Investigate: What question(s) will students be investigating? What process will students follow to collect information that can be used to answer the question(s)?
Students will investigate and discover trigonometric ratios by exploring questions like:
- What is the relationship of a triangle's angles and their corresponding sides?
- What happens when I have several triangles with the same angle measures, but different side measurements?
- How can this help me in real life?
Students will place all measurements and calculations in a table. They will compare their calculations with their peers' and draw conclusions based on their findings. (MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others.)
Analyze: How will students organize and interpret the data collected during the investigation?
- The teacher should distribute the "Discovering Trig" worksheet to each student. Each student should also have a ruler, pencil, scientific or graphing calculator and a half sheet of paper. Students should be placed into groups of 3 or 4 for this activity.
- Students will complete section A on the sheet. The teacher should circulate around the room to ensure that each student is correctly measuring each triangle's lengths.
- When students complete measuring of the triangles, they should complete sections B and C of the handout.
- The teacher should instruct the groups to discuss their findings and draw a conclusion based on their comparisons. (MAFS.K12.MP.2.1 Reason abstractly and quantitatively.) Give each student 15 seconds to share. Let the student with the earliest birthday share first and the student on the left shares next, when the teacher calls time. Repeat this until all students have shared. Then give an additional minute for students to collaborate and summarize their findings. Students should then complete section D on the handout. (See the Discover Trig Key for examples of student responses.) (MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others.)
- For section E, use the "Commit and Toss" strategy as described below. (MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others)
The teacher will then ask students to comment on proposed hypothesis in a contrive manor and help combine correct hypotheses. The teacher then will introduce that there are formal ratios and the terminology of sine, cosine and tangent. Teacher will demonstrate how to find these ratios on a scientific or graphing calculator. (Note: remember calculators must be in degree mode).
Students will record the trigonometric ratios on their sheet under section F using the calculator. NOTE: The teacher should not yet tell the students the actual ratios of each trig function as the students should be concluding that in Part I.
Students will use the measurements from their first chart to complete the chart in Section F. Students can also complete section G.
The teacher can use the method in step 4 to allow students to share with each other and form conclusions. Then, students should answer section H.
Section I: Give students think time to answer sections I and J. For section I, students should be able to conclude that: (MAFS.K12.MP.2.1 Reason abstractly and quantitatively.) The teacher should circulate within the room to see if students are drawing the correct conclusions for this portion. If students are struggling, the teacher can draw the students' attention to the charts and their corresponding answers. Let them know that you want a generic equation for each that you can use with any right triangle. Once all students get this portion, the teacher should be sure to tell them that what they have discovered are called Trigonometric Ratios.
In Part J, students should see the relationship between all of the trig functions and how the relationships between the complementary functions. Students may need guidance through this section. (MAFS.K12.MP.2.1 Reason abstractly and quantitatively.)
Pass out the Applying Trig Function Sheet to the students. This sheet focuses the students on how to find the angle of a triangle given two sides. It extends their understanding of the ratios they just discovered giving meaning and purpose to the trigonometric functions.
Demonstrate to the students the first three problems on the sheet and how to use the inverse trig functions on their calculators to find the missing angle when given two lengths of a right triangle. (Note: remember calculators must be in degree mode). Teacher should model how to determine which trig function is appropriate for the angle and sides given. Labeling the triangle with the words "opposite, adjacent and hypotenuse" is helpful.
Have students practice the next three problems using the first three problems as a guide. Teacher should circulate to make sure the students are correctly identifying which trig function is appropriate for each problem. (MAFS.K12.MP.2.1 Reason abstractly and quantitatively.
Have students compare answers with their partners to verify appropriate trig functions were chosen and confirm answers. If there is disagreement, encourage partners to discuss and help each other through the issue, coming to a consensus. Review the answers as a whole group. (MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others.)
Teacher should pass out the application worksheet to the students and discuss that since they now have discovered the 3 basic trigonometric function for right triangles we can use them to solve problems.
