3 English and 3 math
Here are 6 random questions to try. Takers earning below 75% are unlikely to receive an offer. You have 1 minute 48 seconds per question on the real exam.
Question 1 of 6
2026 Practice Test D English, question 3Passage A Memory Revolution
A Memory Revolution
A high school senior logs on to a computer at the library to double-check the application due date listed on a college admissions website. Nearby, a librarian helps a group of biology students use a database to search for recent studies about mammals. In the past few decades, the Internet has become an integral component of daily life for many people. The seemingly limitless power of search engines made the Internet search extremely common, and today people increasingly rely on the Internet’s vast accumulation of sources to access all types of information. Scientists are beginning to examine how this reliance is modifying the strategies people use to store and prioritize information in their mind.
A Dependable and Valuable Asset
Psychologist Benjamin Storm from the University of California, Santa Cruz, and researchers Sean Stone and Aaron Benjamin devised an experiment to study students’ tendency to depend on the Internet for facts. To begin, the scientists divided sixty participants into multiple groups, including an “Internet” group and a “memory” group, and placed them in front of computers. The Internet group was required to use the search engine Google to answer eight challenging trivia questions. In contrast, participants in the memory group were permitted to use only their personal knowledge to answer the questions. In the second round, the researchers administered notably easier questions. This time, they allowed each group the option of using Google as they answered. Their results showed that 83 percent of the Internet group continued to consult Google in the second round, while only 63 percent of the memory group chose to do so.
People’s growing inclination to rely on the Internet in order to retrieve information, particularly facts and figures, is called cognitive offloading. Canadian researcher Evan F. Risko and British researcher Sam Gilbert, who have written extensively about the topic, say a similar process has been taking place for centuries. In the past, people used resources like encyclopedias to assist their memories; however, today the Internet, serving as a vast extended memory, allows people to digitally access and retrieve much larger volumes of information. Consequently, people’s minds are free for other cognitive feats, such as connecting data, learning new information, or solving problems.
Filing Information Away
The use of the Internet also appears to be modifying the strategies people use to store information in their mind. Researchers Betsy Sparrow from Columbia University, Jenny Liu from the University of Wisconsin-Madison, and Daniel Wegner from Harvard University conducted several studies to discover how people efficiently manage their information intake.
To begin, the researchers examined how people evaluate which information deserves their effort to remember. For this experiment, participants read forty trivia facts, such as “An ostrich’s eye is bigger than its brain,” and typed the statements into a computer file. Half the participants had been previously told the file would be saved, while half believed it would be erased. Next, the participants wrote down every fact they could recall. Those who believed the information would be erased and no longer available could recall 40 percent more facts than those who thought the information would be saved.
In another experiment, the same researchers tested the ability of study participants to remember where to access information. For this trial, participants read and typed trivia statements, which they saved in folders with generic names such as “Facts” and “Items.” After spending ten minutes writing down all the facts they could recall from memory, participants were asked which folder contained a particular fact based on a keyword. For example, “Which folder has the fact about ostriches?” Overall, participants recalled the information’s location more often than the content itself, correctly identifying 49 percent of the folders for specific facts while remembering only 23 percent of the actual trivia. The researchers concluded that our memory is adapting to the Internet age by prioritizing where to locate information even when the specific details are forgotten. According to Sparrow, the Internet has become an important form of transactive memory, an external source of the recollections and associative networks that constitute memory.
As the Internet’s resources continue to expand our “external” memory, some question whether the process may cause people to depend too heavily on technology. However, Steven Pinker, a professor of psychology at Harvard University, says, “Knowledge is increasing exponentially; human brainpower and waking hours are not. Fortunately, the Internet and information technologies are helping us manage, search and retrieve our collective intellectual output at different scales, from Twitter and previews to e-books and online encyclopedias. Far from making us stupid, these technologies are the only things that will keep us smart.”
Question
Read this sentence from paragraph 7.
As the Internet’s resources continue to expand our “external” memory, some question whether the process may cause people to depend too heavily on technology.
How does the sentence contribute to the structure and development of ideas in the passage?
- AIt presents a claim about the risks of relying on the Internet that prompted the research described in paragraphs 2 and 6.
- BIt contrasts a disadvantage of relying on the Internet with the benefits of Internet use that are described in paragraphs 2 and 6.
- CIt signals a shift from a neutral viewpoint in paragraphs 2 and 6 to a presentation of an argument and a counterargument.
- DIt introduces a counterargument and marks a transition from an optimistic tone in paragraphs 2 and 6 to a cautious tone as the counterargument is developed.
Show the answer
Correct answer: C
The question asks how the sentence from paragraph 7 contributes to the structure and development of ideas in the passage.
A. Incorrect. Although the studies showed, in part, that reliance on the Internet is increasing (“participants recalled the information’s location more often than the content itself” [paragraph 6]), the studies were not conducted in order to determine the risks of Internet use. The claim in the sentence did not prompt the research described in paragraphs 2 and 6.
