Finding Similarities and Differences between Linear and Exponential Functions:

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Finding Similarities and Differences between
Linear and Exponential Functions

• Using Robert Marzanos Classroom Instruction that
  Works and technology to frame my mathematics
  teaching

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Some resources I use to teach this lesson (plus
Excel and TinkerPlots!)
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Marzanos Nine Best Teaching Practices
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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.

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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.

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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.

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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.

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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!

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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, Geometers Sketchpad and Excel as
  part of their mathematical studies throughout the
  year.

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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.

Which agreement would you choose?
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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?

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Following discussion, students work to create a
technology-generated T-chart and graph of Jennas
allowance growth.
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Jennas Allowance over 10 years
Teaching point How can we express this graph in
terms of ymxb?
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Next Step Kaylas agreement

• Teacher then asks students to create a T-chart of
  Kaylas allowance growth over ten years.

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Kaylas Allowance over 10 Years
Students then create a graph of this data using a
calculator, TinkerPlots, or Geometers Sketchpad.
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Time for discussion

• How do the results of Jennas and Kaylas
  allowance growth compare with your predictions?
• What do you notice about Jennas and Kaylas
  allowance growth?
• Which allowance agreement would YOU want to
  makeJennas or Kaylas?

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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 Kaylas
  allowance growth over the ten years. How would
  this compare with the formula used to express
  Jennas allowance growth?
• Present your findings to the class using
  technology.

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Possible graphic organizers for student use
Venn Diagram
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Graphic Organizers, continued
T-Chart
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Lesson Closure

• Students present their compare/contrast projects
  and discuss them as a class.
• Final Teaching Point Introduce yabt, the
  exponential growth formula.
• .And now for more work with exponential
  functions!

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Questions or Comments?

• Suggestions on how can I improve this lesson?
• Any tweaking that needs to be done?
• Your reactions?