Lesson 4
Applications
Lesson Plan
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Math
1
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Time: |
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Three
50-minute class periods, or one and a half 90-minute
blocks
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Goals: |
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Distance
Formula Activity
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Students
will apply the
Pythagorean Theorem to find the coordinate distance between two
points. That is, students will develop the distance formula.
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Pythagorean
Triples Activity
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Students
will prove that
{k, (½)(k2-1), (½)(k2+1): k odd},
{2m, m2-1, m2+1: m a natural
number}, and {2uv, u2-v2, u2+v2:
u > v, u and v natural numbers} are sets of Pythagorean triples.
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Students
will review the
concept of greatest common factor and will determine the difference
between primitive and nonprimitive Pythagorean Triples.
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Students
will generate
several Pythagorean Triples in the same family.
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Students
will gain a
historical perspective on the Pythagorean Theorem and its applications.
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Students
will gain
experience using Excel. |
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Other Shapes Activity
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Students
will recall and
extend the geometric meaning of the Pythagorean Theorem.
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Students
will generate and
defend solutions to one of three problems (equilateral triangles,
regular hexagons, or semicircles).
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(Optional)
Students
will prove that
the midpoint of the hypotenuse of a right triangle is equidistant from
the three vertices of the triangle.
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(Optional)
Students
will determine
whether any shape placed on
the edges of a triangle will behave in the same way as semicircles and
regular polygons
do.
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NCTM Content &
Process Standards
Addressed: |
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Algebra
(use symbolic algebra to represent and explain mathematical
relationships)
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Geometry
(analyze properties of two-dimensional objects; use Cartesian
coordinates; investigate conjectures and solve problems involving
two-dimensional objects represented in Cartesian coordinates)
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Measurement
(understand and use formulas for the area of geometric figures)
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Problem
Solving (build new mathematical knowledge through problem solving;
solve problems that arise in mathematics and other contexts)
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Reasoning
and Proof (recognize reasoning and proof as fundamental aspects of
mathematics)
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Communication
(communicate mathematical thinking coherently to peers, teachers, and
others; analyze and evaluate the mathematical thinking and strategies
of others)
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Connections
(recognize and use connections among mathematical ideas; understand how
mathematical ideas interconnect and build on one another to produce a
coherent whole)
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GPS Content &
Process Standards
Addressed: |
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MM1G1d
(understand the distance formula as an application of the Pythagorean
Theorem)
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MM1P1a,b,c,d
(solve mathematical problems)
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MM1P2b,c
(make
conjectures, develop and evaluate mathematical proofs)
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MM1P3a,b,c,d
(communicate mathematically)
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MM1P4a,b,c
(make
connections among mathematical ideas)
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Supplies and Resources:
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Several
12-unit long loops of rope with 12 equally-spaced knots or marks
(optional)
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Excel
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GSP,
and three GSP files (see files) -
Triangles.gsp, RegularHexagons.gsp, and Semicircles.gsp
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Assessment: |
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Solution
presentations
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Overview: |
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Day 1 - Distance
Formula Activity
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Introduction (10 minutes)
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Discussion:
You're visiting Gridville, where the roads are laid out in a perfect
orthogonal grid, with roads like 1st Ave. and 2nd Ave. going East-West,
and roads like A St. and B St. going North-South. Each
block is one mile long. You're at
the corner of 34th and K, and your friend is at 42nd and E. How
far do you have to walk in the East-West direction to get to your
friend? How far do you have to walk in the North-South
direction? If you had a helicopter and could fly directly from
your location to your friend's location, how far would you have to
travel?
Discuss possible representations of the above problem (e.g., coordinate
grid; you're at (34,11), and your friend is at (42,5)).
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Exploration
(30 minutes)
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Students use GSP's
coordinate grid and draw two moveable points anywhere on the
plane. Problem: What is the distance between these two points?
Questions: Does it matter which quadrant the points are in?
What is a general formula for the distance between your two
points? Include a calculation on your GSP file with your general
formula, then compare it to GSP's Measure > Coordinate Distance
measurement.
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Wrap-Up (10 minutes)
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What formula did you
find? Does it matter whether you do xA-xB
or xB-xA? Does it matter whether you have
your y's first or your x's?
How does the distance formula relate to the introductory question about
Gridville? Does the distance formula give you the same answer for
the helicopter ride as the answer you got before?
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Day 2 - Pythagorean Triples Activity
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Introduction (10 minutes)
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The ancient Egyptians used
a twelve-unit loop of rope with twelve equally-spaced knots to make
right angles. (Optional: Give out loops of rope. Question:
Can you figure out how they did this?)
How can we be sure 3-4-5 make the side lengths of a right triangle?
Say we have two perpendicular segments, one of which is 6 feet long,
the other of which is 9 feet long. How long would the hypotenuse
be for the right triangle formed by these two legs? (Answer: An
irrational number). So what's special about 3-4-5? (Answer:
All three lengths are whole numbers). These are called Pythagorean Triples.
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Exploration (30 minutes)
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Students are split into
three groups. Group One works with the ancient Greeks' triples
generator: {k, (½)(k2-1), (½)(k2+1):
k odd}; Group Two works with Plato's generator:
{2m, m2-1, m2+1: m a natural
number}; Group Three works with the nonprimitive version of Euclid's
generator: {2uv, u2-v2, u2+v2:
u > v, u and v natural numbers}.
Question: How do you know your numbers are Pythagorean Triples (Answer:
Each number is a whole number, and the square of the last equals the
sum of the squares of the first two).
Group Questions: Group One - Why does k have to be odd; can k be
1? Group Two - If m2-1 is even, is m2+1
even? Group Three - Why does u have to be greater than v?
Students use an Excel spreadsheet to generate several Pythagorean
Triples from their generator. This task will be slightly more
difficult for Group Three, who will have to have one column for u and
another for v.
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Day 3 - Other Shapes Activity
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Introduction (5 minutes)
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Brief review of area.
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Exploration (25 minutes)
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Students work in three
groups: equilateral triangles, regular hexagons, and semicircles.
Question: Given two [semicircle]s, how can you construct a
[semicircle] whose area is equal to the sum of the other two
[semicircle]s' areas?
Students work on GSP to make the construction. Students may be
given GSP files with the original shapes and Script tools for drawing
those shapes (see files), or
students may be expected to generate the drawing from scratch.
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Wrap-Up (20 minutes)
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What did you find about
these shapes? Students present solutions.
What is special about the shapes you used? Do you think you would
get the same results for any shape? What about an irregular
polygon? How could you decide on the dimensions of irregular
polygons (e.g., for rectangles, would you want all three rectangles to
have equal heights or similar proportions?)
Optional extension: Look at the semicircle group's
solution. What would happen if we made the largest semicircle
into a full
circle? (Answer: The circle seems to go through all three
vertices.) So what could we guess is true? (Possible
answer: The midpoint of a right triangle's hypotenuse is equidistant
from the triangle's three vertices.) Prove your conjecture!
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