Practice Test

Why it is hard

Most students know that an instinct is something an animal does without thinking, and the sentence just before the phrase says hoarding is not “a decision to plan.” The only real pitfall is “merely”: a student who skips it can hear praise of a natural talent, and one who thinks only about hoarding may pick a choice about storing too much. Reading the phrase with its contrast makes the answer plain.

1. Hard words minor Understanding the question

   The phrase pairs “merely,” which shrinks what follows, with “instinctual,” and the sentence before it uses “hoard”; a student unsure of any of the three has little to go on. (Perfetti, 2007; Freedle & Kostin, 1993)

2. The right answer is reworded minor Choosing the answer

   The right choice restates the idea as acting “automatically rather than with true intention,” while the wrong ones stay with the sentence’s own subject, finding, gathering and storing food. (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)

Strategy: Ask what the word “merely” adds to “instinctual,” then reread the sentence before the phrase and use its contrast to decide what the author is saying about hoarding.

Traps

• Dictionary meaning, not meaning here: A common meaning of the word that does not fit its use in this text.
• Outside knowledge: A choice is plausible in the real world but not supported by this text.

The wrong answers, classified

• A Wrong tone or attitude (Graesser, Singer & Trabasso, 1994)

  Reveals partial knowledge Expected pull: moderate

  Hears “instinctual” as a natural gift, as in an instinctive athlete, and turns the phrase into praise of animals’ skill, missing that “merely” makes it a put-down.

• B Outside knowledge (King et al., 2004; Recht & Leslie, 1988)

  Reveals a surface strategy Expected pull: weak

  Brings in the idea of instinct as an urge that overrides everything else, but the passage never ranks food gathering above an animal’s other activities.

• C Dictionary meaning, not meaning here (Perfetti, 2007)

  Reveals a surface strategy Expected pull: weak

  Gives “hoard” its everyday sense of piling up too much, though the passage says animals store food so they “will not run out later,” not beyond their needs.

How the questions are analyzed, and the research behind it

Why it is hard

The paragraph is short, plainly worded and about two named girls, so the one singular pronoun stands out once students look for the person each pronoun stands for. Points are lost mainly by students unsure what “vague” means, who suspect a sentence with several pronouns or a plural “they” instead. Tracing each pronoun back to its noun settles the question in seconds.

1. Something you must already know minor Working it out

   Students must know what makes a pronoun vague: “she” in sentence 3 could mean either Eliza or Brianna, while each plural “they” clearly means both girls. (Anderson & Krathwohl, 2001)

Strategy: Circle every pronoun, draw a line from each to the noun it replaces, and look for one where the line could go to more than one person.

Traps

• Fixes what isn't broken: Changes a part of the sentence that was already correct, while the real error stays; or, where the question asks which sentence has an error or should go, picks one that is correct and belongs.

The wrong answers, classified

• A Fixes what isn't broken (Connors & Lunsford, 1988)

  Reveals a misconception Expected pull: moderate

  Sees two pronouns, “their” and “they,” in the first sentence and suspects confusion, though both are plural and plainly refer to Eliza and Brianna together.

• B Fixes what isn't broken (Connors & Lunsford, 1988)

  Reveals a surface strategy Expected pull: weak

  Thinks “The girls” and “they” are too general and wants the names repeated, though both point clearly back to the two girls just named.

• D Fixes what isn't broken (Connors & Lunsford, 1988)

  Reveals a surface strategy Expected pull: weak

  Wonders whether “they” could include the choir members of the sentence before, though “Both girls” at the start of sentence 4 fixes who “they” are.

How the questions are analyzed, and the research behind it

Why it is hard

The three quoted words all make the same point, and the opening comparison with “scaling Mount Everest” makes it again, so the effect is easy to sense. Points are lost on degree: the sentence holding two of the three words also names “several elite runners,” which makes a choice about only the best runners look supported. Students who hold on to “humanly impossible” rule it out, and the choices about rivalry and training fall away once they check that the words describe the goal, not the runners or their preparation.

1. A very close wrong answer minor Choosing the answer

   The choice about only the most skilled runners and the right one both stress how hard the four-minute mile was; they differ on degree, possible for the best runners or thought impossible for anyone. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. Hard words minor Understanding the question

   “Impenetrable” and “feat” carry the tone, and a student who does not know that impenetrable means impossible to break through may miss how absolute the barrier sounded. (Perfetti, 2007; Freedle & Kostin, 1993)

Strategy: Decide what feeling the three quoted words share, then reread paragraph 1 to see what they describe, and keep only a choice that fits both the feeling and the thing described.

