Carli Sowder
Overview
This Learning Progression is
for the CCSS domain “8th Grade
Geometry” and the cluster
“Understand and apply the
Pythagorean Theorem.” This is
written for the alternative high school,
Excel in Ellensburg, WA. The textbook
that aligns with this progression is
“Geometry” by Siegfried Haenisch.
Standards:

CCSS.Math.Content.8.G.B.6
Explain a proof of the
Pythagorean Theorem and its
converse.

CCSS.Math.Content.8.G.B.7
Apply the Pythagorean
Theorem to determine
unknown side lengths in right
triangles in real-world and
mathematical problems in
two and three dimensions.

CCSS.Math.Content.8.G.B.8
Apply the Pythagorean
Theorem to find the distance
between two points in a
coordinate system.
CCSS.Math.Content.8.G.B.6
Students need to be able to
understand the Pythagorean
Theorem’s proof before they can use it
properly. This standard is crucial for
students to meet the expectation that
they understand a concept before they
begin using it. An example of what the
picture would look like is shown
below along with a proof.
Students must form a proof that
a^2 + b^2= c^2 based on a picture
similar to the one shown above.
Students have been simplifying square
roots and squaring numbers
previously. This standard is assessing
the mathematical reasoning and logic
of the theorem. Once the Pythagorean
Theorem has been discussed, we will
move onto the converse.
The converse of the
Pythagorean Theorem also needs to
be understood for this standard to be
met. In Euclid's Elements (Book I,
Proposition 48) the converse is
defined
"If in a triangle the square on one of the
sides equals the sum of the squares on the
remaining two sides of the triangle, then
the angle contained by the remaining two
sides of the triangle is right."
The converse is useful to solving
various problems. A basic proof of the
converse should look something like
the picture below.
“Let ABC be a triangle with side lengths a,
b, and c, with a2 + b2 = c2. Construct a
second triangle with sides of length a and b
containing a right angle. By the
Pythagorean theorem, it follows that the
hypotenuse of this triangle has length c = √
a2 + b2 , the same as the hypotenuse of the
first triangle. Since both triangles' sides are
the same lengths a, b and c, the triangles
are congruent and must have the same
angles. Therefore, the angle between the
side of lengths a and b in the original
triangle is a right angle.”
Once the converse is proven, students
can use the statements



If a2 + b2 = c2, then the triangle
is right.
If a2 + b2 > c2, then the triangle
is acute.
If a2 + b2 < c2, then the triangle
is obtuse.
Plan to Teach this Standard
To understand both the
Pythagorean Theorem and its
converse we will review the picture
above and the given proof. Students
will follow along and draw this in their
notes. We will discuss parts of it to
further understanding. We will then
look at the converse and how to
determine if a triangle is acute, obtuse,
or right based on the theorem. We will
do example problems on the board for
their notes. Worksheets will be given
to the students to practice. As
homework, students will be asked to
write their own explanation of the
theorem and its converse and prepare
to explain it out loud.
CCSS.Math.Content.8.G.B.7
Students need to be able to
apply the Pythagorean Theorem. This
standard requires students to use the
theorem to find missing lengths of
triangles. For example, a problem may
look like
.
Students need to use the equation
a^2+b^2=c^2 so solve for X. If
students have met the previous
standard, they will know what
numbers replace which variables.
Plan to Teach this Standard
We will begin with reviewing
the student’s explanations of the proof
from the previous homework. They
will watch examples being completed
on the white board. After a few
examples are completed, students will
work on a worksheet with a partner.
The worksheet will have problems
much like the one above.
CCSS.Math.Content.8.G.B.8
Now that the students have be
mastered the first two standards, they
will focus on the last standard in the
domain. The Pythagorean Theorem
can be used in a coordinate plane also.
It is important for students to know
this because they can find distances
between points within a coordinate
plane. This concept could be applied
to the real world and used with maps
and other things involving distance.
Also, it will help engage them because
it will relate to their real life.
Plan to Teach this Standard
These activities and lessons are
designed to be completed in three
days. However, these students have
difficulty comprehending at times.
Therefore, lessons may need to be retaught or more time may need to be
spent on them. The standards are fully
covered by these lessons. By the end
of these lessons, students will
understand the proof of the
Pythagorean Theorem, its converse,
how to apply it to find missing lengths,
and find distances in a coordinate
plane using the theorem.
Students will begin with a
warm up of the last two standards.
This will ensure they are ready to
move on to this standard. We will
review example on page 292 in the
given textbook.
This will give them a good example to
have in their note to refer back to.
They will get into groups to solve #610 on page 294 which are very similar
to the example above. Once the
students have finished those
examples, I will put a coordinate plane
on the overhead that is a map of
Ellensburg. They will work in groups
to find distances between different
place in Ellensburg, or points on the
plane. This will help them practice
using this standard in the real world.
Conclusion
Lesson Title: Pythagorean Theorem
Unit Title: Triangles
Teacher Candidate: Carli Sowder
Subject, Grade Level, and Date: Math, 8th grade, 2/1/14
Placement of Lesson in Sequence
This lesson is the second lesson in the Pythagorean Theorem lesson. Pythagorean Theorem
will take 3 days to complete.
Central Focus and Essential Questions
The central focus of this lesson is to have the students improve their procedural fluency and
conceptual understanding by doing practice problems. They will discuss their explanations
of the proof to their partner so they can talk and work out an accurate explanation together
and confirm they both understand. They will master the mathematical reasoning by doing
real world examples and determining when the Pythagorean Theorem could be used in the
real world as their homework assignment. They will work in pairs today to help each other
master their procedural fluency and help each other.
Content Standards
CCSS.Math.Content.8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.
CCSS.Math.Content.8.G.B.7 Apply the Pythagorean Theorem to determine unknown side
lengths in right triangles in real-world and mathematical problems in two and three
dimensions.
CCSS.Math.Content.8.G.B.8 Apply the Pythagorean Theorem to find the distance between two
points in a coordinate system.
Learning Outcomes
Assessment
Students will be able to:
I will assess their explanation by collecting
 Explain the proof of the Pythagorean the homework from the night before and
grade how well they understood. This will
Theorem and its converse.
be a formative assessment and I will make
 Solve for a missing side of a triangle
notes on their explanation to help them but
using the Pythagorean Theorem.
 Solve a real world problem using the if they tried and turned it in on time, they
will receive full credit. The rest of the
Pythagorean Theorem.
outcomes will be graded as I am walking
around the classroom and assessing their
work ethic with their partner and by
grading their homework for this lesson.
Learning Targets
I will be able to:
 Explain the proof of the Pythagorean
Theorem and its converse.
 Solve for a missing side of a triangle
using the theorem.
 Determine when the theorem can be
used in the real world.
Student Voice
Students will use their voice when they are
explaining their reasoning to their partner.
They will also use their voice when they are
solving the problems with their partner and
I walk around and ask them about their
process.
Prior Content Knowledge and Pre-Assessment
Students need to know how to multiply, and do square roots to solve the Pythagorean
Theorem. They also need to understand the proof from the day before to begin doing
example problems. They also need to have a background of the properties of triangles.
Academic Language Demands
Vocabulary & Symbols
Language Functions