B. Incorrect. While the sentence in paragraph 7 poses a question about the disadvantages of relying on the Internet, paragraphs 2 and 6 present the study data on Internet use and memory in a neutral tone (“correctly identifying 49 percent of the folders for specific facts while remembering only 23 percent of the actual trivia” [paragraph 6]). Neither study describes the availability of information on the Internet as an advantage or a disadvantage.
C. CORRECT. Paragraphs 2 and 6 describe the results of experiments related to the Internet’s effect on memory without commenting on the advantages or disadvantages of relying on that technology, and the sentence in paragraph 7 presents a questioning viewpoint that moves away from the neutral perspective of the broader passage. The opinion in the sentence is directly argued against in the three sentences that follow it, which present an extended quote from Steven Pinker, a psychology professor at Harvard University. Pinker refutes the idea expressed by the sentence, stating that “knowledge is increasing exponentially; human brainpower and waking hours are not.” He adds that technologies such as the Internet “are the only things that will keep us smart.”
D. Incorrect. Although the sentence in paragraph 7 expresses caution about the possible effects of the Internet on memory, the passage does not shift from an optimistic tone to a cautious tone. Actually, paragraph 6 provides the data on Internet use and memory in a neutral tone. The argument expressed by the sentence is also not elaborated on in the sentences that follow, but is immediately argued against by Steven Pinker, a professor of psychology. The last sentence of paragraph 7 in fact refutes the culled sentence: “ ‘Far from making us stupid, these technologies are the only things that will keep us smart.’ ”
Question analysis
This text-structure question quotes the sentence that opens the last paragraph of “A Memory Revolution,” where some people question whether the Internet may make us depend too heavily on technology. Students decide how it contributes to the structure and development of ideas in the article. Every choice also describes the experiments in paragraphs 2 and 6, so students must judge how the article presents them and what follows.
- Grade level
- Grade 8 reading and task
- Compared with grade 7
- Above grade 7 Grade 7 structure questions often ask what one sentence does in its paragraph. This one asks how a sentence turns the whole article, and every choice also characterizes paragraphs 2 and 6, closer to grade 8 work on opposing viewpoints.
- 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
- A very close wrong answer moderate
- Evidence in several places moderate
- Hard words minor
- Evidence
- Answered from several paragraphs or stanzas
- Reading level
- Passage: about grade 14, 763 words; question and choices: about grade 12, 129 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
The quoted sentence is the article’s only worried note, and the question asks what it does there. Students must judge two things at once: how the earlier studies sound, plain numbers such as 49 and 23 percent with no verdict, and what follows, Pinker’s “However” and a confident rebuttal. The near miss rightly calls the sentence an opposing view but hears the studies as cheerful and expects caution to follow, so only students who read on to the end of paragraph 7 can rule it out.
How, not what moderate Working it out
The stem asks what the sentence does in the article, and its job is a turn: from studies reported without judgment to a worry that Pinker’s quotation rebuts in the same paragraph. (Anderson & Krathwohl, 2001)
A very close wrong answer moderate Choosing the answer
The choice about a counterargument and a cautious tone shares the key’s frame, a shift away from paragraphs 2 and 6 toward opposing views; it differs in calling those paragraphs optimistic and the ending cautious. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Evidence in several places moderate Working it out
Every choice also characterizes paragraphs 2 and 6, so students must recall how the Google study and the folder study were reported, as percentages with no praise or blame, before judging the sentence. (Kirsch & Mosenthal, 1990; Gorin & Embretson, 2006)
Hard words minor Understanding the question
The choices judge structure in abstract words, “neutral viewpoint,” “counterargument,” “optimistic tone” and “cautious tone,” which students must match to how the article actually sounds. (Perfetti, 2007; Freedle & Kostin, 1993)
Strategy: Describe in a word or two how paragraphs 2 and 6 present their studies and what the rest of paragraph 7 does with the quoted sentence’s idea, then test both halves of every choice.
Traps
- 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.
- Wrong part of the text: A choice is true of another paragraph, stanza or speaker than the one asked about.
- Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.
The wrong answers, classified
A Reads too much in (Graesser, Singer & Trabasso, 1994)
Assumes the worry about depending on technology launched the research, though it appears only after the studies, and paragraph 2 says its experiment set out to study a “tendency to depend on the Internet.”
B Wrong part of the text (Kirsch & Mosenthal, 1990)
Credits paragraphs 2 and 6 with benefits found elsewhere, the freed-up minds of paragraph 3 under the heading “A Dependable and Valuable Asset,” though the two studies report results without calling them good.
D Wrong tone or attitude (Graesser, Singer & Trabasso, 1994)
Rightly sees an opposing view raised but hears the studies as optimistic and the ending as cautious, when Pinker’s quotation answers the worry at once and closes on “keep us smart.”
Working it out, step by step
Figure out what the question asks
This is a structure question. Structure is how the parts of an article fit together. The question asks what the sentence does at the start of paragraph 7. Each choice also describes paragraphs 2 and 6. So you must check two things: how those paragraphs sound, and what comes after the sentence.
