Course Profile Principles of Mathematics, Grade 10, Academic, Public

Unit 1: Similar Triangles and Trigonometry

Time: 26 hours

Unit Description

Students are introduced to applications of similar triangles and trigonometry through a variety of activities that use concrete materials and allow students to move about inside and outside the classroom. Primary trigonometric ratios, Sine and Cosine Laws are used to solve problems that are modelled by right-angled or acute triangles. As students move from the first unit to the second unit, they investigate how the tangent ratio for the angle of inclination is connected to slope of a line.

Strand(s) and Expectations

Strand(s): Trigonometry

Overall Expectations: TRV.01, TRV.02, TRV.03.

Specific Expectations: TR1.01, TR1.02, TR1.03, TR1.04, TR2.01, TR2.02, TR2.03; TR3.01, TR3.02, TR3.03, TR3.04, TR3.05.

Activity Titles (Time + Sequence)

What follows is a suggested sequence and timing for teaching Unit 1. Mathematical concepts developed through the activities are noted in [square brackets]. Once the stage has been set through an activity, appropriate follow-up skills or homework are identified. The timing shown for an activity includes enough time for the activity itself, extracting the mathematics from the activity, plus development of the suggested follow-up skills. *360 extra minutes have been allotted for practice of essential skills, review, and quizzes. No time has been allotted to skills identified as prior knowledge. If time is needed for review or re-teaching of these skills, it could be taken from the * asterisked time. The teacher might plan for use of these 360 minutes before beginning the unit, or save them for needs as they develop.

*	Time to practise essential skills, review, take and correct quizzes	360 minutes
Activity 1.1	Similar and Congruent Triangles Review: vocabulary ‘similar’ and ‘congruent’, log-on procedures for the computer lab, The Geometer’s Sketchpad™ commands [Identifying conditions for similar and congruent triangles]	150 minutes
Activity 1.2	Reaching New Heights [Learning four methods for finding heights of tall objects using measurement and approximation techniques] Follow-up skills: applying these methods at various stations, and to textbook questions involving similar triangles	150 minutes
Activity 1.3	SOH – CAH – TOA [Introducing primary trigonometric ratios] Follow-up: developing memory aids for the trigonometric ratios; solve for missing sides of a right-angled triangle Extension: Vector Toy (extra 75 minutes needed)	75 minutes
Activity 1.4	In the Clouds! [Finding missing sides and angles in right triangles using SOH-CAH-TOA] Follow-up: doing more practice	150 minutes
Activity 1.5	Sines and Sides – It’s the Law [Relationship between angles and sides using The Geometer’s Sketchpad™] Follow-up skills: solving triangles, with and without real-world context	225 minutes
Activity 1.6	Up, Up, and Not Away [Applying trigonometry to find heights of distant objects]	75 minutes
Activity 1.7	Cos I Said So [Investigating, how, as a triangle changes from right-angled to acute, the hypotenuse shortens] Follow-up skills: solving triangles with and without context	150 minutes
Activity 1.8	Going to the Fair [Presenting and questioning projects on trigonometric applications]	150 minutes
Activity 1.9	Social Climber [Investigating the link between the slope of a line and the tangent ratio]	75 minutes

Prior Knowledge Required

Solving a proportion; substituting then solving a linear equation

Unit Planning Notes

· This unit is recommended as the first one in the Grade 10 Academic program for the following reasons. The material is new to students (so has no associated baggage) and requires very little algebraic manipulation. It allows students to engage in a variety of kinesthetic activities and have some fun.

· While working on this unit with the class, teachers could assign remedial algebraic work to individual students, if it is indicated through the diagnostic testing done in the Preparation and Gearing Up section. That way, student readiness for the algebraic manipulations in Unit 2 can be optimized.

· Activity 1.1 suggests the use of computers and The Geometer’s Sketchpad™ in order to meet the specific expectation TR1.01. Activity 1.7 suggests the use of technology to save time and to allow students to focus on the relationships rather than on the measurements. The teacher may need to book a lab and review the appropriate features of the software. If lab space is a problem, hand tools may be substituted.

· It would be possible to sequence this unit second, after Analytic Geometry, if use of the computer lab cannot be arranged near the beginning of the course. Note that the Applied profile sequences the Trigonometry Unit second, avoiding conflict with the Academic profile for computer lab space, if it is limited.

· For Activity 1.2, gather metre sticks and small mirrors. The five exercises in the student activity section could be set up as stations. If used in this way, the teacher may wish to plan which pairs of students will work together.

· Activities 1.3, 1.5, and 1.6 can be done with or without The Geometer’s Sketchpad™.

· Activity 1.4 requires the following materials: cardboard, string, protractors, paperclips, and drinking straws.

· Vector toys are needed for the extension to Activity 1.3. See the activity for an illustration. These can be very inexpensive. Companies sometimes use these for promotions, in which case students might be able to bring in their own.

· The products and presentations from Activity 1.8 Trig Fair may be worth sharing with other mathematics classes, administrators, and parents.

· It is suggested that students go outside for Activity 1.6, so some flexibility in timing, according to weather conditions, may be a consideration.

· Note that the Policy document requires the application of Sine and Cosine Laws to acute triangles only. A quick extension to obtuse triangles could be included in Activity 1.7 to demonstrate the reasonableness of a negative value for the cosine ratio of an obtuse angle. It is not intended to be taken farther than that and can be omitted altogether.

Teaching/Learning Strategies

· It is intended that this Unit 1 in the Grade 10 Academic profile provide more physical activity, excitement and engagement on the part of students than some of the remainder of the course will likely allow. If teachers are concerned about the readiness of some students to complete the algebraic manipulations needed in Unit 2, this would be the time to do diagnostic and remedial work.

· This Profile suggests applications of the mathematics in the activities to a variety of jobs and hobbies. Teachers should make a point of highlighting these connections for their students.

· Exploration allows students to engage in the process of developing an understanding of the application of mathematical principles. However, it is critically important that students are guided to clear enunciation of the mathematical principles themselves. An important role for the teacher is facilitating this process of bringing closure to an activity (extracting the mathematics from the context). Carefully providing opportunities for students to communicate their developing understanding, partnered with frequent formative feedback, will ensure development of a solid set of knowledge and skills.

Assessment and Evaluation

It is possible to use the assessment activities as teaching opportunities, and the teaching activities as assessment activities. Students should be assessed in much the same way as they learn.

The profile provides teachers with examples of balanced assessment activities. All assessment data collected by the teacher could be recorded in the four Categories of the Achievement Chart. At the time that an evaluation of student performance is needed, a teacher could weight their assessment data to reflect an appropriate balance among Activities and Categories. It may be that teachers will think it appropriate to give only formative feedback on some of the assessment activities in this profile. If this is the case, the teacher could provide additional instruction on the process or content that is not well enough understood. A substitute assessment activity from the textbook or other resource could be used for formal assessment.

Resources

Exploring Geometry With the Geometer’s Sketchpad. Key Curriculum Press.

Web Sites

The Geometer’s Sketchpad™ and related resources
http://archives.math.utk.edu/software/msdos/geometry/geom_sketchpad/.html
http://www.keypress.com/sketchpad/index.html

Activity 1.1: Similar and Congruent Triangles

Time: 150 minutes

Description

Students use The Geometer’s Sketchpad™ to investigate the properties of similar and congruent triangles.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR1.01 – determine the properties of similar triangles (e.g., the correspondence and equality of angles, the ratio of corresponding sides, the ratio of areas) through investigation, using dynamic geometry software;

TR1.02 – describe and compare the concepts of similarity and congruence.

Planning Notes

· The day before this activity, it is suggested that the teacher do some informal review of the terminology of similar and congruent triangles, log-on procedures, and basic The Geometer’s Sketchpad™ commands. For this activity, each student or pair of students should have access to a computer in order to be fully engaged in the activity. With a single computer in the classroom, the teacher could demonstrate the activity and have students follow along by completing the worksheets. Otherwise, students could work in pairs to construct similar and congruent triangles and measure angles and side lengths by hand.

· It would be advisable to have students work in pairs as they begin to use The Geometer’s Sketchpad™ for this course.