Traps

• Twists what the text says: Uses the text's topic and ideas but changes what it says: who did it, how much, when, or why.
• Word match: Repeats striking words or phrases from the text, so it looks supported, but its meaning does not fit the question.
• Wrong part of the text: A choice is true of another paragraph, stanza or speaker than the one asked about.

The wrong answers, classified

• A Twists what the text says (King et al., 2004)

  Reveals partial knowledge Expected pull: moderate

  Sees that the words stress difficulty but scales it down to a feat for the best runners only, though “humanly impossible” says no runner at all was thought able to do it.

• C Word match (Freedle & Kostin, 1993)

  Reveals a surface strategy Expected pull: weak

  Picks up “elite runners aimed to break” from the same sentence and turns it into a rivalry over records, though the three words describe the barrier, not the runners.

• D Wrong part of the text (Kirsch & Mosenthal, 1990)

  Reveals partial knowledge Expected pull: weak

  Carries the intensive interval training of paragraphs 3 and 4 back to paragraph 1, where the words measure the size of the goal, not the effort of preparing for it.

How the questions are analyzed, and the research behind it

Why it is hard

Paragraph 3 states what the training was for, so the task is to match that sentence and ignore details from elsewhere. The rabbits of paragraph 6, who set Bannister’s tempo and save his effort, supply two tempting effects that belong to the race, not the workouts. The two-minute rests sit in the training paragraph itself, so a student skimming for “recover” can mistake part of the method for its result.

1. A very close wrong answer minor Choosing the answer

   The choice about moderating his pace and the right one both concern pace; only the right one matches the training’s stated aim, to “develop endurance and speed.” (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. The right answer is reworded minor Choosing the answer

   The right choice turns “endurance and speed” into stamina and pace, while two wrong ones reuse the passage’s own words, “conserves” effort from paragraph 6 and “recover” from paragraph 3. (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)

Strategy: Find the paragraph that introduces interval training, note what the training was meant to develop, and pick the choice that means the same thing.

Traps

• Wrong part of the text: A choice is true of another paragraph, stanza or speaker than the one asked about.
• Word match: Repeats striking words or phrases from the text, so it looks supported, but its meaning does not fit the question.
• Twists what the text says: Uses the text's topic and ideas but changes what it says: who did it, how much, when, or why.

The wrong answers, classified

• A Twists what the text says (King et al., 2004)

  Reveals partial knowledge Expected pull: moderate

  Sees the sixty-second laps as practice in holding a pace, but the training aimed to build speed and endurance, and in the race his rabbits had to hold his tempo for him.

• B Wrong part of the text (Kirsch & Mosenthal, 1990)

  Reveals a surface strategy Expected pull: weak

  Matches the saving of “about 15 percent of his or her effort,” which paragraph 6 credits to running behind a rabbit in the race, not to interval training.

• D Word match (Freedle & Kostin, 1993)

  Reveals a surface strategy Expected pull: weak

  Spots “recover” in paragraph 3 and picks the choice about recovery time, though the two-minute rest is part of the workout, and nothing says it grew shorter.

What was corrected from the DOE handbook, and why

• Explanation and answer key: The 2026 printable test includes this question (Passage Set 2, Item 6), but the explanations appendix skips it: it explains only five of the six questions on “A Miracle Mile.” The question is question 40 of the DOE’s NYC Guide to the SHSAT for 2022 Admissions (the 2021–2022 guide, page 165), with the same stem and options lettered E–H. The explanation is the guide’s (page 218) and its key is G; both are relettered E–H to A–D here, so the key is C.

How the questions are analyzed, and the research behind it

Why it is hard

The lines move down level by level, molecules, atoms, protons, and then stop at a question, so the speaker’s curiosity is close to the surface. The choice about how the parts work together comes nearest to the right one, helped by the head-note, which says the collider studies how particles “interact.” Students who notice that the lines list what lies beneath what, and never how things interact, settle on curiosity about what matter is made of.

1. Reading between the lines minor Working it out

   The speaker never names a feeling; the curiosity shows in the dash and the question “what, exactly?” that cut off a confident list of layers. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

2. A very close wrong answer minor Choosing the answer

   The choice about how the parts of matter work together and the right one both give the speaker an interest in matter; they part on its object, how parts interact or what matter is made of. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Read lines 5–8 as one sentence, describe in a word or two what the speaker is doing in them, and pick the choice that matches those exact lines.

