Lesson 4: Finding the length of an unknown side in a right-angled triangle
Learning Goals
Lesson Aims
Finding the length of an unknown side in a right-angled triangle
Recognising angles down to minutes
Learning Intentions
- Students will find the lengths of unknown sides in right-angled triangles with a given angle using digital tools such as a calculator
- Recognise that there are 60 minutes in 1 degree
- Use a calculator to compute angles in degrees and minutes
Success Criteria
- Students are able to rearrange equations for the trigonometric ratios sine, cosine and tangent, to find the lengths of unknown sides in right-angled triangles with a given angle.
- Students are able to convert between degrees and minutes either manually using fractions, or by using a calculator.
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
- Apply trigonometry to find the lengths of unknown sides in right-angled triangles with a given angle including angles in degrees and minutes
ICT and LIT
ICT
None in this lesson
LIT
- Match each diagram card with the correct equation card
Prerequisite Knowledge
- Students should be able to solve linear equations of up to 2 steps and quadratic equations of the form $ax^2 = c$ (MA4-EQU-C-01)
- Students should be able to express the trigonometric ratios sine, cosine and tangent, in terms of sides Opposite, Adjacent, Hypotenuse, for a given angle
Resources
Whiteboard/Smartboard, Student worksheet 4, calculators, Diagnostic test 4, Diagram and equation cards, timer
Vocabulary List
- Subject of a formula – The variable that has been rearranged to be on its own on one side of the equation.
- Rearrange an equation – To use algebraic steps (adding, subtracting, multiplying, dividing) to make a different variable the subject.
- Solve for $\boldsymbol{x}$ / make $\boldsymbol{x}$ the subject – To manipulate the equation until $x$ is by itself on one side.
- Degrees (°) – A unit of angle measurement, where one full turn is 360°.
- Minutes (′) – A subdivision of a degree, where 1° = 60 minutes.
- Seconds (″) – A subdivision of a minute, where 1′ = 60 seconds.
- Calculator notation – The way calculators show angles with degrees, minutes, and seconds (DMS) or convert between them.
Lesson Structure
Introduction
10 minutes
- Greet and settle students down
- Give time for students to prepare for the lesson
- This includes giving time to take out resources for the lesson
- Diagnostic Test
- Students complete a short diagnostic test on Equations
- Solve for $x$. $\, 2x - 4 = 8$
- Solve for $x$. $\, 24 = \frac{2}{x}$
- Make $y$ and $z$ the subject. $\, x = \frac{y}{z}$
- Students complete a short diagnostic test on Equations
Body 1
Concept teaching
20 minutes
- Teacher begins by reminding students of the formulas for trigonometric ratios
- $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \, \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \, \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
- Teacher demonstrates how these formulas can be manipulated using algebraic techniques.
- $\sin \theta = \frac{O}{H} \implies H\sin \theta = O$ (By multiplying $H$ on both sides, students can make $O$ the subject)
- $H\sin \theta = O \implies H = \frac{O}{\sin \theta}$ (By dividing by $\sin \theta$ on both sides, students now can make $H$ the subject)
- Students are to copy the procedure in their books, or write in their own words.
- Students are then challenged to manipulate the other two trigonometric ratios, cosine and tangent.
NOTE: Teacher must ensure that all students are able to manipulate the equations for trigonometric ratios.
- Example Questions
- Teacher must ensure that students observe the question and recognise the appropriate trigonometric ratio to use
- Teacher must ensure that students establish the formula for the trigonometric ratio before manipulating the equation
- Teacher instructs students to take out their calculators, and explore conversion between degrees and minutes.
- Teacher poses a question: How many minutes are in a degree, and how many seconds are in a minute?
- Literacy game: Equation Match-Up (LIT)
- Teacher prepares cards: one set with triangle diagrams labelled with sides/angles, another set with equations (e.g. $\sin \theta = \frac{O}{H} \implies H = \frac{O}{\sin \theta}$)
- Students work in pairs or small groups.
- Match each diagram card with the correct equation card.
- Once matched, students explain verbally (30 seconds) why the match works.
Body 2
Independent practice
25 minutes
- Students work independently OR in pairs on worksheet 4.
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
Short discussion (connecting current lesson to the next lesson)
- Students discuss in pairs on this statement
- “We have learnt to find the unknown side for a given angle using trigonometric ratios. Do you believe then that we could find an unknown ANGLE when given values for the ratios?”