When the students arrive in class today, they find the following directions on the board:
Work in your groups to categorize these figures and arrange them into groups based on what you learned in yesterday's lesson. Use your tables to help you with this activity:
When the students have finished their initial classification, I ask each group to talk briefly about how they categorized their figures.
I distribute the handout entitled Family Tree of Quadrilaterals, which I have copied onto colored paper. Beginning at the top of the page, I lead a class discussion.
First, I ask my students “What do you know about all quadrilaterals?” Fill in what you know to be true about all quadrilaterals in the space provided. After students complete this information, I ask, “So what do you know to be true about all of the polygons underneath the quadrilateral?” I make a big deal about requiring them to fill in "4 sides" for all of the others. This helps them to begin to think in terms of the concept of inheritance (MP7).
Teacher's Note: You or the students will need to draw in the “branches” of the family tree, the lines connecting one figure to the next. This can be done in advance, or, sometimes it is interesting to draw in the lines after the worksheet is complete.
I choose to move onto parallelograms next, leaving trapezoids as a separate category rather than the parent of parallelograms. Again, I ask students to refer to their tables from the previous lesson and to tell me what they discovered to be true about all parallelograms. We discuss features as students nominate them and list them on the family tree when agreement is reached. There is a small diagram of a parallelogram provided on the handout. My students usually ask if they should mark the diagrams and I leave this up to them, because the diagrams are so small.
I then ask a series of questions about rhombuses:
I have provided space on the family tree for 8 characteristics of a rhombus. There are definitely more than this that the students have discovered, and I am flexible on the eighth of these features. My students share ideas like "diagonals bisect the angles" or "diagonals form 4 congruent triangles."
Next I move to rectangles, and ask a similar series of questions again: Are all rectangles quadrilaterals? Are all rectangles parallelograms? So what do we fill in on the family tree? What are the distinguishing characteristics of a rectangle? What makes it more than just a parallelogram?
After a detailed discussion of the rhombus, I ask, "Is a rectangle a rhombus or not? What do you think?" Since we have a list of characteristics for a rhombus, this single question can lead in a number of interesting directions. The big difference that I hope to draw out of the students is that rhombuses have all sides congruent, while rectangles have all angles congruent. How quickly we get there often depends on where we start.
I follow a similar line of questioning for the square. Once the students have entered all the characteristics of squares on the family tree, I ask:
The fact that a square is both a rhombus and a rectangle is a difficult concept for some students. One related line of question that I have found works for my students is: What is your mother’s maiden name? What is your father’s last name? Then, I write these names on the board and I ask the class questions like: Is Sam a Smith? Can he go to the Smith family reunion? Is Sam a Jones? Can he go to the Jones family reunion? So is Sam both a Smith and a Jones?
Teacher's Note: Please see my Trust and Respect reflection before using this teaching move.
Finally, I move to trapezoids and isosceles trapezoids, and I pose questions similar to those I asked in the parallelograms. When we have discussed all of the characteristics, I ask, “Is a trapezoid a parallelogram? Can he be part of the parallelogram family reunion? Why not?”
We then proceed similarly with kites.
I remind the students that the information that we used to fill in our family tree of quadrilaterals was gotten by making hypotheses. Have we proven our hypotheses?
I present the class with a parallelogram proof, entitled In-Class Proof. In it, students prove that the opposite sides of a parallelogram are congruent. The proof requires the use of an auxiliary line, which is drawn in for them, but we will briefly discuss, once again, the frequent need for auxiliary lines in both proofs and numerical problems (MP7).
There are blanks for the students to fill in. I help them through the second reason (Opposite sides of a parallelogram are parallel) and explain that this is the definition of a parallelogram. From here I allow the students to work together to finish up the proof and I walk around the room with an eye for any struggling students.
As the students finish with the proof, we fill in the final “therefore” statement. I ask if we could prove anything further from this proof. We discuss that angle U is congruent to angle D because of corresponding parts and that we could prove angle Q is congruent to angle A by simply redrawing the auxiliary line from U to D and repeating the same proof. I make it clear that these two statements, Opposite sides of a parallelogram are congruent and Opposite angles of a parallelogram are congruent are now available for use in future proofs.
Before we leave for the day, I hand out the Homework Proof. I allow students to begin work on it if there is any time remaining. This proof asks that the students prove that the diagonals of a parallelogram bisect each other. We are not going to prove each and every one of the statements on the Family Tree of Quadrilaterals, but I do continue to provide these proofs from time to time throughout the unit. This strategy increases the depth of their understanding, while reviewing the knowledge gained while writing triangle proofs earlier in the course.