concept category 1 analyzing mathematical models i can graph and use key features to analyze...

Concept Category 1 Analyzing Mathematical

ModelsI can graph and use key features to analyze functions expressed symbolically.

Given a graph, table or function expressed symbolically, I can produce an invertible function from a non-invertible function by restricting the domain and verify by

composition that one function is the inverse of another.

Which one is your favorite and why?

Who do we need to assemble?

What do we know?

Task

Approach Independently

Create 2 Plans that might work by communicating with partner

Approach

Create Plans

The function describes the

concentration of a drug in the blood stream over time. In this case, the medication was taken orally. C is measured in micrograms per milliliter and t is measured in minutes. Sketch the graph of C(t) analyze and interpret the graph in the context of this problem.

C(t) 5t

0.01t 2 3.3

I. Rational Function Model

A. Definition:A rational function is the quotient of two polynomial functions

and , where is nonzero.

We can use limits to analyze rational functions for continuity and asymptotes.

Limit: The concept of approaching an x- value in the domain, from both

sides, to see if we approach a y-value

f (x) a(x)b(x)

b(x) 0

b(x)

f (x)

a(x)

b(x)

B. Visual Horizontal AsymptoteEnd behavior can be

described with a limit? How?

Vertical AsymptoteAs x-> a+, y-> -infinityAs x-> a-, y-> infinity

x c

y c

Is infinity a value?

a

Evaluate the limit if it exists.

Example•

What’s happening in our graph if the limit does not exist?

Evaluate the limit if it exists

C. ProcessAnalyzing Behavior

•Domain•Vertical Asymptotes•Points of Discontinuity

f (x) x 2 9

x 2 x 6

C. ProcessAnalyzing Behavior

•Range•Horizontal Asymptotes•Points of Discontinuity

f (x) x 2 9

x 2 x 6

C. ProcessAnalyzing Behavior

• Intercepts• x-intercept• y-intercept

f (x) x 2 9

x 2 x 6

C. ProcessAnalyzing Behavior

•Domain•Vertical Asymptotes•Points of Discontinuity

•Range•Horizontal Asymptotes•Points of Discontinuity

• Intercepts• x-intercept• y-intercept

f (x) x 2 9

x 2 x 6

Now we can sketch our graph

C. ProcessAnalyzing Behavior

•Domain•Vertical Asymptotes•Points of Discontinuity

•Range•Horizontal Asymptotes•Points of Discontinuity

• Intercepts• x-intercept• y-intercept

D. Purpose of Rational Function Models•Some examples of real-world scenarios are:oAverage speed over a distance (traffic engineers)oConcentration of a mixture (chemist)oAverage sales over time (sales manager)oTime to complete a job (engineer/planner)oAverage costs over time (CFO’s)

Practice ChoicesEvaluate the limit if it exists

Active Practice

What can the limit of a function tell you about the behavior of the graph?

What key features are relevant to the function being analyzed?

How do the key features of a rational function connect to real-world applications?

Practice Choices

A ball is tossed upward from a height of 200 feet with an initial velocity of36 ft/sec. If the height of the ball infeet after t seconds is given by s(t) = −16t2 + 36t + 200, find the instantaneous velocity of the ball at t = 2.

Find the derivative of the function

f (x) = x2 − 3x + 5 at x = a.

Active Practice

Why can’t we just plug in x=0 to find the limit?

What is a derivative?

What do we know about an instantaneous velocity?

Practice Choices #1

Practice Choices #2

Practice Choices #3

Task

Approach Independently

Create 2 Plans that might work by communicating with partner

ApproachCreate Plans

The function describes the

concentration of a drug in the blood stream over time. In this case, the medication was taken orally. C is measured in micrograms per milliliter and t is measured in minutes. Sketch the graph of C(t) analyze and interpret the graph in the context of this problem.

C(t) 5t

0.01t 2 3.3

PPT- Brainstorm• Models

• Piecewise• Quadratic/Rational Models• Composition

Task

Approach Independently

Create a PlanIndependently

Approach

Create Plans

Task

Revise your Planscommunicating with partner

Create Plans

Procedural and Reasoning Question Prompts ChecklistProcedural:

• What do I know about ____?• What can we do first? • When do I use ____? • Do we see a pattern? • What does that mean?

Reasoning:• Why ____?• What if we changed ____?• How does this (piece/part/concept) relate/connect to ____?• What else does this remind me of? • How can we be sure the pattern continues? • How do we determine which method to choose? • This is the best/most reasonable solution because ____• This always/sometimes/never works because ____• Can you think of a counterexample?• Could you reach the same result using a different approach? • I am deciding ____ because of ____ and ____, and chose ____ because ____• Have we ever solved a problem like this before?

