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 12
Passage Excerpt from “Scribe Like an Egyptian”

Excerpt from “Scribe Like an Egyptian”

by Hilary Wilson

In ancient Egypt, literacy was the key to success. However, contrary to popular belief, not all Egyptian scribes understood hieroglyphs. Many relied instead on the simpler hieratic script for the multitude of everyday documents generated by the Egyptian bureaucracy.

Hieroglyphs—“the Words of God”—compose a writing system with more than 1,000 distinct characters, the meanings of which were lost for 1,500 years before they were deciphered by Jean-François Champollion in 1822. Including both ideograms (which convey a whole word or idea, either concrete or abstract, in a single sign) and phonograms (representing either an alphabetic sound or a group of consonants), [the writing system] was used in formal inscriptions on tomb and temple walls as well as on elaborate funerary papyri. For everyday purposes, however, scribes used a shorthand version of the hieroglyphic script known as hieratic, which was quicker to write and more economical of space. The two writings existed side by side for at least 2,500 years.

Scraps of ancient hieratic writing, mostly penned by student scribes on limestone flakes called ostraca, suggest that no matter how humble his origins, an educated Egyptian could achieve almost anything. Horemheb (d.1292 B.C.) is a good example. Born of middle-ranking parents, his scribal training led to an army career. From Scribe of Recruits, during the reign of Akhenaten (1353–1336 B.C.), Horemheb rose through the military ranks and, by the rule of Tutankhamun (1332–1323 B.C.), he was commander in chief of the Egyptian forces. As a close adviser of the young Pharaoh, Horemheb was appointed “Deputy of the King throughout the Two Lands,” and might have expected to succeed to the throne should the king die childless. He had to wait a few years, but eventually Horemheb achieved the pinnacle of his career by becoming the last king of the 18th Dynasty, making his mark by instituting dramatic reforms to the organisation of the army, the judiciary and administration in general. The lasting success of these changes owed much to his scribal background. . . .

But education was not available to all. Government departments and major temples supported schools, where boys commenced their training at six or seven, sometimes earlier. To these boarding establishments . . . family or household servants delivered the students’ food and drink rations daily for several years, during which time the student was not contributing to the family’s income. Boys from poorer families could only hope to be educated with support from a wealthier relative or patron, or through apprenticeship to an older scribe, perhaps the local clerk or land agent, who would teach them the basics of the scribe’s craft. This limited the scope for employment but such “on the job” training allowed apprentices to help out at home while learning. . . .

Scribal education began with the elementary principles of the hieratic script. The lowliest scribes, who trained for just five or six years, probably learned only the rudiments of the hieroglyphic script. Students were set exemplar documents and extracts from popular texts to copy, to practice their hieratic handwriting on basic-format letters, reports and contracts, while absorbing the good advice contained in the texts. Surviving examples of copy-work sometimes include tutors’ corrections added in red. Some significant Egyptian literary works survive almost exclusively from student copies.

A schoolboy “dictionary” of hieroglyphs with their hieratic equivalents shows that a knowledge of more than 450 signs was required for everyday writing purposes. Lessons in record-keeping and filing and labelling enabled any half-competent scribe to perform that most essential of all scribal functions: the making and updating of lists. For professions such as those of government official, priest or lawyer, a scribe would train for several more years, increasing his vocabulary to perhaps a thousand or more signs. Those with the best handwriting or drawing skills might follow the craft of creating beautifully illustrated copies of funerary texts, commonly called Books of the Dead. Others could become draughtsmen, artists or architects. Doctors compiled their own collections of medication recipes, treatments and associated incantations, many copied from texts found in the House of Life, the temple library. Lawyers had to be familiar with the corpus of civil and religious laws and precedents found in the official records, which were administered by archivists. Egypt’s bureaucratic society depended on the skills of an army of scribes of all ranks from filing clerk to tax assessor. For young Egyptians, “be a scribe” was the best of career advice.

From “Scribe Like an Egyptian” by Hilary Wilson from HISTORY TODAY, August 8, 2019. Copyright © 2019 by History Today Ltd. Company.

A pyramid diagram titled “ANCIENT EGYPTIAN SOCIAL CLASS STRUCTURE,” divided into five horizontal levels, each with a small drawing of figures in ancient Egyptian style and a labeled box. From top to bottom the levels are labeled: Pharaoh (a single standing figure); Nobility and priests (a kneeling figure and a standing figure); Traders, shopkeepers, scribes, and artisans (two seated figures working); Animal herders and farmers (a man standing beside a group of cattle); and Unskilled workers (three figures bending forward).

bureaucracy: an administrative staff of government officials
funerary papyri: a sheet or scroll of papyrus containing religious images and hieroglyphs meant to help the deceased be reborn in the afterlife
rudiments: basics
schoolboy: slang that refers to materials used during the course of receiving an education
draughtsmen: people who draw plans of machinery or structures
corpus: a collection of writings

Question

Which claim is best supported by the information presented in paragraphs 3 and 4?

