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
Strong writers match their certainty to their evidence. A hedge is a word or phrase such as may, suggests, likely, or most that signals how sure a writer is. Each claim below goes further than its evidence. Rewrite each one so it says only what the evidence supports. Then explain how too many hedges could also weaken a sentence. (These examples are made up for practice.)
- Claim: Music makes plants grow. Evidence: In one small greenhouse trial, plants near a speaker grew slightly taller.
- Claim: Everyone hates the new library hours. Evidence: In a survey, 60% of visitors preferred the old hours.
- Claim: The bridge will fail within ten years. Evidence: Inspectors found cracks and rated the risk as moderate.
Answer and teacher notes
Answers will vary. Samples:
1. In one small trial, plants near a speaker grew slightly taller, which suggests music may affect growth.
2. Most surveyed visitors preferred the old library hours.
3. Inspectors found cracks and rated the bridge’s risk as moderate, so repairs may be needed.
Too many hedges: a sentence such as “It could possibly perhaps be somewhat true” tells the reader nothing and sounds unsure of the evidence.
Common mistake: adding a hedge but keeping the overreach, such as “Music may make all plants grow.” Have students check scope (how many?), strength (how much?), and cause (did one thing really cause the other?). In #1, the trial doesn’t rule out other causes. In #3, “will fail within ten years” adds a time limit that the evidence never gave. Extension: find a headline online that uses “may” or “could,” then explain what the writer is being careful about.
Math
A surveyor needs the distance across a pond, from point A to point B, but can’t measure it directly. From a point C on land, she measures CA = 420 m, CB = 310 m, and angle C = 58°. (These measurements are made up for practice.) (a) Use the law of cosines, c² = a² + b² − 2ab cos C, to find AB to the nearest tenth of a meter. (b) Use the law of sines to find angle A to the nearest tenth of a degree, then find angle B. (c) Find the area of triangle ABC to the nearest square meter using Area = ½ab sin C. (d) A classmate uses the Pythagorean theorem and gets 522.0 m. Why is that method wrong here?
Answer and teacher notes
(a) Let a = CB = 310 and b = CA = 420. c² = 310² + 420² − 2(310)(420) cos 58° = 96,100 + 176,400 − 260,400(0.52992) ≈ 272,500 − 137,991 = 134,509. c ≈ 366.8 m. (b) sin A ÷ 310 = sin 58° ÷ 366.75, so sin A ≈ 0.7168 and A ≈ 45.8°. B ≈ 180° − 58° − 45.8° = 76.2°. (c) ½(310)(420) sin 58° ≈ 65,100(0.84805) ≈ 55,208 m². (d) The Pythagorean theorem works only with a right angle. Angle C is 58°, not 90°, so it overestimates the distance.
Check on (b): the law of cosines gives the same angle B. cos B = (310² + 366.75² − 420²) ÷ (2 × 310 × 366.75) ≈ 0.2384, so B ≈ 76.2°. ✓ Also, the longest side (CA = 420) sits across from the largest angle (B). ✓ Common mistake: subtracting 2ab before multiplying by cos C. The Pythagorean result is what you get when cos C = 0, which is why the two formulas match at 90°. Extension: what would AB be if angle C were 120°? (cos 120° = −0.5, so c² = 272,500 + 130,200 = 402,700, and c ≈ 634.6 m.)
Science
Scientists recently reported all five nucleobases of DNA and RNA (adenine, guanine, cytosine, thymine, and uracil) in samples from the asteroid Ryugu, which Japan’s Hayabusa2 mission returned in 2020. In DNA, A always pairs with T, and C always pairs with G. A-T pairs share two hydrogen bonds, and C-G pairs share three. (a) A double-stranded DNA sample is 22% adenine. Find the percent of T, G, and C. (b) Find the total hydrogen bonds in a 100-base-pair segment that is 60% G-C pairs. Do the same for one that is 40% G-C. (c) Which segment should take more heat to separate? Why? (d) The five bases turned up in roughly similar amounts in the Ryugu samples. Why is the rule from (a) not a good reason to expect A to equal T in a loose mixture? (e) Why doesn’t finding these molecules on an asteroid prove that life existed there?
Answer and teacher notes
(a) T = 22%. A + T = 44%, so G + C = 56%, and G = C = 28%. (b) 60% G-C: 40(2) + 60(3) = 260 bonds. 40% G-C: 60(2) + 40(3) = 240 bonds. (c) The 60% G-C segment, since it has more hydrogen bonds holding its strands together. (d) A equals T in DNA because the bases are paired in a double helix. Loose molecules in a rock are not paired, so nothing forces them to match. (e) Answers will vary: these molecules can form through chemistry without living things, and the report says they may have formed in space.