Look at paragraphs 2 and 6
Paragraph 2 reports that “83 percent of the Internet group continued to consult Google in the second round, while only 63 percent of the memory group chose to do so.” Paragraph 6 reports “correctly identifying 49 percent of the folders for specific facts while remembering only 23 percent of the actual trivia.” These paragraphs give numbers. They do not say the results are good or bad. Writing that takes no side like this is called neutral.
Read the sentence and what follows
The sentence says “some question whether the process may cause people to depend too heavily on technology.” That is a worry, an argument against relying on the Internet. The very next word is “However.” Then Steven Pinker answers the worry. He ends, “Far from making us stupid, these technologies are the only things that will keep us smart.” An answer to an argument is called a counterargument. So the article moves from neutral reports to an argument and a counterargument.
Check each choice
- A does not fit. The worry comes after the studies. Nothing says it caused the research.
- B does not fit. Paragraphs 2 and 6 do not describe benefits. They only report numbers.
- C fits. The sentence shifts from neutral reports to an argument, and Pinker’s answer follows.
- D does not fit. Paragraphs 2 and 6 are not optimistic, or hopeful. And the end is not cautious. Pinker answers the worry with confidence.
Watch out: D is tempting because the sentence does raise an opposing view. But D gets both tones wrong.
Choose the answer
The sentence turns the article from plain reports to a debate. So the answer is C, “It signals a shift from a neutral viewpoint in paragraphs 2 and 6 to a presentation of an argument and a counterargument.”
Why this answer and not the closest wrong one
D is the closest wrong choice. Both C and D see the sentence bringing in an opposing view. But D calls paragraphs 2 and 6 optimistic, when they only give percentages. D also says a cautious tone follows. Instead, Pinker rejects the worry and ends with “keep us smart.” Only C describes both sides of the turn correctly.
More questions like this
The original DOE question
The passage “A Memory Revolution” as printed (pages 3–4)
Question 2 of 6
2024 Test A English, question 11Passage 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
The author’s use of sequence in paragraphs 1 and 2 contributes to the development of ideas in the excerpt by
- Alisting the many steps that are involved in the process of beading in order to explain its difficulty and complexity.
- Bconveying the importance of following the steps of the beading process in a precise order to work most efficiently.
- Cemphasizing the time required to fully prepare for and execute the many large and small tasks in the activity of beading.
- Ddetailing each step in preparation for beading in order to highlight the author’s enthusiasm for the work.
Show the answer
Correct answer: D
The question asks how the author’s use of sequence in paragraphs 1 and 2 contributes to the development of ideas in the excerpt.
A. Incorrect. In paragraph 2, the author does list the steps her mother takes in preparation to do beadwork, such as “untying the long tasseled strings” of the bag of beads, spreading the beads beside her on a mat, and using “a long, narrow blade” to trim the buckskin into shape. However, the language used to describe this sequence does not indicate that preparing to work with beads is difficult or complex.
B. Incorrect. Although the author’s description of untying the bag of beads, spreading the beads upon a mat, smoothing out a sheet of buckskin, and trimming the buckskin into shape in paragraph 2 indicates that the mother follows a certain routine, it does not place any emphasis on working most efficiently by following the steps of the beading process in a precise order.
C. Incorrect. In paragraph 2, the author describes her mother “untying the long tasseled strings that bound a small brown buckskin bag,” “spread[ing] upon a mat beside her bunches of colored beads,” “smooth[ing] 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” to trim the buckskin. Although these descriptions do outline the steps in her process for creating beadwork, they do not emphasize the time required to do so, since the author does not mention in the excerpt how long it took her mother to fully prepare for and complete the large and small tasks in the activity of beading.
D. CORRECT. In paragraph 1, the author describes how the morning begins with her mother rolling up the canvas of the wigwam and allowing the breeze to come in. In paragraph 2, the author uses sensory language and precise sequencing to describe the way her mother prepares the beading materials by untying “the long tasseled strings that bound a small brown buckskin bag,” spreading the “bunches of colored beads” beside her, and smoothing out “a double sheet of soft white buckskin.” This combination of sequence and sensory language highlights the author’s enthusiasm for beadwork by emphasizing that the author pays close attention to her mother’s preparations.
Question analysis
This reading question asks how the step-by-step description of a mother preparing her beadwork, in the opening paragraphs of Zitkala-Sa's memoir, contributes to the excerpt's ideas. Every choice agrees that steps are listed, so students must judge which purpose the text actually supports and resist reasons that only sound likely for a craft.
- Grade level
- Grade 8 reading and task
- Compared with grade 7
- Above grade 7 Grade 7 structure questions usually ask what a paragraph adds to a topic. This one asks what an ordered description achieves, in dense 1900 prose where the purpose is never stated outright, which goes 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 major
- A very close wrong answer moderate
- Reading between the lines moderate
- Complex sentences minor
- Evidence
- Answered from several paragraphs or stanzas
- Reading level
- Passage: about grade 8, 946 words; question and choices: about grade 11, 95 words.
