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
Use who (or whoever) when the word acts as a subject, and whom (or whomever) when it acts as an object. Choose the correct word in each sentence and name its job. Sentence 5 trips up many adults: explain your choice.
- (Who / Whom) left the lights on in the lab?
- To (who / whom) should I address the letter?
- The artist (who / whom) painted the mural will visit on Friday.
- She is the friend (who / whom) I trust most.
- Give the prize to (whoever / whomever) finishes first.
Answer and teacher notes
1. Who (subject of left). 2. whom (object of the preposition to). 3. who (subject of painted). 4. whom (object of trust: I trust her). 5. whoever (subject of finishes).
The he/him test: answer or rearrange the clause with he or him. If he fits, use who; if him fits, use whom (both end in m). In sentence 5, the preposition to tempts students toward whomever, but the object of to is the whole clause whoever finishes first. Inside that clause, the word is the subject of finishes (he finishes first). Discussion: whom is fading in casual speech. When might a writer still choose it? (Formal letters, academic writing, after a preposition.)
Math
Most of the world uses the ISO “A” paper sizes. Every A sheet has a long side to short side ratio of √2 : 1, and the largest size, A0, has an area of exactly 1 square meter. Cutting a sheet in half across its long side gives the next size (A0 → A1 → A2 → A3 → A4). (a) Show that cutting a √2 : 1 rectangle in half produces another √2 : 1 rectangle. (b) Find the dimensions of A0 to the nearest millimeter. (c) What is the area of A4, in square millimeters? (d) To enlarge an A4 page onto A3 paper, a copier is set to 141%, not 200%. Explain why.
Answer and teacher notes
(a) Let the sides be L and W with L/W = √2. Halving the long side gives a rectangle with sides W and L/2. New ratio: W ÷ (L/2) = 2W/L = 2/√2 = √2. ✓
(b) W × √2W = 1, so W² = 1/√2 and W = 2−1/4 ≈ 0.8409 m. L = √2 × 0.8409 ≈ 1.1892 m. A0 ≈ 841 mm × 1,189 mm. Check: 0.8409 × 1.1892 ≈ 1.000 m². ✓
(c) A4 is A0 halved four times: 1/24 = 1/16 m² = 1,000,000 ÷ 16 = 62,500 mm². The official A4 size, 210 × 297 mm, gives 62,370 mm², close because the standard rounds to whole millimeters.
(d) A3 has twice the area of A4, but a copier’s percent scales lengths. To double the area, each length must grow by a factor of √2 ≈ 1.414, or 141%. A 200% setting would make the area 4 times larger. Going back down, 1/√2 ≈ 0.707, which is why copiers offer a 71% setting.
Timely tie-in: the first photocopy was made 88 years ago today. Extension: U.S. letter paper is 8.5 × 11 inches. Does it keep its shape when folded in half? (11 ÷ 8.5 ≈ 1.29, but 8.5 ÷ 5.5 ≈ 1.55, so no.)
Source: Markus Kuhn, University of Cambridge: International standard paper sizes
Science
On October 22, 1797, André-Jacques Garnerin cut himself loose from a balloon above Paris and rode a parachute to the ground, the first parachute jump from a balloon. A falling object reaches terminal velocity when air drag equals its weight: V = √(2W / (Cd · ρ · A)). Assume a jumper plus gear of 80 kg (g = 9.8 m/s²), air density ρ = 1.2 kg/m³, a round canopy 7.0 m across, and a drag coefficient Cd = 1.5. (a) Find the terminal velocity in m/s and mph (1 m/s ≈ 2.237 mph). (b) What canopy diameter would bring the landing speed down to 3.5 m/s? (c) Witnesses said Garnerin’s parachute swung violently on the way down. Suggest a cause and a design fix.
Answer and teacher notes
(a) W = 80 × 9.8 = 784 N. A = π(3.5)² ≈ 38.48 m². V = √(1,568 ÷ (1.5 × 1.2 × 38.48)) = √(1,568 ÷ 69.27) = √22.64 ≈ 4.8 m/s, about 10.6 mph.
