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

2026 Practice Test A English, question 18
Passage Excerpt from A Voice in the Wilderness

Excerpt from A Voice in the Wilderness

by Grace Livingston Hill

With a lurch the train came to a dead stop and Margaret Earle, hastily gathering up her belongings, hurried down the aisle and got out into the night.

It occurred to her, as she swung her heavy suit-case down the rather long step to the ground, and then carefully swung herself after it, that it was strange that neither conductor, brakeman, nor porter had come to help her off the train, when all three had taken the trouble to tell her that hers was the next station; but she could hear voices up ahead. Perhaps something was the matter with the engine that detained them and they had forgotten her for the moment.

The ground was rough where she stood, and there seemed no sign of a platform. Did they not have platforms in this wild Western land, or was the train so long that her car had stopped before reaching it?

She strained her eyes into the darkness, and tried to make out things from the two or three specks of light that danced about like fireflies in the distance. She could dimly see moving figures away up near the engine, and each one evidently carried a lantern. The train was tremendously long. A sudden feeling of isolation took possession of her. Perhaps she ought not to have got out until some one came to help her. Perhaps the train had not pulled into the station yet and she ought to get back on it and wait. Yet if the train started before she found the conductor she might be carried on somewhere and he justly blame her for a fool.

There did not seem to be any building on that side of the track. It was probably on the other, but she was standing too near the cars to see over. She tried to move back to look, but the ground sloped and she slipped and fell in the cinders1^1, bruising her knee and cutting her wrist.

In sudden panic she arose. She would get back into the train, no matter what the consequences. They had no right to put her out here, away off from the station, at night, in a strange country. If the train started before she could find the conductor she would tell him that he must back it up again and let her off. He certainly could not expect her to get out like this.

She lifted the heavy suit-case up the high step that was even farther from the ground than it had been when she came down, because her fall had loosened some of the earth and caused it to slide away from the track. Then, reaching to the rail of the step, she tried to pull herself up, but as she did so the engine gave a long snort and the whole train, as if it were in league against her, lurched forward crazily, shaking off her hold. She slipped to her knees again, the suit-case, toppled from the lower step, descending upon her, and together they slid and rolled down the short bank, while the train . . . ran giddily off into the night.

The horror of being deserted helped the girl to rise in spite of bruises and shock. She lifted imploring hands to the unresponsive cars as they hurried by her—one, two, three, with bright windows, each showing a passenger, comfortable and safe inside, unconscious of her need.

A moment of useless screaming, running, trying to attract some one’s attention, a sickening sense of terror and failure, and the last car slatted itself past with a mocking clatter, as if it enjoyed her discomfort.

Margaret stood dazed, reaching out helpless hands, then dropped them at her sides and gazed after the fast-retreating train, the light on its last car swinging tauntingly, blinking now and then with a leer in its eye, rapidly vanishing from her sight into the depth of the night.

She gasped and looked about her for the station that but a short moment before had been so real to her mind; and, lo! on this side and on that there was none!

The night was wide like a great floor shut in by a low, vast dome of curving blue set with the largest, most wonderful stars she had ever seen. Heavy shadows of purple-green, smoke-like, hovered over earth darker and more intense than the unfathomable blue of the night sky. It seemed like the secret nesting-place of mysteries wherein no human foot might dare intrude. It was incredible that such could be but common sage-brush, sand, and greasewood wrapped about with the beauty of the lonely night.

No building broke the inky outlines of the plain, nor friendly light streamed out to cheer her heart. Not even a tree was in sight, except on the far horizon, where a heavy line of deeper darkness might mean a forest. Nothing, absolutely nothing, in the blue, deep, starry dome above and the bluer darkness of the earth below save one sharp shaft ahead like a black mast throwing out a dark arm across the track.

As soon as she sighted it she picked up her baggage and made her painful way toward it, for her knees and wrist were bruised and her baggage was heavy.

A soft drip, drip greeted her as she drew nearer; something plashing down among the cinders by the track. Then she saw the tall column with its arm outstretched, and looming darker among the sage-brush the outlines of a water-tank. It was so she recognized the engine’s drinking-tank, and knew that she had mistaken a pause to water the engine for a regular stop at a station.

1^1cinders: track bed made from the residue of burnt coal

From A VOICE IN THE WILDERNESS by Grace Livingston Hill—Public Domain

Question

How does Margaret’s experience in paragraph 9 emphasize a theme of the excerpt?

  • AIt confirms Margaret’s understanding that she cannot rely on help from anyone else.
  • BIt leads Margaret to realize that her desire to change her situation is impractical.
  • CIt causes Margaret to believe that her own actions led to an unfavorable outcome.
  • DIt reinforces Margaret’s frustration about her lack of control over her surroundings.
Show the answer

Correct answer: A

The question asks about the relationship between the theme and paragraph 9, where Margaret desperately tries to get someone on the train to notice her.

A. CORRECT. One important theme of this excerpt concerns Margaret’s willingness to take action to ensure that she reaches her destination. This determination, described in Option A, is why she gathers her belongings together without help, gets off the train without being assisted by employees, and then attempts to climb back on while the train employees are busy examining the engine.

B. Incorrect. While the narrator describes her screaming as useless, she does not feel that her efforts overall are useless, and she does not give up until paragraph 10.

C. Incorrect. While Margaret is feeling a sense of “failure” in paragraph 9, she does not condemn herself as the train moves away.

D. Incorrect. Margaret expresses “terror” in paragraph 9 over being unable to catch someone’s attention, not frustration about her lack of control over her surroundings.

Question analysis

This theme question asks how Margaret’s experience in paragraph 9 of A Voice in the Wilderness, as she screams and runs while the train pulls away, emphasizes a theme of the excerpt. It is a lightly reworded version of an earlier handbook question. Every choice describes a believable reaction to being left behind, so the answer depends on the excerpt’s larger pattern.