Traps

• Twists what the text says: Uses the text's topic and ideas but changes what it says: who did it, how much, when, or why.
• Wrong part of the text: A choice is true of another paragraph, stanza or speaker than the one asked about.
• Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.

The wrong answers, classified

• A Twists what the text says (King et al., 2004)

  Reveals partial knowledge Expected pull: moderate

  Keeps the speaker’s interest in matter but shifts it from what lies beneath atoms to how parts interact, the subject the head-note gives the collider, not these lines.

• C Reads too much in (Graesser, Singer & Trabasso, 1994)

  Reveals a surface strategy Expected pull: weak

  Reads the confident list of layers as a case being argued, though the speaker treats them as known and ends on a question, seeking rather than proving.

• D Wrong part of the text (Kirsch & Mosenthal, 1990)

  Reveals partial knowledge Expected pull: weak

  Jumps ahead to stanza 3, where the speaker asks “How do we find it?” and describes tunnels, but lines 5–8 say nothing about tools or ways of studying matter.

How the questions are analyzed, and the research behind it

Why it is hard

The image is built from words about motion, racing cars “flashing along at close to the speed of light,” so most students see its purpose quickly. Each wrong choice seizes one piece of the picture, the cars’ tininess, their shape or the word “light,” and makes it the point. Students who ask what the whole comparison makes them picture, rather than what one word suggests, avoid all three.

1. Figurative or literary language minor Understanding the question

   The comparison sits in parentheses, “like racing cars, only very tiny cars / flashing along at close to the speed of light,” and students must see that racing cars stand for speed. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

Strategy: Name the two things being compared in lines 23–24 and ask what a reader is meant to picture, then choose the purpose that the comparison’s own words support.

Traps

• Taking figurative language literally: A choice reads a figure of speech or image at face value.
• Word match: Repeats striking words or phrases from the text, so it looks supported, but its meaning does not fit the question.
• Dictionary meaning, not meaning here: A common meaning of the word that does not fit its use in this text.

The wrong answers, classified

• A Word match (Freedle & Kostin, 1993)

  Reveals a surface strategy Expected pull: moderate

  Fixes on “very tiny cars” and decides the point is that the particles are too small to see, though racing cars are chosen for their speed, not their size.

• C Taking figurative language literally (Graesser, Singer & Trabasso, 1994)

  Reveals a surface strategy Expected pull: weak

  Takes the comparison at face value, as if particles looked like little cars, when the speaker uses racing to picture how they move, not what shape they have.

• D Dictionary meaning, not meaning here (Perfetti, 2007)

  Reveals a surface strategy Expected pull: weak

  Reads “flashing” and “light” as a glow, like the one in the darkness of lines 27–28, but “the speed of light” measures how fast the particles travel.

How the questions are analyzed, and the research behind it

Why it is hard

Students no longer have to supply the fix, only find the faulty spot, so the question rests on recognizing a comma splice. The error sits in the first sentence behind a long museum name, while the phrases “including Muhammad Ali’s boxing gloves” and “written in 1863” look more suspicious to anyone unsure about phrases set off by commas. Testing each spot for a full sentence on both sides finds it quickly.

1. A rule that doesn't sound wrong moderate Working it out

   A comma alone joins two full clauses at “Institution, the,” and because the long museum name delays the break, the comma reads as a pause rather than a missing period. (Connors & Lunsford, 1988)

2. Complex sentences minor Understanding the question

   The first clause runs 19 words, most of them the museum’s name, so by the time “the museum is already” arrives it sounds like a continuation, not a new sentence. (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)

3. Every choice must be checked minor Choosing the answer

   Each of the four clickable phrases straddles a comma or a period, so students must check, at every one, whether the words on each side could stand as a sentence. (Embretson & Wetzel, 1987)

Strategy: Find the subject and main verb of each sentence and count how many complete thoughts it holds, then check whether the punctuation at each underlined spot suits what comes on either side.

Traps

• Fixes what isn't broken: Changes a part of the sentence that was already correct, while the real error stays; or, where the question asks which sentence has an error or should go, picks one that is correct and belongs.

The wrong answers, classified

• B Fixes what isn't broken (Connors & Lunsford, 1988)

  Reveals a surface strategy Expected pull: weak

  Suspects the break after “year” because two sentences meet there, though a period already separates two complete sentences correctly, each with its own subject and verb.

