UNIT TOPIC: Constructions/Extension and Application
This unit comes after a chapter on
triangles and polygons. The constructions part of the unit will involve
constructing various shapes/figures the students have been working with and
incorporates using triangle congruence to prove their constructions are valid.
The Extension and Application part of the unit provides a review of concepts
that have been previously covered in a new and challenging way. My CT developed
the plans for the constructions part of the unit and I developed the plans for
the Extension/Application part of the unit.
UNIT CONTEXT
Subject/Content Area Mathematics
Course Geometry
Grade Level 912
Length of Unit: This unit will last one
week. There will be four class periods, three that are 55 minutes long and one
that is 115 minutes long.
FACTS ABOUT THE LEARNERS
Whole Class Information
·
Number of students in class: There are 36 students in period 4 and 37
students in period 5.
·
Demographic Information: Period 4: 18 girls, 18 boys.
Ethnicity25 students “Yes, Hispanic or Latino”. English Learners3 students at
the intermediate level, 2 students at the advanced level. 16 students in the
class who have been redesignated. Special Education3 students with IEPs. Period
5: 17 girls, 20 boys. Ethnicity20 students “Yes, Hispanic or Latino”.
English Learners 2 students at the early advanced level, 8 students who have
been redesignated. Special Education 5 students with IEPs.
·
Developmental Needs: The students enjoy and do well in small groups
working on problems at the large whiteboards available for each group. Many of
the students play sports or are involved on campus in some way. The students
respond well to smaller group activities where they are working together,
rather than lecture based or heavy individual practice lessons. There are many visual
learners in the classroom and students who respond better to handson
activities. Creating a safe environment where students feel free to share their
perspectives and ask questions without the fear of being laughed at or told
they are wrong, is important with these classes.
Individual Student Information and Differentiation Strategies
Provide the following information
for 5 specific students
·
What is the students’ name? Elena
·
What is the level of your English Learner? Early Intermediate
English Learner
·
Elena is a 10^{th} grader. She is from Mexico and has a
large extended family. She moved to the U.S one and half years ago and visits
her grandparents and other family members and friends in Mexico during the
summer.
·
What are the student’s individual ed. goals? Reading, writing
& subject levels? Based on her grades, she is highly motivated to succeed,
so her individual goals are to work toward a higher level of English
proficiency, as well as continuing to develop her content knowledge.
·
She is literate in Spanish and often reads Spanish literature. Her
report cards from school in Mexico indicate above average grades. She is shy
but works well in small groups.
·
What can you do to differentiate each student’s
o
Content. I will provide her with a list of any new vocabulary
ahead of time. In our classroom, we have a word wall that includes graphics, so
this will help her with using vocabulary when doing problems in class, as well
as when she completes exit tickets and reflections.
o
Process. Allow her to work with a bilingual partner.
o
Product. Assess her at the i+1 level and focus on her ability to
express her understanding through reflection and answering simple questions.
(Readiness)
o
Affect. Give feedback through written and oral comments to allow
her to understand and track her progress. (Readiness)
o
Learning Environment. It will be best if Elena can sit next to a
bilingual student, so arranging seats in pairs and small groups will help to
(Readiness/Learning Profile)
Based on their developmental needs (readiness,
interests and learning profile)?
·
In
order to obtain evidence of Elena’s learning, I would choose to assess her on
the informal formative assessments in which she would have the ability to show
me what she is working on, as well as use simple communication to tell me the
process she took to get there. This would give her an opportunity to practice
her oral English, and I can adjust the way that I assess her based on her CELDT
level.
·
Elena
is very motivated, so she will work hard to achieve success in the class. The
next steps I would take to help Elena progress would be to encourage her to
keep practicing her English while working in small groups. She tends to be a
shy person, but gets along well with others and since we have created a risk
free environment for our class, students will not criticize her if she does not
explain things correctly right away. If she starts to practice in the small
groups, she can work up the confidence to present her thinking to the class in
the future.
·
What is the students’ name? Larissa
·
What is the level of your English Learner? Intermediate English
Learner
·
Larissa is an 11^{th} grader. She was born in the U.S and
grew up speaking Spanish with her family. Her mother speaks more English now,
so she speaks English primarily at home now.
·
She would like to do well in school and take as many classes as
possible because she would like to be a nurse and knows that she must take and
pass certain classes in order to do this.
·
Readiness: Able to learn and communicate new
ideas and vocabulary with images and clear statements/definitions. She has
received average grades in past math classes. She has retaken Algebra, but is
confident in her ability now. She is motivated and asks many questions for
clarification.
Learning Profile: Visual and auditory preferences. Shy, so
prefers individual help or working in a small group, rather than speaking to
the class/asking questions. Notetaking has helped keep her organized and
allows her to have something to refer back to and understand during group and
individual practice.
Interest: Spending time with
friends and family. Fashion.
·
What can you do to differentiate each student’s
o
Content: Allow her to use notes when working on in class as well
as exit tickets and reflections. Notes really help her to organize her thoughts
and remind her of the vocabulary words used to describe certain mathematical
concepts.
o
Process: Larissa works well in groups, but is shy, so allowing her
to work in small groups helps her to feel more comfortable.
o
Product: Assess her at the i+1 level.
o
Affect: Larissa asks a lot of questions and seeks feedback often.
Giving her frequent oral as well as some written feedback which she can refer
back to, will be helpful in moving her toward the learning goals.
o
Learning Environment. She works well with others so placing her
near a student who is willing to help her is effective. Her younger sister is
in the same class, they do not work particularly well together so keeping them
apart is best. (Learning Profile)
Based on their developmental needs (readiness,
interests and learning profile)?
·
I
would choose the informal formative assessments to assess Larissa’s progress so
that she has the time and ability to explain her thinking to me in the best way
that she can, in a lowpressure situation.
·
Larissa
is giving a lot of effort in this class and would really like to do well,
allowing her to use notes and offering assessments other than formal written
assessments will help her to feel more comfortable and more successful.
·
What is the students’ name? Alex
·
What category does the student qualify for special education
services? Specialized Learning Disabilityliteracy skill, including the
acquisition of sound/symbol relationships and word identification.
·
Alex is a 10^{th} grader who is
·
Alex would like to improve his use of vocabulary.
·
Alex reads at the 7^{th} grade level and struggles with
decoding words. He has asthma. He is a selfisolating person and does not