Check on (a): 22 + 22 + 28 + 28 = 100. ✓ Common mistake: assuming all four bases must be equal. Chargaff’s observation is that A roughly equals T and G roughly equals C, not that all four match. The bond count is a simple model for comparing the two segments. The report says researchers have to use ultra-clean lab conditions, which is why contamination is worth discussing. Extension: if you were reviewing this study, what test would convince you the molecules did not come from the lab?
Sources: ScienceDaily: All 5 Genetic Letters Discovered in an Asteroid (Ryugu findings, Hayabusa2, clean labs); Nature Scitable: Discovery of DNA Structure and Function (Chargaff’s rule, pairing, hydrogen bonds); LibreTexts: Base Pairing in DNA (pairing, bonds)
Social Studies
Journalist Ida Tarbell was born on November 5, 1857, in northwestern Pennsylvania. Her father built wooden oil tanks and later became an oil producer and refiner. By 1879, Standard Oil controlled about 90% of the country’s petroleum business. Tarbell spent almost two years going through public records, including court testimony, government reports, and newspapers. Her 19-part series in McClure’s ran from November 1902 to October 1904 and became a book. Congress passed the Sherman Antitrust Act in 1890, and on May 15, 1911, the Supreme Court upheld an order to dissolve Standard Oil. (a) How many years ago was Tarbell born? How long after the Sherman Act did the 1911 ruling come? (b) About how many months passed, on average, between installments? (c) Tarbell grew up around the oil business. How could that help her work, and how could it raise questions about bias? (d) Why might public records persuade readers more than unnamed sources? (e) What evidence would you need to decide whether a company grew large by offering better products or by shutting out rivals?
Answer and teacher notes
(a) 2026 − 1857 = 169 years. 1911 − 1890 = 21 years. (b) The series covers about 24 months. 24 ÷ 19 ≈ 1.3 months per installment. (c) Answers will vary: she knew the industry and its people, but a family link could shape what she looked for. (d) Answers will vary: readers and critics can check records for themselves. (e) Answers will vary: prices over time, contracts, rivals’ sales, and testimony from customers and competitors.
Britannica says her articles helped define what President Theodore Roosevelt labeled muckraking in 1906 and that her reporting contributed to the breakup. Source check: the ABA Journal says Standard Oil was split into 37 entities, while Market Realist says 34. Ask students why a count might differ. The ABA Journal adds that the Court applied the law only to “unreasonable” restraints, which is a useful detail for strong classes. Keep the discussion on evidence and method rather than on any modern company. Extension: what does a good investigative reporter do to show that a story is fair?
Sources: PBS American Experience: Ida Tarbell (birth, father, research, 19-part series); Britannica: Ida Tarbell (book, muckraking, breakup); ABA Journal: Supreme Court Orders Standard Oil Breakup (90%, 1890, ruling, 37 entities); Market Realist: What Happened to Standard Oil (34 companies)
Current Events
The crown-of-thorns starfish is known as a coral killer, and the Great Barrier Reef has seen four major outbreaks since the 1960s, roughly every 15 years. A new University of Sydney study complicates that picture. Researchers examined feeding scars at One Tree Island, in protected areas with low starfish numbers, and found that living coral tissue survived in nearly 75% of feeding events. Fast-growing corals in the Acropora family did not regrow over the damage, while others, mainly Montipora and Porites, did. The team says that by feeding on fast growers such as Acropora, a small starfish population can make room for slower species. The authors also warn that outbreaks remain a serious threat. (a) If this rate held across 200 feeding events, in about how many would living tissue survive? (b) Do four outbreaks at about 15-year spacing fit the period since the 1960s? (c) The authors say the results are consistent with the intermediate disturbance hypothesis. Using the fast and slow corals, explain why moderate damage could increase variety. (d) Why does the study’s location limit what it can tell us? (e) Write an accurate headline for this study in 12 words or fewer.
Answer and teacher notes
(a) 0.75 × 200 = about 150 events. (b) Yes, roughly: 4 × 15 = 60 years, and the 1960s to the 2020s is about 60 years. (c) Answers will vary: fast growers can crowd out others, so light damage to them opens space for slower species, while heavy damage would wipe out many species. (d) It looked mainly at protected areas with healthy fish populations and few starfish, so it can’t show what happens during an outbreak. (e) Answers will vary. Sample: “Small Numbers of Crown-of-Thorns Starfish May Help Some Reefs, Study Suggests.”
Link this to today’s ELA prompt: ask students which words in the sample headline are hedges, and why. Common mistake: concluding that outbreaks are harmless. The finding applies to low-density populations. The authors say that once an outbreak is suppressed, the goal should not necessarily be to wipe the starfish out entirely. “Consistent with” does not mean “proves,” so ask students what further evidence would strengthen the idea. The study appeared in Ecology and Evolution. Extension: design a follow-up study. What would you compare, and for how long?
Source: ScienceDaily: The Great Barrier Reef’s Most Notorious Coral Killer May Actually Be Helping It (study location, 75%, outbreaks, species, caveats, journal)
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