Pythagorean Theorem
Hypotenuse
Formula
Area
Obtuse
Acute



Describe
Explain
Demonstrate
Precision, Syntax &
Discourse
Mathematical Precision:
Syntax:
Discourse:
Students will be able to
accurately explain their
reasoning out loud with
proper vocabulary and
usage of academic language.
Language Target
Language Support
Students will be able to
explain the proof in their
own words and accurately
using correct vocabulary.
They will also be able to
explain the sides of the
triangles and their lengths
using correct vocabulary.
I will support them by
assisting with the vocabulary
as they say their explanations
to each other. Also, when they
are doing their worksheet I
will assist them in explaining
the triangles.
Assessment of Language
Target
Language will be assessed
formatively when they turn
in their explanations of the
theorem. Also, I will assess
how they are learning the
vocabulary as I walk around
and listen when they are
doing their worksheet.
Lesson Rationale (Connection to previous instruction and Objective Standards)
This lesson follows a lesson that discusses the proof of the theorem and determining area
based on the theorem. This lesson meets the first standard of the cluster,
CCSS.Math.Content.8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.
The lesson that this lesson plan is for follows this by moving into the procedure and solving
problems based on the theorem proved. The lesson that follows this one will have them
practice finding the missing lengths but in a coordinate plane. It will build on this lesson but
add a step with the coordinate plane involved.
Differentiation, Cultural Responsiveness and/or Accommodation for Individual
Differences
Students who are struggling will be successful because I have everybody with a partner.
This way, since there is only one of me, they can help each other before asking me for
assistance. Different cultures and individual differences will be handled individually if
having a partner to work with is not enough of an accommodation.
Materials – Instructional and Technological Needs (attach worksheets used)
Materials needed by the students are their notebooks and a pencil. They are allowed a
calculator if they have one but it is not required. Teacher materials are the worksheets and
examples to give to the students.
Teaching & Instructional Activities
Time
Teacher Activity
Student Activity
Purpose
5 min Walk around and observe
Share their explanation of Help each other better
explanations
proof to peer
their understanding of
the proof
30
Do 3 examples on the board
Write down examples in
Prepare them for their
min
and assign homework
notes and homework
worksheet
assignment
15
Walk around and assist with Work on worksheet with
Practice problems and
questions
a partner until class ends better their
understanding of how
the theorem is used
Learning Progression Formative Assessment
Complete this worksheet to show your planning and the thinking behind your learning
progression. This activity is intended to help you organize your learning progression and
reveal information that your instructor can use to support your learning progression
writing. Prompts 1-3 must be completed by Jan. 24 and prompts 5-6 must be completed by
Jan. 31.
1. Identify a math textbook and grade level for your learning progression
[ The math textbook I have chosen is Geometry by Siegfried Haenisch. I will be doing my
learning progression for the Excel Geometry class which is grade levels ninth through
twelve. ]
2. Identify the CCSS Math domain and cluster for your learning progression
[ The domain I have chosen for this learning progression is “Understand and apply the
Pythagorean Theorem.” The cluster contains the standards



