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

2025 Practice Test A English, question 17
Passage Excerpt from Winter Wheat

Excerpt from Winter Wheat

by Mildred Walker

Ellen, the narrator, is preparing to leave her parents and the family farm for college the next day.

I love Dad’s way of talking that makes him seem different from other ranchers. He’s lived here twenty-three years, but he still says “back East where I come from.” He’s the one who gets excited when I do about spring coming or a serial running in the magazine we’re both reading, but it’s what Mom says that I depend on. When Mom used to say “Don’t worry” about my pet chicken or dog or new calf, it always got well. Dad is always talking of going some place, not now, but next year, maybe. Mom seems to think of nothing farther away than today or perhaps yesterday or tomorrow morning.

Mom folded the ironing board and put it inside their bedroom that was just off the kitchen. She carried in the freshly ironed clothes. Dad went back to his paper. When Mom came back she took beans from the cupboard to soak for tomorrow. Dad always said Mom could make all the dishes he’d had back in Vermont as well as though she were a New Englander herself, instead of a Russian. All of a sudden, I realized that tomorrow when those beans would be ready to eat I’d be going away. It gave me a funny feeling.

“I’ll be taking the train tomorrow night,” I said aloud, more to hear it myself.

“We can drive you into town in the afternoon,” Dad said, dropping his paper on the floor.

“There’s no need to go to town; she can catch the train at Gotham just as well. We haven’t nothing to take us into town for,” Mom said.

“Well, we don’t have to decide tonight,” Dad said, but I knew he wanted to go into Clark City. It wouldn’t be so flat as just seeing me go off on the train from Gotham. My going away was hard on both of them; they were so different—and I was part of them both. It made me uncomfortable to think of leaving them.

While I was getting ready for bed in my room that’s off the front room, I saw how it would be if I left from town. We’d go in right after dinner and go around to the stores, Dad going one way and Mom and I another. Dad would probably have his hair cut at the barbershop and stop in the bank and meet someone he knew to talk to. Then we’d meet at the big store on the corner and go to the cafeteria for supper. The train stops ten minutes or so at the station in town and there are other people and excitement and you have time to wave from the platform and then again from your window by your seat. We went to the station in Clark City to see the Goodals off when they went back to Iowa.

If I left from Gotham, we’d just drive down in the truck and wait till the train came. It only stops long enough for you to get on and you hardly have time to taste the flavor of going away.

I sat on the bed in my pyjamas with my arms around my knees. I couldn’t keep from thinking of that time Dad went back East. I tried to, and then I just sat still and looked straight at it. Sometimes that’s better than working so hard to keep from looking at what’s in your mind.

Dad went all the way back to Vermont. . . . It was in November and it was already dark when the train came through Gotham. Even now, I could feel how cold and dark it was. I held Mom’s hand. Dad was so dressed-up he seemed strange. . . . We stood there without saying anything until Dad told Mom to remember to call Mr. Bardich, our neighbor, if the cow didn’t calve tomorrow.

“I’ll manage,” Mom snapped back.

“I wish you could go, Anna,” Dad said to Mom, “and we could take Ellen.” . . .

“Good-by, Anna Petrovna,” he said, looking at Mom. I had never heard him call her by two names before.

“Good-by,” Mom said, standing still, without smiling.

Then he was gone and the crossroads were darker than ever. The train light shone on the high window in the top of the grain elevator for a moment and then that too was dark. We got into our old Ford and Mom drove back to the house. My throat ached all the way. The name Dad had called Mom kept saying itself in my ears: “Anna Petrovna, Anna Petrovna.” . . .

Our house seemed lonely when we came back to it. It seemed to be hiding under the coulee. I went with Mom to put the truck in the barn that was bigger than the house. I think Mom was prouder of our barn than the house, anyway. We walked back to look at the cow that was going to calve. She was just a big light blob in the dark, waiting. I had thought she was exciting this morning, but now she seemed sad, too.

The wind blew when we walked across the open space to the house and I couldn’t help shivering with the cold. Inside the house it was warm, but empty.

“Bring your nightgown in here and I heat you some milk,” Mom said.

I drank the milk sitting on a stool in front of the stove. It tasted good, but the lonely ache in my throat was still there. I picked up my clothes and hung them neatly behind the stove and put my cup on the sink board. Mom was fixing oatmeal for tomorrow morning.

“Good night, Mom,” I said almost timidly, standing beside her. She seemed wrapped around in a kind of strangeness. Then she turned around and drew me to her. The front of her dress was warm from the stove. I felt the comfortable heat through my gown. She laid her hand against my face and it felt rough and hard but firm. I dared ask her something I wanted to know.

“Mom, was that really your name—what Dad called you?”

Her voice sounded surprised. “Why, Yeléna, you know that; Anna Petrovna. You know I am born in Russia, in Seletskoe.”

“Yes, but I didn’t know your other name,” I said.

“Anna Petrovna Webb.” She pronounced it slowly. “Once I think what a funny name Ben Webb is!” She laughed. Her laugh was warm and low like our kitchen, and comfortable. The house seemed natural and right again. . . .

But now that I am grown, I feel the wall of strangeness between them, more than when I was a child. I wondered how they would get along without me.