  • AEducation was difficult for the common citizens of Egypt to obtain.
  • BOnce students began studying to become scribes, their future held more possibilities.
  • CIf someone wanted to become a scribe, support from outside sources such as the government or a temple was necessary.
  • DBecoming a successful scribe required a student’s family to sacrifice time and money in order to ensure the best education.
Show the answer

Correct answer: B

The question asks for the claim that is best supported by the details in paragraphs 3 and 4.

A. Incorrect. While paragraph 4 does explain that boys from poorer families had more challenges in receiving an education, the paragraph also provides examples of obtaining a patron or getting an apprenticeship as ways that these aspiring scribes could attain an education. Furthermore, the details in paragraph 3 do not provide any support for the claim that “common citizens” had difficulty getting an education because that paragraph focuses on the example of one person who was able to become a king as a result of his background as a scribe.

B. CORRECT. In paragraph 3, the author provides an example of a person who used his skills as a scribe to rise through the ranks of society by describing the achievements of Horemheb to support the idea that “no matter how humble his origins, an educated Egyptian could achieve almost anything.” The author continues in paragraph 4 to explain that with the proper support, some young Egyptians could become educated, thereby opening more opportunities for themselves.

C. Incorrect. Paragraph 4 explains that schools were supported by the government and religious institutions, but it also explains that apprenticeships were a method of learning the craft of the scribe. As the paragraph indicates, boys could also be helped in their education by support from “a wealthier relative or patron.”

D. Incorrect. In paragraph 4, the author details the sacrifices that families made to keep their sons at the schools, while boys from poorer families could only hope for a patron or an apprenticeship. Paragraph 4 also explains that apprentices could help out at home while learning, so the sacrifice of time and money was not required of every family.

Question analysis

This inference question asks which claim is best supported by paragraphs 3 and 4 of “Scribe Like an Egyptian,” which tell the story of the scribe Horemheb and describe who could get a scribal education in ancient Egypt. Students must weigh each claim against both paragraphs together, and several claims echo a single sentence closely enough to look supported.

NYS standards
  • 7R1 textual evidence and inferences
  • 7R8 evaluating arguments and claims
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Choosing the claim a passage best supports is core grade 7 work. Here the claim must fit two paragraphs that do different jobs, one telling a single man’s story and one explaining access to schooling, and some choices fit only part of that.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
Evidence
Answered from several paragraphs or stanzas
Reading level
Passage: about grade 12, 810 words; question and choices: about grade 10, 75 words.
Corrected from the DOE handbook
  • A misdescription of the passage in the explanation of one wrong answer 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

The trap is a claim that paragraph 4 seems to state almost word for word, which makes it feel safer than the right one. But the question asks for support from both paragraphs, and Horemheb’s story points the other way, toward what schooling made possible. The two other wrong claims harden paragraph 4’s details into rules, a necessary sponsor or a required family sacrifice, so words like “necessary” and “required” are the ones to test.

  1. Evidence in several places moderate Working it out

    The claim must fit two paragraphs that do different work: paragraph 3 follows Horemheb from army scribe to king, and paragraph 4 explains who could pay for schooling and what the alternatives were. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)

  2. The right answer is reworded moderate Choosing the answer

    The right claim restates paragraph 3’s “an educated Egyptian could achieve almost anything” in new words, while the most tempting wrong one nearly copies paragraph 4’s opening, “education was not available to all.” (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)

  3. Reading between the lines minor Working it out

    Students must connect Horemheb’s rise with paragraph 4’s account of who got schooling to see that both paragraphs are about education opening doors, a link the author leaves implicit. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

Strategy: Sum up paragraph 3 and paragraph 4 in a sentence each, then test every claim against both summaries, watching for words that make a claim stronger than the text.

Traps

  • Too narrow: A single detail offered as the central idea or a summary, or a summary that keeps minor details and leaves out an essential part.
  • Too broad or too strong: A choice overstates, generalizes, or uses absolute words the text does not support.
  • Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: The DOE’s explanation of D said paragraph 4 details sacrifices “necessary for some poorer families,” and that paragraph 3’s Horemheb, born of middle-ranking parents, shows the sacrifice was not required of all families. In paragraph 4 the rations and lost income are those of families whose sons were at the schools; poorer boys could only hope for a patron or an apprenticeship; and paragraph 3 says nothing about what Horemheb’s family gave up. The explanation now says so: D is wrong because apprentices could help out at home while learning, so not every family made the sacrifice.

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 claim is best supported by paragraphs 3 and 4 together. A claim is a statement that someone says is true. The right claim must fit both paragraphs. Watch for strong words, like “necessary,” that go further than the text.

  2. Sum up paragraph 3

    Paragraph 3 tells about Horemheb. His parents were middle-ranking, not rich or powerful. His training as a scribe led to a job in the army. He rose to lead the army, then to serve the young king, and in the end he became king himself. The author uses him to show that “no matter how humble his origins, an educated Egyptian could achieve almost anything.” Humble origins means an ordinary family background.

