SWBAT derive and use a function modeling the value of a depreciating automobile.

Auto Loan (linear function) + Car Value (exponential decay function) = REAL LIFE APPLICATION!

1 minutes

This lesson allows the students the opportunity to share out their findings from a previous homework assignment where they derived a function for the value of a depreciating automobile. When assigning the problem initially, I allowed the students one day in between problem roll out and this lesson. This allowed the students to ask any clarifying questions over the task. Most the questions that I received actually dealt with coming up with a linear equation for the loan function. Up front, I allowed the students the flexibility to choose their loan term based off of a 20% down payment.

Please also see the **Video Narrative: Extensions and Scaffolds**.

**Tools/Technology/Software****:** projector screen, a poster printer is also helpful and fun!

5 minutes

For this particular “share out activity” I group my students in very diverse groups. During a typical class investigation involving group thinking, I usually like to lump like-ability students together. However, for the sake of learning from others, I like using this activity for the students to learn from each other. When I grouped my students, I used groups ranging from 5 to 7 students. Although 7 sounds like a lot, I have found that all students are still able to contribute to the discussion. Also, the more vehicles and loan terms that are displayed in a particular group - the better!

After the students have been grouped, I display PowerPoint slide #1 and explain the objectives of the day. When I first did this activity the first thing that I said was that the students could pull up a picture of their car on their Ipad for the discussion. However, I learned my lesson the hard way from doing this! It is fine for the kids to pull up a picture, and in fact, it makes the share out even more fun and applicable. What I did wrong was make this my FIRST direction… because every direction that I said after that went “in one ear and out the other” of the students as they feverously searched for a cool picture of their car!

**PLEASE NOTE: **Without restating the activity, please see the *Logs, Loans, and Life Lessons *lesson in this unit plan. It is the foundation for the work the students put together.

25 minutes

Please see the attached VIDEO NARRATIVE: Level/Language of Conversations – Parental Involvement.

Key ideas during this exciting time…

1) As a teacher, I rotate the room and engage myself in the student’s conversations. It is a lot of fun! Usually the students understand the problem really well, and are even very confident in communicating their knowledge about it. However, one area to push the kids in is to get them to compare/contrast the loan functions that they chose. I use questioning like “I see that you, Sarah, elected to pay your car off in 9 years. However, I see that Jack paid his off in just three. What were each of your monthly payments? Do you think this is manageable?”

2) Asking the students to compare depreciation rates of their vehicles is a really cool way to bring the mathematics standards to the forefront. The students compare functions in a way that is high level, yet easy to understand!

3) As I rotate the room, I am very intentional about getting the students to use not only mathematical terminology, but also reference things like an “upside down” car loans and down payments. It is through THIS LANGUAGE that the mathematics will actually stick with the kids. What better way to show them about decay than to apply it to an automobile? Nearly every kid wants one when he/she is in high school!

4) GRADING: I give my students an oral participation grade in my grade book for the activity. They know that the expectation is for them to be active participants in the conversation, and come with their work prepared. I am sure to instruct the students that as I walk around the room, I will not be able to be involved in every conversation – every minute. I tell the students that if they feel that the grade they receive is not an accurate reflection of their level of participation or content knowledge, then they can meet me during a morning session to make up the points. The students have never had an issue with this and it takes care of the “subjectivity” of the assessment. It also allows me to freely rotate and be involved – rather than be tied to a grading rubric for each student.

5) AS A FINAL PIECE: Have the group mutually agree upon a car to purchase. I allow the students to surf the net and find their vehicle choice and its price. Have the group graph the decay scenario with a loan rate that they choose… I USE POSTER SIZED GRAPH GRIDS FOR THIS AND IT IS REALLY COOL! After all groups are complete, post the graphs on the front board so that they can be compared and contrasted mathematically and financially. It also serves as great closure to the lesson.

10 minutes

As you observe the students conversations about the auto loans wrapping up, hand them the second day’s notes over the “Demystifying e” lesson. This allows groups who are still in discussion about the automobile purchase to continue. As I hand out the notes, I ask the students to briefly review and discuss the previous notes session (from Monday) and then jump into breaking down the new notes packet. I provide the students with highlighters. Usually, these discussions do not have time to get far and the students homework for the day is to take them home and dissect them. At the very least, however, they come in the next day prepared to engage in the class conversation about e – this is not entirely different than a college-course and is a new and unique way for the students to be challenged.