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Characteristics of Functions

Characteristics of Functions. Positive and Negative. Graphical Algebraic. A definition of a concept is only possible if one knows, to some extent, the thing that is to be defined. Pierre van Hiele. Concept Attainment.

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Characteristics of Functions

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  1. Characteristics of Functions Positive and Negative Graphical Algebraic

  2. A definition of a concept is only possible if one knows, to some extent, the thing that is to be defined. Pierre van Hiele

  3. Concept Attainment • Concept Attainment is a strategy designed to teach concepts through the presentation of examples and non-examples. Students form, test, and refine hypotheses about the concept as examples and non-examples are presented. Then, they determine the critical attributes of the concepts - the characteristics that make the concept different from all others. Finally, students demonstrate that they have attained the concept by generating their own examples and non-examples. • Retrieved from http://www.glc.k12.ga.us/pandp/critthink/conceptattainment.htm

  4. Concept Attainment • Show students a few examples of the concept, allowing time for them to think about the similarities. • Show students a few non-examples of the concept, again allowing them time to think about the similarities between the non-examples and how they may differ with the examples. • Continue alternating between a few more examples and non-examples of the concept. • Have students formulate a definition/hypothesis of the concept. • Provide more non-examples and examples and have students test out their theories.

  5. Visualizing the Concept

  6. Concept: Examples of the CONCEPT

  7. Concept: Non-Examples of the CONCEPT

  8. Concept: EXAMPLES of the CONCEPT

  9. Concept: NON-EXAMPLES of the CONCEPT

  10. Comparison • EXAMPLE • NON-EXAMPLE

  11. Comparison • EXAMPLE • NON-EXAMPLE

  12. Concept: EXAMPLES or NON-EXAMPLES of the CONCEPT A. B. C. D.

  13. More Practice Connecting the Graphical and Algebraic Representations Identifying where functions are POSITIVE and NEGATIVE

  14. Building towards the Algebraic Representation Let’s take a look at y = x2 – x – 6.

  15. Let’s look at the linear factors of the function y = x2 – x – 6 = (x + 2) ( x – 3) Make a table: Graph the linear functions: • What will students notice?

  16. Let’s look at the linear factors of the function y = x2 – x – 6 = (x + 2) ( x – 3) Fill in the product column: Plot the product points.        • What will students notice?

  17. Let’s look at the product of the linear factors y = (x + 2) ( x – 3) = x2 – x – 6 . • What will students notice?

  18. Another Example

  19. Extension

  20. Places to visit/Articles to Read Concept Attainment Gay, S.A. (2008).Helping teachers connect vocabulary and conceptual understanding . Mathematics Teacher, 102, 218-223. Conceptualizing Polynomial Functions Weinhold, M.W. (2008). Designer functions: Power tools for teaching mathematics. Mathematics Teacher, 102, 28-33. These graphs were created on gcal.net and graphcalc. (http://sourceforge.net/project/downloading.php?group_id=73729&use_mirror=internap&filename=GraphCalc4.0.1.exe&81618777)

  21. Questions?

  22. Thank You for Attending! • Now go- • Make those connections! • Incorporate technology! • Strengthen student understanding!

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