Prior Knowledge Required

Students will require some time to become familiar with The Geometer’s Sketchpad™ and its menu functions, if they did not have an opportunity to do so in Grade 9. Students should also be familiar with such terms as midpoint; corresponding angles, and corresponding sides, and with the concepts of similarity and congruence.

Teaching/Learning Strategies

Teacher Facilitation: Check that the preferences in The Geometer’s Sketchpad™ are appropriate for this activity. Length should be measured in centimetres, accurate to two decimal places. Angles should be measured in degrees, accurate to the nearest degree. In order for students to have a record of their work, teachers may choose to have students print a copy or make a sketch of the final diagram in their notebooks.

The teacher may wish to create a class set of Sketchpad instructions, to be collected and reused. Students could make notes in their notebooks. Although the instructions appear to be long, they are detailed to make them user-friendly for the first activity in the profile.

Student Activity: Investigating Similar Triangles

In this activity you will investigate the relationships of corresponding sides and corresponding angles in both similar and congruent triangles. The following steps guide you through the inquiry process. You will explore the required types of triangles and their measures using a dynamic geometric model that can be manipulated to test many examples quickly. You are encouraged to hypothesize what is special about side, angle, and area measures as you follow the steps outlined. Later steps will help you confirm or deny your hypotheses. At various stages you will be asked questions to summarize your findings and make conclusions.

Create a dynamic geometric model:

1. Open The Geometer’s Sketchpad™.

2. Use the Segment tool to construct a triangle.

3. Use the Label tool to label each of the vertices of the triangle. (Notice that vertices are given capital letter names, A, B, and C)

4. Use the Selection Arrow to select one side of the triangle. Use the Construct menu to create a midpoint. Repeat this process for each of the remaining two sides.

5. Use the Text tool to name each of the midpoints.

Note: If your labels are different from the diagram, change them by double clicking on each label.

6. Use the Segment tool to join the three midpoints to form a new triangle DEF.

7. Construct the midpoints for each side of the new triangle DEF, label the new midpoints and join them. (see steps above if necessary)

8. At this point, you should see three triangles, one inside the other starting with triangle ABC , then triangle DEF, and finally triangle GHI on the inside. As you complete the next section, you will be using The Geometer’s Sketchpad™ to make some observations. Record the data in the chart below.

Before you explore, using Sketchpad, can you form a hypothesis about the lengths of sides and measures of angles in triangles ABC, DEF, and GHI?

Explore:

9. Use the Selection Arrow to select one side of triangle ABC. Use the measure menu to find the length of this side. Record in the table below.

10. Repeat this step for the remaining two sides of triangle ABC. Repeat for triangles DEF and GHI.

11. Use the Selection Arrow and hold the Shift Key down to select points C, A and B. Be sure to select them in the correct order, as The Geometer’s Sketchpad™ keeps track of the order. The middle letter names the vertex and that is the short name used for the angle in the table below. Use the Measure menu to calculate the size of angle CAB. Record this measure as angle A.

12. Repeat this process for the remaining angles in triangle ABC and again for triangles DEF and GHI. Record your information.

Triangle	Side Lengths			Angle Measures
ABC	AB	BC	AC	A	B	C
Area =
DEF	DE	EF	DF	EDF	DEF	EFD
Area =
GHI	GH	HI	GI	HGI	GHI	GIH
Area =

Manipulate the model:

13. Grab a vertex of triangle ABC and drag it. Notice what measures changes and what relationships remain constant in your table. Add a second set of data to your table to illustrate your observations.

14. Answer the following questions to summarize your findings:

a) What do you notice about the measures of corresponding angles in the three triangles?

b) What do you notice about the lengths of corresponding sides?

c) Select one of the sides of the large triangle ABC. While holding the shift key select the corresponding side on the medium triangle DEF. Choose ratio from the Measure menu.
Record the ratio.

d) In a similar manner calculate the ratio of the remaining sides of triangle ABC to the corresponding sides of triangle DEF. Record the ratios.

e) Calculate the ratio of the sides of triangle DEF to the corresponding sides of triangle GHI.

f) What can you say about the ratio of corresponding sides?

Further Investigation

1. Use the Selection Arrow to select one vertex from triangle ABC and drag it to a new position. Observe any changes in the measures of either the angles or side lengths.

2. Use the Selection Arrow and hold down the Shift Key to select points G, H, and I. From the Construct menu, choose Polygon Interior. Use the Measure menu to calculate the area of triangle GHI. Record.

3. Repeat the process to find the areas for triangles DEF and ABC. Record.

4. Examine the ratio of the areas of these triangles.

5. Print a copy of your screen, or draw a neat sketch of your diagram into your notebook. Summarize your findings about areas of similar triangles, referring to your sketch.

Congruent vs. Similar Triangles

1. Name at least two pairs of congruent triangles in the diagram.

2. How many congruent triangles can you find? What criteria did you use to look for congruent triangles?

3. Use Polygon Interior to colour sets of congruent triangles using one colour. Print a copy of your screen, or draw a neat sketch of your diagram in your notebook.

4. You can check to see if they are congruent by:

· selecting the points that create one of the triangles while holding the shift key

· choosing Polygon Interior from the Construct menu

· choosing Copy and Paste from the Edit menu

· moving the “pasted” triangle around the screen. Move it to the other triangles to check if they are congruent.

5. Explain how the pasted triangle test supports the criteria you listed in Question 2.

6. a) Name at least two pairs of similar triangles.

b) For each pair of similar triangles, state which sides correspond and which angles correspond.

c) Colour one pair of similar triangles using a different colour than the colour used in Question 3.

Print a copy of your screen, or draw a neat sketch of your diagram in your journal, than answer the following questions in your journal.

1. Explain in your own words the difference between shapes that are congruent and shapes that are similar.

2. When triangles are similar, what relationship is true about the sides of the triangles?

3. When triangles are similar, what relationship is true about the angles of the triangles?

4. When triangles are similar, what relationship is true about the areas of the triangles?

5. If shapes are congruent would they also be similar? Explain.

6. If shapes are similar would they also be congruent? Explain.

7. Use the information from your chart and the observations you have made to create a definition for similar triangles and for congruent triangles.

Teacher Facilitation: Provide formative feedback on the journal entries that summarize learning in this activity to ensure that all of the appropriate mathematics is drawn out.

Assessment/Evaluation Techniques

While students engage in this activity, the teacher can assess students’ learning skills as pertaining to following written instructions. Assistance should be offered to students who are experiencing problems. Informal feedback on the use of technology will be helpful. Students' journal entries can be informally assessed for completion, correctness, use of markings on diagrams and vocabulary as entries are shared and discussed.

Activity 1.2: Reaching New Heights

Time: 150 minutes

Description

In this activity, students use similar triangles to calculate heights; lengths of, distances between various objects around them. They also use similar triangles and proportions to solve a variety of problems whose context is explained.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR1.02 – describe and compare the concepts of similarity and congruence;

TR1.03 – solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying).

Planning Notes

· For some of the exercises, students will need metre sticks, small mirrors, paper, glue, cardboard or cue cards, depending on the exercises chosen.

Prior Knowledge Required

Students should know the properties of similar triangles – in particular, that the ratios of corresponding sides are equal.

Teaching/Learning Strategies

Teacher Facilitation: Student pairs might be encouraged to share answers first with each other and then with the whole class. The teacher might ask for several definitions for “similar” figures that students have prepared as homework, and then help the whole class to agree on an appropriate one. The teacher may then introduce the concept of using similar triangles to make estimations on distance or height. Many people in occupations such as farming, surveying, or interior design have to make such estimates. Students may do one or more of the exercises outlined below. Several students could calculate the height of the same object from different starting points, then compare data for accuracy. Alternately, students could be paired so that each student pair uses a different method to find the height of an object, and then they compare results with the rest of the class.

Methods 1 and 4 are quick estimation methods, while Methods 2 and 3 involve collecting data (lengths, distances, etc.) and then calculating heights with pencil and paper.

The third method (with the mirror) can be challenging to execute effectively. Students will have to apply the fact that the angle of incidence is equal to the angle of reflection. The value of this method is that it involves pairs of similar triangles that are reflections of one another instead of in the configurations seen more commonly.

