secuencia "with a little help from trigonometry"

1MATEMÁTICAS

OBJECTIVESAt the end of this lesson, you will be able to:1. Apply trigonometry to solve real world problems, such as measuring the height of tall buildings2. Identify the trigonometric ratio that must be used3. Use the goniometer quadrant to measure angles4. Apply Trig ratios to solve right-angled triangle pro-blems5. Use the calculator correctly6. Define and describe processes correctly, in written and oral form.7. Represent each situation geometrically and accurately

1. Trigonometric ratios.2. Use of the calculator to find the trigonometric ratios (sinA, cos A and tanA).3. Use of the calculator to calculate angles.4. Use of formulae and trigonometric ratios to solve right-angled triangle problems.5. Use of trigonometry to calculate heights.6. Use of formuale and trigonometric ratios to solve everyday situations.7. Use of instruments to measure angles

CONTENTS

ASSESSMENT CRITERIA1 Use trigonometric ratios of angles less than 90º correctly.2. Use instruments of measure to calculate heights3.Represent different situations by means of geometric figures4. Use the calculator correctly in order to solve trigonometric ratios and calculate angles, knowing the trigonometric ratio5. Solve problems using the appropriate ratio6. Describe the process followed to solve different situations, accurately and using the specific terminology

Sections:1. Identifying and Defining2. Calculations and Measures. Use of Calculators.3. Applications of Trigonometry4. Writing and Solving Word Problems (Card Game)5. Measuring Heights: Our High school6. FINAL TASK: Using Trig to measure city landmarks

WITH A LITTLE HELP OF

side

2 With a little help of trigonometry

SOLVING EVERYDAY PROBLEMS

1.3 How can we calculate heights?Imagine that you have a tall tree in your yard that needs to be cut down. You want to make sure that the tree won't hit your house when it falls. How can you approximate the height of the tree?

1.4 Watch the following video and see what these two people do.

http://www.learner.org/courses/learningmath/video/measurement/wmp/m5b1.html

1.5 Think-Pair-Share: What steps have been given?

1

2

3

4

5

6

SECTION 1: IDENTIFYING AND DEFINING

1.1 How can you measure or calculate the height of a really tall building? Read the text and find out how these calculations were made in ancient times

The legend says that Egyptian priests wanted to test Thales of Miletus’ wisdom and asked him to solve a really difficult task: to calculate the height of Cheops’ pyramid. There are others who say that it was the Pharao Asmasis who called this geometrician to solve this difficult problem . Apparently, he wished to build a pyramid taller than the pyramid of Cheops -- then, there was the need to know the measurements of this pyramid. However, it was not an easy task: to measure the height of an oblique solid.No matter how or why, the fact is that Thales (690 BC) used the following method to measure the Pyramid

height: he lay on the ground and marked the length of his own height with two sticks. Then, he stood up next to one of these sticks and when he saw that his shadow measured the same as the length marked on the ground, he measured the shadow of the famous pyramid. He told the priests: “Now that my shadow and my height are the same, the length of the pyramid must be the same as its height.” So, he concluded its height was 146.5 m. (488 ft).

As the distance from the Sun to the Earth is very big, the sun rays can be considered parallel when they reach the pyramid and the person.

At present, it is completely unnecessary to wait for the sunset. We can use trigonometry to calculate the height of the pyramid of Cheops. Maybe we won’t ever need to calculate the height of the pyramid of Cheops, but there may be other everyday life situations where we may find it useful to apply the so called trigonometric ratios to calculate the height of very high monuments, objects or buildings. In fact, we will use trig to find out how high our school is.

a) Thales of Miletus measured the height of the pyramid by ____________________b) Trigonometry is really useful to ________________________________________c) In order to calculate heights, we will use_________________________________

1.2 Read again and complete the following statements:

3MATEMÁTICAS

SOLVING EVERYDAY PROBLEMS

1.3 How can we calculate heights?Imagine that you have a tall tree in your yard that needs to be cut down. You want to make sure that the tree won't hit your house when it falls. How can you approximate the height of the tree?

1.4 Watch the following video and see what these two people do.

http://www.learner.org/courses/learningmath/video/measurement/wmp/m5b1.html

1.5 Think-Pair-Share: What steps have been given?

1

2

3

4

5

6

SECTION 1: IDENTIFYING AND DEFINING

1.1 How can you measure or calculate the height of a really high building? Read the text and find out how these calculations were made in ancient times

The legend says that Egyptian priests wanted to test Thales of Miletus’ wisdom and asked him to solve a really difficult task: to calculate the height of Cheops’ pyramid. There are others who say that it was the Pha-raon Asmasis who called this geometrician to solve this difficult problem . Apparently, he wished to build up a pyramid higher than the pyramid of Cheops -- then, there was the need to know the measures of this pyramid. However, it was not an easy task: to measure the height of an oblique façade solid.No matter how or why, the fact is that Thales (690 BC) used the following method to measure the Pyramid

height: he lay on the ground and marked the length of his own height with two sticks. Then, he stood up next to one of these sticks and when he saw that his shadow measured the same as the length marked on the ground, he measured the shadow of the famous pyramid. He told the priests: “Now that my shadow and my height are the same, the length of the pyramid must be the same as its height.” So, he concluded its height was 146.5 m. (488 ft).

