Download Lessons for Geometry - Bemidji State University

Lessons for Geometry
Perimeter, Area, Volume, Measurements in the English and
SI (metric) Systems, Similar Triangles, Pythagorean
Theorem, and Right Triangle Trigonometry
Math 5066 – Summer 2012
Will Baumann
[email protected]
Minnesota State Community and Technical College
1
This document contains lessons to be used in a Technical Mathematics course and a Foundations
of Mathematics course. They were specifically designed to meet the following common course
objectives defined for the classes listed above at Minnesota State Community and Technical
College:

Identify appropriate geometric formulas to solve perimeter, area and volume applications.

Perform English and Metric measurement conversions.

Use English and Metric measurements to solve application problems.

Solve basic right triangle trigonometry application problems.
Students will learn to use appropriate formulas to solve perimeter, area, and volume applications
as they work to answer questions about the size of classroom objects and buildings on a far. The
will perform conversions both within and among the English and SI (Metric) systems of
measurement with applications they might encounter in their daily lives or work. They will
solve basic right triangle trigonometry applications as they investigate the heights of trees or
buildings.
After completing all of the lessons, students will be ready to take an exam to show competency
with the common course objectives indicated above. A sample pre-test/post-test is included after
the lesson plans. This could be given before and after going through the lessons to tell how
much the students have learned.
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Table of Contents
Pages 4 – 5:
Lesson 1 – Perimeter and Area
(1 Day)
Page 6:
Lesson 2 – Surface Area and Volume
(2 Days)
Page 7:
Lesson 3 – Unit Analysis in English and SI (Metric) Systems
(1 Day)
Page 8:
Lesson 4 – Conversions between the English and SI (Metric) Systems
(1 Day)
Page 9:
Lesson 5 – Properties of Triangles and the Pythagorean Theorem
(1 Day)
Pages 10 – 11:
Lesson 6 – Similar Triangles
(1 Day)
Pages 12 – 14:
Lesson 7 – Right Triangle Trigonometry
(3 Days)
Pages 15 – 16:
Pre-test / Post-test
(2 Days)
Page 17:
Lesson 6 Similar Triangle Problem Starter
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Lesson 1 - Perimeter and Area (1 Day)
Materials:
Meter sticks. Picture of the window for projector. Pre-cut triangles that taken
together make up a pre-cut rectangle. Pre-cut half-circles that taken together
make up a circle.
Objectives:
1. Students will review/learn methods and formulas for finding the perimeter of
polygons and circumference circles, and combinations of them.
2. Students will review/learn methods and formulas for finding the area of
polygons and circles.
Launch:
How can we find the perimeter of this room? What is the area of the floor in this room?
What about the clock? Does it have a perimeter and area? How about the area the front
of a house or a stained glass window? What about their perimeter?
Couresy of http://www.freefoto.com
Explore/Share:
Have students work in groups of three to find and record measures for each of the
following objects: perimeter of the room, area of the floor, circumference of the clock.
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Have the groups share with the class what they found and work through any
controversies. Introduce formulas for the circumference and area of a circle.
Have the students get back into their groups to propose methods for finding the area and
perimeter of the front of a house and the stained glass window.
Discuss as a class how we might go about doing this. Use the pre-cut shapes to illustrate
how to find perimeter and area of “combination” shapes. Introduce the formula for area
of a triangle.
Summarize:
Discuss the usefulness of being able to combine shapes to make irregular shapes.
Reinforce the fact that the area of a triangle is half the area of a rectangle.
5
Lesson 2 - Surface Area and Volume (2 Days)
Materials:
Textbook and some sample drawings of simple houses, grain bins, and natural gas
tanks. Computer lab with Elmo projector.
Objectives:
1. Students will learn methods and formulas for finding the surface area of objects
such as rectangular solids, cylinders, spheres, cones, and combinations of them.
2. Students will learn methods and formulas for finding the volume of objects
such as rectangular solids, cylinders, spheres, cones, and combinations of them.
Launch:
Suppose we are going to paint the exterior of our house, grain bin, and natural gas tank,
and fill our grain bin with grain for our animals for the winter. How do we know how
much paint to buy? How much grain should we purchase so we can fit it all in the bin?
Explore/Share:
Work in groups of three to first make a drawing of the house, grain bin, and natural gas
tank on this farm. Every group can use different dimensions, but the gas tank must be a
combination of a cylinder and hemisphere, the grain bin must be a combination of a
cylinder and a cone and the surfaces of the house must be a combination of rectangles
and triangles that include at least one window on each side and at least one side with a
peak to be painted (triangle). The teacher must o.k. the drawing before proceeding.
Have the students use their textbooks to find and use formulas to complete the task.
The students will type up a paper of their findings and include their drawing to turn in.
They will use the Elmo projector to give a brief presentation of their findings.
Summarize:
Discuss with the usefulness of being able to break up shapes into regular shapes that we
know formulas for.
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Lesson 3 – Unit Analysis in the English and SI (Metric) Systems (1 Day)
Materials:
Computer lab.
Objectives:
1. Students will learn how to convert lengths, areas, volumes, and weights within
the English System and the SI (Metric) System.
Launch:
How many square inches are in a football field? How many square feet are in an acre?
How many cubic centimeters are in a milliliter?
Explore/Share:
Have students search the internet for websites with conversions in the English system
Have them work in groups of 3 to create a Microsoft Word document containing a few
examples of conversions between the following: inches, feet, yards, miles.
Have the students add to their document by repeating with the following lists:
-- square inches, square feet, square yards, acres, and square miles
-- cubic inches, cubic feet, cubic yards
-- fluid ounces, pints, quarts, gallons
-- ounces, pounds, and tons
Do the students see any relationships between the different lists?
Now have the students search the internet to find the similar measures of length, area,
volume, and weight in the SI (Metric) system. Have them write include the basic units of
measure for these on their typed document. Have them make a “prefix ladder” for
conversions similar to those found on the web, and include it on their document.
Summarize:
Discuss with the class the efficiency of the SI (Metric) system. Have student work on
workbook unit analysis problems.
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Lesson 4 – Conversions between the English and SI (Metric) Systems (1 Day)
Materials:
Computer lab.
Objectives:
1. Students will learn how to convert between the English System and the SI
(Metric) System.
Launch:
How fast is 90 kilometers per hour? Suppose you go to Canada and want to avoid a
speeding ticket. How could you figure out how fast you can go?
Explore/Share:
Have students search the internet for websites with conversions between the English
system and the SI (Metric) System.
Have them work in groups of 3 to create a Microsoft Word document containing a few
examples of conversions between the following units of similar kinds of measurements:
-- inches and centimeter, feet and meters, miles and kilometers
-- square inches and square centimeters, square meters and square yards,
acres and hectares
-- cubic inches and milliliters, quarts and liters, gallons and liters
-- ounces and grams, kilograms and pounds.
Do the students see any relationships between the different lists?
Have the student work on conversion problems in their workbook
Summarize:
Discuss the similarities between the different conversions within a similar type of
measurement.
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Lesson 5 – Properties of Triangles and the Pythagorean Theorem (1 Day)
Materials:
Computers in computer lab and grid paper.
Objectives:
1. Students will learn the basic properties of triangles.
2. Students will see a “proof” of the Pythagorean Theorem and learn to use it to
solve for missing sides of a right triangle.
Launch:
How long does it take a crow to fly from your house to school? If we know the rate at
which a crow flies, and the roads are on a grid, can we figure this out?
Explore/Share:
Project the grid paper and mark locations for a house, school (not on the same row or
column) and some roads. Draw a right triangle on this picture, using the house and
school as two endpoints connected by the hypotenuse. How long is the hypotenuse?
Have students bring up the following applets to illustrate the Pythagorean Theorem:
http://www.ies.co.jp/math/java/geo/pythasvn/pythasvn.html
http://www.ies.co.jp/math/java/geo/pytha2/pytha2.html
How can we figure out how long it takes the crow to fly the distance (marked by the
hypotenuse)?
Pass out the grid paper and have them diagram their situation. If the roads aren’t straight,
perpendicular, and parallel where they live, approximate it with “square” roads.
Have the students work in groups of three to help each other each figure out how long it
will take for a crow to fly from their home to school.
Summarize:
Discuss with the class the possible uses of the Pythagorean Theorem.
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Lesson 6 –Similar Triangles (1 Day)
Materials:
Picture of a similar triangle problem for projector. Grid paper, meter sticks,
straws, manila folders grid paper with protractor in corner, string, paper clips.
Objectives:
1. Students will learn about properties of similar triangles.
2. Students will learn how to use similar triangles to solve for missing sides of
triangles.
Part A
Launch:
Ask the students how we might figure out the length of the support beam in the picture.
Explore/Share:
Have students work in pairs to come up with a method and solution for finding the
missing length.
Have the students share their method and solution with the class.
Go over properties of similar triangles, having students use grid paper to make similar
triangles.
Part B
Launch:
Ask the students how we might figure out how tall a tree is. Draw a picture on the board
of how the situation might be represented using similar triangles with shadows of a
person and tree for bases, and the heights of the person and tree for heights of triangle.
Explore/Share:
Show the student an inclinometer and demonstrate briefly how it might be used to solve
this problem.
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Have the students work in pairs to make an inclinometer and then go outside to get
measurements to solve the problem.