readily join grouplearning situations.
·
What can you do to differentiate each student’s
o
Content: Provide new vocabulary ahead of time and allow Alex to
see and use the word wall with graphics for reference. Make a sheet similar to
the word wall that he can use at home as well.
o
Process: Since the class is group based, allow Alex to have a more
independent role in the group, suggesting he be the recorder of information.
o
Product: Encourage him to speak to you about his understanding,
and allow him to turn in his work and assess him, knowing about his learning
difficulties.
o
Affect: Written feedback will be best for Alex, as he does not
respond well to interaction.
·
I
would choose to assess Alex’s exit tickets and reflections since he will not
readily share his thinking orally.
·
Encouraging
Alex to take a more active role in groups will help him move forward. We have
created a risk free environment where all students should be accepting of
others ideas.
·
What is the students’ name? John
·
What
category does the student qualify for special education services? Special day
class
·
John is an 11^{th} grader who is involved in a special day
class for students with severe learning disabilities. He has a supportive home
environment, in which his parents support his learning goals.
·
John would like to improve his understanding of geometry, and use
vocabulary to explain what he is saying. He has trouble using the words that he
would like to explain the process that he uses to solve problems.
·
Readiness: Algebra
skills are sufficient to learn new concepts and apply old knowledge. He is in a
special day class.
Learning Profile: Visual
preferences. Pictures and organizers are very helpful. Notes/organizers to
refer back to help to keep thoughts and new ideas organized and are helpful to
refer back to on future assignments. Does not always do well in small group
situations.
Interest: Art, especially drawing, and reading.
·
What can you do to differentiate each student’s
o
Content: Giving organization to notes will help John to make sense
of concepts, and allowing him to have access to vocabulary before a lesson,
will help him.
o
Process: Allowing him extra time to work on things, so that he can
get help in his resources class, will help him feel the support he needs to
succeed.
o
John does well on more formal tests, as he is not great at
verbalizing his understanding and is better at relating concepts on paper.
o
John has a tendency to speak out in class, so keeping him close to
the front of the room, will allow the instructor to speak with him readily.
·
The
formal formative assessments will be best to display John’s progress towards
learning goals. He does not verbalize his understanding as well as he can write
it, so allowing him to display knowledge on the exit tickets and reflection,
will be best for him.
·
Making
sure that John feels comfortable and welcome in the classroom will be the best
way to ensure his success. If he feels that he is not welcome or wanted in the
class, then he will not perform. Making sure to continue the environment
created where all students have a chance to contribute, will help John feel
comfortable.
·
What is the students’ name? McKinley
·
What
category does the student qualify for special education services? McKenna has
an IEP plan based on her Specialized Learning Disability
·
Describe student’s grade level, culture, language, SES, family,
affect. McKenna is an 11^{th} grader who speaks English only. She comes
from a large middle class family where she is always encouraged to read and do
her homework, but is rarely checked in on or receives little help with
homework. She is a social person, who is very involved on campus through
organized activities and ASB.
·
What are the student’s individual ed. goals? Reading, writing
& subject levels? McKenna is a very hard worker and is motivated to succeed
in all subjects. She enjoys English the most, and feels that she struggles in
math because of her past experiences.
·
Describe developmental needs (readiness, interest, & learning
profile) for each student. McKenna’s learning profile suggests that multiple
modalities will be the most successful for her. She will benefit from
directions being clarified.
·
What can you do to differentiate each student’s
o
Content: She has trouble relating concepts, so allowing her to use
notes on assignments is necessary.
o
Process: Clarifying directions is necessary for McKinley’s
success.
o
Product: Informal assessments are best for her. Allowing her to
work on the test in another room or have extended time is necessary.
o
Affect: Both oral and written feedback will be best for her.
Learning Environment: Allowing McKinley to work
in groups will help her to feel comfortable with the material.
·
Informal
formative assessment will be best to check for McKinley’s understanding. She
gets anxiety with more formal tests and often seeks clarification on many
problems because she feels she cannot ask questions. Allowing her to verbalize
her understanding through these less formal assessments will be best for her.
·
McKinley
is highly motivated to succeed and will work very hard to get a good grade in
the class. Allowing her to talk through and verbalize her understanding
whenever possible and assessing her as much in this way, will help her feel
successful.
Unit Rationale: Enduring Understandings & Essential Questions
This unit comes at a great point
in the year, as we have just finished a unit on triangle congruence, which
incorporated a lot of review from the first two units, which covered angles,
segments, properties of parallel lines and polygons. This unit combines
constructions with a review of previous concepts in a way that challenges
students to apply knowledge to more challenging problems than they have seen
before. It is important for the students to truly understand these concepts and
how they affect the real world, as they start to pick up on where they see
these concepts outside of the classroom. This unit gives use a chance to
explicitly link concepts to the student’s lives and challenges them to think
about geometry in a new way. Having students think through problems and label
their thinking by justifying their answers is a goal of the course that is
particularly important in this unit.
Enduring Understandings (EU)
At the end of this unit, I hope that
students have a better understanding of the core concepts of the class by
applying their previous knowledge and thinking to solve complex problems.
Students should be able to label the steps that they took to come to the
conclusion.
Essential Questions
·
What are the ways we apply old knowledge to new concepts?
·
Is it possible to look at a picture and answer the questions we
have by solving mathematically?
·
Where in the real world do we see/notice concepts from class?
·
How can we use real world examples of geometry to model what we
have covered in class?
·
How can we construct what we have been learning and prove that we
know it is correct?
·
What other shapes can we construct using knowledge of how to
construct basic figures?
Reason for the Instructional
Strategies & Student Activities
STANDARDS
Content Standards
5.0 Students prove that triangles are congruent or similar,
and they are able to use the concept of corresponding parts of congruent
triangles.
7.0 Students prove and use theorems involving the properties
of parallel lines cut by a transversal, the properties of quadrilaterals, and
the properties of circles.
12.0 Students find and use measures of sides and of interior
and exterior angles of triangles and polygons to classify figures and solve
problems.
13.0 Students prove relationships
between angles in polygons by using properties of complementary, supplementary,
vertical, and exterior angles.
16.0 Students perform basic
