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
A poster invites people to join a library’s six-month reading challenge. Identify the main rhetorical appeal in each line: ethos (credibility), pathos (emotion), or logos (logic and evidence). Then write a fourth line for the poster that uses a different appeal than the one you think is strongest, and explain your choice in one sentence.
- “Our head librarian has run this challenge for twelve years and has cheered hundreds of readers across the finish line.”
- “Picture turning your last page in the spring, knowing you kept a promise to yourself.”
- “Twenty minutes a day adds up to more than 60 hours of reading over six months.”
Answer and teacher notes
1. Ethos: it builds trust in the person running the challenge. 2. Pathos: it asks the reader to imagine a feeling of pride. 3. Logos: it uses numbers. Check the math: 20 minutes × 182 days = 3,640 minutes ÷ 60 ≈ 60.7 hours, so “more than 60 hours” holds up.
Most real persuasion mixes appeals. Line 1 also carries some pathos (“cheered”), and line 3 leans on a claim readers can verify. Discussion: which appeal works best on teenagers, and does that change depending on whether the audience is students, parents, or a principal? Push students to notice that a logos line is only as strong as its numbers, which is why checking them matters.
Math
A ball is thrown upward from a height of 5 feet. Its height in feet after t seconds is modeled by h(t) = −16t² + 64t + 5. (a) When does the ball reach its maximum height, and what is that height? (b) Find h(1) and h(3). Why are they equal? (c) To the nearest hundredth of a second, when does the ball hit the ground?
Answer and teacher notes
(a) The vertex is at t = −b/(2a) = −64/(2 · −16) = 2 seconds. h(2) = −16(4) + 64(2) + 5 = −64 + 128 + 5 = 69 feet.
(b) h(1) = −16 + 64 + 5 = 53 ft. h(3) = −144 + 192 + 5 = 53 ft. A parabola is symmetric about its vertex, and t = 1 and t = 3 are each 1 second from t = 2, so the ball is at the same height on the way up and on the way down.
(c) Solve −16t² + 64t + 5 = 0. Quadratic formula: t = [−64 ± √(64² − 4(−16)(5))] / (2 · −16) = [−64 ± √4,416] / (−32). √4,416 ≈ 66.45, so the positive root is t = (−64 − 66.45)/(−32) = 130.45/32 ≈ 4.08 seconds. Check: h(4.07) ≈ 0.44 and h(4.08) ≈ −0.22, so the height crosses 0 between them, closer to 4.08. ✓
The other root (about −0.08 s) is a negative time, so it doesn’t fit the situation. Discussion: the −16 comes from gravity (half of 32 ft/s²). What does the model ignore? (Air resistance, spin, wind.)
Science
Living things take in carbon-14 from the air or food. After an organism dies, it stops taking in carbon-14, and the carbon-14 it already has decays with a half-life of about 5,730 years. The fraction left after t years is (1/2)t/5,730. (a) A bone has 25% of its original carbon-14. How old is it? (b) About what fraction of carbon-14 remains in a 1,200-year-old sample? (c) Radiocarbon dating is generally used for samples up to about 50,000 years old. Use the formula to explain why it stops working beyond that.
Answer and teacher notes
(a) 25% = 1/4 = (1/2)², so two half-lives have passed: 2 × 5,730 = 11,460 years.
(b) (1/2)1,200/5,730 = (1/2)0.2094 ≈ 0.865, or about 86% remains.
(c) 50,000 ÷ 5,730 ≈ 8.73 half-lives. (1/2)8.73 ≈ 0.0024, so only about 0.24% of the original carbon-14 is left. That is so little that it becomes very hard to measure accurately, and tiny amounts of contamination could throw off the result.
Extension: solve for the age of a sample with 70% remaining. t = 5,730 × log(0.7)/log(0.5) ≈ 2,950 years. Physicist Willard Libby developed radiocarbon dating around 1946. Connection: archaeologists used radiocarbon dating this month to show a mummified dog found in Peru is at least 1,200 years old.
Source: Britannica: Carbon-14 dating
Social Studies
On October 20, 1803, the U.S. Senate ratified the Louisiana Purchase treaty by a vote of 24 to 7. For about $15 million, the United States gained about 828,000 square miles from France, roughly doubling the country’s size. President Thomas Jefferson worried that the Constitution did not clearly give the federal government the power to buy territory, and he considered asking for a constitutional amendment. He dropped the idea, partly out of fear that France would back out of the deal. (a) The Constitution requires two-thirds of senators present to approve a treaty. Did the treaty pass with room to spare? Show your math. (b) One square mile is 640 acres. About how much did the land cost per acre? (c) In two or three sentences: was Jefferson right to go ahead without an amendment? Explain the tradeoff he faced.
Answer and teacher notes
(a) 24 + 7 = 31 senators voted. Two-thirds of 31 ≈ 20.7, so 21 votes were needed. The treaty got 24, three more than required, about 77% (24 ÷ 31 ≈ 0.774).
(b) 828,000 × 640 = 529,920,000 acres. $15,000,000 ÷ 529,920,000 ≈ $0.028, or about 3 cents per acre.
(c) Answers will vary; grade the reasoning. The tradeoff: following his own careful reading of the Constitution might have cost the nation a huge, time-sensitive opportunity, but going ahead set a precedent for broader readings of federal power. Strong answers name both the benefit and the cost. Context: the treaty had an October 30 deadline for ratification, and France’s leader Napoleon had threatened to void it. Extension: what are the risks when leaders act first and settle legal questions later?
Sources: National Archives: Jefferson Buys Louisiana Territory, and the Nation Moves Westward; Library of Congress: Louisiana Purchase Legislative Timeline, 1803–1804
Current Events
In 1960, cybernetics expert Heinz von Foerster and two students published a paper in Science titled “Doomsday: Friday, 13 November, A.D. 2026.” Using 24 estimates of world population from the past 2,000 years, they argued that growth kept speeding up and projected that population would approach infinity on that date, just over three weeks from now. A historian writing in The Conversation (shared on ScienceDaily last week) notes the forecasts tracked well until around 1993, then population growth fell behind. One reason he gives: global birth rates dropped, and today’s rate is slightly less than half the 1960 level. (a) How many years ahead were the authors projecting? (b) What is risky about extending a past trend far into the future? (c) The authors themselves urged lower birth rates. Can a warning help cause its own prediction to fail? Explain.
Answer and teacher notes
(a) 2026 − 1960 = 66 years.
(b) A model fit to the past assumes the same conditions will continue. People, technology, and choices change, and a curve that predicts infinity is a sign the model breaks down before then. (c) Yes, this is sometimes called a self-defeating prophecy: if people act on a warning, the predicted outcome may never happen. That makes it hard to judge whether a forecast was “wrong” or whether it worked. The historian argues the paper’s shocking title, backed by real data, drew attention to a serious issue. Keep discussion on how models and predictions work, not on policy. Discussion: can you think of another warning that changed behavior, such as a storm forecast that led people to prepare?
Source: ScienceDaily: Scientists Predicted Doomsday for November 13, 2026. Should We Be Worried? (Alan MacEachern, Western University, via The Conversation)
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