From WINTER WHEAT by Mildred Walker, published by University of Nebraska Press. Copyright © 1944 by Harcourt, Brace and Company, Inc. Copyright renewed 1971 by Mildred Walker. All rights reserved.

serial: story published in short segments at regular intervals
coulee: small gulch or ravine

Question

Read these sentences from paragraph 2.

Dad always said Mom could make all the dishes he’d had back in Vermont as well as though she were a New Englander herself, instead of a Russian. All of a sudden, I realized that tomorrow when those beans would be ready to eat I’d be going away. It gave me a funny feeling.

The sentences help develop a theme of the excerpt by

  • Asuggesting that life presents people with many challenges.
  • Bimplying that the stress of major life events can cause confusion.
  • Cdemonstrating that moving on from the familiar is a common human experience.
  • Demphasizing the idea that people can easily learn the routines of being part of a new culture.
Show the answer

Correct answer: C

The question asks how the sentences in paragraph 2 help develop a theme of the excerpt.

A. Incorrect. Although the sentences from paragraph 2 describe Ellen’s “funny feeling” upon realizing that she will no longer be home when the beans that her mother is making are ready to eat, the details do not show that life presents many challenges. Additionally, the theme that life presents people with many challenges is not a theme found in the excerpt.

B. Incorrect. The sentences from paragraph 2 state that Ellen has “a funny feeling” while awaiting a major life event, but they do not indicate that she is confused or stressed. The details in the sentences convey a sense of wistfulness, but the theme that the stress of major life events causes confusion is not found in the excerpt.

C. CORRECT. The sentences from paragraph 2 show that Ellen’s mother has moved on from the familiar and left her childhood home behind, just as Ellen is about to do. This information links the experiences of parent and child and helps develop the theme that moving beyond the familiar is a common human experience.

D. Incorrect. Although the sentences from paragraph 2 show that Ellen’s mother has learned to cook New England food very well (“as though she were a New Englander herself”), they do not provide information about how easy or difficult it was for Ellen’s mother to learn a new way of cooking. The theme that people can easily learn the routines of a new culture is not a theme found in the excerpt.

Question analysis

This theme question quotes three sentences from paragraph 2 of the Winter Wheat excerpt, about Mom’s cooking and Ellen’s sudden realization that she is leaving, and asks how they help develop a theme. Students must connect a small kitchen moment to an idea that runs through the whole excerpt, choosing among theme statements that range from narrow to very general.

NYS standards
  • 7R2 theme or central idea
  • 7R5 structure and genre features
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Explaining how details develop a theme is a grade 7 standard. These sentences never state the idea, so students must infer it from a remark about Mom’s cooking and Ellen’s passing feeling, then link it to the whole excerpt, more than typical grade 7 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 4, 1179 words; question and choices: about grade 7, 117 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 quoted sentences seem to be about cooking and a passing mood, and the link to a theme hides in where the parents come from. Students who miss that Dad and Mom both left homes far away read the passage as being about Mom’s skill or Ellen’s nerves, and pick a theme built on one of those. The vague choice about life’s challenges catches students who sense that leaving is hard but cannot pin the theme to these details.

  1. Reading between the lines major Working it out

    The theme is never stated: students must notice that “back in Vermont” and “instead of a Russian” mean both parents left their first homes, just as Ellen realizes she will be “going away.” (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

  2. Evidence in several places moderate Working it out

    A theme of the excerpt must hold beyond paragraph 2, so the choice has to be checked against Dad’s trip back East in the flashback and Mom’s Russian name in paragraph 22. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

  3. How, not what minor Working it out

    The stem asks how the sentences develop a theme, so a remark about cooking has to be read for what it reveals about the parents’ pasts, not for its subject. (Anderson & Krathwohl, 2001)

Strategy: Note what each quoted sentence tells you about the person in it, then pick the theme that these details and the rest of the excerpt both support.

Traps

  • Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.
  • 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 how three sentences from paragraph 2 help build a theme. A theme is a big idea about life that a story keeps showing. Look at what each sentence tells you about the people in it, not just at its subject, cooking.

  2. Read the sentence about Mom and Dad

    The first sentence says “Dad always said Mom could make all the dishes he’d had back in Vermont as well as though she were a New Englander herself, instead of a Russian.” Vermont is a state far to the east, in New England. Dad grew up there. Mom came from Russia. So both parents left the places where they grew up.

  3. Read the sentences about Ellen

    The next sentences say “All of a sudden, I realized that tomorrow when those beans would be ready to eat I’d be going away. It gave me a funny feeling.” The beans are soaking for tomorrow’s meal. Ellen sees that she won’t be home to eat them. She is about to leave her own home, just as her parents once left theirs.

  4. Find the theme

    Put the two parts together. Dad left Vermont, Mom left Russia, and now Ellen is leaving the farm. Each of them moves on from a familiar place. So the sentences show that leaving what you know is something many people go through. Ellen’s “funny feeling” is wistful, a quiet sadness, like missing home before you go.

  5. Check each choice

    • A does not fit. It is so broad it could fit almost any story. These sentences show leaving home, not many challenges.
    • B does not fit. Ellen’s feeling is quiet and wistful. Nothing shows her stressed or confused.
    • C fits. Parents and daughter all move on from a familiar home.
    • D does not fit. Dad praises Mom’s cooking, but the sentence never says learning it was easy.
  6. Choose the answer

    The sentences link Ellen’s leaving to her parents’ own leaving long ago. So the answer is C.