Student Activity: Reaching New Heights

Method 1: Finding heights of tall objects using a 45°: 45°: 90° triangle

This method is often used to predict where a tree will fall when damaged trees in a wood lot are being cleared, or which trees should be cut down (based on height).

1. Choose a flagpole, tree, or tall building outside the school. (In inclement weather, find the height of the walls in the gymnasium)

2. Hold a portion of a metre stick straight up such that the height of the metre stick above your arm is the same as the length of your arm (as shown). Your arm should be extended parallel to the ground.

3. a) Position yourself (move closer to or farther from the object) so that you just see the top of the object above the metre stick.

b) Measure the distance from where you are standing to the base of the object.

c) Add this distance to the distance from your shoulder to the ground.

d) What does this estimate represent?

4. Consider the simplified diagram of this scenario on the right. Label the sides of the triangles with the appropriate measures (your arm length etc.)

5. Explain why the triangles are similar. (What are the angles in the triangles equal to?)

6. Why is your answer only an estimate? Is the diagram on the right completely accurate? Explain.

7. What do the line segments hanging down represent?

Method 2: Finding heights of tall objects using similar triangles

1. On a sunny day, measure the length of the shadow of a flagpole.

2. Hold a metre stick at right angles to the ground, and have a partner measure the length of its shadow.

3. Draw right triangles on the diagram below, showing the locations of the shadows. Label the triangles with all the necessary information.

4. Explain why the two triangles are similar.

5. Calculate the height of the flagpole. Show your work!

Method 3: Finding heights of tall objects using a mirror

1. Lay a small mirror horizontally on the ground exactly 1 metre in front of the tall object.

2. Slowly walk backwards until you can just see the top of the object in the mirror. Measure this distance from the mirror.

3. Measure the distance from the ground to eye level.

4. A diagram of the scenario and a simplified version are shown below. Label the appropriate measures on the triangles shown and explain why the two triangles are similar.

5. Calculate the height of the tall object.

Method 4: Estimating the distance of an object using a “Rule of Thumb”

Did you know that your arm is about ten times longer than the distance between your eyes?

That fact, together with the use of similar triangles, can be used to estimate the distance between you and any object of approximately known size.

Imagine, for example, that you’re standing on the side of a hill, trying to decide how far it is to the arcade on the other side of the valley. Parked at the arcade is a red sports car that you are sure is about 2 metres wide on the side facing you.

Here’s what you would do:

1. Hold one arm straight out in front of you, elbow straight, thumb pointing up.

2. Close one eye, and align one edge of your thumb with a particular spot on the front of the arcade. Without moving your head or arm, switch eyes, now sighting with the eye that was closed and closing the other. Your thumb will appear to jump sideways as a result of the change in perspective.

3. How far did it move? Let’s say it jumped about six times the width of the sports car, or about 12 metres. Now you multiply that figure by the handy constant 10 (the ratio of the length of your arm to the distance between your eyes), and you get the distance between you and the arcade - 120 metres!

With a little practice, you’ll find that you can perform a quick thumb-jump estimate in just a few seconds, and the result will usually be more accurate than an out-and-out guess. At a minimum, it will provide some assurance that the figure is in the ballpark — which, in many cases, is as close as you need to get.

1. In the diagram shown, how many similar triangles are there?

2. Why are the triangles similar?

3. Where is your thumb in the diagram?

4. Why did a line have to be drawn down the centre of the triangles? (What do those distances represent?)

5. Try this “rule of thumb” method on an object of your choice. How close was your estimate to the known value?

Teacher Facilitation: It may be possible to set up the following applications as stations, and have groups rotate through the stations. Similar questions could be assigned for homework.

For the fourth application on eye charts, the letters have been specially created so that the width of the second E is half that of the first and the width of the third E is one-third that of the first. Ideally, the students should hypothesize that if the lengths are halved, then the distance away will be halved as well, etc.

Student Activity

Application 1: Finding the width of a river

The diagram shows how surveyors can lay out two triangles to find the width of the St. Lawrence River near Brockville. Use the triangles to calculate the width of the river. In your solution, label the triangles, explain why the triangles are similar, and which sides correspond. (Note: The diagram is clearly not to scale)

Application 2: Calculating the diameter of the earth

Erathostenes was a mathematician who lived around 230 BC. While living in Egypt, he learned that at noon on the first day of summer (about June 21 on the modern calendar), the sun shone directly down into a deep well in the city of Syene. This meant the sun was directly overhead. At the same time, in Alexandria, about 800 km almost due north of Syene, the sun’s rays were at an angle of 7.2° to the vertical.

Using this information, Erathostenes calculated the circumference of the Earth.

He set up the proportion

1. Solve the proportion to find the circumference of the Earth.

2. Use your answer from above to calculate the diameter of the Earth.

3. The actual circumference of the Earth is about 40 000 km. How close was Erathostenes estimate?

Application 3: Similar triangles in ironing boards

Explain why ironing boards always remain parallel to the floor and can be adjusted for people of various heights. You may want to consider the following:

1. The legs of the board are built so that AD = BD and DE = DF, and ADF and BDE are straight lines as shown.

2. Which angles are equal? Why?

3. Join EF. Which triangles are similar? Why? Name three other objects that demonstrate this property and explain why they are designed in that way.

Application 4: Eye charts

When testing a patient’s eyes for focussing ability, one doctor uses these three cards. Card 1 is placed 3.0 m from the patient. The other two cards are placed so that the letter E on them will appear to be the same size to the patient as the E on Card 1.

1. Measure the width of the E on each card.

2. Calculate the ratios and .

3. Use these ratios to predict how far Card 2 and Card 3 should be placed from the patient, if the patient’s vision is normal.

4. Create the three cards and carry out the test with a partner. Do the results confirm your predictions?

5. Can you describe how the experiment would change if Card 2 was placed 3 m away to start?

Assessment/Evaluation Techniques

· The teacher could observe students as they work in pairs to complete the stations. Any of teamwork, communication, and independence skills could all be assessed.

· This is also an excellent time for a performance task based on students’ ability to estimate heights and lengths. For example, have the students estimate the height of the ceiling using one of the methods or the distance to an object in the room (using their thumb). If done in the classroom (or other area of the school), estimates can be verified by measuring. This activity is setting the stage for a similar (outdoor) performance task later on.

Resources

Astronomy online
http://hq.eso.org/outreach/spec-prog/aol/market/collaboration/erathostenes

Activity 1.3: SOH - CAH - TOA!

Time: 75 minutes

Description

In this activity, students discover some important properties relating the sides and angles of right-angled triangles, namely, the primary trigonometric ratios. They also use trigonometry to find missing lengths in triangles.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR1.04 – define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles;

TR2.01 – determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios;

TR2.02 – solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation).

Planning Notes

· Teachers have an option of using the activity sheets as pencil and paper exercises, or using The Geometer’s Sketchpad™ to have students construct the same triangles, use the Measure option to determine side lengths and then the Calculate option to determine the trigonometric ratios.

· There is an extension to this activity entitled “The Vector Toy”. If teachers intend to use this portion of the activity, they will need to allocate more time (perhaps as much as 75 more minutes).

Teaching/Learning Strategies

Teacher Facilitation: The teacher may have students work individually or in pairs to make the measurements and complete the chart, then have them compare results. The teacher will need to specify how many decimal places to carry in the ratio to ensure reasonable accuracy. A discussion of possible causes for differences would also be appropriate. The teacher may wish to discuss the use of trigonometry in many fields such as electronics, construction, surveying, computer graphics, and technology as motivation for the activity. At the end of the activity, the teacher will introduce the proper names for the primary trigonometric ratios.

Student Activity: SOH - CAH - TOA!

In this activity you will investigate the relationship between the angles and side lengths of right angle triangles.

When working with right angle triangles it is helpful to name the sides of the triangle. The names are opposite, adjacent, and hypotenuse and are dependent on the perspective of a given angle or the angle you are working with.

Fill these blanks, then follow the instructions.