- Corrected from the DOE handbook
- A misquotation of 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
All four choices agree that the paragraphs lay out steps, so the question turns entirely on purpose, and three of the purposes sound like natural reasons to describe a craft. Nothing in paragraphs 1 and 2 calls the work hard, slow or a matter of efficiency; what they do show is a child following every move and becoming intensely interested. Students who look for the narrator's own reaction find the purpose, while those who reason from the steps alone drift to difficulty or time.
How, not what major Working it out
The stem asks what the ordered account of untying the bag, spreading the beads, and smoothing and trimming the buckskin does for the excerpt, a purpose the author never states. (Anderson & Krathwohl, 2001)
A very close wrong answer moderate Choosing the answer
The difficulty choice shares the right one's frame, steps laid out in order to show something, and differs only in what they show: how hard beading is, or how eagerly the girl watched. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Reading between the lines moderate Working it out
The enthusiasm has to be pieced together from the close attention to each step and from “intensely interested” and “a proud, beaming face” in paragraph 2. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)
Complex sentences minor Understanding the question
Paragraph 2 uses literary word order, as in “my mother spread upon a mat beside her bunches of colored beads,” which slows the reading of the very steps in question. (Abedi & Lord, 2001; Freedle & Kostin, 1993; Messick, 1989)
Strategy: For each choice, look in paragraphs 1 and 2 for words that would prove its purpose; a purpose that no sentence states or shows can be ruled out.
Traps
- Half right: One half of a two-part choice is right and the other is not.
- 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.
- Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.
The wrong answers, classified
A Half right (King et al., 2004; Thissen, Steinberg & Fitzpatrick, 1989)
Rightly sees that paragraph 2 lists many preparation steps, then decides the list must show that beading is difficult and complex, which nothing in these paragraphs says.
B Wrong purpose or structure (Anderson & Krathwohl, 2001)
Reads the sequence as instructions about doing the steps in a precise order for efficiency, a purpose a watching child's memories never suggest.
C Reads too much in (Graesser, Singer & Trabasso, 1994)
Reasons that so many steps must take a long time, perhaps recalling the “many a sunny hour” of paragraph 4, but paragraphs 1 and 2 never mention time.
What was corrected from the DOE handbook, and why
- Explanation: In rejecting choice C, the handbook quotes paragraph 2 as “spread[ing them] upon a mat beside her bunches of colored beads.” In the excerpt, the bunches of colored beads are what the mother spreads (“my mother spread upon a mat beside her bunches of colored beads”). The inserted “them” refers to nothing and turns the beads into a place. The quotation now reads “spread[ing] upon a mat beside her bunches of colored beads,” which is how the handbook itself reads the sentence in its reason for rejecting choice A (“spreading the beads beside her on a mat”).
Working it out, step by step
Figure out what the question asks
Sequence means the order in which things happen. Paragraphs 1 and 2 tell the steps of a morning in order. The question asks why the author tells them step by step. What does that careful order help show?
List the steps
In paragraphs 1 and 2, the mother:
- rolls up the side of the wigwam to let in the breeze,
- unties a bag made of buckskin (soft deer leather),
- spreads her beads on a mat,
- smooths out a sheet of buckskin, and
- trims it into shape with a blade.
Find the author’s reaction
Right after the steps, the author tells how she felt. “Then I became intensely interested in her designing. With a proud, beaming face, I watched her work.” Intensely means very strongly. A beaming face has a big, happy smile. She noticed every little step because she was so excited. The details show her enthusiasm, or eager interest.
Check each choice
- A is wrong. The steps are listed, but nothing says beading is hard or complex.
- B is wrong. Nothing says the steps must go in a set order to save effort.
- C is wrong. Paragraphs 1 and 2 never say how long anything takes.
- D is right. The careful details show how closely the girl watched, with eager interest.
Watch out: A is tempting because there are many steps. But many steps don’t prove a task is hard. The author never calls it difficult.
Choose the answer
The step-by-step order shows a child following every move her mother makes. So the answer is D.
Why this answer and not the closest wrong one
A is the closest wrong answer. Like D, it says the steps are listed to show something. But no word in paragraphs 1 and 2 calls beading hard. The author does say she was “intensely interested” and had “a proud, beaming face.” That shows enthusiasm, so D fits better.
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
2026 Practice Test D English, question 10Passage 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
The photograph mentioned in paragraph 4 is significant because it
- Adocuments that the machine rose into the air as soon as the tethering wire was released.
- Bprovides proof of the critical moment the machine took flight.
- Cdocuments that the wind reduced the speed of the plane at the start of the flight.
- Dprovides proof that the pilot had to gradually increase the height of the plane in the air.
Show the answer
Correct answer: B
The question asks why the photograph mentioned in paragraph 4 is significant.
A. Incorrect. According to paragraph 4, the photograph was not taken immediately after the tethering wire was released but rather after an intervening period during which “Wilbur ran at the side of the machine, holding the wing to balance it on the track.” The picture was not taken until “the machine had reached the end of the track,” which occurred after a “forty-foot run” (paragraph 4).