As the distance from the Sun to the Earth is very big, the sun rays can be considered parallel when they reach the pyramid and the person.

At present, it is completely unnecessary to wait for the sunset. We can use trigonometry to calculate the height of the pyramid of Cheops. Maybe we won’t ever need to calculate the height of the pyramid of Cheops, but there may be other everyday life situations where we may find useful to apply the so called trigonometric ratios to calculate the height of very high monuments, objects or buildings. In fact, we will use trig to find out how high our school is.

a) Thales of Miletus measured the height of the pyramid by ____________________b) Trigonometry is really useful to ________________________________________c) In order to calculate heights, we will use_________________________________

1.2 Read again and complete the following statements:


WRITING AND TALKING ABOUT MATHS

What happens to these quotients? Think-Pair-Share

DEVELOPING SKILLS

2.4 Time to measure, calculate and reach conclusions

a) Choose an angle between 10º and 30 ºb) Draw three right-angled triangles with the chosen anglec) Write the angles, measure the sides of these triangles and fill in the following table. Follow instructions given in section number 2.

SECTION 2: CALCULATIONS AND MEASURES

In this section you will learn:1. how to make simple calculations2. the basics of trigonometry DEVELOPING SKILLS

2.1 Measure the angles and sides of the right-angled triangles. Be careful when you take measurements. If you want to get accurate results, you must take measurements correctly. 2.2 Fill in the table below. In the case of angle values, write just a decimal number; in the case of sides, write two decimal numbers. 2.3 Share the results with your partner. Are they the same? Are they different? Why or Why not?

5MATEMÁTICAS


What happens to these quotients? Think-Pair-Share

DEVELOPING SKILLS

2.4 Time to measure, calculate and reach conclusions

a) Choose an angle between 10º and 30 ºb) Draw three right-angled triangles with the chosen anglec) Write the angles, measure the sides of these triangles and fill in the following table. Follow instructions given in section number 2.

SECTION 2: CALCULATIONS AND MEASURES

In this section you will learn:1. how to make simple calculations2. the basics of trigonometry DEVELOPING SKILLS

2.1 Measure the angles and sides of the right-angled triangles. Be careful when you take measures. If you want to get accurate operation results, you must take measurements co-rrectly. 2.2 Fill in the table below. In the case of angle values, write just a decimal number; in the case of sides, write two decimal numbers. 2.3 Share the results with your partner. Are they the same? Are they different? Why or Why not?



d) Why do you think the three quotients of the triangles are the same? e) What kind of triangles are they?f) Which of the following theorems will you use to demonstrate that the results of the quotients above are the same?

Pythagoras’ Theorem Thales’ Theorem The remainder Theorem The Height Theorem The Sine Theorem The Cosine Theorem

g) What does this Theorem say? Write down its formulation and make a graph to explain it more clearly.

Useful vocabulary:

Straight linesSecant

SegmentRatio

QuotientArea

be equal toHypotenuse

This Theorem says that _____________________________________________________________

________________________________________________________________________________

In this picture, ____________________________________________________________________

If ____________________________________________________________________________, then _____________________________________________________________________________

Remember: Symbols and words.It is important to use words and symbols appropriately. Part of being able to write

Mathematics well is knowing when to use symbols and knowing when to use words.

DEVELOPING SKILLS

2. 5 Theory: naming the sides of a right -angled triangle

In a right triangle there are two acute angles, two legs and one hypotenuse.

a) Read the text below and name the sides of these triangles

If we look at angle B, the two legs of the right triangle will be called adjacent leg (AB leg, next to angle B), and opposite leg (the AC leg, opposite to angle B).

Leg

7MATEMÁTICAS


d) Why do you think the three quotients of the triangles are the same? e) What kind of triangles are they?f) Which of the following theorems will you use to demonstrate that the results of the quotients above are the same?

Pythagoras’ Theorem Thales’ Theorem The remainder Theorem The Height Theorem The Sine Theorem The Cosine Theorem

g) What does this Theorem say? Write down its formulation and make a graph to explain it more clearly.

Useful vocabulary:

Straight linesSecant

SegmentRatio

QuotientArea

be equal toHypotenuse

This Theorem says that _____________________________________________________________

________________________________________________________________________________

In this picture, ____________________________________________________________________

If ____________________________________________________________________________, then _____________________________________________________________________________

Remember: Symbols and words.It is important to use words and symbols appropriately. Part of being able to write

Mathematics well is knowing when to use symbols and knowing when to use words.

DEVELOPING SKILLS

2. 5 Theory: naming the sides of a right -angled triangle

In a right triangle there are two acute angles, two legs and one hypotenuse.

a) Read the text below and name the sides of these triangles

If we look at angle B, the two legs of the right triangle will be called adjacent leg (AB leg, next to angle B), and opposite leg (the AC leg, opposite to angle B).

Leg


The ratio formed by the opposite leg of an acute angle and the hypotenuse is called the sine of the angle.

