Sinusoidal Modeling Project – Daylight and Temperatures for Cities around the World

Students and their partners will choose cities to compare the periodic behavior of their daylight and temperatures. While hours of daylight is perfectly periodic and predictable each year, temperatures are more erratic and curves will be based on averages. Students will identify the unique geographical characteristics of their cities that determine their hours of daylight and influence their temperatures. Students will then observe how these differences are revealed in the sinusoidal equations for their data.

Subject/Course:

Calculus

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Significant Content

The circular functions sin x , cos x and tan x : their domains and ranges; amplitude, their periodic nature; and their graphs. Composite functions of the form f (x) = asin (b(x + c) ) + d . Transformations

21st Century Competencies

Collaboration:

must work with partner on technical material

Communication:

Written report and oral presentation of findings.

Critical thinking:

Critical Thinking

Creativity and innovation:

choice of cities and presentation medium

Driving Question

How are differences in cities’ daylight hours and temperatures reflected in the sinusoidal equations used to model these characteristics?

Entry Event

Class discussion of where they have visited around the globe – what was the weather like? Have you been very far north or very far south in summer or winter? When did the sun rise, set?

Products

Individual:

Team:

Poster, powerpoint, video detailing the outcome of their research on their cities.

Specific content and copetencies to be assessed:

Specific content and competencies to be assessed:

Public Audience

Students might seek out local weathermen to discuss the influences on the climates of their chosen cities.

Resources Needed

On-site people facilities:

On-site people, facilities:

Equipment:

Internet access to obtain data on cities.

Materials:

Community resources:

Community Resources:

Learning outcomes targets:

I can write circular (trig) functions for sine and cosine.

I can analyze trig functions and identify their characteristics such as domain, range, amplitude, and period.

I can compare trig functions and understand the effects of transformations in equations and on the function graphs.

I understand some basic applications of trig functions to real world data or situations.

I can evaluate trig functions for a given input or solve trig functions for a specific output.

Checkpoints formative assessments:

1. journal; progress logs 2. class discussions on problems encountered. 3. exit ticket – write your equations.

1. exit ticket – identify city with largest amplitude or range in daylight, temperatures 2. draft of report

1. class discussion – how are the equations for cities nearer the poles be different from cities nearer the equator? 2. draft of report.

1. journal – why would daylight and temperature be considered periodic? 2. draft of report

1. Quick quiz on substitution and solving sample problems. 2. journal, progress logs

Instructional strategies for all learners:

Direct instruction – lesson on writing and graphing sine and cosine functions.

Modeling – identify two cities and compare their daylight and temperatures – graphed first and then in equations.

Outside speaker – the effects of geography on weather. Which characteristics produce extreme weather?

Class discussion – brainstorming for other behaviors that would be periodic (tides, amount of air in lungs, etc. for example). Have examples of trig functions that model these behaviors?

Direct instruction - In class examples of how to evaluate or solve trig functions.

Journal learning log:

describe progress made

Whole-class discussion:

what problems are being encountered?

Survey:

Survey

Focus group:

Focus Group

Fishbowl Discussion:

Fishbowl Discussion

Other:

Existing group(s)

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