Five warm-ups for grades 9–12. Each takes three to five minutes. Answers and teacher notes are in the collapsed box under each prompt.
ELA
Some word pairs look or sound alike but mean different things. Choose the correct word in each sentence, then write a one-line rule or memory trick for each pair.
- The cold weather did not (affect / effect) the team’s performance.
- From her silence, I (implied / inferred) that she disagreed.
- This line is for customers with ten items or (less / fewer).
- The red scarf (complements / compliments) her green coat.
- Honesty is a (principal / principle) I try to live by.
Answer and teacher notes
1. affect. Usually, affect is the action (verb) and effect is the end result (noun).
2. inferred. The speaker implies; the listener infers. (She implied it; I inferred it.)
3. fewer. Use fewer for things you can count (items, people) and less for amounts you can’t (time, water). Many store signs get this wrong.
4. complements. A complement completes something; a compliment is praise (“I like your scarf”).
5. principle. A principle is a rule or belief; a principal is a person in charge or the main thing (“principal reason”).
Note the exceptions for item 1: effect can be a verb meaning “to bring about” (“effect change”), and affect can be a noun in psychology (a person’s displayed emotion). Spell-checkers won’t catch these errors because both words are real. Extension: write one sentence that correctly uses both words in a pair.
Math
A z-score tells how many standard deviations a value is from the mean: z = (x − μ) ÷ σ. A student scored 84 on a history test (class mean 76, standard deviation 5) and 88 on a chemistry test (class mean 82, standard deviation 8). (a) Find the z-score for each test. (b) On which test did the student do better compared with the class? (c) What chemistry score would have matched the student’s history z-score? (d) Assume the history scores are roughly normal. Using the 68–95–99.7 rule, about what percent of the class scored between 71 and 81? About what percent scored above 86?
Answer and teacher notes
(a) History: z = (84 − 76) ÷ 5 = 8 ÷ 5 = 1.6. Chemistry: z = (88 − 82) ÷ 8 = 6 ÷ 8 = 0.75.
(b) History. Even though 88 is the higher raw score, the history score is 1.6 standard deviations above the mean, compared with 0.75 for chemistry.
(c) x = μ + zσ = 82 + 1.6 × 8 = 82 + 12.8 = 94.8.
(d) 71 and 81 are one standard deviation below and above 76, so about 68%. A score of 86 is two standard deviations above the mean (76 + 2 × 5). About 95% of scores fall within two standard deviations, leaving 5% split between the two tails, so about 2.5% scored above 86.
Check (c) by working backward: (94.8 − 82) ÷ 8 = 12.8 ÷ 8 = 1.6. ✓ Common mistake: comparing raw scores, or forgetting to subtract the mean before dividing. Discussion: why would a college admissions office, a coach, or a scientist want to compare z-scores rather than raw numbers? (The scores in this problem are made up for practice.)
Science
The Gateway Arch in St. Louis was completed on October 28, 1965. It is 630 feet (192 m) tall, with a stainless-steel skin. Metals expand when heated, by ΔL = αLΔT, where α is the coefficient of linear expansion. Use α = 17 × 10−6 per °C for stainless steel and 23.1 × 10−6 per °C for aluminum. (a) Suppose a straight 192 m stainless-steel beam warms from −15 °C on a winter night to 38 °C on a summer afternoon. How much longer does it get, in centimeters? (b) How much would a 192 m aluminum beam grow over the same range? (c) The National Park Service says the Arch can sway up to 18 inches in 150 mph winds. What percent of its height is that? (d) Why do engineers design tall structures to flex and leave room for expansion instead of making them perfectly rigid?
Answer and teacher notes
(a) ΔT = 38 − (−15) = 53 °C. ΔL = (17 × 10−6)(192 m)(53) ≈ 0.173 m ≈ 17 cm.
(b) ΔL = (23.1 × 10−6)(192)(53) ≈ 0.235 m ≈ 24 cm, about 36% more than steel.
(c) 630 ft = 7,560 in. 18 ÷ 7,560 ≈ 0.0024, or about 0.24% of its height.
(d) A rigid structure that can’t move builds up huge internal stresses as temperature changes and wind push on it, which can crack welds or concrete. Allowing small, controlled movement lets the structure absorb those forces safely.