Why this answer and not the closest wrong one

B is the closest wrong choice, because a “funny feeling” before a big change could sound like stress. But Ellen is not mixed up or panicked. She simply realizes she will be gone tomorrow. The sentences tie that moment to her parents’ moves from Vermont and Russia. That shared experience of leaving is C.

The original DOE question
Question 17 as shown on the screen of the DOE’s digital test.
As shown on the screen of the DOE’s digital test: 2025 Practice Test A English, question 17.
The passage “Excerpt from Winter Wheat” as shown on the testThe passage “Excerpt from Winter Wheat” as shown on the DOE’s digital test, screen 1 of 7.The passage “Excerpt from Winter Wheat” as shown on the DOE’s digital test, screen 2 of 7.The passage “Excerpt from Winter Wheat” as shown on the DOE’s digital test, screen 3 of 7.The passage “Excerpt from Winter Wheat” as shown on the DOE’s digital test, screen 4 of 7.The passage “Excerpt from Winter Wheat” as shown on the DOE’s digital test, screen 5 of 7.The passage “Excerpt from Winter Wheat” as shown on the DOE’s digital test, screen 6 of 7.The passage “Excerpt from Winter Wheat” as shown on the DOE’s digital test, screen 7 of 7.

Question 2 of 6

2024 Test B English, question 40
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

How does the map provide additional support for a central idea of the passage?

  • Aby demonstrating that Morse’s telegraph system greatly exceeded the limitations of previous long-distance communication systems
  • Bby indicating that Morse was lacking foresight by seeking a patent only in North America
  • Cby proving that Gale’s advice helped Morse extend the telegraph’s range much farther than first thought possible
  • Dby showing that the Morse telegraph had a surprisingly immense influence across the world
Show the answer

Correct answer: A

The question asks how the map provides additional support for a central idea of the passage.

A. CORRECT. The map shows the telegraph line crossing continents in 1902, just forty-four years after the first transatlantic message was sent in 1858. This is a much greater distance than the first line Morse built between “the Supreme Court chamber of the Capitol building and the railroad station in Baltimore” (paragraph 7).

B. Incorrect. The map shows the distance the line covers; it does not indicate Morse’s lack of foresight in seeking patents only in North America.

C. Incorrect. While Gale’s advice was crucial to Morse’s success, the map shows the physical line stretching across continents many years later, after their initial work.

D. Incorrect. While it can be inferred that the telegraph had influence across the world, the amount of influence cannot be determined.

Question analysis

This question pairs “Invention of the Telegraph” with a map of the British All-Red Telegraph Line in 1902 and asks how the map adds support for a central idea of the passage. Students must read a world map with a key and a caption, connect what it shows to the story of Samuel Morse’s invention, and sort through four claims about the map that vary in how far they go.

NYS standards
  • 7R1 textual evidence and inferences
  • 7R2 theme or central idea
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Linking a graphic to a text is familiar grade 7 work. Here students must pin down a central idea of the passage and work out how a 1902 world map, made decades after Morse’s first line, bears on it, a step beyond typical demand.
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 11, 910 words; question and choices: about grade 11, 75 words.
Corrected from the DOE handbook
  • An inaccurate detail in the explanation 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

Lines on a map can prove how far the telegraph reached, and the question turns on how much more than that each choice claims. The key depends on a link back to paragraph 1, where a message went only as far as the next hilltop tower, read many minutes before the map. The influence choice skips that link and states the map’s message in bigger terms, so students who do not ask what a map of routes can show are drawn to it.

  1. A very close wrong answer major Choosing the answer

    The influence choice, a telegraph with “a surprisingly immense influence across the world,” sounds like a fair reading of lines circling the globe; it differs from the key in claiming impact and surprise rather than reach beyond earlier systems. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  2. Evidence in several places moderate Working it out

    The earlier systems the key measures against are in paragraph 1, towers on distant hills passing flag and light signals from station to station, far from the map at the end of the passage. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

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

    The map shows a 1902 British line from London to Canada and Vancouver, south through Bermuda and Barbados to Cape Town, and on to India, Australia and New Zealand, with Washington left unconnected. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

Strategy: Look at what the map and its caption actually show, sum up a central idea of the passage in your own words, and keep only a choice that both of them back up.

Traps

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: The handbook says the 1902 line came “forty-four years after the first message was sent in 1858.” The first telegraph message was Morse’s in 1844 (paragraph 1); 1858 is the year of the first transatlantic message, as the map’s caption says. The site now says “the first transatlantic message.”

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Figure out what the question asks

    This question asks how a graphic adds to the passage. You need two things. First, what does the map really show? Second, which central idea, a main point the whole passage builds, does it back up?

  2. Read the map and its caption

    This map is a picture of one British network of telegraph wires, as it stood in 1902. In its key, a thick black line stands for an electric telegraph line. From London, a line crosses the Atlantic Ocean and Canada to Vancouver. Other lines reach Bermuda, Barbados, Cape Town, India, Australia and New Zealand. According to the caption, by 1902 this network “connected most of the world.”

