##
* *Reflection: Real World Applications
Modeling with Functions - Section 4: Closure

This ended up being a really cool closing activity. I could not believe the different ideas that students were coming up with. You can see from the attached student work that the students were able to get very creative with their graphs. Unfortunately, we ran somewhat short on time so the descriptions left a little to be desired. That said, if a student was able to construct a graph with the axes labeled they had at least envisioned the situation in their head and thought about how the graph might appear.

This lesson in general was quite challenging for students. Most students were able to get the first two models in the investigation that were linear. Student's lack of experience with non-linear models held them back from even being able to envision what "losing half of its value each year" would look like graphically. I think it will be very interesting to revisit an assignment like this after students have worked with exponential and quadratic functions to see how their level of thinking changes.

*Real World Applications: Closure-Reflection*

# Modeling with Functions

Lesson 9 of 18

## Objective: SWBAT determine a function that would model a particular situation based on a verbal description or a graph.

*40 minutes*

#### Warm Up

*5 min*

**see modeling_w_functions.pptx slide #2**

Ask students to individually write down the domain and range for each function. Once students have completed the domain and range for each function they can check their answers with a partner to determine how they did and look for any discrepencies in the work.

**Teaching Point: **With functions that increase at a rapid rate (such as quadratic functions when a > 1), students may not realize that the domain is actually all real numbers. Make a point to show students how the graph continues to increase without bound.

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#### Opening

*10 min*

**See Opening.pdf or Slides 3 and 4 on the .pptx**

This opening activity allows students to investigate non-linear functions in context. The opening activity scaffolds the work to be done in the investigation, when students will not be given a graph. Depending on the amount of time allotted, students can either write down their description for each of these scenarios or discuss them with a partner. In either case, students should make note of important features of each graph. Both of these graphs require students to interpret a mathematical model (MP4) set in context.

Personally, I like to have students do a **think pair share **around each of these graphs where, rather than writing a story, they make a list of bullet points to discuss the key features. This tends to speed up the process and allow more time for students to discuss and critique each others interpretation of the graph.

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#### Investigation

*15 min*

Students will work on this investigation with partners. However, as a first step, I ask all the students to make the first sketch individually. Partner work begins with sharing these sketches. This helps students to build a shared understanding of the task and it helps ensures students encounter different observations than they would make on their own.

**Note:** The purpose of this investigation is not for students to have graphs that are an exact match for the equations given on the second page. You should be looking for two important points: (1) are the students sketching graphs that are increasing or decreasing correctly. (2) Are the students constructing graphs that are linear/non-linear correctly. For graph "B" some students will realize that the graph is not continuous (discrete) others will not. For graph "C", students will not draw a hyperbola but should draw a graph that appears more like an exponential.

Also in regards to the equation for graph C with is hyperbolic, while this function is outside the scope of the algebra course the function is simple enough that students can still evaluate it using their knowledge of functions.

#### Resources

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#### Closure

*10 min*

see Closure.pdf or modeling_w_functions_ideas.pdf slide 5

In this ticket out the door, students will make up their own scenario and sketch a function to go along with it. Students should pay particular attention to putting key features on the graph. It may be easier for students to include numerical values on both the x and y axes as a point of reference.

Some students may have difficulty getting started with this closing so I have included some ideas for scenarios that students can sketch.

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- LESSON 1: PRE-ALGEBRA: Evaluating Expressions
- LESSON 2: Defining Functions Recursively
- LESSON 3: Tower Task: Exploring Explicit Formulas
- LESSON 4: Function Notation
- LESSON 5: Understanding Domain and Range
- LESSON 6: Multiple Representation of Functions
- LESSON 7: Piecewise and Step Functions
- LESSON 8: Mirror Task: Understanding Equivalent Functions
- LESSON 9: Modeling with Functions
- LESSON 10: Functions Practice and Assessment
- LESSON 11: Introduction to Piecewise Functions: Dance-a-Thon Question
- LESSON 12: More with Piecewise Functions
- LESSON 13: Evaluating Functions Day 2
- LESSON 14: Transformation of Functions Day 1
- LESSON 15: Transformation of Functions Day 2
- LESSON 16: Transformations "How To" Guide
- LESSON 17: Functions Review Assignment
- LESSON 18: Functions Unit Assessment