Six SHSAT Questions

3 English and 3 math

Here are 6 random questions to try. Takers earning below 75% are unlikely to receive an offer. You have 1 minute 48 seconds per question on the real exam.

Question 1 of 6

2024 Test A English, question 46
Passage Excerpt from The Highest Tide

Excerpt from The Highest Tide

by Jim Lynch

The narrator’s summer job is collecting marine specimens in Puget Sound, Washington. In this excerpt, he is looking for specimens in the mudflats before daybreak. Mudflats, or flats, are coastal landforms created when sediment and silt from tides are deposited as the tides recede.

I rounded the oyster beds, to the far side. . . . It was low tide by then, and I saw the water hesitating at its apex, neither leaving nor returning, patiently waiting for the gravitational gears to shift. Dozens of anxious clams started squirting in unison like they did whenever vibrating grains of sand warned them predators were approaching. I stopped and waited with them, to actually see the moment when the tide started returning with its invisible buffet of plankton for the clams, oysters, mussels and other filter feeders. It was right then, ankle deep in the Sound, feet numbing, eyes relaxed, that I saw the nudibranch.1

In all my time on the flats I’d never seen one before. I’d read about them, sure. I’d handled them at aquariums but never in the wild, and I’d never even seen a photo of one this stunning. It was just three inches long but with dozens of fluorescent, orange-tipped hornlike plumes jutting from the back of its see-through body that appeared to be lit from within.

Nudibranchs are often called the butterflies of the sea, but even that understates their dazzle. Almost everything else in the northern Pacific is dressed to blend with pale surroundings. Nudibranchs don’t bother, in part because they taste so lousy they don’t need camouflage to survive. But also, I decided right then, because their beauty is so startling it earns them a free pass, the same way everyday life brakes for peacocks, parade floats. . . .

The dark mudflats loomed like wet, flattened dunes stretching deep into Skookumchuck Bay in front of our house. From a distance, they looked too barren to support sea life. Up close, they still did, unless you knew where to find the hearty clams, worms and tiny creatures that flourish in mud. . . . I’m not sure why I decided to take a look. It was still an hour before sunrise, and I knew exactly what the bars2 looked like in the moonlight, but for some reason, I couldn’t resist.

I heard it long before I saw it. It was an exhale, a release of sorts, and I instantly wondered if a whale was stranded again. We had a young minke3 stuck out there two summers prior, and it made similar noises until the tide rose high enough for rescuers to help free it. . . . I looked for a hulking silhouette but couldn’t find one. I waited, but there were no more sounds. Still, I went toward what I thought I’d heard, avoiding stepping into the mud until I had to. I knew the flats well enough to know I could get stuck just about anywhere. The general rule was you didn’t venture out past the shells and gravel with an incoming tide. I sank up to my knees twice, and numbing water filled my boots.

. . . I kept stepping toward the one sound I’d heard, a growing part of me hoping I’d find nothing at all. When I stopped to rest and yank up my socks, my headlamp crossed it. My first thought? A giant octopus.

Puget Sound has some of the biggest octopi in the world. They often balloon to a hundred pounds. Even the great Jacques Cousteau4 himself came to study them. But when I saw the long tubular shape of its upper body and the tangle of tentacles below it, I knew it was more than an octopus. I came closer, within fifty feet, close enough to see its large cylindrical siphon5 quiver. I couldn’t tell if it was making any sounds at that point, because it was impossible to hear anything over the blood in my ears. . . .

The creature’s body came to a triangular point above narrow fins that lay flat on the mud like wings, but it was hard to be sure exactly where it all began or ended, or how long its tentacles truly were, because I was afraid to pry my eyes off its jumble of arms for more than half a second. I didn’t know whether I was within reach, and its arms were as big around as my ankle and lined with suckers the size of half-dollars. If they even twitched I would have run. So, I was looking at it and not looking at it while my heart spangled my vision. I saw fragments, pieces, and tried to fuse them in my mind but couldn’t be certain of the whole. I knew what it had to be, but I wouldn’t allow myself to even think the two words. Then I gradually realized the dark shiny disc in the middle of the rubbery mass was too perfectly round to be mud or a reflection.

It was too late to smother my scream. Its eye was the size of a hubcap.6

From THE HIGHEST TIDE by Jim Lynch. Copyright © 2005 by Jim Lynch. Used by permission of Bloomsbury USA. All rights reserved.

1nudibranch: type of sea slug

2bars: sandbars

3minke: species of small whale

4Jacques Cousteau: renowned French oceanographer

5siphon: tube-like organ that is used for drawing in or removing fluids

6hubcap: removable cap over the end of a wheel axle

Question

How does the setting affect the plot of the excerpt?

  • AThe darkness of the mudflats before sunrise causes the narrator to pay close attention to his surroundings.
  • BThe rugged terrain of the mudflats makes it more difficult for the narrator to do his job.
  • CThe slow return of the tide along the mudflats forces the narrator to be patient during his journey.
  • DThe barren appearance of the mudflats creates a sense of mystery that unsettles the narrator.
Show the answer

Correct answer: A

The question asks how the setting affects the plot of the excerpt.

A. CORRECT. In paragraph 4, the narrator says that it is an hour before sunrise. It is difficult to see because the mudflats are dark, so the narrator, while familiar with the terrain of the mudflats, is cautious.

B. Incorrect. While the terrain of the mudflats is challenging because of the “wet, flattened dunes” (paragraph 4), the narrator is familiar with the terrain and moves carefully. In addition, the excerpt focuses on the narrator’s curiosity and enjoyment as he explores the mudflats, not on his work collecting specimens.

C. Incorrect. In paragraph 1, the narrator watches the water “hesitating at its apex” and “patiently waiting for the gravitational gears to shift.” This illustrates that the narrator is already familiar with the changes in his surroundings, that the tide is slow to move in, and that he can move carefully through the mudflats without the threat of rising water.