constructions with a straightedge and compass, such as angle bisectors,
perpendicular bisectors, and the line parallel to a given line through a point
off the line.
ELD Standards
Listening and Speaking: Cluster 7 EA. Respond to messages
by asking questions, challenging statements, or offering examples that affirm
the message. Cluster 9EA. Prepare and deliver brief oral presentations/reports
on historical investigations, a problem and solution, or a cause and effect.
Writing Strategies and Applications: Cluster 5EA. Write
reflective compositions that explore the significance of events.
UNIT OBJECTIVES
Constructions:
1. After seeing an example
modeled, while following along, students will be able to individually construct
an equilateral triangle, copy of a segment, copy of an angle, an angle
bisector, the midpoint, a perpendicular bisector (through any point, a point on
a line, or a point not on the line) using a straight edge and compass, and will
be assessed on their ability to construct these through both formative and
summative assessments. (Standard 16.0, L&S Cluster 7) (Cognitive)
2. After the previous unit on
triangle congruence, and following along with instructor led “labeling of
thinking” students will be able to write a proof of various constructions
including, angle bisector, copying an angle and perpendicular bisector and will
be assessed on their ability to understand and apply their knowledge to
complete the proof on their own. (Standards 5.0,16.0) (Cognitive)
Extension/Application:
3. Using previous knowledge of
theorems and properties of lines, triangles and polygons, students will discuss
with group members, and label the steps taken to solve for missing parts in complex
figures and all group members will be asked to explain their thinking to group
members or to the class, in order to assess their deeper understanding.
(Standards 7.0, 12.0, L&S Cluster 7 and 9) (Cognitive)
4. After working together in small
groups to come to conclusions about missing information using a “pass the pen”
method to allow all students to give input, students will individually display
their knowledge on a smaller version of these types of problems, indicating the
vocabulary and knowledge they needed to solve. (Standards 7.0, 12.0, 13.0,
L&S Cluster 7 and WS&A Cluster 5)
ASSESSMENT PLAN
·
Name
of Assessment: Constructions
Understanding (Objectives 1 and 2)
·
Formality: Informal
·
Type: Formative
·
Purpose: To assess student understanding and
ability to explain process. Instructors will walk around checking for
understanding by looking at student work and asking students to explain the
steps they took.
·
Implementation
Method: Written
(Constructions) and verbal (Explanations)
·
Communication
of Expectations: Modeling
by instructor to show expectation of steps, student samples to show
expectations of results.
·
Evaluation
Criteria: Correct
markings indicate steps taken to get result, formal proof indicates understanding
of thinking by whole class.
·
Feedback
Strategies: Oral
feedback and supplemental instruction individually
·
Student
SelfAssessments:
Students will check their construction with the student sample provided.
·
Name
of Assessment: Constructions
Quiz
·
Formality: Formal (30 points)
·
Type: Formative
·
Purpose: To check for student’s
understanding when doing constructions individually, and how well they can
apply knowledge of proofs to prove their construction is valid.
·
Implementation
Method: Written
(Constructions)
·
Communication
of Expectations: Modeling
and practice in class.
·
Evaluation
Criteria: Correct
markings indicate steps taken to get result, formal proof indicates
understanding of applying previous knowledge to constructions.
·
Feedback
Strategies: Written
feedback on quiz to be returned to students.
·
Name
of Assessment: Extension/Application
Understanding
·
Formality: Informal
·
Type: Formative
·
Purpose: Assess students on ability to
provide correct answers for missing parts as well as justifications for their
answers. Instructors will walk around to groups or individuals to check for
correctness and ask students to explain and write their thinking. If appropriate,
have students present to the class or in smaller groups to have them practice
using vocabulary.
·
Implementation
Method: Written and
verbal.
·
Communication
of Expectations: Examples
of student work shown to model the methods of “labeling thinking”.
·
Evaluation
Criteria: Rubric
provided in resources.
·
Feedback
Strategies: Oral
feedback
·
Student
SelfAssessments:
Rubric provided in resources.
·
Name
of Assessment: Extension/Application
Exit Ticket
·
Formality: Formal (Credit is given on the
rubric scale)
·
Type: Formative
·
Purpose: Check for student’s use and
understanding of labeling thinking in order to justify their answers for
missing parts.
·
Implementation
Method: Written
·
Communication
of Expectations: Group
work done in class provides expectations, as well as the rubric created for the
extension/application portion of the unit.
·
Evaluation
Criteria: Rubric is
provided in resources.
·
Feedback
Strategies: Written
feedback on exit ticket to be returned to students.
·
Student
SelfAssessments:
Rubric is provided in resources.
Into: (Monday, 10/15)
Objectives/Standards
Students will use prior knowledge
to look at a complicated figure and record things that they notice and things
that they wonder to get oriented with the picture and contribute to class
discussion about what students recorded, then students will work in groups to
talk through and label their thinking, providing reasons for the conclusions
they make. (Standards 7.0, 13.0, L&S Cluster 7)
Student Activity
·
Hook: Challenge students to think about the figure (provided in
resources) mathematically, asking them to record what they notice and what they
wonder.
·
This activity will be directly related to things that they have
seen in the past. There are many connections to previous concepts that students
will notice. The notice/wonder introduction activates this prior knowledge by
giving them the freedom to recall what they want to.
·
To begin the overall into lesson, students will record what they
notice/wonder. Then, the instructor will provide numbers for five of the angles
and students will work in groups to fill in the rest of the angles. They will “pass
the pen” between group members filling in one angle at a time and writing in
their justifications in the order in which they solve.
·
The room is arranged with white boards around the room that groups
of four to six are assigned to work at for group work. Seating is changed every
two weeks, so groups are switched up regularly.
·
Transitions have been practiced, so students know that they have
only a certain amount of time before they will be expected to be at their boards
starting the problem. If student behavior becomes an issue, I will address it
in the small groups, so that the behavior issues do not distract other groups.
If it will help, I will rotate groups or switch members to create a more
productive environment.
·
Questions to prompt learning: What specifically are you struggling
with? What is the next step you should take? How do you know that (point to
something specific)? Justify your thinking, how can you explain what you have
done so far, and how can you use this information to continue.
·
Unit Preview: A short discussion about using mathematics to prove
things that we notice about figures or things in the real world will be
conducted as a short conclusion before students are told to complete this for
homework so that we can come to final conclusions about what students were
wondering from the picture.
Assessment
The homework will be to finish
this star problem, labeling and solving, and it will be checked for
completeness the next class. Also, a discussion about what students had
wondered will be debriefed, using the answers that they found on their
homework.
Through:

