Today my students will discover the Law of Cosines which will give them another tool to solve oblique triangles. Students start by analyzing three pieces of information for possible triangles. I'll let my students work for 3-4 minutes. Once they seem to be ready, I plan to ask, "How many of you drew a diagram and labeled the diagram with the given information? Why did you draw a diagram?" A point of emphasis today will be drawing diagrams as an important step in solving problems.
As we discuss the benefit of a visual model, I will ask a student to put a diagram of the first problem on the board. Once the diagram is complete, I'll ask, "To solve this triangle, what would you find first?" Since the first problem is a right triangle, many students find the 3rd side. Some would use the Law of Sines to find one of the acute angles. As we discuss this, I'll ask, "Is the Law of Sines the best method to use?" To help advance this point, I'll ask students to find the Law of Sines activity from earlier in the unit in their notes. Together, we'll read the box on the first page. I'll ask, "Can we use the Law of Sines when solving right triangles?" As I state in my reflection, I let students know they can use the Law of Sines, but they also choose to use more efficient methods.
On page 2, we continue with the next problem. I plan to begin by asking, "Is it important to draw an accurate sketch? If you are given the information that there is an obtuse angle, should you draw an obtuse triangle?" This discussion revolves around how the diagram helps us reason (MP4, MP6). If I draw an acute angle when there is an obtuse angle, then I may not realize that the side opposite angle C in this case has to be the largest side.
Often, my students are not sure how to solve the last two problems in this opening section. Students make comments like:
If my students need a hint, I have them focus on the last problem first. I ask "Do you remember a theorem from geometry that relates the sum of the lengths of the two shorter segments to the lengths of the largest side in a triangle?" If necessary I will provide a sentence frame with blanks for the important information. As a final intervention, I may ask if it is possible to construct a triangle with sides measuring 1 inch, 2 inches and 3 inches. This last intervention is usually enough to allow my students to see that the last problem gives information that will make a triangle.
Now that my students recognize that there are oblique triangles we cannot solve with the Law of Sines, I introduce my students to the Law of Cosines. For this introduction I give my students the Law of Cosines activity (from NCTM Illuminations). I ask my students to work in groups to complete the activity.
There are several parts of this activity that my students struggle through. For example, in Step 4 my students are not sure what is meant by expand (c-x)^2. I wait until numerous students ask about this question before discussing the meaning with the class. I want the students to think on their own and try to solve the problem. If I decide that intervention is necessary, I ask a student to tell me their response to Step 3. I then write (c-x)^2 on the board and ask, "How do you think we can expand this binomial?" The first suggestion is usually "(c-x)(c-x)." This helps everyone to get the ball rolling. Individual students begin to think about multiplying the factors together, using well practiced routines from Algebra and Algebra 2. Once students identify the correct expansion, I remind students that this new expression is equivalent to (c-x)^2 so we can substitute the new expression for (c-x)^2.
Like the above example, many issues arise from a shallow reading of the directions that makes it difficult to determine what is to be done. In order to promote student ownership of the process of discovery, when a student ask me about directions I try to rephrase the step into terms the students will understand without reducing the cognitive challenge.
Step 6 provides a challenge when the directions say, "we will eliminate x." When students ask about this step I read the entire step out loud and then say:
Oh, it says to write an equation involving cos B and x. What is the first part of the step telling us?
If students are unsure I will explain that it is giving us the reason we need the equation involving cos B and x. Modeling reading techniques is an important intervention that helps students learn how to read mathematics. Many of my students are not proficient with technical reading and need to be regularly reminded of how to read mathematics. I consider this important enough that I continue to model it until my students become proficient.
To end the day's lesson, we will go back to Question 3 from the bell work. I will ask my students to work in groups to set up the problem using the Law of Cosines. As a class, we will discuss how we are given enough information for the Law of Cosines, but not the Law of Sines.