THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS Grade 8 The Pythagorean Theorem 8.G.6-8 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES
NEWARK PUBLIC SCHOOLS Office of Mathematics Math Tasks 8.G.6-8 The Pythagorean Theorem Analyze proportional relationships and use them to solve real-world and mathematical problems. Goal:. In this module students will discover the Pythagorean relationship through an exploration of squares drawn on the sides of a right triangle. Students will then use the Pythagorean Theorem to solve everyday problems in two and three dimensional geometry. The student will then develop strategies for finding the distance between two points on the coordinate grid. Prerequisite Skills: Fluency with operations involving square roots and square numbers Solving One-Step equations Graphing on a coordinate grid Essential Questions: How can spatial relationships be described by careful use of geometric language? How do geometric relationships help in solving problems and/or make sense of phenomena? Is it appropriate and useful to use the Pythagorean Theorem in his situation? How do I know this? Embedded Mathematical Practices MP.1 Make sense of problems and persevere in solving them MP.2 Reason abstractly and quantitatively MP.3 Construct viable arguments and critique the reasoning of others MP.4 Model with mathematics MP.7 Look for and make use of structure LESSON 5 8.G.6-8 Golden Problem LESSON 4 8.G.8 Finding Distances on a Coordinate Grid LESSON 1 8.EE.1-2 Square Numbers LESSON 2 8.G.6 Finding Pythagoras LESSON 3 8.G.7 Applying the Pythagorean Theorem Lesson Structure: Introductory Task Guided Practice Collaborative Work Journal Questions Skill Building Homework Page 2 of 23
Lesson 1: Introductory Task The Tarpon Prerequisite Competencies Introductory Task 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 =p and x 3 =p, where p is positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that the is irrational Introductory Task 1. To estimate the weight, in pounds, of a large fish, fishermen square the girth, multiply by the length, divide by 800, and then add 1/10 of that number. What is the weight of the tarpon below? Girth = 47 in. 6 ft 6 in The Illustrative Mathematics Project: http://illustrativemathematics.org/standards/k8 Page 3 of 23
Lesson 1: Multiple Representations Framework Using Pattern Block Concrete Representations Graphical Representation Pictorial Representations O Abstract Representations Using The Pythagorean Theorem a 2 + b 2 =c 2 Page 4 of 23
Lesson1 Guided Practice Guided Practice 1) Ignoring air resistance, the distance d in feet an object falls in t seconds is. The Sears tower is 1,450 ft tall. If a window washer at the top of the tower drops his squeegee, about how much time passes before the squeegee hits the sidewalk below? Find the square roots of each number. 2) 49 3) 900 4) 5) 6) 7) Moesha has 196 pepper plants that she wants to form in square formation. How many pepper plants should she plant in each row? Determine whether each statement is sometimes, always, or never true. Explain or give a counter example to support your answer. 8) The y-coordinate of a point in quadrant II is negative 9) The x-coordinate of a point on the y-axis is zero 10) In quadrants I and III, the x-coordinate of a point is positive : Course 3 Common Core Series, Prentice Hall Page 5 of 23
Lesson1 Collaborative Work 1) Graph and connect the points (3,2), (-2,2), (-2,7) (3,7) and (3,2) in order. Then graph and connect the points(3,- 2), (-2,-2), (-2,-7), (3,-7) and (3,-2) in order. How are these two figures related? 2) Marika had to draw ABC that fit several requirements a. It must fit in the box shown b. The end points of have coordinates A(-2,0) and B(2,0) c. Point C must be on the y-axis and its y-coordinate an integer Name all the points that could be point C 3 2 1-3 -2-1 O -1 1 2 3 4-2 -3 3) The area of a square postage stamp is. What is the side length of the stamp? 4) The formula represents the distance in miles d you can see from h feet above ground. On the London Eye Ferris Wheel, you are 450 ft above ground. To the nearest tenth of a mile, how far can you see? 5) A student evaluated the expression and got the answer 5. What error did the student make? 6) A tile is shown at the right. The area of the larger square is 49 Find the area of the smaller square. 2in 2in 2in Journal Question 2in Explain how an ordered pair locates a point in the coordinate plane Page 6 of 23