The ratio formed by the adjacent leg of an acute angle and the hypotenuse is called the cosine of the angle.

Note: In a right triangle the measurements of the legs are less than the hypotenuse, so both trigonometric ratios are less than one, 0 < cos B < 1 and 0 < sin B < 1 The ratio formed by the opposite leg of an acute angle and the adjacent leg is called the tangent of the angle.

2. 6 Theory: Trigonometric ratios

Measure the sides and angles of the right triangle in section 2.5 and calculate the sine, cosine and tangent of the two acute angles B and C.

sine of the angle B= sin B = ------------------------- =-------------=---------------------=opposite leg

hypotenuseACBC

cosine of the angle B= cos B = ------------------------- =-------------=---------------------=adjacent leg

hypotenuseABBC

tangent of the angle B= tan B = ------------------------- =-------------=---------------------=opposite leg

adjacent legACBC


hypotenuseACBC


hypotenuseABBC


adjacent legACBC

sine of the angle C= sin C = ------------------------- =-------------=---------------------=opposite leg

hypotenuse

cosine of the angle C= cos B = ------------------------- =-------------=---------------------=adjacent leg

hypotenuse

tangent of the angle C= tan C = ------------------------- =-------------=---------------------=opposite leg

adjacent leg

A calculator will also find the measure of <A when sin A or cos A is given. To do this we use the keys labeled SIN-1

and COS-1. These are the second function of SIN and COS and are accessed by first pressing SHIFT. We can think of the meaning of sin-1 as “the angle whose sine is”. Therefore, if SIN A= 0.2588, then sin-1 (0.2588) can be read as “the angle whose sine is 0.2588.”To find the measure of angle A ,<A, from the calculator, we use the following sequences of keys:

ENTER: SHIFT S IN -1 O.2588 ) ENTERDISPLAY

S IN-1(0 .2588) 14 .998870

APPLYING SKILLS

2.7 Using a Calculator

a) Read the text and follow the instructions

On a calculator, we use the keys labelled SIN and COS to display the values of the sine and cosine of an angle. The sequence of keys that a calculator requires for tangent will be the same as the sequence for sine and cosine.

For example, to find sin 50º and cos 50º, we use the following:

E N T E R S I N 5 0 ) E N T E R

E N T E R C O S 5 0 ) E N T E RDISPLAY

SIN (50)0.766044

COS (50) 0.642787

Note: To make these calculations, make sure that the letter D (degrees) appear on the top of the calculator screen; if not, press or type SHIFT MODE 3 (Deg.) . Now, the letter D will appear on the screen and you will able to check the results shown in the example

b) Given the angles B and C which appeared on the previous section, use the calculator to find the value of sin B, cos B, tan B, sin C, cos C and tan C. Important: Check that the results coincide with the values obtained measuring the two sides of the triangle.

C) Use the calculator to fill in the table below, if possible.

sin A = 0.5 A = cos A = 0.345 A =cos A = 0.6 A = tan A = 34.6 A =tan A = 1 A = cos A = 1.54 A =

Given an angle measure, find the sine or cosine ratio

Given a sine or a cosine ratio, find the angle measure

Note: If we press the key o,,, we get the angle in degrees, minutes and seconds; so

in this case we will get 14º 59’ 56’’(the calculator shows 14º 59’ 55.93’’)

9MATEMÁTICAS

The ratio formed by the opposite leg of an acute angle and the hypotenuse is called the sine of the angle.

The ratio formed by the adjacent leg of an acute angle and the hypotenuse is called the cosine of the angle.

Note: In a right triangle the measurements of the legs are less than the hypotenuse, so both trigonometric ratios are less than one, 0 < cos B < 1 and 0 < sin B < 1 The ratio formed by the opposite leg of an acute angle and the adjacent leg is called the tangent of the angle.

2. 6 Theory: Trigonometric ratios

Measure the sides and angles of the right triangle in section 2.5 and calculate the sine, cosine and tangent of the two acute angles B and C.


hypotenuseACBC


hypotenuseABBC


adjacent legACBC


hypotenuseACBC


hypotenuseABBC


adjacent legACBC

sine of the angle C= sin C = ------------------------- =-------------=---------------------=opposite leg

hypotenuse

cosine of the angle C= cos B = ------------------------- =-------------=---------------------=adjacent leg

hypotenuse

tangent of the angle C= tan C = ------------------------- =-------------=---------------------=opposite leg

adjacent leg

A calculator will also find the measure of <A when sin A or cos A is given. To do this we use the keys labeled SIN-1

and COS-1. These are the second function of SIN and COS and are accessed by first pressing SHIFT. We can think of the meaning of sin-1 as “the angle whose sine is”. Therefore, if SIN A= 0.2588, then sin-1 (0.2588) can be read as “the angle whose sine is 0.2588.”To find the measure of angle A ,<A, from the calculator, we use the following sequences of keys:

ENTER: SHIFT S IN -1 O.2588 ) ENTERDISPLAY

S IN-1(0 .2588) 14 .998870

APPLYING SKILLS

2.7 Using a Calculator

a) Read the text and follow the instructions

On a calculator, we use the keys labelled SIN and COS to display the values of the sine and cosine of an angle. The sequence of keys that a calculator requires for tangent will be the same as the sequence for sine and cosine.

