TPACK Lesson
Title: Slope of Lines using a CBR
Subject Area: Algebra, Math 1
Grade Level: 9-12
Statement of exploration
Mathematics is embodied in the nature of our bodies, language and cognition. Bodily activities
may be involved in conceptualization processes in mathematics. Perception and action are central
to higher cognition. Subsequently, there is growing interest in the embodied nature of mathematics
thinking. It is even suggested that embodied thinking can be a means of moving students from a
point-wise view of representations to a global view of representations. The question thus arises as
to the relationship between the body as a locus of constitution of subjective mathematical meanings
and the meanings conveyed by mathematical signs? What is the relationship between bodily
actions and mathematical activities? In this artifact you will explore this relation using a motion
detector (CBR) and a Ti-84 plus graphing calculator.
Description
In this lesson students will investigate the relationship between the body as a locus of
constitution of subjective mathematical meanings and the meanings conveyed by mathematical
signs. The students will investigate the relationship between bodily actions and mathematical
activities as well. The students will explore this relation using a motion detector (CBR) and a Ti84 plus graphing calculator.
Standards

Use function notation, evaluate functions for inputs in
their domains, and interpret statements that use function notation in terms of a
context.
 CCSS.MATH.CONTENT.HSF.IF.B.4 For a function that models a relationship between two
quantities, interpret key features of graphs and tables in terms of the quantities,
and sketch graphs showing key features given a verbal description of the
relationship. Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.*
 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.

CCSS.MATH.CONTENT.HSF.IF.A.2
Also the mathematical content that we could address are slope and interpreting graphs. Slope
was important during this exploration because I had to judge how the original graph was
constructed and then I had to use my prior knowledge about slope to determine how I would
make the representation of the graph from my walking speed/motion. Also, I needed to know
how to determine what direction I needed to walk to make a positive slope or a negative slope
and what movements to do to make a horizontal line, vertical line, or a curved line. Interpreting
graphs was another aspect that I needed to know prior to this exploration because I needed to
know that a linear graph is produced by a constant slope. So I had to make connections about a
constant slope equals a constant walking speed and the same for other different types of slopes.
These are some of the mathematical content from the Common Core that were addressed in this
exploration.
Mathematical Practices

Make sense of problems and persevere in solving them
I used this standard during the activity by analyzing the graph given to me and making conjectures about
how to construct this type of graph. So I planned a solution pathway rather than just jumping into a
solution for each graph. Also by using this mathematical practice standard I was better able to
understand/conceptualize in order to solve the problem/task at hand.

Reason abstractly and quantitatively
I used this standard during the activity by making sense of the relationships between time and distance.
This standard helped me to think of solutions that would make the graph that I needed and when it is
impossible to make a graph like the graphs g and h. Also it allowed me to determine how I needed to
make changes to my graph whenever the graph didn’t turn out right the first time.

Model with mathematics
I used this standard during the activity by doing the activity. I applied the mathematics that I already
knew to solve problems and model the graphs that I needed. I used the CBR to interpret my
mathematical results and reflected on whether the results made sense or if I needed to improve the
model if it didn’t serve/reach its purpose.

Use appropriate tools strategically
I used this standard during the activity by using the CBR. The CBR helped me to explore the
mathematical task and deepen my understanding of concepts such as slope. Also, the CBR helped me to
visualize the results of my assumptions and explore consequences and compare my predictions with the
data.

Look for and make use of structure
I used this standard during the activity by looking at the graphs and breaking them down into multiple
parts. This helped me to understand the graph better by determining what I would have to do to make
that part of the graph instead of trying to make the whole graph at once. So this standard allowed for
me to closely look at the graph’s pattern or structure and see it in parts instead of seeing the graph as a
complicated line/graph.