(b) V is proportional to 1/√A, and A is proportional to D², so V is proportional to 1/D. D = 7.0 × (4.76 ÷ 3.5) ≈ 9.5 m. Check: A = π(4.76)² ≈ 71.1 m²; √(1,568 ÷ (1.8 × 71.1)) ≈ √12.25 = 3.5 m/s. ✓
(c) Answers will vary; reward physical reasoning. A strong answer: trapped air spills out unevenly from one edge of the canopy and then the other, rocking the jumper like a pendulum. A small vent at the top lets air escape steadily through the center, a feature many later round parachutes use. The mass, canopy size, and Cd here are assumptions for the problem, not Garnerin’s actual numbers. Sources even disagree on how high he jumped (figures from about 2,000 to 3,200 feet appear), a good reminder to check claims against more than one reference.
Sources: Air & Space/Smithsonian: The First Parachute Jump; Britannica: André-Jacques Garnerin; NASA Glenn Research Center: Terminal Velocity
Social Studies
On October 22, 1938, in Astoria, Queens, New York, Chester Carlson and engineer Otto Kornei made the first xerographic copy. It read simply “10-22-38 Astoria.” Carlson offered a share of his invention to many companies, and none was interested. In 1944, the Battelle Memorial Institute, a research organization in Ohio, agreed to help develop it. Later, Haloid, a small photo-paper company in Rochester, New York, that needed a new product after World War II, took the gamble. Its first commercial xerographic copier came out in 1950, and in 1961 the company renamed itself Xerox. (a) How many years passed from the first copy to the first commercial copier? (b) Give two reasons established companies might pass on a promising invention. (c) Why might a small company be more willing to take that risk?
Answer and teacher notes
(a) 1950 − 1938 = 12 years. From the first copy to the Xerox name: 1961 − 1938 = 23 years.
(b) Answers will vary; grade the reasoning. Strong ideas: early prototypes are crude and it is hard to picture the finished product; development is expensive with no guaranteed payoff; companies already earning money from existing methods may not want to compete with themselves; and timing mattered, since World War II pulled resources elsewhere. (c) A company that badly needs a new product has less to lose and more to gain, while a large, successful company may be more cautious. Push students to treat this as a pattern to test with other examples, not a rule. Extension: what does this story suggest about the difference between inventing something and innovating (getting it into wide use)?
Sources: The Henry Ford: Light and Static: The Origins of the Xerox Copy; EBSCO Research Starters: Carlson and Kornei Make the First Xerographic Photocopy
Current Events
An 1871 article in the journal Nature described a “very intense magnetic disturbance” on October 18, 1841, that disrupted railway telegraph signals and delayed the 10:05 p.m. train from Exeter, England, by 16 minutes. For years it was cited as the earliest known case of space weather affecting technology. A team led by Lancaster University reported this week in the journal Space Weather that the date can’t be right: the Exeter-to-Starcross rail line didn’t open until 1846. By comparing railway timetables, newspaper reports, aurora sightings, and old magnetic-field measurements, they concluded the event happened on October 18, 1848. (a) What time did the delayed train leave? (b) How does the 1846 opening date disprove 1841? (c) Why is checking several independent kinds of records stronger than relying on one source?
Answer and teacher notes
(a) 10:05 p.m. + 16 minutes = 10:21 p.m. (b) A train can’t be delayed on a line that doesn’t exist yet, so any date before 1846 is impossible. That is an anachronism, one of the strongest clues that a source contains an error.
(c) Each record type could be wrong on its own, but timetables, newspapers, aurora reports, and magnetic measurements were made by different people for different reasons. When they all point to the same date, a coincidence is unlikely. The redating means Exeter is no longer the earliest known case; that title now goes to telegraph interference on Britain’s Midland Railway in March 1847. Science link: a solar storm rapidly changes Earth’s magnetic field, and a changing magnetic field induces electric currents in long wires such as telegraph lines. Extension: power grids, satellites, and rail signals are still vulnerable today. Why might a 21st-century solar storm cause bigger problems than one in 1848?
Source: ScienceDaily: A Train Was Delayed by a Solar Storm. But Something Didn’t Add Up. (RMIT University; Space Weather, 2026)
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