Introduction to the Slope Ratio (Tangent)
Lesson 4 of 11
Objective: Students will be able to use similarity and to make sense of the meaning of the slope ratio (tangent ratio).
In Side-Angle Relationship Warm-Up students are given four triangles for which they must determine whether the missing leg should be greater than, less than, or equal to the given side. Since this lesson is our first on trigonometry, I developed this warm-up to motivate my students to focus on sense making with respect to reasonable side lengths (MP2).
I expect some of my students will feel that they do not have enough information to do the warm-up. To overcome this obstacle, I will encourage students to talk to each other to justify their ideas about the problems. Many of my students will use arguments like, “the longest side is always opposite the biggest angle,” but some will also bring in other ways to defend their reasoning, for example, by drawing the equilateral triangle around the triangle in (c) or the square around the triangle in (d). At this point, I feel it is important to highlight multiple ways of making sense of sides and angles in triangles because this will foster students’ understanding of the many tools they can integrate in trigonometry.
Before introducing trigonometry formally, I want my students be able to recall what they have learned about triangle similarity so they can begin building their conceptual understanding of trigonometric ratios. In this quick activity, I pair students up, post a definition of similar, and ask them to analyze two sets of triangles that have been drawn to scale on graph paper. Pairs of students discuss whether the triangles are similar and what they notice about the triangles’ corresponding pairs of angles.
After a few minutes I will call on student volunteers to share out with the whole class. This sharing session should be quick, since this is a review of the last three lessons on similarity.
Now, we will move onto the Slope Ratio Task.
The goal of the Slope Ratio Task is for students to see that all slope triangles on the same line have the same slope angle and are similar. This activity is particularly important because it lays the foundation for students to understand why all trigonometric ratios (in this case, slope ratios) are always the same given a particular reference angle, no matter the size of the triangle.
In this task, each partner uses a different color to draw different-sized slope triangles for the line y=(1/5)x. In examining several slope triangles, students will see that every slope triangle has a slope angle of ~11° and a slope ratio of 1/5. Pairs will check in with me when they can explain the height of a slope triangle with a base 1 unit long and how they know.
To debrief the Slope Ratio Task with the class, I select a pair of students to present their findings, which ideally includes explanations like, “The slope angle for all of the slope triangles is 11° and all the slope ratios are equal since they all equal 1/5” or “The height of a slope triangle with slope angle 11° is 1/5 since the height is always 1/5 the length of the base.”
Ultimately, I want students to see that given any slope angle (the “reference angle”), the slope ratios (“tangent ratios”) will always be equal. I tell students that since we know the slope ratio for 11° can be written as 1/5, 2/10, 3/15, 4/20, etc., we can now write this as “tan(11°) = 1/5.”
I then model for students how they can use their scientific calculators to show them that tan(11°)=0.19438, which is very close to 1/5. Additionally, I show them tan(36.87°)=¾, which is the slope ratio for the triangles in Set A in the first partner task in this lesson. I then give students a few examples for which they can practice using their scientific calculators—this gives me time to circulate the room to help students who are struggling or who need help switching their calculator’s mode from radians to degrees.
It is extremely important in this discussion to build in time for students’ questions. Nearly every time I have taught this lesson, a student has asked me if it is possible to determine the slope angle if you know the lengths of the legs of a right triangle—this creates an opportunity to use the triangles from Set B from the first pair task, to show that tan-1(8/15) = 28.07° and to open a discussion around the idea of tan inverse. If this kind of question does not emerge, a good way to use any leftover time would be to ask students to explain in their own words what “tan (70)=2.75” means.