The teacher should model the first three problems on the "Applying Trig" worksheet, introduce the Arc sine, Arc cosine and Arc tangent buttons (inverse trig functions) on the calculator. Discussion should occur as to what and how these functions are useful in finding the angle measure when two sides are known in a right triangle.
Once the teacher has modeled how to use these functions, the students should work in teams to complete the next three problems, discussing and comparing answers. (MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others.) The teacher should walk around, listen to discussions and correct any misconceptions that are forming by addressing the group or having a full group discussion.
The students should complete the rest of the worksheet for independent practice.
- The teacher should instruct each student to take out a half sheet of paper.
- Give the students 10 seconds to think about the group discussion and the data that they have seen and formulate a hypothesis.
- After 10 seconds, each student will write their hypothesis on a half sheet of paper.
- Students will crumple their papers into a ball and stand by their desk.
- When the teacher signals to start, each student can throw the paper until the teacher signals again to stop.
- When the teacher signals to stop, each student should pick up one paper ball and stand by their desk.
- The teacher should then call on several students to share the hypothesis that is on the paper and then sit down.
- If a student reads a hypothesis that other students have on their papers as well, those students may sit down. The teacher can use this strategy to formatively assess the students.
Closure: What will the teacher do to bring the lesson to a close? How will the students make sense of the investigation?
The teacher can show the Triangle Side review portion of the YouTube video "Trigonometric Ratios" to review (1 min 50 sec to 4 min 35 sec, video after 4:35 is more appropriate for when finding a missing side is needed. (See Prior Knowledge for the link and directions for this.) Teacher can also have students share their answers for the second three questions on the board from the Applying Trig worksheet.
The teacher will use the attached Applying Trig worksheet to assess student understanding of the lesson. The teacher can choose to assign one problem or several problems. The answer key is included in the attached worksheet along with a rubric for grading. Specific guidelines for mastery are suggested in the formative assessment section of the lesson.
At the beginning of the lesson, students will discuss what they know about triangles and why triangles are important. Students will identify how they have seen triangles used in their communities. The teacher can use questions from the "Guided Questions" section to engage students and assess their understanding of triangles before the lesson.
Throughout the lesson, the teacher should circulate and observe the students. The teacher can formatively assess the students based on student conversations about the assignment, ideas that are formed based on the assignment and the conclusion students make. The teacher should ask probing questions to gauge the students' level of understanding at several points throughout the lesson.
In addition, the rubric at the end of the Applying Trig worksheet could be used as formative assessment or summative assessment depending on the needs of the teacher. The teacher can evaluate the student's subcategory points as well as overall score to determine at what step they seem to be having most difficulty (if any).
Applying Trig Problems: Can they determine the appropriate trig function, set up the ratio and get the correct answer? Mastery 13+points; Progressing towards Mastery 9-12 points; Needs additional practice 5-8 points and needs remediation 0-4 points
Calculator Use Problems: Look at the ability for the students to use the calculator correctly and know when to use the inverse trig functions. Did they get all 8 Points? Questions 7-10 regular trig functions, Questions 11-14 inverse trig. Mastery 6+ points; needs practice and remediation
Overall Progress: Mastery 20+points; Progressing towards Mastery 14-19 points; Needs additional practice 9-14 points and needs remediation 0-8 points
Feedback to Students
Students will receive feedback from their peers when they are working in groups. Students will also receive feedback from the teacher while the teacher circulates around the classroom throughout the lesson. Once the rubric is scored, the student can also see what parts of the concept they are struggling in by analyzing the subcategory scores.
SOURCE AND ACCESS INFORMATION
Name of Author/Source: Keisha Wallace, Carrie Meyers
District/Organization of Contributor(s): Volusia
Is this Resource freely Available? Yes
Access Privileges: Public
* Please note that examples of resources are not intended as complete curriculum.