Characteristics of Functions Positive and Negative Graphical Algebraic.

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Presentation transcript:

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. 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

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 Retrieved from

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.

Visualizing the Concept

Concept: Examples of the CONCEPT

Concept: Non-Examples of the CONCEPT

Concept: EXAMPLES of the CONCEPT xy

Concept: NON-EXAMPLES of the CONCEPT xy E

Comparison EXAMPLE EXAMPLE NON-EXAMPLE

Comparison EXAMPLE EXAMPLE NON-EXAMPLE xf(x) xf(x) -10E E30

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

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:

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

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 Make a table:Graph the linear functions: What will students notice?

x y 1 = x +2y 2 = x -3 y = (x+2)(x-3)        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)

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

Another Example

Extension

Places to visit/Articles to Read Concept Attainment Concept Attainment Gay, S.A. (2008). Helping teachers connect vocabulary and conceptual understanding. Mathematics Teacher, 102, 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, mathematics. Mathematics Teacher, 102, These graphs were created on gcal.net and graphcalc. ( rnap&filename=GraphCalc4.0.1.exe& ) These graphs were created on gcal.net and graphcalc. ( rnap&filename=GraphCalc4.0.1.exe& ) rnap&filename=GraphCalc4.0.1.exe& rnap&filename=GraphCalc4.0.1.exe&

Questions?

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