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1 EXAMPLES: EXAMPLES: EXAMPLES:

2 CYLINDER CONE SPHERE

3 NAME DATE PERIOD VOLUME OF A CYLINDER Volume = 4. Volume = 5. Volume = 6. Volume = 6908 mm 3 Volume = km 3 Volume = Height = Radius = Find the missing value, and remember to draw a picture to assist you with your understanding. 7. Find the amount of wax required to make a candle with radius 22 mm and height 61 mm. 8. Find the height of a cylinder with a volume of 30 in 3 and a radius of 1 in. 9. Find the height of a cylinder with a volume of 100 cm 3 ad a radius of 2 cm. 10. Find the radius of a cylinder with a volume of 208 cm 3 a height of 4 cm.

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6 NAME NAME PACKAGING PLAN You have opened up your own business of soda distribution. In order to reduce cost, your team of experts agrees it is ideal for you to package, load, and ship your product, until the brand has grown stability and outsourcing becomes a requirement. You decide to sell your product in individual, double, and family sized cans shaped like cylinders. INDIVIDUAL Diameter of can is 4 inches and height is 6 inches A. Find the volume of each size can. DOUBLE Diameter of can is 6 inches and height is 9 inches FAMILY Diameter of can is 9 inches and height is 12 inches. INDIVIDUAL DOUBLE FAMILY B. How many times larger is the volume of the Family can compared to that of the individual can? C. If you are shipping the cans in large boxes for distribution, how many of each size can you fit in a 36 inches CUBED box? INDIVIDUAL DOUBLE FAMILY D. Which of the above will allow you to distribute the GREATEST volume of your product? (Must use numbers/words to explain)

7 PERFORMANCE TASK: CROSS COUNTRY TEAM FUNDRAISER The cross country team is raising money for new uniforms. As a team you decide to sell ice cream cups for $1.50 each. Now you must choose which of the following containers you will serve your product in. As a group, you must make a viable argument with a CLEAR rationale for your recommendation. Use your mathematical knowledge and business sense to support your claim, and remember to use proper unit of measures. MATH TO SUPPORT: ARGUMENT:

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9 NAME DATE PERIOD USING SQUARE UNITS TO FIND RIGHT TRIANGLES Using the precut squares, find 5 combinations that will create right triangles. Write the area of each square in their correlating location, as well as the side measure. (See example below) DO NOT duplicate the measures i.e. 1, 2, 3 and 2,1,3.

10 Now let s make some MATHEMATICAL observations. 1. What relationship between each side of the triangle and the CORRELATING squares do you notice? 2. Write a conjecture or equation about how the area of two squares could be used to find the area of the largest square. 3. Write a conjecture or equation about how the side measures of the two smaller squares could be used to find the measure of the side of the largest square. 4. Use your conjecture or equation to find the longest side of a triangle that has smaller lengths of 9cm and 40cm.

11 PYTHAGOREAN THEOREM GRAPHIC ORGANIZER The Right Triangle Hypotenuse: Leg: What is Pythagorean Theorem: A statement about triangles containing a right angle. The Pythagorean Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." In laymen terms: In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. This can also be written as a simple equations which we discovered last week. a 2 + b 2 = c 2

12 We can use the equation a 2 + b 2 = c 2 to solve countless problems involving triangles and other polygons, even CIRCLES but that s for another math at another time. Let s start with the basics by proving Pythagorean Theorem. Do the following measures make a right triangle? a = 8 b = 6 c = 10 Plug them into the diagram to the right to help you make a determination. Write an equation for the triangle demonstrating Pythagorean Theorem. Try this out! Is the following proof of Pythagorean Theorem? What are the measures of each side if the given values are the areas of each square? Write an equation for the triangle demonstrating Pythagorean Theorem. What is the area of the missing square? What are the measures of each side of the triangle? Write an equation for the triangle demonstrating Pythagorean Theorem.