  3. Recall the older ways of signaling

    Next, compare that with paragraph 1. Before electricity, people signaled with flags or lights. An observer would “decipher a signal from a high tower on a distant hill and then send it on to the next station.” So a message could only jump as far as a person could see. Even Morse’s first line, in paragraph 6, reached Baltimore, just “forty miles away.”

  4. Connect the map to that idea

    The caption says the first message across the Atlantic was sent in 1858. Only 44 years later, in 1902, lines crossed several oceans and linked much of the globe. Paragraph 8 adds that telegraph lines “connected the continents of Europe and America” before Morse died. So the map supports a central idea: electric telegraph lines went far beyond the reach of older signal systems.

  5. Check each choice

    • A fits. Lines that span oceans go far past the limits of hilltop towers.
    • B does not fit. A map of lines shows nothing about patents. Also, paragraph 5 says Morse sought patent protection in Europe, too.
    • C does not fit. The map names no inventors, so it cannot prove what Gale’s advice did decades before.
    • D does not fit. The map shows where lines went, not how much they changed the world, and nothing on it shows surprise.
  6. Choose the answer

    The map shows telegraph lines reaching around the globe, far past what flags and towers could do. So the answer is A.

Why this answer and not the closest wrong one

D is the closest trap, because lines around the world seem to prove a huge effect. But a map of routes shows only how far messages could travel. It cannot measure influence, and it cannot show that anyone was surprised. A sticks to what a map of lines can prove: the telegraph reached much farther than earlier systems.

The original DOE question
Question 40 as printed on page 188 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 188, question 40. Open page 188 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 “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 3 of 6

2026 Practice Test D English, question 8
Passage Excerpt from “How We Made the First Flight”

Excerpt from “How We Made the First Flight”

by Orville Wright

In 1903 brothers Wilbur and Orville Wright conducted experiments related to flying machines. These experiments would eventually lead to air travel becoming a reliable form of transportation.

During the night of December 16, 1903, a strong cold wind blew from the north. When we arose on the morning of the 17th, the puddles of water, which had been standing about camp since the recent rains, were covered with ice. The wind had a velocity of 10 to 12 meters per second (22 to 27 miles an hour). We thought it would die down before long, and so remained indoors the early part of the morning. But when ten o’clock arrived, and the wind was as brisk as ever, we decided that we had better get the machine out and attempt a flight. We hung out the signal for the men of the Life Saving Station.1 We thought that by facing the flyer into a strong wind, there ought to be no trouble in launching it from the level ground about camp. We realized the difficulties of flying in so high a wind, but estimated that the added dangers in flight would be partly compensated for by the slower speed in landing.

Final Preparations

We laid the track on a smooth stretch of ground about one hundred feet north of the new building. The biting cold wind made work difficult, and we had to warm up frequently in our living room, where we had a good fire in an improvised stove made of a large carbide2 can. By the time all was ready, J. T. Daniels, W. S. Dough and A. D. Etheridge, members of the Kill Devil3 Life Saving Station; W. C. Brinkley of Manteo, and Johnny Moore, a boy from Nags Head,4 had arrived.

We had a “Richard” hand anemometer with which we measured the velocity of the wind. Measurements made just before starting the first flight showed velocities of 11 to 12 meters per second, or 24 to 27 miles per hour. . . .

Audacity—and Calculation

Wilbur having used his turn in the unsuccessful attempt on the 14th, the right to the first trial now belonged to me. After running the motor a few minutes to heat it up, I released the wire that held the machine to the track, and the machine started forward in the wind. Wilbur ran at the side of the machine, holding the wing to balance it on the track. Unlike the start on the 14th, made in a calm, the machine, facing a 27-mile wind, started very slowly. Wilbur was able to stay with it till it lifted from the track after a forty-foot run. One of the Life Saving men snapped the camera for us, taking a picture just as the machine had reached the end of the track and had risen to a height of about two feet. The slow forward speed of the machine over the ground is clearly shown in the picture by Wilbur’s attitude. He stayed along beside the machine without any effort.

Flight

The course of the flight up and down was exceedingly erratic, partly due to the irregularity of the air, and partly to lack of experience in handling this machine. The control of the front rudder was difficult on account of its being balanced too near the center. This gave it a tendency to turn itself when started; so that it turned too far on one side and then too far on the other. As a result the machine would rise suddenly to about ten feet, and then as suddenly dart for the ground. A sudden dart when a little over a hundred feet from the end of the track, or a little over 120 feet from the point at which it rose into the air, ended the flight. As the velocity of the wind was over 35 feet per second and the speed of the machine over the ground against this wind ten feet per second, the speed of the machine relative to the air was over 45 feet per second, and the length of the flight was equivalent to a flight of 540 feet made in calm air. This flight lasted only 12 seconds, but it was nevertheless the first in the history of the world in which a machine carrying a man had raised itself by its own power into the air in full flight, had sailed forward without reduction of speed and had finally landed at a point as high as that from which it started.

From “How We Made the First Flight” by Orville Wright—Public Domain/Federal Aviation Administration

1Life Saving Station: one of the rescue stations along the Atlantic coastline that provided assistance to mariners in distress

2carbide: a very hard material composed of carbon and other heavy metals

3Kill Devil: the town of Kill Devil Hills in eastern North Carolina

4Nags Head: a town in eastern North Carolina

Question

How does paragraph 1 introduce the idea that the Wright brothers knew that their flight attempt was risky?