The longest side of each right-angled triangle is called the _______________. It is easily found since it is always the side across from the _______________ angle.

The side across the triangle from the given angle is called the _______________ side.

The side that helps form, or is next to, the given angle is called the _______________ side.

1. a) Label the sides of each triangle. The first one is done for you.

b) Use a ruler to measure all three sides of each triangle and record them in the chart below.

Use your calculator to find the ratios of the sides and record them in the chart, correct to three decimal places.

Triangle	Hypotenuse	Opposite	Adjacent	Opposite Hypotenuse	Adjacent Hypotenuse	Opposite Adjacent
A
B
C
D

d) Note that each column of ratios is _______________. This is because we used the same _______________ in each triangle.

This means that for any triangle with a 30° angle, the ratio:

Opposite

Hypotenuse

= ________

Adjacent

Hypotenuse

= ________

Opposite

Adjacent

= ________

Example:

Question: A hydro pole is to be secured to the ground 10 m away with a guy wire as shown. Using a device known as a clinometer, a person measures the angle of elevation to the top of the pole to be 30º. a) How high is the hydro pole? b) What length of wire will be needed?

Solution:

a) How high is the hydro pole?

From our previous work, we know that = 0.577 for 30° angles (approx.).

In this case, the opposite side is h and the adjacent side is 10 m, so = 0.577. Solving this equation will yield an answer of h = _______ m.

b) What length of wire will be needed?

We know that = 0.866 for 30° angles (approximately).

In this case, the adjacent side is 10 m and the hypotenuse is L.

(Solve the equation that has been set up for you and find the value of L).

= .866

so =

Teacher Facilitation: In the next question the triangles have angles of 15°, 30°, 45°, and 60°. If you use different triangles, make sure one of the triangles you use has a 60° angle as the data collected will be used in Question 3. The instructions for Question 3 are very prescriptive. If your class can handle an open question, you could cut out the step by step instructions.

Student Activity

2. a) Use a protractor to measure the angles listed in the chart below and record the measure of the angles in the appropriate cell in the chart.

b) Label the opposite, adjacent, and hypotenuse for each triangle relative to the angles you just measured.

c) Use a ruler to measure all three sides of each triangle and record them under the appropriate headings in the chart below.

d) Use your calculator to find the ratios of the sides and record them in the chart below. Round your answers to three decimal places.

Angle Measure	Hypotenuse	Opposite	Adjacent	Opposite Hypotenuse	Adjacent Hypotenuse	Opposite Adjacent
mÐA =
mÐD=
mÐG=
mÐJ=

3. In earlier work, you used similar triangles to solve for missing side lengths. Now you will use the information in the table and trigonometry to solve for a missing side length.
Example: A surveyor needs to know the height of a cliff. He measures 100 m horizontally out from the base of the cliff and then looks up and measures the angle to the top of the cliff to be 60°. How high is the cliff?

Try to solve this problem on your own. If you have difficulties, follow the procedure below.

Procedure (hints):

a) Draw a diagram and label the sides of the triangle.

b) Set up a ratio involving the side we want and the side we know.

c) In this case, the ratio needed is opposite over adjacent.

d) According to your chart, this ratio for 60° should be approximately __________

e) Use this information to finish setting up the equation and solve to find the height, h.

Teacher Facilitation: At this time, provide other examples where students use the values for the ratios collected in the chart to find missing side lengths in right triangles containing those specific angles.

Student Activity

4. We can now solve any problem involving a right-angled triangle containing one of the angles in our chart. What do we do if we have a different angle like 57° or 39°? We certainly do not want to make a chart with every single angle on it and all the ratios! In the past, these tables did actually exist and were included in most math textbooks. Students looked up the values whenever they needed them. Today, we use our calculators. If you look at your calculators, you will not find a button that says opposite/hypotenuse or the other ratios on it. That is because these special ratios have been given specific names.

Opposite

Hypotenuse

is called the sine of the angle,

Adjacent

Hypotenuse

is called the cosine of the angle and

Opposite

Adjacent

is called the tangent of the angle.

These three ratios are called the Primary Trigonometric Ratios.

To simplify things, mathematicians, scientists, engineers, and anyone else using trigonometry use Greek letters like θ (pronounced “theta”) as the angle and write:

sin(θ) =

Opposite

Hypotenuse

cos(θ) =

Adjacent

Hypotenuse

and tan(θ) =

Opposite

Adjacent

In order to remember these ratios, many people just use the term SOH-CAH-TOA.

How does this memory aid work?

Teacher Facilitation: The teacher will need to do some typical examples to find missing side lengths in right triangles where an angle is known. It might be beneficial to remind the students that there are often several ways to determine the same length. (Use the complementary angle, a different trigonometric ratio, Pythagorean theorem, etc.). Assign problems from the textbook for homework.

Extension: The Vector Toy!

Teacher Facilitation: Teachers may want to present a brief introduction to forces and how to break them down into horizontal and vertical components before proceeding with this extension.

Description: The Vector Toy!

This toy is an interesting application of force. It illustrates the vector components used to make the toy walk forward and to make it stop at just the right time. A mass is attached by a string to the front of the walking toy and then hung over the side of a desk. The mass causes the toy to walk forward and amazingly come to a halt right at the edge of the desk. The string itself essentially forms the vectors that make the toy work!

Materials: Walking vector toy (these are available at specialty stores), board, protractor, balance, paper clips to use as weights.

Part A

1. Attach a mass by string to the front of the walking toy.

2. Watch the toy walk forward. Resist the temptation to stop the toy when it gets to the edge.

3. Why doesn’t the toy fall off the desk? How does it “know” when to stop?

Part B

1. Remove the mass and string from the front of the toy.

2. Place the toy on the board.

3. Raise the board to an angle that just causes the toy to walk down the ramp.

4. Use the angle of the ramp and the mass of the toy to calculate how much weight would have to be attached to the string to get it to move forward.

Lesson

· For the toy to move, you must apply a force that is at least as great as the frictional force trying to stop it. The weight of the mass pulls along the string and provides the force that results in the toy’s motion. The string pulls diagonally, though, and only the horizontal component of the force makes the toy move forward. We will have to break up the force vector in order to isolate the horizontal component.

· As the toy moves closer to the edge of the desk, the angle of the pull changes. The component of force pulling forward gets relatively weaker, and the component pulling down gets relatively stronger.

· At the edge of the table, there is no component of force pulling the toy forward, so it stops! When the toy stands on the ramp, the force that causes its motion comes from its weight. The weight vector, however, acts perpendicular to the tabletop, not the ramp. So again we break the vector into its components. The vector that is parallel to the ramp is responsible for the forward motion of the toy. The triangles created by the ramp and table and by the toy and ramp, as shown in the diagram, are similar, and the angles indicated are equal.

Enlargement

Measure the angle of elevation, q, of the ramp (between the ramp and the desk)

· Find the mass of the toy.

· Find the mass of a small paper clip.

· Perform the necessary calculations to determine how many paper clips should be needed to make the toy move (as outlined in the example below).

· Test to see if your calculations were correct, experimentally, with the vector toy.

For example,

[Recall that ]

Rearrange the equation above and solve for Force (of Pulling)

Force (of Pulling) = Force (of Weight of Toy) ´ sin(θ)

So if the toy has a mass of 0.021 kg and the angle of elevation is 4°, we get:

mass of toy = 0.021kg Angle of elevation = 4°

Force (of weight of toy) = 0.021kg ´ 10N/kg sin(4°) = 0.06975

= 0.21 N

Then, Force (of Pulling) = Force (of Weight of Toy) ´ sin(θ)

Force (of Pulling) = 0.21 N ´ (0.06976)

= 0.015 N

Now, calculating the force caused by the weight of one paper clip:

mass of one paper clip = 0.00043 kg

Force (paper clip weight) = 0.00043 kg ´ 10 N/kg

= 0.0043 N

Finally, we will need to divide the Force (of Pulling) by the weight of one paper clip to find the number of paper clips needed to move the toy forward:

# of paper clips = Force (of Pulling) ¸ Force (caused by the weight of one paper clip)

= 0.015 N ¸ 0.0043 N

= 3.48 or 4 paper clips.