Adapted from The Art of Questioning in Mathematics, NCTM Professional Teaching Standards 8/20/15 31© Clemmer, K., Laskasky, K., & Mirzaian, T.

Task

How are you monitoring the effectiveness of your plan?Ex

ecut

e

What (exactly) are you doing? Can you describe _______ precisely? Why are we doing _______? How does _______ fit into the solution? How does _______ help us with _______? What will we do with _______ when we obtain it?

Schoenfeld, A., 2005

8/20/15 33© Clemmer, K., Laskasky, K., & Mirzaian, T.

Math Writing ChecklistState givensState what is asked for (goal/want/claim)State any assumptions you are making: “We assume that ___.”Define each variable: “We will let x denote ___, we will let the

quantity (whatever) be denoted by ___, or we will represent the (whatever it is…height, distance, time, etc.) by x.”

Give reasons to logically support each step in explanation: “It follows that ___; we see that ___; therefore ___; so/since/because; from ___ we get ___; in comparing ___ and ___.”

Clearly state final answer or conclusion

8/20/15 © Clemmer, K., Laskasky, K., & Mirzaian, T. 34

Modified from Jacqueline Dewar, Professor emerita of mathematics, Loyola Marymount University

Concept Category 1: LT 3Analyzing Mathematical

ModelsI can determine and create the most appropriate

mathematical model to describe relationships between two quantities and use the model to analyze the

relationships between the two quantities. I can explain why the model is an accurate representation of the rate

of change in the data over time.

What makes Tony Stark a superhero?

Task: Write a single function for the height of a rocket with respect to time shown below.A rocket is launched vertically at a constant thrust of 4 meters per sec. After a 2 sec burn, the rocket shuts off and glides in a parabolic arc till it lands on a platform, 16 secs later. The 8 m landing platform is put into position 7 secs after the launch. The rocket reaches a max height of 16 m.

Did you …• Annotate the problem?• Identify givens?• Create an alternative representation? (Diagram?)• Restate the goal?

• Identify models and relationships and/or concepts that might apply?

Task: Write a single function for the height of a rocket with respect to time shown below.

Individually: Create an Approach

Did you …• Annotate the problem?• Identify givens?• Create an alternative representation? (Diagram?)• Restate the goal?

• Identify models and relationships and/or concepts that might apply?

III. Using Models as ToolsA. Definition

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling.

B1. Visual for Projectile Motion

Credit: http://mathspig.wordpress.com/2014/08/19/3-stuntman-maths-motorbike-jump/

Quadratic

B2. Visual for Box, Garden & Fence

Quadratic/Polynomial

C. ProcessA rectangular field is to be fenced off along the bank of a river and no fence is required along the river. The material for the fence costs $8 per foot for the two ends and $12 per foot for the side parallel to the river. Your budget for the fence is $3600. Find the dimensions and area of the largest possible field that can be enclosed.

C. ProcessA tree stands on a hillside of slope 28° from the

horizontal. From a point 75 feet down the hill (from person’s feet) the angle of elevation to the top of the tree is 45°. Find the tree’s height.*

D. Application• Science

• Biology – growth rates, life spans, environmental equilibriums• Physics - (Projectile motion)

Active Practice: Choose One Goal Problem

Sketch the piecewise Graph

A 200-foot tall monument is located in the distance.From a window in a building, a person determines that the angle of elevation to the top of the monument is 15°, and that the angle of depression to the bottom of the tower is 2°. How far is the person from the monument?

Active Practice

The person is 660.35 ft from the monument.

Practice to Choose from.• P. 77 #37-90

• P. 243 #21-27 odd

• P. 458 #49

Goal Problem: Making ChoicesIf you are ready to move on:• What key concepts led to your

success?• (Post-it)

Write your evidence in knowing you are ready to move on.

• What should you practice now?• How much time should you spend

in the next subject matter?

If you need more practice:• Did you find what is missing in your

process or understand?• (Post-it)

Write all questions you still need answered.

• How much time will you spend on this subject matter before moving on?

Practice ChoicesThe height of a cylinder is four times its radius. Find a function that models the volume V of the cylinder in terms of its radius.

From a point on the ground 500 ft from the base of a building, an observer finds that the angle of elevation to the top of the building is 24º and that the angle of elevation to the top of the flagpole atop the building is 27º. Find the height of the building and the length of the flagpole.

A Ferris wheel has a radius of 10 m, and the bottom of the wheel passes 1 m above the ground. If the Ferris wheel makes one complete revolution every 20 s, find an equation that gives the height above the ground of a person on the Ferris wheel as a function of time.

Practice Choices

𝑉=4𝜋𝑟 3 222.6ft, 32 ft

𝑦=11+10 sin ( 𝜋 𝑡10 )