  3. Sum up paragraph 4

    Paragraph 4 opens by saying that not everyone could get an education. Government offices and big temples ran schools. Families sent food and drink for years, while the boy earned nothing for them. Poorer boys could hope for help from “a wealthier relative or patron.” A patron is someone who pays for another person’s training. Or they could learn on the job from a working scribe. That path led to fewer jobs.

  4. Put the two paragraphs together

    Both paragraphs are about what schooling could do for a boy. Paragraph 3 shows a man rising all the way to king because he trained as a scribe. Paragraph 4 shows that, with the right support, boys could get that schooling. It even notes that the on-the-job path led to fewer jobs. So education opened doors. That is what choice B says.

  5. Check each choice

    • A does not fit. It matches only the start of paragraph 4. Paragraph 4 also gives ways in for poorer boys, and paragraph 3 shows a man from an ordinary family rising to the top.
    • B fits. Both paragraphs show schooling leading to more possibilities.
    • C does not fit. A government or temple school was not “necessary.” A relative, a patron or an apprenticeship could also work.
    • D does not fit. Apprentices could help out at home while they learned, so not every family gave up time and money. And the paragraphs never mention “the best education.”
  6. Choose the answer

    The claim that fits both paragraphs is that schooling gave a boy more possibilities. So the answer is B.

Why this answer and not the closest wrong one

A is the closest wrong choice, because paragraph 4 begins by saying not everyone could get an education. But the question asks about paragraphs 3 and 4. Paragraph 3 tells of a scribe from an ordinary family who became king. Paragraph 4 shows ways around the cost. Both point to what education made possible, which is B.

The original DOE question
Question 12 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 12. This is the DOE’s version, before the corrections listed in the analysis.
The passage “Excerpt from “Scribe Like an Egyptian”” as shown on the testThe passage “Excerpt from “Scribe Like an Egyptian”” as shown on the DOE’s digital test, screen 1 of 6.The passage “Excerpt from “Scribe Like an Egyptian”” as shown on the DOE’s digital test, screen 2 of 6.The passage “Excerpt from “Scribe Like an Egyptian”” as shown on the DOE’s digital test, screen 3 of 6.The passage “Excerpt from “Scribe Like an Egyptian”” as shown on the DOE’s digital test, screen 4 of 6.The passage “Excerpt from “Scribe Like an Egyptian”” as shown on the DOE’s digital test, screen 5 of 6.The passage “Excerpt from “Scribe Like an Egyptian”” as shown on the DOE’s digital test, screen 6 of 6.

Question 2 of 6

2026 Practice Test D English, question 12
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

Read this sentence from paragraph 5.

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.

How does the sentence help convey Orville Wright’s perspective about this first flight?

  • AIt suggests that he was frustrated by the poor flying conditions on the day of the flight.
  • BIt emphasizes that he believed the flight was successful despite its short distance.
  • CIt provides a comparison between flight distances under calm and high wind conditions.
  • DIt highlights the importance of such calculations in the success of future flights.
Show the answer

Correct answer: B

The question asks how the sentence from paragraph 5 helps convey Orville Wright’s perspective about the first flight.

A. Incorrect. Although the sentence from paragraph 5 refers to the velocity of the wind during the flight, there is no indication that Orville felt a sense of frustration with the windy conditions. Instead, the sentence emphasizes the “speed of the machine relative to the air” and presents the equivalent flight length if the machine had flown through calm air (“the length of the flight was equivalent to a flight of 540 feet made in calm air”), details that emphasize the importance of the Wrights’ accomplishment.

B. CORRECT. In the sentence from paragraph 5, Orville uses the wind velocity and the machine speed to highlight the distance that the plane covered while in the air and to determine the distance that it would have flown on a day with calm winds, which emphasizes the magnitude of the accomplishment.

C. Incorrect. Since Orville does not suggest in the sentence from paragraph 5 how far the plane traveled under the actual conditions of high winds, the comparison is incomplete. Additionally, this statement does not address the reason that he would make such a comparison, which allowed him to describe the flight in more impressive terms (540 feet versus 120 feet).

D. Incorrect. Unlike the wind velocity calculations made in preparation for the flight, which might help in determining the success of future flights, the comparative calculations that Orville provides in the sentence from paragraph 5 help interpret the results of the first flight (by describing how fast and far the machine flew relative to the flying conditions) and thereby communicate its success. The calculations of the first flight’s relative speed and equivalent distance would not affect the success of future flights, but the information does provide context for understanding the Wright brothers’ accomplishment.

Question analysis

This point-of-view question quotes the long sentence from paragraph 5 in which Orville Wright works out the machine’s speed through the air and the length of an equivalent flight in calm air. Students must decide how this sentence helps convey his perspective on the first flight. The sentence is mostly numbers, from 35 feet per second to 540 feet, which makes its purpose easy to miss.

