Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 130044
Proving quadrilaterals algebrically using slope and
distance formula
Working in groups, students will prove the shape of various quadrilaterals using slope, distance formula, and polygon properties. They will then
justify their proofs to their classmates.
Subject(s): Mathematics
Grade Level(s): 9, 10
Intended Audience: Educators
Suggested Technology: Computer for Presenter, LCD
Projector
Instructional Time: 90 Minute(s)
Keywords: parallel slope, perpendicular slope, coordinate proof, rectangle, rhombus, parallelogram, isosceles
trapezoid, square
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
proofs key pg 3.jpg
proofs key pg 4.jpg
proofs key pg 5.jpg
Proofs key pg 5 additional work for problem 9.docx
proofs key pg 6.jpg
proofs key pg 7.jpg
Warm up - Shapes brainstorm.xlsx
Warm up - Shapes brainstorm answers.xlsx
Using parallel and perpendicular slopes.pptx
group worksheet.docx
Using parallel and perpendicular lines homework.pdf
Using parallel and perpendicular lines homework key.pdf
proofs key pg 1.jpg
proofs key pg 2.jpg
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?
Using coordinates for slope and the distance formula, along with properties of quadrilaterals, students will algebraically prove that a figure is a square, rhombus,
rectangle, trapezoid, or parallelogram.
Students will justify their solutions.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should know:
page 1 of 5 how to use the slope formula:
relationships between the slopes of parallel and perpendicular lines
parallel lines have same slope
perpendicular lines have opposite reciprocal slopes
properties of various quadrilaterals (warm-up answer sheet in attachments)
how to use the distance formula:
how to use the format of two-column proofs: Statement/Reason columns
Guiding Questions: What are the guiding questions for this lesson?
How do we describe the relationship between two lines?
Where can we see examples of parallel and perpendicular lines in the real world? Are they always vertical or horizontal?
Teaching Phase: How will the teacher present the concept or skill to students?
Before class:
Place student desks in groups of 3 to 4 with student names on each grouping.
Pre-arrange groups by ability level, if preferred.
It is suggested that posters be placed around the room for students to write answers during group work.
Place brainstorm work sheet and one marker at each desk (be sure each group has a different color).
Suggested ways to pre-group students - Create groups of 3 to 4 students of similar mathematical understanding.
Chapter pre-test
Teacher observation
Previous year's testing data (least helpful)
Another method you choose
During class:
Display slide number 2 as students come into class as Warm up.
Have one brainstorm worksheet already placed at each group of desks.
Students should sit in assigned groups.
Direct students to list, as a group, as many different properties for each shape as they can remember. Use the brainstorm worksheet.
Give them 2 -3 minutes.
Groups should be instructed to travel to posters around the room and record one property on each poster that has not already been recorded. Give students about 5 to
7 minutes.
The teacher should monitor to ensure all groups have written on each poster. This is easy to do since groups should have different colored markers.
The teacher brings the class together to discuss the posters as a whole group (10 minutes).
Review each poster.
Be sure that all needed properties are listed (see brainstorm key in attachments).
Be sure the concept of slope is discussed when discussing parallel and perpendicular sides.
Parallel sides should be on all posters. Ask students how they know that lines are parallel (they have the same slope).
Ask the same question about perpendicular (their slopes are opposite reciprocals).
Ask students to explain what opposite reciprocal means. (fraction is flipped and has the opposite sign)
Next, have students prove slide 3 is a rectangle using these properties. It is suggested that you use a two-column proof, but any type of proof that you are
emphasizing in this lesson will work. It is also fine to allow students to use a variety of different types of proofs. (Answers Attached: Proofs Key pg1) (10 minutes): Ask
students:
Make a conjecture for the shape that you see. (rectangle)
Are all the properties of a rectangle needed to prove that this is a rectangle? (no - only the properties in red on the brainstorm key are needed)
Which property can we prove first? (students should suggest slopes)
The teacher continues working through the two column roof on proof key pg 1. Be sure to ask for input from students from each group.
Once this is competed, students will move on to guided practice.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Day 1:
Group proof keys begin at the bottom of Proof Key pg 1 and end on Proof Key pg. 4.
Give each group one of the handouts (groups worksheet) with the various shapes.
Instruct students to make their conjecture first by just looking at the shape.
They should write this conjecture in the blank provided before they begin the proof.
Walk around to be sure that each group has an answer in the blank on the worksheet.
The teacher should be aware of any incorrect conjectures. Be sure to circulate back around to check that they amend their conjecture after they find all the
information.
Encourage them not to erase, just write the new answer at the bottom of the page or to the side of the conjecture line.
A good side lesson here is that mistakes are okay; that is why we go through the proof phase (or check our work).
page 2 of 5 some common mistakes:
Group 1 is likely to pick rhombus instead of square. Be sure to circulate back around to check that they amend their conjecture after finding that the slopes are
opposite reciprocals.
Group 2 is likely to choose rectangle. The slopes are not opposite reciprocals, however, so it is a parallelogram, but not a rectangle.
Groups 3 and 4 are more straight forward, therefore a little easier.
Group 5 may say trapezoid, but may miss the isosceles part initially. Be sure they prove that the non-parallel sides are the same length.
Students should be encouraged to use properties used in the class example. (slope and distance formulas) Circulate, allowing them time to complete their proof.
the slope formula is used to prove parallel and perpendicular sides
the distance formula is used to show congruent and non-congruent sides
Use the posters to prompt groups if they get stuck. Ask leading questions to ensure that students have all needed steps to solve the actual shape, especially if their
conjecture was off.
