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 logical fallacy is a flaw in reasoning that can make a weak argument sound strong. Match each statement with one fallacy from this bank: ad hominem, bandwagon, false dilemma, hasty generalization, slippery slope. Then rewrite statement 4 so that it makes a fair argument.
- Everybody in our grade uses this app, so it must be the best one.
- Don’t listen to Sam’s fundraiser idea. He couldn’t even parallel park on his driving test.
- Either we go to the beach this weekend, or our whole summer will be wasted.
- I read one science fiction novel and found it dull, so all science fiction must be dull.
- If the team skips one practice, soon nobody will show up, and then the whole program will fall apart.
Answer and teacher notes
1. Bandwagon: popularity doesn’t prove quality.
2. Ad hominem: attacks Sam instead of his idea.
3. False dilemma: presents only two options when others exist.
4. Hasty generalization: draws a big conclusion from one example.
5. Slippery slope: claims one small step will lead to disaster with no evidence for each link.
Rewrite: Sample answer: “The one science fiction novel I read was dull, so I’d like a recommendation before I try another.”
Quick test: ask what evidence actually connects the reason to the conclusion. Point out that a fallacy doesn’t prove a conclusion is false. The app in statement 1 might be great, but the argument doesn’t show it. Common mistake: confusing false dilemma with slippery slope. A false dilemma limits the choices; a slippery slope predicts a chain of events. Extension: find one of these fallacies in an advertisement and explain how it works on the reader.
Math
An email filter is tested on 1,000 messages. Of these, 20% are spam. The filter flags 95% of spam messages, and it wrongly flags 2% of legitimate messages. (These numbers are made up for practice.) (a) Fill in a two-way table with rows Spam and Legitimate and columns Flagged and Not Flagged. (b) Find P(flagged). (c) Find P(spam | flagged). (d) Now suppose only 1% of messages are spam, but the filter works the same way. Find the new P(spam | flagged). Why does it change so much?
Answer and teacher notes
(a) Spam: 200 total, so 190 flagged and 10 not flagged. Legitimate: 800 total, so 16 flagged and 784 not flagged. (b) (190 + 16) ÷ 1,000 = 0.206. (c) 190 ÷ 206 ≈ 0.922, about 92%. (d) Of 1,000 messages, 10 are spam and 990 are legitimate. Flagged spam: 10 × 0.95 = 9.5. Flagged legitimate: 990 × 0.02 = 19.8. P(spam | flagged) = 9.5 ÷ 29.3 ≈ 0.324, about 32%. When spam is rare, false alarms from the huge group of legitimate messages outnumber the true catches.
Check on (c) with Bayes’ rule: (0.95 × 0.20) ÷ (0.95 × 0.20 + 0.02 × 0.80) = 0.19 ÷ 0.206 ≈ 0.922. ✓ Common mistake: thinking P(spam | flagged) equals P(flagged | spam), or 95%. Those are different questions with different denominators. The idea is called the base rate: how common something is to begin with. In (d), fractions of a message are fine because these are expected counts. Extension: suppose you could change only the false-alarm rate. What rate would you need in (d) to get P(spam | flagged) above 90%? (About 0.1%: 9.5 ÷ (9.5 + 0.99) ≈ 0.906.)
Science
Mariner 10 launched on November 3, 1973, and became the first mission to use a gravity assist, using a planet’s gravity to change a spacecraft’s speed and path. It flew past Venus on February 5, 1974, then reached Mercury on March 29, 1974, becoming the first spacecraft to visit that planet. NASA says a direct trip to Mercury would have needed extra fuel and a large, expensive rocket. Mercury orbits about 0.4 AU from the Sun, where 1 AU is the Earth-Sun distance. (a) How many days did the trip take from launch to Venus? From Venus to Mercury? (b) Light intensity follows an inverse-square law: I is proportional to 1/d2. About how many times stronger is sunlight at 0.4 AU than at 1 AU? (c) NASA says sunlight from Mercury’s surface can be “as much as seven times brighter” than from Earth. Why might that differ from your answer? (d) To reach Mercury, did the spacecraft need to gain or lose orbital energy relative to the Sun? Explain how a flyby of Venus could help without burning much fuel.
Answer and teacher notes
(a) November 3 to February 5 = 94 days. February 5 to March 29 = 52 days (23 days left in February 1974 + 29 in March). (b) 1 ÷ 0.42 = 1 ÷ 0.16 = 6.25 times stronger. (c) 0.4 AU is a rounded average, and Mercury’s distance from the Sun changes as it orbits. A slightly smaller distance gives a larger value: at 0.387 AU, 1 ÷ 0.3872 ≈ 6.7. (d) It needed to lose orbital energy so it could fall inward toward the Sun. As the spacecraft passes Venus, gravity pulls on both, and the much larger planet barely changes. The spacecraft leaves on a new path with less speed relative to the Sun, so it needs little or no extra fuel.