D. Incorrect. While the narrator is on the mudflats for the entire excerpt, the phrase “eyes relaxed” in paragraph 1 indicates that the narrator feels comfortable there. The narrator does not show fear until paragraph 6, when a growing fear about the unknown creature arises.

Question analysis

Set on the mudflats of Puget Sound before daybreak, this excerpt from the novel The Highest Tide is the basis for a question about how setting affects plot. Students must link features of the place and time to what the narrator does across the whole excerpt, and each choice pairs a real feature of the setting with an effect that may or may not be in the text.

NYS standards
  • 7R3 plot elements and ideas
  • 7R1 textual evidence and inferences
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Analyzing how setting shapes plot is a grade 7 literary standard, but the excerpt never states the link: the effect must be inferred from scattered details, and every choice names a genuine feature of the flats, which is beyond typical work.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Evidence
Answered from the whole text
Reading level
Passage: about grade 6, 902 words; question and choices: about grade 8, 77 words.
Corrected from the DOE handbook
  • A statement in the explanation that contradicted the excerpt’s introduction was corrected.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Every choice names a real feature of the mudflats, so the question turns on which feature the excerpt actually shows shaping events. The darkness never gets a sentence of its own; it is felt in a sound heard before anything is seen, a silhouette that cannot be found and a headlamp beam. The barren look of the flats is the strongest pull, because the scene is eerie, but the narrator knows life hides in the mud and his unease comes from the creature.

  1. Reading between the lines major Working it out

    The excerpt never says the darkness changes anything; the effect must be assembled from the hour before sunrise, the sound heard “long before I saw it”, the missing silhouette and the headlamp that finally finds the creature. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

  2. The right answer is reworded moderate Choosing the answer

    The right choice uses no words from the text, while the wrong ones echo it: the “barren” look of the flats, the water “patiently waiting”, the narrator's summer job. (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)

  3. Evidence in several places moderate Working it out

    The darkness is established in paragraph 4 and felt in paragraphs 5, 6 and 8, so no single passage shows the setting at work. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

Strategy: Collect the details of time, place and conditions across the excerpt, then for each choice ask whether the text shows that feature of the setting actually changing what the narrator does.

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.

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: In rejecting choice B, the handbook says “the excerpt shows that the narrator explores the mudflats for curiosity and enjoyment, not work.” But the introduction says the narrator’s summer job is collecting marine specimens and that “in this excerpt, he is looking for specimens in the mudflats before daybreak.” The explanation now says the excerpt focuses on his curiosity and enjoyment as he explores the mudflats, not on his work collecting specimens. That keeps the handbook’s point (the excerpt is not about the terrain making his job harder) without contradicting the introduction.

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Know what setting and plot mean

    The setting is where and when a story happens. The plot is the chain of events. This asks which part of the setting changes what happens or what the narrator does.

  2. Find the time of day

    Paragraph 4 says, “It was still an hour before sunrise.” The narrator is out on the mudflats in the dark. The only light comes from the moon and his headlamp.

  3. See how the dark shapes events

    Because it is dark, he must rely on sounds and small clues. He says, “I heard it long before I saw it.” He looks for “a hulking silhouette,” a dark outline, but can’t find one. He spots the creature only when “my headlamp crossed it.” Then he stares hard, trying to make out its shape. The darkness keeps him alert and watching closely.

  4. Check each choice

    • A fits: the dark makes him pay close attention to everything around him.
    • B is wrong because the excerpt follows a strange sound and never shows the ground making his job harder.
    • C is wrong because in paragraph 1 he chooses to stop and watch the tide, and nothing forces him.
    • D is wrong because he knows life hides in the mud, and what scares him is the creature.

    Watch out: D feels right because the scene is eerie. But the flats’ empty look never troubles him.

  5. Choose the answer

    The darkness before sunrise is what keeps the narrator listening and looking so carefully. So the answer is A.

Why this answer and not the closest wrong one

D comes closest, because the dark flats do feel spooky. But the narrator says they only look “too barren to support sea life” from far away, and he knows better. His fear comes from the creature. The darkness, though, shapes every step: he hears before he sees and must watch closely.

The original DOE question
Question 46 as printed on page 66 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 66, question 46. Open page 66 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D). This is the DOE’s version, before the corrections listed in the analysis.
The passage “Excerpt from The Highest Tide” as printed (pages 64–65)The passage “Excerpt from The Highest Tide” as printed on page 64 of the DOE handbook.The passage “Excerpt from The Highest Tide” as printed on page 65 of the DOE handbook.

Question 2 of 6

2026 Practice Test C English, question 11
Passage Excerpt from “Niagara Falls”

Excerpt from “Niagara Falls”

by Rupert Brooke

The real secret of the beauty and terror of the Falls is not their height or width, but the feeling of colossal power and of unintelligible disaster caused by the plunge of that vast body of water. If that were taken away, there would be little visible change, but the heart would be gone.

The American Falls do not inspire this feeling in the same way as the Canadian. It is because they are less in volume, and because the water does not fall so much into one place. By comparison their beauty is almost delicate and fragile. They are extraordinarily level, one long curtain of lacework and woven foam. Seen from opposite, when the sun is on them, they are blindingly white, and the clouds of spray show dark against them. With both Falls the colour of the water is the ever-altering wonder. Greens and blues, purples and whites, melt into one another, fade, and come again, and change with the changing sun. Sometimes they are as richly diaphanous1^1 as a precious stone, and glow from within with a deep, inexplicable light. Sometimes the white intricacies of dropping foam become opaque and creamy. And always there are the rainbows. If you come suddenly upon the Falls from above, a great double rainbow, very vivid, spanning the extent of spray from top to bottom, is the first thing you see. If you wander along the cliff opposite, a bow springs into being in the American Falls, accompanies you courteously on your walk, dwindles and dies as the mist ends, and awakens again as you reach the Canadian tumult. And the bold traveller who attempts the trip under the American Falls sees, when he dare open his eyes to anything, tiny baby rainbows, some four or five yards in span, leaping from rock to rock among the foam, and gambolling beside him, barely out of hand’s reach, as he goes. One I saw in that place was a complete circle, such as I have never seen before, and so near that I could put my foot on it. It is a terrifying journey, beneath and behind the Falls. The senses are battered and bewildered by the thunder of the water and the assault of wind and spray; or rather, the sound is not of falling water, but merely of falling; a noise of unspecified ruin. So, if you are close behind the endless clamour, the sight cannot recognise liquid in the masses that hurl past. You are dimly and pitifully aware that sheets of light and darkness are falling in great curves in front of you. Dull omnipresent foam washes the face. Farther away, in the roar and hissing, clouds of spray seem literally to slide down some invisible plane of air.

