Principles of Calculus · SHS1 Additional…

Lesson notes

Learning Objectives

Indicator: 1.3.1.LI.5 - Use technology or any innovative ways to investigate the rate of change of a function, h(u), with respect to u.

By the end of the lesson, learners can:

Calculate the average rate of change of a function h(u) over an interval using graphical and algebraic methods.
Investigate how the rate of change of a function varies at different points on its graph using GeoGebra, graphing software, or graph sheets.
Estimate the instantaneous rate of change at a specific input value by examining the behaviour of average rates over shrinking intervals.
Interpret the significance of the sign (positive or negative) and magnitude of the rate of change at a point in practical terms.
Explain the link between the rate of change of a function and its derivative, building on the limit concept from the previous lesson.

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Curriculum details

Strand: Calculus (Strand 3)
Sub-strand: Principles of Calculus (3.1)
Content standard: 1.3.1.CS.1 - Demonstrate understanding of the limit of a function, investigate the behaviour of a function near a value in its domain and establish the derivative of a function. 1.3.1.LO.1 Describe graphically and algebraically the behaviour of the function about an input value and determine its derivative.
Indicator: 1.3.1.LI.5 - Use technology or any innovative ways to investigate the rate of change of a function, h (u), with respect to u.
Suggested placement: Semester 2, Week 10 (Week 30 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

exemplars - p.200: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 200

Exemplars (from the NaCCA curriculum)

Collaborative learning, Experiential Learning, and initiate Talk for Learning.
Learning Experience: Learners working in mixed-ability groups investigate how the limit of a function relates to its derivative.
Activity 1: Learners working in mixed-ability groups and deliberate on the link between the rate of change of a function and derivative. For example, in pairs, discuss the rate of change at a point for a given function ℎ(𝑢) and compare the rate of change at different points using a technological tool, software or any innovative way.
Example 1: On a graph sheet/technological tools/software or graphical calculator, draw the graph ℎ(𝑢) = 𝑢 ) − 6𝑢 + 5 use u values between 0 and 8 inclusive and determine the rate of change of the function at u= 0.5,3,4 and 2.5.
Example 2: Assuming you are walking along the curve below, then ask learners to discuss the rate of change at a point on the curve. What point will it be the same?

A coordinate graph with labelled axes, plotted curves or points, and the annotations visible in the crop.

Teaching and Learning Resources:
- GeoGebra
- PhET
- Technology tools
- Mathematical sets
- Calculators
- Learners textbooks
- Graph sheets
Assessment (1.3.1.AS.5). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.