Summary
Students will investigate and discover the trigonometric ratios through observations about right triangles.
Essential Question(s)
How are the angles and sides of a right triangle related?
Snapshot
Engage
Students complete a card sort using triangle vocabulary.
Explore
After seeing the need for another way to solve right triangles, students investigate the ratios of side lengths in similar right triangles.
Explain
Students define sine, cosine, and tangent based on the relationships discovered in the exploration.
Extend
Students use a clinometer to investigate angles in inclination and declination.
Evaluate
Students demonstrate their knowledge of trigonometric ratios.
Materials
Rulers
Protractors or similar
Scientific calculator
Right Triangle Relationships Observation Sheet handout (attached; one per student)
Sine Cosine Tangent Observation Sheet handout (attached; one per student)
Triangle Card Spot handout (attached; one per group)
Triangle Examples handout (attached; one per student)
Clinometers
Engage
10 Minute(s)
Use slides 3-4 to introduce the essential questions and lesson objectives as you see fit. Display slide 5. Form students into groups of 2-3 and provide them with a set of the Triangle Card Sort.
Allow students 5-15 minutes to sort the cards in their groups, allowing them to sort them in any manner that they find reasonable. You should not help students sort them, but instead question students about the sorting scheme they used and have them justify it. Choose a few groups to explain/share their sorting with the class.
If necessary, discuss other sorting schemes that become obvious after students have shared.
Explore
30 Minute(s)
Display slide 6. Display for the students a right triangle that is easily solved using the Pythagorean Theorem, and have the students walk through finding the solution. Make sure to have them explicitly state the information they're using and how they're using it.
Display slide 7. After this discussion, display a right triangle that cannot be solved using only the Pythagorean Theorem. Pose the question to the class: "How can we solve this one?" The students will be puzzled, but they may have some idea or agree that they need more information. It cannot be solved with what is given. (An example has been provided under Attachments)
Display slide 8. Provide each student the attached Right Triangle Relationships Observation Sheet. Explain that by investigating right triangles, the class can figure out a way to solve the mysterious triangle. Allow students to work in their groups on this investigation. Provide help if needed, but try not to guide students explicitly. Instead, question the students about their process and thinking to help them come to their own conclusions.
It may be prudent to pause here and wait to begin the next portion during the next class period, as students will likely need more than 30 minutes to complete the investigation.
Explain
30 Minute(s)
Display slide 9. Bring the class back together for a discussion.
Display slide 10. Return to the "unsolvable" triangle and have students puzzle out how to solve it as a class, now that they have trigonometric ratios in their toolkit.
Have several students share their hypotheses about the ratio relationships they observed during their investigation.
Then, carefully facilitate a discussion about these relationships (it might be helpful first to discuss the origin of error(s) in their measurements) by asking students to demonstrate and explain how they formed their conclusions and how they tested their hypothesis using 30-60-90 triangles.
Ask the students leading questions: "Could your new understanding of right triangles help you solve for a missing side?" Have the students show you how. "Can we do this for any triangle with a 21 degree angle?" Yes, of course. "And then you figured out that it also works if your angle is 30 degrees or 60 degrees or 45 degrees?" Yes, they should have.
Find the missing side length in another example for a 30-60-90 or 45-45-90 triangle that the students used in their investigation using the ratios they agreed upon, or have a student draw and work it out while the class guides them.
Draw a triangle on the board with another angle measure (25 degrees in this example) and ask them to help with the equation: "Now that we know we can find the missing sides for a right triangle with a measure of 21 degrees, and even one with 30 degrees, what if the measure was 25 degrees? What would we have to do?"
Ask the students if there could be a shortcut and if so, what it would look like?
Inform the students there is a shortcut, that there are established functions to show us the relationships between angles and side measures in a right triangle. These are called sine, cosine and tangent. Provide students the attached Sine, Cosine, and Tangent Observation Sheet and allow them to work it out. After students have determined the correct ratios, have them share with the class.
Return to the "unsolvable" triangle and have students puzzle out how to solve it as a class, now that they have trigonometric ratios in their toolkit.
Extend
10 Minute(s)
Display slide 11. Now with the understanding of trigonometric functions, students should create their own real world trigonometric function scenario for their peers to solve. They are not solving the problem, but creating it. On the bottom of the scenario, they need to write their definitions of sine, cosine, and tangent that they originally produced earlier in the lesson.
Evaluate
10 Minute(s)
Display slide 12. Students just completed making their trigonometric function word problem. Now, have students find a partner and solve their real world scenario using the knowledge they gained previously in the lesson on a separate sheet of paper. They should also analyze their peer’s definition of sine, cosine, and tangent. They should check if their definitions are similar to theirs and determine how can they combine the two definitions to make one concrete definition for future reference. If they do not believe their peer’s definition is accurate, they can examine that as well while creating a final draft of the definitions.
Resources
"Finding Trigonometric Ratios." CPALMS lesson title used for inspiration. (2013-2015). Florida State University. http://www.cpalms.org/Public/PreviewResource/Preview/46546
K20 Center. (n.d.). Card Sort. Strategies. https://learn.k20center.ou.edu/strategy/147