CCSS.Math.Content.8.G.B.6 Explain a proof of the Pythagorean Theorem and its
converse.
CCSS.Math.Content.8.G.B.7 Apply the Pythagorean Theorem to determine unknown
side lengths in right triangles in real-world and mathematical problems in two and
three dimensions.
CCSS.Math.Content.8.G.B.8 Apply the Pythagorean Theorem to find the distance
between two points in a coordinate system. ]
3. Use the CCSS Math resources (Standards, Published Learning Progression, math
textbook, and web) to write an outline of math activities and benchmark
assessments for each CCSS Math in the CCSS Math cluster.
[ Day 1:
 Draw the picture of the Pythagorean Theorem as a square. Discuss how the triangles
within it explain the theorem. Show the basic proof on the board and go through it
with the class. (20 min).
 Do 2 examples of how to apply the Theorem together on the board.
 Give them the Workbook Activity 86 to do in class. What they don’t finish is
homework.
 HOMEWORK: Finish in class worksheet. Write their own explanation of the proof to
say to a peer and turn into me as a formative assessment of how well they
understood it.
Day 2:
 Students will pair up and read their explanations from the day before to their peer.
(5 min)
 We will go through three examples of how to apply the Pythagorean Theorem
including application problems.
 With the same partner they will complete a worksheet together.

HOMEWORK: Finish worksheet and write down three examples of how the
Pythagorean Theorem could be used in the real world.
Day 3:
 Students will begin with a warm up on the board of the last 2 days.
 I will put the Example on page 292 under the doc cam and ask them to talk with
their group on the steps to solve it. We will go through the steps one at a time as a
class and apply the Pythagorean Theorem.
 They will work in their groups to solve #6-10 on page 294 in their notebooks. Once
they are checked off, they can move on to the next activity.
 I will put a map of Ellensburg with a coordinate system on it. They will get a sheet
of questions about distances on the map and they will use the Pythagorean Theorem
on a coordinate plane to solve it.]
4. Write the learning progression narrative in the same format as the Published
Learning Progression: The narrative is an explanation about how the conceptual
understanding, procedural fluency, and math reasoning aspects of the CCSS Math
will be taught in a connected way using math activities, leading questions, and
benchmark assessments. The explanation should explain the purpose of the
activities and how the benchmark assessment will be used in the progression of
activities. Similar to the Published Learning Progression, your learning progression
should have the narrative on the left –hand side and details about the math
activities, benchmark assessments, and CCSS Math on the right-hand side.
5. Identify one activity in your progress and write a lesson plan for implementing that
activity.
[ Below is the lesson plan for day 2. ]
6. Steps for planning a formative assessment process in your lesson plan:
a. Select a formative assessment technique.
[ I will look at the students personal explanation of the Pythagorean Theorem and its proof.
I will use it to determine how well they understood the formula and how its proof. If they
seem like they understood the background of the theorem I will continue with the
conceptual understanding and procedural fluency with activities and worksheet. ]
b. How will the formative assessment technique be used to support student
learning of the CCSS Math?
[ The first standard in the cluster requires students to understand the proof of the theorem.
By asking them to explain it in their own words I can directly check if they understood my
explanation of the theorem. By having it be formative, students will not feel like I am testing
them after one day of material but I am still getting information on their learning process. ]
c. How will the formative assessment technique be implemented? (How will
students be introduced to the technique and what materials are needed?)
[ I will explain the technique as a homework assignment that needs to be completed alone
so I can monitor how well they learned the material of the day. I will assure them they are
not being graded on how well they explain it but rather how much effort they put into the
assignment. This will hopefully eliminate them getting online and getting perfect
explanations which would ruin the purpose of the formative assessment. They will only
need a paper and pencil to complete this assessment. It will be assigned as homework so
they can time to think about it alone. ]
d. What adjustments will be needed for special populations of students?
[ If students are unable to do the assignment as assigned, they can type it if they wish. Any
other adjustments that need to be made will be made with the student privately. If they can
not complete this as homework, they can come in during the lunch hour or before school
and talk to be about their explanation instead. If a student does not speak English and are
unable to explain their explanation to their partner, an adjustment will be made. ]
e. How will the formative assessment data be analyzed to support student
learning and guide instruction?
[ I will read the explanations and determine if the students wrote the key parts down. I will
use this information to decide if they understand the background of the theorem. Further
instruction will be based on if they understood the reasoning behind it. I do not want to
focus solely on computations. The students need to understand the proof. ]