ma. 8.g. 2. 4 next generation sunshine state standards subject area: grade level: supporting idea/...
TRANSCRIPT
Welcome to the
This 6-Hour Module focuses twoConcepts from the Next Generation
Sunshine State Standards
http://www.floridastandards.org/index.aspx
8
Big Idea #2Geometry
MA. 8 .G. 2. 4
Next Generation Sunshine State Standards
Subject Area:Grade Level:
Supporting Idea/Big Ideas:
Benchmark:
Mathematics
Benchmark
MA. 8. G. 2. 4
Validate and apply Pythagorean theorem to find
distances in real world situations or between points in the coordinate plane .
8
Supporting Idea #6Number and Operations
MA. 8 .A. 6. 2
Next Generation Sunshine State Standards
Subject Area:Grade Level:
Supporting Idea/Big Ideas:
Benchmark:
Mathematics
Benchmark
MA. 8. A. 6. 2
Make reasonable approximations of square roots and mathematical expressions that include square roots, and use them to estimate solutions to problems and to compare mathematical expressions involving real number s and radical expressions.
3 – Two Hour Parts
This lesson will be divided into 3 parts
Pythagorean’s Theorem and Square Roots by Hand (2 Hours)
Calculators in the Middle School Classroom (2 Hours)
Validate, Explore, Practice (2 Hours)
Before continuing with this Power Point…
Please take the
Content PretestContent Pretest for Block 13.docx
Because triangles are seen everywhere!
Why Do I Have to Learn This?
The Flatiron Building
• Located in New York
• One of the first skyscraper (1902)
• Sits on a triangular island
http://www.prometheanplanet.com/server.php?show=ConResource.20665
The Golden Gate Bridge
•San Francisco Bay
•Originally the longest suspension bridge in the world when it was completed during the year of 1937
•Since its completion, the span length has been surpassed by eight other bridgeshttp://www.prometheanplanet.com/server.php?show=ConResource.20665
The Louvre Pyramid
•Paris, France
•Glass and Metal Pyramid
•Main entrance to the museum
•Completed in 1989
•Landmark for the city of Parishttp://www.prometheanplanet.com/server.php?show=ConResource.20665
The Epcot
A theme park at Walt Disney World
Second part built
In 2007, Epcot was ranked the third-most visited theme part in the United States, and the sixth-most visited in the world.
http://www.prometheanplanet.com/server.php?show=ConResource.20665
One Way to Validate Pythagorean’s
Theorem
http://users.ucom.net/~vegan/images/Pythagoras_6.jpg
Things You Will Need
You will need a
•Partner
•3 Sheets of Graph/Grid Paper
•Scissors
•Ruler
•Crayons
Start with any right triangle
Draw any size right triangle in the middle of the page. You and your partner should have the same triangle.
½ab
a
bc
-b
a
Constructing Three Squares
Draw three squares so that each square corresponds to each side of the triangle.
The squares must be connected to the triangle.
½ab
a
b
a
a²
c²
b b²
a
-b
c
Cut and Color
Cut out the three squares and the triangle.
½ab
a
b
a
a²
c
c²
b b²
a
-b
Construct, Cut and Color
Take out another sheet of graph paper and construct three more triangles with area ½ab. Write ½ab on the three triangles.
Cut them out.
½ab
a
bc
½ab
a
bc
½ab
a
bc
Construct, Cut and Color
Finally construct a square whose length is (a-b).
To do this, put you a² and b² on the graph paper.
Then cut it out and color it.
a²b²
a-b
(a-b)²
Fit the 4 triangles into c².
½ab
½ab
½ab
½ab
c²
Validating Pythagorean’s Theorem Geometrically
There is a square left in the middle to fill?
Is it a²?
Is it b²?
Or is it (a-b)²
a
a²
b²
(a-b)²
Yes (a-b)² fits!
Validating Pythagorean’s Theorem Geometrically
Now take a² + b².Check to see if the same five objects fit into this configuration.
a
b²
(a-b)²
½ab
½ab
½ab
½ab
a²
They do!!