Beyond the foot of the Falls the river is like a slipping floor of marble, green with veins of dirty white, made by the scum that was foam. It slides very quietly and slowly down for a mile or two, sullenly exhausted. Then it turns to a dull sage green, and hurries more swiftly, smooth and ominous. As the walls of the ravine close in, trouble stirs, and the waters boil and eddy. These are the lower rapids, a sight more terrifying than the Falls, because less intelligible. Close in its bands of rock the river surges tumultuously forward, writhing and leaping as if inspired by a demon. It is pressed by the straits into a visibly convex form. Great planes of water slide past. Sometimes it is thrown up into a pinnacle of foam higher than a house, or leaps with incredible speed from the crest of one vast wave to another, along the shining curve between, like the spring of a wild beast. Its motion continually suggests muscular action. The power manifest in these rapids moves one with a different sense of awe and terror from that of the Falls. Here the inhuman life and strength are spontaneous, active, almost resolute. . . . A place of fear.

One is drawn back, strangely, to a contemplation of the Falls, at every hour, and especially by night, when the cloud of spray becomes an immense visible ghost, straining and wavering high above the river, white and pathetic and translucent. The Victorian lies very close below the surface in every man. There one can sit and let great cloudy thoughts of destiny and the passage of empires drift through the mind; for such dreams are at home by Niagara. I could not get out of my mind the thought of a friend, who said that the rainbows over the Falls were like the arts and beauty and goodness, with regard to the stream of life—caused by it, thrown upon its spray, but unable to stay or direct or affect it, and ceasing when it ceased. In all comparisons that rise in the heart, the river, with its multitudinous waves and its single current, likens itself to a life, whether of an individual or of a community. A man’s life is of many flashing moments, and yet one stream; a nation’s flows through all its citizens, and yet is more than they. In such places, one is aware, with an almost insupportable and yet comforting certitude, that both men and nations are hurried onwards to their ruin or ending as inevitably as this dark flood. Some go down to it unreluctant, and meet it, like the river, not without nobility. And as incessant, as inevitable, and as unavailing as the spray that hangs over the Falls, is the white cloud of human crying. . . . With some such thoughts does the platitudinous2^2 heart win from the confusion and thunder of a Niagara peace that the quietest plains or most stable hills can never give.

From LETTERS FROM AMERICA by Rupert Brooke—Public Domain

1^1diaphanous: sheer, translucent

2^2platitudinous: clichéd, common

Question

In paragraph 4, the idea that human life and history ultimately end in the same way is illustrated mainly through

  • Athe discussion of how the rainbows visible in the Falls are like the art and beauty created by humankind.
  • Bthe comparison between the movement of water in the Falls and the human experience.
  • Cthe inclusion of details that show that every observer’s experience with the Falls is different.
  • Dthe acknowledgment that contemplating the Falls at night sparks an awareness of humankind’s destiny.
Show the answer

Correct answer: B

The question asks how paragraph 4 of the excerpt mainly illustrates the idea that human life and history ultimately end in the same way.

A. Incorrect. In paragraph 4, the author compares the rainbows visible in the Falls to “the arts and beauty,” which share the rainbows’ dependence on “the stream of life”—that is, they are “caused by it, thrown upon its spray, but unable to stay or direct or affect it, and ceasing when it ceased.” This comparison does not illustrate the idea that human life and history ultimately end in the same way but rather that art and beauty, like human beings, are unable to “stay or direct or affect” their ultimate fate.

B. CORRECT. The author concludes in paragraph 4 that the movement of the Falls can be compared to people and history when he writes, “both men and nations are hurried onwards to their ruin or ending as inevitably as this dark flood.” This comparison illustrates the idea that human life and history ultimately end in the same way.

C. Incorrect. Although in paragraph 4 the author states that “One is drawn back, strangely, to a contemplation of the Falls” and provides an account of “a friend, who said that the rainbows over the Falls were like the arts and beauty and goodness, with regard to the stream of life,” the author does not provide details to show that every observer’s experience with the Falls is different. Rather, the author explains that “the Victorian lies very close below the surface in every man” (paragraph 4).

D. Incorrect. While the setting at night does provide a place of contemplation, the author notes the “dark flood” as the overall illustration of the human experience.

Question analysis

In this question on the closing paragraph of Rupert Brooke’s “Niagara Falls,” students identify the main way the author illustrates the idea that human life and history ultimately end in the same way. It is a lightly reworded version of a 2018 Test B question. The paragraph is long and reflective, mixing several images with abstract thoughts, so students must decide which of them actually illustrates this particular idea.

NYS standards
  • 7R1 textual evidence and inferences
  • 7L5 figurative language and word relationships
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Most grade 7 work asks how an idea is shown in plain modern prose. This paragraph is a century-old reflection in long sentences with words like “certitude” and “unavailing,” and several of its images could each seem to carry the idea.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Evidence
Answered from one paragraph or stanza
Reading level
Passage: about grade 9, 987 words; question and choices: about grade 11, 81 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Paragraph 4 offers three pictures, the spray as a night ghost, the friend’s rainbows and the river as a life, and only one of them shows people and nations meeting an end. The 2026 stem’s “ultimately end” makes the rainbows’ “ceasing when it ceased” more tempting, and the night-time “thoughts of destiny” sound like the idea itself. Students who ask what the paragraph uses to show the idea, rather than where it mentions endings or destiny, land on the dark flood.

