This warm-up gives students the opportunity to use ideas about rhombuses and cubes, trigonometry, the Pythagorean Theorem, and special right triangles to solve two problems. Students must bring in their precise understanding of the properties of rhombuses and cubes and take risks to draw a “good picture” to make sense of problem #2.
Because neither problem is very straight forward, I encourage students to “put a ‘because’ on that,” with the “that” referring to the claims they make. For example, in problem #1, I want students to justify their reasoning by saying something like, “I can use the cosine ratio to solve for the length of x because I know the diagonals of a rhombus bisect each other perpendicularly, making right triangles.” This is the kind of math talk I want to hear and the kind I try to elicit during the debrief of the warm-up.
At this time in the unit, students need a chance to use triangle similarity and trigonometry to construct arguments and write proofs (MP3). Making use of structure, assigning variables, and using these variables flexibly, however, are not easy tasks. I launch this task by working through problem #1 with the whole class—this is important because it gives me time to model the kind of thinking and work I expect from my students; it also allows me to create a safe space for confusion, for example, by saying things like, “the first time I looked at this proof, I felt stumped since we are talking about triangle area and there is no mention of height!” After taking questions, I show them an example with numbers, then ask students to work on the remaining two problems in their groups with the rule that all members of the group stick together.
As groups work, I circulate the room listening in on how students justify how they know. I try to select at least two different students to present their proofs for problem #2 and #3 using the document camera. It is really important to select a student who can clearly articulate his/her argument for #3 and answer questions from students who are confused. I have found that simply asking the presenting student questions like “how did you know that?” or “why did you do that?” can help them reflect more clearly, which helps other students to follow along as the presenter shares his/her work.
Since this is the last lesson before the unit assessment, I think it’s important to give students time to look back at their work from the unit for the purpose of choosing two problems that helped them to develop their understanding of trigonometry. For each problem, students are to write out the problem and solution, then explain in 2-3 sentences why this problem matters to their understanding.