Have the students share their solutions with the rest of the class.
Summarize:
Discuss how similar triangles can be used to make ratios and proportions useful for
finding unknown side lengths.
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Lesson 7 - Right Triangle Trigonometry (3 Days)
Materials:
Computers in the computer lab, grid paper, rulers, and inclinometers made in
Lesson 5.
Objectives:
1. Students will learn how sine, cosine, tangent and their inverses can be defined
using ratios of sides of a right triangle.
2. Students will learn how to use their calculators to get values for the
trigonometric functions and their inverses.
3. Students will learn how to find missing sides or angles of a right triangle.
4. Students will learn how to use the freeware GeoGebra.
Part A (1 Day)
Launch:
When we have similar triangles, we know the angles are congruent. What about the
sides? Are there any relationships amongst the sides of a triangle between two similar
right triangles?
Explore/Share:
Have the students go to http://www.geogebra.org/cms/ and download GeoGebra. Show
the students how to use GeoGebra to make triangles and measure their angles.
Have the students make a few similar right triangles and lead a discussion about some
possible relationships.
Are the ratios of pairs of sides in similar right triangles congruent?
Hand out grid paper and rulers to groups of three students. Have them carefully draw
three similar right triangles and prepare to report their findings to the class through
illustrations drawn at the board.
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Are these congruent ratios related to the congruent angles? Does the size (irrespective of
shape) of a triangle influence its angles? Does the size (irrespective of shape) of a
triangle influence the ratios of its side lengths?
Part B (1 Day)
Launch:
Draw a triangle on the board from class yesterday that had a known angle (other than
the right angle) and known side lengths. Draw a triangle with that same angle, but
different side lengths, leaving the hypotenuse and one side length unknown. How might
we find that unknown side length?
Explore/Share:
Introduce the definitions of sine, cosine, and tangent in terms of sides of right triangles.
Use the mnemonic device SohCahToa. Show the students the keys on their calculators
and have them calculate the sine, cosine, and tangent of the known angle with the
unknown sides.
Introduce the inverse trigonometry functions, and illustrate how they can be used to solve
for missing angles.
Can we use these definitions to find missing sides or angles of a triangle? Show how this
can be done for the triangle in questions.
Draw three right triangles with missing sides and/or angles on the board. Have students
work in pairs to solve for the unknowns. After letting them work, call on a few groups to
share their processes and solutions.
Have students work in on problems with missing sides and angles in their workbook.
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Part C (1 Day)
Launch:
How can we find the height of a tree or building? Can it be done differently than before?
Explore/Share:
Hand out the inclinometers and have the students get in groups and use the inclinometers
to measure the height of a tree or building using a method other than that used two days
ago.
Have students share their findings, and help those groups who need it.
Summarize:
Trigonometric functions are very useful for being able to find unknowns in a triangle.
Discuss the kinds of triangles that can and can’t be solved (i.e. not enough information)
using the methods of this last day of the lesson. Show some applications to conduit
bending.
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Pre-test / Post-test
1. Find the area of a triangle with a base of 3 ft. and a height of 4 ft.
2. Find the perimeter of a rectangle that is 5 in. long and 6 in. wide.
3. Find the circumference of a circle with diameter 10 ft.
4. Find the area of a circle with diameter 10 ft.
5. Find the volume of a rectangular solid with length 4 in., width 3 in., and height 2 in.
6. For a cylinder, volume =
and surface area =
. Find the volume and
surface area of a cylinder with a diameter 4 ft. and a height of 10 ft.
7. Change the following units of measurement:
a) 3.5 km to cm
b) 3.5 km to miles
8. Change the following units of measurement:
a) 10 square yards to square feet
b) 10 square yards to square meters
7. Change the following units of measurement:
a) 2 quarts to gallons
b) 2 liters to gallons
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8. Change the following units of measurement:
b) 5 kilograms to grams
b) 5 kilograms to pounds
9. An electrician must bend a pipe to make a 4 ft. rise in a 5 ft. horizontal distance. What is the
measure of the angle at each bend? (Hint: Draw a picture representing the situation.)
10. A ladder 8 ft. long leans against the side of a building and makes an angle of 65° with the
ground. How high up the building does the ladder reach? (Hint: Draw a picture
representing the situation.)
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Lesson 6 Similar Triangle Problem Starter
Example: Vertical Beam Length
Two triangles are called similar if each internal angle of one triangle is equal to each
internal angle of the other triangle.
When this occurs, their corresponding sides are proportional.
Suppose you are framing a roof and the cross section of one half corresponds to the
following picture and associated lengths.
What should the length of the shorter vertical beam be?
20 ft.
9 ft.
9 ft.
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