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Question 1 of 6
2026 Practice Test D English, question 12Passage 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.
- 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.
- Thinking it demands
- Depth of knowledge 3, strategic thinking · mainly analyze
- What makes it hard
- How, not what moderate
- Reading between the lines moderate
- Complex sentences minor
- 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.
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)
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)
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
A Wrong tone or attitude (Graesser, Singer & Trabasso, 1994)
Links the high wind to the cold, hard morning of paragraphs 1 and 2 and assumes Orville resented it, though here he uses the wind’s speed to make the flight count for more.
C Wrong purpose or structure (Anderson & Krathwohl, 2001)
Describes what the sentence contains, a flight measured against one in calm air, without saying what that comparison shows about Orville’s view of the flight, which is what the question asks.
D Outside knowledge (King et al., 2004; Recht & Leslie, 1988)
Assumes careful calculations must be meant to guide later flights, a reasonable idea about early aviation, but this sentence measures the flight just made and looks no further ahead.
Working it out, step by step
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.
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.”
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.
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.
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.
More questions like this
The original DOE question
The passage “Excerpt from “How We Made the First Flight”” as printed (pages 13–14)
Question 2 of 6
2024 Test A English, question 16Passage 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
| Date | Event |
|---|---|
| 1500s | The Dakota use beads made from bones, shells, stones, and animal teeth. European traders bring glass beads to North America. |
| Late 1600s | The Dakota begin trading with the French in Minnesota. |
| 1830s | The Dakota begin using glass beads in clothing, artwork, and decorations in place of Native-made beads. |
| 1862–1865 | The Dakota are expelled from their homelands in Minnesota as a result of the U.S.-Dakota War. |
| 1900s–1920s | The 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.
- 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.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly understand
- What makes it hard
- Every choice must be checked major
- A lot to read moderate
- Hard words minor
- 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.
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)
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)
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
A Too narrow (King et al., 2004)
Keeps the familiar palette image but leaves out trimming the buckskin, piercing it with the awl and twisting the thread, and even moves the smoothing onto a table.
B Too narrow (King et al., 2004)
Matches paragraph 3's piercing and twisting closely but leaves out the preparation, laying out the beads and smoothing the buckskin, and starts the process with cutting.
D Too narrow (King et al., 2004)
Accepts a tidy, well-ordered outline that keeps most steps but leaves out smoothing the buckskin and piercing it with the awl, and piles every tool onto the mat.
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.)
Working it out, step by step
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.
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.”
- Spread the beads on a mat.
- Smooth out a double sheet of buckskin on a lapboard (a board held on the lap).
- Trim the buckskin into shape with a blade.
- Pull out a single sinew, a strong thread.
- Pierce (poke a hole in) the buckskin with an awl, a pointed tool.
- Thread the sinew through the buckskin.
- String the beads, twisting the thread after every stitch.
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.
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.
More questions like this
The original DOE question
The passage “Excerpt from “Impressions of an Indian Childhood”” as printed (pages 42–43)
Question 3 of 6
2025 Practice Test A English, question 12Passage 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.
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.
- 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.
- Thinking it demands
- Depth of knowledge 3, strategic thinking · mainly evaluate
- What makes it hard
- Evidence in several places moderate
- The right answer is reworded moderate
- Reading between the lines minor
- 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.
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)
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)
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
A Too narrow (King et al., 2004)
Rests on paragraph 4’s opening, “education was not available to all,” and ignores paragraph 3, whose Horemheb shows an educated man rising from modest origins to the throne.
C Too broad or too strong (King et al., 2004)
Makes the government and temple schools of paragraph 4 “necessary,” overlooking its other routes: help from “a wealthier relative or patron,” or apprenticeship to a working scribe.
D Reads too much in (Graesser, Singer & Trabasso, 1994)
Turns the rations and lost income that some families bore in paragraph 4 into a requirement for every successful scribe, and adds “the best education,” which the paragraph never mentions.
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.
Working it out, step by step
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.
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.
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.
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.
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.”
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.
More questions like this
The original DOE question
The passage “Excerpt from “Scribe Like an Egyptian”” as shown on the test
Question 4 of 6
2024 Test B Math, question 57Question
If x and y are both negative integers with , which expression has the least value?
- A
- B
- C
- D
Show the answer
Correct answer: A
Since x and y are both negative integers, both and are positive.
Therefore, both and are negative, and both and are positive.
Because and both x and y are negative, and . So .
Since and x and y are integers, .
So .
Therefore, .
