Slope & Rate of Change

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Objective

SWBAT calculate the rate of change in a linear equation.

Big Idea

Students will interpret the rate of change in the context of a problem, and use it to make predications about a situation that shows linear growth.

Do-Now

10 minutes

Students will complete a Do Now. While students are working, I will circulate around the room passing back the graded exit cards from our last class. After about 4 minutes, I will ask three students to deeply discuss their responses to the Do-Now aloud.  

Next a student volunteer will read today's lesson objective,  "SWBAT calculate the rate of change in a linear function."

I will ask students to share out what they already know about the rate of change, and to recall where we have used it before. 

Guided Notes + Practice

45 minutes

Students will be sorted into groups of 2-3 students and will follow along with this Presentation. Each group will also have white boards and markers to write on in lieu of guided notes.

Most of my students are familiar with slope formula from their middle school math classes, so today's lesson aims to further their understanding of this concept, and to show students the connection to the linear functions we have been studying in class. (My students already knew how to calculate the slope between two points, but didn't have a firm understanding of what it meant)

After a student reads Slide 2 aloud, I will show students the table on Slide 3. I will tell students that we will assume that the two celebrities featured in the example are moving at a constant rate of change, and that their movement can be represented using a linear function. 

I will ask students to decide which Rapper won the first leg of the race. I'll give students time to work independently, using their own mathematical reasoning and to calculate an answer. After a few minutes, most students came to the conclusion that Jay Z one the first leg of the race because he was able to run at a speed of 1 mile every 8 minutes. I will illustrate the math used to calculate this answer on the board: 

(40min - 16min)/(5miles - 2miles) = 8 minutes per mile.

Next, I will write the values in the table as coordinate points. Then I will label the points: (x1, y1) (x2, y2). Lastly, I will show students that the math that we used to calculate Jay'z's speed was really just the formula for slope, but applied in a different context.  

(y2 - y1) / (x2 - x1)

Before moving on, I will ask students to write two linear equations that model both Jay-Z and Kanye's speed. We will analyze what each component of the equation represents as a review from our last class.

Slide 7: Students will be expected to calculate the rate of change in the context of a function. I will show students this variation of the formula, and discuss the similarities and differences between using f(x1) vs y1. 

By today's class students should be familiar with function notation. After this point, I will only represent the rate of change using f(x). 

Slide 8: Students should calculate the rate of change the the savings account balance. Students should also create an equation to model this situation. Discuss whether the slope is positive or negative how this can be verified.

f(10) – f(0) / 10 - 0

Slide 9: Students should calculate the rate of change of the height of the maple tree. Students should also create an equation to model this situation. Discuss whether the slope is positive or negative how this can be verified.

 f(60) – f(20) / 60 - 20

Group Activity: White Board Relay

15 minutes

We will spend the next 15 minutes to practice calculating the rate of change through a relay race. Students will form 4-5 teams, and then stand in lines in the back of the classroom. There will be whiteboards on the other side of the classroom on a desk that is directly across from where each line in standing.

When I say "GO!", two students from each line will race to the other side of the classroom to complete a rate of change problem together. They will hold up their board to me when finished. When it is correct, they will run back to the other side of the room to tag two more people on their team who will then repeat the process. The pair with the most points at the end of 15 minutes will be the winner.

I used questions from this Handout for the relay race. I laminated and cut up the handout and left the cards in a cup at each desk next to the whiteboards (Keep the numbers on the cards when you cut them in order to quickly check each problem when students are showing you a response).

Closing

10 minutes

I will ask students to turn to a neighbor and to briefly summarize what we learned today in class. I will then ask a few pairs to share out for the group.

Students will then complete an Exit Card.