Tiffany N. Keys
& Gayle GilbertÕ s Instructional Unit:

Concepts
of Measurement

**DAY
1:**

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**Goal: Students will derive
the formula for area using GSP sketches.**

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Open
discussion and lecture:

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Discuss
area.

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Students
will begin a series of activities using GSP to find formulas for the areas of
various figures.

Sample
Activity:

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Students
will be given a variety of shapes to look at on the SMART Board and will be
required to find the area of each of the figure shown.

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Students
should be able to find the areas for all the figures given the unit of area.

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Students
will also be given specific areas and asked to create a rectangular,
triangular, and a figure of their choice on GSP.

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The
link to the GSP Area Discovery Activity.

**DAY
2:**

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**Goal: Students will extend
their knowledge of area.**

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Open
discussion and lecture:

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Students
will be introduced to formal definitions and formulas of area rectangles,
triangles, parallelograms, and trapezoids.

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Students
will also be introduced to the altitude of a triangle.

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The
formula for the area of a triangle will be proved.

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Allow
students to make conjectures for the formulas for the areas of parallelograms
and trapezoids.

Proof:

Given a triangle with altitude h and base a,
construct a rectangle inscribing the triangle with length a and width h.

This rectangle is made up of two smaller
rectangles of areas and . The legs of
the triangle, b and c, bisect the areas of the rectangles, so the area of these
triangles is and . Thus, the
total area of the given triangle is the sum of these two smaller triangles: . Thus, the area
of a triangle is given by the formula area is equal to half the base times the
height.

After demonstrating this proof, ask students
what would happen if the triangle was obtuse instead of acute.

**DAY
3:**

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**Goal: Students will review
Trig ratios and use them to solve problems. **

** **

Open
discussion and lecture:

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Today we are going to
work with right triangles.

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Review
trigonometric ratios (SOH CAH TOA).

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o

o

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Students
will also review the cotangent, secant, and cosecant ratios.

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o

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Discuss
properties of the trigonometric ratios.

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Individually,
students will find the altitude of some given triangles trigonometric
ratios. Some examples are given:

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Students
will work on application problems in groups. Students should display their work on a poster and present
their results to the class.

o
A
group of students were using shadows to find the heights of trees near their
school. They used the diagram
shown here to represent the general situation. Let represent the angle of elevation of the
sun and represent
the length of the treeÕs shadow.

¤
In
one case, the students found and . What is the
height of the tree?

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Later
that day, with a different tree, they got and . What is the
height of that tree?

¤
Develop
a general expression for the height of the tree in terms of and . Make sure to draw a diagram to help
explain.

o
A
sailboat is in trouble, and the people on board are considering trying to swim
to shore. A lookout station on
shore is able to tell them they are 2.3 miles from the station and the line
from the station to the boat forms an angle of with the
shoreline. Assume the shoreline is
straight.

¤
If
the people are capable of swimming 1.5 miles, will they be able to make it to
shore or should they call for help?
Explain.

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The
lookout station officer would like to be able to tell people in such situations
their actual distance from shore.
Find a general formula the officer can use to find this distance. Your formula should express this
distance in terms of the distance from the station to the boat and the angle
between the line to the boat and the shoreline.

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**DAY
4:**

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**Goal: Students will derive
and prove the Pythagorean Theorem given a few GSP sketches.**

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Open
discussion and lecture:

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Students
will be introduced to the Pythagorean Theorem.

o
Given
a right triangle, the area of the square on its hypotenuse is equal to the sum
of the areas of the squares on its other sides.

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Students
will be asked to derive a formula from the below GSP sketch.

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For
further clarification, the students will be shown two additional GSP sketches
that provide proof of the Pythagorean Theorem.

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Students
will be asked to explain why these sketches prove the theorem.

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Students
will also be asked how the below sketches are either similar or different from
the above sketch.

**DAY
5:**

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**Goal: Students will apply the
Pythagorean Theorem to real world applications.**

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Class
Activity:

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Students
will be put into five groups of four to work on the following real world
problems.

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1.
An
8-foot ladder is leaning against a wall, as shown in this diagram. The
bottom of the ladder is 2 feet from the wall. How high up from the wall
does the ladder reach?

2.
Marlene
wants to check that her doorframe makes right angles at the corners. The
door is 2.5 meters high and 1.5 meters wide. How long should the diagonal
of the door be if the corners are right angles?

3.
Bonny
was doing one of her favorite trick billiard shots. Her shot started at
one corner of the table, hit the exact center of the back cushion, and
rebounded into the other corner. Draw an accurate diagram of BonnyÕs
shots on the pool table. How far
did her billiard ball travel if the table is 8'x5'?

**4.
**Corinne
and Deanna decide to race from one corner of an open field to the other.
The field is rectangular in shape, 60 meters long and 80 meters wide.
Since Corinne is older and faster, she's going to run along the outside
of the field, and Deanna will take the diagonal route. If Deanna can run
at a rate of 5 meters per second, how fast will Corinne have to run to get
there at the same time as Deanna?

