Understanding By Design Unit Template
Title of Unit
Pythagorean Theorem and other Properties
of Right Triangles
Mathematics - Geometry
Lizzie Nash, Phil Ewing
Curriculum Area
Developed By
Grade Level
9th grade
Time Frame
January 28th – February 14th
Identify Desired Results (Stage 1)
Content Standards
Common Core State Standards:
G-SRT: 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
G-SRT: 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-GPE: 1.Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle
given by an equation.
Understandings
Essential Questions
Overarching Understanding
Overarching
Students will understand…
U1. How to apply the Pythagorean Theorem to problems.
U2. The combinations of side lengths that create a right triangle (by using the PT)
U3. How to use the Pythagorean Theorem to solve multi-step problems involving
area, perimeters and other applications.
U4. How to justify the Pythagorean Theorem by using triangle similarity
U5. How to find the distance between two points by using the PT and the
coordinates
U6. How to find missing side lengths on special triangles
Related Misconceptions
M1. The students might not understand what side is designated as the hypotenuse
(also known as c^2)
M2. Students might not understand what makes a theorem a theorem (why is it
any different then other true statements?)
M3. Students might forget that a right triangle has to have a 90 degree angle
M4. How to plot points on a coordinate plane (x,y) vs. (y,x)
M5. Confusion between perimeter and area
M5. Applying the Pythagorean theorem in more complex problems (word problems,
not knowing when to plug in the numbers)
OQ1. When is the Pythagorean
Theorem used in real life?
OD2. How do we know the
Pythagorean Theorem is true?
OD3. How will we find these
formulas with the information
given?
OD4. How do you know a right
triangle is a right triangle?
OD5. What is a theorem? What
is the point of a theorem?
Objectives
Topical
TQ1. How do you use the
Pythagorean Theorem?
TQ2. How can I use the
information I know about
triangles to see if it is a right
triangle?
TQ3. What is difference
between triangles and right
triangles? Does the PT still work
for triangles with no right angle?
Why doesn’t it?
TQ4. What makes these
triangles special?
TQ5. How do we know the PT
works? What makes it that way?
TQ6. What are the different
aspects of right triangles that
are important to knowing it’s
side lengths work?
Knowledge
Skills
Students will know…
Students will be able to…
K1. How to state the Pythagorean Theorem
K2. How to determine whether three side lengths form a right triangle
K3. How to use the Pythagorean Theorem to solve multi-step problems
involving area, perimeters, and other applications
K4. How to justify the Pythagorean Theorem
K5. How to find the distance between two points using their coordinates and
the Pythagorean Theorem
K6. How to find missing side lengths of special triangles (30-60-90 and 45-4590) using only one side length.
S1. State and use the Pythagorean Theorem in problems to find
side lengths of right triangles
S2. Define theorem
S3. Explain the uses of theorems
S4. Identify right triangles
S5. Determine whether three sides lengths form a right triangle
S6. Explain the phrase “if and only if” to others
S7. Interpret when “if and only if” would be used
S8. Create converse statements from a given statement
S9. Use the Pythagorean Theorem in problems involving multiple
steps to find the solution.
S10. Use the Pythagorean Theorem to find area and perimeter of
triangles or quadrilaterals
S11. Teach others how to prove the Pythagorean Theorem
S12. Find the distance between two points using their coordinates
and the Pythagorean Theorem
S13. Find side lengths of special triangles (given one side length)
Assessment Evidence (Stage 2)
Performance Task Description
Goal
Role
Audience
Situation
Product/Performance
Standards
To have students understand each learning objective in this unit by applying the Pythagorean theorem in complex
ways, learn more about (special) right triangles, finding missing side lengths with given information, finding the
distance between two coordinates by using the Pythagorean theorem, learning how to write converse statements
and understand what a phrase means by “if and only if”.
To accurately assess student understanding on each learning objective, to facilitate productive class discussions, to
help students discover aspects of the Pythagorean Theorem
9th Grade (Regular/Non-Honors) Geometry Students
Skill Checks after each Learning Objective is completed and the students have a good understanding for the
learning objective
When students receive a score “I can teach someone” on a skill check (none wrong) I can assess that they should
be able to help students around them who may be struggling. If most students do not receive above a yes I can
assess that we might need to take more time reviewing the lesson objective instead of moving on to the next
objective.