The 3.48 paper clips would sustain the motion if the force were applied horizontally at the base of the toy, as shown in the first diagram. Place the toy on a level surface, attach the string and four paper clips to the toy (hanging freely off the edge) and check to see if (with a gentle push to start) it continues moving.

Teacher Facilitation: As a follow-up, the teacher could discuss component vectors and take a look at textbook questions or other activity requiring similar calculations: boats and currents; airplanes and wind, etc.

Follow-up Activity: Students could be asked to come up with a creative way to remember the primary trigonometric ratios (SOH-CAH-TOA). Some examples might be a song, a rhyme, a riddle, or an interesting acronym.

Assessment/Evaluation Techniques

Standard questions involving missing side lengths could be given as an assignment or quiz with an objective marking scheme. The teacher could ask students to create a journal entry that summarizes their learning. For the vector toy extension, the teacher may want to assess some of the learning skills such as teamwork as the students complete the activity.

Resources

For more information regarding the vector toy, visit
http://www.exploratorium.edu/snacks/vector_toy/index.html

Activity 1.4: In The Clouds!

Time: 75 minutes

Description

Students use trigonometry and their calculators along with the Pythagorean theorem to determine missing lengths and angles in right-angled triangles from a given context. Students also build and use clinometers to determine heights of inaccessible objects around the school.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR2.01 – determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios;

TR2.02 – solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation);

TR2.03 – determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles.

Planning Notes

· This activity can be used as an introduction to solving triangles or after a lesson has been taught on solving triangles.

· Instructions are provided for the students to build their own clinometers. Students will need the following materials: cardboard, string, protractors, paperclips or other weight and drinking straws.

· The third part of this activity involves going outside to measure tall objects. If the weather does not permit this, students could work in the gym, cafeteria, or stairwell to measure ceiling height, height of a basketball net, etc.

Teaching/Learning Strategies

Teacher Facilitation: The first two parts of the activity should require little guidance from the teacher, but for the third part of the activity the teacher may find it helpful to have a sample clinometer ready to use as an example. It might be motivating to have students look up and discuss the history of the clinometer and its use in measuring inaccessible distances. If a set of clinometers is already available, the teacher may choose to use these instead of having the students make their own (and skip “Making A Clinometer” in Part 3).

Student Activity: In The Clouds!

Part 1: Doing Trigonometric Calculations On Your Calculator

In many situations you will need to calculate the value of the various trigonometric ratios (sine, cosine and tangent) for a particular angle. Your calculator does this very easily. For example, to calculate sin(33°) you either type “33 [sin]” or for some calculators you type “[sin] 33 =”. [Make sure your calculator is in degree mode]. It is very important to remember what the values you calculate really mean. The following chart might help to remind you. The first one is filled in for you; you fill in the other two.

Key	Angle	Answer	What The Answer Means
sin	37°	0.602	In any right triangle with a 37° angle, the ratio is equal to approximately 0.602 [i.e., The opposite side is about 60% as long as the hypotenuse.]
cos	71°	0.326
tan	22°	0.404

Sometimes you may need to determine the value of a missing angle instead of a side length, as in the following example:

A wheelchair ramp is usually set up with a slope of 1:12. That is, for every 12 units of distance covered horizontally, the ramp can rise 1 unit. At what angle is a wheelchair ramp inclined?

You can get the angle measure on your calculator. For most calculators, the button looks like [tan^-1] and you might have to use your 2^nd or shift button to use it. In this case, you would type “1 ¸ 12=” and then “tan^-1 =”or for some calculators, you would type “tan^-1 (1 ¸ 12) =”. You say “inverse tan” for tan^-1.

Try it on your calculator to make sure you are doing it correctly (you should get 4.76°)

There are also [sin^-1] and [cos^-1] buttons. Once again it is very important to understand what is going on when you use your calculator. Fill in the following chart:

Button	Ratio	Answer	What The Answer Means
[sin^-1]	0.656	41°	In any right triangle, if the ratio is equal to 0.656, the reference angle is approximately 41°.
[cos^-1]	0.515
[tan^-1]

The following is an example where it is necessary to determine missing angles as well as missing side lengths.

A ladder that is 5 m long is leaning on a window ledge 4 m above the ground as shown.

a) What acute angle does the ladder form with the ground?

b) What angle does the ladder form with the wall? (Did you need to use a trigonometric ratio for this?)

c) How far out from the base of the wall is the foot (bottom) of the ladder?

d) Verify your answer for part c) by calculating the distance using a different method.

Part 2: Finding the Heights of the Clouds

At airports a sweeping light beam is sometimes used with a light-source detector to determine the height of clouds directly above a detector on the ground as in the diagram. With the axis of the detector vertical, the light beam is allowed to sweep from the horizontal to the vertical. When the beam illuminates the base of the clouds directly over the detector, the angle is read from a protractor scale on the light source. In the diagram, h represents the height of the clouds, d is the distance between the light source S and the detector D and θ is the angle that is measured.

Answer the following questions using trigonometry. The first one is done for you.

1. The light source S at Sky High Airport is 300 m from the detector D. Calculate the height of the clouds if the angle created by the light beam is 35°.
Solution:

2. Calculate the height of the clouds at Sky High Airport if the angle created by the light beam is 30°.

3. On a certain day the ceiling, that is, the height of the clouds at Sky High Airport is 500 m. Calculate the angle shown on the protractor scale of the light source.

4. Aircraft are not allowed to fly at Sky High Airport if the angle is less than 80°. Calculate the minimum ceiling for flights at Sky High.

Part 3 – The Clinometer

Making A Clinometer

In this activity, you will build a clinometer, a device that will allow you to calculate the heights of various objects.

Materials: drinking straw, a semi-circle of cardboard, tape, piece of string, and weight (several paper clips will suffice )

Procedure:

· Draw a baseline along the bottom edge of the cardboard, and mark the centre.

· Use protractor to mark the cardboard into degrees from 90° to 0° to 90° with zero at the bottom of the curve (see diagram). Make sure that the centre of the protractor meets the centre of the cardboard.

· Tape the straw along the straight edge (top) of the semicircle.

· Tape the string to the centre of the straight edge of the semicircle. Attach a weight to the string.

Using A Clinometer

1. Find a flagpole or a tall tree, or a basketball net, or a light fixture high up in a ceiling.

2. Hold the clinometer at eye level and sight the top of the pole by looking through the straw.

3. Ask a partner to read the angle on the clinometer (where the string is touching).

Record this value. Angle of elevation ___________

4. Measure the distance between where you are standing and the base of the flagpole.
Record this value ______________

5. Measure the distance between your eyes and the ground. Record this value _____________

6. Complete the diagram below of you and the flagpole by adding the values you recorded.

7. Use these values to calculate the height of the flagpole. Which trig ratio will you need to use? Why?

8. Why was it necessary to calculate the distance between your eyes and the ground?

9. Use the clinometer to find the height of at least three different objects. Compare your results with a classmate.

Teacher Facilitation: After completing this activity students should be able to determine any missing angles or side lengths in right-angled triangles. The teacher will probably wish to assign a number of questions on solving triangles for practice.

Assessment/Evaluation Techniques

For the first two parts of this activity, the students could be assessed with a pencil and paper assignment or quiz testing knowledge and understanding and possibly application (given a different context). The third part of the activity involving the clinometer is a good opportunity for a performance task assessment. For example, after having practised with the clinometer, the students could be asked to go and individually determine the height of a specific object on school property. In order to complete the assessment in one class, teachers could send three or four students at a time, each measuring different objects, with only a limited amount of time for data collection. The write-up of the performance task could be assessed for Communication using criteria: clarity of explanation and use of labelled diagrams. Application could be assessed using an objective marking scheme.

