Problem Solving with Quadrilaterals
Lesson 3 of 5
Objective: Students will be able to apply their knowledge of quadrilaterals to numeric and algebraic problem solving.
To begin today's class, I give my students a problem set entitled Guided Practice. This handout begins with a chart that asks the students to recall what they have learned about the sides, angles, and diagonals of the members of the parallelogram family. I expect my students will refer to their Family Tree of Quadrilaterals as they fill in the chart.
The problem set then includes one or two numerical problems for each type of the quadrilaterals These problems are not particularly challenging. They are a warmup designed to reinforce the students’ basic understanding of these figures. One of the problems (#5b) offers the possibility of applying knowledge with regard to the sides of an isosceles right triangle. My students may choose to pursue this strategy, and if so, it will be a point of discussion to prepare for what follows in the lesson. Overall, I want this Guided Practice to go quickly.
To help students work precisely, I ask that they work in groups, one problem at a time. I check in with the whole class, one problem at a time, when it appears each group has completed a problem. As we go over each problem, I am careful to ask “How do you know…?”. I expect my students will start to follow this practice in their group work:
In problem #2a, it is important to ask, “How do you know a = 8 and b = 10?” and I expect my students to answer, “Opposite sides of a parallelogram are congruent.”
I think it is important to ask students to justify over and over again in this unit – it helps them with their work on quadrilateral proofs and, by frequently writing and speaking the justifications, also helps them to remember the different features of all the different quadrilaterals.
Independent Problem Solving
Now it’s time to allow the students to work at their own pace with the handout entitled Quadrilateral Practice. As students work I circulate around the room, watching for any students who might struggle, and also continue to challenge individual students with that question, “How do you know?”
Teacher's Note: Before using this problem set, it is really important that you, the teacher, spend some time looking at it beforehand. I have indeed made the mistake of just grabbing it out of my file and not looking at it ahead of time. It created lots of confusion!
Many of the problems read, “Fill in all possible information.” These problems can be approached from a very basic level or, more ambitiously, using Pythagorean triples, Special Right Triangles, and trigonometry. Sides can be found in either simplest radical form, or rounded to the nearest tenth. These are all decisions that you need to make before setting the students loose on the problems.
Each section of the problem set includes one problem in which the students must draw their own diagram. I have found that it is very important to include this skill, and I will address this skill in a future lesson, as well. But it seems like each and every year that I teach this unit, I am caught off guard at least once by a student who becomes stumped when faced with having to draw his or her own diagram. This year, for example, a young lady raised her hand during the unit test and asked if I could show her what a trapezoid looks like. Yet there were two diagrams of isosceles trapezoids on the very next page of the test. Guess I need to keep working at it!
Wrap-Up + Homework
It is typical that many of my students are not finished with the Quadrilateral Practice problems at the end of this lesson. Instead of reviewing the answers, I ask them to stop what they are doing to ask for feedback:
- How are the problems going?
- Are there any concepts in particular with which they are struggling?
- Which kinds of problems are hard and which are easy?
I let them know that we will continue with this problem set in the next class meeting, and I hand out tonight's Homework Proof.