MthEd 377 Lesson Plan
Cover Sheet
Name: Jefferson Hall / Date: 10/28/05Section Title: The SSS Similarity Theorem (UCSMP Geometry 13.1)
Big Mathematical Idea(s): Proportionality between all the sides of two triangles implies similarity of those triangles (this is the opposite of similarity implies proportionality). When there are more than 3 sides, proportionality of sides does not imply similarity.
Why are these BMIs important mathematically? Because this principle only works for triangles, this is one of the several principles that make them unique. This and similar principles unique to triangles make trigonometry possible. Understanding the uniqueness of triangles in this respect is integral to understanding how trigonometry works.
How does this lesson fit into the overall unit? (i.e., How does this lesson build mathematically on the previous lessons and how do subsequent lessons build mathematically on it?) The previous unit deals with what similarity is and what similarity implies. This lesson builds on one of the principles of similarity i.e. similarity implies proportionality and this lesson changes the one way implication, in the case of triangles only, into an if and only if statement. The subsequent lessons explore more principles unique to triangles and, once the student is properly prepared, build trigonometry on those principles.
Grading rubric (for Keith’s use)
5 The Big Mathematical Idea addresses core mathematical concepts and is clearly articulated
5 Description of the importance of the topic is well thought out and relevant
5 There is a clear, insightful discussion of how this lesson fits in to the mathematical content of the overall unit
5 Lesson sequence is well thought out and detailed
5 Students' thinking is anticipated with forethought and detail
5 Reactions to students' thinking is mathematically oriented, insightful and detailed
10 3-5 reflection paragraphs demonstrate thoughtful reflection and are clearly articulated / 10 Met with Dr. Leatham and made appropriate revisions based on this discussion
30 3-5 page reflection paper demonstrates thoughtful reflection and is clearly articulated
Lesson Sequence: Learning activities, tasks and key questions (what you will do and say, what you will ask the students to do) / Time / Anticipated Student Thinking and Responses / Your response to student responses and thinking / Formative Assessment, Miscellaneous things to remember /
Launching the Lesson
Introduce the and explain the activity for the students to use straws of varying lengths to construct shapes of their choice. (Provide the straws and paper.) / 3 min / Students may not understand that their partner isn’t supposed to see their shape. / Don’t wait for the concern to arise. Make sure that all of the students know not to show what their shape is. / Don’t explain at this point that their shape needs to be duplicated.
Orchestrating the Task
1. Have the students draw their shapes. (Pay close attention to make sure that their partners can’t see their shapes.
2. Explain that the partner needs to replicate a similar shape with sides half the length of the original and that, if they do it properly, they get a starburst.
3. Call the class back to order and see who properly replicated the similar shape. Give out the proper rewards.
4. Repeat the process. / 12 min / 1. Students may not use the straws for length.
2. Students will probably give a mix of triangles quadrilaterals and pentagons on the first attempt (hexagons and bigger will probably be rare).
3. Students may have trouble figuring out how to cut the length in half.
4. On the second attempt, once students understand how the game works, they will most likely all use triangles. / (1)Tell them to use the straws for lengths as this is integral to the activity.
(3)Tell them that they can just estimate half the length. If necessary, make a general class announcement that an approximation will be sufficient. / Make sure that no students try to describe to their partners what the shape looks like.
Everyone should earn a starburst on the second round. Be sure to give some to the flies, cameraperson, and keith.
Facilitating the Discussion
1. Ask students why they used triangles on the second round.
2. Ask them why shape with more than three sides didn’t work very well on the first round.
3. Remind the students that they learned that similarity implies proportionality.
4. Ask for someone to make a conjecture based on what they have learned.
5. Ask if anyone in the class can explain why proportionality implies similarity. (proof)
6. Allow some discussion on the proof but keep it directed and shut down anyone who tries to tell you how to teach it. / 13 min / 1. Students should see that, if the sides of a triangle are proportional, then the triangle has to be similar.
2. Most students will see that proportionality doesn’t imply similarity.
3. Hopefully students will remember this.
4. Various wordings of the same conjecture will probably come out.
5. There is an algebraic way of proving this by constructing a triangle similar to one and congruent to the other. There is also a simple proof with a compass and strait edge. Student could use either one. / 1. If students fail to see this, let is wait till later when proof is discussed.
2. Ask a student to show why SSSS… doesn’t work. Most likely, they will use a counter example.
3. Prompt students to explain this for you.
4. Write down what conjectures are made and lead the discussion to eliminate any false conjectures.
5. Try and lead students to the compass and straight edge proof as it is easier to see.
Debriefing the Lesson
Restate that similarity implies proportionality and then that it works the other way for triangles and write the conclusion in formal mathematical language. / 2 min / Be sure to also restate that SSSS… doesn’t work.