Thomas Allen

Dr. Adu-Gyamfi

12/4/13

Artifact 3

This artifact is about using Ti-Nspire software to help demonstrate the translations that a quadratic function moves through depending on which value is replaced as a variable. We also see how a quadratic function keeps it’s the parameter where the roots stay 3 and 5 while the function is being manipulated by outside factor.

1. I noticed while I varied the value of (a) the slope of the parabola locus varied along with the value of the variable (a).

2. I noticed while I varied the value of (b) the parabola translated along the (c) value.

3. I noticed while I varied the value of (c) the parabola translated up and down according to the value of (c).

A) What happens to the graph as a varies and b and c are held constant? When a varies from negative to positive the direction of the parabola switches from downward to upward.

B) Is there a common point to all the graphs? What is it?There is a common point is at 3 which is the constant for variable c in the function a*x^(2)+2*x+3

C) What is the significance of the graph where a=0?When a is zero the expression losses a degree and transforms into a linear function.

A) What happens to the graph as b varies and a and c are held constant?The graph translates across the c variable constant 3.

B) Is there a common point to all the graphs? What is it?Yes the c variable which took the value of 3 in the function x^(2)+b*x+3.

C) What is the significance of the graph where b=0? When b is equal to zero the reflection point of the function x^(2)+b*x+3 lies on the y axis and intersects at point (0,3)

A) What happens to the graph as c varies and a and b are held constant?The function x^(2)+2*x+c translates upward and downward according to the varying c variable.

B) Is there a common point to all the graphs? What is it?There is no common point of all the graphs share with respect to intersection points.

C) What is the significance of the graph where c=0?When c is equal to 0 the y coordinate lies on the x-axis.

1. What do you notice about the roots of all 15 graphs?The roots stay the same as long as the constants 3 and 5 are not altered by a computation due to the order of operations.

2. What do you notice about the intercepts of these graphs?All of the intercepts are through x coordinates (3,0) and (5,0) with the exception of when a is equal to 0 of the function (x-3)*(x-5)*a

3. What do you notice about the intersection points. The points are all through x values 3 and 5.

4. What do you notice about the Orientation or Position of the graphs.The graphs are all scalar multiples of each other and as the variable a varies from negative to positive the orientation of the graphs switch from opening downward to upward.

5. Do they have common points? What can you say about their common points. They have the points (3,0) and (5,0) in common.

6. What do you notice about the correlation between the orientation of the graphs and the sign or coefficient of the x^2 term.

The orientation of the graph opens upward when the value of the coefficient of the x^2 is positive and it opens downward when it takes on a negative value.

7. What do you notice about the locus of the vertex of each of these graphs? The locus of the vertex lies on the axis of symmetry.

IDP TPACK TEMPLATE (INSTRUCTIONAL DESIGN PROJECT TEMPLATE)

NAME: ___Thomas Allen_____ DATE:____12/4/12_____

Content. Describe: content here. (COMMON CORE STANDARDS)

CCSS.Math.Content.HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

Describe: Standards of mathematical Practice

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Use appropriate tools strategically

1. Attend to precision.

Look for and express regularity in repeated reasoning.

Pedagogy. Pedagogy includes both what the teacher does and what the student does. It includes where, what, and how learning takes place. It is about what works best for a particular content with the needs of the learner.

1. Describe instructional strategy (method) appropriate for the content, the learning environment, and students. This is what the teacher will plan and implement.

This lesson will be exploration based. The teacher will go over the basic topics such as the standard form of an equation and the basic techniques that manipulate said equation with use of TI-Inspire

Additionally the teacher will present the class with an appropriate worksheet to guide the students along.

Walk around the class during the student’s investigation and ask any probing questions.

2. Describe what learner will be able to do, say, write, calculate, or solve as the learning objective. This is what the student does.

The student will be able to explore transformations in the quadratic equation based on the varying coefficients by utilizing sliders as well as using multiple functions and their graphs on the same plane in order to gain an understanding of each coefficient and its respective effect on the graph.

3. Describe how creative thinking--or, critical thinking, --or innovative problem solving is reflected in the content.

In this lesson the sliders will help show what the effects that each coefficient has on its function but will not give an explicit answer as to why. This implores the student to discretely figure out what is going on with the function in relation to its varying coefficients.

Technology.

1. Describe the technology

TI-Inspire is a computer software that combines various elements of mathematics that enables its user to gain a deep conceptual understanding of the properties and concepts in question. As in this case the relationship between algebraic and graphical representations of quadratic functions.