For example, to find sin 50º and cos 50º, we use the following:

E N T E R S I N 5 0 ) E N T E R

E N T E R C O S 5 0 ) E N T E RDISPLAY

SIN (50)0.766044

COS (50) 0.642787

Note: To make these calculations, make sure that the letter D (degrees) appear on the top of the calculator screen; if not, press or type SHIFT MODE 3 (Deg.) . Now, the letter D will appear on the screen and you will able to check the results shown in the example

b) Given the angles B and C which appeared on the previous section, use the calculator to find the value of sin B, cos B, tan B, sin C, cos C and tan C. Important: Check that the results coincide with the values obtained measuring the two sides of the triangle.

C) Use the calculator to fill in the table below, if possible.

sin A = 0.5 A = cos A = 0.345 A =cos A = 0.6 A = tan A = 34.6 A =tan A = 1 A = cos A = 1.54 A =

Given an angle measure, find the sine or cosine ratio

Given a sine or a cosine ratio, find the angle measure

Note: If we press the key o,,, we get the angle in degrees, minutes and seconds; so

in this case we will get 14º 59’ 56’’(the calculator shows 14º 59’ 55.93’’)



2.8 Using the definition below, as a model, define the rest of trigonometric ratios, cosine and tangent of the right angle

REMEMBER:a) When talking about legs and/or hypotenuses, the triangles always have a right angle. So, in the definitions of the trigonometric ratios, we must say “given a right triangle…”b) When solving problems using trigonometric ratios, we must make sure that we are applying them to right triangles.

In a right triangle ABC, B = 90º, the sine of the angle A is defined as the quotient between the opposite leg of the angle and the hypotenuse

sine of the angle A=sin A= ----------------------=--------opposite leghypotenuse

ab

In right triangles, angles and their opposite sides are labelled with the same letter; the only difference is that angles are named with uppercase and sides with lowercase

SECTION 3: APPLICATIONS OF TRIGONOMETRY

3.2 Listen to a talk on applications of trigonometry and find out the relationship existing between these fields

3.1 Match the following fields to the corresponding definition.

Geometry

Physics

Topography

Trigonometry

3.3 Listen again and complete the following mindmap. Add as many boxes as necessary.

Trigonometry: its applications

Now, we will use trigonometry to solve some of the questions mentioned before

3.4. Listen to the following word problems. You will hear them three times. Use the works-heet provided to take down notes, draw a sketch and solve them.

It is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids and higher dimensional analogues

It is the branch of science concerned with the nature and properties of matter and energy.

It is a detailed description or representation on a map of the physical featu-res of an area.

11MATEMÁTICAS


2.8 Using the definition below, as a model, define the rest of trigonometric ratios, cosine and tangent of the right angle

REMEMBER:a) When talking about legs and/or hypotenuses, the triangles always have a right angle. So, in the definitions of the trigonometric ratios, we must say “given a right triangle…”b) When solving problems using trigonometric ratios, we must make sure that we are applying them to right triangles.

In a right triangle ABC, B = 90º, the sine of the angle A is defined as the quotient between the opposite leg of the angle and the hypotenuse

sine of the angle A=sin A= ----------------------=--------opposite leghypotenuse

ab

In right triangles, angles and their opposite sides are labelled with the same letter; the only difference is that angles are named with uppercase and sides with lowercase

SECTION 3: APPLICATIONS OF TRIGONOMETRY

3.2 Listen to a talk on applications of trigonometry and find out the relationship existing between these fields

3.1 Match the following fields to the corresponding definition.

Geometry

Physics

Topography

3.3 Listen again and complete the following mindmap. Add as many boxes as necessary.

Trigonometry: its applications

Now, we will use trigonometry to solve some of the questions mentioned before

3.4. Listen to the following word problems. You will hear them three times. Use the works-heet provided to take down notes, draw a sketch and solve them.

It is the branch of mathematics concerned with the properties and relation-ships between points, lines, surfaces, solids and higher dimensional analo-gues.

It is the branch of science concerned with the nature and properties of matter and energy.

It is a detailed description or representation on a map of the physical featu-res of an area.

What can trigonometry do for you?


Listening Comprehension Worksheet

Name: Group:

Example nº ____

Data/details given Sketch

Question(s):o _______________________________________o _______________________________________

Process and operations:

Solution/result:

Procedure:1. Establish groups of 2 or 3 students2. Shuffle the cards.3. Distribute the cards among the groups but make sure that the teacher has got one of them.4. Each group must have a different card, with two different pieces of information: an answer to a word problem and certain details which must be used to write their own math problem or question. First, you must write a word problem including that information (data, phrases, words, etc…) The phrase in bold is the question to be calculated.

SECTION 4: WRITING AND SOLVING WORD PROBLEMS

A Card Game

QUESTION Roof of a building, angle of elevation = 45º, 5m closer the new angle of elevation = 55º, Michael is 175cm, height of the building.