• C Fixes what isn't broken (Connors & Lunsford, 1988)

  Reveals a misconception Expected pull: moderate

  Believes a phrase opening with “including” after a comma is a fragment, though it correctly attaches examples to “more than 30,000 objects” within a complete sentence.

• D Fixes what isn't broken (Connors & Lunsford, 1988)

  Reveals a misconception Expected pull: moderate

  Thinks “written in 1863” cannot describe the copy without “which was,” though the sentence already has its verb “is” and the participial phrase is correctly set off.

What was corrected from the DOE handbook, and why

• Explanation: The DOE’s explanation of choice 3 calls “including Muhammad Ali’s boxing gloves and a dress sewn by Rosa Parks” a “nonrestrictive appositive phrase.” An appositive renames a noun; this phrase gives examples of the objects, and “including” followed by a noun phrase works as a preposition, so it is a prepositional phrase. The explanation of choice 4 calls “written in 1863 during the presidency of Abraham Lincoln” a “nonrestrictive clause”; it has no subject or finite verb of its own, so it is a participial phrase. The site now says “nonrestrictive prepositional phrase” and “nonrestrictive participial phrase.” Both parts are still correctly constructed, and the answer, “Institution, the,” is unchanged.

How the questions are analyzed, and the research behind it

Why it is hard

The shift to “Then she studies” stands out in a story told in the past tense with “spent,” “did” and “recited,” and finding either half of the right option is enough to choose it. That makes this one of the easiest editing questions. Students who miss it usually trust a half-remembered comma rule, such as a comma before “so,” and stop checking the verb in the same option.

1. A rule that doesn't sound wrong minor Working it out

   Only two nouns, “emotions” and “motivations,” follow “the,” so the comma between them has no job, yet a habit of putting commas before “and” in lists hides it. (Connors & Lunsford, 1988)

2. Every choice must be checked minor Choosing the answer

   The four options each pair a tense change with a comma change, one option per sentence, and an option is right only when both of its edits repair real errors. (Embretson & Wetzel, 1987)

Strategy: Find the tense of the paragraph's first verb, check every underlined verb against it, and then confirm that the comma change in the same choice is needed too.

The wrong answers, classified

• A Verb tense or form (Connors & Lunsford, 1988)

  Reveals a misconception Expected pull: weak

  Thinks “had spent” is needed because the preparing came before the audition, and accepts a comma after “play” that would cut off “at the community theater.”

• B Punctuation (Connors & Lunsford, 1988)

  Reveals a misconception Expected pull: moderate

  Applies the rule of a comma before “so” joining two clauses, missing that “so that” opens a purpose clause with no comma, and accepts the present-tense “does.”

• D Verb tense or form (Connors & Lunsford, 1988)

  Reveals partial knowledge Expected pull: weak

  Notices that the tenses clash but takes the present-tense “studies” as the standard, shifts “recited” to match, and drops the comma that sets off “making slight adjustments.”

How the questions are analyzed, and the research behind it

Why it is hard

Adding the four counts to 20 and writing 8 over 20 is routine, so most students succeed. The losses come from the number line: its decimal labels make it easy to read the 8 pears as 0.8, and a part-to-part fraction such as 8/12 falls between marks and has to be rounded, which should prompt a second look. Simplifying 8/20 to 2/5 and then to 4/10 makes the mark plain.

1. Something you must already know minor Working it out

   The question does not say that a probability is the favorable count over the whole count, here the 8 pears over all 20 pieces of fruit rather than over the other 12. (Anderson & Krathwohl, 2001)

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

   The number line is labeled only in tenths from 0 to 1, so 8/20 has to be rewritten as a decimal before the right mark can be chosen. (Enright, Morley & Sheehan, 2002)

Strategy: Add up all the fruit for the total, write the number of pears over that total, rewrite the fraction in tenths, and select that mark on the line.

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.

The wrong answers, classified

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

  Reveals a surface strategy Expected pull: strong

  Takes the 8 pears straight to the mark labeled 0.8, never dividing by the 20 pieces of fruit in the basket.

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

  Reveals a misconception Expected pull: moderate

  Compares the 8 pears with the 12 other pieces of fruit, 8/12 or about 0.67, a part-to-part ratio, and plots the nearest tenth.

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

  Reveals a slip Expected pull: weak

  Puts the 12 apples, oranges and bananas over the 20 pieces of fruit, plotting the chance of not getting a pear instead of getting one.