NYS standards
  • 7R2 theme or central idea
  • 7R3 plot elements and ideas
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Typical grade 7 theme questions trace a given theme through a text. Here students must decide which theme one frantic paragraph supports, and all four choices describe reactions that fit the scene, so the decision draws on the whole excerpt.
Estimated difficulty
Among the hardest on the SHSAT: a non-obvious approach, heavy reading, or a well-hidden trap.
Evidence
Answered from the whole text
Reading level
Passage: about grade 8, 981 words; question and choices: about grade 10, 65 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

Each choice names a feeling a stranded traveler might have, and two lean on the paragraph’s own words, “useless” and “failure,” so the paragraph alone cannot decide it. Paragraph 9 must be read as the last link in a chain of trainmen, passengers and a departing train that all leave her to herself. Students who judge from that one paragraph often decide she has given up, though she soon picks up her bags and walks toward the dark shaft by the track.

  1. Reading between the lines major Working it out

    The theme is never stated; it must be drawn from paragraph 9’s cries that no one answers and from the passengers “comfortable and safe inside, unconscious of her need” in paragraph 8. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

  2. Evidence in several places moderate Working it out

    The pattern begins in paragraph 2, when no trainman helps her off, and runs through the train lurching away “as if it were in league against her” in paragraph 7. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

Strategy: Sum up in a phrase what keeps happening to Margaret from the start of the excerpt to its end, then check which choice paragraph 9’s own words most clearly add to.

Traps

  • Too broad or too strong: A choice overstates, generalizes, or uses absolute words the text does not support.
  • Wrong tone or attitude: Misjudges the tone, attitude or point of view, such as reading admiration as criticism, or taking a character's or subject's feelings for the author's.

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 a theme: a big idea a story keeps coming back to. How does what happens in paragraph 9 point to a theme? The right choice should fit a pattern across the whole excerpt.

  2. Read paragraph 9

    Paragraph 9 describes “A moment of useless screaming, running, trying to attract some one’s attention, a sickening sense of terror and failure.” Margaret screams and runs, trying to get someone to notice her. No one does. The last car rolls past “as if it enjoyed her discomfort.”

  3. Find the same thing earlier

    This is not the first time. In paragraph 2, none of the workers “had come to help her off the train.” In paragraph 8, the passengers sit “comfortable and safe inside, unconscious of her need.” That means they don’t know she needs help. Paragraph 9 is one more time when nobody comes. She will have to count on herself.

  4. Check each choice

    • A fits. Her cries go unanswered, just as before. She cannot rely on anyone else.
    • B does not fit. Impractical means not sensible or workable. Her screaming fails, but she never decides that trying to change things is pointless. In paragraph 14 she “picked up her baggage” and set off again.
    • C does not fit. She feels “failure,” but she never blames herself.
    • D does not fit. The paragraph shows terror at being ignored, not frustration about having no control over her surroundings.
  5. Choose the answer

    Paragraph 9 adds one more moment where no one helps Margaret. So the answer is A.

Why this answer and not the closest wrong one

B is the closest wrong choice, because paragraph 9 calls her screaming “useless.” But one failed moment does not mean she gives up. Soon she takes her bags and walks toward the tall dark shape ahead. What stays true from paragraph 2 to paragraph 9 is that no one helps her. That is A.

The original DOE question
Question 18 as shown on the DOE’s digital test, page 26 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test A English, page 26, question 18.
The passage “Excerpt from A Voice in the Wilderness” as printed (pages 128–129)The passage “Excerpt from A Voice in the Wilderness” as printed on page 128 of the DOE handbook.The passage “Excerpt from A Voice in the Wilderness” as printed on page 129 of the DOE handbook.

Question 2 of 6

2024 Test A English, question 28
Passage Champion of the Channel

Champion of the Channel

In 1926 an editor at the London Daily News predicted that Gertrude Ederle, an American swimmer with eighteen world records and three Olympic medals, would fail in her attempt to swim across the English Channel. He claimed that “even the most uncompromising champion of the rights and capacities of women must admit that in contests of physical skill, speed and endurance they must remain forever the weaker sex.” Yet, at only nineteen years old, Ederle not only became the first woman to accomplish this feat, she also broke the men’s record by two hours. Gertrude Ederle’s triumphant swim across the English Channel was a testimony to her determination, innovative spirit, and passion for swimming.

Crossing the English Channel is a daunting task for any swimmer. At its narrowest point, the channel measures twenty-one miles across. Its icy waters hover around sixty degrees Fahrenheit, and its unruly tides and currents toss swimmers about like bobbing corks. Stinging jellyfish, seaweed, and floating debris from shipwrecks and lost cargoes present added hazards.

For decades the channel’s perils have defeated countless swimmers. Ederle, too, failed in her first attempt to cross the channel in 1925. Just six miles short of finishing, she became ill, and her coach had to haul her out of the water. Undeterred, Ederle decided to try again. Ederle knew that if she did not complete the challenge this time, she might never get the opportunity to set this record, because a rival female swimmer was preparing to make her second attempt at the crossing as well.

To prepare for the marathon swim, Ederle found ways to improve her equipment. She and her sister Meg discovered that melted candle wax perfectly sealed goggle edges, effectively waterproofing Ederle’s goggles against hammering waves. The sisters also designed a two-piece silk swimsuit for Ederle. During her first channel-crossing attempt she had worn a standard one-piece swimsuit that, after the lengthy hours of swimming across the channel, had stretched out, filling with water and creating drag, making an already challenging task almost insurmountable. Unlike the cumbersome typical bathing suit, this silk invention weighed little and allowed for easy movement.

