Today's concept development section is relatively straightforward for my students, but I enrich it by touching upon broader themes like classification systems, definitions, and properties. My intent is to give students time to discuss the concepts, so that they can rehearse mathematical talk. I find that this helps students to internalize concepts and learn to use more precise language (MP6).
I will lead the class through the Parallelogram Definition and Properties Presentation. As I do, I will be monitoring student progress towards the following learning targets:
1. Recite the definition of a parallelogram
2. Recognize the four properties of parallelograms
3. Explain the definition and the four properties using a diagram
4. Explain the difference between the definition of and a property of a geometric object like a parallelogram
By the end of this section of the lesson, my aim is for all of my students to internalize the definition and properties of parallelograms. As we pursue this goal, students will apply their knowledge to justify statements and write proofs. Developing students' proof skills is an ongoing effort. In an upcoming lesson we'll prove the properties of parallelograms.
Using the Parallelogram Definitions and Properties Graphic Organizer, I instruct my students to:
1. Write the definitions and properties using precise language
2. Illustrate the definitions and properties on the given diagrams provided using appropriate geometrical symbols and notations
3. Use mathematical language to record an explanation of what can be inferred from the definition and properties
My basic plan for enacting this lesson segment is:
Part 1: My students individually complete steps 1 through 3 for a given section of the Graphic Organizer
Part 2: A-B partners compare papers and discuss any inconsistencies
Part 3: I model answers using the document camera (or ask a student to show their paper on the document camera for the class to review. As we review the answers, we discuss the work so that I can clarify misconceptions
After our discussion of the Graphic Organizer, I give students the Parallelogram Statements and Reasons handout. It contains 3 items that ask students to supply statements and reasons to complete a mathematical proof. As students complete the work on this set of practice problems, we will use a process similar to the discussion of the graphic organizer to make sure that all students are on track.
The third proof problem is relatively challenging. Depending on the students' progress, I may choose to model this problem for students, rather than have them explore it on their own. In the next section, students will have an opportunity to write proofs similar to the one in item 3.
The goal of this section is for students to apply what they've learned. In the previous section, students had a proof modeled for them that is similar to the ones they will write as they complete Triangle Congruence using properties of parallelograms. After my students complete this proof, they will move on to Parallelogram Properties Problem Solving. This handout requires students to use the four properties of parallelograms to model and solve problems involving unknown measurements.
I have provided an answer key with Parallelogram Properties Problem Solving (p. 3-4) that illustrates the precision of language and the level of reasoning that I expect from my students (MP2, MP6). As students are working on the proof and the measurement problems, I walk around the classroom, possibly using a random student generator to select students with whom to check in. When I encounter students who are stuck, I ask them to articulate what they know about parallelograms. Then, I ask them if they can think of a way to use their knowledge to take a next step in the proof or a problem.
As the lesson comes to a close, I plan to summarize the learning that has taken place and sets the stage for the next lesson by offering a summary like:
We've learned the definition of parallelogram and the four properties that all parallelograms have. We've also used these to make and justify statements, solve problems, and to prove other geometric relationships. We are now prepared for our next lesson in which we will actually prove why each of the properties of parallelograms must be true.