SWBAT define trigonometric ratios using similar triangles

Students build on prior knowledge about similar triangles to investigate trigonometric ratios.

5 minutes

As the students walk in the room, they are given a slip of paper with a triangle on it. The triangle is split into four different triangles. Students are instructed to measure the angles and the sides of each triangle. Rather than have each student measure each angle and side, I assign different students to measure different parts. I make sure to assign more than one student the same parts to ensure accuracy. I ask students to measure the sides using centimeters in order for their measurements to be in decimals (MP6). They use rulers and protractors that are kept in a box on their tables. If students have questions about using the protractor or measuring in centimeters, I have other students at their table explain how to measure.

10 minutes

At the beginning of the Mini-Lesson, I call on students to give their measurements. Since more than one student measured each part, there may be more than one answer. If there is a large discrepancy between measurements, I have all of the students remeasure the part. It is important that the measurements be as accurate as possible for the activity in this lesson.

This is a good time to have a discussion about precision in measurements. I usually allow a tolerance of plus or minus one millimeter in length and plus or minus two degrees in the measurements of angles. Each student writes down all of the measurements in their notebooks.

After all of the measurements are recorded, I ask the students what they notice about the measurements of the angles. They recognize that the corresponding angles in each of the triangles are congruent. We then answer the question on the presentation, "What do the measurements of the angles tell us about the relationships between sides FB, GC, HD, and IE?" In previous lessons, we have discussed the theorem that states, "If two lines intersected by a transversal form corresponding angles that are congruent, the lines are parallel." Although students may not be able to explicitly state this theorem, they are able to recognize that the sides are parallel.

Next I ask the students what we know about the relationships between triangles ABF, ACG, ADH, and AEI. I call on a student to give a response. Since corresponding pairs of angles are congruent, the triangles are similar. I further elicit that corresponding sides are in proportion. We use this relationship to write ratios for the activity.

23 minutes

A shift in the common core standards is to first introduce right triangle trigonometry to students in the geometry curriculum. This activity helps students see the relationship between similar triangles and trigonometry.

Students begin the activity by writing ratios of sides in each of the triangles using angle A to write the ratios. They write ratios for the adjacent side to the hypothesis, the opposite side to the hypothesis, and the opposite side to the adjacent side. There are twelve ratios in total. I group students in threes and assign each student a different ratio to write and simplify to four decimal places.

After about 6 minutes, students share their results with other students in their group. They come up with a conclusion about each set of ratios. The corresponding ratios for each triangle are approximately the same depending on how accurate the measurements of the sides are. Most students are able to identify this. Students repeat the process using the other acute angle in the triangles to verify their conclusion (MP7, MP8).

We gather back together as a group after about 15 minutes. I ask a student from each group to give his or her conclusion. The groups usually have the same results. Then we discuss why the ratios are not necessarily exactly the same. Students can identify that the accuracy of the measurements can affect the accuracy of the ratio due to human error.

At this point, I introduce the trigonometry ratios to the students. I instruct them to make sure their calculators are in degree mode and then have them find and write down the sine, cosine, and tangent for each of the acute angles. We discuss how the trigonometric ratios relate to their ratios. Students identify that the ratios correspond to each of their ratios.

7 minutes

As a summary, I have students write an explanation about how to find the sine, cosine, and tangent of an acute angle in a right triangle (G.SRT.6). They use the results of their activity to write the explanation. After about three minutes, I call on a few students to read their explanation. We come up with the trigonometric ratios and I have each student write them in their notebooks before leaving the room.

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