On August 6, 1926, Ederle waded into the channel near Cape Gris-Nez, France. At first she shivered in the bone-chilling water even though she had covered her body in eight layers of grease for insulation. Her limbs felt stiff. Her strokes were irregular. Driving forward, she fought to clear her mind and find what she called her “sphere,” a place where the sea became her only companion and the shrieks of gulls and the humming of boat engines faded away. Using a new overhand stroke called the American crawl, Ederle eventually settled into a steady pace, briskly breaking through waves.

Throughout Ederle’s swim, two tugboats accompanied her. One carried newspaper reporters who wired dispatches of her progress to shore. The other, displaying a sign that read “This way, ole kid!” with an arrow pointing forward, transported her coach, family, and friends. Her coach played songs, such as “Yes, We Have No Bananas,” on a phonograph so that Ederle could time her strokes to the rhythm. Using a net, her coach also passed her baby bottles of broth for nourishment.

For hours Ederle swam, dodging debris with an amused smile. However, as she neared the English shore, a sudden fierce storm erupted. The tides and waves forced Ederle backward, and she fought the stubborn swells for several hours. The salty water caused her tongue to swell and inflamed her ears. Yet Ederle felt indescribably happy as she churned through the sea. Finally, as she neared the English shore, the storm abated, and the tide turned. No longer fighting against her, the sea pushed her toward the shore and victory.

After fourteen hours and thirty-one minutes, Ederle, on wobbly legs, stepped onto the English shore. The waiting crowd roared, honked their automobile horns, blasted their tugboat whistles, and set off flares that flashed in the sky. Ederle had swum into history.

When Ederle returned to New York, she received a parade, where thousands of people shouted “Trudy!” Not only were everyday American citizens proud of Ederle, but she also inspired them to be more active. Over the next few years, more than 60,000 people credited her with motivating them to earn their American Red Cross swimming certificates. Gertrude Ederle’s accomplishment proved to the world that with determination and passion, it was possible for a person to achieve his or her goals.

SOLO SWIMS ACROSS THE ENGLISH CHANNEL

Earliest Speed Records

Year Swimmer Time
1875 Matthew Webb (M) 21 hours, 45 minutes
1923 Enrico Tiraboschi (M) 16 hours, 33 minutes
1926 Gertrude Ederle (F) 14 hours, 31 minutes
1926 Arnst Vierkotter (M) 12 hours, 38 minutes
Current Speed Records
Year Swimmer Time
2012 Trent Grimsey (M) 6 hours, 55 minutes
2006 Yvetta Hlavacova (F) 7 hours, 25 minutes

Question

Which statement describes how the author’s use of a problem-and-solution structure in paragraph 5 contributes to the development of ideas in the passage?

  • ADetailing the challenges presented by the cold channel waters highlights the effectiveness of Ederle’s training.
  • BDescribing Ederle’s physical difficulties during her swim provides evidence of the team effort required to ensure her safety.
  • CExplaining the difficulties that arose early in the effort helps predict the additional obstacles that occurred during Ederle’s attempt.
  • DIllustrating Ederle’s process of blocking out her discomfort shows that swimming the channel was both a mental and a physical challenge.
Show the answer

Correct answer: D

The question asks how a problem-and-solution structure in paragraph 5 contributes to the ideas presented in the passage.

A. Incorrect. The details in paragraph 5 do not provide a connection between the cold temperature of the water and the effectiveness of Ederle’s training.

B. Incorrect. The difficulties Ederle encountered were only partially relieved by her team, and her team’s efforts to ensure Ederle’s safety are not explained in the passage.

C. Incorrect. The paragraph does not focus on the relationship between the problems Ederle encountered at the start of her swim and the problems she encountered during or near the end of her swim.

D. CORRECT. The paragraph describes how the water’s temperature made Ederle uncomfortable and made it difficult for her to regulate her stroke. The paragraph continues with the explanation of how she overcame these issues by focusing her mind on the sea and tuning out the distractions in her surroundings (“the sea became her only companion and the shrieks of gulls and the humming of boat engines faded away” [paragraph 5]).

Question analysis

Gertrude Ederle's cold, stiff first hours in the English Channel are told as problem and solution, and this text-structure question asks what that pattern adds to the passage's ideas. Students must spot the troubles she met in the cold water and how she handled them, then pick the abstract statement that accounts for both halves of the pattern.

NYS standards
  • 7R5 structure and genre features
  • 7R3 plot elements and ideas
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 A problem-and-solution question about one paragraph is familiar grade 7 fare, yet these four answers are general claims about the swim, each tied to a real hardship, so students need a clear map of the paragraph before judging them.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Evidence
Answered from one paragraph or stanza
Reading level
Passage: about grade 9, 790 words; question and choices: about grade 14, 95 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 opening halves of the choices are all accurate, which moves the whole decision onto their second halves. Readers who notice only the physical side of the paragraph, the cold and the stiffness, reach for training, teamwork or foreshadowing to explain it. The paragraph's turning point is a mental one, clearing her mind until the noise faded, and only the statement that names both a mental and a physical challenge accounts for the problem and the solution together.

  1. How, not what major Working it out

    The question asks what paragraph 5's pairing of problems (bone-chilling water, stiff limbs, irregular strokes) with solutions (finding her “sphere”, the American crawl) does for the passage, so retelling the hardships earns nothing. (Anderson & Krathwohl, 2001)

  2. Every choice must be checked moderate Choosing the answer

    Each choice opens with a fair description of paragraph 5, the cold water, her physical difficulties, the early struggle, so only what each claims that description shows can be tested. (Embretson & Wetzel, 1987)

  3. Figurative or literary language minor Understanding the question

    The mental half of the solution is told in images, a place where “the sea became her only companion” and gulls and engines “faded away”, which must be read as blocking out discomfort. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

Strategy: Sort paragraph 5 into two lists, what went wrong for Ederle early in the swim and what she did about it, then ask what that pairing shows about the swim as a whole.