Common mistake on (d): assuming a trip toward the Sun needs more speed. Falling toward the Sun takes less orbital energy, not more, so the spacecraft has to slow relative to the Sun. Students can compare a gravity assist to a tennis ball bouncing off a moving racket: the ball’s speed relative to the court changes even though the racket barely does. A flyby can add speed or remove it, depending on the path. Extension: Mariner 10 passed Mercury three times, coming within 327 km on the last pass. Why might engineers want several flybys?
Sources: NASA JPL: Mariner 10 (launch, first gravity assist, first Mercury visit); NASA: 45 Years Ago, Mariner 10 Off to Venus and Mercury (flyby dates and distances, why a gravity assist was used); NASA: Mercury Facts (0.4 AU, sunlight comparison)
Social Studies
On November 3, 1957, the Soviet Union launched Sputnik 2, weighing 1,120 pounds and carrying a dog named Laika. Sputnik 1, launched on October 4, had weighed 183 pounds. Americans had already feared that Soviet rockets could carry nuclear weapons. On December 6, a U.S. Vanguard test booster rose about three feet and burst into flames on the pad. Britannica reports that Soviet accounts said Laika lived in orbit for six or seven days, but a Russian scientist revealed in 2002 that she had died within hours from overheating and panic. (a) How many days apart were the two Sputnik launches? About how many times heavier was Sputnik 2? (b) Why might that weight have worried Americans more than the satellite itself? (c) Why might a second launch so soon have changed how people viewed the first one? (d) What might explain the gap between the Soviet account and the 2002 account? What does it teach about using government sources from the Cold War?
Answer and teacher notes
(a) 30 days apart. 1,120 ÷ 183 ≈ 6 times heavier. (b) A rocket that can lift a heavy payload into orbit might also carry a heavy weapon a long distance. (c) Answers will vary: one launch might have seemed like a single surprise, while two in a month suggested a steady, growing lead. (d) Answers will vary: governments can control information during a rivalry, and a success story helps a country’s image. Sources created in that setting need checking against records released later.
Some students may find Laika’s story upsetting, so you may wish to give a short warning first. Britannica says pressure to meet a launch deadline left too little time to perfect life support. After these events, Explorer 1 reached orbit on January 31, 1958, and NASA was created on October 1, 1958. Note that the two NASA pages listed below give the masses in different units: 1,120 pounds and 508.3 kg agree. Extension: what kinds of evidence might historians use to check an official Cold War account?
Sources: NASA: Sputnik and the Origins of the Space Age (launch dates, weights, Vanguard test); NASA: Sputnik (fears about rockets, Explorer 1, creation of NASA); NASA: Sputnik and the Dawn of the Space Age (508.3 kg); Britannica: Laika (Soviet account, 2002 report, launch deadline)
Current Events
NASA Earth Observatory recently published Landsat 9 images of Vega Island, east of the Antarctic Peninsula, where rocky and icy features look like a grinning skull seen from the side. A rocky peak forms an eye socket, and a glacier forms the lower jaw. One image is from March 22, 2025 and the other from October 10, 2026. The report says unusually warm winter conditions in 2026 exposed large areas of darker glacier ice. Glaciologist Mauri Pelto notes that exposed ice absorbs more solar energy, which could speed melting. Albedo is the fraction of sunlight a surface reflects. (a) What season is each image from in Antarctica? Why does that matter when comparing them? (b) Suppose snow has an albedo of 0.8, ice has an albedo of 0.4, and 200 watts of sunlight reach each square meter. How many watts does each surface absorb per square meter? (c) Explain why exposing ice could lead to more melting in a repeating cycle. (d) What data would you want before saying one warm winter shows a long-term trend?
Answer and teacher notes
(a) March is autumn and October is spring in the Southern Hemisphere. Seasons change snow cover, so differences between images might come from the time of year as well as from unusual weather. (b) Snow absorbs (1 − 0.8) × 200 = 40 W/m². Ice absorbs (1 − 0.4) × 200 = 120 W/m², or three times as much. (c) Darker ice absorbs more energy, which can melt more surface ice or remove more snow, which exposes more dark ice. This is a positive feedback. (d) Answers will vary: many years of images, temperature records, snowfall measurements, and glacier mass measurements.
The albedo values in (b) are assumptions for practice. Real albedo depends on the condition of the surface, so don’t treat the numbers as measurements. The report notes that monitoring of Devil’s Bay Glacier began in 1998–1999 and shows an overall reduction in mass. Common mistake on (c): thinking “positive” means “good.” In science, it means the change reinforces itself. The skull is also a good example of how people find faces in patterns. Extension: why would a satellite be better for this work than a camera on the ground?
Sources: ScienceDaily: NASA Reveals a Haunting Skull Hidden in Antarctica’s Ice (NASA Earth Observatory report, image dates, features, monitoring); National Snow and Ice Data Center: Albedo (definition)
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