Listed above in content standards
Other Evidence
Students teaching other students will help me assess their high-level performance and I should be able to expect these students to perform well on the
unit test. The students who struggle on the skill checks and do nothing to retake them, will most likely struggle on the unit test and I will have to watch
them closely to make sure they receive help from the students who are teaching others (when working in small groups). Strategically making groups will
be essential.
Learning Plan (Stage 3)
Day in Unit
Lesson Topic
Lesson Learning
Objective
January 29
Introduce the PT
by introducing the
pattern
Have students understand the
pattern behind the PT and start
to practice using the
Pythagorean Theorem
How to state the Pythagorean
Theorem
January 30 (1/2
day)
Justifying the PT
Help the students understand
why this theorem works.
How to justify the Pythagorean
Theorem
January 31
Practice using the
PT
Review problems using the
theorem
February 1
After SC1, Look at
side lengths that
work for right
triangles
Have students understand when
side lengths work for a right
triangle and when side lengths
do not work.
February 4
Learning how to
apply the PT to
problems involving
perimeters
Applying the PT to
problems involving
area
To have students understand
the ways to apply the PT to
problems involving more then
just finding a side length
See how the students can use
the PT to find area
How to state the Pythagorean
Theorem, How to justify the
Pythagorean Theorem, (Using the
Pythagorean Theorem to solve
multi-step problems involving
whether three side lengths form
a right triangle)
How to state the Pythagorean
Theorem, How to justify the
Pythagorean Theorem, Using the
Pythagorean Theorem to solve
multi-step problems involving
whether three side lengths form
a right triangle.
Using the Pythagorean Theorem to
solve multi-step problems involving
area, perimeters and other
applications.
Using the Pythagorean Theorem to
solve multi-step problems involving
area, perimeters and other
applications.
February 5
Description of how lesson
contributes to unit-level
objectives
Assessment activities
Self-Assessment Game
Skill Check 1: Justifying the PT and
stating the PT
Favorite no on a story problem
about perimeter to see how they
understand the material
February 6
(1/2 day)
Apply their
knowledge of the
PT to the distance
between two
points
Apply their
knowledge of the
PT to the distance
between two
points
Discovering special
triangle properties
by using
geogebra?
Students should understand
how to find the distance
between two coordinates by
using the PT
February 11
Applying special
triangle properties
to problems
Students should utilize what
they know about special
triangles to solve problems
about special triangles
February 12
Discovering the
idea of the formula
of the circle (see
how the PT helps
you find the
formula)
Applying the
knowledge to use
in the formula of
the circle
To start to think about why the
formula works and how the PT
relates.
February 7
February 8
February 13
Students should understand
how to find the distance
between two coordinates by
using the PT
Students should start to think
about special triangles and
discover what makes them
special
To understand the formula and
why it works
Using the Pythagorean Theorem to
solve multi-step problems involving
the distance between two points
using their coordinates and the
Pythagorean Theorem.
Using the Pythagorean Theorem to
solve multi-step problems involving
the distance between two points
using their coordinates and the
Pythagorean Theorem.
Using the Pythagorean Theorem to
solve multi-step problems involving
finding missing side lengths of
special triangles (30-60-90 and 4545-90) using only one side length.
Using the Pythagorean Theorem to
solve multi-step problems involving
finding missing side lengths of
special triangles (30-60-90 and 4545-90) using only one side length.
Derive the equation of a circle given
the center and the radius using the
Pythagorean Theorem. (honors
will reach this, but regular may
need more time to learn the
other skills)
Derive the equation of a circle given
the center and the radius using the
Pythagorean Theorem.
Favorite no on the distance
between two points using
coordinates
Skill Check #2: G2. I can use the
Pythagorean Theorem to solve multistep problems involving…
a. area, perimeters and other
applications.
b. the distance between two points
using their coordinates and the
Pythagorean Theorem.
c. whether three side lengths form a
right triangle.
If they are ready for d, if not
they will be checked on skills ac(d. finding missing side lengths of
special triangles (30-60-90 and 4545-90) using only one side length.)
Favorite no at the end of class
February 14
Review
Review over unit learning
objectives
February 14
Unit Test
Unit Test Covers learning
objectives
This will help students get their
misconceptions about topics of
circles answered before they have to
take the test. Their questions will
help us assess their overall
knowledge for the topic.
Assessment over the entire unit
Favorite no about the biggest
misconception students have, and
maybe have a few problems that we
do this with from the review.
Test covering all of the learning
objectives from the Unit