes from the shipment. Each wrong choice matches one way of mishandling the two conditions, so checking the result against both words, undamaged and not, is what the question rewards.

1. Much to keep track of moderate Working it out

   Two conditions point away from the obvious numbers: the question wants undamaged boxes while the stem counts the 8 damaged, and it leaves out the 20 sampled, so 150 boxes, not 170. (Sweller, 1988; Carpenter, Just & Shell, 1990)

2. Many steps minor Working it out

   Three steps follow in order, the undamaged share 12/20, the 150 boxes outside the sample, and their product, and each wrong turn lands on a listed choice. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Find the fraction of the sample that is undamaged, count the boxes that were not sampled, and multiply; then check that the answer describes the group of boxes asked about.

Traps

• Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
• 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

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

  Reveals partial knowledge Expected pull: moderate

  Uses the damaged share, 8 of 20 or 0.4, instead of the undamaged share, taking it of the 150 unsampled boxes: 0.4 × 150 = 60.

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

  Reveals partial knowledge Expected pull: weak

  Takes the damaged share of the whole shipment, 0.4 × 170 = 68, answering for damaged boxes and keeping the 20 sampled ones as well.

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

  Reveals partial knowledge Expected pull: strong

  Finds the undamaged share, 0.6, but takes it of all 170 boxes, missing the bold not that leaves the 20 sampled boxes out: 0.6 × 170 = 102.

How the questions are analyzed, and the research behind it

Why it is hard

This is a recall question dressed in a short story: the softball team adds nothing but names for the letters. The one idea needed is that proportional means multiplied by a constant, so doubling the players doubles the bottles. Students who think of a constant as something added, or as a fixed total, are drawn to the two sums that use the same letters.

1. Something you must already know minor Working it out

   The general form of a proportional relationship, one amount equal to the constant times the other, has to be recalled; the stem names p but never says it multiplies. (Anderson & Krathwohl, 2001)

2. Variables instead of numbers minor Understanding the question

   No numbers are given, only b, n and p, so a choice cannot be checked by computing unless students make up a case, such as 2 bottles per player for 10 players. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)

Strategy: Recall how two proportional quantities are linked through the constant of proportionality, then test each choice with sample numbers: if the players double, the bottles should double too.

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.

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: moderate

  Treats proportional as adding a fixed amount, so the bottles would be the players plus p, rather than the players multiplied by a constant.

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

  Reveals a misconception Expected pull: weak

  Takes the constant of proportionality as the total of bottles and players, confusing a constant sum with a constant ratio.

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

  Reveals a surface strategy Expected pull: weak

  Puts all three letters into one sum equal to 1, an equation that uses the right letters but matches no idea of proportion, so it mostly draws guesses.

How the questions are analyzed, and the research behind it

Why it is hard

The question rewards turning the percent into millimeters before doing anything else. Students who treat ±4% as ±4 millimeters find the two whole-number options at either end of the menus, 68.00 and 76.00, which look like a natural pair. Those who compute 2.88 correctly can still lose the item on the maximum by rounding to 3 and picking 75.00. Multiplying 72 by 0.96 and by 1.04 is a quick double-check.

1. Error-prone arithmetic moderate Working it out

   Four percent of 72 is 0.04 × 72 = 2.88, a decimal product, and taking it from 72 means borrowing across the decimal point to reach 69.12. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

2. Turning words into math minor Understanding the question

   The phrase “a variation of ±4% in the width” has to become 4% of 72 millimeters, a length of 2.88 mm, before anything is added or subtracted. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

3. A very close wrong answer minor Choosing the answer

   The second menu lists 75.00 just 0.12 above 74.88, so a rounded estimate of 72 + 3 lands on the wrong option, and estimating cannot separate the two. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

4. Many steps minor Working it out

   Both menus must be right to score, so the minimum and the maximum each need their own subtraction or addition after the shared percent is found. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Find 4% of the 72-millimeter design width first, then subtract that amount for the minimum and add it for the maximum, before looking at the menus.

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.
• Rounds too early: Rounds a value partway through, or rounds the wrong way, so the final answer drifts from the exact one.

The wrong answers, classified

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

  Reveals a misconception Expected pull: strong

  Reads ±4% as ±4 millimeters, so the widths run from 72 − 4 = 68.00 to 72 + 4 = 76.00.

