Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 130889
To Be or Not to Be a Parallelogram
A lesson where students apply parallelogram properties and theorems to solve real world problems. The teacher models a problem solving strategy,
which involves drawing a picture, highlighting important information, estimating and/or writing equation, and solving problem (P.I.E.S.).
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Interactive Whiteboard, LCD
Projector, Overhead Projector
Instructional Time: 1 Hour(s)
Keywords: parallelograms, rectangles, rhombi, squares
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
Warm-up.docx
Warm-up (Answer Key).docx
Exit Ticket.docx
Exit Ticket (Answer Key).docx
Solving Problems Worksheet.docx
Solving Problems Worksheet (Answer Key).docx
P.I.E.S. - Word Problem Solving Strategy.docx
Parallelogram Family Visualizer.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to:
Use and apply theorems about parallelograms to solve real-world problems
Use a problem-solving strategy called P.I.E.S.
P stands for picture representing the problem
I stands for important information
E stands for estimate the answer and write an equation to solve the problem
S stands for solve the problem
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should be able to:
Solve multi-step algebraic equations
Use angle addition postulate to find angle measures
page 1 of 4 Apply line and angle relationships, i.e. supplementary angles
Use slopes of parallel and perpendicular lines to prove sides of quadrilaterals parallel or perpendicular
Describe and explain properties and theorems of special quadrilaterals
Apply the Pythagorean Theorem
Convert between inches and feet
Guiding Questions: What are the guiding questions for this lesson?
Is a rectangle a parallelogram? Is a parallelogram a rectangle? Why or why not?
Is a rhombus a parallelogram? Is a parallelogram a rhombus? Why or why not?
Is a square a rectangle? Is a rectangle a square? How do you know?
What are similarities and differences between rhombi, rectangles, and squares? What makes each special?
What are the minimal requirements to justify that a parallelogram is a rectangle?
What is special about consecutive angles of a parallelogram? Explain.
Teaching Phase: How will the teacher present the concept or skill to students?
As a preparation to this lesson, the teacher will write guiding questions on the board before students enter the classroom, and there will be a poster explaining the
word solving problems strategy P.I.E.S. (or could be written on the board). The students will pick up their materials and "Solving Problems Worksheet" before sitting in
their seats.
Warm-up activity (10-15 min.) - Students will review parallelogram properties and theorems by a structured True/False response board activity lead by the
teacher. Students are not allowed to speak to each other. Each student has a white board, marker, paper towel, and "Parallelogram Family Visualizer" (optional).
Students will listen to the "Warm-up" questions (attached) and answer on the board and wait for teacher direction to show boards. After each question is read by the
teacher (which can be displayed on a document camera or overhead projector one at a time), a few minutes is given for the students to respond, then says "Boards
up!" and quickly looks around the room, saying "Yes, the answer is..." giving the correct answer and positive praise. Usually students will realize why they were wrong,
but if too many students get the wrong answer, then this is a time to address any questions/misconceptions the students may have. This process is repeated until all
the questions have been answered by the students, on their boards.
Guided Practice (30-40 min.) - Transition to this part of the lesson should be smooth. Students will place their boards, marker, and paper towel under the desk,
and they should already have the handout on the desk ready to begin. The teacher can display the "Solving Problems Worksheet" (attached) via document camera,
overhead projector, or interactive board to be viewed by all, as teacher facilitates problem solving. This portion of the lesson is teacher-directed so students will be
participating in a structured whole group discussion.
For question #1, the teacher will ask a student to volunteer to read the directions, then he/she will ask;
"What is the first step when solving this problem using the P.I.E.S. strategy?"
The teacher says, "Notice that there is a picture, so we can proceed to..." Allows students to finish the teacher's sentence.
Students should then highlight/underline information necessary to solve the problem with teacher guidance and feedback.
The teacher proceeds to ask another volunteer to re-read the question - this time making the appropriate marks on the picture to indicate the given information.