NYS standards
  • 7R6 point of view and author position
  • 7R1 textual evidence and inferences
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Grade 7 point-of-view questions usually rest on words that show feeling. This sentence is a chain of speeds and distances with no feeling words, so students must work out why Orville bothers with the arithmetic, a step beyond typical grade 7 work.
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 7, 816 words; question and choices: about grade 9, 132 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

Point-of-view questions usually turn on feeling words, and this sentence has none, only speeds and a distance. Students must ask why Orville bothers with the arithmetic, and the answer is that 540 feet in calm air makes a 12-second hop look like the achievement he believed it was. The strongest pull describes the sentence’s content accurately, a comparison of calm and windy flight, without saying what it reveals, so students who answer what instead of why lose the point.

  1. How, not what moderate Working it out

    The question asks why Orville includes this calculation, so students must see that converting a short hop into “a flight of 540 feet made in calm air” is his way of showing that it counted. (Anderson & Krathwohl, 2001)

  2. Reading between the lines moderate Working it out

    The sentence holds no word of feeling or judgment; his view that the flight succeeded must be read from his choice to restate about 120 feet over the ground as 540 feet through the air. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)

  3. Complex sentences minor Understanding the question

    The 61-word sentence chains four measurements, a wind over 35 feet per second, a ground speed of ten, an airspeed over 45 and the 540-foot equivalent, joined by “As,” “and” and commas. (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)

Strategy: Work out what the speeds and distances in the sentence describe, then ask why Orville includes them at this point and what that choice tells you about his view of the flight.

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.
  • 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.
  • Outside knowledge: A choice is plausible in the real world but not supported by this text.

The wrong answers, classified

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Figure out what the question asks

    This asks about Orville Wright’s perspective, or point of view. How did he feel about the first flight? The quoted sentence is full of numbers, not feelings. So ask why Orville bothered to do this math.

  2. Sort out the numbers

    The wind blew against the machine at over 35 feet per second. The machine moved over the ground at 10 feet per second. It was flying into the wind, so the air rushed past it even faster. Add them: it moved through the air at over 45 feet per second. The flight lasted 12 seconds. Over the ground, that is 10 × 12 = 120 feet. Through the air, it is 45 × 12 = 540 feet. That is how Orville gets “a flight of 540 feet made in calm air.”

  3. Ask why Orville does this math

    Paragraph 5 says the flight ended “a little over 120 feet from the point at which it rose into the air.” That sounds short. But Orville shows it was really much more. In calm air, the same flight would have gone 540 feet. He wants you to see that the short hop was a real flight. So his view is that the flight was a success.

  4. Check each choice

    • A does not fit. There is no sign of frustration. Orville uses the wind to make the flight count for more.
    • B fits. The math turns a short flight into an impressive one.
    • C does not fit. The sentence does compare calm and windy air. But that only says what the sentence contains, not what Orville thinks.
    • D does not fit. The math measures the flight that just happened. It does not look ahead to future flights.

    Watch out: C is tempting because it describes the sentence correctly. But the question asks about his perspective. C never says how he felt about the flight.

  5. Choose the answer

    Orville does the math to show that his short flight was a real success. So the answer is B, “It emphasizes that he believed the flight was successful despite its short distance.”

Why this answer and not the closest wrong one

C is the closest wrong choice. It is true that the sentence compares a windy flight with one “made in calm air.” But C only tells what the sentence says. The question asks why Orville says it. He turns about 120 feet into 540 feet to show the flight was a real achievement. B gives that reason.

The original DOE question
Question 12 as shown on the DOE’s digital test, page 19 of its printable version.
As printed in the DOE’s printable version of 2026 Practice Test D English, page 19, question 12.
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 3 of 6

2024 Test A English, question 16
Passage Excerpt from “Impressions of an Indian Childhood”

Excerpt from “Impressions of an Indian Childhood”

by Zitkala-Sa

Zitkala-Sa (Gertrude Simmons Bonnin) was a Native American writer, musician, teacher, and political activist who was raised on the Yankton Sioux Reservation in South Dakota. In 1900 she published “Impressions of an Indian Childhood” (the term Indian was commonly used at the time to refer to Native American people) to expose readers to what life is like on a reservation.

Soon after breakfast Mother sometimes began her beadwork. On a bright, clear day, she pulled out the wooden pegs that pinned the skirt of our wigwam1 to the ground, and rolled the canvas part way up on its frame of slender poles. Then the cool morning breezes swept freely through our dwelling, now and then wafting the perfume of sweet grasses from newly burnt prairie.

Untying the long tasseled strings that bound a small brown buckskin2 bag, my mother spread upon a mat beside her bunches of colored beads, just as an artist arranges the paints upon his palette. On a lapboard she smoothed out a double sheet of soft white buckskin; and drawing from a beaded case that hung on the left of her wide belt a long, narrow blade, she trimmed the buckskin into shape. Often she worked upon small moccasins for her small daughter. Then I became intensely interested in her designing. With a proud, beaming face, I watched her work. In [my] imagination, I saw myself walking in a new pair of snugly fitting moccasins. I felt the envious eyes of my playmates upon the pretty red beads decorating my feet.

