Characteristics of Functions Positive and Negative Graphical Algebraic.

Characteristics of Characteristics of FunctionsFunctions

Positive and Positive and NegativeNegative

Graphical Algebraic

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

Pierre van Hiele Pierre van Hiele

Concept AttainmentConcept Attainment Concept Attainment is a strategy designed to Concept Attainment is a strategy designed to

teach concepts through the presentation of teach concepts through the presentation of examples and non-examples. Students form, test, examples and non-examples. Students form, test, and refine hypotheses about the concept as and refine hypotheses about the concept as examples and non-examples are presented. Then, examples and non-examples are presented. Then, they determine the critical attributes of the they determine the critical attributes of the concepts - the characteristics that make the concepts - the characteristics that make the concept different from all others. Finally, students concept different from all others. Finally, students demonstrate that they have attained the concept demonstrate that they have attained the concept by generating their own examples and non-by generating their own examples and non-examples.examples.

Retrieved from Retrieved from http://www.glc.k12.ga.us/pandp/critthink/conceptattainment.htm

Concept AttainmentConcept Attainment Show students a few examples of the concept, Show students a few examples of the concept,

allowing time for them to think about the allowing time for them to think about the similarities.similarities.

Show students a few non-examples of the Show students a few non-examples of the concept, again allowing them time to think concept, again allowing them time to think about the similarities between the non-about the similarities between the non-examples and how they may differ with the examples and how they may differ with the examples.examples.

Continue alternating between a few more Continue alternating between a few more examples and non-examples of the concept.examples and non-examples of the concept.

Have students formulate a Have students formulate a definition/hypothesis of the concept.definition/hypothesis of the concept.

Provide more non-examples and examples and Provide more non-examples and examples and have students test out their theories.have students test out their theories.

Visualizing the Visualizing the ConceptConcept

Concept: Concept: 0)( xf

Examples of the CONCEPT

Concept:Concept: 0)( xf

Non-Examples of the CONCEPT

Concept:Concept: 0)( xf

EXAMPLES of the CONCEPT

x y

-6 7-5 5-4 3-3 1-2 -1-1 -30 -51 -72 -9

Concept:Concept: 0)( xf

NON-EXAMPLES of the CONCEPT x y

-1.5 2.4986

-1.0 1.9994

-0.5 1.4998

0 1

0.5 .49985

1.0 -6E-4

1.5 -.5014

2.0 -1.002

2.5 -1.504

ComparisonComparison EXAMPLEEXAMPLE NON-EXAMPLENON-EXAMPLE

ComparisonComparison

EXAMPLEEXAMPLE NON-EXAMPLENON-EXAMPLEx f(x)

-2.5 -22.25

-2.0 -14

-1.5 -7.25

-1.0 -2.0

-0.5 1.75

0 4.0

0.5 4.75

1.0 4.0

1.5 1.75

x f(x)

-10E10 -1

-10 -1.001

-5 -1.031

-1 -1.5

0 -2

1 -3

5 -33

10 -1025

100 -1E30

Concept: Concept: 0)( xf

EXAMPLES or NON-EXAMPLES of the CONCEPTA

.

D.

C.

B.

More Practice Connecting the Graphical and More Practice Connecting the Graphical and Algebraic RepresentationsAlgebraic Representations

Identifying where functions are POSITIVE and NEGATIVEIdentifying where functions are POSITIVE and NEGATIVE

For what values of For what values of xx is: is:

the graph below the the graph below the xx – – axis?axis?

the graph above the the graph above the xx--axis?axis?

For what values of For what values of xx is: is:  

For what values of For what values of xx is: is:

221 xy 62 xxy 2)1)(2(

4

1 xxy

?0)1)(2(4

1 2 xx

?0)1)(2(4

1 2 xx

062 xx 062 xx

Building towards Building towards the Algebraic the Algebraic

RepresentationRepresentationLet’s take a look at Let’s take a look at y y = = xx22 – –

x – 6.x – 6.

Let’s look at the linear factors of Let’s look at the linear factors of the function the function

y y = = xx22 – – x – 6 x – 6 = (= (xx + 2) ( + 2) ( xx – 3) – 3)

x y1= x +2 y2= x - 3

-4 -2 -7

-3 -1 -6

-2 0 -5

-1 1 -4

0 2 -3

1 3 -2

2 4 -1

3 5 0

4 6 1

5 7 2

Make a table: Graph the linear functions:

• What will students notice?

x y1= x +2 y2= x -3y = (x+2)(x-3)

-4 -2 -7 14

-3 -1 -6 6

-2 0 -5 0

-1 1 -4 -4

0 2 -3 -6

1 3 -2 -6

2 4 -1 -4

3 5 0 0

4 6 1 6

5 7 2 14

Fill in the product column:

Plot the product points.

• What will students notice?

Let’s look at the linear factors of Let’s look at the linear factors of the function the function

y y = = xx22 – – x – 6 x – 6 = (= (xx + 2) ( + 2) ( xx – 3) – 3)

Let’s look at the product of the Let’s look at the product of the linear factors linear factors

y y = (= (xx + 2) ( + 2) ( xx – 3) = – 3) = xx22 – – x – 6 .x – 6 .

• What will students notice?

Another ExampleAnother Example

)5)(1( xxy51 21 xyxy

ExtensionExtension

)2)(5)(1( xxxy

Places to visit/Articles to Places to visit/Articles to ReadRead

Concept AttainmentConcept AttainmentGay, S.A. (2008).Gay, S.A. (2008). Helping teachers connect vocabulary and Helping teachers connect vocabulary and conceptual understanding . conceptual understanding . Mathematics Teacher, 102 Mathematics Teacher, 102, 218-, 218-223.223.

Conceptualizing Polynomial FunctionsConceptualizing Polynomial Functions Weinhold, M.W. (2008). Designer functions: Power tools for Weinhold, M.W. (2008). Designer functions: Power tools for

teaching teaching mathematics. mathematics. Mathematics Teacher, 102, Mathematics Teacher, 102, 28-33.28-33.

These graphs were created on gcal.net These graphs were created on gcal.net and graphcalc. and graphcalc. ((

http://sourceforge.net/project/downloading.php?group_id=73729&use_mirror=internap&filename=GraphCalc4.0.1.exe&81618777))

Questions?Questions?

Thank You for Attending!Thank You for Attending!

Now go-Now go-Make those connections!Make those connections!Incorporate technology!Incorporate technology!Strengthen student Strengthen student understanding!understanding!