Question analysis
Two negative integers x and y, with x less than y, appear in four expressions built from squares and minus signs. Students find the expression with the least value. It tests multiplying negative numbers and comparing their sizes, and the difficulty is tracking how each minus sign and each square changes both the sign and the size of the result.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- Above grade 7 Grade 7 students multiply negative numbers, but this asks them to compare four variable expressions for any pair of negative integers. Deciding how the squares compare when x is less than y goes beyond typical grade 7 sign practice.
- Estimated difficulty
- Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly analyze
- What makes it hard
- Variables instead of numbers moderate
- A very close wrong answer moderate
- Every choice must be checked minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 8, 20 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
Two facts settle the question: squares of negatives are positive, and the more negative number has the larger square. Students who track the minus signs carelessly think a leading minus makes an expression negative, and students who assume the larger y has the larger square choose the wrong negative expression. Substituting x = −3 and y = −2 gives −18, −12, 18 and 12, which orders all four at once.
Variables instead of numbers moderate Understanding the question
No values are given, only that x and y are negative integers with x < y, so students must reason about signs or invent a pair such as −3 and −2. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)
A very close wrong answer moderate Choosing the answer
x²y and xy² use the same letters with the exponent moved; both are negative, and only comparing x² with y² separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Every choice must be checked minor Choosing the answer
All four expressions have to be placed in order, because the minus signs in front of the last two turn those expressions positive. (Embretson & Wetzel, 1987)
Strategy: Pick two negative integers with x less than y, such as −3 and −2, evaluate all four expressions, and confirm the order with a second pair.
Traps
- Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)
- Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
The wrong answers, classified
B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Knows x²y and xy² are both negative but takes y² as the larger square because y > x, though the more negative x has the bigger square.
C Negatives and direction (Brown & Burton, 1978)
Reads the leading minus as making −x²y negative, missing that y is negative, so −x²y is in fact positive.
D Negatives and direction (Brown & Burton, 1978)
Assumes the leading minus makes −xy² negative, missing that x is negative, so −xy² is in fact positive.
Worked solution, step by step
Formulas used
- Signs in multiplication
- a and b are positive numbers, so −a and −b are negative. A negative times a negative is positive. A negative times a positive is negative. So a product is positive when it has an even number of negative factors.
- Squaring a number
- x is any number. x² (x squared) means x times itself. The square of a number other than 0 is always positive, because two negatives or two positives multiply to a positive.
Sort out the question
x and y are negative integers, such as −3 and −2. Since x < y, x is to the left of y on the number line. So x is more negative, farther from 0.
You need the expression with the least value. That is the most negative one.
Find the sign of each expression
Squares are positive, so and are positive.
- is a positive times a negative, so it is negative.
- is a negative times a positive, so it is negative.
- and are the opposites of these, so they are positive.
So the least value must be A or B. Watch out: The minus sign in front of C and D does not make them negative. It turns a negative into a positive.
Compare A and B with numbers
Try x = −3 and y = −2, since −3 < −2.
Since −18 < −12, A is smaller.
See why it always works
Both expressions contain x × y, which is positive. and . Multiplying the same positive number by the more negative x gives the more negative result.
Watch out: y is bigger than x, but is smaller than : 4 is less than 9. Thinking y² is bigger leads to choice B.
So the answer is A, .
Check your answer
Try another pair, x = −5 and y = −1. Then A is , B is , C is 25 and D is 5. A is still the least.
More questions like this
The original DOE question
Question 5 of 6
2018 Test A Math, question 1Question
In the figure above, PQRS is a parallelogram. What is the value of ?
Grid-in question: work out the answer and enter it as a number.
Show the answer
Correct answer: 162
First, find the measure of angle PQR. The measure of angle PQR is equal to the measure of angle PSR.
m∠PSR = 180 − 72 = 108.
So, the measure of angle PQR is also 108.
108 + 90 + x = 360
198 + x = 360
x = 162
Question analysis
A grid-in geometry question about angles around a parallelogram PQRS. A 72° angle sits outside corner S on the extended base, and at corner Q a right angle and the unknown angle x complete the turn around the point. Students chain three angle facts, and the angle asked for lies outside the shape, not inside it.
- NYS standards
- Grade level
- Grade 8 content, by NYS standard
- Compared with grade 7
- Above grade 7 Grade 7 angle work usually applies one or two facts, such as supplementary or vertical angles, to find an angle. This chains three, and needs the equal opposite angles of a parallelogram, which rest on grade 8 parallel-line facts.