Close beside my mother I sat on a rug, with a scrap of buckskin in one hand and an awl in the other. This was the beginning of my practical observation lessons in the art of beadwork. From a skein3 of finely twisted threads of silvery sinews my mother pulled out a single one. With an awl she pierced the buckskin, and skillfully threaded it with the white sinew. Picking up the tiny beads one by one, she strung them with the point of her thread, always twisting it carefully after every stitch.

It took many trials before I learned how to knot my sinew thread on the point of my finger, as I saw her do. Then the next difficulty was in keeping my thread stiffly twisted, so that I could easily string my beads upon it. My mother required of me original designs for my lessons in beading. At first I frequently ensnared many a sunny hour into working a long design. Soon I learned from self-inflicted punishment to refrain from drawing complex patterns, for I had to finish whatever I began.

After some experience I usually drew easy and simple crosses and squares. These were some of the set forms. My original designs were not always symmetrical nor sufficiently characteristic, two faults with which my mother had little patience. The quietness of her oversight made me feel strongly responsible and dependent upon my own judgment. She treated me as a dignified little individual as long as I was on my good behavior; and how humiliated I was when some boldness of mine drew forth a rebuke from her!

In the choice of colors she left me to my own taste. I was pleased with an outline of yellow upon a background of dark blue, or a combination of red and myrtle-green. There was another of red with a bluish-gray that was more conventionally used. When I became a little familiar with designing and the various pleasing combinations of color, a harder lesson was given me. It was the sewing on, instead of beads, some tinted porcupine quills, moistened and flattened between the nails of the thumb and forefinger. My mother cut off the prickly ends and burned them at once in the centre fire. These sharp points were poisonous, and worked into the flesh wherever they lodged. For this reason, my mother said, I should not do much alone in quills until I was as tall as my cousin Warca-Ziwin.

Always after these confining lessons I was wild with surplus spirits, and found joyous relief in running loose in the open again. Many a summer afternoon a party of four or five of my playmates roamed over the hills with me. We each carried a light sharpened rod about four feet long, with which we pried up certain sweet roots. When we had eaten all the choice roots we chanced upon, we shouldered our rods and strayed off into patches of a stalky plant under whose yellow blossoms we found little crystal drops of gum. Drop by drop we gathered this nature’s rock-candy, until each of us could boast of a lump the size of a small bird’s egg. Soon satiated with its woody flavor, we tossed away our gum, to return again to the sweet roots.

From “Impressions of an Indian Childhood” by Zitkala-Sa—Public Domain

A BRIEF HISTORY OF BEADWORK IN SOUTH DAKOTA

DateEvent
1500sThe Dakota use beads made from bones, shells, stones, and animal teeth. European traders bring glass beads to North America.
Late 1600sThe Dakota begin trading with the French in Minnesota.
1830sThe Dakota begin using glass beads in clothing, artwork, and decorations in place of Native-made beads.
1862–1865The Dakota are expelled from their homelands in Minnesota as a result of the U.S.-Dakota War.
1900s–1920sThe Dakota create items to sell outside their community.

Source: MNopedia

1wigwam: hut with an arched framework of poles covered with bark, mats, or animal hides

2buckskin: leather made from the skin of a male deer

3skein: coiled length of yarn or other thread loosely wound on a reel

Question

Which sentence best summarizes the process of beading that is described in the excerpt?

  • ATake a buckskin bag full of beads and spread them out on a mat in different colors like a paint palette; take a double sheet of buckskin and smooth it out on a table; take a sinew and awl and thread the beads onto the buckskin in a desired pattern.
  • BCut the double sheet of buckskin into a shape; take a skein of sinew and pierce the buckskin with an awl; thread the sinew with beads of many different colors in a simple or complex pattern; twist the sinew to keep it tight after every stitch into the buckskin.
  • CArrange the beads into groups of colors on a mat; smooth out a double sheet of buckskin and cut it to shape; take a single thread of sinew; pierce the buckskin with an awl; thread the buckskin with the sinew and string it with beads, carefully twisting after every stitch.
  • DGather beads, buckskin, sinew, and awl and place them on a mat; cut the buckskin into the desired shape; decide on a pattern for the beads and create it using the sinews and the awl; thread the beads onto the sinew in the desired pattern and twist it tight.
Show the answer

Correct answer: C

The question asks which sentence best summarizes the process of beading that is described in the excerpt.

A. Incorrect. In paragraphs 2 and 3, the author describes how at the beginning of the process her mother “smoothed out a double sheet of soft white buckskin” on a lapboard (paragraph 2) and how she strings the beads into a pattern. However, this sentence does not describe the steps that take place between the preparation and the stringing of the beads. Thus, this sentence does not best summarize the beading process.

B. Incorrect. In paragraphs 2 and 3, the author describes the process her mother uses to string the beads with “the point of her thread, always twisting it carefully after every stitch” (paragraph 3). While this sentence includes some of the steps in the beading process, it does not describe the necessary preparation, namely spreading the beads out on a mat and smoothing out the buckskin on a lapboard. Thus, this sentence does not best summarize the beading process.