5. In 5 minutes, each group will have to INVENT and WRITE the word problem corresponding to the card given. Here is an example

Michael, a photographer for the National Geographic, is trying to take a picture of a really beautiful buil-ding but he needs to know its exact height to adjust the tripod. From where he is standing, he can see the roof of this building with an angle of elevation of 45º; if he moves 5m closer, the new angle is 55º. Knowing that he is 175cm tall, calculate the height of the building

The problem must include not just the raw details. You must invent a situation, saying who, what, where and/or why.Here are some useful phrases and expressions:Knowing that ______, calculate the ____________Given _____, calculate the ________________How far_______?If ________________, how tall/high_______________?

6. The first player is the teacher; he or she will read a text for the question which is on his or her card.7. You will try to solve the problem and see if you have the right answer on your card, by comparing the result you obtained and the solution appearing in your card.8. If your group has the right solution in the card, you will be asked to solve the problem for the whole class. After explaining to your partners how you made the calculations, you will read your own problem.

ANSWER

Height = 13.74cm Unequal side= 14.62cm

Aims: 1. to work cooperatively in order to solve problems as quickly as possible2. to be able to explain the process followed3. to invent a word problem including the information provided APPLYING SKILLS

13MATEMÁTICAS

Listening Comprehension Worksheet

Name: Group:

Example nº ____

Data/details given Sketch

Question(s):o _______________________________________o _______________________________________

Process and operations:

Solution/result:

Procedure:1. Form groups of 2 or 3 students2. Shuffle the cards.3. Distribute the cards among the groups but make sure that the teacher has got one of them.4. Each group must have a different card, with two different pieces of information: an answer to a word problem and certain details which must be used to write their own math problem or question. First, you must write a word problem including that information (data, phrases, words, etc…) The phrase in bold is the question to be calculated.

SECTION 4: WRITING AND SOLVING WORD PROBLEMS

A Card Game

QUESTION Roof of a building, angle of elevation = 45º, 5m closer the new angle of elevation = 55º, Michael is 175cm, height of the building.

5. In 5 minutes, each group will have to INVENT and WRITE the word problem corresponding to the card given. Here is an example

Michael, a photographer for the National Geographic, is trying to take a picture of a really beautiful buil-ding but he needs to know its exact height to adjust the tripod. From where he is standing, he can see the roof of this building with an angle of elevation of 45º; if he moves 5m closer, the new angle is 55º. Knowing that he is 175cm tall, calculate the height of the building

The problem must include not just the raw details. You must invent a situation, saying who, what, where and/or why.Here are some useful phrases and expressions:Knowing that ______, calculate the ____________Given _____, calculate the ________________How far_______?If ________________, how tall/high_______________?

6. The first player is the teacher; he or she will read a text for the question which is on his or her card.7. You will try to solve the problem and see if you have the right answer on your card, by comparing the result you obtained and the solution appearing in your card.8. If your group has the right solution in the card, you will be asked to solve the problem for the whole class. After explaining to your partners how you made the calculations, you will read your own problem.

ANSWER

Height = 13.74cm Unequal side= 14.62cm

Aims: 1. to work cooperatively in order to solve problems as quickly as possible2. to be able to explain the process followed3. to invent a word problem including the information provided APPLYING SKILLS


QUESTION:

A barge is moving at a constant speed along a straight canal. The angle of elevation of a bridge is 10°. After 10 minutes the angle of elevation is 15°. After how much longer does the barge reach the bridge? Give your answer to the nearest second.

9. Go on until all the cards have been read.

h10v+ tv

Since both the data and the unknown are on the legs of the right-angled triangle, we must use the tangent ratio to solve this problem.

tan 10=-------------; tan15= -------------

First, we find the value of “h” in both equations and match them h= (10v+tv) tan 10= tv tan 15 a 10v tan 10+ tv tan10= tv tan 15

After that, we simplify v, and we obtain:10 tan 10+ t tan 10= t tan 15

Finally, we find the value of the unknown t

10 tan 10= t (tan 15- 10) a t= -------------- = 19.245 min a t= 19 min 15 sec

htv

10 tan10tan15-tan10

EXPLANATION: AN EXAMPLE

If we call the speed in minutes “v”, then the distance travelled by the barge in 10 minutes is 10v metres, since d= speed*time=v *10= 10 v m

If we call “t” to the time it will take the barge to pass across the bridge, then the distance travelled by the barge in t minutes is t v metres.

The graph or sketch of the situation is:

Remember: you will have to explain the steps followed, so use linkers such as first, then, after that, finally.

15MATEMÁTICAS

QUESTION:

A barge is moving at a constant speed along a straight canal. The angle of elevation of a bridge is 10°. After 10 minutes the angle of elevation is 15°. After how much longer does the barge reach the bridge? Give your answer to the nearest second.