  • Athrough the mention of a signal to notify lifesaving experts that the flight attempt was about to begin
  • Bby providing specific details about the speed of the wind and the Wright brothers’ response to the windy conditions
  • Cby suggesting that a slower landing would be necessary at the end of the flight in order to maintain safety
  • Dthrough the indication that the Wright brothers waited indoors for most of the morning because of the poor weather
Show the answer

Correct answer: A

The question asks how paragraph 1 introduces the idea that the Wright brothers knew that their flight attempt was risky.

A. CORRECT. Paragraph 1 describes the Wright brothers hanging out “the signal” to notify the lifesaving crew to stand by as they attempted flight; the need for lifesaving experts to be available in case of an accident introduces the idea that the attempted flight might crash.

B. Incorrect. The specifics on the speed of the wind by themselves do not emphasize the danger of the flight. While the brothers admitted that the strong wind posed certain risks, they also noted that the windy conditions might make landing safer (“estimated that the added dangers in flight would be partly compensated for by the slower speed in landing” [paragraph 1]).

C. Incorrect. A slower landing was expected as a result of the windy conditions, but it was not part of a plan to increase the safety of the flight.

D. Incorrect. While the Wright brothers’ initial decision to wait to see whether the wind would die down does suggest they were concerned about the safety of the flight, the explanation of their decision to proceed with the flight shows that they expected one benefit from the poor weather: “the slower speed in landing” (paragraph 1).

Question analysis

This question on Orville Wright’s account “How We Made the First Flight” asks how the opening paragraph, which describes a cold, windy December morning at the brothers’ camp, introduces the idea that the Wright brothers knew their flight attempt was risky. Students must judge which of four details from the paragraph does that work, and the paragraph mentions both dangers and reasons for confidence.

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 Finding how a paragraph introduces an idea is grade 7 work, but this paragraph is long and old-fashioned, every choice points to something it mentions, and students must decide which detail most clearly signals that the brothers saw a risk.
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 7, 816 words; question and choices: about grade 10, 93 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 1 states the danger outright only in its last sentence, and that sentence then weighs it against a gentler landing, so the choice built on the brothers’ response to the wind is the strongest pull. The key rests on a quieter detail whose meaning comes from a footnote: men from a rescue station were signaled before the attempt. Students must decide which detail shows the brothers preparing for trouble, not just noticing the weather, and that test is narrower than most expect.

  1. Reading between the lines moderate Working it out

    The key never says danger: students must connect “the signal for the men of the Life Saving Station” to footnote 1, rescue stations for “mariners in distress,” and see that summoning rescuers means expecting trouble. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

  2. Every choice must be checked moderate Choosing the answer

    All four details come from paragraph 1, and its plainest admission of danger, “We realized the difficulties of flying in so high a wind,” sits inside a wrong choice, so each detail must be weighed for what it shows. (Embretson & Wetzel, 1987)

  3. Complex sentences minor Understanding the question

    The paragraph’s last sentence sets danger against benefit, “the added dangers in flight would be partly compensated for by the slower speed in landing,” and students must untangle it to see the brothers playing the risk down. (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)

Strategy: Locate each of the four details in paragraph 1, reread the sentence it sits in, and ask what that detail on its own reveals about what the brothers feared could go wrong.

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.
  • Twists what the text says: Uses the text's topic and ideas but changes what it says: who did it, how much, when, or why.

The wrong answers, classified

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 paragraph 1 first shows that the Wright brothers knew their flight was risky. Risky means it could end badly. All four choices name details from paragraph 1. Your job is to find the detail that shows them getting ready for trouble.

  2. Find the signal in paragraph 1

    Paragraph 1 says “We hung out the signal for the men of the Life Saving Station.” The small number 1 after “Station” points to footnote 1 at the end of the excerpt. Read it. A Life Saving Station was “one of the rescue stations along the Atlantic coastline that provided assistance to mariners in distress.” Mariners are sailors. Distress means danger or trouble.

  3. Work out what the signal means

    The brothers called rescue workers before they tried to fly. People call rescuers only when someone might get hurt. So the signal shows the brothers knew the flight could go wrong. They wanted help close by in case of a crash.

  4. Check each choice

    • A fits. Signaling rescuers shows the brothers planned for trouble.
    • B does not fit. The wind speeds are only numbers. The brothers’ response was to fly anyway, thinking “there ought to be no trouble in launching.”
    • C does not fit. The slower landing was a side effect of the wind. They counted it as a plus, not as a safety step they needed.
    • D does not fit. They stayed inside only “the early part of the morning,” hoping the wind would die down. Then they went out in the same wind.

    Watch out: B is tempting because the paragraph says “We realized the difficulties of flying in so high a wind.” But that same sentence weighs those dangers against a slower landing. It plays the risk down.

  5. Choose the answer

    The call to the rescue station is the detail that shows the brothers preparing for danger. So the answer is A, “through the mention of a signal to notify lifesaving experts that the flight attempt was about to begin.”