Look for and express regularity in repeated reasoning
I used this standard during the activity by explaining what I did to create each representation and
looking for anomalies in each graph. By using this standard I continually evaluated the reasonableness of
the results from the CBR while attending to the details. I also looked to see if I could repeat some of the
calculations to create shortcuts for other graphs.
Objectives



Content: After this lesson, students will be able to construct graphs based on analyzing
time, distance, and speed of body motions using a CBR.
Technology: Students will be able to use a CBR and a Ti-84 plus graphing calculator to
gain a greater understanding of the relationship between time/distance/speed through
bodily actions.
Pedagogy: I will assist students in discovering the relationships between
time/distance/speed of bodily actions by facilitating appropriate questions and giving
appropriate graphs throughout the entire lesson.
Technology Standards



HS.TT.1.1 Use appropriate technology tools and other resources to access information.
HS.TT.1.2 Use appropriate technology tools and other resources to organize information.
HS.TT.1.3 Use appropriate technology tools and other resources to design products to
share information with others.
Technology Requirements




Internet access/Laptops
CBR
Ti-84 plus graphing calculator
Ti-Connect software
Materials and Supplies Needed






Laptops/Computers
CBR
Ti-84 plus graphing calculator
Ti-Connect software
Pencil/Paper or Microsoft Word
Instructions to the Exploration
Components of the Lesson
Phase 1: Problem Posing (10-15 minutes)
I will begin class by asking students what they already know about slope and interpreting graphs.
Then I will have students discuss how slopes/graphs change based on direction, time, distance
and speed. After that, I will ask students to predict how to create/construct the graphs given
based on what they already know about time/speed/distance/direction of slope and interpreting
graphs. Once the students have made their predictions, I will give them their task using the CBR
to explore the relationship between bodily actions and mathematical activities.
Phase 2: Small- Group Investigation (30-40 minutes)
I will pass out the Embodied Math with the Calculator Based Ranger (CBR) worksheet with
instructions/questions that have been pre-made to guide students in their investigation. Students
will work in groups of 2-3 to complete this task. I will walk around the classroom and monitor
student progress and pose guiding questions as needed. For example, if I notice that a group is
struggling creating a graph with the CBR then I will ask the students if their assumptions have
held true for all circumstances for slope. I will also ensure each student is participating and
writing down the answers to each question instead of just one group member doing all the work.
Phase 3: Whole Class Discussion of Investigation (20-30 minutes)
When all the groups have completed the investigation, I will ask one group what they
constructed/created from one of the 12 graphs that they had to create using the CBR. After that
group has finished sharing their construction/discoveries, I will ask the rest of the groups if they
agree with everything that group said or if they think they discovered something else that the
group had left out. If there appears to be a disagreement among the groups/students, I will have
them spend a little extra time investigating the graph until the whole class has reached an
agreement/consensus on the graph. Then I will repeat this process by picking different groups
until all 12 graphs have been discussed.
Phase 4: Summarizing and Extending (5-10 minutes)
Now that all the students agree on the 12 different graphs using the CBR, I will restate our
findings of the relationships so that the students can record these conclusions in their notebooks.
For homework, I will have the students give a brief overview of the relationships of bodily
actions and mathematical activities and have the students come up with their own activity using
the CBR.
Worksheet with Directions and Investigation Questions on other attachment
INSTRUCTIONAL DESIGN PLAN
NAME: _ David Broome
Content.
DATE: 11/21/14
Describe: content here. (COMMON CORE STANDARDS)
CCSS.MATH.CONTENT.HSF.IF.A.2
Select content from the
Common Core for
Mathematics go to page 79
Use function notation, evaluate functions for inputs in their
domains, and interpret statements that use function notation
in terms of a context.
CCSS.MATH.CONTENT.HSF.IF.B.4
For a function that models a relationship between two
quantities, interpret key features of graphs and tables in terms
of the quantities, and sketch graphs showing key features
given a verbal description of the relationship. Key features
include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and
periodicity.*
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
MP 1: Make sense of problems and persevere in solving them
MP 2: Reason abstractly and quantitatively
MP 4: Model with mathematics
MP 5: Use appropriate tools strategically
MP 7: Look for and make use of structure
MP 8: 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.
Include 21st Century thinking
skills: creative, critical,
innovative, problem solving
www.p21.org You may focus
on just critical, or just creative,
or both critical thinking and
innovative problem solving.
See:
Creativity and Innovation
Critical Thinking and Problem
Solving