Day 1: M (10/15)

Day 2/3: T/W (10/16 and
10/17)

Day 4: Th (10/18)

Day 5: F (10/19)

Content Standards

Geo 7.0 Students prove and
use theorems involving the properties of parallel lines cut by a transversal,
the properties of quadrilaterals, and the properties of circles.
Geo 13.0 Students prove
relationships between angles in polygons by using properties of
complementary, supplementary, vertical, and exterior angles.
Geo 16.0 Students perform
basic constructions with a straightedge and compass, such as angle bisectors,
perpendicular bisectors, and the line parallel to a given line through a
point off the line. Listening and Speaking: Cluster 7 EA. Respond to
messages by asking questions, challenging statements, or offering examples
that affirm the message.

Geo 7.0 Students prove and
use theorems involving the properties of parallel lines cut by a transversal,
the properties of quadrilaterals, and the properties of circles.
Geo 16.0 Students perform
basic constructions with a straightedge and compass, such as angle bisectors,
perpendicular bisectors, and the line parallel to a given line through a
point off the line.
Listening and Speaking:
Cluster 7 EA. Respond to messages by asking questions, challenging
statements, or offering examples that affirm the message.

Geo 5.0 Students prove that
triangles are congruent or similar, and they are able to use the concept of
corresponding parts of congruent triangles.
Geo 16.0 Students perform
basic constructions with a straightedge and compass, such as angle bisectors,
perpendicular bisectors, and the line parallel to a given line through a
point off the line.
Listening and Speaking:
Cluster 7 EA. Respond to messages by asking questions, challenging
statements, or offering examples that affirm the message.