How the questions are analyzed, and the research behind it

Why it is hard

One subtraction solves the inequality, so the points are lost in the count that follows. The boundary 16 is the last number the solving produces and is easy to type in, or to count up to; checking that 16 + 7 is 23, which is not less than 23, rules it out. Listing 1 through 15, the positive integers below the boundary, settles the count.

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

   Solving gives x < 16, but the box asks how many positive integers make the inequality true, a count that still has to be made after the solving is done. (Enright, Morley & Sheehan, 2002)

2. Much to keep track of minor Working it out

   Two conditions set the ends of the count at once: the word positive starts it at 1 rather than 0, and the strict < stops it below 16. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Undo the addition to find the boundary for x, then count the positive whole numbers below it, checking the largest one and the boundary in the original inequality.

Traps

• Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).
• 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

• 16 Counting the ends (Brown & Burton, 1978)

  Reveals partial knowledge Expected pull: strong

  Solves to x < 16 and then enters 16, either the boundary itself or a count of 1 through 16, though x = 16 makes x + 7 equal 23, not less than it.

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

  Reveals a misconception Expected pull: weak

  Moves the 7 to the right side without changing its sign, 23 + 7 = 30, and counts the 29 positive integers below 30.

How the questions are analyzed, and the research behind it

Why it is hard

The opening lines are short and concrete, and their tools, measuring tapes and sketchbooks, plainly say why the men came. The wrong choices each add something: men too busy to talk, a hidden motive, or a fascinated speaker, and the last borrows the speaker’s later delight in “swallowing the sun.” Checking each against the rest of the poem, rather than against first impressions of strangers arriving, is what the question rewards.

1. How, not what minor Working it out

   The stem asks what lines 1–2 do in the poem, so students must see the measuring tapes and sketchbooks as setting up the visit’s purpose, not only as a description of the men. (Anderson & Krathwohl, 2001)

2. Evidence in several places minor Working it out

   Ruling out two wrong choices takes later lines: a museum man offers the speaker an idea in lines 32–33, and the speaker leads the men in line 7 because Ma says to. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

Strategy: Say in your own words what lines 1–2 tell you about the men, then reject any choice that depends on something the poem never shows.

Traps

• Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.
• Wrong part of the text: A choice is true of another paragraph, stanza or speaker than the one asked about.

The wrong answers, classified

• A Reads too much in (Graesser, Singer & Trabasso, 1994)

  Reveals partial knowledge Expected pull: weak

  Pictures men absorbed in measuring who ignore the child, though nothing says so and one of them later suggests the mound looks “as if swallowing the sun.”

• C Reads too much in (Graesser, Singer & Trabasso, 1994)

  Reveals a surface strategy Expected pull: weak

  Reads a secret aim into the visit, but lines 1–2 show the men carrying measuring tools in the open, and the poem never hints at any other purpose.

• D Wrong part of the text (Kirsch & Mosenthal, 1990)

  Reveals partial knowledge Expected pull: moderate

  Jumps to line 35, where the speaker likes a museum man’s phrase; in lines 1–2 the men have only arrived, and the speaker guides them because Ma says to.

How the questions are analyzed, and the research behind it

Why it is hard

Both phrases are quoted with their lines, and the sentence that holds them goes on to list the body, the coiled tail and the open mouth, so its subject is what can be seen. The wrong choices slide from seeing to knowing: who knows where the mound is, who wants it hidden, and what it means. “Less obvious” is the slippery phrase, since obvious can also mean easy to understand.

1. Hard words moderate Understanding the question

   “All but” is an idiom meaning almost, so “all but / invisible” means nearly unseen; a student who takes it as the opposite, plainly visible, loses the sense that the humps are hard to see. (Perfetti, 2007; Freedle & Kostin, 1993)

2. Reading between the lines minor Working it out

   Students must keep both phrases about sight and connect them to lines 24–25, where only climbing a tree gives “some idea of the whole thing.” (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

Strategy: Put both phrases into plain words, starting with the idiom “all but,” then test each choice against what lines 16–25 actually describe.

Traps

• Twists what the text says: Uses the text's topic and ideas but changes what it says: who did it, how much, when, or why.
• Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.
• Dictionary meaning, not meaning here: A common meaning of the word that does not fit its use in this text.

