##
* *Reflection: Intervention and Extension
Sorting out the Change - Section 1: Warm-up

Since many of my students had the same misconception on the isotope problem (see below for a snippet of yesterday's reflection on student work). I am going to do a **Favorite No** activity at the start of class today using this student’s answer: Counting the change, isotope question 2, student 4. I mainly want to demonstrate to students how this equation won’t work for t. It is in terms of half-lives. Then I want to open it up to the class to help me write an equation that relates t and d. Then I also want students to take time to solve this using a system. Approaching this problem using a system of equations brings up some opportunities for discussion of exponential function properties and the rules of exponents. Help students to see that the cube root of one half is the same as writing one half to the power of one third.

**Student Work on Isotope Problem:**

Overall, my students were not very successful on the isotope problem they turned in at the end of class yesterday. I had a few students do well, but a majority of my students struggled. Most of my students solved the first problem correctly using the patterns they identified, such as this student: Counting the Change, Isotope question 1, student 1. I had a few students attempt to use a system of equations. Some were successful with this method: Counting the change, isotope question 1, student 3. Some were not successful: Counting the change, isotope question 1, student 2.

Not one of my students got the correct answer for question 2, part b. As I thought may happen, I got the same incorrect answer repeatedly. This was definitely the most popular wrong answer: Counting the change, isotope question 2, student 4. Most of my students wrote the equation as a function of half-lives, not as a function of time (or didn’t even attempt part b at all!). I did have a few students try a system of equations approach, but were just unsuccessful: Counting the change, isotope question 2, student 5.

*Intervention and Extension: Warm-up Changed*

# Sorting out the Change

Lesson 7 of 14

## Objective: SWBAT analyze the rate of change to determine whether a relationship is linear or exponential and write the corresponding equation.

## Big Idea: Students identify the type of function (linear or exponential) which represents a situation & writes equations to model these situations.

*50 minutes*

#### Warm-up

*10 min*

Depending on where class left off and how students responded to the Thumbs Assessment yesterday, I will have students complete the *Isotope Problem *and/or review the various solution methods. I may have students present their approaches. Hopefully I can find an example of the three approaches that I am seeing (and maybe even ones that I’m not thinking about) and have those students just slip their work under the document camera with a quick explanation.

The three main approaches I will be looking for to be demonstrate (or I will demonstrate) are:

1) Finding from two points (like page 19 of flipchart)

2) Writing the exponential function in terms of number of half-lives and then substituting in the half lives as a function of time.

3) Identifying the rate of change in the table and using the initial value to write the equation.

#### Resources

*expand content*

#### Sorting Out the Change

*35 min*

The next task I want students to work through is the Student Handout Sorting out the Change.docx. Check out this video, Sorting the Change, video narrative, middle.MP4, for more details on how I like to run this activity.

This activity was adapted from a task written by The mathematics Vision Project. The original lesson can be found at www.mathematicsvisionproject.org.

*expand content*

Technically, students have been exposed to everything of extreme importance in this unit at this point. Over the next week students will be practicing a bit more with finding equations of exponential functions and real world applications and will also take a more in-depth look at stretching exponentials. So at this point I would like to know how students feel that they are progressing. I want students to complete a **3-2-1 Assessment** for this unit. I will ask students for 3 things they’ve learned, 2 questions they still have, and 1 thing they want to tell me.

#### Resources

*expand content*

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Cat Island (Day 1 of 2): Cats can’t add but they do multiply!
- LESSON 2: Cat Island (Day 2 of 2): Cats can’t add but they do multiply!
- LESSON 3: Graphing Exponential Functions
- LESSON 4: Shifting Exponential Functions
- LESSON 5: Counting the Change: Linear, quadratic, or exponential? (Day 1 of 2)
- LESSON 6: Counting the Change: Linear, quadratic, or exponential? (Day 2 of 2)
- LESSON 7: Sorting out the Change
- LESSON 8: Stretching Exponential Functions (and your mind)
- LESSON 9: Credit Card Investigation: What is interest? (Day 1 of 4)
- LESSON 10: Credit Card Investigation: What is interest? (Day 2 of 4)
- LESSON 11: Credit Cards Investigation: How is interest really calculated? (Day 3 of 4)
- LESSON 12: Credit Card Investigation: A Dastardly Scheme (Day 4 of 4)
- LESSON 13: Exponential Functions Test Review
- LESSON 14: Exponentials Functions Unit Test