Pizza, Hot Chocolate and Newton's Law of Cooling: Adding Constants to Exponential Functions

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SWBAT interpret and apply Newton's Law of Cooling as an exponential function. SWBAT combine standard exponential functions using arithmetic. SWBAT paraphrase complex arguments.

Big Idea

Students investigate applications of Newton's Law of Cooling and create their own exponential functions modeling other contexts.

Entry Ticket

15 minutes

For today's Entry Ticket: Pizza, Hot Chocolate and Newton's Law of Cooling  I have students practice their skills at interpreting exponential functions. I want to activate students' prior knowledge from this unit, and be used to interpreting different aspects of exponential functions because in class today we will be looking at how adding a constant changes exponential functions. 


Explicit Instruction and Active Note-Taking

20 minutes

After the Entry Ticket, class turns to an interactive discussion on Newton's Law of Cooling. I chose this particular law/formula because it lends itself to comparing and contrasting the effects of adding/subtracting constants for exponential functions. 

I go through the Class Notes: Pizza, Hot Chocolate and Newton's Law of Cooling slides. As part of the presentation, I show short videos on Euler, Newton and Newton's Law of Cooling to provide Multiple Means of Representation for different types of learners to access the material. 

During this time, students are engaged in an Active Note-taking process, where they are expected to be writing, reading, listening and speaking to be active participants in their learning. 

Euler Biography video:


Newton Law of Cooling Video


Newton Biography Video



Guided Practice

20 minutes

After reviewing Newton's Law of Cooling and providing some background information for students, we complete the Guided Practice: Pizza, Hot Chocolate and Newton's Law of Cooling on cooling pizza as a class for Guided Practice. The guided practice work is on slides 5-8 of the powerpoint slides (and a handout with the practice problems is attached as a resource in this section). 

In this problem, we want to find out how long we have to wait for a pizza right of the oven takes to cool to a temperature that won't burn our mouths. One tip for this lesson is to take your time in reviewing the first practice problem. These types of problem lend themselves beautifully to Math Practice 1 MP.1 as students, including my highest level honors students, with all of the variables and unknowns in the equation. 

I start by asking students to simply identify which of the variables we know and which we do not know. 

I do like to show students how to solve for the constant k in the equation, but for the remaining practice problems and exit ticket I provide the value of k to students (it is a great extension/challenge for gifted students to calculate k for each of the examples). In the pizza problem k is approximately equal to -0.16. 

Collaborative Work

20 minutes

After completing the Guided Practice, I arrange students in small groups of 3-5 students to work on the Collaborative Work: Pizza, Hot Chocolate and Newton's Law of Cooling.where the surrounding temperature of the pizza changes. I then ask students to work on a scenario where the end goal temperature changes. The reason for keeping the original problem is to keep the context (a cooling pizza) and the same and allow students to zoom in and focus on what happens when we adjust the surrounding temperature (changing a constant to the exponential function). 

I like having students work in groups and develop important collaborative skills. I am looking for groups to be more and more self-sustaining as the year goes by, and continue to rely more and more on fellow group members for generating knowledge and understanding, and less and less reliance on me for assistance. 

In terms of the constant k, I provide it to groups and have them assume it continues to stay at approximately -0.16

Note: the collaborative practice is included in this section as a handout, and can also be found on slides 9 and 10 of the powerpoint slides for this lesson.

Exit Ticket: Hot Chocolate

15 minutes

To conclude class, students work individually on the day's Exit Ticket: Pizza, Hot Chocolate and Newton's Law of CoolingIn this example, I ask students to apply Newton's Law of Cooling to estimate the amount of time someone has to drink hot chocolate before it gets too cold. 

I like this type of assignment to assess student learning because it is problem-based and flexible. I want students to not only know that they have to take the log of both sides of an equation to eliminate e, but perhaps more importantly I want them to be able to apply the process and answer to the scenario being modeled. 

This lesson is heavy both procedurally and conceptually, so I often will assign the exit ticket as homework due to time constraints. If I assign the exit ticket as homework, then I use the last few minutes of class to review the group work practice problems and have students summarize what they learned in class today with a Think-Pair-Share.  

Homework: Changing an Exponential by an Arithmetic Constant

For the Homework: Changing Exponential Functions by a Constant I have students complete a problem where they have to compare and contrast three exponential functions, all of which only differ by a constant. 

The intent of the homework is to focus on an important underlying concept associated with the lesson on Newton's Law of Cooling with problems that are more manageable for students to complete independently at home. 

I like to reinforce the concept and check for student understanding the next class by reviewing this homework assignment and having a recap discussion with the whole class to try and generalize patterns for how exponential functions are affected by adding or subtracting a constant to the function.