B. CORRECT. The photograph mentioned in paragraph 4 is significant because it provides proof that the machine did, in fact, take flight: “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.”
C. Incorrect. The photograph mentioned in paragraph 4 is described as showing the plane moving forward through the air at a height of two feet, but while the effect of the wind may be visible, it is not why the photo is significant. Instead, the photograph provides proof that the plane is, in fact, moving through the air (“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”).
D. Incorrect. Although the plane is shown at a height of two feet in the photograph mentioned in paragraph 4, the picture documents only one moment of the plane’s flight and does not provide proof that the pilot had to gradually increase the height of the plane in the air.
Question analysis
This question on Orville Wright’s account of the first airplane flight asks why the photograph mentioned in paragraph 4, taken by one of the Life Saving men, is significant. The photograph itself is not printed with the excerpt, so students must work from Orville’s description of the moment it was taken. Several choices describe the machine’s movement, so each must be checked against that description.
- Grade level
- Grade 8 reading and task
- Compared with grade 7
- Above grade 7 Grade 7 readers often explain why a detail matters. This photograph is not printed, so students must picture it from Orville’s description, which uses “attitude” in an old sense meaning posture, and then judge why the picture mattered.
- 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
- Reading between the lines moderate
- The right answer is reworded moderate
- Hard words minor
- Evidence
- Answered from one paragraph or stanza
- Reading level
- Passage: about grade 7, 816 words; question and choices: about grade 7, 68 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 photograph is not printed, so students work from three sentences of description, and the last two tell what Orville thinks the picture shows: the machine’s slow speed, visible in Wilbur’s easy pace. That stated detail is the strongest lure. The question asks for significance instead, which the excerpt leaves unstated; students must link the moment captured, the machine just off the track at the end of its run, to paragraph 5’s closing claim that this was the first such flight in history.
Reading between the lines moderate Working it out
Orville never says why the picture matters; students must notice it was taken “just as the machine had reached the end of the track and had risen to a height of about two feet,” the instant of lift-off. (Graesser, Singer & Trabasso, 1994; Kintsch, 1998)
The right answer is reworded moderate Choosing the answer
The key’s “critical moment the machine took flight” appears nowhere, while the wrong choices echo the paragraph’s released wire, the slow speed Orville says the picture shows, and the “height of about two feet.” (Freedle & Kostin, 1993; Kirsch & Mosenthal, 1990)
Hard words minor Understanding the question
Orville says the slow speed shows “by Wilbur’s attitude,” using attitude in an old sense, Wilbur’s posture as he keeps pace, which students must decode to follow the description of the picture. (Perfetti, 2007; Freedle & Kostin, 1993)
Strategy: Reread the three sentences about the picture in paragraph 4, note when it was taken and what Orville says it shows, then weigh which choice explains why such a picture would matter.
Traps
- True but not the answer: A choice says something accurate about the text that does not answer this question.
- Twists what the text says: Uses the text's topic and ideas but changes what it says: who did it, how much, when, or why.
- Reads too much in: A reasonable-sounding conclusion that goes further than the text allows.
The wrong answers, classified
A Twists what the text says (King et al., 2004)
Collapses the release of the wire and the lift-off into one moment, skipping Wilbur’s run beside the machine; it rose only after a forty-foot run, when the picture was taken.
C True but not the answer (King et al., 2004; Embretson & Wetzel, 1987)
Takes Orville’s own remark that the picture shows “the slow forward speed of the machine” as its significance, but that is a detail in the picture, not why a picture of this moment matters.
D Reads too much in (Graesser, Singer & Trabasso, 1994)
Treats the two-foot height as one step in a steady climb, though a single picture catches one instant and paragraph 5 describes sudden rises to ten feet and darts toward the ground.
Working it out, step by step
Figure out what the question asks
This asks why the photograph in paragraph 4 matters. The photograph is not printed with the excerpt. You have to picture it from Orville’s words. Find when it was taken. Then ask why a picture of that moment would be important.
Read the description of the photograph
Paragraph 4 says “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 next sentences say “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.”
Work out what the picture shows
The machine had just left the track and was two feet in the air. So the picture caught the moment it lifted off. Here “attitude” means the way Wilbur held his body. He kept up with the machine easily, so it was moving slowly. Paragraph 5 says this flight was “the first in the history of the world” of its kind. A photo of that lift-off is proof that it really happened.
Check each choice
- A does not fit. The machine did not rise at once. It lifted “after a forty-foot run.”
- B fits. The picture caught the moment the machine first took flight.
- C does not fit. The picture does show the slow speed. But that is a detail in the photo, not why the photo matters.
- D does not fit. One picture shows one moment. It cannot show a steady climb. In fact, paragraph 5 says the machine rose and dropped suddenly.
Watch out: C is tempting because Orville himself says the slow speed is “clearly shown in the picture.” But the question asks why the picture is significant, which means important.
Choose the answer
The photo captured the very moment the first flight began. So the answer is B, “provides proof of the critical moment the machine took flight.”
