Transcript Unit 3: Pythagorean Theorem

Unit 3: Pythagorean Theorem
Miss Wheaton
Math 8
NEMS
Warm Up
1. 5 1/2+p=6
2. Estimate to the nearest tenth or hundredth.
√40
3. Multiply. 73 x 72
4. Solve the equation. 32 + 42 = x
Pick A Job
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•
•
•
Reader
recorder 1
recorder 2
materials manager
• Put the sign on your desk
1. READER reads article to the group
2. RECORDER 1 takes the sheet of paper with
questions about the article and reads them out
loud to the group one at a time and writes down
the group’s answer.
Now we will discover the Pythagorean theorem.
3. MATERIALS MANAGER gets cardboard outline
and one cup of stones.
4. RECORDER 2 gets sheet of paper with stone
questions on them down to the dashed line.
5. Place stones back in your cup and MATERIALS
MANAGER pass your cardboard mat one group
clockwise. Repeat these steps with another mat
and answer the rest of the questions
6. MATERIALS MANAGER puts stones back in
cup and returns stones and mat. Everyone else
gets out notebook for some notes
Closure
1. What is the Pythagorean theorem?
2. What does it help us do?
3. What kind of shape must it be used with?
Warm Up
1.
2.
3.
4.
m-1 ½=-5/4
Simplify. 3√27
Divide. 8012/8015
Who is credited with discovering the
Pythagorean theorem?
5. What is the Pythagorean theorem? Draw
and label a triangle.
We use the Pythagorean theorem to find the
missing side length of a right triangle when we
know two of the side lengths.
Exit Ticket
Warm Up
1. -3/4b=2
2. Write 2 rational numbers and 2 irrational
numbers.
3. Simplify. 4,329,4040
4. Find the missing hypotenuse.
QR Codes
• If you have a phone that can read QR codes
bring it to class tomorrow.
• You can only use it in this class and only for
the purpose that we are going to be using it!
Would you need to find a leg or the
hypotenuse? Which variable would you solve
for in the Pythagorean theorem?
Amusement Park
Use the Pythagorean theorem to calculate your
distances. Write your total distance at the
bottom of the page. What is the shortest
distance you can come up with? You and your
partner need to check each other’s work to
make sure it is correct!
• Each unit on the grid is 10 yards
Closure
• How do you know if you are finding the leg or
the hypotenuse?
• How does this change your mathematical
process when using the Pythagorean
theorem?
Warm Up
1.
2.
3.
4.
x-3=-5½
Estimate √33 to the nearest tenth or hundredth
Write in exponential form. 24-99
Determine if you would need to find a missing leg or
hypotenuse. Then find the missing side length. Remember
a2 + b2 = c2.
A Right Triangle Puzzle
• VERY CAREFULLY cut out shapes – both people
cut
• Answer #2 at the top in a complete sentence.
Use math vocabulary!
• Work on #3
• #4
• #5
QR Codes
• You need your own piece of paper, a pencil,
and a calculator
• Go to your assigned station and work the
problem – show all work!
• When I say to check your work, scan the QR
code. You will have 30 seconds to scan the
code and make any changes to your work.
• Rotate to the next station when I say to!
Sandra has made a conjecture that when you
have a right triangle the area of the two smaller
squares is equal to the area of the larger square.
Work with your team to verify or disprove her
conjecture.
Warm Up
1.
2.
3.
4.
x- ½ =1 ¼
Find 3√8
Write in exponential form. (453)6
Determine if you would need to find a missing leg or
hypotenuse. Then find the missing side length. Remember
a2 + b2 = c2.
Pythagorean Triples
http://prezi.com/ehllqcrcsb-x/pythagoreantriples-activity/
Problem Solving Stations
1. Draw a picture
2. Label the picture with the given information
3. Solve the problem! Be sure to LABEL your
answer with appropriate units.
Closure
What makes a triangle a Pythagorean triple?
Homework:
p. 202
Regular #8, 24, 29
Advanced #16, 23, 29
Warm Up
1. Lisa is cooking muffins. The recipe calls for 7 cups of sugar.
She has already put in 2 cups. Write an equation and solve
for how many more cups she needs to put in.
2. Put the following in order on a number line. √64, √49, 3√64,
3√8
3. Add. Write the answer in scientific notation. (8.3 x 104) +
(3.4 x 103)
3
4. Find the missing side length.
4
• Bring graph paper tomorrow!
Converse of the Pythagorean Theorem
We have been learning the Pythagorean
theorem which states that if we have a right
triangle, then a2 + b2 = c2. Today we will study
the reverse – if a2 + b2 = c2, do we have a right
triangle?
• What is the difference between these two
statements?
• When we reverse statements, the reverse is
called the converse of the original statement.
1. All dogs are animals.
2. All oceans are bodies of water.
3. All carrots are vegetables.
4. When it snows it is cold.
5. If you are at least 16 years old, then you can
get a drivers license. (in Tennessee)
The converse of the Pythagorean theorem states
that…
… If a2 + b2 = c2 then the side lengths will make a
right triangle.
http://www.nsf.gov/news/special_reports/footb
all/pythagorean.jsp
Exit Ticket
Roger Goodell is the commissioner of the NFL (National
Football League). A defensive player approached him to ask
about the angle of pursuit. The player wanted to understand
why he could run just as fast as the receiver (who has the ball)
but not reach the receiver in time to tackle him before he
scored a touchdown. Mr. Goodell has turned the question
over to you.
Let’s say the receiver catches the ball on the 30 yard line (30
yards until he scores a touchdown). The defensive player is
also on the 30 yard line but 10 yards away from the receiver.
