Finding Similarities and Differences between Linear and Exponential Functions:

1. Finding Similarities and Differences between Linear and Exponential Functions: Using Robert Marzano’s Classroom Instruction that Works and technology to frame my mathematics teaching

2. Some resources I use to teach this lesson (plus Excel and TinkerPlots!):

3. Marzano’s Nine Best Teaching Practices:

4. Background: Finding Similarities and Differences Identifying Similarities and DifferencesThe ability to break a concept into its similar and dissimilar characteristics allows students to understand (and often solve) complex problems by analyzing them in a more simple way. Teachers can either directly present similarities and differences, accompanied by deep discussion and inquiry, or simply ask students to identify similarities and differences on their own. While teacher-directed activities focus on identifying specific items, student-directed activities encourage variation and broaden understanding, research shows. Research also notes that graphic forms are a good way to represent similarities and differences. Applications: * Use Venn diagrams or charts to compare and classify items.* Engage students in comparing, classifying, and creating metaphors and analogies.

5. Lesson Objectives: Students will: Explore real life examples of linear and exponential functions. Analyze problem, predict a reasonable outcome, and generate accurate solutions that make sense to them. Communicate and share their ideas on how to solve the problem given. First make sense of and then apply the standard mathematical formulas for linear and exponential functions to the given situation. Employ technology to assist and reflect their understanding of the problems and solutions. Compare and contrast linear and exponential functions in terms of concept, graphic illustration, outcome and mathematical formulas.

6. Assessments: Students will produce and present a technology-generated presentation demonstrating their understanding of how linear and exponential functions are similar and different. Students will generate and share formulas representing both linear and exponential growth. Teacher will observe students working in groups to assess conceptual understanding.

7. Washington State Conceptual GLEs addressed in this lesson: 1.4.5: Understand and apply data techniques to interpret bivariate data. 1.4.6: Evaluate how statistics and graphic displays can be used to support different points of view. 1.5.1: Apply understanding of linear and non-linear relationships to analyze patterns, sequences, and situations.

8. Additional 8th grade GLEs: 1.5.2: Analyze a pattern, table, graph, or situation to develop a rule. 1.5.4: Apply understanding of concepts of algebra to represent situations involving single-variable relationships. Plus, this lesson involves problem-solving, analyzing/reasoning, communicating, and relating mathematics to the real world! Wow!

9. Some assumptions for this lesson: This lesson takes place in the latter half of the 8th grade year, over the course of several class sessions. Students have already worked with linear functions. Students have actively used graphing calculators, TinkerPlots, Geometer’s Sketchpad and Excel as part of their mathematical studies throughout the year.

10. The Allowance Dilemma Jenna is negotiating her allowance with her parents. Now in the third grade, she earns $5 per week. She proposes that her allowance should increase by $1.00 each year until she reaches her senior year of high school. Kayla is also proposing an allowance agreement with her parents. She wants to start earning $5 per week in the third grade and increase her allowance by 50% each year until her senior year.

11. Lesson Flow: Ask students to predict which agreement would generate more allowance. Students then work in partners or groups to calculate the amount of allowance each girl would receive in her senior year. Students share their ideas with the class. Teaching point: What does it mean to increase by 50% each year?

13. Jenna’s Allowance over 10 years

14. Next Step: Kayla’s agreement Teacher then asks students to create a T-chart of Kayla’s allowance growth over ten years.

15. Kayla’s Allowance over 10 Years

16. Time for discussion: How do the results of Jenna’s and Kayla’s allowance growth compare with your predictions? What do you notice about Jenna’s and Kayla’s allowance growth? Which allowance agreement would YOU want to make—Jenna’s or Kayla’s?

17. Assignment: Work with a partner or group to compare and contrast the data and graphs of the two allowance growth patterns: Why are they different? How are they the same? Find an algebraic formula to represent Kayla’s allowance growth over the ten years. How would this compare with the formula used to express Jenna’s allowance growth? Present your findings to the class using technology.

18. Possible graphic organizers for student use:

19. Graphic Organizers, continued:

21. Lesson Closure: Students present their compare/contrast projects and discuss them as a class. Final Teaching Point: Introduce y=ab^t, the exponential growth formula. ….And now for more work with exponential functions!

22. Questions or Comments? Suggestions on how can I improve this lesson? Any tweaking that needs to be done? Your reactions?