Why this answer and not the closest wrong one
C is the closest wrong choice. Orville does say the picture shows the machine’s slow speed. But slow speed is not what makes this photo special. It was taken just as the machine “had risen to a height of about two feet” at the end of the track. That is proof of the first moment of flight, as B says.
More questions like this
The original DOE question
The passage “Excerpt from “How We Made the First Flight”” as printed (pages 13–14)
Question 4 of 6
2026 Practice Test B Math, question 24Question
A chemical decays in such a way that the amount left at the end of each week is 20% less than the amount at the beginning of that same week. What percentage of the original amount is left after two weeks?
- A40%
- B60%
- C64%
- D80%
Show the answer
Correct answer: C
Let x be the original amount of the chemical. It loses 20% after each week, which means 80% of the chemical remains at the end of each week. End of first week:
At the end of the second week, 80% of the amount left at the end of the first week remains.
End of second week: or 64%
Question analysis
A chemical loses 20% of the amount present at the start of each week, and students find what percentage of the original amount is left after two weeks. It tests applying a percent decrease more than once. The question turns on which amount each week's loss is measured against.
- NYS standards
- 7.RP.3 multistep ratio and percent problems
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- Above grade 7 A typical grade 7 percent task applies one increase or decrease to one amount. This one applies a weekly decrease twice in a row, an early form of exponential decay that goes beyond typical grade 7 work.
- Estimated difficulty
- Hard: combines ideas, needs a plan, and has more than one tempting wrong answer.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- A very close wrong answer major
- Turning words into math moderate
- Variables instead of numbers minor
- Solution
- 2 steps
- Reading level
- Question and choices: about grade 8, 40 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
Two equal decreases in a row feel like one decrease of twice the size, and that is the trap: 20% and 20% suggest 40% gone. The stem states the rule, each loss measured from that week’s start, in one quick phrase. Working week by week from 100 shows the second loss is 16, not 20, which separates the answer from the choice four points away, while stopping after one week or reporting the loss costs the point more simply.
A very close wrong answer major Choosing the answer
The choices 60% and 64% sit four percentage points apart, so a rough estimate cannot choose between them; only seeing that the second 20% is taken from 80, not 100, does. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Turning words into math moderate Understanding the question
The phrase “the amount at the beginning of that same week” sets each week’s base, so the second week’s 20% is measured from what was left after the first week. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Variables instead of numbers minor Understanding the question
No starting amount is given, so students must pick one, such as 100 grams, or work with a percent of a percent, 0.8 × 0.8. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)
Strategy: Use a starting amount of 100 so that percents become units, find what remains after the first week, then apply the same 20% loss to that remainder.
Traps
- Percent of the wrong amount: Finding the whole from a part, or a percent change measured from the wrong starting amount.
- Stops one step early: A choice is a number from partway through the solution.
- 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
A Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Adds the weekly losses, 20% + 20% = 40%, and gives that figure, which describes the amount lost rather than the percentage still left.
B Percent of the wrong amount (Parker & Leinhardt, 1995)
Subtracts 20% of the original amount twice, 100 − 20 − 20 = 60, though the second week’s 20% comes from the 80% left, not the starting amount.
D Stops one step early (King et al., 2004; Gierl et al., 2017)
Finds that 80% remains after the first week and stops, never taking the second week’s 20% from that remainder.
Worked solution, step by step
Formulas used
- Percent of an amount
- p is the percent, and amount is the whole you take the percent of. Percent means out of 100, so 20% is 20/100. Multiply to find the part. Be sure the amount is the right whole. Here, each week uses the amount at the start of that week.
- Percent left after a loss
- The whole amount is 100%. If 20% goes away, the rest stays. So 80% of each week’s starting amount is left at the end of that week. Keeping 80% is the same as multiplying by 0.8.
Sort out the question
Each week the chemical loses 20% of what it has. The key words are “the amount at the beginning of that same week.” So each week’s 20% comes from that week’s starting amount. No amount is given. You need the percent of the original amount that is left after two weeks.
Pick an easy starting amount
Pretend the chemical starts at 100 grams. With 100, the number of grams left is the same as the percent left. Any starting amount gives the same percent, so choose the easiest one.
Work out the first week
Use the percent formula. The first week starts with 100 grams:
The chemical loses 20 grams, so grams are left. That is 80% after one week.
Watch out: 80% is only the first week. Stopping here gives choice D.
Work out the second week
The second week starts with 80 grams, not 100. So take 20% of 80:
This time the chemical loses only 16 grams, so grams are left.
Watch out: taking away 20 grams again uses the original amount. That gives 60%, choice B. Choice A, 40%, adds the two losses. But the question asks what is left, and only 36% was lost.
Read the answer as a percent
Week Start (grams) Loss (grams) Left (grams) 1 100 20 80 2 80 16 64 Another way: each week keeps 80%, or 0.8, of its start. Two weeks gives , which is 64%.
64 grams out of the original 100 grams is 64%. So the answer is C, 64%.
Check your answer
Think about the size of the second loss. The second week starts with less, so it loses less than 20 grams. That means more than 60% is left. It does lose something, so less than 80% is left. Of the four choices, only 64% is between 60% and 80%.