Activity 1.5: Sines and Sides - It’s the Law

Time: 225 minutes

Description

This activity is separated into three parts. In Part A, The Geometer’s Sketchpad™ could be used to construct and measure triangles to investigate the relationship between the length of a side and its opposite angle [e.g., opposite the largest angle is the longest side]. In Part B, students investigate the relationship between the length of the side on an acute triangle and the sine of its opposite angle. [i.e., the Sine Law]. In Part C, students look at a formal proof of the Sine Law.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR3.01 – determine, through investigation, the relationships between the angles and sides in acute triangles (e.g., the largest angle is opposite the longest side; the ratio of side lengths is equal to the ratio of the sines of the opposite angles), using dynamic geometry software;

TR3.02 – calculate the measures of sides and angles in acute triangles, using the sine law and cosine law;

TR3.03 – describe the conditions under which the sine law or the cosine law should be used in a problem;

TR3.04 – solve problems involving the measures of sides and angles in acute triangles.

Planning Notes

· Students will work in pairs throughout the activity. Using The Geometer’s Sketchpad™ will save time on construction and measurement, and give more time for analysis. Alternatively, students could work in pairs to construct a triangle and measure the side lengths and angles by hand. The results of each group would then be recorded in a chart on the board or overhead projector so that the students could form conclusions regarding the sine law relationship.

Prior Knowledge Required

Students should:

· be comfortable using The Geometer’s Sketchpad™;

· be able to solve a proportion.

Teaching/Learning Strategies

Teacher Facilitation: The teacher may need to review the definition of an acute triangle. Students may investigate the relationship in obtuse triangles as an extension. Check that the preferences in The Geometer’s Sketchpad™ are appropriate for this activity. Length should be measured in centimetres, accurate to two decimal places. Angles should be measured in degrees, accurate to the nearest degree.

Introduce the convention of naming angles with single upper case letters and their opposite sides with the same lower case letter.

This activity has been structured using the actions suggested in the Inquiry/Problem solving Chart.

Student Activity

Part A: Opposite Sides and Angles

We want to find a relationship between side lengths and angle measures in a triangle.

Explore

1. Open The Geometer’s Sketchpad™.

2. Use the Segment tool to construct an acute triangle.

3. Use the Label tool to name the vertices A, B, and C.

4. Use the Label tool to name the opposite sides of the triangle a, b, c as shown in class. You will need to double click on the labels to change them to a, b, c.

5. Measure ÐCAB.

6. Measure the length of side a.

7. Measure ÐABC.

8. Measure the length of side b.

9. Measure ÐACB.

10. Measure the length of side c.

Hypothesize

Form a hypothesis about the relationship between the length of a side and the measure of its opposite angle in any triangle. Record this hypothesis in your notebook.

Transform/Manipulate

1. Record your observations for Triangle 1 in the following table.

Observations
Triangle	ÐA (mÐCAB)	a	ÐB (mÐABC)	b	ÐC (mÐACB)	c
1
2
3
4
5

2. Use the indicator arrow to select and drag one of the vertices. Record the data for this Triangle 2 in the table.

3. Repeat steps 1 and 2 three more times for Triangles 3, 4, and 5.

4. Save this sketch for the second half of this activity.

Infer/Conclude

Answer these questions in your notebook.

1. For each triangle, how do the side lengths compare to their corresponding opposite angle measures? State any trends that you have observed. State specific examples to justify your answer.

2. Was your hypothesis supported by the results of this investigation? Why or why not?

3. How might you use the trends that were observed in this investigation when solving a problem?

Student Activity

Part B: Sines and Sides: It’s The Law

In Part A you investigated the relationship between the size of an angle and the length of the side opposite it. In this activity you will investigate the relationship between the length of the side of an acute triangle and the sine of the angle opposite to it.

1. Open The Geometer’s Sketchpad™.

2. Open the sketch made for Part A.

3. From the Measure menu, select Calculate. A calculator will appear on the screen. Select Functions and select Sine. Now select m ÐCAB where it is recorded on your sketch. It should appear on the calculator screen. Select the] key to close the brackets and select OK.

4. Now set up the ratio of the length of a side to the sine of the angle opposite that side. Select Measure and then Calculate. Select the length of side a on the sketchpad, then the “/” symbol on the calculator, and sinÐCAB on the sketchpad. Select OK. The sketchpad should show the ratio of these two measures.

5. Repeat steps 3 and 4 for side b and ÐABC.

6. Repeat steps 3 and 4 for side c and ÐACB.

7. Record your results in the following table in the row for triangle 1.

Triangle	or	or	or
1
2
3
4
5

8. Use the Selection arrow to choose one of the vertices of the triangle and drag the vertex to form a different acute triangle. Record your results in the table. Repeat this step three more times.

Questions

1. What do you observe about the ratios that were calculated for each triangle?

2. Write a mathematical expression that summarizes the relationship.

3. What would happen to the ratios if their reciprocals were formed?

4. Write a mathematical expression that summarizes the relationship.

5. When would it be convenient to use the relationship in order to solve a problem? (What pieces of information would you need?)

Teacher Facilitation: A follow-up discussion may help the students form conclusions about their observations. Students may need some help in working with three equal ratios. Students will need to see some examples of how to use the Sine Law to solve for missing information in triangles, with and without real-world contexts. Provide practice in using the Sine Law by assigning problems from the textbook.

Part C: Proof of Sine Law

Teacher Facilitation: It is not intended that students will be able to reproduce the formal proof of the Sine Law but rather be exposed to it.

Students should work in heterogeneous groups. Teachers will photocopy the steps of the Sine Law for each group and cut them into separate pieces. Students will work with their group to put these steps in order.

Student Activity: Proof of Sine Law

The pieces provided contain the proof of the Sine Law, which was investigated earlier. Put the pieces in a logical order.

Construct altitude h.

and

rearrange these equations to get: c sinA = h and a sinC = h

since c sinA = h and a sinC = h

c sinA = a sinC

dividing both sides by ac gives us

Teacher Facilitation: Discussion should follow to show how rotating this triangle and completing similar steps could prove the remaining parts of the Sine Law. Students could be asked to write the steps to show . The teacher should discuss with the students how it can be deduced that if p = q and q = r then p = q = r.

The student activities have been organized around key actions of a mathematical inquiry: explore, hypothesize, transform/manipulate, infer/conclude. The model action has been laid out for students in the table template. This would be a good time to help students begin a note on the Inquiry/Problem solving actions. As new knowledge and applications are acquired, they can be added as a tool for the explore action

Assessment/Evaluation Techniques

Have students, working in pairs, peer and self-assess the journal entries in their notebooks, using the following criteria: completeness, clarity of communication, appropriateness of labels on diagrams, integration of narrative, and symbolic forms. Pencil and paper quizzing of application of Sine Law could be administered quickly. An objective marking scheme could be used.

Activity 1.6: Up, Up and Not Away

Time: 75 minutes

Description

In this activity, students use trigonometry to calculate the height of distant objects.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR1.04 – define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles;

TR2.01 – determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios;

TR2.02 – solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation).

Planning Notes

· Teachers have an option of using the activity sheets as paper and pencil exercises, or using The Geometer’s Sketchpad™ to have students draw the same triangles, use the Measure menu to determine side lengths and then the Calculate menu to determine the trigonometric ratios.

· Each pair of students will need two clinometers, a helium filled balloon, a measuring tape, a 5 m length of string and two position markers (e.g., stakes, fluorescent tape tied to spikes)

· This activity could be used, either as a tool to teach connections in an application problem, or as an assessment tool. No new concepts are introduced; connections need to be explored.

Prior Knowledge Required

Students will need to review the names of the sides of a triangle from the perspective of a reference angle (i.e., opposite, adjacent, hypotenuse).

Teaching/Learning Strategies

Teacher Facilitation: It is suggested that the teacher have students work in pairs to make the measurements and complete the chart. At that time, students could be asked to answer the questions individually, or to continue to work in pairs. The teacher will need to specify how many decimal places to carry in the ratio to ensure reasonable accuracy. A discussion of possible causes for differences would also be appropriate.

The teacher should pre-mark the string that attaches to the balloon with three or four colours, at pre-determined lengths. Different pairs of students could be assigned to work with strings whose lengths are marked with different colours.

Student Activity: Up, Up, and Not Away

You have already used a clinometer to determine the height of a tall object like a tree or a building. You measured the distance from your position to the base of the object and assumed that the object was perpendicular to the ground. What if it was impossible to measure that distance because of an obstacle such as a river? Determining the horizontal distance to a point under an aircraft is an example of a scenario where that is the case.