C. CORRECT. This sentence best summarizes the process of beading that is described in the excerpt because it includes details about each step in the process. In paragraph 2, the author explains how her mother “spread upon a mat beside her bunches of colored beads” and how “on a lapboard she smoothed out a double sheet of soft white buckskin; and drawing from a beaded case that hung on the left of her wide belt a long, narrow blade, she trimmed the buckskin into shape.” The author continues her description in paragraph 3, stating, “From a skein of finely twisted threads of silvery sinews my mother pulled out a single one. With an awl she pierced the buckskin, and skillfully threaded it with the white sinew. Picking up the tiny beads one by one, she strung them with the point of her thread, always twisting it carefully after every stitch.”

D. Incorrect. In paragraphs 2 and 3, the author describes the process her mother uses as she “smoothed out a double sheet of soft white buckskin” (paragraph 2) on a lapboard before stringing the beads. While this sentence includes most of the steps in the beading process, it excludes the step of piercing the buckskin. This sentence also provides few details about the process of beading and uses vague wording to describe each step. Therefore, the sentence does not best summarize the beading process.

Question analysis

This reading question asks which sentence best summarizes the beadwork process Zitkala-Sa watches her mother carry out, from laying out the materials to stringing the beads. Each choice is a long list of steps, so students must compare them detail by detail with the excerpt and notice what each one adds, leaves out or misplaces.

NYS standards
  • 7R2 theme or central idea
  • 7R1 textual evidence and inferences
Grade level
Grade 8 reading and task
Compared with grade 7
Above grade 7 Summarizing a process is grade 7 reading, but the four summaries are long and nearly alike, and checking them against two paragraphs of 1900 prose with words like awl, sinew and skein is heavier than usual.
Estimated difficulty
Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
What makes it hard
Evidence
Answered from several paragraphs or stanzas
Reading level
Passage: about grade 8, 946 words; question and choices: about grade 16+, 212 words.
Corrected from the DOE handbook
  • An inaccurate detail about the excerpt 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

This question is won by bookkeeping, not insight: list the mother's steps from the two paragraphs, then hold each long summary against the list. The most tempting wrong summary reads like a sensible craft routine in a sensible order, yet it never has the buckskin pierced and puts every tool on the mat. Others open with a familiar image or close with precise stitching while skipping steps, so students who check only the beginning or only the end are caught.

  1. Every choice must be checked major Choosing the answer

    Each summary is a chain of four to six steps, so all four must be checked step by step against paragraphs 2 and 3 for steps missing, added or out of order. (Embretson & Wetzel, 1987)

  2. A lot to read moderate Understanding the question

    The four choices together run to about 200 words of semicolon-linked steps, and the text they summarize is two paragraphs of dense 1900 prose. (Kintsch, 1998; Gorin & Embretson, 2006; Freedle & Kostin, 1993)

  3. Hard words minor Understanding the question

    Words like awl, sinew, skein and lapboard name the tools and materials, and a student unsure of them cannot tell which steps a summary has dropped. (Perfetti, 2007; Freedle & Kostin, 1993)

Strategy: List the mother's steps from paragraphs 2 and 3 in order, then check each choice against your list for a missing step, an invented detail or a step out of place.

Traps

  • Too narrow: A single detail offered as the central idea or a summary, or a summary that keeps minor details and leaves out an essential part.

The wrong answers, classified

What was corrected from the DOE handbook, and why

  • Explanation: In rejecting choice B, the handbook says the choice leaves out the preparation, “namely spreading the beads and the buckskin out on a table.” The excerpt mentions no table: in paragraph 2 the mother spreads the beads on a mat and smooths out the buckskin on a lapboard. The explanation now says so. (Choice A’s “on a table” is part of a wrong answer and is left as printed.)

How the questions are analyzed, and the research behind it

Working it out, step by step

  1. Figure out what the question asks

    You need the summary that best matches the beading steps in the excerpt. A summary is a short retelling of the main points. The best one keeps every main step, in order, and adds nothing new.

  2. List the steps from paragraphs 2 and 3

    Paragraphs 2 and 3 give the mother’s steps in order. The last one ends with her “always twisting it carefully after every stitch.”

    1. Spread the beads on a mat.
    2. Smooth out a double sheet of buckskin on a lapboard (a board held on the lap).
    3. Trim the buckskin into shape with a blade.
    4. Pull out a single sinew, a strong thread.
    5. Pierce (poke a hole in) the buckskin with an awl, a pointed tool.
    6. Thread the sinew through the buckskin.
    7. String the beads, twisting the thread after every stitch.
  3. Check each choice against the list

    • A is wrong. It skips trimming, piercing and twisting. It also says “on a table,” but she used a lapboard.
    • B is wrong. It starts with cutting. It skips spreading the beads and smoothing the buckskin.
    • C is right. It has every step, in the same order.
    • D is wrong. It never says to smooth the buckskin or pierce it. It also puts every tool on the mat, but only the beads went there.