How far will the defensive player have to run to tackle the
receiver just before he scores a touchdown?
Warm Up
Write a letter to someone explaining how to tell
if you are looking for a missing leg or the missing
hypotenuse.
What difference does this make in your
calculations?
Draw and label pictures to aid your
explanations.
Yesterday’s Exit Ticket
Roger Goodell is the commissioner of the NFL (National
Football League). A defensive player approached him to ask
about the angle of pursuit. The player wanted to understand
why he could run just as fast as the receiver (who has the ball)
but not reach the receiver in time to tackle him before he
scored a touchdown. Mr. Goodell has turned the question
over to you.
Let’s say the receiver catches the ball on the 30 yard line (30
yards until he scores a touchdown). The defensive player is
also on the 30 yard line but 10 yards away from the receiver.
How far will the defensive player have to run to tackle the
receiver just before he scores a touchdown?
Plot the points and find all side lengths
Plot the points and find all side lengths
Stations
• Use graph paper
• Show all work
• Compare work and answer with others in
group
With your partner, record your responses to
these questions:
1. How many different ways did people solve
this?
2. How is another pair’s thinking/reasoning
different than yours?
3. How did you use the Pythagorean theorem to
solve this problem?
Closure
Create your own problem naming 3 ordered
pairs that would form a right triangle and with
your partner find the length of the hypotenuse.
HW: p. 206
Regular: #1, 3, 5-6
Advanced: #9-10, 13-14
Warm Up
1. At a restaurant, Mike and his three friends
decided to divide the bill evenly. If each person
paid $13, then what was the total bill? Write
and solve an equation.
2. Subtract. Write the answer in scientific notation.
(4 x 106) - (3 x 103)
3. If a is 13 inches long and c is 25 inches long, find
b.
4. Is the triangle in #4 a Pythagorean triple? How
do you know?
You want to put a fence around your yard. You have found
two companies who will do the job.
Company 1
http://www.youtube.com/watch?v=T-zAU-DMiFA
Company 2
http://www.youtube.com/watch?v=7eEekBZqcrg
Your challenge today is to find which company will give you a
better deal.
The companies need to know the perimeter of your yard. You
do not know the perimeter of your yard but you do know the
information listed on the sheet. You will need to use this
information to find the perimeter and then the amount that
company 1 would charge you and the amount that company 2
would charge you so that you can make an informed decision
as a customer.
Jobs
• Red – Group leader. Reads the material out loud
as the group follows along; keeps group on task.
• Green – Materials Manager – get 1 ruler for
everyone in the group
• Yellow – Recorder. EVERYONE does rough draft of
graph and recorder does final copy. Then turn in
stapled together.
• Blue – Presenter/Evaluator – make sure all 5 steps
are completed with work checked
• Graph the lines representing the cost for
company 1 and company 2 on the same
coordinate axis.
• Would there ever be a time when it would be
cheaper to go with company 1? What would
this situation be?
Warm Up
1. How many packages of energy drinks can you
buy with $40 if one package costs $8? Write
and solve an equation.
2. Multiply. Write the answer in scientific
notation. (2 x 102) (6 x 103)
3. If a is 9 inches long and b is 2 inches long,
find c.
4. Is the triangle in #4 a Pythagorean triple?
How do you know?
Reflect
1. How did you know which operation to use
when calculating the costs for the different
companies?
2. How did you know which operations to use
when calculating sales tax?
3. How did you know you could use the
Pythagorean theorem to solve this problem?
Closure
The rectangular bottom of a box has an interior length
of 19 inches and an interior width of 14 inches. A stick
is placed in the box along the diagonal of the bottom of
the box. Which measurement is closest to the longest
possible length for the stick?
F
G
H
J
12.8 in.
16.3 in.
23.6 in.
33.0 in.
Warm Up
1. Last Friday Trevon had $29. Over the weekend
he received some money for cleaning the attic.
He now has $41. How much money did he
receive? Write and solve an equation.
2. Divide. Write the answer in scientific notation.
(9 x 102) (3 x 103)
3. Identify if the triangles are right triangles.
a) a = 3 b = 4 c = 5
b) a = 1 b = 5 c = 7
c) a = 6 b = 8 c = 10
Warm Up
1. r/10+4=5
2. Determine if each number is rational or
irrational: 3.14, π, 1.583060…, 1.58306, ½
3. A 6 foot ladder is leaning against the side of a
house. The bottom of the ladder is 4 feet from
the base of the house. How high up the house
does the ladder reach?
a) Draw and label a diagram.
b) Use the Pythagorean Theorem to solve. Show all
work and label your answer!
To derive an equation means to start with some
given information, and use it to discover another
relationship/equation.
Mathbook Review
• Spend 5 min brainstorming and listing
everything you can think of to do with your
topic.
• Include vocab, examples, questions, etc.
Topics: Pythagorean theorem, converse of the
Pythagorean theorem, Pythagorean triples, real
life applications, proofs
Tomorrow is converse day!
Warm Up
1.
2.
3.
4.
5.
Draw and label a right triangle a, b, and c
A crane is needed to change a streetlamp that is 30 feet tall. If the crane
is positioned 10 feet from the base of the streetlamp, how tall must the
crane be?
a) Draw and label a diagram.
b) Use the Pythagorean Theorem to solve. Show all work and label
your answer!
A 20-foot ladder is leaning against a wall. The bottom of the ladder is 9
feet from the base of the wall. How high up the wall does the ladder
reach?
a) Draw and label a diagram.
b) Use the Pythagorean Theorem to solve. Show all work and label
your answer!
Determine if the triangle is a right triangle. A = 21, b = 15, c = 64
Who discovered the Pythagorean theorem?