13 USE SQUARE UNITS AND EQUATION TO PROVE OF PYTHAGOREAN THEOREM DUAL DEMO I DO: The three side measures are 37, 12, and 35 1 st : Plug in the side measures (REMEMBER: the largest measure is ALWAYS the hypotenuse) 2 nd : Find the area of each square a 2 = b 2 = c 2 = 3 rd : Write the equations and determine if the statement is TRUE. WE DO: The three side measures are 7, 41, and 40 a 2 = b 2 = c 2 = YOU DO: The three side measures ae 15, 17, and 8 a 2 = b 2 = c 2 =

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15 NAME DATE PERIOD PROOF OF PYTHAGOREAN THEOREM AND ITS CONVERESE For each of the three diagrams at the top of the next page: (i) Calculate the area of square A, (ii) Calculate the area of square B, (iii) Calculate the sum of area A and area B, (iv) Calculate the area of square C, (v) Check that: area A + area B = area C

16 Using the method shown in Example 1, verify Pythagoras' Theorem for the right-angled triangles below: YES or NO YES or NO YES or NO The whole numbers 3, 4, 5 are called a Pythagorean triple because = 5 2. A triangle with sides of lengths 3 cm, 4 cm and 5 cm is right-angled. Use Pythagoras' Theorem to determine which of the sets of numbers below are Pythagorean triples: (a) 15, 20, 25 (b) 24, 26, 10 YES or NO YES or NO (c) 11, 30, 22 (d) 9, 8, 6 YES or NO YES or NO

17 NAME DATE PERIOD USING PYTHAGOREAN THEOREM EXIT TICKET Determine if the following triangles are right triangles. 1. Triangle ABC has the measures of 10cm, 5 cm, and 15cm. Is it a right triangle? 2. Determine if the measures create a right triangle. YES / NO YES / NO 3. Determine if the measures create a right triangle. 4. Triangle RST has the side measures of 8 in, 17 in, and 15 in. Is this a right triangle? YES / NO YES / NO 5. Find the measures of the missing side of the right triangle using Pythagorean Theorem equation. 6. If B is equal to or greater than 20, which value COULD NOT be the area of the largest square? A. 64 B. 113 C. 49 D. 100

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19 NAME DATE PERIOD FINDING THE HYPOTENUSE Calculate the length of the hypotenuse of each of these triangles: Calculate the length of the hypotenuse of each of the following triangles, giving your answers correct to 1 decimal place.

20 1. A rectangle has sides of lengths 5 cm and 10 cm. How long is the diagonal of the rectangle? 2. Calculate the length of the diagonal of a square with sides of length 6 cm. Calculate the length of the side marked x in each of the following triangles: 1. The diagonal of a rectangle is 61meters long. If the length of the rectangle is 60 meters, what is the width? 2. Find the measure of the height of the triangle to located to the right

21 Calculate the length of the side marked x in each of the following triangles giving your answer correct to 1decimal place.

22 NAME DATE PERIOD FINDING THE MISSING SIDE EXIT TICKET 1. What is the value of b? 2. What is the value of, f, the missing side? A. 46 B. 136 C. 64 D. 8 A. 25 B C D A triangle has the side measures of 12, and 5. What is the measure of the triangle s hypotenuse? 4. The area of the square correlating to the hypotenuse is 160 sq cm. One of the triangle s side lengths is 9 cm. What is the length of the third side of the triangle rounded to the nearest tenth? (Use the illustration below to assist you.) 5. A square has a diagonal that is 20 m long. What is the measure of the side lengths? 6. EXPLAIN the process that must be used to find the measure of a side of a right triangle if given the measure of the hypotenuse and one side.

23 WHO DID IT? A burglary took place today on Atlanta Road in Campbell Apartment Community. The victim lives on the third floor of building. There was no forced entry, and the only finger prints located on the door belong to the tenant. With this knowledge the investigators believe the perpetrator(s) used access from the window above the lower levels balconies to obtain entry. **Note: There is a 4 foot deep thorny bush that is along the ENTIRE edge of the 1 st level. First floor railing can ONLY support 12½-foot or shorter ladders. The community is completely gated with NO access in or out with the exception of the main entry. This area is manned 24-hour with a minimum of 3 certified security officers who keep a detailed log of resident and guest entering and exiting the community. This is also supported by a state of the art perimeter video security system. There is NO interior community surveillance. As the chief investigator taking over, you must determine what took place using your knowledge of Pythagorean Theorem. Use the information and illustrations located below and on the back to help put together the pieces of the puzzle and determine who actual burglarized Ms. Ellis s apartment. AERIAL OF THE COMMUNITY VISUAL OF CRIME SCENE

24 POSSIBLE SUSPECTS LIST PAINTER: Has 12½-foot tall ladder Working in apartments 111 and 512 CARPENTER: Has 15-foot ladders Working in apartments 111, 512, and 833 MAINTENANCE WORKER: Has 18½-foot ladder Working in building , RENOVATOR: Has 20½ -foot ladder Working on tennis court TREE CUTTER: Has 40-foot ladder attached to vehicle and 18-foot detached ladder HANDYMAN: Has 25½-foot ladder Working in apartments 112, 422, 1212, and 1511 BRICK MASON: Has a 24½-foot ladder Working on tennis court ASPHALT LABORER No access to ladder Working on parking area around tennis and basketball courts

