Lesson 3: Trigonometric ratios in similar right-angled triangles
Learning Goals
Lesson Aims
Trigonometric ratios in similar right-angled triangles
Learning Intentions
- Students familiarise themselves with identifying trigonometric ratios for a given angle
- Students compare trigonometric ratios for a given angle in similar right-angled triangles, and verify that they are equal for corresponding angles.
Success Criteria
- Students are able to communicate that $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ with fluency.
- Students are able to verify that trigonometric ratios for the corresponding angles in similar right-angled triangles are equal.
- Students are able to explain that trigonometric ratios are consistent in similar right-angled triangles because all sides are proportional.
Syllabus Links
MAO-WM-01
- develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
MA5-TRG-C-01
- Applies trigonometric ratios to solve right-angled triangle problems
Content descriptor
- Verify the constancy of the sine, cosine, and tangent ratios for a given angle by applying knowledge of similar right-angled triangles.
ICT and LIT
ICT
- Students record angle sizes, side lengths, and trigonometric ratios in a spreadsheet to verify that ratios are equal for corresponding angles in similar triangles.
LIT
- Explain why trigonometric ratios remain constant in similar triangles.
- Write step-by-step observations in spreadsheet and describe results.
Prerequisite Knowledge
- Students should be able to know and use relative proofs for similar triangles, including equal-angles and proportional corresponding sides.
Resources
Whiteboard/Smartboard (markers), Diagnostic test, students’ devices, student workbooks, Student worksheet 3
Vocabulary List
- Similar triangles – Triangles that have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are proportional.
- Proportional sides – When the ratios of the lengths of corresponding sides in two triangles are equal.
- Corresponding angles – Angles in two triangles that are in the same relative position.
- Constancy of trigonometric ratios – The fact that sine, cosine, and tangent of a given angle stay the same in all similar right-angled triangles.
- Scale factor – A number that describes how much one figure is enlarged or reduced compared to another similar figure.
Lesson Structure
Introduction
15 minutes
- Greet and settle students down
- Give time for students to prepare for the lesson
- This includes giving time to take out relevant resources for the lesson
- Diagnostic Test
- Question 1: What proofs for similar triangles do you know? Do they have acronyms?
- Draw two right-angled triangles that are similar, and label the features that indicate the similarity.
- Teacher discusses the answers to the questions with the whole class.
- Teachers must check students’ understanding either by asking for thumbs up/down for understood/confused signal.
- Ensure students are familiar with the concept of similar triangles before proceeding with the rest of the lesson.
Body 1
Concrete learning + Learning through Technology
25 minutes
- Students are given papers to cut out a right-angled triangle of any dimension. Students are then to consider similar triangles of the first right-angled triangle, draw and cut them out too
- The enlargement/reduction factor is to be chosen freely by each student, but it is to be written in their books for record.
- With their rulers and protractors, confirm that their triangles are similar, by both AAA and proportional sides.
- Students open their devices and access a spreadsheet. They are to record the following each in new columns that are clearly labelled. (ICT)
- Angle sizes (degrees), chosen ‘given angle’ size (Theta), Length of sides (Opposite, Adjacent, Hypotenuse), Trigonometric ratios (Sine, Cosine, Tangent)
- Teacher demonstrates how to write equations or compute trigonometric ratios in an Excel spreadsheet
- Spreadsheet Observation Writing (LIT): Students record angles, sides, and trigonometric ratios in spreadsheet and describe patterns in writing sentences.
- After all measurements are taken, students are to form groups in their tables for discussion.
- Discussion questions
- How are the values of trigonometric ratios related in similar triangles? Why?
Explanation Writing (LIT): In groups, students write sentences explaining why ratios are equal for corresponding angles. - Discuss the truth of this statement. “Two triangles are always similar if they share a common angle and their trigonometric ratios are equal”
- How are the values of trigonometric ratios related in similar triangles? Why?
- Discussion questions
Body 2
Independent practice
15 minutes
- Students work independently on worksheet 3.
Students are expected to complete the exercise for homework if they have not finished in class
Teacher roams around the classroom to help individually students with questions.
- If many require explanation on a specific question, teacher brings attention from whole class to solve together the question on the board
Conclusion
5 minutes
Student learning reflection Checklist
- Student learning reflection questions
- I am able to define trigonometric ratios in terms of Opposite, Adjacent, and Hypotenuse (Confident, ok, need practice, or no idea)
- I understand that similar triangles have equal ratios for corresponding given angles (Confident, ok, need explanation, no idea)
- I understand why similar triangles have equal trigonometric ratios for corresponding given angles (Confident, Somewhat, confused)
NOTE: This checklist survey is to be collected by the teacher to grasp an understanding of students’ learning progress