Impact of Constraints on Inputs & Outputs 2D I can apply piecewise defined functions to a given set of data and explain why the model is appropriate to the context (include GIF) 2E I can explain how to model limiting behavior (asymptotic behavior, min/max) and explain restrictions (domain)

What question comes to mind?

What Mathematical Model predicts Attaining Buzz’s Dream?

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

B. Visual Horizontal Asymptote End behavior can be described with a limit? How? Horizontal Asymptote End behavior can be described with a limit? How? Vertical Asymptote As x-> a+, y-> -infinity As x-> a-, y-> infinity Vertical Asymptote As x-> a+, y-> -infinity As x-> a-, y-> infinity Is infinity a value? c

C. Process Analyze Behavior Find domain (identify values for possible V.A.) Find intercepts Evaluate symmetry (even/odd/neither) Investigate end behavior (horizontal asy.) Find local maximums and local minimums (using calculator) and identify max/min. values Identify increasing/decreasing intervals Sketch Find zeros of b(x) How will we know if the zeros of b(x) be are vertical asymptotes? Compare degree of a(x) and b(x) If n < m, horizontal asymptote at y = 0 If n = m, horizontal asymptote at (ratio of leading coefficients) If n = m, horizontal asymptote at (ratio of leading coefficients) If n > m there is no horizontal asymptote Find y-intercept by evaluating f(0) Find x-intercept by finding zeros of a(x) Simplify f(x) if possible f(x) = f(-x) f(-x) = -f(x) f(x) = f(-x) f(-x) = -f(x)

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

C. 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.

E: Connections between Fundamental Concepts Contrast rational functions to polynomial functions Write everything I know about polynomial functions

Goal Problems LT 2E Recall & Reproduction Finding asymptotes and Domain What questions do you have after working on the goal problems? Routine Sketching and Analyzing graphs Non-Routine Analyze behavior using limits

Sense-Making Active Practice: Learning Targets 2D & 2E What question(s) did you need answered to clear up confusion? What strength did you discover? Select one problem from targeted practice to complete independently

Active Practice Insights Teacher insights added to notes based on questions/clarifications posed by students At home: Re-do Goal problems  solve independently Based on evidence from home: Identify questions/strengths & practice choices for next active practice in class

FYI Targeted Practice Recall & Reproduction R & R: 2 problems Multiple Opportunity Guidance: 1 routine problem from each previous concept category Routine Routine: 2 problems Non-Routine Non-Routine: 1 problem Based on the questions that you have from the goal problems, choose: 5 problems from the current practice 1 problem from a previous concept category

Find the Mathematical Model

Goal Problems LT 2E Recall & Reproduction Sketching and Analyzing graphs What questions do you have after working on the goal problems? Routine Application Non-Routine Sdfa afd

FYI Targeted Practice Recall & Reproduction R & R: 2 problems Multiple Opportunity Guidance: 1 routine problem from each previous concept category Routine Routine: 2 problems Non-Routine Non-Routine: 1 problem Based on the questions that you have from the goal problems, choose: 5 problems from the current practice 1 problem from a previous concept category

Level Up Mixed Active Practice: Purpose: Clear up all areas of confusion and deepen areas of strength --Teach peers --Get questions answered by peers/teacher Goal: Write up a solution with a peer to non-routine problem and critique At home: Re-visit Non-Routine problem  solve independently

Quick Check: Concept Categories Student Scores Initial Evidence: Use Rubric & assess “right or wrong”: If recall is wrong, level 1, begin analysis If recall is right, routine partially right level 2 If recall is right, routine is right, non- routine is partially wrong, level 3 All correct, level 4 Another Opportunity: 1.Analyze notes to refine/verify solution pathways (2 nd color) 1.Given the key, analyze evidence and annotate on Q.C. (third color) Isolate, explain, and resolve areas of confusion (without copying how to do the problem) Identify & explain areas of strength Analyze reasoning (Approach for all, Plan for R and NR) Analyze written responses to routine and non-routine problems, are explanations sufficient and clear 3. Annotate notes based on evidence and “re-score”

Analyze Quick Check Results: Answer questions posed by teacher in green, using notes Select more practice from Learning Targets based on QC evidence Add to/revise concept map or annotate notes as you practice problems Clear up ALL areas of confusion by asking the teacher for support Take out your calendar & Learning Targets. 4 days from now write down the 4 problems you will complete to analyze retention. Place evidence in portfolio and hand in reflection on what is being retained, why, and next steps on day 5.

More Possible Mathematical Models 2F I can apply the concept of minimum and maximum to applied problems (box, garden, fence problems) that involve quadratic and cubic functions and use domain to determine if my solution is viable. 2G I can determine when equations, inequalities, systems of equations and/or inequalities (mixture/projectile) should represent a problem and its constraints. 2H I can create direct, inverse, and joint variation equations (physics) in two or more variables to represent relationships between quantities.

Observe the materials What properties do both have? Describe all of the information you know about each

PROBLEM Approach Independently Create 1 Plan independently that might work Compare plans by communicating with partner Draft a proposal Approach Create Plans How can you minimize the cost and maximize the height for building a structure using spaghetti and marshmallows that will support a tennis ball. Write a proposal that will convince the manufacturing company to accept your plan

II. More Mathematical Models A.Types of Models 1.Box, Garden & Fence Problems (LT 2F) 2.Mixture (LT 2G) 3.Projectile (LT 2G) 4.Direct, Indirect and Joint Variation (LT 2H)

B1. Visuals for Box, Garden & Fence

B2. Visual for Mixture Models Credit:

B3. Visual for Projectile Motion Credit:

B4. Visual for Variation

C1: Process for Box, Garden & Fence Prior Knowledge: Perimeter/Circumference and Area formulas How to find maximum/minimum of a polynomial function Most problems require you to use the givens to write a system of equations in order to find a polynomial that must be maximized/minimized in order to solve the problem.

C2: Process for Mixture Models Prior: You can determine how much “stuff” is in a container by multiplying its concentration (%) by the volume. To create an algebraic model, add the concentrations.

C3: Process for Projectile Motion The algebraic model for the height h(t) after t seconds of an object launched with initial velocity v 0 from an initial height h 0 is given by: h(t) = -gt 2 + v 0 t + h 0 Prior from Earth Science: Gravity = g = -9.8 m/s 2 = -32 ft/s 2.

C4: Process for Variation a.“x varies directly with y” becomes x = ky b.“x varies indirectly with y” becomes x = k/y c.“x and y vary jointly with z” becomes xy = kz d.Math is a matching game: match the sentence framework with the equation EXACTLY to obtain your algebraic model.

Goal Problems LT 2FGH Recall & Reproduction Write Algebraic models for the following: "F varies as x“ "F varies jointly as x and y“ "F varies as x + y“ "F varies inversely as x“ You Decide: Which 5 problems from the Learning Targets from this category do you choose? What 1 problem from previous concept category do you choose? Routine A park contains a flower bed 50 m long and 30 m wide with a path of uniform width around it having an area of 600m 2. Determine the width of the path. Non-Routine To contain radiation, a closed rectangular safe is to be made of lead of uniform thickness on the top, bottom and sides. The inside dimensions are 4ft x 4ft x 6ft and 450ft 3 of lead is to be used. Find the thickness of the sides.