The wrong answers, classified

• B Twists what the text says (King et al., 2004)

  Reveals partial knowledge Expected pull: moderate

  Turns “invisible” into “unknown,” helped by Ma sending the speaker to guide the men, but the phrases say how hard the mound is to see, not who knows where it is.

• C Reads too much in (Graesser, Singer & Trabasso, 1994)

  Reveals a surface strategy Expected pull: weak

  Finds a wish for secrecy in the hidden humps, though the speaker leads the museum men straight to the mound and never tries to keep it from anyone.

• D Dictionary meaning, not meaning here (Perfetti, 2007)

  Reveals partial knowledge Expected pull: moderate

  Takes “less obvious” in its sense of harder to understand, but the sentence goes straight on to list the body, the coiled tail and the open mouth, so the word is about sight.

How the questions are analyzed, and the research behind it

Why it is hard

The factoring itself is quick: halving twice and dividing by 3 three times leaves 7. The question is built so that multiplying back proves nothing, since every choice equals 756, and three of them simply stop the factor tree at a composite number. Students who check only the product, or who take an odd factor such as 21 or 27 for a prime, pick a factorization that is unfinished.

1. A very close wrong answer moderate Choosing the answer

   2² · 3² · 21 shares the right answer's 2² and differs only in 3² · 21 against 3³ · 7, and both equal 756, so neither an estimate nor a product check separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. Something you must already know minor Working it out

   Students must know that a prime factorization may hold only primes, and must spot that 21, 9 and 27 are composite even though none of them is even. (Anderson & Krathwohl, 2001)

Strategy: Divide 756 by 2 while the result is even, then by 3 while it divides evenly, and keep going until every factor left is prime; then compare factor by factor, not by product.

Traps

• Stops one step early: A choice is a number from partway through the solution.
• 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 Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: strong

  Divides out 2 twice and 3 twice, then stops at 21, which is 3 × 7, so one composite factor remains in a product that still equals 756.

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

  Reveals a misconception Expected pull: weak

  Splits off a 3 and a 7 but keeps 9 as if it were prime, perhaps because it is odd, though 9 = 3 × 3.

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

  Reveals partial knowledge Expected pull: moderate

  Splits 756 into 4 × 189 and 189 into 7 × 27, then stops, not seeing that 27 is 3 × 3 × 3.

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

One division finds the boundary, so the question is really about direction. Two things pull students the wrong way: the −12 on the left makes the inequality look like one that must be flipped, and the result −2 > x has x on the right, where it is easy to read backward. Testing x = −3, which gives −12 > −18, shows that values below −2 work.

1. A very close wrong answer moderate Choosing the answer

   x < −2 and x > −2 share the boundary −2 and differ only in direction, so finding the right number settles nothing until the direction is read correctly. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. Something you must already know minor Working it out

   Students must know that the symbol flips only when both sides are multiplied or divided by a negative number, and that −2 > x says the same as x < −2. (Anderson & Krathwohl, 2001)

Strategy: Divide both sides by 6, asking whether you divided by a negative number, then rewrite the result with x on the left so that the symbol still opens toward the larger side.

Traps

• Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
• 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 surface strategy Expected pull: weak

  Adds 6 to −12 instead of dividing by 6, getting −6, and also reverses the direction, so both the boundary and the symbol are wrong.

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

  Reveals a misconception Expected pull: weak

  Undoes the 6 by adding it to −12 rather than dividing, as if 6x meant x − 6, getting −6 > x and so x < −6.

• D Negatives and direction (Brown & Burton, 1978)

  Reveals a misconception Expected pull: strong

  Finds −2 > x but flips the symbol because −12 is negative, or reads −2 > x as x > −2, though dividing by a positive 6 keeps the direction.

How the questions are analyzed, and the research behind it

Why it is hard

Three of the four right fractions, 3/4, 75/100 and 6/8, are familiar forms of 0.75, so the item comes down to the fourth. Students who stop after the familiar ones must decide between 15/20, which needs dividing by 5 to recognize, and 8/10, which sits close to 0.75 on a number line. Dividing each numerator by its denominator, or scaling 3/4 up by 5, settles it.

1. Every choice must be checked moderate Choosing the answer

   Six fractions with six different denominators have to be tested one by one, by dividing or by simplifying, since 15/20 and 8/10 cannot be judged by how they look. (Embretson & Wetzel, 1987)

2. A very close wrong answer minor Choosing the answer

   8/10 is 0.8, only 0.05 above 0.75, and it follows 6/8 by adding 2 to the top and the bottom, so it looks like one more member of the family. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

3. Much to keep track of minor Working it out

   Four of the six must be chosen and all four must be right for the item to count, so each verdict has to be kept while the remaining fractions are tested. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Turn each choice into a decimal by dividing its numerator by its denominator, or reduce it to lowest terms, and keep only the ones whose value is exactly the decimal given.