Why this answer and not the closest wrong one

B is the closest wrong choice. The paragraph does admit the wind made flying harder. But the brothers then argue the danger down: a slower landing would partly make up for it. A shows them acting on the risk. They called in rescuers before they flew. You only do that if you think something could go wrong.

The original DOE question
Question 8 as shown on the DOE’s digital test, page 15 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test D English, page 15, question 8.
The passage “Excerpt from “How We Made the First Flight”” as printed (pages 13–14)The passage “Excerpt from “How We Made the First Flight”” as printed on page 13 of the DOE handbook.The passage “Excerpt from “How We Made the First Flight”” as printed on page 14 of the DOE handbook.

Question 4 of 6

2024 Test B Math, question 14

Question

Karen reads at the rate of 60 pages per day. Martina reads 25% more pages per day than Karen. If the two students begin 1,500-page reading assignments on the same day, how many days sooner will Martina complete the assignment than Karen?

  • A4
  • B5
  • C7
  • D15
Show the answer

Correct answer: B

First, determine the rate at which Martina reads, in pages per day, using the rate at which Karen reads, 60 pages per day.

60+60∙25100=60+60\bullet\frac{25}{100}=

60+60∙0.25=60+60\bullet 0.25=

60+15=7560+15=75

Next, determine the number of days it will take each person to complete the 1,500-page reading assignment.

Karen: 1,500 pagesk days=60 pages1 day\frac{1{,}500\text{ pages}}{k\text{ days}}=\frac{60\text{ pages}}{1\text{ day}}

60k=1,50060k=1{,}500

(60÷60)k=1,500÷60(60\div 60)k=1{,}500\div 60

k=25 daysk=25\text{ days}

Martina: 1,500 pagesm days=75 pages1 day\frac{1{,}500\text{ pages}}{m\text{ days}}=\frac{75\text{ pages}}{1\text{ day}}

75m=1,50075m=1{,}500

(75÷75)m=1,500÷75(75\div 75)m=1{,}500\div 75

m=20 daysm=20\text{ days}

Finally, determine the difference between the two numbers of days.

25−20=525-20=5

So Martina will complete the assignment 5 days sooner than Karen.

Question analysis

A reading-rate word problem: Karen reads 60 pages a day, and Martina reads 25% more pages a day. Both start a 1,500-page assignment on the same day, and students find how many days sooner Martina finishes. It combines a percent increase with rates and times, and asks for a difference in days.

NYS standards
  • 7.RP.3 multistep ratio and percent problems
  • 6.RP.3b unit rate problems
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 percent problems usually stop after one increase or decrease. This one turns the percent into a new rate, finds two finishing times and compares them, four steps that go beyond a typical grade 7 task.
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 6, 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

Each step is easy on its own; the difficulty is in carrying the chain to the end. Three of the choices are numbers a student meets or makes along the way: the extra pages per day, a quotient of two rates, or a rate built on 25 pages instead of 25%. Writing both finishing times down before subtracting keeps the target, a number of days, in view.

  1. Many steps moderate Working it out

    Four operations chain together: 25% of 60, Martina's 75 pages, the two finishing times 1,500 ÷ 60 and 1,500 ÷ 75, and finally their difference. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

  2. Combines ideas moderate Working it out

    A percent increase in a daily rate has to be turned into a change in days, and 25% more pages a day does not mean 25% fewer days. (Embretson, 1998; Daniel & Embretson, 2010)

  3. Turning words into math minor Understanding the question

    “25% more pages per day than Karen” makes Martina's rate 1.25 × 60, and “how many days sooner” asks for a difference of two times, not of two rates. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Increase 60 by 25% to get Martina's daily pages, divide 1,500 by each girl's daily rate to get her number of days, then subtract the two numbers of days.

Traps

  • Stops one step early: A choice is a number from partway through the solution.
  • Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
  • Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Percent increase
new=old+p100×old\text{new} = \text{old} + \frac{p}{100} \times \text{old}
Old is the starting amount, and p is the percent it grows by. First find p percent of the old amount. That is the extra. Then add the extra to the old amount to get the new amount.
Time from a rate
days=total pages÷pages per day\text{days} = \text{total pages} \div \text{pages per day}
Total pages is the whole assignment. Pages per day is how fast someone reads. Dividing tells how many days of reading fit into the whole job.
  1. Sort out what you know

    Karen reads 60 pages per day. Martina reads 25% more pages per day than Karen. Each girl must read 1,500 pages. You need how many days sooner Martina finishes. That is the difference between their numbers of days.

  2. Find Martina’s pages per day

    25% more means Karen’s 60 pages plus 25% of 60.

    new=old+p100×old\text{new} = \text{old} + \frac{p}{100} \times \text{old}

    60+25100×60=60+15=7560 + \frac{25}{100} \times 60 = 60 + 15 = 75

    Martina reads 75 pages per day. Watch out: 15 is extra pages, not days. Stopping here gives 15, choice D. Adding 25 pages instead of 25% gives 85 pages and leads to 7, choice C.

  3. Find each girl’s number of days

    Divide the whole assignment by each girl’s pages per day.

    days=total pages÷pages per day\text{days} = \text{total pages} \div \text{pages per day}

    Karen: 1,500÷60=25\text{Karen: } 1{,}500 \div 60 = 25

    Martina: 1,500÷75=20\text{Martina: } 1{,}500 \div 75 = 20

    Karen needs 25 days, and Martina needs 20 days.