9. Go on until all the cards have been read.

h10v+ tv

Since both the data and the unknown are on the legs of the right-angled triangle, we must use the tangent ratio to solve this problem.

tan 10=-------------; tan15= -------------

First, we find the value of “h” in both equations and match them h= (10v+tv) tan 10= tv tan 15 a 10v tan 10+ tv tan10= tv tan 15

After that, we simplify v, and we obtain:10 tan 10+ t tan 10= t tan 15

Finally, we find the value of the unknown t

10 tan 10= t (tan 15- 10) a t= -------------- = 19.245 min a t= 19 min 15 sec

htv

10 tan10tan15-tan10

EXPLANATION: AN EXAMPLE

If we call the speed in minutes “v”, then the distance travelled by the barge in 10 minutes is 10v metres, since d= speed*time=v *10= 10 v m

If we call “t” to the time it will take the barge to pass across the bridge, then the distance travelled by the barge in t minutes is t v metres.

The graph or sketch of the situation is:

Remember: you will have to explain the steps followed, so use linkers such as first, then, after that, finally.

SECTION 5: MEASURING HEIGHTS

Aim:1. To follow instructions and make a goniometer quadrant2. To apply trigonometry to the calculation of heights

5. 2 Instruments to measure heights.Throughout history, several instruments have been used to calculate the height of things. Here are just a few examples, Can you recognize any of them? Match the names below to the corresponding picture. What’s the missing name?

5.1 The Processa) But, first let’s watch the following video and learn about trigonometry and its functions. What is SOH-CAH-TOA? What does it stand for? http://www.youtube.com/watch?v=jwGD5bY7dEA? b) Watch again and Think-Pair-Share: What steps have been given to measure the height of the highest tree?

1

2

3

4

5

6

a goniometer quadrant a hypsometeran astrolabe

Jacob’s sticka theodolite


TASK: BUILDING AND USING GONIOMETER QUADRANTS

You will learn: 1. how to build a goniometer quadrant2. how to use this instrument to measure the height of buildings (for example, the height of your school)

Background information:

A goniometer is an instrument that allows us to measure angles. The term “goniometry” is derived from two Greek words, “gonia”, meaning “angle” and “metron”, meaning “measure”.This quadrant is a tool used to measure angles in vertical. It consists of a graduated arc and a plate in the shape of a quarter of a circle. In one of the sides there are two peepholes, to direct it towards the right point.Hanging from the vertex, there is a weight hanging, which shows the vertical direction. The position of the string of the weight on the graduated arc shows the measurement.

What is it? What is it like?

How to construct a working goniometer model:Follow the instructions carefully and build a goniometer

Things you will need:• A 20X20 cm square of thin wood• A cup and a nail or a push pin• Something heavy, which can be used as a weight (for example, a nut or a screw)• A graduated semicircle• A hollow thin tube or two peepholes

Instructions:

Finally, place/ attach the tube or the two peepholes on the top of the square (Points B and B´).

1

2

3

4

First, make an enlarged photocopy of the graduated semicircle (you can also draw it on paper and cut out the pattern). Remember: we only need a quadrant.

Secondly, paste or glue the cutout on the square of thin wood. See figure 1.

Firmly attach the nail right in the center of the graduated arc (Point A). Now, you can attach a weight to one end of a string Attach the other end of the string to the push pin or nail

Figure 1

17MATEMÁTICAS

TASK: BUILDING AND USING GONIOMETER QUADRANTS

You will learn: 1. how to build a goniometer quadrant2. how to use this instrument to measure the height of buildings (for example, the height of your school)

Background information:

A goniometer is an instrument that allows us to measure angles. The term “goniometry” is derived from two Greek words, “gonia”, meaning “angle” and “metron”, meaning “measure”.This quadrant is a tool used to measure angles in vertical. It consists of a graduated arc and a plate in the shape of a quarter of a circle. In one of the sides there are two peepholes, to direct it towards the right point.Hanging from the vertex, there is a weight hanging, which shows the vertical direction. The position of the string of the weight on the graduated arc shows the measurement.

What is it? What is it like?

How to construct a working goniometer model:Follow the instructions carefully and build a goniometer

Things you will need:• A 20X20 cm square of thin wood• A cup and a nail or a push pin• Something heavy, which can be used as a weight (for example, a nut or a screw)• A graduated semicircle• A hollow thin tube or two peepholes

Instructions:

Finally, place/ attach the tube or the two peepholes on the top of the square (Points B and B´).

1

2

3

4

First, make an enlarged photocopy of the graduated semicircle (you can also draw it on paper and cut out the pattern). Remember: we only need a quadrant.

Secondly, paste or glue the cutout on the square of thin wood. See figure 1.

Firmly attach the nail right in the center of the graduated arc (Point A). Now, you can attach a weight to one end of a string Attach the other end of the string to the push pin or nail

Figure 1

When the object has an inaccessible base Procedure: • First, we will measure the distance from the object base to the place where we will be standing to measure the angle; besides, we will measure the angle (A) using the goniometer quadrant the angle. Be careful: you must wait till you see the top of the ob-ject to be measured.• Next, we will walk away from the object a specific distance whose length (d) is known. Then, we will use again the quadrant to measure the angle B, observing the highest part of the object.• After that, we will use the trigonometric tangent ratio with angles A and B to obtain a system of equations. Once solved, it will give us the length value (y).• Finally, we must add the value y and the height value of the person who is using the goniometer quadrant (y +ho).

Read the instructions carefully and find out how to use your goniometer. Underline the key differences between the two procedures described.