Materials: Send the materials person in your group to gather: two clinometers, a helium filled balloon, measuring tape, two position markers (e.g., stakes or fluorescent tape tied onto spikes), a length of string at least 5 m long.

Procedure

1. Place the markers roughly ten metres apart. Record the actual distance between the markers, accurate to one centimetre, in the table below.

2. A student with a clinometer should stand at each marker. Both students are going to measure the angle of inclination to the balloon from their position. (One student will measure ÐA and the other will measure ÐB).

3. A third student (or the teacher) will release the tied balloon, playing out string until the requested colour mark on the string is in hand. (Three or four pre-measured coloured marks should be made on the string.) Keep track of the colour of the mark which will correspond to how high above the ground the balloon was flying.

4. When the balloon is in position, C, each student should measure the angle of inclination to the balloon. Record the measures of ÐA and ÐB in the table below.

Data Table
Distance Between Markers (m)
ÐA
ÐB
ÐC

Teacher Facilitation: The teacher may wish to have students complete the following questions individually, or in pairs.

Questions

1. Sketch this scenario. Make sure that you label the diagram with angle names and measures and side names and any known lengths.

2. What is the measure of ÐC? How did you know?

3. Express in your own words how the sine law will apply to this scenario.

4. Use the sine law to calculate distances a and b. Show your steps.

5. How could you use this information to determine the height of the balloon? Hint: On your diagram, draw a vertical line from the balloon to the ground. Use the definition of the sine function in right angle triangles to form the parts of your equation.

6. Describe other scenarios where you think this technique would be useful.

Teacher Facilitation: A follow up discussion may be needed to help the students set up the sine law equation. The teacher should reinforce for the students how to properly set up a solution.

Assessment/Evaluation Techniques

Teamwork and the ability to interpret written mathematical communications could be assessed while students engage in this activity. An objective marking scheme could be used to assess individual written submissions of answers to the questions posed. Or, a rubric could be developed since a range of performances is likely. Knowledge, Application, and Communication Categories could be assessed. Under Communication, headings in the rubric could be:

· ability to read and interpret mathematics;

· quality of reporting on the application of the Sine Law;

· correctness of use of mathematical symbols, labels, and conventions.

Activity 1.7: Cos I Said So

Time: 150 minutes

Description

Students discover how the Pythagorean theorem evolves into the Cosine Law when dealing with acute angles (rather than right angles). The development of the Cosine Law is general at first, and then only with very specific guidance is the Cosine Law itself be introduced. Applications follow.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR3.02 – calculate the measures of sides and angles in acute triangles, using the sine law and cosine law;

TR3.03 – describe the conditions under which the sine law or the cosine law should be used in a problem;

TR3.04 – solve problems involving the measures of sides and angles in acute triangles.

Planning Notes

· The Geometer’s Sketchpad™ is needed for this activity. Students will work in pairs throughout the activity. Alternatively, students may work in pairs to construct a triangle and measure the side lengths and angles by hand. The results of each group would then be recorded in a chart on the board or overhead projector so that the students could form conclusions regarding the Cosine Law relationship.

Teaching/Learning Strategies

As the students begin to form hypotheses regarding the Cosine Law, the teacher will need to reassure them that only a generalization is required (e.g., As the angle changes from a right angle to an acute angle, what used to be the hypotenuse becomes shorter and shorter. Some length has to be subtracted.). The students will not be able to suggest a specific modification (i.e., Reduce the former hypotenuse by the quantity 2abcosC) at this time.

Student Activity: Cos I Said So!

Students will work in pairs to carry out the following investigation.

First, let’s revisit the Pythagorean theorem. In your notebook, state the Pythagorean theorem in your own words. Include a labelled diagram if that would help.

What condition must be met before this theorem can be used to calculate the length of a side of a triangle? (Write the answer in your notebook.)

Form a hypothesis about the modifications that would have to be made to the Pythagorean relationship to compute what used to be the hypotenuse as the right angle is rotated to become smaller. Record the hypothesis in your notebook.

Now, explore - gather some data!

1. Open The Geometer’s Sketchpad™ .

2. Use the Segment tool to construct an acute triangle.

3. Use the Label tool to name the vertices A, B, and C.

4. Also use the labelling tool to name the corresponding (opposite) sides a, b, and c.

5. Measure ÐACB (often called simply ÐC, but Sketchpad labels it mÐACB).

6. Measure the lengths of sides a, b, and c.

7. Select point C. Drag point C until ÐACB is 90°.

8. What is the Pythagorean relationship for this triangle? Record the equation that you would use to calculate the length of the hypotenuse in your notebook.

Model - Set up the Pythagorean relationship on the Sketchpad

9. Next you will have Sketchpad calculate the value of a² + b².
Select the displayed lengths of a and b. From the Measure menu, choose Calculate. When the calculator appears, choose length (a) from the values menu, select ^ , 2, and + on the calculator, then choose length (b) from the values menu, and select ^ and 2 on the calculator. Click on OK. The Sketchpad should show the sum of the squares of the two shorter sides of the triangle.

Move this calculation to a clear spot on your sketch.

10. Next you will have Sketchpad calculate the value of c².
Select the displayed lengths of c. From the Measure menu, choose Calculate. From the Values menu (on the calculator), click on length (c), select ^ and 2 on the calculator and click on OK. The Sketchpad should now show the value of c².
Move this value to a spot underneath the calculation for a² + b².

11. Is a²+ b² = c²? If not, are they close? Why might they be off by a little bit?

12. What happens if you move vertex C further away from line segment AB? (Name three things that change.)

13. Does the Pythagorean theorem work anymore? Why not?

14. Now we need to construct the expression a² + b² - c² (to determine how much the Pythagorean theorem is off by). To do this, select the various components and from the Measure menu choose Calculate and fill in the operations. (Refer to steps 9 and 10 for help.)
Place this expression by itself in a clear spot on your sketch.

Manipulate/transform the geometric model and record changes to the numeric model

15. Move point C to five other locations and record the data in the table below. Make sure one of your triangles has ÐACB = 90°

Observations
Triangle	a	b	c	ÐACB (or ÐC)	a² + b²	c²	Missing Part (a² + b² - c²)
1
2
3
4
5

16. Construct the expression 2abcos(mÐABC). Here’s how: Select (using the shift key) the displayed lengths for a and b, as well as the displayed angle measure for ÐACB. From the Measure menu, select Calculate and once in the calculator, type 2 * [length(a)] * [length(b)] * cos(m ÐACB) and then choose okay. Note: length(a), length(b) and m ÐACB are all found in the values menu, cos is found in the functions menu. Place this near your expression for a² + b² - c² from above.

Infer/conclude

17. Are they the same? Do you think it was just a lucky guess?

18. Compare your answers to Question 12 to the expression in Question 16. Do you see anything in common? Does it make sense? Explain.

19. Based on your observations, does the Pythagorean relationship apply to acute triangles?

20. What modification would be needed to make the Pythagorean theorem work for acute triangles? (What expression do we need to include on the right side of the equation c² = a² + b² so that it works for acute triangles?)

Sample Screen

Teacher Facilitation: Be sure that the students realize that by moving point C the three things that change are a, b, and the angle at C – all of the components of the extra term in the cosine law. Assign completion of the activity for homework. On the second day of this activity the teacher should begin with a full class discussion of results. Ask the students to present their suggestions for modification to the class and summarize these on the board or overhead projector. Show the derivation of the Cosine Law and work through some examples of its application. Be aware that the formal derivation of Cosine Law usually involves the solution of a system of equations, a process not yet expected of students working on their own. Students should, however, be able to follow the teacher’s development. Discuss when to use the Cosine Law in a question. Pose this question to the students after giving them a number of examples where it is and is not needed. Assign problems from the textbook for homework.

Assessment/Evaluation Techniques

Students should be given a variety of questions where they have to decide whether to use Sine Law, Cosine Law, Pythagorean theorem, or SOH-CAH-TOA. Students could create a journal entry that provides an explanation of when it is appropriate to use each of these. Encourage use of a flow chart or statements written in “If…, then…” format. Teachers could assess their solutions to application questions using an objective marking scheme.