  1. Complex sentences major Understanding the question

    The idea sits inside a hedged sentence, “one is aware, with an almost insupportable and yet comforting certitude, that both men and nations are hurried onwards to their ruin or ending,” and the comparison comes last: “as inevitably as this dark flood.” (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)

  2. The right answer is reworded moderate Choosing the answer

    The key sums up the paragraph’s river-and-life likeness in its own words, “movement of water” and “human experience,” while the other choices borrow the paragraph’s vocabulary: “rainbows,” “art and beauty,” “night” and “destiny.” (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)

  3. Hard words minor Understanding the question

    “Multitudinous,” “certitude,” “insupportable” and “unavailing” all sit in the lines that carry the idea, and the footnoted “platitudinous” closes the paragraph, so the comparison has to be read through them. (Perfetti, 2007; Freedle & Kostin, 1993)

Strategy: Find the sentence in paragraph 4 that actually states the idea about life and history ending, then ask how that very sentence makes its point and match a choice to it.

Traps

  • Supports only part of the claim: In a question about evidence: a line that backs one half of the claim but not the other, or a related point rather than the claim itself.
  • Outside knowledge: A choice is plausible in the real world but not supported by this text.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Figure out what the question asks

    This asks how an idea is shown in paragraph 4. The idea is that people’s lives and whole nations end in the same way: they all come to an end. You need the picture the author mainly uses to show that idea. Paragraph 4 has several pictures, so find the one tied to this idea.

  2. Find the sentence with the idea

    Paragraph 4 says “both men and nations are hurried onwards to their ruin or ending as inevitably as this dark flood.” Men means people. Nations means countries and their history. Inevitably means in a way that can’t be stopped. The “dark flood” is the river’s dark water rushing over the Falls.

  3. See how the comparison works

    Just before, the author says “the river, with its multitudinous waves and its single current, likens itself to a life.” Multitudinous means very many. A current is water flowing one way. Likens itself to means seems like. A life has many moments but moves one way, like the river. And just as the water must rush on, people and nations rush toward their end. So the idea is shown by comparing moving water with human life.

  4. Check each choice

    • A does not fit. A friend compares the rainbows to “the arts and beauty and goodness.” That picture is about art, not about lives and nations ending.
    • B fits. The river’s movement stands for human lives and history.
    • C does not fit. The paragraph never says that each visitor sees the Falls differently.
    • D does not fit. The night view and the “thoughts of destiny” tell when and where the author thinks. They do not show how lives end.

    Watch out: D is tempting because destiny, or what happens in the end, sounds like the idea itself. But the question asks what illustrates the idea, and that is the river.

  5. Choose the answer

    The author pictures people and nations rushing to their end like the river’s water. So the answer is B, “the comparison between the movement of water in the Falls and the human experience.”

Why this answer and not the closest wrong one

D is the closest wrong choice. Paragraph 4 really does say the Falls at night stir “great cloudy thoughts of destiny.” But that only tells you the author was thinking about such things. The picture that shows lives and nations ending is the river. They are “hurried onwards to their ruin or ending as inevitably as this dark flood.”

The original DOE question
Question 11 as shown on the DOE’s digital test, page 19 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test C English, page 19, question 11.
The passage “Excerpt from “Niagara Falls”” as printed (pages 134–135)The passage “Excerpt from “Niagara Falls”” as printed on page 134 of the DOE handbook.The passage “Excerpt from “Niagara Falls”” as printed on page 135 of the DOE handbook.

Question 3 of 6

2026 Practice Test B English, question 22
Passage Excerpt from In Search of the Unknown

Excerpt from In Search of the Unknown

by Robert W. Chambers

It was at that time the policy of the trustees and officers of the Zoological Gardens neither to employ collectors nor to send out expeditions in search of specimens. The society decided to depend upon voluntary contributions, and I was always busy, part of the day, in dictating answers to correspondents who wrote offering their services as hunters of big game, collectors of all sorts of fauna, trappers, snarers, and also to those who offered specimens for sale, usually at exorbitant rates.

To the proprietors of . . . mangy lynxes, moth-eaten coyotes, and dancing bears I returned courteous but uncompromising refusals—of course, first submitting all such letters, together with my replies, to Professor Farrago.

One day towards the end of May, however, just as I was leaving Bronx Park to return to town, Professor Lesard, of the reptilian department, called out to me that Professor Farrago wanted to see me a moment; so I . . . retraced my steps to the temporary, wooden building occupied by Professor Farrago, general superintendent of the Zoological Gardens. The professor, who was sitting at his desk before a pile of letters and replies submitted for approval by me, pushed his glasses down and looked over them at me with a whimsical smile that suggested amusement, impatience, annoyance, and perhaps a faint trace of apology.

“Now, here’s a letter,” he said, with a deliberate gesture towards a sheet of paper impaled on a file—“a letter that I suppose you remember.” He disengaged the sheet of paper and handed it to me.

“Oh yes,” I replied, with a shrug; “of course the man is mistaken—or—”

“Or what?” demanded Professor Farrago, tranquilly, wiping his glasses.

“—Or a liar,” I replied.

After a silence he leaned back in his chair and bade me read the letter to him again, and I did so with a contemptuous tolerance for the writer, who must have been either a very innocent victim or a very stupid swindler. I said as much to Professor Farrago, but, to my surprise, he appeared to waver.

“I suppose,” he said, with his near-sighted, embarrassed smile, “that nine hundred and ninety-nine men in a thousand would throw that letter aside and condemn the writer as a liar or a fool?”

“In my opinion,” said I, “he’s one or the other.”

“He isn’t—in mine,” said the professor, placidly.