  4. Subtract the days

    “How many days sooner” asks for the difference.

    25−20=525 - 20 = 5

    Martina finishes 5 days sooner. Watch out: dividing 60 by 15 mixes two rates and gives 4, choice A.

    So the answer is B, 5.

Check your answer

Multiply back. Karen: 25×60=1,50025 \times 60 = 1{,}500 pages. Martina: 20×75=1,50020 \times 75 = 1{,}500 pages. Both read all 1,500 pages, and 25 days is 5 days more than 20 days.

The original DOE question
Question 71 as printed on page 202 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 202, question 71. Open page 202 of the handbook (PDF)

Question 5 of 6

2024 Test B Math, question 44

Question

Which inequality is equivalent to 6x−20<5x−2(5−3x)6x-20<5x-2(5-3x)?

  • Ax<16x<16
  • Bx<52x<\frac{5}{2}
  • Cx<−2x<-2
  • Dx>−2x>-2
Show the answer

Correct answer: D

6x−20<5x−2(5−3x)6x-20<5x-2(5-3x)

Distribute the coefficient −2 on the right-hand side:

6x−20<5x+(−2)(5)+(−2)(−3x)6x-20<5x+(-2)(5)+(-2)(-3x)

6x−20<5x−10+6x6x-20<5x-10+6x

Subtract 6x6x from both sides:

−20<5x−10-20<5x-10

Add 10 to both sides:

−10<5x-10<5x

Divide both sides by 5:

−2<x-2<x

This is equivalent to

x>−2x>-2

Question analysis

Students choose the simplest equivalent form of a linear inequality that has x on both sides and a negative number multiplied into parentheses. It tests distributing, combining like terms and solving inequalities. Sign errors in the distribution, and reading the direction of the final inequality, are where most mistakes happen.

NYS standards
  • 7.EE.4b two-step inequalities
  • 8.EE.7b linear equations with rational coefficients
  • 7.EE.1 linear expressions with rational coefficients
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 inequalities have the form px + q < r. This one has x on both sides and a negative factor to distribute across a subtraction, steps that belong to grade 8 equation work, so it is harder than typical grade 7 practice.
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 16+, 10 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 move is standard algebra, but the inequality is built so that small slips change the answer completely. The −2 has to reach both terms in the parentheses, and once the boundary −2 is found its direction still has to be read correctly, since the variable ends up on the right as −2 < x. Rewriting that as x > −2, or substituting x = 0 into the original, catches the trap.

  1. Error-prone arithmetic moderate Working it out

    Distributing −2 over (5 − 3x) gives −10 + 6x, two sign decisions in one move, and the x terms then sit on both sides of the inequality. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  2. A very close wrong answer moderate Choosing the answer

    x < −2 and x > −2 share the boundary and differ only in direction, which depends on reading −2 < x correctly or flipping the sign when dividing by a negative. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

  3. Many steps minor Working it out

    Four moves follow one another, distributing, collecting 5x + 6x, moving the x terms and dividing by 5, and a slip at any of them changes the result. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Distribute the −2 to both terms in the parentheses, combine the x terms on the right, then gather the x terms on the side that keeps their coefficient positive.

Traps

  • Order of operations: Working left to right, or distributing a minus sign wrongly, gives a choice.
  • 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.)
  • A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distributive property
a(b+c)=a×b+a×ca(b + c) = a \times b + a \times c
a is the number outside the parentheses. b and c are the terms inside. Multiply a by each one. A minus sign inside counts as a negative term, so 5 − 3x is 5 + (−3x). A negative times a negative is positive.
Keeping an inequality true
if a<b, then a+c<b+c\text{if } a < b \text{, then } a + c < b + c
a and b are the two sides, and c is any number. Adding or subtracting the same number on both sides keeps the inequality true. Dividing both sides by a positive number also keeps it true. Only multiplying or dividing by a negative number flips the sign.
  1. Sort out the question

    An inequality compares two sides with a sign like <, which means is less than. To solve it, you get x alone on one side.

    You need the short inequality that has the same answers as the long one.

  2. Distribute the −2

    Multiply −2 by both terms inside the parentheses, 5 and −3x.

    −2(5−3x)=(−2)×5+(−2)×(−3x)=−10+6x-2(5 - 3x) = (-2) \times 5 + (-2) \times (-3x) = -10 + 6x

    A negative times a negative is positive, so the second part is +6x. Watch out: If the −2 multiplies only the 5, you end up with x<52x < \frac{5}{2}, choice B.

  3. Combine the x terms

    The right side is now 5x − 10 + 6x. Put the x terms together: 5x + 6x = 11x.

    6x−20<11x−106x - 20 < 11x - 10

  4. Gather the x terms on one side

    Subtract 6x from both sides. That leaves the x terms on the right, where they stay positive.

    6x−20−6x<11x−10−6x6x - 20 - 6x < 11x - 10 - 6x

    −20<5x−10-20 < 5x - 10

  5. Get x by itself

    Add 10 to both sides. Then divide both sides by 5.