Lesson1 Homework PROBLEM SOLVING 1) John bought a bag of lawn fertilizer that will cover 400 square feet. What are the dimensions of the largest square plot of lawn that the bag of fertilizer will cover? 2) The time t in seconds for an object dropped from a height of h feet to hit the ground is given by the formula How long will it take an object dropped from a height of 500 feet to hit the ground? Round to the nearest tenth. 3) A cardboard envelope for a compact disc is a square with an area of 171.61 square centimeters. What are the dimensions of the envelope? Skill Practice Plot the following points on the coordinate grid 4) (4,-5) 5) (3,4) 6) (5,0) 7) (0,-3) O Glencoe Course 2 Math Word Problems Page 7 of 23
8.G.6 Lesson 2: Introductory Task Looking for Pythagoras 8.G.6-Explain a proof of the Pythagorean Theorem and its converse Introductory Task Each leg of the right triangle on the left below has a length of 1 unit. Suppose you draw squares on the hypotenuse and legs of the triangle, as shown on the right. How are the areas of the three squares related? For each row of the table: Draw a right triangle with the given leg lengths on dot paper Draw a square on each side of the triangle Find the areas of the squares and record the results in the table. Length of Leg 1 (units) Length of Leg 2 (units) Area of Square on Leg 1 (square units) Area of Square on Leg 2 (square units) Area of Square on Hypotenuse (square units) 1 1 1 1 2 1 2 2 2 1 3 2 3 3 3 3 4 Connected Mathematics 2, Looking For Pythagoras Prentice Hall Page 8 of 23
Lesson2 Guided Practice Guided Practice: 1) Different students came up with different ways to show how the Pythagorean Theorem is true. Can you explain how each works? 2) Find the missing leg length. If necessary, round to the nearest tenth. R a) b) B c) 24 in S 7 in 8 cm R 4 in S T T 15 cm D 3) Find the missing leg length. a and b represent the lengths of the two legs and c represents the length of the hypotenuse. If necessary, round to the nearest tenth. 1) a= 7, b=24 b) a=11, b=14 c) a=18, b=22 6 in. L 4) A ramp is 1 ft high. The base of the ramp extends 14 ft along the side of a building. How long is the sloped part of the ramp to the nearest hundredth of a foot? 5) An architect drew the sketch of a bridge shown below. The Bridge has 12-ft-long horizontal members and 24-ft-long vertical members. What is the length in feet of each diagonal member? Round to the nearest foot. 24 ft 12 ft. Course 3 Common Core Series, Prentice Hall Page 9 of 23
Lesson2 Collaborative Work Collaborative Work 1) To the right is a student s representation of the Pythagorean Theorem, explain how it works. 2) Find the perimeter of a right triangle with legs of 6 cm and 8 cm. 3) The television is measured by the diagonal dimension of its screen. For example, a 24-in. television has a diagonal measure of 24in. a. A television screen is 16 in. high and 22 in wide. What is its diagonal dimension to the nearest integer? b. Find the dimensions of a television screen with the same diagonal measure as the one in part (a) but with a different height and width 4) Two hikers start a trip from a camp walking 1.5 km due east. They turn due north and walk 1.7 km to a waterfall. To the nearest tenth of a kilometer, how far is the waterfall from the camp? 5) A stair case is 20 ft. high. The horizontal distance from one end of the stair case to other end is 24 ft. What is the distance from the top of the stair case to the bottom of the stair case? Round to the nearest foot 6) A book is leaning with one end at the top edge of a bookend. The bookend is 6 in. high. The distance along the shelf from the edge of the book to the bottom of the bookend is 4in. How long is the book? Round to the nearest inch 7) Sarah walks across a rectangular field as shown. What is the distance she walks? path 40 ft 60 ft 8) A circus performer walks on a tightrope 25 feet above the ground. The tightrope is supported by two beams and two support cables. If the distance between each beam and the base of its support cable is 15 ft, what is the length of the support cable? Round to the nearest foot. Journal Question: 9) Explain how you find the distance AB across the lake at the right. Then find AB to the nearest foot. 100 ft 50 ft A 200 ft B Page 10 of 23