Geometry 12.0 Students find and use measures of
sides and of interior and exterior angles of triangles and polygons to
classify figures and solve problems.
Geo 16.0 Students perform
basic constructions with a straightedge and compass, such as angle bisectors,
perpendicular bisectors, and the line parallel to a given line through a
point off the line.
Listening and Speaking:
Cluster 7 EA. Respond to messages by asking questions, challenging
statements, or offering examples that affirm the message. Cluster 9 EA. Prepare and deliver brief oral
presentations/reports on historical investigations, a problem and solution,
or a cause and effect.

Learning Objectives

After following along with
instructor led examples, students will be able to use a straight edge and
compass to construct an equilateral triangle and copy a segment, using notes
taken during example.
Students will use prior
knowledge to look at a complicated figure and record things that they notice
and things that they wonder to get oriented with the picture and contribute
to class discussion about what students recorded, then students will work in
groups to talk through and label their thinking, providing reasons for the
conclusions they make.

After following along with
instructor led examples, students will be able to use a straight edge and
compass to copy and angle, construct an angle bisector and determine the
location of a midpoint, using notes taken during example. Students will also
follow along while instructor labels thinking on a proof of how we know the
methods of construction for copying an angle and constructing an angle
bisector are valid, then students will be able to individually write up their
formal proofs for this to include in their notes.
Students will use prior
knowledge of parallel lines and properties of a triangle to record their
observations about a complicated figure and solve for certain missing pieces
while providing justifications on their individual whiteboards while the
instructor monitors for correct answers and reasoning.

After following along with
instructor led examples, students will be able to use a straight edge and
compass to construct a perpendicular line and perpendicular bisector, in
order to prove that their constructions are right, they will use the thinking
labeled on their board and create their own proof.
After working in groups to
complete a problem that requires finding values of missing angles and
justifying thinking, students will be able to individually complete a similar
task as an exit ticket.

After a week of taking notes
on and practicing constructions, students will be able to complete a quiz
covering three topics they have seen previous days.
After learning about
properties of polygons and having some time to think about and note important
aspects of the figure presented, students will be able to answer the question
”How many sides does the polygon that is cut off have?” using discussion with
peers, direct questioning and a stepbystep approach to solving.

Student Activity

Students will use the
straight edge and compass to follow along while the instructor models how to
copy a segment and construct an equilateral triangle. Then students will try
these constructions on their own.
Students will record what
they notice/wonder about the figure. Then, the instructor will provide
numbers for five of the angles and students will work in groups to fill in
the rest of the angles. They will “pass the pen” between group members
filling in one angle at a time and writing in their justifications in the
order in which they solve.

Students will use the
straight edge and compass to follow along while the instructor models how to
copy an angle and construct a angle bisector, and determine the location of a
midpoint. Then students will try these constructions on their own.
Students will briefly write
down what they notice and wonder about the figure on the board. Then, the
instructor will indicate which angle they should give the value for. Students
will write the value of this angle and how they know that the answer they
have written is true, based on information given in the figure.

Students will use the
straight edge and compass to construct a perpendicular line after seeing an
example from the instructor.
Students will split up into
groups to find the missing angles of a figure given by the instructor. They
will discuss and “pass the pen” so that all members of the group have a
chance to contribute to the solution. One group will be asked to present
their findings and justify them in order to allow the whole class to assess
their knowledge as a group. Then, students will copy down a sketch of the
homework and will be told to write three mathematical facts that they know
about the shape in the picture. Finally, students will complete an exit ticket
as an assessment.

Students will write on the
board the ideas that they came up with for homework, investigation of the
pentagon. Then students will complete a short quiz on constructions.
Students will write on their
papers the things that they notice and wonder about the figure that is
presented. Then, they will share some of the things that they came up with to
contribute to the class discussion. Then students will split up into groups
to answer the question, “How many sides does the cut off polygon have?” Once
students have come to a conclusion in groups, one groups will present the way
that they solved the problem.

Assessment

The homework will be to
finish this star problem, labeling and solving, and it will be checked for
completeness the next class. Also, a discussion about what students had
wondered will be debriefed, using the answers that they found on their
homework. (Student samples of this homework are provided in the resources,
graded with rubric)

Whiteboard answers will be
assessed for correctness through an informal formative assessment. Instructor
will be informed of overall understanding and misconceptions and students
will be able to self assess when correct answer is presented. Also, students
will complete an exit ticket that is a simpler version of the parallel lines
problem, in which they will solve for missing angles and “label their
thinking” using vocabulary. (graded with rubric)

Students will complete an
exit ticket in which they must solve for missing angles and justify how they
got their answer by “labeling their thinking”, using vocabulary. (graded with
rubric)

Students will reflect on the
concepts and knowledge that they needed in order to answer this question.
Students will use vocabulary words in their reflections. (Student samples are
provided in resources.)