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

  Reveals a slip Expected pull: moderate

  Finds the 69.12 minimum, then estimates the maximum by rounding the 2.88 change up to 3, so 72 + 3 = 75.00 is picked over 74.88.

How the questions are analyzed, and the research behind it

Why it is hard

The numbers could hardly be friendlier, a $2 raise on $10, so the question checks what to divide by what. Dividing $12 by $10 gives 1.2, the new pay as a percent of the old, and stopping there is tempting because the division feels like the whole job. Reading the $2 itself as 2% is the other quick error, since dollars of change are not percent of change.

1. Something you must already know minor Working it out

   The rule for percent change, the change divided by the starting amount, is not given, and each wrong choice comes from a different comparison of the $10 and the $12. (Anderson & Krathwohl, 2001)

2. Turning words into math minor Understanding the question

   “Percent increase in Malik’s hourly pay” has to be read as the $2 rise measured against the $10 he earns now, not as the new wage compared with the old. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Find how many dollars the hourly pay goes up, then divide that change by the pay Malik earns now and write the result as a percent.

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.
• 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.
• 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

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

  Reveals a misconception Expected pull: moderate

  Finds the $2 increase and reads it as 2%, as if a change in dollars were a percent, without dividing it by the original $10.

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

  Reveals a misconception Expected pull: weak

  Divides the old pay by the new, 10 ÷ 12 ≈ 0.83, turning the ratio of the wages upside down and reporting 83%.

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

  Reveals partial knowledge Expected pull: strong

  Divides the new pay by the old, 12 ÷ 10 = 1.2, which gives the new wage as a percent of the old, not the increase.

How the questions are analyzed, and the research behind it

Why it is hard

One division gives the constant, and the question itself names the order by asking for millimeters per day. The tempting slip is dividing the smaller number by the larger, 8 by 20, out of habit, which gives a rate in days per millimeter. A plant that grows 20 millimeters in 8 days grows more than 2 millimeters a day, a quick sense check on any result.

1. Something you must already know minor Working it out

   Students must know that the constant is the height divided by the days; the units in the question, millimeters per day, name the order of the division. (Anderson & Krathwohl, 2001)

Strategy: Use the units in the question, millimeters per day, as the instruction: divide the millimeters grown by the number of days.

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

  Reveals partial knowledge Expected pull: strong

  Divides the days by the millimeters, 8 ÷ 20 = 0.4, which gives days per millimeter, the rate turned upside down.

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

  Reveals a surface strategy Expected pull: weak

  Subtracts the days from the millimeters, 20 − 8 = 12, combining the two numbers without thinking about a rate.

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

  Reveals a surface strategy Expected pull: weak

  Gives the 20 millimeters the plant grew over all 8 days, the total growth, rather than the growth for each single day.

How the questions are analyzed, and the research behind it

Why it is hard

Two quick tests decide proportionality, and the tables are built so that each test alone lets a wrong table through. The last table starts at (0, 0) like the right one, while the middle two grow by equal steps like the right one. Only checking that y ÷ x is the same in every row with x not 0 separates the one table whose ratio stays at 3/8.

1. Every choice must be checked moderate Choosing the answer

   Each table has to be tested row by row, and two of them start at (0, 0), so the opening row settles nothing; only y ÷ x in the other rows separates them. (Embretson & Wetzel, 1987)

2. Error-prone arithmetic minor Working it out

   The y-values include 3/2, 7/2 and −1/2, so the ratios mean dividing fractions by 4, such as (3/2) ÷ 4 = 3/8, and comparing them with 3 ÷ 8. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: For each table, divide y by x in every row where x is not zero, and check that the quotient never changes and that y is 0 when x is 0.

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.
• Checks only one number: Tries one value in each choice and takes the first that works, when that value happens to work for a wrong choice too; a second value would rule it out.

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: moderate

  Sees y rise by the same 5/2 each time x rises by 4 and takes the steady change as proportional, missing that y is 1 when x is 0.

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

  Reveals a slip Expected pull: weak

  Spots a 0 in the y column and a steady rise of 1/2, but that 0 belongs to x = 8, while at x = 0 the table gives −1.

• D Checks only one number (Schoenfeld, 1985)

  Reveals partial knowledge Expected pull: strong

  Checks only the row (0, 0), which every proportional table has, or only 12 ÷ 4 = 3, without testing 16 ÷ 8 = 2 in the last row.

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