Traps

  • 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.
  • Outside knowledge: A choice is plausible in the real world but not supported by this text.
  • Wrong part of the text: A choice is true of another paragraph, stanza or speaker than the one asked about.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Learn what the question asks

    A problem-and-solution structure tells about a difficulty and then how someone deals with it. The question asks what this pattern in paragraph 5 adds to the ideas of the passage. So first sort the paragraph into problems and solutions.

  2. List the problems in paragraph 5

    When Ederle waded in, “she shivered in the bone-chilling water.” Next, “Her limbs felt stiff. Her strokes were irregular.” Limbs are arms and legs. Irregular means uneven, not steady. All of these are troubles with her body.

  3. List her solutions

    Ederle “fought to clear her mind.” She looked for what she called her “sphere.” There, “the sea became her only companion and the shrieks of gulls and the humming of boat engines faded away.” In other words, she shut out the noise and her discomfort. Then, using the American crawl, she “settled into a steady pace.”

  4. Connect problems and solutions

    The problems were physical, which means of the body. A big part of the fix was mental, which means of the mind. She calmed her thoughts so her body could keep working. So the paragraph shows that the swim tested her mind as well as her body.

  5. Check each choice

    • D fits: blocking out her discomfort shows the swim was “both a mental and a physical challenge.”
    • A is wrong because paragraph 5 never mentions her training.
    • B is wrong because her team on the tugboats appears in paragraph 6, and her safety is never discussed.
    • C is wrong because the early troubles are solved inside this paragraph instead of setting up later ones.

    Watch out: A storm does hit in paragraph 7, so C may look right. But paragraph 5 does not point ahead to it.

  6. Choose the answer

    Only D explains both the problem and the way Ederle solved it. So the answer is D.

Why this answer and not the closest wrong one

C is tempting, since a storm really does strike in paragraph 7. But paragraph 5 never hints at it. The paragraph ends with its problem solved, as Ederle is “briskly breaking through waves.” D explains both halves of the pattern: troubles of the body, eased by the mind.

The original DOE question
Question 28 as printed on page 54 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 54, question 28. Open page 54 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).
The passage “Champion of the Channel” as printed (pages 56–57)The passage “Champion of the Channel” as printed on page 56 of the DOE handbook.The passage “Champion of the Channel” as printed on page 57 of the DOE handbook.

Question 3 of 6

2018 Test B English, question 56
Passage Invention of the Telegraph

Samuel Morse, an American inventor, is credited with creating the electronic telegraph, a communication device that allows users to send messages using a system of short and long pulses that represent letters, numbers, and punctuation. In 1843 the United States Congress passed the Telegraph Bill, which provided Morse with the funds to build an electric telegraph system.

Invention of the Telegraph

Earlier Signal Systems

Long before Samuel F. B. Morse electrically transmitted his famous message “What hath God wrought?” from Washington to Baltimore on May 24, 1844, there were signaling systems that enabled people to communicate over distances. Most were visual or “semaphore” systems using flags or lights. In the eighteenth century, such systems used an observer who would decipher a signal from a high tower on a distant hill and then send it on to the next station. The young American republic wanted just such a system along its entire Atlantic coast and offered a prize of $30,000 for a workable proposal. The framers of this legislation1^1 had no way of knowing that when they used the word “telegraph” to refer to this visual semaphore system, they would be offered an entirely new and revolutionary means of communication—electricity.

The Growth of an Idea

The idea of using electricity to communicate over distance is said to have occurred to Morse during a conversation aboard ship when he was returning from Europe in 1832. Michael Faraday’s recently invented electromagnet was much discussed by the ship’s passengers, and when Morse came to understand how it worked, he speculated that it might be possible to send a coded message over a wire. While a student at Yale College years before, he had written his parents a letter about how interesting he found the lectures on electricity. Despite what he had learned at Yale, Morse found when he began to develop his idea that he had little real understanding of the nature of electricity, and after sporadic attempts to work with batteries, magnets, and wires, he finally turned for help to a colleague at the University of the City of New York, Leonard D. Gale.

Gale was a professor of chemistry and familiar with the electrical work of Princeton’s Joseph Henry, a true pioneer in the new field. Well before Morse had his shipboard idea about a telegraph, Henry rang a bell at a distance by opening and closing an electric circuit. In 1831, he had published an article, of which Morse was unaware, that contained details suggesting the idea of an electric telegraph. Gale’s help and his knowledge of this article proved crucial to Morse’s telegraph system because Gale not only pointed out flaws in the system but showed Morse how he could regularly boost the strength of a signal and overcome the distance problems he had encountered by using a relay system Henry had invented. Henry’s experiments, Gale’s assistance, and, soon after, hiring the young technician Alfred Vail were keys to Morse’s success.

Obstacles and Opportunities

By December 1837, Morse had enough confidence in his new system to apply for the federal government’s appropriation, and during the next year he conducted demonstrations of his telegraph both in New York and Washington.

However, when the economic disaster known as the Panic of 1837 took hold of the nation and caused a long depression, Morse was forced to wait for better times. It was during this period that Morse visited Europe again and tried not only to secure patent protection overseas but to examine competing telegraph systems in England. . . .

By 1843, the country was beginning to recover economically, and Morse again asked Congress for the $30,000 that would allow him to build a telegraph line from Washington to Baltimore, forty miles away. The House of Representatives eventually passed the bill containing the Morse appropriation, and the Senate approved it in the final hours of that Congress’s last session. With President Tyler’s signature, Morse received the cash he needed and began to carry out plans for an underground telegraph line.

