Lesson 2 — Approximation and the Shape of Change

Summary

Learners turn the limit idea into a rate of change: the average slope over an interval (a secant) shrinks toward the slope at a single instant (a tangent) — the informal derivative. They connect it to speed at a moment, a plant’s growth, and a population’s changing rate, and to the long human habit of approximating quantities ever more finely, from Babylonian root tables to the calculus written down in Europe.

Objectives

• Apply the idea of a limit and of continuity to describe change and approximation, and use average rates of change to approach an instantaneous rate. (D05.S1.12.01)

Connection

A speedometer reads “60” at a single moment, yet any measured speed is distance over time — an average. The number on the dial is the limit of those averages as the time interval shrinks toward zero. The same move — shrink the interval, watch what the slope approaches — describes a plant’s fastest growth week, a river’s peak flow, or a fever’s peak. Approximating ever more finely is one of the oldest and most universal human moves in mathematics.

Materials

• Rate-of-change diagram sheet
• Math journal
• Low-tech: curved track or rolling object, string and ruler

Preparation

• Copy or draw the curve with secant and tangent; prepare the shrinking-interval table.
• Retrieval: from Lesson 1, “closer and closer” (the limit idea); from Grade 11, slope as rise over run. Today we shrink the run to zero and watch the slope settle.
• Prepare the worked example (a distance-time curve) to model first (S-011).

Facilitator note

This lesson is written to the learner (“you”). The idea to land: an average rate of change (secant slope) becomes an instantaneous rate (tangent slope) in the limit as the interval shrinks to a point — this is the informal derivative, and it needs the curve to be continuous and smooth. Teach the worked example explicitly — build the table of shrinking intervals, then let the number settle (S-011). The limit is what makes “at an instant” meaningful rather than mystical.

The ethics lens: honesty about what a speedometer is — a limit, not a direct measurement — is honesty about what numbers can and cannot claim. The egalitarianism lens: the derivative needs no special talent, only the shared move of shrinking an interval and watching. The global lens: the habit of approximating a value ever more finely is ancient and worldwide — the Babylonian tablet YBC 7289 (c. 1800–1600 BCE) carries √2 correct to about six decimal places (S-430), and the calculus we inherit was written up in 17th-century Europe by Newton and Leibniz (S-302) on top of much older, multi-rooted work (S-272, S-429). The technology lens: the derivative is the working language of physics, engineering, and the simulation engines behind every screen. The environment lens: a population’s growth rate is a derivative — it rises, then slows toward zero as the population nears its carrying capacity (S-436). Preview: Lessons 3–4 turn to reasoning about the size of large systems.

Procedure

1. Recall (5 min). From Lesson 1, name what a limit is. From Grade 11, slope is rise ÷ run. Today the run shrinks toward zero.
2. Average speed hides the moment (8 min). A ball rolls 3 m in 3 s: average speed is 1 m/s — but it was faster at the start and slower at the end. An average rate of change over an interval is one number; it cannot see the instant. To see the instant, shrink the interval.
3. Worked example — shrinking the interval (15 min). A distance-time curve passes through (2, 4) and, later, (5, 13). The secant slope between them is (13 − 4)/(5 − 2) = 3. Now take points closer to (2, 4): at t = 3, 2.5, 2.1, 2.01, compute the slope each time. The slopes settle toward a single number — that number is the instantaneous rate, the slope of the tangent at (2, 4). Write: the limit of the secant slopes is the tangent slope.
4. Name it (6 min). The instantaneous rate is the derivative (informally): how fast a quantity changes at a moment. A smooth curve has one; a corner does not (there is no single tangent).
5. Guided practice (12 min). With a partner, take a curve and two points; compute the secant slope; move the second point closer; predict the limit; compare with the tangent drawn on the sheet. Check each other’s arithmetic.
6. Retrieve and connect (8 min). In your journal, write one real quantity whose rate of change matters (a river’s flow, a plant’s height, a battery draining) and one sentence on why “average over a long stretch” is not enough.
7. Close (2 min). Say — or write, sign, gesture, or use AAC to express — what the derivative approximates and what it is the limit of.

Differentiation

• Support: Compute only the secant slope for one wide interval, then one narrow one, and say which is closer to “the moment.”
• Extension: Explain why a graph with a sharp corner has left-hand and right-hand secant slopes that settle on different numbers, so no single tangent exists.
• Number access (dyscalculia): Pre-fill the step-3 shrinking-interval table (t = 3, 2.5, 2.1, 2.01) with the slope column left blank, so the learner copies the worked pattern rather than re-deriving each rise and run; offload the rise-over-run division to a partner or calculator; and take the verbal route — say where the ball is fastest and slowest, then describe in words or a sketch how the average slope “settles” on the slope at a single instant, without computing any quotient.
• Communication access: Every spoken step — saying, naming, reading aloud, discussing, or closing — can be done in writing, sign, gesture, or AAC instead. Learners who are non-speaking, d/Deaf, hard-of-hearing, or who use AAC complete every task in their preferred mode; no step requires producing or hearing sound.

Assessment

• Formative (peer + self): Can the learner compute a secant slope, shrink the interval, and state (in speech, writing, sign, or AAC) what the slope is approaching (the tangent slope)?
• Portfolio artifact (unit): The shrinking-interval table with the limit named, added to the math journal.

Home connection

Watch something change at home — a kettle heating, a plant growing, a phone charging. Describe in one sentence where the rate is fastest and one where it is slowest, and what “at a moment” would mean for it.

Resources

• On worked examples and guided practice: Kirschner, Sweller & Clark (2006) (S-011).
• On the Babylonian √2 approximation (YBC 7289): S-430; on the calculus of Newton and Leibniz and its multi-rooted history: Newton, Principia (S-302); Boyer & Merzbach, A History of Mathematics (S-429); Joseph, The Crest of the Peacock (S-272).
• On a population’s growth rate slowing near carrying capacity: Britannica — Carrying capacity (S-436).