Closure/Beyond:
The lesson plan provided below is
the closure/beyond lesson.
The objectives, standards,
activities and assessment are presented just above here in the unit calendar,
and in more detail in the lesson plan below.
·
In this lesson students will make meaning of the concepts that
have been covered in the unit by developing their own questions about a figure,
through a notice and wonder activity, and then they will use mathematics to
answer the question that they developed. Throughout the unit students have been
using vocabulary and including this in their exit tickets, as well as
throughout the lesson by “labeling their thinking”. The quiz on constructions
will also allow students to summarize what they have learned so far and they
must include a proof in order to incorporate their prior knowledge.
·
The “product” that I will collect from students as evidence of
their learning is the reflection that they write at the end of the class that
summarizes what they needed to know in order to solve the problem that they
worked on in class. For constructions, the quiz is collected, graded and
returned in order to get an idea of where students are in understanding
constructions and give them feedback about how they are meeting the
expectations of the course.
·
The next unit will be on quadrilaterals. Students have been using
their knowledge and vocabulary to find the values of missing angles and label
the process that they took to get those answers by including justification.
This will carry over into the next unit, as students will be finding missing
angles and sides of quadrilaterals and will be using a similar approach.
MATERIALS/RESOURCES
All materials that are needed for
the unit are included. It is also necessary to have a class set of compasses as
well as rulers or some kind of straight edge.
Lesson Plan:
1. TITLE OF THE LESSON
Assessment of
constructions and polygon extension
as a part of a unit on constructions, and extension and application of
previously covered topics.

2. CURRICULUM AREA &
GRADE LEVEL
Geometry, grades 912

3A. STUDENT INFORMATION:
English Language Learners
Lupe
1.) Readiness Level: Intermediate
level. 11^{th} grader. Able
to learn and communicate new ideas and vocabulary with images and clear statements/definitions.
2.) Learning Profile: Visual
and auditory preferences. Shy, so prefers individual help or working in a
small group, rather than speaking to the class/asking questions. Notetaking
has helped keep her organized and allows her to have something to refer back
to and understand during group and individual practice.
3.) Interest: Spending time
with friends, fashion. Wants to be a nurse when she is older.
Ana
1.) Readiness Level: Intermediate
level. 12^{th} grader. She speaks Spanish at home with her family.
She has previously taken SEI classes. In 10^{th} grade she took SEI
Algebra 1 and failed both semesters. In 11^{th} grade she took
Algebra 1 and passed one semester and failed the other. She has the skills
necessary to pass this class, she need to focus and not get worked up when it
comes time to take a test.
2.) Learning Profile: She is
a visual/auditory learner and works well in groups.
3.) Interest: She spends a
lot of time with her friends and family and she would like to get a job after
she graduates so that she can pay for college.
Melissa
1.) Readiness Level: Intermediate
level. 11^{th} grader. She speaks both English and Spanish at home.
In 10^{th} grade she took Algebra 1 and passed the first semester and
failed the second semester. This class is a challenge for her because she
doesn’t have a very strong background in Algebra, which is necessary to apply
the concepts in geometry. She will benefit from some extra Algebra help.
2.) Learning Profile: She is
a Kinesthetic learner and prefers to have problems explained slowly while she
follows along and then trying one on her own.
3.) Interest: She likes
spending time at the beach with her friends and she would like to go to
college for 8 years to become a dentist.
Martin
1.) Readiness Level: Early
Advanced level. 11^{th} grader. He speaks Spanish with his family. He
passed two semesters of Algebra 1 in 8^{th} grade, failed two
semesters of Geometry in 9^{th} grade, and then took Algebra 1 again
in 10^{th} grade and passed one semester and failed the other. Since
this class is a repeat of what he has seen, he will benefit from clear
definitions to reach his prior knowledge and clear explanation of
expectations so that he can be successful.
2.) Learning Profile: Visual/auditory
learner. He will benefit from clear visuals of figures and their properties.
3.) Interest: In his free
time he enjoys drawing and lifting weights. He would like to be a firefighter
or go into the military when he graduates from high school.
Eduardo
1.) Readiness Level: Early
Advanced level. 12^{th} grader. Starting learning English in
preschool. Continues to speak both Spanish and English at home. In 9^{th}
grade he took SEI Algebra and did not pass either semester. In 10^{th}
grade he continued on to Geometry, but failed both semesters, so in 11^{th}
grade he took Transitional/Basic math and passed both semesters. He is
passing the class so far this year, but will need to work hard to keep his
grade.
2.) Learning Profile: He is a
visual learner, and he is motivated to do well. He works well individually,
and benefits from being able to ask clarifying questions.
3.) Interest: After school,
he enjoys playing soccer and boxing. He is close with his family and enjoys
spending time with them.
Christian
1.) Readiness Level: Advanced
level. 12^{th} grader. He speaks Spanish at home with his family. In
9^{th} grade he took SEI Algebra and failed both semesters. In 10^{th}
and 11^{th} grade he took Algebra 1 and he passed one out of four
semesters. He seems to have developed a good understanding of Algebra through
taking those classes, and he is determined to pass this class in order to
graduate.
2.) Learning Profile: He is a
visual learner. He works well in groups and is a good leader.
3.) Interest: He enjoys
hanging out with his friends and is working hard to graduate at the end of
this year.
Daniela
1.) Readiness Level: Advanced
level. 11^{th} grader.
She speaks Spanish at home with her family. In 9^{th} and 10^{th}
grade she took Algebra 1 and failed all four semesters. She has trouble with
the Algebra concepts that are needed for this class, so she will benefit from
extra help on making sure she keeps her process of solving organized.
2.) Learning Profile: She is
a visual learner and benefits from pictures and clear, repeated explanations.
3.) Interest: She likes to
play soccer and draw in her free time and she would like to join the Navy
after high school.