- Estimated difficulty
- Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Reading a figure, table or graph major
- Combines ideas moderate
- An answer in an unexpected form minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 7, 14 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
The arithmetic is two subtractions; the difficulty is seeing which angles the marks refer to. Students lose points by treating the 72° as the parallelogram's own angle at S, or by stopping at angle PQR, the most natural number to grid. The question rewards tracing a route from S to Q before computing anything, and checking at the end that x is the angle outside the shape that the arc marks.
Reading a figure, table or graph major Understanding the question
The 72° is marked outside the parallelogram, between side SR and the base extended past S, and x sits outside corner Q, between QP and a vertical ray that meets QR at a right-angle mark. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Combines ideas moderate Working it out
Three angle facts have to be chained: a straight line gives the angle inside S, opposite angles of a parallelogram carry it to Q, and the angles around Q make a full 360° turn. (Embretson, 1998; Daniel & Embretson, 2010)
An answer in an unexpected form minor Choosing the answer
As a grid-in there are no choices to check against, so a student who stops at angle PQR, inside the parallelogram, gets no signal that the job is unfinished. (Enright, Morley & Sheehan, 2002)
Strategy: Use the straight line at S to find the parallelogram's angle there, carry it to the opposite corner Q, then take that angle and the right angle away from 360.
Traps
- Stops one step early: A choice is a number from partway through the solution.
The wrong answers, classified
108 Stops one step early (King et al., 2004; Gierl et al., 2017)
Finds angle PSR = 180 − 72 = 108 and carries it to angle PQR, then grids that interior angle instead of the angle x outside it.
198 Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Reads the 72° as the parallelogram's interior angle at S, so takes angle PQR as 72 and grids 360 − 90 − 72 = 198.
252 Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Overlooks the right-angle mark at Q and subtracts only the 108° angle PQR from 360, giving 252.
Worked solution, step by step
Formulas used
- Angles on a straight line
- a and b are two angles side by side on a straight line. Together they make a half turn, so they always add up to 180 degrees. Use it when an angle you need shares a straight line with an angle you know.
- Opposite angles of a parallelogram
- The symbol ∠ means angle. ∠PQR is the angle at corner Q, between sides QP and QR. ∠PSR is the angle at the opposite corner, S. In a parallelogram, opposite corners always have equal angles, because the shape leans the same way at both.
- Angles around a point
- a, b and c are angles that fit around one point with no gaps and no overlaps. Going all the way around is a full turn, which is 360 degrees. So the angles add up to 360.
Sort out what the figure shows
PQRS is a parallelogram. That is a four-sided shape whose opposite sides are parallel, which means they never meet. You need the value of x, in degrees.
Look closely at the marks. The 72° angle is outside the shape, at S. It sits between side SR and the base line, which keeps going past S. At Q, a vertical line meets side QR. The small square means a right angle, 90°. The angle x is outside the shape too. It runs from that vertical line around to side QP.
Find the angle inside corner S
Angle PSR is the shape’s own angle at S. It sits right next to the 72° angle, on the same straight base line. So the two angles add up to 180°.
So angle PSR is 108°. Watch out: 72° is not the inside angle at S. Using 72 as the inside angle leads to the wrong answer 198.
Carry the angle over to Q
Corners Q and S are opposite each other. In a parallelogram, opposite angles are equal. So angle PQR matches angle PSR.
So angle PQR is 108°. Watch out: this is not x. Angle PQR is inside the shape, and x is outside it. Stopping here and gridding 108 is the most common mistake.
Add the angles around Q
Three angles fill the space all the way around point Q. They are the right angle (90°), angle PQR (108°) and x. Together they make a full turn.
Watch out: don’t skip the right angle at Q. Leaving out the 90 gives 252.
Solve for x
Subtract 198 from 360 to find what is left for x.
So x is 162°. So you would grid 162.
Check your answer
Add the three angles at Q: . That is a full turn. Also, the arc for x is wider than a right angle but not as wide as a straight line. So x should be between 90° and 180°, and 162° is.
More questions like this
The original DOE question
Question 6 of 6
2024 Test A Math, question 18Question
Each number in a sequence is formed by doubling the previous number and then adding 1. If the 9th number in the sequence is 63, what is the 10th number minus the 7th number?
- A96
- B111
- C112
- D127
Show the answer
Correct answer: C
First, write an equation based on the rule described in the item. Let x be the previous number in the sequence and y be the number in the sequence. The ninth number in the sequence is 63.