    Watch out: D sounds like a sensible plan in a sensible order. But a summary must match the excerpt, not just sound right. Check it step by step.

  4. Choose the answer

    Only C follows the mother’s steps from start to finish. So the answer is C.

Why this answer and not the closest wrong one

D is the closest wrong answer. It is neat and keeps most of the steps. But it skips smoothing the buckskin and never says to pierce it with the awl. It also piles every tool onto the mat. C matches the excerpt step for step, so it is the best summary.

The original DOE question
Question 16 as printed on page 45 of the DOE handbook.
As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 45, question 16. Open page 45 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D). This is the DOE’s version, before the corrections listed in the analysis.
The passage “Excerpt from “Impressions of an Indian Childhood”” as printed (pages 42–43)The passage “Excerpt from “Impressions of an Indian Childhood”” as printed on page 42 of the DOE handbook.The passage “Excerpt from “Impressions of an Indian Childhood”” as printed on page 43 of the DOE handbook.

Question 4 of 6

2018 Test A Math, question 30

Question

Today, Tien’s age is 14\frac{1}{4} of Jordan’s age. In 2 years, Tien’s age will be 13\frac{1}{3} of Jordan’s age. How old is Jordan today? 

  • A4 years old
  • B6 years old
  • C12 years old
  • D16 years old
Show the answer

Correct answer: D

First, set up an equation to express Tien’s age (TT) and Jordan’s age (JJ) today:

T=14JT=\frac{1}{4}J

Two years from now, Tien’s age will be T+2T+2, and Jordan’s age will be J+2J+2.

Set up an equation about the relationship between Tien’s age and Jordan’s age in two years:

T+2=13(J+2)T+2=\frac{1}{3}(J+2)

Solve the above equation for TT:

T=13(J+2)−2T=\frac{1}{3}(J+2)-2

Now set the two equations equal to each other and solve for JJ:

14J=13(J+2)−2\frac{1}{4}J=\frac{1}{3}(J+2)-2

14J=13J−43\frac{1}{4}J=\frac{1}{3}J-\frac{4}{3}

−112J=−43-\frac{1}{12}J=-\frac{4}{3}

J=−43(−121)J=-\frac{4}{3}\left(-\frac{12}{1}\right)

J=16J=16

Question analysis

This age word problem says Tien is one fourth of Jordan's age today and will be one third of Jordan's age in two years, then asks for Jordan's current age. Students must turn two fractional relationships into equations and solve them together. Both people age by the same two years, and it is easy to solve for the wrong person.

NYS standards
  • 8.EE.8c systems in real-world problems
  • 8.EE.7b linear equations with rational coefficients
Grade level
Grade 8 content, by NYS standard
Compared with grade 7
Above grade 7 Grade 7 equations have one unknown on one side, like 3x + 5 = 20. This one links two unknown ages with fractional coefficients and leads to a variable on both sides, which is grade 8 algebra.
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 1, 31 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 difficulty is in the setup: two fractional relationships at two different times, with both people aging. Once the equations are right, the algebra is short, but it ends with two ages and only one is asked for. Testing the choices against both conditions is a reliable alternative that avoids the algebra entirely.

  1. Turning words into math major Understanding the question

    The second sentence must become T + 2 = (J + 2)/3; writing J/3, or forgetting that Jordan also ages two years, breaks the equation. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

  2. Error-prone arithmetic moderate Working it out

    Substituting J/4 for T leaves J/4 + 2 = (J + 2)/3, an equation with fractions on both sides that must be cleared, usually by multiplying by 12. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

Strategy: Let Jordan's age be J, write Tien's age as J/4, and set J/4 + 2 equal to one third of (J + 2); or test each choice against both conditions.

Traps

  • Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

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 \times (b + c) = a \times b + a \times c
a, b and c are any numbers. To multiply a sum by a number, multiply each part of the sum, then add. It works because a groups of (b + c) is the same as a groups of b plus a groups of c.
  1. Sort out what you know

    Today, Tien’s age is 14\frac{1}{4} of Jordan’s age. That means Jordan is 4 times as old as Tien.

    In 2 years, both of them will be 2 years older. Then Tien’s age will be 13\frac{1}{3} of Jordan’s. So Jordan will be 3 times as old as Tien. You need Jordan’s age today.

  2. Name the ages with a letter

    Let t stand for Tien’s age today. Jordan is 4 times as old, so his age is 4t, which means 4 times t. Add 2 years to each age:

    PersonTodayIn 2 years
    Tienttt+2t + 2
    Jordan4t4t4t+24t + 2

    Watch out: Jordan gets 2 years older too. Do not forget to add 2 to his age.

  3. Write an equation for later

    In 2 years, Jordan’s age will be 3 times Tien’s age. Write that as an equation:

    4t+2=3×(t+2)4t + 2 = 3 \times (t + 2)

    Use the distributive property on the right side. Multiply both t and 2 by 3:

    4t+2=3t+64t + 2 = 3t + 6

  4. Solve for Tien’s age

    Take 3t away from both sides. Doing the same thing to both sides keeps them equal:

    t+2=6t + 2 = 6

    Now take 2 away from both sides:

    t=4t = 4

    So Tien is 4 years old today. Watch out: 4 is Tien’s age, not Jordan’s. Choice A, 4 years old, is a trap.