3B. STUDENT INFORMATION: Students
w/ Special Needs
Victor
1.) Readiness Level: Algebra
skills are sufficient to learn new concepts and apply old knowledge. He is in
a special day class.
2.) Learning Profile: Visual
preferences. Pictures and organizers are very helpful. Notes/organizers to
refer back to help to keep thoughts and new ideas organized and are helpful
to refer back to on future assignments. Does not always do well in small
group situations.
3.)
Interest: Art, especially drawing,
and reading.
Ernesto
1.) Readiness Level: He has a
bleeding heart problem in which he takes medicine for. He has trouble
focusing on the tasks that he is assigned and can often be a distraction to
himself and the other people around him.
2.) Learning Profile: He has
visual and auditory preferences, with an emphasis on auditory, as he does
better when there is constant prompts about new vocabulary and the use of new
vocabulary in context.
3.) Interest: He enjoys
playing baseball and hanging out with his friends. He would like to continue
school in order to become a mechanic.
Sydney
1.) Readiness Level:
2.) Learning Profile: She is
a visual learner as well as has a preference for hands on activities in which
she can make connections to the real world.
3.) Interest: She is very
interested in baking and would like to go to culinary school in order to own
and run her own bakery which she would like to travel with. She enjoys
hanging out with her friends and she is also involved in the drama department
as a behind the scenes technical manager.
Chyanne
1.) Readiness Level: 10^{th}
grader. Has severe vision problems, instead of 20/20 she has 4/4 and easily
gets dizzy when she must concentrate to read or understand something that she
cannot easily see. She will need larger print documents and may need
assistance seeing certain aspects of the lesson.
2.) Learning Profile: She is
a kinesthetic learner and learns best when she is able to use manipulatives
and participate in hands on activities.
3.) Interest: She enjoys
making origami and art in general. She also loves animals, especially horses.
Javier
1.) Readiness Level:
2.) Learning Profile: He is a
visual and auditory learner and benefits from clear, repeated explanation.
3.) Interest: In his free
time, he wrestles, runs and plays football.
Isaac
1.) Readiness Level: He is
. He has behavior issues in
other classes, so it will be best to make sure that he continues to be an
exemplary student in this class.
2.) Learning Profile: He is
an auditory learner. He benefits from listening to others use vocabulary in
order to master it himself. He will also do well when using the vocabulary in
group situations, as he is a social person who likes to talk, but knows how
to stay on task.
3.) Interest: He would like
to own his own shoe company and he likes to play football and basketball in
his free time.

4. RATIONALE
A. Enduring
Understanding: Students will make
connections to previous knowledge and concepts covered in class by applying
it to a more abstract problem. Students will understand that using a
stepbystep approach, which they have used for proofs, they will be able to
solve even more challenging problems.
B. Essential
Questions: How can we apply our old
knowledge to new concepts? Is it possible to look at a picture and answer the
burning questions that we have by solving mathematically? How can we use real
world examples to model concepts we have covered in class?
C. Reason for
Instructional Strategies and Student Activities: The unit created on constructions was only meant to
last half of each class period, so that students don’t get too overwhelmed.
The other half of each class was planned to cover previous topics. It was decided
that these “minilessons” would be a good opportunity to use some more
challenging extension problems that involve many topics we have covered, but
also get students to step back and really think about what they have learned.
By allowing the students to think about the problem during the notice and
wonder activity of this day’s lesson, students will gather their own thoughts
and ideas before splitting up in groups to work on the problem. Having them
work in groups to discuss and challenge each other’s ideas as well as
encouraging students to ask very specific questions of the instructors and
their peers, will help them to think about the challenging task.


5.
CONTENT STANDARD(S)
Geometry
12.0 Students find and use
measures of sides and of interior and exterior angles of triangles and
polygons to classify figures and solve problems.
Geo 16.0 Students perform
basic constructions with a straightedge and compass, such as angle bisectors,
perpendicular bisectors, and the line parallel to a given line through a
point off the line.

6.
ELD STANDARD(S)
Listening
and Speaking: Cluster 7 EA. Respond to messages by asking questions,
challenging statements, or offering examples that affirm the message.
Cluster 9 EA. Prepare and
deliver brief oral presentations/reports on historical investigations, a
problem and solution, or a cause and effect.

7. LEARNING GOAL(S) 
OBJECTIVE(S)
A. Cognitive
After learning about
properties of polygons and having some time to think about and note important
aspects of the figure presented, students will be able to answer the question
”How many sides does the polygon that is cut off have?” using discussion with
peers, direct questioning and a stepbystep approach to solving.

8. ASSESSMENT(S)
A. Diagnostic/Entry
Level: A class discussion/debrief
about the homework assigned for the previous night, an investigation of what
is known about the pentagon, will give a good idea of what students were able
to recall about polygons and the measures of their angles.
B.
FormativeProgress Monitoring: As
students are working on the problem, we will monitor their conversations and
the questions they are asking in order to get an idea of what they understand
and what they still need work on.
C. Summative: A final class discussion will determine if students
were able to solve the problem. Also, individual reflections on what was
needed to solve this problem will be collected. These will both give a good
idea of the understanding of the whole class.