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

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

  Reveals a surface strategy Expected pull: weak

  No usual working links 2/5 to 0.75, since 2 ÷ 5 = 0.4, so it mostly catches students filling the fourth slot by guessing.

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

  Reveals a misconception Expected pull: strong

  Builds 8/10 from 6/8 by adding 2 to the top and the bottom, as if equal additions kept a fraction's value, or accepts 0.8 as close enough.

How the questions are analyzed, and the research behind it

Why it is hard

The numbers are small, but the speed is per hour and the answer is wanted in minutes, so students have to choose a route through the units. Finding the minutes per mile first, 4 at 15 miles an hour, turns the rest into a single multiplication. The wrong choices come from using a number in the stem as the answer, from mixing the 60 minutes with the 6 miles, or from rounding 0.4 hour to a half hour.

1. Something you must already know minor Working it out

   Nothing on the page says that time is distance divided by speed or that an hour has 60 minutes, and the bold word minutes requires both. (Anderson & Krathwohl, 2001)

2. Error-prone arithmetic minor Working it out

   6 ÷ 15 is less than 1, 0.4 hour, and turning it into minutes means 0.4 × 60; the cleaner route, 60 ÷ 15 = 4 minutes a mile, avoids decimals. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Find how many minutes Maria needs to ride one mile at 15 miles per hour, then multiply by the number of miles to school.

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.
• 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

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

  Reveals a surface strategy Expected pull: weak

  Divides the 60 minutes in an hour by the 6 miles, 60 ÷ 6 = 10, as if the ride took a full hour, and ignores the 15 mph speed.

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

  Reveals a surface strategy Expected pull: moderate

  Takes the 15 in 15 miles per hour as the number of minutes, a number copied from the question without any calculation.

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

  Reveals partial knowledge Expected pull: moderate

  Sees that 6 miles is a little under half of 15, rounds the time of 0.4 hour up to half an hour, and gives 30 minutes.

How the questions are analyzed, and the research behind it

Why it is hard

The plan is clear from the stem: count the stamps, then take away what the six new albums hold, so the question tests careful arithmetic. Its choices punish the estimating that saves time elsewhere, since rounding either factor shifts the result by only a few tens, onto another listed number. Students who skip the album counts altogether and compare one album with one album find 198 waiting.

1. A very close wrong answer moderate Choosing the answer

   612, 620 and 632 lie within 20 of one another, so rounding 576 and 378 to friendly numbers before multiplying lands on a wrong choice; only exact products separate them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. Error-prone arithmetic minor Working it out

   Two products of three-digit numbers, 5 × 576 = 2,880 and 6 × 378 = 2,268, must be done by hand and then subtracted with regrouping. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Multiply to find the total number of stamps, multiply again to find how many the six new albums hold, subtract, and check the last digit of each product.

Traps

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

  Subtracts 378 from 576, comparing one old album with one new album, and ignores that there are 5 old albums and 6 new ones.

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

  Reveals partial knowledge Expected pull: strong

  Rounds 576 to 580 and 378 to 380 before multiplying, so 2,900 − 2,280 = 620, an estimate rather than the exact number left over.

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

  Reveals partial knowledge Expected pull: weak

  Rounds only 576 up to 580, getting 2,900 stamps, then subtracts the exact 2,268, so the count comes out 20 too high.

How the questions are analyzed, and the research behind it

Why it is hard

The steps spell out how to round 135.28 to the nearest tenth, but nothing says so, and they have to be followed exactly. The decisive moment is step 2, where adding 0.5 pushes 1,352.8 past 1,353; skip it and the digits are simply cut, not rounded. Students who recognize the routine can check the result directly, since 135.28 lies closer to 135.3 than to 135.2.

1. A very close wrong answer moderate Choosing the answer

   135.3 and 135.2 differ only in the tenths digit, and 135 shares their whole-number part, so a rough sense of the result's size rules out nothing. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

2. Many steps minor Working it out

   Four listed steps move the decimal point twice, and doing them in another order, or skipping one, still produces a believable number near 135. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

3. Error-prone arithmetic minor Working it out

   Adding 0.5 to 1,352.8 carries into the ones place, since 8 tenths and 5 tenths make 1.3, which turns 1,352 into 1,353 before the decimals are dropped. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Write down the number after each step instead of doing them in your head, and carry out the steps exactly in the order listed.

