Evaluating Logarithmic Expressions

Math Talk routine — reveal one problem at a time, collect strategies before moving on. Students work silently for ~30 seconds per problem, then share reasoning aloud.

The decimals (#3, #4) seed the idea that logs of numbers less than 1 are negative — the structural insight that makes Activity 1 (Compare Without Computing) possible.

Strategy to surface: watch for students who rewrite \(0.1 = 10^{-1}\) and \(0.001 = 10^{-3}\). That rewrite is the move that turns a hard mental computation into a one-step exponent read — name it explicitly in synthesis (slide 3).

No ACB on warm-up (discussion only). Answers and the sign rule are formalized on slide 3 synthesis.

End the warm-up with the sign rule explicit on the board: positive when the argument is greater than 1, zero when it equals 1, negative when it is between 0 and 1. Students will use this rule in the next ten minutes to compare log expressions without a calculator (slide 6).

The decimal items from the Do Now (\(\log 0.1\), \(\log 0.001\)) are the structural payoff — surface any student who rewrote \(0.1 = 10^{-1}\) or \(0.001 = 10^{-3}\). That move is the strategy.

If anyone says “log of 1 is zero in any base” — capture it on the board and reference it on slide 4 when the MASL row introduces \(\log_b 1 = 0\) as a callout.

No ACB on synthesis (discussion only).

Read the We Say card aloud, slowly, pointing at each symbol as you say it. Then trace the equivalence on the Math card: “log base b of x — the question — equals n — the answer — exactly when b raised to n gives back x.” The two callouts below the row are the sign-rule anchors from the warm-up: \(\log_b 1 = 0\) (any valid base) and logs of numbers between 0 and 1 are negative. Students will lean on both in the next ten minutes on slide 5.

Launch (~3 min). Open with the violet funds-of-knowledge prompt — 1–2 minutes of partner or whole-class talk about real comparisons made without measurement. Invite 1–2 student responses, then transition into the priming example.

If no one volunteers: share your own. “When I am in a grocery aisle and have to pick a checkout line, I do not count people — I look at carts. A full cart with one person is slower than three people each holding two items. That is a comparison without computing.”

Bridge to the math: Read the priming example aloud and trace the reasoning — sign of the log, integer bracket, and closeness within a bracket are the three moves students will use on the next slide. Reference the warm-up sign rule (\(\log_b x < 0\) when \(0 < x < 1\)) explicitly — it is the easiest of the three moves and the entry point for problem 1.

Optional · Log War extension. If your class has printed Log War decks, you can run the IM card-game version of Activity 1 here (20 min) — give pairs the first 20 cards, have them order least-to-greatest, then play War. Re-join the deck at slide 7. Otherwise proceed with slide 6 as the slide-based version.

Synthesis (3 min). Name the three reasoning moves explicitly — sign, integer bracket, closeness — and write them on the board in that order of refinement. Each move handles a harder case than the last: sign separates by category, bracket separates by integer range, closeness separates within a bracket.

The fourth click is the bridge to the next slide: students saw their estimate for \(\log 980\) come out “just under 3,” and the calculator confirms log(980) ≈ 2.991. Read the input aloud as “log open-paren nine-eighty close-paren” — students will need to hear this syntax again on slide 8 when they enter it themselves. The estimate is the check on the calculator, never the other way around. If a student says “why not just always use the calculator?” — that is the question slide 9 (Activity 2 Synthesis) answers explicitly.

Quick discussion (~3 min). The big takeaway: the estimate is the check on the calculator, not the other way around. Calculators round, drop digits, and accept whatever base-10 keystroke you give them — even when you meant a different base.

Q1 normalizes precision differences (Tyler vs. Priya). Many students assume one of two slightly-different calculator readings must be “wrong” — surface that both are correct, and that the bracket between 1 and 2 confirms both.

Q2 is the “is this reasonable?” check that the lesson summary depends on. The 2.675 answer is the classic base-mistake artifact: it is what you get when you compute \(\log_{10} 18\) on a calculator that has no base-2 button. Without the estimate, the student would accept it. With the estimate, they catch it instantly.

End with this rule on the board: “When the calculator answer and your estimate disagree by more than a tenth or so — believe the estimate first and re-enter the calculation.”

No ACB on synthesis (discussion only).

Lesson Synthesis (~3 min). Cold-call the question and let students reason out loud before clicking the answer. The point is that the two recall facts (\(\log_b 1 = 0\), \(\log_b b = 1\)) plus the sign check (is the input above, equal to, or below 1?) handle a surprising number of logs without any computation at all. Estimation by bracketing and the calculator's log button cover the rest.

Read the SWBAT block aloud after the discussion concludes — this is confirmation of what students just did today, not a forecast of what they will do. SWBAT belongs here, not on the title slide.

Lesson Summary slide. Optional read-aloud or self-read at the end of class. The two-step rhythm — estimate first, then verify — is the habit we want students to carry forward; the calculator is a confirmation tool, not a first move. Preview tomorrow in one sentence: “Tomorrow we will see what happens when the base is not 10 — the calculator will still help, but we will need log(x)/log(b), the change-of-base form.” Do not assess change of base today; just plant the seed.