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Modelling Summative Project: Instructions & Rubric

Overview

Using our understanding of sinusoidal and exponential functions, we will analyze data sets to create models for the data.

Steps in the Project:

Project:

● PART 1: Find data that follows a sinusoidal and an exponential trend

○ Sinusoidal - Think about a sinusoidal that does not involve temperature (tide

levels, number of daylight hours in a particular location, height of rider on Ferris wheel, sunset times, mm of rain, population of an animal over a year, seasonal spending/sales, etc.), for which you can find real data for at least 3 cycles. You may obtain data by looking at online sources, making sure to reference them by their URL.

○ Exponential - Think about a situation where a quantity grows (or decays) over

time. For example, population growth, economic growth, spreading/dying out of a virus, financial investments, radioactive decay, price of something increasing/decreasing, etc. You may obtain data by looking at online sources, making sure to reference them by their URL. Make sure you have at least 6 data points.

○ Organize your data in a spreadsheet or table

○ For each, define the independent and dependent variables.

■ Let x represent (___) in (units).

■ Let y represent (___) in (units).

■ Be specific. For example, if a variable represents time, indicate what that number means. Does 1 mean 1 hour from a certain event, 1 o’clock in the morning, etc.? If a variable is height, what is it relative to?

○ Note assumptions you have made about the data, if any

● PART 2: Model and Graph your data

○ Graph your Sinusoidal and Exponential data in Desmos

○ Sinusoidal - Find an appropriate maximum, minimum, period and phase shift. Use these values to calculate the parameters for your equation algebraically.

Create at least two models for your data by making different assumptions when finding your maximum/minimum/period/phase shift. Show how you created your models and explain what assumptions you made.

○ Exponential - State an appropriate horizontal asymptote. Use this to calculate the parameters for your equation algebraically. Try to use as much of your data as possible in your calculations by averaging calculated parameters.

Create at least two models for your data by making different assumptions when finding your horizontal asymptote/other parameters. Show how you created your models and explain what assumptions you made.

○ Graph your data and both models in Desmos for your Sinusoidal and Exponential functions

○ Model by inspection by using Sliders in Desmos. Write down the models you created by adjusting sliders visually.

● PART 3: Analyze Models

○ From your work in Part 2, you will have at least 3 models for each data set

○ Comment on how well each model fits the data. Choose one model you think fits the data best and justify why. Use this model for the remaining parts.

○ For each model, show an example of how you can solve algebraically for the dependent variable (y), given the independent variable (x). Interpret your answer.

○ For each model, show an example of how you can solve algebraically for the independent variable (x), given the dependent variable (y). Interpret your answer.

In Class Assessment:

● PART 4: Follow up questions (completed in class after Parts 1-3 are done)

○ In class on May 20th/21st

Optional Extension:

● PART 5: Extension opportunities (completed prior to in class assessment)

○ Due May 20th/21st (with parts 1-3)

Rubric

Learning Goals		Success Criteria
FAC[K] Facts & Knowledge Identify, Use and Describe key facts, properties, and concepts, to develop a mathematical understanding of functions	Part 2	State and describe the key features of a given function. 口 Explain and demonstrate mathematical ideas, relationships, and terminology. 口 Identify function characteristics (amplitude, period, maximum, minimum, sinusoidal axis, horizontal asymptote, growth/decay rate, domain, range, etc.)
ALG[K]: Algebra Apply algebraic rules and procedures, to simplify expressions and solve equations.	Part 2/3	口 Rearrange algebraic expressions to equivalent forms using appropriate techniques for the type of expression. 口 Evaluate expressions algebraically. 口 Solve equations and inequalities algebraically.
RJ[T]: Reasoning & Justification Reason mathematically, to connect and justify facts, algebra, representations, and problem solving strategies.	Part 3/4	口 Justify mathematical concepts, strategies and processes.
REP[A]: Representations Represent functions in a variety of forms by connecting numbers, algebra, graphs, and words.	Part 1/2/4	口 Draw the graph of the various types of functions. 口 Represent a function that is in one form, using another form (including numbers, equations, graphs, and words). 口 Use graphs or diagrams to represent ideas (values, solutions).
COM[C]: Communication Communicate mathematical concepts and meaning in explanations & interpretations, form & presentation, and graphing & diagrams.	Part 1/2/3/4	Explain and interpret using clear and precise language. 口 Express ideas using mathematical conventions. 口 Present processes clearly and logically. 口 Present graphs and diagrams clearly and with necessary labels.

The learning goals were demonstrated in what instances and with what level of consistency?

Level 1: Limited

Level 2: Some

Level 3: Many/Most

Level 4:

Almost All/Always

Communication Criteria

Explanations & Interpretations (CE)

Explanations and interpretations are clear and concise: Proper vocabulary use, Relevant examples, Natural flow of thought, Efficient explanation (concise, not redundant), Appropriate final statements, Units wherever appropriate

Form & Presentation (CF)

Algebra steps and symbols are used properly, following conventions; Work is presented clearly and appropriately: Clear steps provided to show the process, Clear and proper use of notation and mathematical symbols, Logical order to work presented, Appropriate

rounding

Graphing & Diagrams (CG)

Graphical work is presented clearly and with necessary labels, markings, conventions: Use of ruler where appropriate, Labels where appropriate, Appropriate scale on axes, Arrows to indicate trends, Effective use of grid space

PART 4:

After, during a class period, you will complete some questions related to your project (so you need to make sure Parts 1-3 are done!), then hand it all in.

The questions will use your work from Parts 1-3 and ask conceptual questions about what you did in your project. You will have access to your project work but it will be closed book otherwise.

PART 5: Extension (Optional)

You also have the option to do some extension problems if you are interested.

These problems are designed to be open ended. I can clarify the prompt, or give you a hint about where to get started, but I will not tell you exactly what I am looking for or how to do it. I want you to interpret these yourself and do what you think is right!

a. Use a combination of functions (e.g. sinusoidal and exponential, polynomial and sinusoidal, etc.) to model one of your data sets or another data set of your choosing. Describe your process for modeling the data. How does the fit of your more complex model compare to the simpler model in your project?

b. Create a spreadsheet to automate the calculations for your procedure as you adjust your assumptions or data. (For an example, see the spreadsheet in the 5.4 CYU #3). Make sure to explain/justify how you created your spreadsheet.

c. Calculate average and instantaneous rates of change using the models from your project at various points in the domain. Interpret the meaning of these calculations and describe how they might be useful for someone using your model.

If you get help with your spreadsheet (Stack Overflow, ChatGPT, a friend, a parent, etc.), please make sure to include who helped you in the project document.