25 OFFICIAL AFFIDAVIT POLICE REPORT NUMBER REPORTING AGENCY DATE 10/19/2015 REFERENCE NUMBER SMYRNA CITY POLICE AMATEUR POLICE DEPARTMENT REPORT TYPE/STATUS EVALUATION COMPLAINT/OFFENSE/INCIDENT (SEE REVERSE SIDE FOR SUMMARY) BURGLARY COMPLAINT RECEIVED BY NAME OF VICTIM LOCATION CAMPBELL APARTMENT COMMUNITY TIME RECEIVED 16:00 AGE DATE/TIME 10/19/15 14:30 15:30 DATE RECEIVED 10/19/15 D.O.B. BONESHEQUA ELLIS ADDRESS 123 ATLANTA ROAD APT # 531 SMYRNA, GEORGIA RACE BLACK WEIGHT 110 LBS COOPERATIVE AGENCY REPORTING COMPLAINT COLOR HAIR BLACK WITH BLONDE AND AUBURN STREAKS AGE 25 IDENTIFYING MARKS NONE 25 PHONE NUMBER (678) COLOR EYES DARK BROWN SEX ADDRESS 03/04/1995 SSN HEIGHT 4 11 COMPLEXION FAIR N/A N/A CONTACT PERSON PHONE NUMBER N/A N/A ADDITIONAL PERSON(S) RELATED TO REPORT NAME PHONE NUMBER ADDRESS

26 PROPERTY DAMAGE/LOST (ADDITIONAL ITEMS LISTED IN SUMMARY) 1. CAT- FLUFFERS 4. CROCK POT WITH CHICKEN SOUP 2. CAT LITTER BOX 5. A CAN OF RED BEANS AND RICE 3. CAT FOOD 6. MICROWAVE POPCORN Summary of Complaint/Offense/Incident and Statements: The victim stated she left her home at approximately 7:30 and returned at approximately 13:00 to feed the cat. She left no more than an hour later, and returned again (approximately 30) after she realized she left her wallet. This is when she noticed the missing items. Resident from Apt# 512 stated that the painter and carpenter entered his home at approximately the same time. Both did work on the balcony, and both used their ladder. The maintenance worker was also noticed outside of the balcony located at apartment 512 with his ladder around the time of the criminal act. No other workers with ladders were near apartment # 512. The tree cutter entered the community, but left soon after to cut trees located outside the community fence which is 9 feet tall. The asphalt laborer began working on the parking area after the renovator and brick mason finished work on the 9-foot tennis court wall, accidently blocking them in. There was a new layer of asphalt laid (19 feet wide) that could not be disturbed. A resident who chose to remain anonymous stated that she recalled one or maybe both of the gentlemen trying to use their ladder to exit the courts, but she did not stay around to see if either made it out. An unknown individual was noticed by an observer with their ladder hanging outside of Apt# 422 window approximately 30 minutes before the observer noticed the same from the handyman. The observer was unsure of the purpose.

27 SECURITY LOG DAY MONDAY DATE OCTOBER, 19, 2015 Guest Time In Destination Officer Initials Time Out MAINTENANCE WORKER 7:00 CLOCK IN CJ PAINTER 8:03 BUILDING 111 AND :13 CJ TL Officer Initials HANDYMAN 10:00 APTS # 112, 422, 1212, and 1511 CJ RENOVATOR 10:15 TENNIS COURTS 15:17 CJ TL BRICK MASON 10:38 TENNIS COURTS CJ TREE CUTTER 11:12 FRONT OFFICE TO CHECK IN 11:21 CJ TL CARPERNTER 11:36 APTS # 111, 512, and 833 CJ TL ASPHALT LABORER 12:37 PARKING AREA BY TENNIS COURT TL 14:58 TL

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29 ELIMINATION OF SUSPECTS Explain which 4 individuals you eliminated as suspects. Your explanation must be DETAILED and have MATHEMATICAL supporting evidence (use of Pythagorean Theorem). Be sure to use information you learned about the suspect from the Possible Suspect List, the Official Affidavit, the Security Log, AND, of course, your understanding of Pythagorean Theorem to support you findings. SUSPECT AND EXPLANATION OF ELIMINATION

30 YOUR FINAL CONCLUSION Explain IN DETAIL using mathematical reasoning why you think your remaining suspect committed the crime. You MUST provide as much supporting evidence as possible for the culprit to be convicted and placed behind bars. Use information from the Suspect List, Official Affidavit, Security Log, and Pythagorean Theorem to support your case.