The teacher continues with the following guiding questions: "If this picture is a parallelogram, what do you know about the consecutive angles? Opposite angles?
Opposite sides? Why?"
"What equation can we derive to find the measure of angle NOP? How do we know?" The teacher will accept and discuss responses. If students are having
difficulties answering, the teacher may ask probing questions like, "What two angles make up angle NOP? So, together those two angles make up what angle? What
postulate will assist us in finding the measure of angle NOP?" After discussing the angle addition postulate students should realize that measure of angle NOM +
measure of angle MOP = measure of angle NOP.
Then students will substitute values and solve-showing all steps required as the teacher solves it, as well. The teacher may then ask if there are any questions that
need to be addressed and discuss how to check the answer (if it is numerical).
For #2, which is to find the measure of angle MPO, teacher will ask;
"What is the relationship between angle NOP and angle MPO? Are they opposite? Consecutive? Explain." Since students have labeled the congruent angles, they
should notice that the angles are consecutive, which makes them supplementary, therefore measure of angle NOP + measure of angle MPO = 180 degrees. After
substituting the values in the equation, then students will solve and check.
For #3, which is to find the measure of angle MNO, the teacher will state:
"Now that we know the measure of angle MPO and measure of angle NOP, which theorem will help us find the measure of angle MNO? Explain."
Listen to students' responses and scaffold if necessary. Some students may use the theorem that states that
opposite angles in a parallelogram are congruent; therefore, measure of angle MPO = measure of angle MNO.
For #4, which is to find the length of segment MN, the teacher asks;
"Now, what theorem will help in finding the length of MN? How do you know?" Students should use the theorem that states that opposite sides of a parallelogram
are congruent, therefore MN=PO. Since PO is given, then the length of MN is equal to the length of PO.
For #5, which is to find the length of MP and NO if the perimeter of the figure is 100, the teacher states;
"Perimeter is..." and allows students to respond. "If the perimeter is 100 and the length of PO=30, what theorem will help us write an equation to solve this
problem?" Students should realize that opposite angles of a parallelogram are congruent, so if PO=30, then MN=30.
The teacher continues to guide and says;
"We have the lengths of a pair of opposite sides-now what other lengths do we need to include in order to make up that perimeter of 100? Are those sides
opposite? Are they congruent? Which theorem can you apply to assist you in solving this problem?" After scaffolding, students should derive that the equation is MN
+ PO + MP + NO = 100. Students will then substitute all the values to solve and check.
For #6, teacher will ask a volunteer to read the problem, then another to re-read. Using the P.I.E.S word problem solving strategy and teacher prompting, students
will sketch a picture to visually represent the problem. Then teacher will guide discussion on what information to highlight, underline, or circle. Students will state
page 2 of 4 relationships of the missing angles with the given angle along with an explanation as to how they know, which will result in an equation to solve and check.
Instruct the students to continue working on the "Solving Problems Worksheet" until the last few minutes of the class period (allow 3-5 minutes for the Exit Ticket).
They may work with a partner. To ensure everyone's participation, instruct students to show all work. The teacher will circulate around the room, checking for
progress and giving feedback.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
During the guided practice, students will be completing 6-7 problems from the "Solving Problems Worksheet." Discuss as a class the problems that students struggled
with. When the teacher asks a question to the group, students should raise their hand and wait to be called on. Then for the next question, the student that previously
answered picks someone else to answer and that continues until all students have given some type of input. For those students who don't get it, and they are chosen,
it's acceptable for them to ask a question or the teacher will ask guided questions to this student, to get them thinking in the right direction. In addition, a word
problem solving strategy will be modeled when completing this worksheet where students sketch a picture to represent a situation in the word problem; highlight or
underline relevant information; make an estimate of the answer; derive an equation; then solve (P.I.E.S.).
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
(20-30 min.)
The students will continue working on the same worksheet independently or with a partner, as the teacher walks around the room and assists each student with
questions as needed.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
(4-8 min.)
