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. A definition of a concept is only possible if one knows, to some extent, the thing that is to be defined. Pierre van Hiele 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. 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 Retrieved from http://www.glc.k12.ga.us/pandp/critthink/conceptattainment.htm 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 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. 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. Continue alternating between a few more examples and non-examples of the concept. Have students formulate a definition/hypothesis 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. 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 xy -67 -55 -43 -31 -2 -3 0-5 1-7 2-9

9. Concept: NON-EXAMPLES of the CONCEPT xy -1.52.4986 1.9994 -0.51.4998 01 0.5.49985 1.0-6E-4 1.5-.5014 2.0-1.002 2.5-1.504

10. Comparison EXAMPLE EXAMPLE NON-EXAMPLE

11. Comparison EXAMPLE EXAMPLE NON-EXAMPLE xf(x) -2.5-22.25 -2.0-14 -1.5-7.25 -2.0 -0.51.75 04.0 0.54.75 1.04.0 1.51.75 xf(x) -10E10 -10-1.001 -5-1.031 -1.5 0-2 1-3 5-33 10-1025 100-1E30

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

13. More Practice Connecting the Graphical and Algebraic Representations Identifying where functions are POSITIVE and NEGATIVE For what values of x is: the graph below the x – axis? the graph above the x-axis? For what values of x is:

14. Building towards the Algebraic Representation Let’s take a look at y = x 2 – x – 6.

15. Let’s look at the linear factors of the function y = x 2 – x – 6 = (x + 2) ( x – 3) xy 1 = x +2y 2 = x - 3 -4-2-7 -3-6 -20-5 1-4 02-3 13-2 24 350 461 572 Make a table:Graph the linear functions: What will students notice?

16. x y 1 = x +2y 2 = x -3 y = (x+2)(x-3) -4-2-714 -3-66 -20-50 1-4 02-3-6 13-2-6 24-4 3500 461 6 572 14        Fill in the product column:Plot the product points. What will students notice? Let’s look at the linear factors of the function y = x 2 – x – 6 = (x + 2) ( x – 3)

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

18. Another Example

19. Extension

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

21. Questions?

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