Lesson2 Homework 1) A baseball diamond is really a square 90 feet on a side. How far is second base from home plate? 2) Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure below shows some of the dimensions but is not drawn to scale. Is the shaded triangle a right triangle? Explain how you found your answer. 3) Explain how the picture below represents the Pythagorean Theorem. Skill Building Find the length of the hypotenuse of each triangle; a andb represent the lengths of the two legs. If necessary, round to the nearest tenth. 4) a= 3, b= 4 5) a=9, b=12 6) a=6, b=4 7) a=11, b=14 http://jwilson.coe.uga.edu/emt668/emt668.student.folders/headangela/essay1/pythagorean.html http://www.learner.org/courses/learningmath/geometry/session6/part_h/homework.html The Illustrative Mathematics Project: http://illustrativemathematics.org/standards/k8 Page 11 of 23
8.G.7 Lesson3: Introductory Task The Umbrella 8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Introductory Task 1) Doug is shipping a 32-inch long umbrella. Will the umbrella fit in a box that is 24 inches long, 18inches wide and 16 inches tall? 16 in. 24 in. 18 in. The Illustrative Mathematics Project: http://illustrativemathematics.org/standards/k8 Page 12 of 23
Lesson3 Guided Practice Guided Practice 1) The hypotenuse of right triangle is 20.2 ft long. One leg is 12.6 ft long. Find the length of the other leg to the nearest tenth. 2) The bottom of an 18-ft ladder is 5ft from the side of a house. Find the distance from to the top of the ladder to the ground. Round to the nearest tenth. 3) An artist is measuring a rectangular canvas. Its length is 30 in. The distance from one corner of the canvas to the other (along the diagonal) is 34 in. What is its width? 4) Stephen is constructing a ramp to test his model car. The ramp is a triangular prism. The wood costs $3.12 per square meter. (1 m 2 =10,000 cm 2 ). How much does the wood cost? Find the length of the hypotenuse for the right triangle Find the surface area of the ramp. Convert to square meters What is the total cost of the wood? 20 cm 24 cm 45 cm Course 3 Mathematics Common Core, Prentice Hall Page 13 of 23
Lesson3 Collaborative Work Collaborative Work: 1) A diver swims 20m under the water to the anchor of buoy that is 10m below the surface of the water. On the surfaces, how far is the buoy located from the place where the diver started? Round to the nearest meter. 2) Gillian is building a portable pet ramp with the dimensions shown on the right. She wants to cover all the faces of the ramp with carpet, which costs $1.59 per square foot (1 square foot = 144 square inches). How much does the carpeting cost? 14 in 18 in 48 in 3) The top of the badminton net is 5ft high. Ropes connect the top of each pole to stakes in the ground. The ropes are 8.5 feet long. What is the distance from the stake to the base of the pole? 4) You stand at the edge of a 4-m high diving platform. A beach ball is exactly 8 m from the base of the platform. To the nearest tenth of a meter, what is the distance d from the top of the platform to the beach ball? 5) Will a pen that is 14 cm long fit into a 3-cm by 4-cm by 12 cm box? 6) Satellites that relay television signals to Earth cruise at a distance of about 22,200 miles above Earth s surface. The radius of Earth is about 4,000 miles above Earth s surface. The radius of the Earth is about 4,000 miles. Find the distance a from the satellite to point T in the diagram below. Round to the nearest hundred miles. T a 22,200 mi 4,000 mi Earth Journal Question: 1) One leg of a right triangle is 3 cm and the hypotenuse is 4 cm. A student evaluates to find the length of the other leg. What error did the student make? The Illustrative Mathematics Project: http://illustrativemathematics.org/standards/k8 Course 3 Mathematics Common Core Series, Prentice Hall Page 14 of 23
Lesson 3: Homework 1) A tree forms a right angle with the ground. If you place the base of a 12-ft ladder 3 ft from the tree, how high up the tree will it reach? 2) A jogger runs around the city park shown below. Her friend cuts through the park on a diagonal. In miles far does each jogger run on a five-lap jog? Start/Finish Jogger s Path 500 ft Friend s Path Both 1,000 ft 3) The hypotenuse of a right triangle is 5 cm. The lengths of both legs are equal. Find the lengths of the legs. Round to the nearest tenth. 4) A computer screen has a diagonal length of 17 in and a height of 9 in. To the nearest tenth, what is the area of the screen? 5) A 10-ft-long slide is attached to a deck that is 5 ft high. Find the distance from the bottom of the deck to the bottom of the slide to the nearest tenth. Skills Practice Find the length of the diagonal. 8 cm 12 cm 24 cm 8 cm 10 cm 10 cm 10 cm 10 cm 9 cm Page 15 of 23