Accommodations

Students who have trouble following written instructions should be paired with a partner who works through each step of the investigation with them.

Activity 1.8: Going to the Fair

Time: 150 minutes

Description

In this activity, students recognize and describe how and where trigonometry exists in the world around them. They share their findings with peers in a "fair" environment, after doing their preparations as part of homework time over a couple of weeks.

Strand(s) and Expectations

Strand(s): Trigonometry

Specific Expectations

TR2.02 – solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation);

TR3.04 – solve problems involving the measures of sides and angles in acute triangles;

TR3.05 – describe the application of trigonometry in science or industry.

Planning Notes

· The teacher could speak with the teacher-librarian in advance to co-ordinate use of the library and arrange for a brief review of research techniques.

· Internet access should be booked at this time if possible.

· Guests could be invited to Fair day.

Prior Knowledge Required

Unit 1

Teaching/Learning Strategies

Teacher Facilitation: The teacher should set out clear timelines for the completion of predetermined phases of the project. Progress should be closely monitored to ensure that students stay on track and complete their projects on time. No class time need be used for completion of the projects beyond a first research day in the Library/Resource Centre.

During the fair, one member of each pair should remain with the project for the first half of the class, while their partner goes to the fair. Partners exchange roles at half time.

Student Activity: Trigonometry Applications Mathematics Fair

Instructions

1. Working in pairs, students will research the use of trigonometry in everyday life.

2. Students will spend one period in the library to participate in a workshop on the use of specialized sources, including the Internet, to research their topic.

3. Each group will prepare a project that will have both a visual and an oral component. Projects will be displayed on bristol board and should clearly demonstrate the importance of trigonometry in their applications.

4. Each student must create a problem based on their findings that requires the use of trigonometry to solve. The problem should be part of the visual presentation.

5. Information will be displayed and presented in a “fair” atmosphere. All presentations must be ready at the beginning of the period. Oral presentations should last no longer than five minutes. Each student will be required to present the project, alone, for half of the class.

Assessment/Evaluation Techniques

· Knowledge can be assessed through the oral presentation and the written solution to the problem posed. Communication can be assessed, by considering the depth and clarity of the visual and oral parts of the presentation. Application can be assessed by examining the quality of the connection made to trigonometry through the project.

· Students can be asked to do peer evaluations of their classmates’ projects and self-evaluation of their efforts. Individual assessment data can be assigned for the Knowledge and Communication aspects of the project, while shared assessment data might be more appropriate for the Application component. To facilitate appropriate sharing for Application, the teacher might ask the partners to decide on the part of the mark that goes to each of them. A more meaningful discussion, between the partners, might be encouraged, by ensuring that they work with an odd number of marks, so that they cannot default to a 50:50 split.

Activity 1.9: Social Climber

Time: 75 minutes

Description

In this activity, students relate the concepts of slope, equation of a line and the tangent ratio by investigating the construction of staircases. The main motivation for this activity is to provide some review of the equation of lines and to provide a transition from the Trigonometry unit to the Analytic Geometry unit.

Strand(s) and Expectations

Strand(s): Trigonometry, Analytic Geometry

Specific Expectations

TR2.01 – determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios;

TR2.02 – solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation);

TR2.03 – determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles.

Planning Notes

On the day before doing this activity in class, assign the students the task of measuring the tread depth and riser height of a staircase at home. Students who do not have a staircase at home may measure a staircase at school or another public building. Graph paper will be necessary for a scatter plot and a diagram.

Prior Knowledge Required

Students will need to know how to use and , and how to write the equation of a line in y = mx + b form

Teaching/Learning Strategies

Teacher Facilitation: Give the first part of this activity (obtaining tread depth and riser height) to students for homework a day or two before you want to use the data. Collate the data on a chart or spreadsheet, then provide it to the students in compiled form to complete this activity.

Student Activity: Social Climber

When a builder constructs a staircase, careful attention must be paid to the riser height and tread depth to prevent accidental falls. In this activity you will investigate the dimensions of a set of stairs in your school or home and then explore ways to describe the stairs mathematically.

Common guidelines for building staircases state that:

· All risers must have the same height and all treads must have the same depth for a particular staircase.

· The depth of the treads must be between 25 cm and 30 cm.

· The depth of the treads plus two times the height of the risers should be approximately 64 cm.

Part 1

1. Locate a staircase in your school or home and measure the riser height and tread depth (as shown in the diagram). Record your observations here.

Riser Height: ____________________ Tread Depth: _________________________

2. Set up a table as illustrated below, collecting all of the class data for riser height and tread depth. Calculate the slope for each stair. Remember that .

Tread Depth(cm)	Riser Height (cm)	Slope

3. Construct a scatter plot using tread depth as the independent variable and riser height on the vertical axis.

4. What is the range of values for riser height?

5. What is the range of values for tread depth?

6. Locate the point on the scatter plot that represents the staircase that has the steepest slope. Find the angle of elevation, θ, to the nearest degree for this staircase. [See diagram]

7. Locate, in the data table, the measures for the step from the staircase that has the lowest slope. Find the angle of elevation to the nearest degree for this staircase.

8. What trigonometric ratio did you use to calculate the angles of elevation? Why?

9. What do you notice about the value of this trigonometric ratio and the slope of the stair?

10. Using the guidelines provided at the beginning of this activity, identify the combination of tread depth and riser height that would result in the steepest slope allowable. Plot this point on your scatter plot and connect it to the origin.

11. Calculate the slope of the line in Question 10.

12. Recall that the equation of a line can be written in the form y = mx + b where m is the slope and b is the y-intercept. Use this to write the equation of this line.

13. Using the guidelines provided, identify the combination of tread depth and riser height that would result in the gentlest slope. Plot this point on your scatter plot and connect it to the origin.

14. Calculate the slope of the line in Question 13. Write the equation of this line.

15. Do all of the points in your scatter plot represent staircases that fall within the safety guidelines? If not, identify the ones that do not, and suggest reasons why these points exist.

Part 2

You have been hired as a computer consultant by a construction company to describe, mathematically, the slope and position of staircases.

1. Choose an appropriate riser height and tread depth for your staircase design.

2. Sketch a diagram of your staircase, with the bottom of the first riser placed at the origin. Choose a scale that will allow you to show ten stairs.

3. When constructing staircases, a “stringer” is usually installed, as shown. Determine the equation of the line for the “stringer” for your staircase.

4. Determine the angle of elevation of your staircase.

5. A railing is to be installed parallel to the stringer and 1 m above the height of the “stringer”. Draw a line to represent the railing in your diagram. Determine the equation of the line for the railing.

6. Compare the equations of the lines for the stringer and the railing. How are they the same? How are they different?

7. If the stringer went along the top of the stairs, how would the stringer equation change?

8. What would happen to the equations of the lines if the staircase faced in the opposite direction?

9. Describe one situation where you might find a set of steps which do not conform to the guidelines given earlier.

10. What considerations would you have to take into account if you had to replace a staircase with a wheelchair ramp?

Follow-up Questions

1. In this activity, we found that the slope of the staircase, m, was equal to the value of tan(θ), where θ was the angle of elevation [in other words, m = tan(θ)]. Do you think this relationship holds true for other situations? Why or why not?

2. Describe three other situations where you could verify this phenomenon (i.e., m = tan(θ))

3. What kind of occupations would be concerned that staircases are being constructed to code?

Assessment/Evaluation Techniques

The follow-up questions could be asked in a journal entry or could provide an opportunity for class discussion. As a performance task, teachers could have the students measure tread depth and riser height for a specific staircase in the school and complete a subset of the questions from the activity. In particular, finding the slope of the staircase and the angle of elevation and verifying the relationship.

Students could be asked to investigate why t + 2r = 64 is set as a standard for step construction, or why 25 £ t £ 30 cm, where t is tread depth and r is riser height. This investigation could be assessed for Inquiry/Problem Solving using explore, hypothesize, model, manipulate, and conclude as the headings in a rubric.