The class will come together for a brief discussion (students are listening during this part of the lesson) by asking the class to think about parallelogram properties and
theorems and ask, "How did these help in solving word problems?" After a few minutes of discussion, the teacher will ask a student to restate and describe the
strategy that helped in solving the word problems.
Summative Assessment
The summative assessment will be a two question exit ticket to be turned in for a grade before class ends to determine if students have demonstrated mastery of this
concept. The worksheet given as guided and independent practice may be finished as homework for a grade as well.
Formative Assessment
The teacher will lead the warm-up activity, which is a response board activity where students will review parallelogram properties and theorems through a series of
true/false questions, replacing the false word to make the statement true. The questions can be displayed via a document camera or overhead projector, one at a
time.
The teacher will model and guide students through solving real world problems using several questions from the worksheet while discussing guided questions to
ensure understanding.
During guided practice, after solving various problems with teacher, students will have an opportunity to solve one problem with a partner on their response board
to show comprehension.
The teacher will circulate around the room to facilitate completion of the worksheet as students work in pairs and answer questions as needed.
Feedback to Students
Feedback is given to the students on an on-going basis.
Beginning with the response board activity where the teacher can give immediate verbal feedback as students recall and discuss the learned concept.
During the guided practice, the teacher facilitates a question and answer session where feedback is immediate. The students demonstrate that they can apply
parallelogram properties and theorems to solve problems from the worksheet and use the P.I.E.S. word problem solving strategy.
The graded exit ticket will be returned to the students as feedback the next day. Questions or concerns are addressed at that time.
Any misconceptions will be addressed. For example, stating that all rectangles are squares, when the converse statement is the true statement. Rectangles are
sometimes squares. Another misconception is that in some parallelograms the diagonals appear to bisect the opposite angles, but this is not a property of
parallelograms. Students should be cautioned that diagonals do bisect angles in rhombi and squares.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Pair struggling students with a peer
Enlarge worksheets for visually impaired students
Have students with visual difficulties get a blank paper and cover every other problem to focus on the one they are working on
Shorten assignment(s) at your discretion (ie. instead of 20 of the same type of question, 8-10 can be used to show mastery)
Let students use a calculator
Students that finish early can complete enrichment assignment or can peer tutor at teacher's discretion
Extensions:
Find the area and perimeter of parallelograms from the worksheet
Make tessellations in GeoGebra (http://www.geogebra.org/) using parallelograms
Use a Gizmos lesson (https://www.explorelearning.com/) so students can manipulate parallelograms to form rectangles, rhombi, and squares and compare angle
and side measures
Make a collage on a poster board with pictures of real life objects that are parallelograms, rectangles, squares, and rhombi - with labels and properties that make
them what kind of parallelogram they are.
Suggested Technology: Document Camera, Computer for Presenter, Interactive Whiteboard, LCD Projector, Overhead Projector
Special Materials Needed:
page 3 of 4 Copies of "Solving Problems Worksheet" and "Exit Ticket" (one per student)
White response boards (one per student)
Dry erase markers (one per student)
Paper towel to wipe the boards (one per student)
Rulers/straightedges (one per student)
Students can write responses on an index card if there are no boards. Also, if the teacher does not have an interactive board, worksheets can be viewed and
problems worked out on an overhead projector.
Further Recommendations:
For higher-level classes the teacher can assign a project involving this unit, area, and perimeter.
"Parallelogram Family Visualizer" is included as additional review worksheet or reference.
Additional Information/Instructions
By Author/Submitter
This lesson may take more or less than one hour depending on your class. The teacher can adapt/modify as needed.
This lesson aligns with the following Standard for Mathematical Practice:
MAFS.K12.MP.1.1 - Make sense of problems and persevere in solving them.
SOURCE AND ACCESS INFORMATION
Contributed by: Mary Gonzalez
Name of Author/Source: Mary Gonzalez
District/Organization of Contributor(s): Miami-Dade
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.3.11:
Description
Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include:
opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.
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