9A. EXPLANATION OF
DIFFERENTIATION FOR
ENGLISH LANGUAGE LEARNERS
1.) Content/Based on Readiness, Learning Profile or
Interest
This
student works well in groups. She is shy, but she benefits from hearing what others
have to say and asking questions in order to clarify concepts. Having the
students work in groups to solve this difficult problem, will be a helpful
differentiation strategy for her.

9B. EXPLANATION OF
DIFFERENTIATION FOR
STUDENTS WITH SPECIAL
NEEDS
1.) Content/Based on Readiness, Learning Profile or
Interest
This
student tends to want to do his own thing when he is in a group setting. He
will have difficulty with the complexity of this problem. By encouraging all
students to record what they know about the figure beforehand and then
suggesting they focus on one aspect at a time, will help him to focus energy
and knowledge. I will also keep a close eye on his group during this activity
in order to make sure he is on task and not getting too overwhelmed. I will
have hints to give him when he is able to tell me what he is stuck on, or I
will allow him to quickly “spy” on another group.

10. INSTRUCTIONAL STRATEGIES
Constructions:
A.
Anticipatory
Set/Into
As students are walking in, ask
them to write some of the mathematical things that they noticed about the
pentagon investigation that they did for homework the previous night.
(included in the resources) Pass out materials needed for constructions while
students are putting their work on the boards. (35 minutes)
B.
Independent Practice/Through
Instruct students to use the
straight edge and compass to complete the quiz. (included in the resources)
Tell students who finish early to write down more things that they know about
pentagons on the back of their quiz. (10 minutes)
Extension
and Application:
A.
Anticipatory
Set/Into
a. As a class, debrief the
information that students have put up on the board. Specifically focus on the
mathematical things that were noticed by students. Lead this short discussion
to offer a short background relating to the next problem. (35 minutes)
b. Put up the problem
(provided in the resources) on the document camera for students to make sense
of. Say nothing except “What do you notice? What do you wonder?” Allow them a
short time to write down these things (3 minutes)
B. Instruction/Through
Conduct a short discussion
to highlight some of the main points that were noticed about the figure.
Also, allow students to share what they are wondering about the picture. If
the discussion does not naturally progress to students talking about how many
sides the cutoff polygon has, then prompt this discussion. The goal is to
end with directing them to work in groups to find the answer to the question “How
many sides does the cut off polygon have” and provide concrete justification
for their conclusion. (5 minutes)
C. Guided Practice/Through
Before allowing them to
start, emphasize the importance of following the stepbystep approach that
we have been using in order to keep track of thinking and make the problem
more manageable. Tell them to start with what they know, or noticed in the
figure. Monitor groups while they are working on this question, making sure
they “pass the pen” and encouraging them to ask specific questions when they
get stuck. If students feel that they don’t know how to proceed with the
problem, they can send a “spy” to one of the other groups to just take a look
and see if they can add anything that might help them. These “spies” cannot
bring a paper or pencil and they must be able to justify any information that
they gather. Tell students about this option only if they seem that they need
it. (1015 minutes)
D.
Closure
Debrief the polygon problem
as a whole class in order to develop certainty as a group about the number of
sides that the cutoff polygon has. (8 minutes)
E. Beyond
Have students write a
reflection including three things that they needed to know to answer the
question that was posed today.
Collect these responses to
get an idea of the concepts students connected to the problem. (3
minutes)

11. STUDENT ACTIVITIES
Constructions:
A.
Anticipatory
Set/Into
Students will put their
findings from the previous night’s homework on the board as they come into
class. (35 minutes)
B.
Independent
Practice/Through
Students will use straight
edge and compass to complete the quiz. When they are finished they will write
down more things that they know about pentagons on the back of their paper. (10
minutes)
Extension and
Application:
A. Anticipatory Set/Into
a. Students will listen and
contribute to a conversation about what they wrote about the pentagon
picture. (35 minutes)
b. Students will write down
things that they notice and things that they wonder about the figure that is
put up on the document camera. (3 minutes)
B.
Instruction/Through
Students will contribute to
the class discussion about things that they noticed about the figure that was
presented. They will also share what they are wondering about the picture as
well. (5 minutes)
C.
Guided Practice/Through
Students will work in groups
by discussing and “passing the pen” to answer the question of the day that is
a result of the class discussion. If they are stuck, they will need to ask
their peers or the instructors, very specific questions. The students may
send a “spy” from their group if prompted to get more information. The spy
must not bring a pencil or paper. Students must have justification for all
claims made to get their final answer. (1015 minutes)
D.
Closure
Students in each group will
share what they got for their final answer with a short justification of how
they know it is true. The class will agree on a conclusion for the number of
sides that the cut off polygon has. (8 minutes)
F.
Beyond
Students will write a
reflection in which they write three things that they needed to know in order
to answer the question that was investigated in class. (3 minutes)

12. RESOURCES
Constructions: Compass, Ruler, Plain Paper
Extension and Application:
Previous night’s homework
Pentagon investigation (First picture below), Polygon Problem (Second picture
below)

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