The beam in (a) is a simplified model; the real Arch is curved and anchored at both ends, so its expansion shows up as tiny changes in shape. The temperatures are illustrative. Stainless steels vary: reference tables list α from about 10 to 17 × 10−6 per °C, depending on the alloy, so 17 gives an upper estimate. Connection: expansion joints on bridges and the small gaps between sidewalk slabs solve the same problem.
Sources: Wikipedia: Thermal expansion (expansion coefficients); National Park Service: The Gateway Arch Celebrates 50th Year of Completion (completion date, height, and sway)
Social Studies
On October 28, 1636, the Massachusetts General Court voted to found “a schoale or colledge,” setting aside £400. It became Harvard College, the oldest institution of higher learning in what is now the United States. It was located in Newetowne, which was renamed Cambridge in 1638. That year, a young minister named John Harvard died and left the college his library and about £800, roughly half his estate; in 1639 the college was named for him. The Puritan founders wanted to avoid leaving their churches with “an illiterate Ministry.” The first nine students graduated in 1642. (a) How many years ago was the college founded? (b) How did John Harvard’s gift compare with the colony’s grant? (c) What does the goal of avoiding “an illiterate Ministry” tell you about Puritan values? (d) Why might a young colony spend scarce money on a college?
Answer and teacher notes
(a) 2026 − 1636 = 390 years. (b) £800 ÷ £400 = 2: his bequest was about twice the colony’s original grant, which helps explain why the college took his name. (c) Answers will vary: Puritans believed people should read and understand scripture themselves, so they valued education and an educated clergy who could interpret texts carefully; learning was tied to religious and community life.
(d) Answers will vary: the colony could not rely on England to keep sending trained ministers and leaders; training its own would help the colony survive and govern itself over the long term; a college signaled that the settlement was permanent. Discussion: how is the purpose of a college today similar to or different from its purpose in 1636? Extension: the first graduating class had 9 students. What can that number tell you about who had access to higher education in the colonies?
Source: EBSCO Research Starters: Establishment of Harvard College
Current Events
A team led by Lisa Dang of the University of Waterloo used the James Webb Space Telescope to watch the rocky “lava world” TOI-561 b, about 280 light-years away, through nearly four complete orbits in 37 hours (reported this week by ScienceDaily). The planet reflects about 75% of the starlight that hits it, similar to Venus. Computer models suggest clouds of silicate minerals (the materials in sand and rock) may form over its dayside and act like a “giant sunscreen.” A bare rock this close to its star would be expected to reach about 2,700 °C, but an earlier Webb study measured the dayside at about 1,800 °C. (a) If the planet reflects 75% of incoming starlight, what fraction does it absorb? How much less energy is that than a surface that absorbed all of it? (b) Nearly four orbits took 37 hours. What does that tell you about the length of one orbit? NASA reports the orbit takes less than 11 hours. Are the two facts consistent? (c) The light Webb collected left the planet’s star about 280 years ago. In about what year? (d) Explain how a reflective atmosphere could make the dayside much cooler than expected.
Answer and teacher notes
(a) It absorbs 100% − 75% = 25%, one quarter, so it takes in about 4 times less energy than a perfect absorber would. (b) If four full orbits took 37 hours, each would last 37 ÷ 4 = 9.25 hours. Since the team saw nearly (slightly fewer than) four orbits, one orbit must be a little longer than about 9.25 hours. That fits NASA’s “less than 11 hours.” ✓ A whole “year” on this planet is shorter than half a day on Earth. (c) 2026 − 280 ≈ 1746, during the colonial era.
(d) Bright clouds reflect much of the starlight back into space before it can reach and heat the surface, much as white clothing stays cooler in the sun than black clothing. The study also found that the atmosphere seems to carry heat from the dayside to the nightside, which spreads the heat around. Emphasize the hedged language: the “sunscreen” idea comes from models, and the atmosphere’s gas makeup is still unknown (possibilities include carbon dioxide or sulfur dioxide). The team plans more Webb observations. Discussion: why does a 2,700 °C prediction versus a 1,800 °C measurement count as evidence for an atmosphere?
Sources: ScienceDaily: James Webb Discovers a Scorching Lava Planet with a “Giant Sunscreen” (University of Birmingham; The Astronomical Journal; October 27, 2026); NASA: Webb Detects Thick Atmosphere Around Broiling Lava World (orbital period and temperatures)
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