What does this meanGeometrically?
c² a²b²
a
b²a²
What does this meanGeometrically?
c²a²
b²
a
b²
It means that c² has the same area as a² + b².
We have now validated that a² + b² = c² !!
What does this mean algebraically?
It means that c² = (a – b)² + ½(ab)• 4Simplifying we get,
c² = a² + b²We have now validated Pythagorean’s
Theorem algebraically.
Another Way to Validate Pythagorean’s
Theorem
http://users.ucom.net/~vegan/images/Pythagoras_6.jpg
Cut and Color
Again begin with a², b², and c². ½ab
a
b
a
a²
c
c²
b b²
a
-b
Validating Pythagorean’s Theorem Geometrically
Construct a square with your four triangles and c².
This time the triangles don’t go inside.
Can you do it?
½ab
½ab
½ab
½ab
c²
Validating Pythagorean’s Theorem Geometrically
c²
Validating Pythagorean’s Theorem Geometrically
How about with your four triangles,a² and b²?
Can you do it?
½ab
½ab
½ab
½ab
a² b²
Validating Pythagorean’s Theorem Geometrically
a²
b²
Validating Pythagorean’s Theorem Geometrically
a²
b²
c²
What does this mean geometrically?
Validating Pythagorean’s Theorem Geometrically
a²
b²
c²
It means that without the triangles,
c² = a² + b², and geometrically validates Pythagorean’s Theorem.
What does it mean algebraically?
c²
a b
b
a
b
a
a b
It means that
(a + b)² = ½ab•4 + c²
a² + 2ab + b² = 2ab + c²
Subtracting 2ab from each side, we get
a² + b² = c² This validates Pythagorean’s Theorem algebraically.
Benchmark
MA. 8. G. 2. 4
Validate and apply Pythagorean theorem to find
distances in real world situations or between points in the coordinate plane .
How will you explainValidating Pythagorean’s Theorem
to your eighth grade students?
Will you do it algebraically or geometrically or both?
Discuss this with your partner.
http://eppsnet.com/images/math-problems.gif
The Next Benchmark states
MA. 8. A. 6. 2
Make reasonable approximations of square roots and mathematical expressions that include square roots, and use them to estimate solutions to problems and to compare mathematical expressions involving real number s and radical expressions.
Making Reasonable Approximations of Square
Roots
http://www.coverbrowser.com/image/bestselling-comics-2007/2502-1.jpg
Approximating the Square Root of a Number by Hand
There are several ways that you can approximating the square root by hand.
1. Using the number line to round to the
nearest integer
2. Using a Percentage Approximation
to round to the nearest tenths
place.
How do you approximate to the nearest integer using the number
line?
52
0 7 8
52
is between and . 52 749 864
Which integer is it closer to? 7
752
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
7852
How do you approximate to the nearest integer using the number
line?
33
0 5 6
33
is between and . 33 525 636
Which integer is it closer to? 6
633
25 26 27 28 29 30 31 32 33 34 35 36
5 633
Using percentage approximations, how do you approximate to the
nearest tenths place.52
You know now that is closest to 7. Count how many spaces it is away from 7.
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
7852
52
Now count how many integer square roots there are between 7 and 8.
3
15
Your approximation is 3/15 = 1/5 = 0.2. So your percentage approximation to the nearest tenth is 7.2.
Using percentage approximations, how do you approximate to the
nearest tenths place.33
25 26 27 28 29 30 31 32 33 34 35 36
5 633
You know now that is closest to 6. Count how many spaces it is away from 5.
33 8
Now count how many integer square roots there are between 5 and 6.
11Your approximation is 8/11 ≈ 0.72… So your percentage approximation to the nearest tenth is 5.7.
Now you try!!
Approximate to the nearest tenths place.
146Approximately 12.1
23Approximate to the nearest tenths place.
Approximately 4.8
How will you teach your eighth students to approximate the
square root of a number?1.Using the number line
to round to the nearest integer
2. Using a percentage approximation to round to the nearest tenths place.
http://eppsnet.com/images/math-problems.gif
There is still another method called the Classical Approach that you will learn in the next unit.