Understanding Tangent as a Function

Present the lesson opener.  The opener for this lesson (found in the slideshow) asks students to find missing sides of two right triangles given an acute angle and the measure of one leg.   Because the location of the unknown side relative to the given angle differs, it is correct to multiply the known length by the tangent ratio in one case, but the known length must be divided by the tangent ratio in the other. 

Students can avoid this potential source of confusion by writing an algebraic equation using the definition of tangent and the measures given in the problem and then solving this equation for the missing side length.   Many students will not like to write an equation, however, preferring to find the missing side length by performing operations on the given measures.   The purpose of the opener is to recognize valid methods of finding the missing sides of a triangle using the tangent ratio and to steer students away from methods that are likely to lead to trouble.   To hear my thoughts on which student-developed methods are valid and which are flawed, see the video that accompanies this lesson:  ApplyingTangentandArctantent_Reflection.wmv.  

The lesson opener follows the Team Warm-Up classroom routine, which is described in my Strategy folder.  I ask students to work in their teams to agree on an answer.   I tell them up front that I will choose a member of each team at random to write the team’s solution on the white board.  I predict that some students will multiply the known length of the second triangle by the tangent ratio, while others will divide.  The students’ solutions on the white board and the disagreement in their answers stimulate a short class discussion.  I demonstrate the method that I recommend and recognize other approaches that are valid.  

 
While students are working on the lesson opener, I take attendance and circulate around the classroom noting who does not have their homework or their materials.

After we hold a discussion about the opener, I display the agenda and learning goals for the lesson while distributing the resource for the next activity.

Have students work in pairs and teams to practice applying the tangent and arc tangent function (15-20 minutes).   Before class, print the resource for this activity.  Make one copy for every two students and cut into half-sheets.  You may want to make a few extra copies, in order to allow students to start over with a fresh sheet if they go down the wrong path.  Distribute the half-sheets to the teams.   Tell the students that they are to work in pairs to complete the problems. 

I used a Kagan Structure (Rally Coach).  My students learn the rules and roles for this mini-activity at the beginning of the school year, so it is a classroom routine.  The instructions are in the slide show.  More information on my use of this structure can be found in my Strategy folder.

Each student should have an opportunity to use the tangent ratio to find the missing sides of a right triangle twice before they are asked to use the relationship in reverse: finding the missing angle measure of a right triangle when they are given the lengths of its legs.  As students are working, circulate around the classroom.  Common problems to look for:

• Students may ignore the instructions and complete both parts (a and b) of a problem before changing roles (problem-solver and coach) with their partner.  The sequence of problems is designed so that, when students switch roles after each part, all students get to see the same types of situations (albeit in a different order).  
• Students may be confused when using the given side lengths to find the measure of an unknown acute angle, because they cannot find the actual ratio of sides that they calculated in the table.  I tell student to use the nearest value in the table (MP6).  This can lead to a short conversation on the fact that trigonometric ratios are often irrational numbers.  

At the end of this activity, I collaborate with students in awarding ‘team points’.  More information on my use of team points can be found in my Strategy folder.

Have students complete the guided notes while summarizing what they have learned as a class (10-15 minutes).   This activity follows a class routine.  More information on my use of guided notes can be found in my Strategy folder.  Before class, print the guided notes.  Distribute the notes to the class, and lead the class in a summary of what they have learned about trigonometry and the tangent ratio as students complete the notes.  

Have students reflect on the lesson in pairs (3 minutes).  Display the closing question and ask students o brainstorm answers in pairs, then as a team, before writing an answer.  Students write their answers in their earning journals.

Homework as assigned in the syllabus for this unit.