  5. Find Jordan’s age

    Jordan is 4 times as old as Tien:

    4×4=164 \times 4 = 16

    So the answer is D, 16 years old.

Check your answer

Put 16 back into the story. Today, Tien is 16 ÷ 4 = 4. In 2 years, Tien is 6 and Jordan is 18. Is 6 one third of 18? Yes, because 18 ÷ 3 = 6. Choice C fails: Tien would be 3, then 5, and 5 is not a third of 14.

The original DOE question
Question 87 as printed on page 87 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 87, question 87. Open page 87 of the handbook (PDF)

Question 5 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 6 of 6

2018 Test A Math, question 40

Question

In the set of consecutive integers from 12 to 30, inclusive, there are four integers that are multiples of both 2 and 3. How many integers in this set are multiples of neither 2 nor 3?

  • A5
  • B6
  • C13
  • D15
Show the answer

Correct answer: B

The question asks for integers from 12 to 30 that are not divisible by 2 or 3. The set of consecutive integers is

{12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}

Since all even numbers are divisible by 2, eliminate all even numbers, leaving the odd numbers in the set:

{13, 15, 17, 19, 21, 23, 25, 27, 29}

Eliminate those integers that are multiples of 3 (15, 21, and 27). The remaining integers are:

{13, 17, 19, 23, 25, 29}

Therefore, there are 6 numbers in the set that are multiples of neither 2 nor 3.

Question analysis

This number properties question takes the integers from 12 to 30, inclusive, and asks how many are multiples of neither 2 nor 3. Students must count the whole set, remove multiples of 2 and of 3 without removing any number twice, and count what remains. Inclusive endpoints and the overlap make it easy to miscount.

NYS standards
  • 6.NS.4 common factors and multiples
Grade level
Grade 6 content, by NYS standard
Compared with grade 7
Above grade 7 Multiples are sixth-grade content, but grade 7 problems rarely ask students to count a set under two overlapping conditions. The inclusive range and the numbers that are multiples of both make this longer and trickier than usual.
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 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

The question gives the overlap, four multiples of both 2 and 3, which invites a subtraction method with three places to go wrong: the count of the set, the overlap, and the final complement. Each of those errors has its own choice. Listing the 19 numbers and crossing out evens and then odd multiples of 3 is slower to describe but faster and safer to do.

  1. Much to keep track of moderate Working it out

    Each of the 19 integers must be checked against two conditions, and the four multiples of 6 meet both, so they must be removed once, not twice. (Sweller, 1988; Carpenter, Just & Shell, 1990)

  2. A very close wrong answer moderate Choosing the answer

    The choices 5 and 6 differ by one, the result of counting 30 − 12 = 18 integers instead of 19, so only counting the inclusive set exactly separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Cross out every even number first, then cross out the odd multiples of 3 that remain, and count the survivors; listing all nineteen numbers is quick enough.

Traps

  • Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.
  • Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Counting whole numbers in a list
n=b−a+1n = b - a + 1
a is the first number and b is the last. n is how many whole numbers there are, counting both ends. You add 1 because subtracting counts the gaps, and there is one more number than gaps.
  1. Sort out the question

    The set is every integer (whole number) from 12 to 30. “Inclusive” means 12 and 30 are both in.

    A multiple of 2 is an even number. A multiple of 3 is a number in the 3 times table. You need the numbers that are neither: not even, and not in the 3 times table.

  2. Count the numbers in the set

    n=30−12+1=19n = 30 - 12 + 1 = 19

    There are 19 integers. Watch out: 30 − 12 = 18 forgets one end. That slip leads to choice A, 5.

  3. Cross out the even numbers

    Every even number is a multiple of 2, so remove them all. The odd numbers are left:

    13, 15, 17, 19, 21, 23, 25, 27, 29

    That is 9 numbers.

  4. Cross out the multiples of 3

    Check each odd number for 3. A quick test: add its digits. If the sum is in the 3 times table, so is the number. That removes 15, 21 and 27. These are left:

    13, 17, 19, 23, 25, 29

    Watch out: only 12, 18, 24 and 30 are multiples of both. Removing just those gives 19 − 4 = 15, choice D. “Neither” means you remove every multiple of 2 and every multiple of 3.

  5. Count what is left

    Six numbers are left. Watch out: 13 is how many you crossed out, choice C. The question asks for the ones left. So the answer is B, 6.

Check your answer

Count another way. Of the 19 numbers, 10 are even. Of the 9 odd ones, 3 are multiples of 3. So 10 + 3 = 13 are crossed out, and 19 − 13 = 6 are left. Same answer.

The original DOE question
Question 97 as printed on page 88 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 88, question 97. Open page 88 of the handbook (PDF)

Sample #387495