Bell Ringers for Grades 9–12: Tuesday, October 27, 2026

Sample revisions (others can work):
1. Because it was raining, the game was postponed. (“Due to the fact that” → “because”; “postponed” already means “moved to a later date.”)
2. I found the novel’s ending surprising. (“In my personal opinion, I think” says the same thing three times; “unexpected” and “surprising” overlap.)
3. Many students arrive before school starts. (“There are… who” is an empty opening; “early in the morning” is implied by “before school starts.”)
4. The outcome matched our prediction. (“Final,” “initial,” “at the beginning,” and “basically” add nothing.)
5. The club is planning a fundraiser. (“At this point in time” → “now,” or nothing; “in the process of planning” → “planning.”)

Have students count words before and after; item 4 drops from 17 words to 5. Point out that concise doesn’t mean choppy; the goal is to cut words that carry no meaning, not to strip out useful detail. Extension: take one paragraph from a recent essay and cut it by 20% without losing a single idea.

(a) Order doesn’t matter (pepperoni-mushroom-onion is the same pizza in any order), so use combinations: C(10, 3) = 10! ÷ (3! × 7!) = (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120 pizzas.
(b) Order matters (gold ≠ silver), so use permutations: P(8, 3) = 8 × 7 × 6 = 336 ways.
(c) Order matters and repeats are allowed: 40 × 40 × 40 = 64,000. (There are 40 numbers from 0 to 39, not 39.)
(d) Each topping is either on or off: 2¹⁰ = 1,024 pizzas.

Check the link between (a) and (b): dividing permutations by 3! = 6 removes the order. For the pizzas, P(10, 3) = 720, and 720 ÷ 6 = 120. ✓ For the runners, 336 ÷ 6 = 56 would count groups of three finishers with no medals assigned. Fun fact for discussion: a “combination lock” is badly named, since order matters on the dial. Mathematically it’s closer to a permutation lock. Common mistake in (c): counting 39 choices instead of 40.

(a) The pH values differ by 7.0 − 2.2 = 4.8, so the ratio is 10^4.8 ≈ 63,000 times. (b) [H⁺] = 10^−2.9 ≈ 1.3 × 10⁻³ mol/L (0.0013 M). (c) Seawater is more basic (higher pH). Blood has 10^{8.5 − 7.4} = 10^1.1 ≈ 13 times more H⁺ than seawater. (d) pH 3. Diluting 10-fold divides [H⁺] by 10 (from 10⁻² to 10⁻³ M), and −log(10⁻³) = 3.

The big idea: each step of 1 on the pH scale is a factor of 10, so a difference that looks small (2.2 vs. 7.0) is huge in concentration. Common mistake: saying lemon juice is “4.8 times” or “about 3 times” as acidic. Note that the dilution rule in (d) is a good approximation for strong acids, and it breaks down near pH 7, where the water’s own H⁺ matters. Diluting an acid can never make it basic. Typical values vary with brand and conditions. Connection: the Richter scale and decibels are also logarithmic.

Source: LibreTexts Chemistry: The pH Scale (definition and typical values)

(a) 51 ÷ 85 = 0.60, about 60%. (Check: 51 + 29 + 5 = 85. ✓) Today is the 239th anniversary (2026 − 1787). (b) Answers will vary: readers would judge the arguments rather than the men; it presented three writers as one consistent voice; it followed a common custom of the time of writing under classical pen names (Publius Valerius Publicola was an early leader of the Roman Republic).

(c) Sample: “Can people sit down, think things through, and choose their own form of government, or is government always the result of luck, war, or conquest?” It mattered because the United States was testing the first idea: writing a constitution on purpose and asking citizens to approve it. (d) Newspapers reached many readers quickly and cheaply while the ratification debate was happening; essays could respond to opponents’ arguments as they appeared. (The essays were later collected into a book in 1788.) Note: the Constitution had opponents, often called Anti-Federalists, who also wrote newspaper essays. This prompt asks students to analyze how the Federalists made their case, not to pick a side.

Sources: Wikipedia: The Federalist Papers (date, newspaper, and authorship counts); Teaching American History: Federalist No. 1 (text)

(a) 20,264 ÷ 3,600 ≈ 5.63 hours; 21,204 ÷ 3,600 ≈ 5.89 hours. (b) 5.63 × 15° ≈ 84° and 5.89 × 15° ≈ 88°, so roughly 84° to 88° of longitude, nearly a quarter of the way around the globe. That’s why a record of where a total eclipse was seen lets scientists measure how much Earth’s rotation has changed. The eclipse was 2,734 years ago (709 + 2026 − 1, since there was no year 0).

(c) The Moon’s gravity raises tidal bulges in Earth’s oceans. Earth’s rotation drags the bulges slightly ahead of the Moon, and the friction of moving all that water acts like a gentle brake on Earth’s spin. (The same exchange slowly pushes the Moon farther away.) (d) The band where an eclipse is total can be fairly narrow, so a small location error can decide whether the eclipse was total at that spot, which shifts the range of ΔT values that fit the record. The study also suggests an ancient description of the eclipsed Sun may refer to the solar corona, but the authors call that interpretation uncertain. Discussion: what makes a 2,700-year-old written record trustworthy enough to use as scientific data?

Source: ScienceDaily: A 2,700-Year-Old Solar Eclipse Just Revealed Something Surprising About Earth (Nagoya University; The Astrophysical Journal Letters; October 26, 2026)