Fostering Geometric Thinking – 3.3 Proving Lines Parallel

Unit # __3.3______

Group Members: Julianne Meier, Kelli Schoening, ShukriaGharzai, Kyle Schumann, Kristin Johnson, Matt James

Big Idea / Unit Topic Write Proof for Parallel and Perpendicular Lines

I. Lesson Information

Name of the Lesson/Objective : Proving Lines Parallel
Common Core Standards: CC.9-12.G.CO.9 Prove Theorems about lines and angles
Goals of Lesson
Mathematical:Know what angle relationships prove lines are parallel
Know why those angle relationships prove lines are parallel
Habits of Mind:Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Questions to Consider and Time Allotments / Detailed Lesson Components / Teacher Questions to Guide Learning / Expected Student Misconceptions/ Student Response / Method of Evaluation
Lesson Introduction / "Hook"
"The Hook" – an overarching problem or activity that helps point students toward the core idea of the lesson.
  • How are you launching the problem or activity?
  • What prior knowledge do your students need and how will you make that connection?
  • What advantages or difficulties do you foresee?
  • How are you making it personal and relevant to your students
/ ______minutes
  • Present students with the 1010 Office Building (optical illusion picture)
/
  • Are the lines in the picture parallel?
  • What are parallel lines? (Don’t tell students the answer yet!)
/
  • Student response: “Just look at the lines! They are not parallel, duh!”

Questions to Consider and Time Allotments / Detailed Lesson Components / Teacher Questions to Guide Learning / Expected Student Misconceptions / Method of Evaluation
Lesson Development
  • What mathematical connections are you making?
  • How are you using multiple representations, visual diagrams, symbols, problems, technology?
  • How are you engaging students in conceptual ideas that underlie the procedures to complete tasks and develop understanding?
  • How are you involving students in non-algorithmic thinking (e.g. more than just telling students the rule and students copying examples)?
  • What are you doing to provide students with opportunities to discuss mathematics?
  • How are you providing students with opportunities for metacognition (reflection)?
/ ______minutes
Materials needed:
-Patty paper
  • Have students partner up
  • Ask students to draw an example of parallel lines and non-parallel lines on patty paper
  • Ask students to discuss for 2 minutes why their lines are parallel or not parallel
  • Teacher will focus on the discussion of non-parallel lines to start
  • Use patty paper to get students to fold the paper one of two ways to get conversation started about a transversal (Fold either on top of itself or the line on top of the other line)
  • Now focus discussion on parallel lines
/
  • How can you be certain the lines you drew are parallel? Non-parallel?
  • Could you convince someone else that your lines are parallel using math language?
  • (When looking at non-parallel lines) What about looking at the lines make you think they are not parallel?
  • What makes lines parallel?
  • Teacher response to lines looking closer: “How can we figure out if the lines are actually closer on one side or not?”
  • Teacher response: “How can you measure the distance between lines? How do you have to hold the ruler compared to the lines?”
  • Teacher response: “Does it have to form a right angle with both lines or just one? Can it be at a right angle to both lines at the same time?”
  • Teacher response: “Do we want to take the time to extend the lines until they do meet? Efficient method?”
  • How can we show the distances between the parallel lines are the same without using a ruler?
  • How are you certain that you drew parallel lines?
  • How do you know? Do we know that if we extend the lines that they will never meet?
  • How do we actually extend lines forever?
  • How do we find the distance between your “parallel” lines?
  • How many times do we have to measure the distances between the lines to say that the lines are parallel? (Bring up with non-parallel lines we would only have to measure twice)
  • Why does this work?
  • Why does this prove the lines are parallel? (Could this be one that is a postulate? We like to have the postulate be the most general case so that it can prove as much stuff as possible.) Is there anything more general that we could do with the transversal?
  • Why does this work?
  • Why does this work? We call it a postulate because we can’t really prove that is works but we all agree that it does.
  • Besides corresponding, what are the other angle relationships that we have studied? Can we use these other angle relationships to prove the lines are parallel?
/
  • Students may say, “To be parallel, they have to never intersect and stay the same distance apart from each other. Those lines look like they are getting closer.”
  • Student response: “The lines look closer on one side than the other.”
  • Student response: “Measure the distance between the lines in two different places.”
  • Student response: “The ruler needs to form a right angle with the line.”
  • Student response: “At least one. If the lines are not parallel then you can’t make a line perpendicular to both lines.”
  • Student response: “If we extend the segments into lines then they will eventually meet.”
  • Student response: “If we fold the paper to try and make each line fold on top of itself, then they are parallel.”
  • Student response: “They will never intersect.”
  • Student response: “The distances between the lines are the same.”
  • Student response: “The ruler must be perpendicular to both lines”
  • Student response: “I can fold the paper to make each line fold on top of itself.
  • We created a perpendicular transversal.
  • We could fold the paper and make one line fold on top of the other line.
  • I can draw a transversal and match up some congruent (corresponding angles).
  • There are some! Alternate interior angles, consecutive interior angles, alternate exterior angles.

Questions to Consider and Time Allotments / Detailed Lesson Components / Teacher Questions to Guide Learning / Expected Student Misconceptions / Method of Evaluation
Guided Practice
  • How are you organizing the students to practice the mathematics?
  • How are you maintaining student engagement?
  • What are you doing to ensure that all students are involved?
  • What are you doing to provide students with opportunities to discuss the mathematics?
/ ______minutes
Materials needed:
- Angle pair cards (one set per group)
- Big sticky poster paper (one for each group)
- Markers
  • Get students in groups of 3 or 4
  • Teacher needs to draw on board:

  • Ask students to make a t-chart on the poster paper that has headings: Can prove parallel and Can’t prove parallel
  • Have students arrange the angle pair cards on the poster paper according to if the information on each card is enough information to prove the lines parallel or not
  • After students have agreed in their group on the placement of the cards, have groups put their poster on the board
  • Talk about posters that are different and have students justify the placement of the cards
  • Come to a classroom agreement on justifications/reasonings
  • Guided class discussion after activity
/
  • Which cards prove the lines are parallel? Justify your answer.
  • Which cards prove the lines are not parallel? Justify your answer.
Guided Discussion Questions:
  • Which angle pairs prove lines parallel and why?
  • IF you switched a card from a column, why did you switch? What convinced you?
  • Can we write conditional statements to represent what we have learned about proving lines parallel and angle relationships?
  • Will our conditional statements work in all pictures?
/
  • Students may think that an angle pair will prove lines are parallel when they don’t
  • They think that their angle pair does not prove lines are parallel when they do
/
  • Teacher monitor classroom group work followed by a class discussion
  • Looking at posters to check for understanding

Questions to Consider and Time Allotments / Detailed Lesson Components / Teacher Questions to Guide Learning / Expected Student Misconceptions / Method of Evaluation
Closure/Summary
  • How are you orchestrating the discussion/activity so that all students are able to summarize their thinking?
  • What are the mathematics and processes you are drawing out and emphasizing from the lesson?
/ ______minutes
  • Present students with a 3-2-1 exit ticket
  • Choose student to share parts of their exit ticket to the class
  • Assign students homework worksheet and if time present students with a challenge problem
/
  • Collect exit tickets as students leave the classroom