Traps

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

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

  Reveals a slip Expected pull: weak

  Skips step 1, so adds 0.5 to 135.28, drops the decimals to get 135, and then divides by 10 to reach 13.5.

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

  Reveals a slip Expected pull: moderate

  Divides by 10 before dropping the digits after the decimal point, so 1,353.3 becomes 135.33 and then 135, doing steps 3 and 4 in reverse order.

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

  Reveals partial knowledge Expected pull: strong

  Leaves out step 2, or adds 0.5 without carrying, so 1,352 is kept and 135.2 results: 135.28 cut down instead of rounded up.

How the questions are analyzed, and the research behind it

Why it is hard

The arithmetic is two one-step sign calculations and a division by 6. The danger is reading the top and the bottom as alike: they differ in a single sign, and treating them the same way gives 1 or −1. Students who rewrite 3 − (−3) as 3 + 3 and 3 + (−3) as 3 − 3 see 0 over 6 at once, and zero divided by a nonzero number is zero.

1. Error-prone arithmetic minor Working it out

   The top adds −3 and the bottom subtracts −3, signs that look nearly the same but give 0 and 6, so each has to be rewritten before anything is divided. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Simplify the top and the bottom separately, rewriting the subtraction of a negative as an addition, and only then divide the top by the bottom.

Traps

• Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)

The wrong answers, classified

• A Negatives and direction (Brown & Burton, 1978)

  Reveals a misconception Expected pull: moderate

  Adds the sizes on top and keeps the minus, taking 3 + (−3) as −6, so the fraction becomes −6/6 = −1.

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

  Reveals a surface strategy Expected pull: weak

  No sign rule gives a top of 1, so 1/6 mostly draws students who find the bottom, 6, and are unsure what a top of 0 means.

• D Negatives and direction (Brown & Burton, 1978)

  Reveals a slip Expected pull: moderate

  Treats the top like the bottom, adding 3 and 3 in both places, so the fraction becomes 6/6 = 1.

What was corrected from the DOE handbook, and why

• Explanation: The DOE’s explanation says “3 − (−3) is equivalent of adding 3 to 3”, without the article that the same sentence uses in “3 + (−3) is the equivalent of subtracting 3 from 3”. The site now says “is the equivalent of adding 3 to 3”.
• Explanation: The DOE’s explanation says “The zero property of division states that zero divided by any number equals zero.” Zero divided by zero is undefined, not zero, so the rule holds only when the divisor is not zero. The site now says “zero divided by any nonzero number equals zero”. Here the divisor is 6, so 0/6 = 0, as before.

How the questions are analyzed, and the research behind it

Why it is hard

The numbers divide evenly, since 16/48 is 1/3, so the work is in the setup. Writing the two ratios in the order the sentence gives them makes the apples a third of the oranges; writing either one upside down makes them three times the oranges. A quick sense check helps: cherries are fewer than walnuts, so apples must be fewer than oranges.

1. Turning words into math minor Understanding the question

   The sentence pairs apples with oranges and cherries with walnuts in the same order, so a/6 = 16/48; reading either pair the other way turns one ratio upside down. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

2. Much to keep track of minor Working it out

   Four kinds of fruit are named and three counts given, and the unknown apples must be matched with the smaller count, the cherries, not the larger, the walnuts. (Sweller, 1988; Carpenter, Just & Shell, 1990)

Strategy: Write apples over oranges equal to cherries over walnuts, simplify the known ratio, and scale it to match the 6 oranges.

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).
• Stops one step early: A choice is a number from partway through the solution.
• 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 Stops one step early (King et al., 2004; Gierl et al., 2017)

  Reveals partial knowledge Expected pull: moderate

  Divides the 48 walnuts by the 16 cherries to get the factor 3 and stops before using it to divide the 6 oranges.

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

  Reveals a misconception Expected pull: strong

  Turns one ratio upside down, setting apples over oranges equal to 48 over 16, so the apples come out as 3 × 6 = 18.

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

  Reveals a misconception Expected pull: weak

  Reasons with differences instead of ratios, taking the gap of 32 between walnuts and cherries and adding it to the 6 oranges.

How the questions are analyzed, and the research behind it