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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 • • • • 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?