2. Describe how the technology enhances the lesson, transforms content, and/or supports pedagogy.

This technology in this lesson enables the students to manipulate the various coefficient values. They can easily manipulate the coefficient value and receive an instant image that represents the change that was made to the function as opposed to having to graph each graph individually. This also allows the students to quickly make and test conjectures about the changes made to the function. The geometry trace function in TI-Inspire is also useful in that it will allow the user to trace a certain point of the graph and show its translation over the plane according to the changes made to the function.

3. Describe how the technology affects student’s thinking processes.

Tracing the vertex of the quadratic equations the students will be able to create a conjecture about how each of the coefficients makes divers transformations to the parabola. This application is useful in that it shows the previous changes to the quadratic equation.

Reflect—how did the lesson activity fit the content? How did the technology enhance both the content and the lesson activity?

Reflection

The lesson reflects what the content was based which was the common core standards. Students weren’t necessarily picking out different pieces of the graph but they are using those pieces to create an understanding of the transformations of the quadratic equation. The technology made it feasible to put a plethora of graphs on one graph and be able to look at them at once and see the change according to the changes made to the respective variable.

Lesson Plan Template MATE 4001 (2013)

Title: Quadratic Transformations

Subject Area: Math 2

Grade Level: Secondary

Concept/Topic to teach: Transformations of Quadratics

Learning Objectives:

Content objectives (students will be able to……….)Know each coefficients effect on the graph and how they interact with each other.

Essential Question

What question should student be able to answer as a result of completing this lesson?

What are the effects of the variables (a), (b), and (c) on the quadratic equation and its graph?

Standards addressed:

Common Core State Mathematics Standards:

CCSS.Math.Content.HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★

Common Core State Mathematical Practice Standards:

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Use appropriate tools strategically

Attend to precision.

Look for and express regularity in repeated reasoning.

Technology Standards:

HS.TT.1.1: Use appropriate technology tools and other resources to access information (multi-database search engines, online primary resources, virtual interviews with content experts).

HS.TT.1.2: Use appropriate technology tools and other resources to organize information (e.g. online note-taking tools, collaborative wikis).

Required Materials:

Computers, Paper /Pencil, Projector

Notes to the reader:

Students already have a basic knowledge of the quadratic function, and how to use TI-Inspire.

Time: Assume 90 minutes

Time Teacher Actions Student Engagement

I. Focus and Review (Establish prior knowledge)

Review basic part of parabola. Draw a parabola and have students call out parts of the graph.

As an open class discussion students will come to the board and label and define the graph with the aid of the class if necessary.

II. Statement (Inform student of objectives)

Teacher will introduce the basic steps to graphing a quadratic equation and instruct students to use TI-Inspire to create their own quadratic function.

Students will use TI-Inspire to look at the quadratic function.

III. Teacher Input (Present tasks, information, and guidance)

Teacher will supply a worksheet that students who are put into small groups would take them through basic procedures to different steps of creating a quadratic function. As well as write down their conjectures of a given graph, procedure or case.

Pick up the techniques that will be needed to complete the requirements in TI-Inspire. Follow along on their own computers or calculators and record observations and conjectures that each variant made on the effects of the graph and discuss the validity of their conjectures.

IV. Guided Practice (Elicit performance, provide assessment and feedback)

Circulate and ask questions where necessary.

The students will then have to move on to b, c with the sliders. Then the students will overlay graphs with only a changing and likewise for b and c and record their observations about each.

V. Independent Practice -- Seatwork and

Circulate and ask questions where necessary. Provide assistance if necessary for students to be able to create 10 equations in a timely

Students will create 8 equations that have the roots 2 and 6 and overlay them on one graph and see the changes that occur in those graphs

Homework (Retention and transfer)

manner. and their similarities.

VI. Closure (Plan for maintenance)

When a/b/c change what happens to the graph?

Are there any common points to the graphs?

What is the significance when a/b/c=0?

When all equations have roots of 3 and 5:

What do you notice about the roots of all 15 graphs

What do you notice about the Intercepts of these graphs

What do you notice about their Intersection points

What do you notice about the Orientation or Position of the graphs

Do they have Common points? What can you say about their common points

What do you notice about the correlation between the orientation of the graphs and the Sign or coefficient of the x^2 term?

What do you notice about the Locus of the vertex of each of these graphs?

Present findings in a whole class discussion.

Reflection

TI-Inspire is very useful in that you can utilize the application of the geometry trace. It was interesting to find while traveling through this exploration that as the b value varied the various vertex’s created with the trace application created the parabola with the negated a value. The technology also really helps with being able to input lots of graphs simultaneously quickly, without this benefit the conceptual learning that the class period would have would be reduced dramatically due to listless hand computations. By being able to see all the graphs on one page and being able to utilize a slider the students will gain a better and deeper conceptual understanding of the lesson and its objective