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 byutilizingsliders 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 isa computer software that combinesvarious 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 manipulatethevarious

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 transformationsto 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: Transformationsof 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

createtheir 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

conjecturesthat each variant made on

the effects of the graph and discuss

the validityof 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

Homework

Circulate and ask questions where

necessary. Provide assistance if

necessary for students to be able to

create 10 equations in a timely

manner.

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

and their similarities.

(Retention

and transfer)

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 isvery 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 utilizea slider the

students will gain a better and deeper conceptual understanding of the lesson and its objective