Realizing a Great Invention

Morse had hired the ingenious construction engineer Ezra Cornell to lay the pipe carrying the wire, and although Cornell did his job superbly, one of Morse’s partners, Congressman F. O. J. Smith, had purchased wire with defective insulation. Too much time had been wasted laying bad wire, and with the project on a rigid deadline, something had to be done quickly. Cornell suggested that the fastest and cheapest way of connecting Washington and Baltimore was to string wires overhead on trees and poles. The desperate Morse gave the go-ahead, and the line was completed in time for the dramatic and spectacularly successful link between the Supreme Court chamber of the Capitol building and the railroad station in Baltimore.

Soon, as overhead wires connected cities up and down the Atlantic coast, the dots-and-dashes method2^2 that recorded messages on a long moving strip of paper was replaced by the operator’s ability to interpret the code in real time. . . . Telegraph lines soon extended westward, and within Morse’s own lifetime they connected the continents of Europe and America.

“Invention of the Telegraph”—Public Domain/Library of Congress

1^1legislation: Telegraph Bill

2^2dots-and-dashes method: the short and long pulses of Morse code that are sent and received by telegraph operators

World map titled “British All-Red Telegraph Line, 1902,” with a compass rose (N, S, E, W) and a key showing a thick black line as an electric telegraph line. The line runs from London across the Atlantic Ocean and Canada to Vancouver, then west into the Pacific Ocean off the map’s left edge; from London and from Canada’s east coast to Bermuda, then to Barbados, across the Atlantic to Cape Town, and across the Indian Ocean to Perth, which a second line links to India; and from Perth across Australia to Brisbane and Norfolk, with branches to New Zealand and east into the Pacific off the map’s right edge. Washington, D.C., in the United States, is marked with a dot but no line reaches it.

The first transatlantic electric telegraph message was sent in 1858, and by 1902 the British All-Red Line connected most of the world.

Bar graph titled “Telegraph Messages Transmitted, 1870–1920,” with the number of messages transmitted, in millions, on the vertical axis (0 to 180 in steps of 20) and the years 1870, 1880, 1890, 1900, 1910 and 1920 on the horizontal axis. The bars show about 10 million messages in 1870, 30 million in 1880, 55 million in 1890, 65 million in 1900, 75 million in 1910 and 155 million in 1920.

Source: U.S. Bureau of the Census. Historical Statistics of the United States, Colonial Times to 1970, Bicentennial Edition, Part 2. Washington, D.C., 1975.

Question

The idea that, in the mid-nineteenth century, the United States was mostly unaware of the possibilities of electricity is illustrated in the passage mainly through the

  • Adescription of the government’s initial desire to expand a semaphore signaling system that used either flags or lights along the Atlantic Coast.
  • Bdelay by the House of Representatives to pass the bill funding Morse’s telegraph line six years after he first applied for the appropriation.
  • Cdiscussions of the newly invented electromagnet that sparked the idea of sending codes through wires.
  • Ddescription of how an electric circuit could be closed to ring a bell at a distance.
Show the answer

Correct answer: A

The question asks for information that conveys the idea that the United States was mostly unaware of the possibilities of electricity in the mid-nineteenth century. The passage states that most signaling systems at the time were “visual or ‘semaphore’ systems using flags or lights” (paragraph 1) and that the United States was looking for “just such a system along its entire Atlantic coast” (paragraph 1).

Option A is correct because waving lights and flags reveals how limiting and simple the system of the time was, and Option A states that the United States wanted to expand this system, not improve upon it.

Option B is incorrect because the delay in funding had to do with the Panic of 1837 and the “long depression” (paragraph 5), not because the House of Representatives was not aware or ready for innovation in the use of electricity.

Option C is incorrect because it focuses on what prompted Morse to consider an electronic wire-based communications system.

Option D is incorrect because it describes a basic aspect of an early electrical discovery that demonstrated the interest of scientists in the possibilities of electricity, not the United States as a whole.

Question analysis

Students identify how “Invention of the Telegraph” mainly illustrates the idea that, in the mid-1800s, the United States was largely unaware of what electricity could do. The idea is implied rather than stated, so students must decide which detail truly reveals the nation’s thinking at the time.

NYS standards
  • 7R1 textual evidence and inferences
  • 7R3 plot elements and ideas
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Grade 7 evidence questions often match a stated idea to a sentence. This idea is only implied, so students must infer what a historical detail reveals about a whole nation’s understanding, closer to grade 8 inference.
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 11, 910 words; question and choices: about grade 12, 102 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 stem asks about a whole nation’s understanding, so details about individuals with electrical interests, the ship’s passengers and Henry, show the opposite of what is needed. The real trap is the funding delay, which looks like official indifference but has an economic cause the passage states. Students who remember that the semaphore plan came with a remark about not knowing electricity was coming find the answer.

  1. Reading between the lines moderate Working it out

    The idea is implied: wanting a semaphore system shows unawareness only alongside paragraph 1’s comment that its framers “had no way of knowing” they would be offered electricity. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

  2. Evidence in several places moderate Working it out

    The choices point to four separate places, paragraph 1’s semaphore plan, the funding delay of paragraphs 4 to 6, the shipboard talk in paragraph 2 and Henry’s bell in paragraph 3, and each must be checked there. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

  3. Hard words minor Understanding the question

    “Semaphore” and “appropriation” are uncommon words, and the passage defines “semaphore” only in passing, as systems “using flags or lights.” (Perfetti, 2007; Freedle & Kostin, 1993)

Strategy: For each choice, find the detail in the passage and ask whether the author uses it to show what the country did or did not understand about electricity.

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.
  • 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.

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 which part of the passage best shows an idea. The idea is that in the mid-1800s, the United States mostly did not know what electricity could do. Look for a detail about the country as a whole, not about one scientist.