How to use the goniometer quadrantWe will hold the quadrant in such a way that the string hangs vertically, and we will place it so that the highest part of what we want to measure can be seen through the peepholes. Just then we will look at the angle measure given by the string on the graduated arc.The procedure used to calculate heights with a go-niometer quadrant varies depending on whether the object to be measured has an accessible base (for example, a tower) or not (a mountain, for ins-tance).When the object has an accessible base

Procedure:

• First, we measure the distance from the object base to the place where we will measure the angle using the goniometer quadrant.• Next, we will use the quadrant to measure the an-gle A, that is the one corresponding to the highest part of the object.• After that, we’ll use the trigonometric ratio (tan A), so as to get the length value (y).• Finally, to obtain the height of the object, we must add the height of the person who is using the quadrant to the calculated figure (y +ho).

Important: when we are calculating measures, bear in mind that the three

points on the ground must be on a straight line.

tan A= ----------

y= d. tan Ah= y+ ho

yd


Which one is correct? Why?

Task description: Calculate the height of our school, and prepare a report to show your findings.a) Driving question(s):How tall is our school?How can we measure its height?What can Trig do for you?b) Process:In small groups,1. Use the goniometer quadrant to measure the height of our school. Make the calculations both for an accesible and an inaccesible base.Important: Pay attention to the measurement of the angles, as the more accurate these measure-ments are, the more accurate the height calculations will be.

2. Try to think of other ways to calculate its height, but this time without using trigonometry. Try to describe three other ways and apply them to measure the height of the school.3. What advantages do you think trigonometry has to calculate the height of objects, compared to the other me-thods?4. Check if the results you obtain using the different methods are approximate.5. Finally, prepare a report to show your calculations. Don’t forget to include tables and figures. (For dossier)

Extra: Click on the link below and practice measuring the height of important landmarks such as the Eiffel Tower http://highered.mcgraw-hill.com/sites/0098840016/student_view0/chapter1/math_in_action_interactive.html

Criteria Points

Explanation

Use of Visuals

Mechanics

Demonstrated knowledge

Requirements

A complete response with a

detailed explanation

Clear diagram or sketch with some

detail

No math errors

Shows complete understanding

of the questions, and processes.

Goes beyond the requirements of

the problem.

Good solid response with

clear explanation

Clear diagram or sketch

No major math errors or serious

flaws in reasoning.

Shows substantial understanding of

the problem, ideas, and processes

Meets the requirements of

the problem

Explanation is unclear.

Inappropriate or unclear diagram

May be some se-rious math errors

or flaws in reasoning.

Response shows some understan-

ding of the problem.

Hardly meets the requirements of

the problem.

TOTAL:

Misses key points.

No diagram or sketch.

Major math errors or serious

flaws in reasoning.

Response shows a complete lack of understanding for the problem

Does not meet the requirements of the problem.

4 3 2 1

ASSESSMENT CRITERIA

19MATEMÁTICAS

Which one is correct? Why?

Task description: Calculate the height of our school, and prepare a report to show your findings.a) Driving question(s):How high is our school?How can we measure its height?What can Trig do for you?b) Process:In small groups,1. Use the goniometer quadrant to measure the height of our school. Make the calculations both for an accesible and an inaccesible base.Important: Pay attention to the measurement of the angles, as the more accurate these measure-ments are, the more accurate the height calculations will be.

2. Try to think of other ways to calculate its height, but this time without using trigonometry. Try to describe three other ways and apply them to measure the height of the school.3. What advantages do you think trigonometry has to calculate the height of objects, compared to the other me-thods?4. Check if the results you obtain using the different methods are approximate.5. Finally, prepare a report to show your calculations. Don’t forget to include tables and figures. (For dossier)

Extra: Click on the link below and practice measuring the height of important landmarks such as the Eiffel Tower http://highered.mcgraw-hill.com/sites/0098840016/student_view0/chapter1/math_in_action_interactive.html

Criteria Points

Explanation

Use of Visuals

Mechanics

Demonstrated knowledge

Requirements

A complete response with a

detailed explanation

Clear diagram or sketch with some

detail

No math errors

Shows complete understanding

of the questions, and processes.

Goes beyond the requirements of

the problem.

Good solid response with

clear explanation

Clear diagram or sketch

No major math errors or serious

flaws in reasoning.

Shows substantial understanding of

the problem, ideas, and processes

Meets the requirements of

the problem

Explanation is unclear.

Inappropriate or unclear diagram

May be some se-rious math errors

or flaws in reasoning.

Response shows some understan-

ding of the problem.

Hardly meets the requirements of

the problem.

TOTAL:

Misses key points.

No diagram or sketch.

Major math errors or serious

flaws in reasoning.

Response shows a complete lack of understanding for the problem

Does not meet the requirements of the problem.

4 3 2 1

ASSESSMENT CRITERIA


FINAL TASK: Using Trig to measure city landmarks

Driving question(s):How high is that famous landmark building?What calculations and measurements will you have to do?

Process:In groups (3 to 4 people), you will prepare a video or a presentation showing the procedure followed, the calculations made and results obtained, when calculating the height of a building of your city.1. First, choose a landmark building or monument (tower, church, block of flats, etc.)2. Second, use the goniometer to calculate the height of this building.