“What!” I exclaimed. “Here is a man living all alone on a strip of rock and sand between the wilderness and the sea, who wants you to send somebody to take charge of a bird that doesn’t exist!”

“How do you know,” asked Professor Farrago, “that the bird in question does not exist?”

“It is generally accepted,” I replied, sarcastically, “that the great auk has been extinct for years. Therefore I may be pardoned for doubting that our correspondent possesses a pair of them alive.”

“Oh, you young fellows,” said the professor, smiling wearily, “you embark on a theory for destinations that don’t exist.”

He leaned back in his chair, his amused eyes searching space for the imagery that made him smile.

“Like swimming squirrels, you navigate with the help of Heaven and a stiff breeze, but you never land where you hope to—do you?”

Rather red in the face, I said: “Don’t you believe the great auk to be extinct?”

“Audubon saw the great auk.”

“Who has seen a single specimen since?”

“Nobody—except our correspondent here,” he replied, laughing.

I laughed, too, considering the interview at an end, but the professor went on, coolly:

“Whatever it is that our correspondent has—and I am daring to believe that it is the great auk itself—I want you to secure it for the society.”

When my astonishment subsided my first conscious sentiment was one of pity. Clearly, Professor Farrago was on the verge of dotage—ah, what a loss to the world!

I believe now that Professor Farrago perfectly interpreted my thoughts, but he betrayed neither resentment nor impatience. I drew a chair up beside his desk—there was nothing to do but to obey, and this fool’s errand was none of my conceiving.

Together we made out a list of articles necessary for me and itemized the expenses I might incur, and I set a date for my return, allowing no margin for a successful termination to the expedition.

“Never mind that,” said the professor. “What I want you to do is to get those birds here safely. Now, how many men will you take?”

“None,” I replied, bluntly; “it’s a useless expense, unless there is something to bring back. If there is I’ll wire you, you may be sure.”

“Very well,” said Professor Farrago, good-humoredly, “you shall have all the assistance you may require. Can you leave to-night?”

The old gentleman was certainly prompt. I nodded, half-sulkily, aware of his amusement.

“So,” I said, picking up my hat, “I am to start north to find a place called Black Harbor, where there is a man named Halyard who possesses, among other household utensils, two extinct great auks—”

We were both laughing by this time. I asked him why on earth he credited the assertion of a man he had never before heard of.

“I suppose,” he replied, with the same half-apologetic, half-humorous smile, “it is instinct. I feel, somehow, that this man Halyard has got an auk—perhaps two. I can’t get away from the idea that we are on the eve of acquiring the rarest of living creatures. It’s odd for a scientist to talk as I do; doubtless you’re shocked—admit it, now!”

But I was not shocked; on the contrary, I was conscious that the same strange hope that Professor Farrago cherished was beginning, in spite of me, to stir my pulses, too.

“If he has—” I began, then stopped.

The professor and I looked hard at each other in silence.

“Go on,” he said, encouragingly.

But I had nothing more to say, for the prospect of beholding with my own eyes a living specimen of the great auk produced a series of conflicting emotions within me which rendered speech profanely superfluous.

From IN SEARCH OF THE UNKNOWN by Robert W. Chambers—Public Domain

Audubon: John James Audubon, an ornithologist and artist who created scientific illustrations of birds
dotage: a loss of reasoning brought about by old age

Question

How does the author develop the contrast between the narrator’s point of view and the professor’s point of view?

  • Aby providing both the narrator’s and professor’s thoughts on how age and experience influence each other’s reasoning
  • Bby using the conversation between the narrator and the professor to emphasize their reactions to the letter
  • Cby describing the professor’s persistent efforts to change the narrator’s mind about the letter
  • Dby including dialogue that explains why the professor is the supervisor and the narrator is his subordinate
Show the answer

Correct answer: B

The question asks how the author develops the contrast between the narrator’s point of view and the professor’s point of view.

A. Incorrect. The excerpt does not describe the narrator’s thoughts about how age and experience influence the professor’s reasoning in enough detail to fully contrast them against the thoughts about the narrator that the professor expresses in paragraphs 15–17. The narrator’s idea that the professor might be “on the verge of dotage” (paragraph 24) is meant in a humorous way.

B. CORRECT. The primary conflict of the excerpt revolves around the narrator’s disagreement with the professor about whether the letter about the great auks could be true. As the two men converse, their contrasting attitudes toward the letter become clear: the narrator dismisses the possibility of great auks out of hand (“ ‘of course the man is mistaken’ ” [paragraph 5]; “ ‘here is a man . . . who wants you to send somebody to take charge of a bird that doesn’t exist!’ ” [paragraph 12]), while the professor calmly admits that although “ ‘nine hundred and ninety-nine men in a thousand would throw that letter aside and condemn the writer’ ” (paragraph 9), he himself believes that the writer could be telling the truth (“ ‘How do you know,’ asked Professor Farrago, ‘that the bird in question does not exist?’ ” [paragraph 13]).

C. Incorrect. The professor explains why he disagrees with the narrator (paragraphs 19–21), but he does not make persistent efforts to convince the narrator to change his mind about the letter. Instead he states that the narrator will go on an expedition to retrieve “ ‘whatever it is that our correspondent has’ ” (paragraph 23), thereby requiring the narrator to go collect the great auks whether the narrator believes in them or not.

D. Incorrect. The point of the dialogue is not to explain the subordinate-supervisor relationship between the narrator and the professor but rather to contrast what each character believes about the contents of the letter.

Question analysis

This point-of-view question asks how Robert W. Chambers develops the contrast between the narrator’s outlook and Professor Farrago’s in the excerpt from In Search of the Unknown. Students must step back from single lines and judge the author’s method across the whole excerpt. The choices describe techniques in general terms, and several connect to real moments in the story.