31 Name Date Period EXIT TICKET: REAL WORLD APPLICATION OF PYTHAGOREAN THEOREM Solve the following word problems. Use of a calculator is permitted. 1. Jackie leans a 17-foot ladder against the side of her house so that the base of the ladder is 8 feet from the house. How high up the side of the house does the ladder reach? Round your answer to the nearest tenth if necessary. Diagram: Steps: Solution 2. A garden has a length of 24 feet and a width of 18 feet. A fence will extend diagonally from the southwest corner of the garden to the northeast corner of the garden. How long does the fence need to be? Draw your diagram, label the sides, show all the steps of your work, and write the solution below. Round your answer to the nearest tenth if necessary. Draw your Diagram: Steps: Solution

32 3. Stephanie is planning a right triangle garden. She marked two sides that measure 24 feet and 25 feet. What is the length of side n? Round your answer to the nearest tenth if necessary. Diagram: Steps: Solution 4. A builder needs to add diagonal support braces to a wall. The wall is 16 feet wide and 12 feet high. What is the length of each brace? Round your answer to the nearest tenth if necessary. Draw your Diagram: Steps: Solution 5. The bases on a softball diamond are 60 feet apart. How far is it from home plate to second base? Round your answer to the nearest tenth if necessary. Draw your Diagram: Steps: Solution

33 NAME DATE PERIOD FINDING DISTANCE USING A COORDINATE PLANE Plot the missing places on the graph and be sure to LABEL them. City Hall (0, 0) Library (10, 9) Johnson s House (4, -5) Burress s House (9, -7) Campbell Apt. (-2. 4) Your House (8, -8) Recreation Center (-9, 7) Felicia s House (3, -4) 1. Draw a purple right triangle to find the shortest distance from Lee s house to City Hall. Show your math work in the allotted space on the back. 2. Draw a blue right triangle to find the shortest distance from the library to Campbell Apt. Show your math work in the allotted space on the back. 3. Draw a green right triangle to find the shortest distance from the Recreation Center to your house. Show your math work in the allotted space on the back.

34 4. Draw an orange right triangle to find the shortest distance from your house to Ware s House. Show your math work in the allotted space on the back. 5. Draw a yellow right triangle to find the shortest distance from Johnson s house to the Library. Show your math work in the allotted space on the back. 6. Draw a red right triangle to find the shortest distance from the Library to the Recreation Center. Show your math work in the allotted space on the back. 7. To get from point A to point B you must avoid walking through a pond. To avoid the pond, you must walk 34 meters south and 41 meters east. To the nearest meter, how many meters would be saved if it were possible to walk through the pond? 8. Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point? Draw a diagram to assist you with solving. 9. John leaves school to go home. He walks 6 blocks North and then 8 blocks west. How far is John from the school? Draw a diagram to assist you with solving.

35 Find the distance between the two points. Use the graph above and a calculator for assistance

36 NAME DATE PERIOD EXIT TICKET: FINDING THE SHORTEST DISTANCE Find the distance between each pair of points. Use a calculator if necessary During a football play, Jermaine runs a straight route 40 yards up the sideline before turning around and catching a pass thrown by Miqueen. On the opposing team, Mary who started 20 yards across the field from Jermaine saw the play setup and ran a slant towards Jermaine. What was the distance the Mary had to run to get to the spot where Jermaine caught the ball? Illustrate the problem: Set Up the problem: Solve the problem:

37 NAME DATE PERIOD USING THE PYTHAGOREAN THEOREM CONSTRUCTED RESPONSE Draw and label a picture to represent the situation and then write an equation to represent the situation. Solve your equation and write your answer in a complete sentence. 1. Draw a right triangle. One of the legs measures 8cm. The other leg measures 6cm. What is the length of the hypotenuse? ILLUSTRATE SHOW WORK 2. A 25 foot ladder is resting against a wall. The base of the ladder is 15 feet from the base of the wall. How high up the wall will the ladder reach? ILLUSTRATE SHOW WORK 3. A television screen measures 18 tall by 24 wide. All televisions are advertised by giving the approximate length of the diagonal of the screen. (For example: A 48 television means that the diagonal of the television measures 48.) How should the television in this example be advertised? ILLUSTRATE SHOW WORK 4. Carl lives 12 miles east of the school. Bill lives five miles north of the school. What is the shortest distance between the two houses? ILLUSTRATE SHOW WORK

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39 NAME DATE PERIOD UNIT 3 CULMINATING TASK Find the exact area (in square units) of the figures below. Explain your method(s) Remember to show your work! Explain your method(s). Find the areas of the squares on the sides of the triangle below.