Determine the 10th number in the sequence, y, by making and solving:
Determine the 8th number in the sequence, x, by making and solving:
Subtract one from both sides of the equation.
Determine the 7th number in the sequence, x, by making and solving:
Subtract the 7th number in the sequence from the 10th number:
The difference is 112.
Question analysis
A sequence question: each number is found by doubling the one before it and adding 1, and the 9th number is 63. Students find the 10th number minus the 7th. That means running the rule forward once and backward twice, while keeping track of which position each number holds.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- Above grade 7 Grade 7 students solve two-step equations but rarely have to run a rule backward through a sequence. This mixes forward and backward steps with position counting, so it asks for more planning than a typical grade 7 task.
- Estimated difficulty
- Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- No obvious method moderate
- Many steps moderate
- Much to keep track of minor
- Solution
- 4 steps
- Reading level
- Question and choices: about grade 7, 32 words.
- Corrected from the DOE handbook
- A misplaced parenthesis in the explanation’s working 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
Applying the rule forward is routine; running it backward is not, and the question needs both. Students must also count positions carefully, since the 9th number is given and the 7th is two steps back, not one. Each listed wrong answer stops or slips at a different stage, so writing the four numbers in a row, 15, 31, 63 and 127, before subtracting is the reliable way.
No obvious method moderate Working it out
The rule is stated only forward, double and add 1, but the 7th and 8th numbers come before the given 9th, so the rule must be undone: subtract 1, then halve. (Schoenfeld, 1985)
Many steps moderate Working it out
One forward step reaches the 10th number, 127, two backward steps reach the 8th and 7th, 31 and 15, and a final subtraction finishes. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)
Much to keep track of minor Working it out
Four positions and their values, 7th to 10th, have to be kept straight, and the 8th number is easy to mistake for the 7th. (Sweller, 1988; Carpenter, Just & Shell, 1990)
Strategy: Apply the rule to 63 to get the 10th number; to go back a position, subtract 1 and halve, doing it twice to reach the 7th; then subtract.
Traps
- Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).
- Stops one step early: A choice is a number from partway through the solution.
The wrong answers, classified
A Counting the ends (Brown & Burton, 1978)
Steps back only once from the 9th number and subtracts the 8th, 127 − 31 = 96, one position short of the 7th.
B Misreads the question (Hegarty, Mayer & Monk, 1995)
Doubles 63 to 126 but forgets to add 1 on that step, then subtracts the correct 7th number: 126 − 15 = 111.
D Stops one step early (King et al., 2004; Gierl et al., 2017)
Finds the 10th number, 2 × 63 + 1 = 127, and stops without subtracting the 7th number from it.
What was corrected from the DOE handbook, and why
- Explanation: Finding the 7th number, the handbook writes “2x + (1 − 1 = 31 − 1)”, with the closing parenthesis after the whole equation, so the line reads as an equation inside an expression (and 1 − 1 = 31 − 1 is false). The site now reads 2x + (1 − 1) = 31 − 1, the same step the handbook writes correctly for the 8th number (2x + (1 − 1) = 63 − 1). It leads to 2x = 30 and x = 15 as before.
Worked solution, step by step
Formulas used
- The sequence rule
- A sequence is a list of numbers that follows a rule. x is a number in the list, and y is the next number. To get the next number, double x and then add 1.
- Going back one number
- y is a number in the list, and x is the one just before it. To go back, undo the rule in reverse order. The rule doubles and then adds 1. So first subtract 1, then divide by 2.
Sort out the question
Each number is “formed by doubling the previous number and then adding 1”. Previous means the one just before. The 9th number is 63. You need the 10th number minus the 7th number. So you go forward one number and back two numbers.
Find the 10th number
Use the rule with x = 63.
The 10th number is 127. Watch out: remember to add 1. Doubling only gives 126 and leads to 111, choice B.
Plan how to go back
The 7th number comes before the 9th, so you must undo the rule. The rule doubled, then added 1. Undo the last step first: subtract 1. Then undo the doubling: divide by 2.
Find the 8th number
The 8th number is 31. Watch out: 31 is the 8th number, not the 7th. Using it gives , choice A.
Find the 7th number
Step back once more, from the 8th number. Use the same rule with y = 31.
The 7th number is 15.
Subtract the 7th from the 10th
Watch out: 127 alone is choice D. You still have to subtract the 7th number. So the answer is C, 112.
Check your answer
Go forward from 15 using the rule. , then , then . You land on 63 as the 9th number, so the list is right. And .
More questions like this
The original DOE question