  2. Find the detail in paragraph 1

    Paragraph 1 says most early systems were visual, or semaphore, “systems using flags or lights.” Then it says, “The young American republic wanted just such a system along its entire Atlantic coast.” The next sentence explains that its lawmakers “had no way of knowing” they would be offered “an entirely new and revolutionary means of communication—electricity.”

  3. Explain what the detail shows

    A semaphore system sends signals people can see, like waving flags. The country asked for that old kind of system. It did not ask for an electric one, because it did not know one was possible. “Revolutionary” means bringing a huge change. So the government’s plan shows how unaware the nation was of what electricity could do.

  4. Check each choice

    • A fits. The government wanted to spread flags and lights, not electric signals.
    • B is wrong. The delay had a money cause. The Panic of 1837 “caused a long depression.” It was not about electricity.
    • C is wrong. Passengers talking about the electromagnet shows interest in electricity, the opposite of not knowing about it.
    • D is wrong. Henry ringing a bell shows one scientist exploring electricity. It says nothing about the whole country.
  5. Choose the answer

    The country’s wish for a flags-and-lights system shows it did not yet see what electricity could do. So the answer is A, the description of the government’s plan for a semaphore system along the Atlantic Coast.

Why this answer and not the closest wrong one

B is the closest trap, because a long wait for money can look like doubt about electricity. But paragraph 5 gives the real cause, the Panic of 1837 and the hard times after it. A is backed by the passage’s own words: the lawmakers “had no way of knowing” electricity was coming.

The original DOE question
Question 56 as printed on page 151 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 151, question 56. Open page 151 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).
The passage “Invention of the Telegraph” as printed (pages 146–148)The passage “Invention of the Telegraph” as printed on page 146 of the DOE handbook.The passage “Invention of the Telegraph” as printed on page 147 of the DOE handbook.The passage “Invention of the Telegraph” as printed on page 148 of the DOE handbook.

Question 4 of 6

2026 Practice Test C Math, question 15

Question

A grocery store sells 1-gallon containers of milk for $3.99. The store also sells orange juice in a 6-pack of 5.5-fluid-ounce bottles for $1.79. Suppose the store wants to sell its orange juice in gallon containers instead. To the nearest dollar, how much more would a gallon of orange juice cost than a gallon of milk?

Enter your answer in the space.

Fill-in question: work out the answer and type it as a number.

Show the answer

Correct answer: 3

To determine the cost of a gallon of orange juice, begin by finding the total number of ounces in the 6-pack of bottles:

5.5 oz×6=33 oz5.5\text{ oz} \times 6 = 33\text{ oz}

So, 33 oz of orange juice costs $1.79.

There are 128 oz in 1 gallon. Set up a proportion using the $1.79 cost per 33 oz of orange juice and the unknown cost (x) per 1 gallon of orange juice:

33 oz$1.79=128 ozx\frac{33\text{ oz}}{\$1.79} = \frac{128\text{ oz}}{x}

Cross multiply to solve for x in the proportion:

(33)×(x)=($1.79)×(128)(33) \times (x) = (\$1.79) \times (128)

33x=$229.1233x = \$229.12

x≈$6.94x \approx \$6.94

So, the orange juice costs $6.94 per gallon, and the milk costs $3.99 per gallon.

Subtract the cost per gallon of milk from the cost per gallon of orange juice.

$6.94−$3.99=$2.95\$6.94 - \$3.99 = \$2.95

Round the difference to the nearest dollar.

$2.95 rounds to $3.

Question analysis

A store sells milk by the gallon for $3.99 and orange juice in six-packs of 5.5-fluid-ounce bottles for $1.79. Students work out what a gallon of the juice would cost at the same price per ounce and how much more that is than the milk, to the nearest dollar. The ounces in a gallon must be known, and several steps involve decimals.

NYS standards
  • 7.RP.3 multistep ratio and percent problems
  • 6.RP.3d converting measurement units
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Unit prices are grade 6 and 7 work, but this chains five steps with decimals and needs a fact the question does not give, the number of fluid ounces in a gallon. Typing a rounded difference, with no choices, adds risk.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Solution
5 steps
Reading level
Question and choices: about grade 5, 60 words.
Corrected from the DOE handbook
  • A rounded value in the explanation is now marked as approximate (≈ instead of =).
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 calculation is long, and its key fact is missing from the page. Students who know the 128 fluid ounces still face a division by 33 that does not come out evenly, which tempts them to round the price per ounce to 5 cents and lose about 50 cents a gallon. Keeping the exact quotient to the end, or scaling 33 ounces up by about 3.9, keeps the difference near $2.95, which rounds to 3.

  1. Something you must already know moderate Working it out

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

  2. Many steps moderate Working it out

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

  3. Error-prone arithmetic minor Working it out

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

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

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

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

Traps

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

The wrong answers, classified

What was corrected from the DOE handbook, and why

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

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Fluid ounces in a gallon
fluid ounces=128×gallons\text{fluid ounces} = 128 \times \text{gallons}
A gallon holds 128 fluid ounces. This is a fact you must know, because the question does not give it. A fluid ounce measures an amount of liquid. Multiply gallons by 128 to change them into fluid ounces.
Cross-multiplying a proportion
ab=cd means a×d=b×c\frac{a}{b} = \frac{c}{d} \text{ means } a \times d = b \times c
A proportion says two ratios are equal. Here each ratio is ounces over dollars. a and c are the tops, and b and d are the bottoms. Multiply each top by the other bottom, and the two products are equal. That turns the proportion into an equation.
  1. Sort out the question

    Milk costs $3.99 for 1 gallon. Orange juice comes in a 6-pack of 5.5-fluid-ounce bottles for $1.79. You need how much more a gallon of juice would cost than a gallon of milk. Round to the nearest dollar.

    The question doesn’t say how big a gallon is. You must know this fact: 1 gallon is 128 fluid ounces.