40 a. How do the areas of the smaller squares compare to the area of the larger square? b. If the lengths of the shorter sides of the triangle are a units and b units and the length of the longest side is c units, write an algebraic equation that describes the relationship of the areas of the squares. c. This relationship is called the Pythagorean Theorem. Interpret this algebraic statement in terms of the geometry involved (Write an equation). 4. What is the relationship between the areas of the regular hexagons constructed on the sides of the right triangle below?

41 5. Does the Pythagorean relationship work for other polygons constructed on the sides of right triangles? Under what condition does this relationship hold? 6. Why do you think the Pythagorean Theorem uses squares instead of other similar figures to express the relationship between the lengths of the sides in a right triangle?

42 Criterion B: Investigating patterns Maximum: 8 At the end of year 3, students should be able to: i. select and apply mathematical problem-solving techniques to discover complex patterns ii. describe patterns as relationships and/or general rules consistent with findings iii. verify and justify relationships and/or general rules. Achievement Level descriptor level 0 The student does not reach a standard described by any of the descriptors below. 1 2 The student is able to: i. apply, with teacher support, mathematical problem-solving techniques to discover simple patterns ii. state predictions consistent with patterns. 3 4 The student is able to: i. apply mathematical problem-solving techniques to discover simple patterns ii. suggest relationships and/or general rules consistent with findings. 5 6 The student is able to: i. select and apply mathematical problem-solving techniques to discover complex patterns ii. describe patterns as relationships and/or general rules consistent with findings iii. verify these relationships and/or general rules. 7 8 The student is able to: i. select and apply mathematical problem-solving techniques to discover complex patterns ii. describe patterns as relationships and/or general rules consistent with correct findings iii. verify and justify these relationships and/or general rules. IB SCORE: GRADING SCALE SCORE TEACHER COMMENT:

43 Name Date Period UNIT 3 APPLICATIONS OF EXPONENTS STUDY GUIDE MCC8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. 1). Use the Pythagorean Theorem to find the approximate distance between (0,1), (5,4). a) 5.2 b) 5.8 c) 6.1 d) 6.4 2). Plot (-3,-3), (-1,5) and then find the shortest distance between the two points.. MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 3) The farmers market sells handmade quilts. The quilts are rectangles 9 feet wide and 10 feet long. What is the length of the diagonal of a quilt to the nearest tenth of a foot? 9 ft 10 ft a) 12 feet b) 13.5 feet c) 15 feet d) 15.4 feet

44 4) The sides of A, B, and C meet to form a right triangle, as shown below. A B C If square A has an area of 35 square centimeters and square B has an area of 85 square centimeters, what is the area of C? a) 45 square centimeters b) 50 square centimeters c) 60 square centimeters d) 120 square centimeters MCC8.G.6 Explain a proof of the Pythagorean Theorem and its converse. Side a =5 cm; b=12 cm and c=13cm. 5) Calculate the areas of the three squares above. 6) How does the area of the largest square (c) relate to the two smaller squares? 7) Complete the following table: a Area of A b Area of B Area of C c

45 MCC8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.) In the figure above, AB and CD are perpendicular. What is the measure of side DB? What is the measure of side AC? What is the perimeter of triangle ABC? 9.) Sophia used an 8 foot rope to secure a 6 foot tent pole as shown above. Approximately how far from the base of the pole is the rope tied? a) 5 feet b) 7 feet c) 10 feet d) 14 feet MCC8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems

46 10.) A cylindrical glass vase is 6 inches in diameter and 12 inches high. There are 3 inches of sand in the vase, as shown below. Which of the following is the closest to the volume of sand in the vase? a) 54in 3 b). 85 in 3 c). 254 in 3 d). 339in 11 and 12. Find the volume of the cone and sphere below. a= 6mm a = b = 10 mm 14 mm

Pythagorean Theorem. 2.1 Soon You Will Determine the Right Triangle Connection The Pythagorean Theorem... 45

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