Students should put their proof on something large enough to bring to the front of the room when justifying their proof. Ideas:
Poster paper is good for this.
If you have individual white boards, these could work but groups may need more than one depending on the size of the board. Storage of student work may be
needed overnight.
Groups could write on paper and transfer to the board when explaining their work. If you use this option, you may not have time for each group to present.
Another option is to have groups verbally explain, using the pictures to guide their explanations.
This will probably bring you to the end of class, so you will need to use materials that will be available the following day when students are asked to justify to the
group.
Day 2:
Display slides 4-8.
Have groups explain their proof as their polygon is displayed.
Encourage the other groups to ask questions of their peers. I sometimes write questions on scraps of paper and assign groups to ask these, forcing the presenting
group to clarify specific points.
Once each group has presented (about 25 minutes) move to independent practice (slide 9).
This lesson could be shortened to one day if the teacher identifies one proof that is well done and highlights it for the group, or if students briefly describe their proofs
before moving to independent practice.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
After presentations, students should complete the problem on slide 9 individually. Students can remain in groups but work individually. The teacher should circulate
during this time to see that students have all steps (about 5 to 7 minutes). Answer: Opposite sides are parallel, opposite sides are congruent, but slopes are not
opposite reciprocals so shape is a parallelogram but not rectangle (proof on Proofs Key pg 7).
A homework worksheet, Using Parallel and Perpendicular Lines is also included in the attachments. It includes some review practice with slope as well as problems
from this lesson. It would be fine to assign the review on day 1, and the remaining for the second night. Along with an answer key, work for problems 8-11 can be
found on Proofs Key pg 5, 5 additional work, and 6.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher should call attention to the front of class and ask for volunteers to review important points of the lesson and activities.
The teacher should project slide 10 for a Summative assessment.
Exit ticket - explain how you can use slope and distance formula to algebraically prove the shape on PowerPoint slide 10 (about 5 minutes).
students should find the slope of each side (2/3, 2/3, -3/2, -3/2) and state that opposite sides are parallel and adjacent sides are perpendicular so the shape is
either a square or a rectangle.
The student should use the distance formula to find the lengths of the sides (all sides are the square root of 13) and determine which it would be (square).
Summative Assessment
Students individually complete the problem on slide 10.
Exit Ticket - Explain how you can use the slope and the distance formula to algebraically prove the shape on PowerPoint slide 10.
Students should find the slope of each side (2/3, 2/3, -3/2, -3/2) and state that opposite sides are parallel and adjacent sides are perpendicular so the shape is
either a square or a rectangle.
The student should use the distance formula to find the lengths of the sides (all sides are the square root of 13) and determine which it would be (square).
Formative Assessment
The teacher will assess prior knowledge during the Warm-Up.
The teacher uses questioning during the teaching phase to see what misconceptions may still remain.
During the guided practice, the teacher should monitor groups to ensure conjectures are correct and assist as needed.
During the independent practice problem (slide 9), the teacher circulates and monitors for errors.
Feedback to Students
There will be a class discussion on the properties listed on the posters during the round-robin.
page 3 of 5 During group work, the teacher circulates and poses questions and listens to student discussions, correcting misconceptions when needed.
During individual practice, the teacher circulates, looking for slopes of two sets of parallel lines, checking for perpendicular lines (none), and checking the length of
congruent line segments.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Some shapes are easier to prove than others. Strategically group students so they are working with a shape at their level of ability.
Group 1 is likely to pick rhombus instead of square. Be sure to circulate back around to check that they amend their conjecture after finding that the slopes are
perpendicular.
Group 2 is likely to choose rectangle. The slopes are not opposite reciprocals, however, so it is a parallelogram, but not a rectangle.
Groups 3 and 4 are more straightforward, so they are a little easier.
Group 5 may say trapezoid, but may miss the isosceles part initially. Be sure they prove that the non-parallel sides are the same length.
Extensions:
Give students 3 points and ask them to drawba fourth in order to make a given quadrilateral.
(0,4), (4,7), (7,3) square - answer: (3,0)
(0,3), (2,0), (0,-2) isosceles trapezoid - answer: (-3,0)
(0,0), (2,0), (3,3) parallelogram - answer: (1,3)
Ask students to choose 4 points that will create a quadrilateral and then name the quadrilateral they created.
Suggested Technology: Computer for Presenter, LCD Projector
Special Materials Needed:
Groups worksheet or graph paper and list of ordered pairs
PowerPoint
Homework worksheet
Further Recommendations:
If wanting to practice plotting points, you could give students graph paper and ordered pairs instead of pictures.
Additional Information/Instructions
By Author/Submitter
Applicable Math Practices:
MP 1.1 Make sense of problems and persevere in solving them.
MP 6.1 Attend to precision.
SOURCE AND ACCESS INFORMATION
Contributed by: Michelle Faulkner
Name of Author/Source: Michelle Faulkner
District/Organization of Contributor(s): Leon
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
Description
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined
by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle
centered at the origin and containing the point (0, 2).
MAFS.912.G-GPE.2.4:
Remarks/Examples:
Geometry - Fluency Recommendations
page 4 of 5 Fluency with the use of coordinates to establish geometric results, calculate length and angle, and use geometric
representations as a modeling tool are some of the most valuable tools in mathematics and related fields.
page 5 of 5