    −10<5x-10 < 5x

    −2<x-2 < x

    You divided by a positive 5, so the sign does not flip.

  6. Read it with x first

    −2<x-2 < x says −2 is less than x. That is the same as saying x is greater than −2, just as 3 < 5 means 5 > 3.

    x>−2x > -2

    Watch out: x<−2x < -2, choice C, has the same number but points the wrong way.

    So the answer is D, x>−2x > -2.

Check your answer

Test x = 0, which is greater than −2. The left side is 6(0)−20=−206(0) - 20 = -20. The right side is 5(0)−2(5−0)=−105(0) - 2(5 - 0) = -10. And −20 < −10 is true. Now test x = −3. The left side is −38 and the right side is −43. That is false, as it should be.

The original DOE question
Question 101 as printed on page 211 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 211, question 101. Open page 211 of the handbook (PDF)

Question 6 of 6

2018 Test B Math, question 53

Question

At North High School, a survey asked two questions, Question A and Question B. For each question, students could answer either “yes” or “no.” Of the 800 students who responded to the survey, 720 answered “yes” to Question A, and 640 answered “yes” to Question B. What is the least possible number of these students who could have answered “yes” to both questions?

  • A80
  • B160
  • C560
  • D640
Show the answer

Correct answer: C

None of the 80 students (800 − 720) who answered “no” to Question A could have answered “yes” to both questions.

Therefore, the least possible number of students who could have answered “yes” to both questions can be found by subtracting the 80 who answered “no” to Question A from the 640 who answered “yes” to Question B or 640 − 80 = 560.

Question analysis

In a survey of 800 students at North High School, 720 said yes to the first question and 640 said yes to the second. Students must find the smallest possible number who said yes to both. It tests reasoning about overlapping groups, and asking for the least possible overlap rather than an exact count makes it unusual.

NYS standards
  • 7.SP.8b sample spaces for compound events
  • 7.EE.4 equations and inequalities from problems
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 students work with Venn diagrams and two-way tables when all counts are given. Finding the smallest possible overlap requires reasoning about extremes rather than computing, which is rare in grade 7 and new to most students.
Estimated difficulty
Among the hardest on the SHSAT: a non-obvious approach, heavy reading, or a well-hidden trap.
Solution
2 steps
Reading level
Question and choices: about grade 8, 60 words.
Corrected from the DOE handbook
  • A repeated phrase and a stray comma in the explanation were removed.
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

Most students have never been asked for a smallest overlap, so they fall back on a picture of nested groups and give the largest one. The key move is to push the two yes groups apart as far as possible: only the 80 students who said no to Question A can absorb some of Question B's yes answers. Once that is seen, the arithmetic is one subtraction, so the question rewards reasoning far more than computation.

  1. No obvious method major Working it out

    Nothing in the wording points to the method: the smallest overlap appears only when all 80 students who said no to Question A are imagined saying yes to Question B. (Schoenfeld, 1985)

  2. Turning words into math moderate Understanding the question

    "The least possible number of these students who could have answered “yes” to both questions" asks for an extreme case, which has to be turned into an arrangement of the 800 students. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  3. A very close wrong answer minor Choosing the answer

    640 is only 80 above the right 560, so a rough sense of size cannot choose between them; only asking whether the overlap is the least or the greatest separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Add the two yes counts and compare the sum with the number of students; any amount above the total must come from students counted in both groups.

Traps

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: The handbook prints “(800 − 720)” twice in its first sentence, after “80 students” and again after “Question A,” and puts a comma between subject and verb in “the least possible number of students who could have answered “yes” to both questions, can be found.” The site keeps the first “(800 − 720)” and drops the comma.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Least overlap of two groups
least overlap=a+b−t\text{least overlap} = a + b - t
a is the size of the first group and b is the size of the second. t is the total number of people. If a + b is more than t, some people must be in both groups. The extra amount is the smallest number that can be in both.
  1. Sort out what you know

    There were 800 students. Of them, 720 said yes to Question A and 640 said yes to Question B. You need the least number who could have said yes to both. The overlap is the group in both yes groups. You want the smallest overlap.

  2. Find who said no to A

    Students who said no to Question A cannot be in the overlap.

    800−720=80800 - 720 = 80

    So 80 students said no to Question A.

    Watch out: 80 is not the answer. It is only a step, and it is choice A.

  3. Keep the overlap small

    To make the overlap as small as you can, put as many B yes answers outside it as possible. The only students outside it are the 80 who said no to A. So pretend all 80 of them said yes to B.

  4. Count the rest of the yes group

    The other yes answers to B must come from students who also said yes to A.

    640−80=560640 - 80 = 560

    So at least 560 students said yes to both.

    Watch out: 640 is the most that could say yes to both, not the least. That is choice D.

  5. Give the answer

    The fewest students who could say yes to both questions is 560. So the answer is C, 560.

Check your answer

Use the overlap formula: 720+640−800=560720 + 640 - 800 = 560. There are 1,360 yes answers but only 800 students, so 560 students must be counted twice. You can also check the groups. 560 said yes to both, 160 to A only and 80 to B only. That makes 560+160+80=800560 + 160 + 80 = 800.

The original DOE question
Question 110 as printed on page 163 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 163, question 110. Open page 163 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.

Sample #818171