1. Describe instructional strategy (method) appropriate for the
content, the learning environment, and students. This is what the
teacher will plan and implement.
Build on students’ prior knowledge of slope, interpreting graphs,
and relationships between time, distance, and speed. I will do this
by having an Exploratory/Discovery Lesson using a CBR to display
the relationships stated above.
The learning environment is in a classroom with
laptops/computers.
The content will be an investigation worksheet with
questions/investigations that build upon each graph/relationship
between time, distance, and speed before continuing to the next
graph. Students will be able to use the CBR and Ti-Connect
software to assist in the learning.
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 learner will be able to explain the relationships between time,
distance, and speed and what these factors do to the
graph/representation.
The learner will identify what each constraint does to the graph by
looking at the graph the CBR gives them and determining what the
time, distance, and speed of the body did to create the graph.
The learner will be able to apply what they learned about the
relationship of bodily actions and mathematical activities to make
real-world connections and see mathematics in everyday life.
The learner will be able to apply the CBR to find new relationships
between different factors.
3. Describe how 21st Century skill is addressed.
The 21st Century Skill is addressed in students being able to exercise
sound reasoning in understanding.
Another skill addressed is identifying and asking significant
questions that clarify various points of view and lead to better
solutions.
Lastly, articulating thoughts and ideas clearly and effectively
through speaking and writing and demonstrate the ability to work
effectively with diverse teams (people).
Technology. Digital tools using
computers, Internet, and
related technologies
1. Describe the technology
CBR, Ti-84 plus calculator, and Ti-Connect Software. The CBR is a
motion detector that tracks how fast and far you are away over a
time period. Then the Ti-84 plus calculator is a calculator used to
graph what the CBR has detected. Then the Ti-Connect Software is
a program on your computer/laptop and it allows you to make a
screenshot of the data or capture the data from the calculator so
you don’t lose it when you start another graph.
The CBR, Ti-84 plus calculator, and Ti-Connect Software are tools
that will help students create constructions of different types of
relationships by making graphs and capturing the graphs/data by
putting the information on the Ti-Connect Software in the
computer.
2. Describe how the technology; i. Enhances the lesson, ii.
Transforms content, and iii. Supports pedagogy.
The CBR, Ti-84 plus calculator, and Ti-Connect Software enhances
the lesson by allowing students to construct graphs from bodily
actions that otherwise can’t be done by hand. These programs
make the lesson better by allowing students to make the graphs
that show the relationship between bodily actions and
mathematical activities with time, distance, and speed. Without the
CBR, Ti-84 plus calculator, and Ti-Connect Software this
investigation/lesson couldn’t be done. By integrating these 3
technology tools into the lesson students will have a better
understanding of the content, multiple representations of the data
from graphs, and see mathematics that wouldn’t be seen otherwise
through bodily actions.
The CBR, Ti-84 plus calculator, and Ti-Connect Software transforms
the content by making it more personal and interactive for the
students. The CBR allows students to track the relationships
between time, distance, and speed and then the Ti-84 plus
calculator allows you to see this data by constructing a graph. The
Ti-Connect Software allows you to take this data and save it so you
can compare different graphs/relationships and expand on the
ideas/thoughts that you constructed/created.
The CBR, Ti-84 plus calculator, and Ti-Connect Software supports
pedagogy by assisting students in constructing graphs that are
provided by making graphs from their body movements.
The CBR, Ti-84 plus calculator, and Ti-Connect Software supports
pedagogy by assisting students in making graphical representations
from their bodily actions and trying to make a copy of the graph
given to them by using their body’s actions.
3. Describe how the technology affects student’s thinking
processes.
The CBR, Ti-84 plus calculator, and Ti-Connect Software affects the
students’ thinking processes by allowing them to see relationships
between time, distance, and speed of their bodily actions in a
graphical representation using the CBR and Ti-84 plus calculator.
The CBR, Ti-84 plus calculator, and Ti-Connect Software allows the
students to focus on conceptual understanding rather than focusing
on other details that aren’t necessary.
The CBR, Ti-84 plus calculator, and Ti-Connect Software allows the
students to change the format of the graph by changing the
speed/distance of their body movements, which adds different
perspectives to the students’ understanding by directly showing
them what is being manipulated and how it is effecting the graph.
The CBR, Ti-84 plus calculator, and Ti-Connect Software also gives
control to the students to control the time/distance/speed by
allowing them to create new graphs and saving them and
comparing the graphs to see the differences or changes they have
been making.