More questions like this
The original DOE question
Question 5 of 6
2026 Practice Test D Math, question 36Question
Leah has set a goal to read a total of 500 pages of a textbook. The table shows the book page numbers for the reading she did during three days.
DAILY READING LOG
| Day | Pages Read |
|---|---|
| 1 | pages 45–90, inclusive |
| 2 | pages 100–135, inclusive |
| 3 | pages 145–190, inclusive |
What percentage of her 500-page goal did Leah read on these three days? Round your answer to the nearest tenth.
Enter your answer in the space.
Fill-in question: work out the answer and type it as a number.
Show the answer
Correct answer: 25.6%
First, determine the number of pages read each day. Since the pages per assignment are inclusive, subtract the first page from the last page assigned and then add 1 to determine the pages read for each day.
Day 1: pages
Day 2: pages
Day 3: pages
Leah read a total of 128 pages, since .
To determine the percent, express 128 pages out of 500 pages as a fraction.
Question analysis
Leah's daily reading log lists three ranges of textbook pages, each marked inclusive, and her goal is 500 pages. This percent question asks what percentage of the goal she read over the three days, rounded to the nearest tenth and typed in. Counting the pages in each inclusive range correctly is the tricky step before the percent.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- Above grade 7 Finding what percent one number is of another is grade 6 and 7 work. This adds three inclusive page ranges to count, which takes more care than subtracting, then a rounding step, all typed with no choices to check against.
- 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
- Something you must already know moderate
- Many steps moderate
- An answer in an unexpected form minor
- Solution
- 4 steps
- Reading level
- Question and choices: about grade 3, 66 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
Counting pages in a range is the whole difficulty; the percent at the end is routine once the total is right. Subtracting first from last feels natural, and here it even rewards the student with a round 25%, so nothing looks wrong. Checking the rule on a tiny range, pages 1 to 3, which is 3 pages and not 2, fixes the count, and the instruction to round to the nearest tenth hints that the answer is not a whole number.
Something you must already know moderate Working it out
Every row says inclusive, but nothing says that pages 45 to 90 hold 90 − 45 + 1 = 46 pages; subtracting alone loses one page on each of the three days. (Anderson & Krathwohl, 2001)
Many steps moderate Working it out
Three page counts, a sum, a division by 500 and a change to a percent come in sequence, and a one-page error in each count carries through to the end. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)
An answer in an unexpected form minor Choosing the answer
The answer is typed with no choices, and the miscounted total, 125 pages, gives a tidy 25% that looks more finished than the right value with its tenth. (Enright, Morley & Sheehan, 2002)
Strategy: Count each day's pages carefully, testing your method on a short range such as pages 1–3, add the three days, divide by 500, and write the result as a percent to the nearest tenth.
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.
- Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
The wrong answers, classified
25 Counting the ends (Brown & Burton, 1978)
Subtracts the first page from the last for each day, 45 + 35 + 45 = 125 pages, forgetting that an inclusive range counts both end pages, so 125 ÷ 500 = 25%.
0.256 Stops one step early (King et al., 2004; Gierl et al., 2017)
Counts the 128 pages and divides by 500 correctly, then enters the decimal 0.256, one step short of multiplying by 100 to make a percent.
26 Misreads the question (Hegarty, Mayer & Monk, 1995)
Finds 25.6% correctly but rounds it to the nearest whole percent, overlooking the instruction to round to the nearest tenth.
Worked solution, step by step
Formulas used
- Counting pages in an inclusive range
- Inclusive means the first and last pages are both read. Subtracting counts the steps from the first page to the last. That misses one page, so add 1. Test it: pages 1 to 3 are 3 pages, and 3 − 1 + 1 = 3.
- Percent of a total
- The part is the amount you have, here the pages read. The whole is the amount you compare with, here the 500-page goal. Dividing gives a decimal. Multiplying by 100 turns it into a percent, a number out of 100.
Sort out the question
Leah’s goal is 500 pages. The log shows three days:
- Day 1: pages 45 to 90
- Day 2: pages 100 to 135
- Day 3: pages 145 to 190
Each range is inclusive. You need the percent of 500 pages she read, rounded to the nearest tenth.
Count the pages each day
Use the rule for an inclusive range: last page minus first page, plus 1.
Watch out: subtracting alone gives 45, 35 and 45 pages. That is 125 pages, which is exactly 25%. A round answer like that can hide the mistake.
Add the three days
Add the pages from all three days to get her total:
Leah read 128 pages in all. This is the part you compare with the goal.
Change to a percent
Divide the pages read by the goal, then multiply by 100:
Leah read 25.6% of her goal. That already ends in tenths, so no rounding is needed.
Watch out: 0.256 is a decimal, not a percent. And 26 rounds to a whole number, not to the nearest tenth.
Enter the answer
The box has no % sign, but the question asks for a percentage, so the number in the box is the percent. Type it with its decimal point. Typing 25.6% also counts on this site. So you would enter 25.6.