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**5.
**Leslie,
the landscape architect, has made a design for a flowerbed for a very important
client. The flowerbed will be in the shape of a triangle, with sides of
lengths 13, 14, and 15 feet. Leslie needs to know the area of the
flowerbed so she can order the correct amount of fertilizer. Suppose you
are Leslie's assistant. In order to find the area, you need to find the
length of the altitude.

(a.) Find the length of the altitude.

(b.) Calculate the area for Leslie.

Open
discussion and lecture:

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Upon
completion of the problems, students will be asked to present the solution to
one of the problems and also address any questions their classmates have for
that particular problem. They will also ask their classmates if they came upon
the correct answer using a different strategy.

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**DAY
6:**

** **

**Goal: Students will work on
maximizing area.**

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Class
Activity:

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Students
will investigate the following scenario in groups:

o
Farmer
Greene is building a corral to keep his horses in. He decides that he can afford to buy 300 feet of
fencing. For aesthetic reasons, he
decides that the corral should be built in the shape of a rectangle. He wants to build the corral such that
there is as much space as possible inside it for the horses to move around. Students should find a formula to find
the area of the corral using *w* as the width.

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After
students work on the problem, a graphical representation should be shown.

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Ask
students what triangle with perimeter of 300 feet has the biggest area.

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Now,
have a discussion about what shape the corral should be if Farmer GreeneÕs only
concern is that the horses have the most area with a perimeter of 300. After discussion, have students fill
out a chart for the area with various regular polygons. Finally, students should develop a
formula for the area of the corral in terms of n, where n is the number of
sides of a polygon.

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**DAY
7:**

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**Goal: Students will construct
a box with the biggest volume possible.**

Class
Activity

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As
students come in they will be put into groups of pairs and given scissors, tape
and construction paper.

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We
explain that a box is a container with four rectangular sides including a
rectangular bottom and an open top.

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Students
will use cereal to judge how much their box can hold. In order to measure the
amount of cereal in their boxes, the students will use a separate container in
which they will pour the cereal from their boxes into it and mark the level.

Class
discussion:

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Have
students line up finished boxes from.

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Have
students to compare boxes and discuss why oneÕs box can hold more or less than
anotherÕs.

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Does
the shape of a container determine how much it can hold?

Homework:

Look
around your home for containers of different shapes and sizes. Take note of the
units of measurement that are used to describe the contents inside these
containers.

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**DAY 8: **

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**Goal: Students will compare
and contrast different shapesÕ volumes.**

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Open
discussion and lecture:

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Compile
a list of the different types of container that the students observed in their
homes.

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Discuss
the different units of measurement in the different containers and how the
units matched the product inside the container.

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Ask
what criteria students think are important when companies are designing
containers.

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Students
will be introduced to the formal definition and formulas of volume and surface
area.

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Students
will then explore the volumes of the different containers they observed at
home, and also of other shapes.

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After
the class activity, students will discuss what they think is the best shape for
maximizing volume.

Class
Activity:

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Students
will work toward solving the following problems.

o
Farmer
Greene has a drinking trough for his animals. The trough is in the shape of a triangular prism. The triangle that forms the base of
this prism is a right triangle whose legs are 1 foot long. The trough is 5 feet long. How much water will the trough hold
when it is full?

o
The
farmer wants to build a barn. The
barn will be in the form of a prism with a base thatÕs a regular polygon. Of course, the sides will be vertical. He wants the base of the prism to have
a perimeter of 300 feet. He is
trying to decide if he should make the barn floor in the shape of a regular
octagon, decagon, or dodecagon. He
wants to paint the outside of the barn, so he decides to look for the shape
that gives the barn the least lateral surface area. The barnÕs walls will be 10 feet tall. What would you suggest to Farmer
Greene?

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**DAY
9:**

** **

**Goal: Students will clarify
their knowledge of the material that has been covered.**

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Open
discussion and lecture:

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Students
will take part in an oral and written review of the different types of
measurement concepts that will be highlighted in their future assessment.

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As
students are completing the review of measurement concepts, they will have to
opportunity to ask their peers and the teacher questions in order to clarify
any misconceptions or misunderstandings they have had.

**DAY
10:**

**Goal: Students will display
their knowledge of the material that has been covered.**

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Assessment

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Students
will be given an assessment that includes different measurement concepts. In
addition to multiple choice and fill-in-the-blank questions, students will also
be asked to complete words problems applying these concepts to the real world
applications.

**References:**

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Fendel, D., Resek,
D., Alper, L., & Fraser, S. (1998). Do Bees Build It Best? *Interactive
Mathematics Program*.
Berkeley, CA: Key Curriculum Press.

Gay,
D. (1998). Geometry by Discovery. John Wiley & Sons, Inc.