1.
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.
2. I noticed while I varied the value of (b) the parabola translated along the (c) value.
3.
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.
4.
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)
5.
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.
6.
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.
7.
IDP TPACK TEMPLATE (INSTRUCTIONAL DESIGN PROJECT TEMPLATE)
NAME: ___Thomas Allen_____ DATE:____12/4/12_____
Describe: content here.
Content.
(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.
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.
Look for and express regularity in repeated reasoning.
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.
8.
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.
Technology.
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.
9.
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.
10.
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.
11.
(Retention
and transfer)
VI. Closure
When a/b/c change what happens to
(Plan for
the graph?
maintenance)
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.
12.
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
Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.
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