Remember: a) You will have to take photos during all the process and present data, calculations and conclusions clearly.b) Every member of the group must take part both in the making and the presentation in class or video.c) You will have to hand in both the video or the presentation and the script used.

Duration of presentation: 6-8 minute

Team members Landmark chosen Explanation (1-10) Use of visuals (1-10)

During the presentation, you will have to fill in the following form for each of the groups

A FEW TIPS FOR YOUR PRESENTATION

Step 1 Planning and PreparationPrepare! Prepare! Prepare!,

because preparation is everything.

Step 2 Structure A good presentation has a clear structure, like a good book or film:

IntroductionBody

Conclusion Step 3 Visual aids

(photos, images, tables, etc)“A picture is worth 1,000 words”

Golden rule: use one image to give one message

Step 4 SignpostingWhen you read a book, you know where you are (the chapter, the page, the section,..) but when you give a presentation, your audience does not know where they are- UNLESS

you tell themHere are just a few useful expressions

My presentation is about … First, I’d like to talk about ..

Second, …

This picture/slide/photo shows That’s the end of my pre-

sentation. Thank you for listening. Have you got any

questions

Function LanguageIntroducing the subject Let’s begin by

First, …Starting one topic Now, we’ll move on to

Next,… Let’s look now at..

Giving an example For example A good example of this is

Summarising and concluding In conclusion, Let’s summarise

Ordering First, … second Firstly, then, next, after that, finally

Step 5 DeliveryTalk to the audience, don’t read and

make sure your audience can hear you clearly. Remember:

use a loud and clear voicespeak naturally

make eye contactvary pitch and tone

use appropriate words and grammar

A few more useful words and phrases

21MATEMÁTICAS

FINAL TASK: Using Trig to measure city landmarks

Driving question(s):How high is that famous landmark building?What calculations and measurements will you have to do?

Process:In groups (3 to 4 people), you will prepare a video or a presentation showing the procedure followed, the calculations made and results obtained, when calculating the height of a building of your city.1. First, choose a landmark building or monument (tower, church, block of flats, etc.)2. Second, use the goniometer to calculate the height of this building.

Remember: a) You will have to take photos during all the process and present data, calculations and conclusions clearly.b) Every member of the group must take part both in the making and the presentation in class or video.c) You will have to hand in both the video or the presentation and the script used.

Duration of presentation: 6-8 minute

Team members Landmark chosen Explanation (1-10) Use of visuals (1-10)

During the presentation, you will have to fill in the following form for each of the groups

A FEW TIPS FOR YOUR PRESENTATION

Step 1 Planning and PreparationPrepare! Prepare! Prepare!,

because preparation is everything.

Step 2 Structure A good presentation has a clear structure, like a good book or film:

IntroductionBody

Conclusion Step 3 Visual aids

(photos, images, tables, etc)“A picture is worth 1,000 words”

Golden rule: use one image to give one message

Step 4 SignpostingWhen you read a book, you know where you are (the chapter, the page, the section,..) but when you give a presentation, your audience does not know where they are- UNLESS

you tell themHere are just a few useful expressions

My presentation is about … First, I’d like to talk about ..

Second, …

This picture/slide/photo shows That’s the end of my pre-

sentation. Thank you for listening. Have you got any

questions

Function LanguageIntroducing the subject Let’s begin by

First, …Starting one topic Now, we’ll move on to

Next,… Let’s look now at..

Giving an example For example A good example of this is

Summarising and concluding In conclusion, Let’s summarise

Ordering First, … second Firstly, then, next, after that, finally

Step 5 DeliveryTalk to the audience, don’t read and

make sure your audience can hear you clearly. Remember:

use a loud and clear voicespeak naturally

make eye contactvary pitch and tone

use appropriate words and grammar

A few more useful words and phrases


Cooperative work. Participation.

Demonstrate knowledge

(1,2,3, 4).

(1,2,3, 4)

(5, 6, 7)

(5,6, 7)

(8,9,10)

(8,9,10)

TOTAL:

.

Criteria Process

Properly organized to

complete project

Managed time wisely

Acquired needed knowledge

(1,2,3, 4)

(1,2,3, 4)

(1,2,3, 4)

(5, 6, 7)

(5, 6, 7)

(5, 6, 7)

(8,9,10)

(8,9,10)

(8,9,10)

Below Avg. Satisfactory Excellent Points

ASSESSMENT CRITERIA

Criteria Product

Explanation. Me-chanics of writing

and speaking.

Requirements of the project.

Organization and structure

Maths mechanics

Use of visuals. Creativity.

(1,2,3, 4)

(1,2,3, 4)

(1,2,3, 4)

(1,2,3, 4)

(1,2,3, 4)

(5, 6, 7)

(5, 6, 7)

(5, 6, 7)

(5, 6, 7)

(5, 6, 7)

(8,9,10)

(8,9,10)

(8,9,10)

(8,9,10)

(8,9,10)

Below Avg. Satisfactory Excellent Points

secuencia "with a little help from trigonometry"

Documents