  2. Find the ounces in a 6-pack

    Each of the 6 bottles holds 5.5 fluid ounces:

    6×5.5=336 \times 5.5 = 33

    So $1.79 buys 33 fluid ounces of juice.

  3. Set up a proportion

    Let x be the cost in dollars of a gallon, 128 fluid ounces, of juice. The price per ounce stays the same. So ounces over dollars is the same for the pack and the gallon:

    331.79=128x\frac{33}{1.79} = \frac{128}{x}

    Cross-multiply:

    33×x=1.79×128=229.1233 \times x = 1.79 \times 128 = 229.12

  4. Find the cost of a gallon

    Divide both sides by 33:

    x=229.12÷33≈6.94x = 229.12 \div 33 \approx 6.94

    The sign ≈ means about equal. The exact value is 6.943…, so a gallon of juice costs about $6.94.

    Watch out: don’t round too early. One ounce costs about $0.054. Rounding that to $0.05 makes a gallon $6.40, and your answer becomes 2.

  5. Subtract the milk’s price

    Find how much more the juice costs:

    6.94−3.99=2.956.94 - 3.99 = 2.95

    A gallon of juice costs $2.95 more than a gallon of milk.

    Watch out: entering 7, the juice’s price, forgets to subtract the milk.

  6. Round and enter it

    The question asks for the nearest dollar. $2.95 is between $2 and $3. Its cents, 95, are 50 or more, so it rounds up to $3. Do not enter 2.95.

    Type only the number in the box. So you would enter 3.

Check your answer

Estimate. 128 ounces is almost 4 packs, since 4×33=1324 \times 33 = 132. Four packs cost 4×1.79=7.164 \times 1.79 = 7.16 dollars, so a gallon costs a bit less than that. Milk costs about $4. About $7 minus $4 is $3, which matches.

The original DOE question
Question 15 as shown on the DOE’s digital test, page 88 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test C Math, page 88, question 15. This is the DOE’s version, before the corrections listed in the analysis.

Question 5 of 6

2024 Test A Math, question 16

Question

A truck rental company charges a one-time fee of $40 plus $1 per mile driven. Dalia rented a truck and used a coupon for 20% off the total rental cost. After the coupon was applied, she spent a total of $60. How many miles did she drive?

  • A8
  • B20
  • C32
  • D35
Show the answer

Correct answer: D

Let m equal the miles driven during Dalia’s rental. The total cost, c, can be expressed as an equation.

c=40+mc = 40 + m

Dalia spent a total of $60, which included a 20% off coupon. Find the cost before the coupon by dividing the final cost ($60) by the percentage paid (80%):

6080%=600.80=75\frac{60}{80\%} = \frac{60}{0.80} = 75

Use $75 as the total cost to find m.

40+m=7540 + m = 75

m=35m = 35

Dalia drove 35 miles.

Question analysis

A multistep word problem about a truck rental that costs a $40 fee plus $1 per mile, with a 20% off coupon applied to the total. Given that Dalia paid $60 after the coupon, students find how many miles she drove. Undoing the discount correctly before removing the fixed fee is the hard part.

NYS standards
  • 7.EE.4a two-step equations
  • 7.RP.3 multistep ratio and percent problems
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students solve two-step equations and work with discounts, usually separately. This combines them and makes students undo a 20% discount starting from the price paid, a reversal many grade 7 students have not practised.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Solution
3 steps
Reading level
Question and choices: about grade 5, 43 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 ordinary skills, undoing a discount and solving a two-step equation, are stacked, and the first runs in reverse. The natural way to undo 20% off is to add 20% back, which gives $72 instead of $75 and a choice three miles from the answer. Students who check by applying the coupon to the full cost of 40 + 35 = 75 dollars see at once that 80% of it is 60.

  1. Turning words into math moderate Understanding the question

    The sentence "After the coupon was applied, she spent a total of $60" makes 60 the price after 20% off, so it is 80% of the full cost, 40 + m. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  2. Combines ideas moderate Working it out

    A percent discount has to be undone before the fee-plus-mileage equation can be solved: 60 ÷ 0.8 = 75, then 75 − 40 = 35 miles. (Embretson, 1998; Daniel & Embretson, 2010)

  3. A very close wrong answer moderate Choosing the answer

    The choices 32 and 35 are three miles apart, too close to tell apart by estimate; only dividing 60 by 0.8, rather than adding 20% of 60, separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Since $60 is 80% of the full cost, divide 60 by 0.80 to find the cost before the coupon, then subtract the $40 fee to get the miles.

Traps

  • Percent of the wrong amount: Finding the whole from a part, or a percent change measured from the wrong starting amount.
  • Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Rental cost
c=40+1×mc = 40 + 1 \times m
c is the full cost in dollars, before any coupon. 40 is the one-time fee, paid once no matter how far you drive. m is the number of miles, and each mile costs 1 dollar.
Price after a discount
price paid=full cost×(1−p100)\text{price paid} = \text{full cost} \times \left(1 - \frac{p}{100}\right)
The full cost is the price before the discount. p is the percent taken off. You pay what is left: 100% minus p%. For 20% off you pay 80%, so you multiply by 0.80.
  1. Sort out the question

    The truck costs a $40 fee, plus $1 for each mile. Dalia’s coupon took 20% off the total. After the coupon, she paid $60. You need how many miles she drove. The $60 is the price after the discount, not the full cost.

  2. Write what she paid

    20% off means she paid what was left of the full cost.

