What we like about this lesson
- Addresses standards 8.F.A.3, 8.F.B.4, and 8.F.B.5,
- Reinforces multiple representations of a function through the use of input/output tables, graphs, and equations
- Requires students to interpret unit rate as slope (rate of change)
- Deepens students’ understanding of slope, $y$-intercept, and domain by having them apply it in a real-world situation
- Allows students to explore plausibility of answers and connect the meaning of mathematical models to a situation in the introduction of the lesson
In the classroom:
- Captures student attention by using an engaging context
- Provides robust opportunities for students to discuss mathematical concepts; includes guiding questions for teachers to use to facilitate discussion
- Uses technology to create a graph and illustrate the mathematics of the lesson
- Offers multiple opportunities for student practice by analyzing pizzas of various sizes
Making the Shifts
How does this lesson exemplify the instructional Shifts required by CCSSM?
Focus Belongs to the major work of eighth grade Coherence Builds on unit rates from seventh grade (7.RP.A) and students' understanding of representing situations with equations (7.EE.B.4a); lays foundations for work students will do in high school building functions to represent more complex situations (F-BF.A) Rigor
Conceptual Understanding: secondary in this lesson
Procedural Skill and Fluency: not addressed in this lesson
Application: primary in this lesson
This lesson is designed as an introduction to eighth grade work with functions, which is part of the major work of the grade. Students build on major work of seventh grade with proportional relationships on graphs and in equations. The lesson introduces a context that allows students to consider many ideas that they will study in-depth in their ongoing work with linear functions (recognizing equations that determine linear functions, modeling relationships with functions, identifying the input- and output-values for a given function). Students are then given time to connect features of the function to the real-world situation. It is not intended for students to meet the full expectations of the grade-level standards through only this lesson.
In this lesson, students learn to calculate the price of a pizza by interpreting the unit rate (per topping cost) as slope and by considering the initial cost of a pizza based on its size ($y$-intercept). Scaffolded questions lead students to calculate slope and $y$-intercept before ultimately finding the corresponding linear equation. There are short answer and open-ended questions, with opportunities for students to justify their answers. The lesson continues by giving students practice creating equations to represent the cost of additional pizzas with an unknown number of toppings. Students then compare the graphs for all three pizza sizes, both qualitatively and quantitatively.
The lesson concludes by presenting the actual costs of various pizzas on a graph. This allows students to consider a situation where the slope is equal to zero. Because the graphs represent continuous functions, they provide an opportunity to explore real-world constraints and discuss the qualitative aspects of graphs (8.F.B.5). Extension activities are included to allow students to apply other mathematical ideas (ratios and proportional reasoning, area of circles and surveying) to the same context.
The work with functions in this lesson lays the groundwork for functions and modeling, which is a significant focus of high school mathematics. For more insight on the grade-level concepts addressed in this lesson and how they relate to later work, read pages 5 and 6 of the progressions document, Grade 8, High School, Functions.