NYS standards
  • 7R6 point of view and author position
  • 7R5 structure and genre features
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Analyzing how an author contrasts two characters’ points of view is the grade 7 standard itself, but here students must judge the main method across the whole excerpt, and each choice names an abstract method that touches something in the text.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Evidence
Answered from the whole text
Reading level
Passage: about grade 9, 1066 words; question and choices: about grade 11, 84 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The right choice is plain enough to seem too obvious, so students look for something more specific. The excerpt offers a candidate: the professor teases “you young fellows,” and the narrator decides the old man is near “dotage,” which makes age look like the method. Those remarks are passing jokes, while the men’s opposite reactions to Halyard’s letter run through the whole conversation. The question rewards judging what carries the contrast from start to end rather than what is most striking.

  1. How, not what major Working it out

    The question asks for the author’s method across the whole excerpt, and every choice names an abstract technique, so students must step back from single lines to see that the dialogue carries the contrast. (Anderson & Krathwohl, 2001)

  2. A very close wrong answer moderate Choosing the answer

    The choice about age and experience and the right one both describe the two men’s opposed views as they come out in the excerpt; they part on the subject, their ages or the letter. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  3. Evidence in several places moderate Working it out

    The contrast builds from the narrator’s “of course the man is mistaken” in paragraph 5 to the professor’s appeal to “instinct” in paragraph 33, so no single passage shows the method. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

Strategy: For each choice, find where the excerpt does what the choice describes, and ask whether that method carries the contrast through the whole excerpt or appears only once or twice.

Traps

  • Too narrow: A single detail offered as the central idea or a summary, or a summary that keeps minor details and leaves out an essential part.
  • 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 purpose or structure: Names a purpose, structure or effect that the part does not serve, or describes what a part says when the question asks what it does.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Figure out what the question asks

    This asks about the author’s method. How does the author show that the two men see things differently? Look at how the whole excerpt is built, not just one line.

  2. Notice how the excerpt is told

    Most of the excerpt is a conversation. The two men talk back and forth about the letter. Each answer shows how one man reacts to it. So their reactions to the letter are where the contrast shows up. A contrast is a clear difference between two things.

  3. Find the two reactions

    The narrator says “of course the man is mistaken” (paragraph 5). He laughs at the idea of “a bird that doesn’t exist!” (paragraph 12). The professor stays open. In paragraph 13 he asks how the narrator can be sure “that the bird in question does not exist.” In paragraph 23, he is “daring to believe” it is the great auk.

  4. Check each choice

    • A does not fit. Age comes up only twice: “you young fellows” and the narrator’s thought about “dotage.” Two short remarks do not build the contrast.
    • B fits. The talk about the letter shows their opposite reactions.
    • C does not fit. The professor asks a few calm questions and then simply sends the narrator. He does not keep trying to change his mind.
    • D does not fit. No line explains how either man got his job.

    Watch out: A is tempting because those remarks about age are real. But they are passing moments, not the author’s main method.

  5. Choose the answer

    The author builds the contrast through the two men’s talk about the letter. So the answer is B, “by using the conversation between the narrator and the professor to emphasize their reactions to the letter.”

Why this answer and not the closest wrong one

A is the closest wrong choice. The professor does tease “you young fellows,” and the narrator does think of “dotage.” But those are two quick remarks. From paragraph 5 to paragraph 33, the two men keep reacting to the letter in opposite ways. That steady back-and-forth builds the contrast, so B fits.

The original DOE question
Question 22 as shown on the DOE’s digital test, page 30 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test B English, page 30, question 22.
The passage “Excerpt from In Search of the Unknown” as printed (page 21)The passage “Excerpt from In Search of the Unknown” as printed on page 21 of the DOE handbook.

Question 4 of 6

2026 Practice Test B Math, question 24

Question

A chemical decays in such a way that the amount left at the end of each week is 20% less than the amount at the beginning of that same week. What percentage of the original amount is left after two weeks?

  • A40%
  • B60%
  • C64%
  • D80%
Show the answer

Correct answer: C

Let x be the original amount of the chemical. It loses 20% after each week, which means 80% of the chemical remains at the end of each week. End of first week: 0.80x0.80x

At the end of the second week, 80% of the amount left at the end of the first week remains.

End of second week: 0.80(0.80x)=0.64x0.80(0.80x) = 0.64x or 64%

Question analysis

A chemical loses 20% of the amount present at the start of each week, and students find what percentage of the original amount is left after two weeks. It tests applying a percent decrease more than once. The question turns on which amount each week's loss is measured against.

NYS standards
  • 7.RP.3 multistep ratio and percent problems
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 A typical grade 7 percent task applies one increase or decrease to one amount. This one applies a weekly decrease twice in a row, an early form of exponential decay that goes beyond typical grade 7 work.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Solution
2 steps
Reading level
Question and choices: about grade 8, 40 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

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

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Percent of an amount
part=p100×amount\text{part} = \frac{p}{100} \times \text{amount}
p is the percent, and amount is the whole you take the percent of. Percent means out of 100, so 20% is 20/100. Multiply to find the part. Be sure the amount is the right whole. Here, each week uses the amount at the start of that week.
Percent left after a loss
percent left=100%−percent lost\text{percent left} = 100\% - \text{percent lost}
The whole amount is 100%. If 20% goes away, the rest stays. So 80% of each week’s starting amount is left at the end of that week. Keeping 80% is the same as multiplying by 0.8.
  1. Sort out the question

    Each week the chemical loses 20% of what it has. The key words are “the amount at the beginning of that same week.” So each week’s 20% comes from that week’s starting amount. No amount is given. You need the percent of the original amount that is left after two weeks.

  2. Pick an easy starting amount

    Pretend the chemical starts at 100 grams. With 100, the number of grams left is the same as the percent left. Any starting amount gives the same percent, so choose the easiest one.

  3. Work out the first week

    Use the percent formula. The first week starts with 100 grams:

    20100×100=20\frac{20}{100} \times 100 = 20

    The chemical loses 20 grams, so 100−20=80100 - 20 = 80 grams are left. That is 80% after one week.