    100%−20%=80%100\% - 20\% = 80\%

    As a decimal, 80% is 0.80.

    price paid=full cost×(1−p100)\text{price paid} = \text{full cost} \times \left(1 - \frac{p}{100}\right)

    60=full cost×0.8060 = \text{full cost} \times 0.80

  3. Undo the discount

    To undo times 0.80, divide by 0.80. Move the decimal point one place in both numbers to make it easier.

    full cost=60÷0.80=600÷8=75\text{full cost} = 60 \div 0.80 = 600 \div 8 = 75

    The full cost was $75. Watch out: the 20% was taken off the full cost, not off $60. Adding 20% of $60 gives $72 and leads to 32 miles, choice C. Taking 20% off $60 again leads to 8 miles, choice A.

  4. Find the miles

    c=40+1×mc = 40 + 1 \times m

    75=40+m75 = 40 + m

    m=75−40=35m = 75 - 40 = 35

    She drove 35 miles. Watch out: do not skip the coupon. Using $60 as the full cost gives 20 miles, choice B. So the answer is D, 35.

Check your answer

Go forward. 35 miles cost $35. Add the $40 fee to get $75. The coupon takes off 20% of $75, which is 0.20×75=150.20 \times 75 = 15 dollars. Then 75−15=6075 - 15 = 60. That is exactly what she paid.

The original DOE question
Question 73 as printed on page 79 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 79, question 73. Open page 79 of the handbook (PDF)

Question 6 of 6

2018 Test A Math, question 5

Question

The sum of two consecutive integers is −15. If 1 is added to the smaller integer and 2 is subtracted from the larger integer, what is the product of the two resulting integers?

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

Show the answer

Correct answer: 63

If x is the smaller consecutive integer, then

x+1x+1

is the larger consecutive integer. Use their sum (−15) to find x:

x+(x+1)=−15x+(x+1)=-15

2x+1=−152x+1=-15

2x=−162x=-16

x=−8x=-8

The two consecutive integers are −8 and −7. One is added to the smaller integer:

−8+1=−7-8+1=-7

and 2 is subtracted from the larger integer:

−7−2=−9-7-2=-9

Find the product:

(−7)(−9)=63(-7)(-9)=63

Question analysis

A grid-in problem about two consecutive integers whose sum is −15. Students find the two integers, add 1 to the smaller and subtract 2 from the larger, then multiply the results. Deciding which negative integer is the smaller one, and keeping signs straight through every step, are what make it tricky.

NYS standards
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 A typical grade 7 task solves one two-step equation or multiplies signed numbers. This links both: set up a consecutive-integer equation, solve it with negatives, adjust each integer, and multiply, with no choices to check against.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Solution
4 steps
Reading level
Question and choices: about grade 9, 30 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

Every step is routine, yet the question collects most of the ways students lose points with negatives. The crucial judgment is which of −8 and −7 is the smaller; a student who picks −7 still produces a tidy, wrong product. Gridding the product of the original pair, or giving a negative product, are the other usual losses. Writing each intermediate pair down, rather than holding them in mind, protects against all three.

  1. Error-prone arithmetic moderate Working it out

    Every number in play is negative: −8 and −7, then −7 and −9, and a product of two negatives that must come out positive. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. Turning words into math moderate Understanding the question

    "1 is added to the smaller integer and 2 is subtracted from the larger" depends on knowing that −8, not −7, is the smaller, since it lies farther below zero. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  3. Many steps moderate Working it out

    Four linked steps: set up x + (x + 1) = −15, solve for x, adjust each integer, then multiply; an error early in the chain carries through to the product. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Halve the sum to see which two integers sit on either side of it, remember that the smaller negative is farther from zero, then adjust each and multiply.

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

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Consecutive integers
n,n+1n, \; n + 1
Integers are whole numbers and their opposites, like −2, −1, 0, 1 and 2. Consecutive integers come one right after another. If n is the smaller one, the next one is n + 1, because each step up the number line adds 1.
Multiplying two negative numbers
(−a)×(−b)=a×b(-a) \times (-b) = a \times b
a and b are positive numbers, so −a and −b are negative. When you multiply two negative numbers, the answer is positive. The two minus signs cancel out.
  1. Sort out the question

    Two consecutive integers add up to −15. Integers are whole numbers and their negatives. Consecutive means one comes right after the other, like 4 and 5. You need to do three things:

    1. Find the two integers.
    2. Add 1 to the smaller one. Subtract 2 from the larger one.
    3. Find the product of the two new numbers. The product is the answer when you multiply.
  2. Find the two integers

    Call the smaller integer n. Then the larger one is n + 1. Their sum is −15.

    n+(n+1)=−15n + (n + 1) = -15

    2n+1=−152n + 1 = -15

    Subtract 1 from both sides. Then divide both sides by 2.

    2n=−162n = -16

    n=−8n = -8

    So the integers are −8 and −7.

  3. Decide which one is smaller

    On a number line, −8 is to the left of −7. Numbers to the left are smaller. So −8 is the smaller integer, and −7 is the larger one. Watch out: 8 is bigger than 7, but −8 is smaller than −7. Mixing them up gives −6 and −10, and the wrong answer 60.

  4. Change each integer

    Add 1 to the smaller integer.

    −8+1=−7-8 + 1 = -7

    Subtract 2 from the larger integer.

    −7−2=−9-7 - 2 = -9

    The two new numbers are −7 and −9.

  5. Multiply the new numbers

    Both numbers are negative, so their product is positive.

    (−7)×(−9)=63(-7) \times (-9) = 63

    Watch out: the answer is not −63, because two negatives make a positive. Also, don’t multiply the first pair, −8 and −7. That gives 56 and skips the changes. So you would grid 63.

Check your answer

Check the pair: −8+(−7)=−15-8 + (-7) = -15, and they are one apart. Half of −15 is −7.5, and −8 and −7 sit on either side of it. Then 7×9=637 \times 9 = 63, and the sign is positive.

The original DOE question
Question 62 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 62. Open page 81 of the handbook (PDF)

Sample #31