Check your answer
Estimate. 10% of 500 pages is 50 pages. 128 pages is a bit more than 2 × 50 = 100, so the answer is between 20% and 30%. Then check exactly: pages. It matches.
More questions like this
The original DOE question
Question 6 of 6
2026 Practice Test D Math, question 48Question
The perimeter of square W is twice the perimeter of equilateral triangle X. What is the ratio of the length of one side of square W to the length of one side of equilateral triangle X?
- A3:8
- B2:3
- C3:2
- D2:1
Show the answer
Correct answer: C
First, set up the ratio of the perimeter of the square to the triangle as 2:1 because the question states that the perimeter of the square is twice the perimeter of the triangle.
Recognizing that the square has 4 equal sides and that an equilateral triangle has 3 sides, use division to find the ratio of the sides.
Perimeter:
Sides:
Question analysis
The perimeter of square W is twice the perimeter of equilateral triangle X, and students find the ratio of a side of the square to a side of the triangle. This ratio question uses perimeters of regular shapes with no lengths given. Turning a comparison of perimeters into one of sides, and keeping the ratio in the order asked, are the tricky parts.
- NYS standards
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- Above grade 7 Grade 7 ratio problems usually supply numbers to work with. This one gives only a relationship between two perimeters, so students must bring in the number of sides of each shape themselves and keep the ratio in the order asked.
- 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
- Turning words into math moderate
- Variables instead of numbers moderate
- A very close wrong answer minor
- Solution
- 3 steps
- Reading level
- Question and choices: about grade 9, 36 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 turns a statement about perimeters into a ratio of sides, and the natural guess, that the sides keep the perimeters’ 2 to 1, ignores that the square shares its perimeter among 4 sides and the triangle among 3. Choosing a triangle perimeter such as 12 makes the comparison concrete. The order of the ratio still has to be checked at the end, because its reverse is listed too.
Turning words into math moderate Understanding the question
“The perimeter of square W is twice the perimeter of equilateral triangle X” must become 4s = 2 × 3t, with the 2 on the triangle’s side; written backward, it gives the triangle the larger perimeter. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Variables instead of numbers moderate Understanding the question
No side length is given, so students must either work with letters for the two sides or choose a convenient perimeter themselves, such as 12 for the triangle. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)
A very close wrong answer minor Choosing the answer
The ratios 3:2 and 2:3 use the same two numbers, so only keeping the square’s side first, as the question orders it, separates the answer from its reverse. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Strategy: Pick an easy side length for the triangle, find its perimeter, double that for the square, divide by the square's number of sides, and write square to triangle.
Traps
- Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.
- Wrong operation: Adds instead of multiplying, divides the wrong way round, turns a ratio or rate upside down, or undoes a step in the wrong direction (subtracts what should be added back).
- Equation written backwards: Writes a comparison the wrong way round, word for word: "six times as many students as professors" as 6S = P. The bigger quantity gets the bigger number.
The wrong answers, classified
A Equation written backwards (Clement, Lochhead & Monk, 1981)
Writes the comparison backward, 2 × 4s = 3t, giving the triangle the larger perimeter, so the ratio of the sides comes out 3:8.
B Wrong operation (Hegarty, Mayer & Monk, 1995; King et al., 2004)
Finds the right relationship but turns the ratio upside down, giving the triangle’s side to the square’s instead of the square’s side first.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Assumes the sides are in the same ratio as the perimeters, 2:1, ignoring that the square has 4 sides and the triangle only 3.
Worked solution, step by step
Formulas used
- Perimeter of a square
- P is the perimeter, the distance around the shape. s is the length of one side. A square has 4 equal sides, so its perimeter is 4 times one side.
- Perimeter of an equilateral triangle
- An equilateral triangle has 3 equal sides. t is the length of one side. So its perimeter is 3 times one side.
- Ratio
- A ratio compares two amounts in order. a : b means a compared with b. The order matters, so write the amounts in the order the question names them.
Sort out the question
Square W has a perimeter twice as long as the perimeter of triangle X. Equilateral means all 3 sides of X are equal.
You need the ratio of one side of the square to one side of the triangle, in that order. No lengths are given, so you can pick a size to try.
Pick a size for the triangle
Choose an easy side length for the triangle, such as 2 units. All 3 sides are 2, so its perimeter is:
Find the square’s side
The square’s perimeter is twice the triangle’s:
The square has 4 equal sides, so divide by 4:
Each side of the square is 3 units.
Write the ratio in order
The square comes first, then the triangle. Their sides are 3 and 2, so the ratio is 3 : 2.
Watch out: 2 : 1, choice D, compares the perimeters, not the sides. The square splits its perimeter into 4 sides, the triangle into only 3. Choice B, 2 : 3, puts the triangle first. Choice A, 3 : 8, comes from giving the triangle the bigger perimeter.
So the answer is C, 3:2.
Check your answer
Try a different size. Give the triangle sides of 4. Its perimeter is 12, so the square’s is 24. Each side of the square is . The ratio 6 : 4 simplifies to 3 : 2. The size does not change the ratio.
More questions like this
The original DOE question