    Watch out: 80% is only the first week. Stopping here gives choice D.

  4. Work out the second week

    The second week starts with 80 grams, not 100. So take 20% of 80:

    20100×80=16\frac{20}{100} \times 80 = 16

    This time the chemical loses only 16 grams, so 80−16=6480 - 16 = 64 grams are left.

    Watch out: taking away 20 grams again uses the original amount. That gives 60%, choice B. Choice A, 40%, adds the two losses. But the question asks what is left, and only 36% was lost.

  5. Read the answer as a percent

    WeekStart (grams)Loss (grams)Left (grams)
    11002080
    2801664

    Another way: each week keeps 80%, or 0.8, of its start. Two weeks gives 0.8×0.8=0.640.8 \times 0.8 = 0.64, which is 64%.

    64 grams out of the original 100 grams is 64%. So the answer is C, 64%.

Check your answer

Think about the size of the second loss. The second week starts with less, so it loses less than 20 grams. That means more than 60% is left. It does lose something, so less than 80% is left. Of the four choices, only 64% is between 60% and 80%.

The original DOE question
Question 24 as shown on the DOE’s digital test, page 96 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test B Math, page 96, question 24.

Question 5 of 6

2024 Test B Math, question 53

Question

Which graph represents the solution to |x+1|≥2|x+1|\geq 2?

  • ANumber line from −3 to 3 with a tick at every integer. Closed dots at −3 and 1. The line is shaded from −3 to the left, ending in an arrow, and from 1 to the right, ending in an arrow.
  • BNumber line from −3 to 3 with a tick at every integer. Closed dots at −1 and 1. The line is shaded from −1 to the left, ending in an arrow, and from 1 to the right, ending in an arrow.
  • CNumber line from −3 to 3 with a tick at every integer. Closed dots at −3 and 1, with the segment between them shaded.
  • DNumber line from −3 to 3 with a tick at every integer. Closed dots at −3 and 3. The line is shaded from −3 to the left, ending in an arrow, and from 3 to the right, ending in an arrow.
Show the answer

Correct answer: A

Since |x+1|≥2|x+1|\geq 2, either x+1≤−2x+1\leq -2 or x+1≥2x+1\geq 2.

Solving x+1≤−2x+1\leq -2 for x gives x≤−3x\leq -3.

Solving x+1≥2x+1\geq 2 for x gives x≥1x\geq 1.

A filled circle on a number line represents “or equal to.” So the correct number line representation has filled circles at −3 and 1, with the number line filled to the left of −3 for “less than” and to the right of 1 for “greater than.”

Question analysis

Students choose the number line graph that shows every solution of an inequality in which x + 1 sits inside absolute value bars. It tests absolute value as distance and graphing inequalities. The tricky part is turning one absolute value statement into two pieces and placing each boundary point and shaded ray correctly.

NYS standards
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students solve and graph two-step inequalities and know absolute value as distance. An inequality with an absolute value has to be split into two cases, which is outside the grade 6 to 8 standards and well above typical grade 7 work.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Solution
3 steps
Reading level
Question and choices: about grade 9, 7 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

Reading |x + 1| as the distance from x to −1 makes this quick: the points 2 units away are −3 and 1, and greater than or equal means outside them. Students who solve only one case and mirror it across 0 land on graphs centered on 0, and students who find both endpoints may still shade between them. The graphs differ in just one dot or one shading choice, so each must be read carefully.

  1. Something you must already know moderate Working it out

    Solving |x + 1| ≥ 2 needs two cases, x + 1 ≤ −2 or x + 1 ≥ 2, joined by or; absolute value inequalities go beyond grade 7, and the stem gives no hint. (Anderson & Krathwohl, 2001)

  2. A very close wrong answer moderate Choosing the answer

    The right graph and the segment from −3 to 1 share both filled endpoints, so only deciding between outside and between the boundaries separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

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

    Four number lines must be read for filled dots and arrow directions, and two of them differ from the right one only in where a single dot sits. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

Strategy: Read the absolute value as distance from −1 on the number line, find the points exactly 2 units away, and decide whether the solutions lie between or outside them.

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.
  • 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

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Absolute value as distance
|x−a|=distance from x to a|x - a| = \text{distance from } x \text{ to } a
x and a are points on the number line. The bars mean absolute value. |x − a| is how far apart the two points are, which is never negative. Since x + 1 = x − (−1), |x + 1| is the distance from x to −1.
  1. Sort out the question

    You need the graph of every x that makes |x+1|≥2|x + 1| \geq 2 true. The sign ≥ means greater than or equal to. The bars mean absolute value.

    On a number line, a filled dot means that point counts. A thick line with an arrow means all the numbers that way count too.

  2. Read it as a distance

    Adding 1 is the same as subtracting −1, so x+1=x−(−1)x + 1 = x - (-1). By the formula, |x+1||x + 1| is the distance from x to −1.

    So the inequality says that x is at least 2 units away from −1.

  3. Find the two boundary points

    Start at −1 and go 2 units each way.

    −1−2=−3-1 - 2 = -3

    −1+2=1-1 + 2 = 1

    At −3 and at 1, the distance is exactly 2. The words or equal to let them count, so they get filled dots.

  4. Decide which side to shade

    Being at least 2 away means being farther out. Numbers between −3 and 1 are too close to −1. For example, 0 is only 1 unit away. So shade to the left of −3 and to the right of 1.

    In algebra, that is x+1≤−2x + 1 \leq -2 or x+1≥2x + 1 \geq 2. These give x≤−3x \leq -3 or x≥1x \geq 1.

  5. Match the graph

    Choice A has filled dots at −3 and 1, with arrows going outward. That matches.

    Watch out: Choice C shades between −3 and 1. Those are the points within 2 of −1, the answer to |x+1|≤2|x + 1| \leq 2. Choices B and D are centered on 0, but the center here is −1.

    So the answer is A, the graph shaded outward from −3 and 1.

Check your answer

Test a point from each part. For x = 2, |2+1|=3|2 + 1| = 3, and 3 is at least 2. For x = 0, |0+1|=1|0 + 1| = 1, which is less than 2. For x = −3, |−3+1|=2|-3 + 1| = 2, which counts. Graph A agrees with all three.

The original DOE question
Question 110 as printed on page 214 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 214, question 110. Open page 214 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).

Question 6 of 6

2018 Test A Math, question 1

Question

Parallelogram PQRS with side PS on a horizontal line that extends past P and S, and side QR at the top. At S, the angle between side SR and the line extended to the right of S is marked 72°. At Q, a vertical line rises above the figure at a right angle to side QR (right-angle mark), and angle x° is marked on the left of Q, between that vertical line and side QP.

In the figure above, PQRS is a parallelogram. What is the value of xx?

Grid-in question: work out the answer and enter it as a number.

Show the answer

Correct answer: 162

First, find the measure of angle PQR. The measure of angle PQR is equal to the measure of angle PSR.

m∠PSR = 180 − 72 = 108.

So, the measure of angle PQR is also 108.

108 + 90 + x = 360

198 + x = 360

x = 162

Question analysis

A grid-in geometry question about angles around a parallelogram PQRS. A 72° angle sits outside corner S on the extended base, and at corner Q a right angle and the unknown angle x complete the turn around the point. Students chain three angle facts, and the angle asked for lies outside the shape, not inside it.

NYS standards
  • 7.G.5 angle relationships and equations
  • 8.G.5 angle sums, transversals, angle-angle similarity
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 angle work usually applies one or two facts, such as supplementary or vertical angles, to find an angle. This chains three, and needs the equal opposite angles of a parallelogram, which rest on grade 8 parallel-line facts.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Solution
3 steps
Reading level
Question and choices: about grade 7, 14 words.
Students, last 30 days
Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The arithmetic is two subtractions; the difficulty is seeing which angles the marks refer to. Students lose points by treating the 72° as the parallelogram's own angle at S, or by stopping at angle PQR, the most natural number to grid. The question rewards tracing a route from S to Q before computing anything, and checking at the end that x is the angle outside the shape that the arc marks.

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

    The 72° is marked outside the parallelogram, between side SR and the base extended past S, and x sits outside corner Q, between QP and a vertical ray that meets QR at a right-angle mark. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  2. Combines ideas moderate Working it out

    Three angle facts have to be chained: a straight line gives the angle inside S, opposite angles of a parallelogram carry it to Q, and the angles around Q make a full 360° turn. (Embretson, 1998; Daniel & Embretson, 2010)

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

    As a grid-in there are no choices to check against, so a student who stops at angle PQR, inside the parallelogram, gets no signal that the job is unfinished. (Enright, Morley & Sheehan, 2002)

Strategy: Use the straight line at S to find the parallelogram's angle there, carry it to the opposite corner Q, then take that angle and the right angle away from 360.

Traps

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Angles on a straight line
a+b=180a + b = 180
a and b are two angles side by side on a straight line. Together they make a half turn, so they always add up to 180 degrees. Use it when an angle you need shares a straight line with an angle you know.
Opposite angles of a parallelogram
∠PQR=∠PSR\angle PQR = \angle PSR
The symbol ∠ means angle. ∠PQR is the angle at corner Q, between sides QP and QR. ∠PSR is the angle at the opposite corner, S. In a parallelogram, opposite corners always have equal angles, because the shape leans the same way at both.
Angles around a point
a+b+c=360a + b + c = 360
a, b and c are angles that fit around one point with no gaps and no overlaps. Going all the way around is a full turn, which is 360 degrees. So the angles add up to 360.
  1. Sort out what the figure shows

    PQRS is a parallelogram. That is a four-sided shape whose opposite sides are parallel, which means they never meet. You need the value of x, in degrees.

    Look closely at the marks. The 72° angle is outside the shape, at S. It sits between side SR and the base line, which keeps going past S. At Q, a vertical line meets side QR. The small square means a right angle, 90°. The angle x is outside the shape too. It runs from that vertical line around to side QP.

  2. Find the angle inside corner S

    Angle PSR is the shape’s own angle at S. It sits right next to the 72° angle, on the same straight base line. So the two angles add up to 180°.

    a+b=180a + b = 180

    ∠PSR+72=180\angle PSR + 72 = 180

    ∠PSR=180−72=108\angle PSR = 180 - 72 = 108

    So angle PSR is 108°. Watch out: 72° is not the inside angle at S. Using 72 as the inside angle leads to the wrong answer 198.

  3. Carry the angle over to Q

    Corners Q and S are opposite each other. In a parallelogram, opposite angles are equal. So angle PQR matches angle PSR.

    ∠PQR=∠PSR=108\angle PQR = \angle PSR = 108

    So angle PQR is 108°. Watch out: this is not x. Angle PQR is inside the shape, and x is outside it. Stopping here and gridding 108 is the most common mistake.

  4. Add the angles around Q

    Three angles fill the space all the way around point Q. They are the right angle (90°), angle PQR (108°) and x. Together they make a full turn.

    a+b+c=360a + b + c = 360

    90+108+x=36090 + 108 + x = 360

    198+x=360198 + x = 360

    Watch out: don’t skip the right angle at Q. Leaving out the 90 gives 252.

  5. Solve for x

    Subtract 198 from 360 to find what is left for x.

    x=360−198=162x = 360 - 198 = 162

    So x is 162°. So you would grid 162.

Check your answer

Add the three angles at Q: 90+108+162=36090 + 108 + 162 = 360. That is a full turn. Also, the arc for x is wider than a right angle but not as wide as a straight line. So x should be between 90° and 180°, and 162° is.

The original DOE question
Question 58 as printed on page 81 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 81, question 58. Open page 81 of the handbook (PDF)

Sample #28