8.G.8 Lesson4: Introductory Task 8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system Introductory Task The library is 5 miles north of your house. The post office is 6 miles east of your house. To the nearest mile, how far is the library from the post office? Hint: Use the coordinate plane below to map the location of the post office and the library. 8 6 4 2 2 4 6 8 Page 16 of 23
Lesson4 Guided Practice Guided Practice 1) On the coordinate grid plot and label the points below: A (-2,4), B (-2,-1), and C (3, -1). Find the length of the hypotenuse to the nearest tenth 3-1 1 1 4 2) On a graph, the points (4, -2), (7,-2), (9, -5), and (2, -5) are connected in order to form a trapezoid. To the nearest tenth, what is its perimeter 3) A softball diamond has a shape of a square. The distance from home plate to second base is about 85 ft. Find the distance a player would run going from first base to second base. 4) Your school is 3 miles south of your house. The general store is 5 miles east of your school. To the nearest mile, how far is your house from the general store? Arizona State Department of Education Page 17 of 23
Lesson4 Collaborative Work 1) Scott, a freshman at Michigan State University, needs to walk from his dorm room in Wilson Hall to his math class in Wells Hall. Normally, he walks 500 meters east and 600 meters north along the sidewalks, but today he is running late, so he decides to take the shortest possible route through the Tundra. a. How many meters long is Scott s shortcut? b. How much shorter is the shortcut than Scott s usual route? 2) Mrs. Kidd likes to invite the neighbors for a cookout and then hid the food in various places around the backyard. Guests start at the center of the yard and the follow her clues to find their food. Here is one set of clues: Meat at (3,3). Vegetables at (-5,-12). Beverages at (5,-12). All Measurements are in yards. What is the distance in yards from the meat to the vegetables? 3) Tom is looking at a map of Great Adventures. The map is laid out in a coordinate system. Tom is at (2,3). The roller coaster is at (7,8) and the water ride is at (9,1). Is Tom closer to the roller coaster or the water ride? 4) Jade has crash-landed in the desert. There is village nearby, but Jade does not know the direction. Jade comes up with a cunning plan. She decides to fill up a water bottle from the plane, and to take a compass. Jade decides to walk east, south, west, north in a pattern. Jade s plan is mapped below on the coordinate grid: N W E S = 1 square mile Jade knows he will find the village no matter what direction it is in, and can (hopefully) find his way back to the plane for fresh water and shade when he needs it. But she needs to know, at the end of each stage: The NPS ACCeSS to Mathematics How far he walked altogether. How far (in a straight line) back to the plane. http://www.nps.k12.nj.us Page 18 of 23
Journal Question List 3 coordinate pairs that are 5 units away from the origin in the first quadrant. Describe how to find the points and justify your reasoning (Note: Points on the axes are not in the quadrant) The Illustrative Mathematics Project: http://illustrativemathematics.org/standards/k8 Page 19 of 23
Lesson4 Homework PROBLEM SOLVING 1) Town A is 90 miles due South of Town B, and 18 miles due East of Town C. Smith Road goes directly from Town B to Town C. Sketch the route and find the length of Smith Road. 2) Using the Pythagorean Theorem, find the distance between (4,2) and (7,10) 3) April is an avid chess player. She sets up a coordinate system on her chess board so she can record the position of the pieces during a game. In a recent game, April noted that her king was at (4,2) at the same time that her opponent s king was at (7,8). How far apart were the two kings? Round to the nearest tenth of a unit if necessary. 4) The coordinates of points A,B, and C are (5,4), (-2,1) and (4,-4), respectively. Which point, B or C, is closer to point A 5) Corey makes a map of his favorite park, using a coordinate system with the units of yards. The old oak tree is at position (4,8) and the granite boulder is at position (-3,7). How far apart are the old oak tree and the granite boulder? Round to the nearest tenth if necessary. SKILL BUILDING Graph each pair of ordered pairs. Then find the distance between the points. Round to the nearest tenth if necessary. 6) (-3,0), (3,-2) 7) (-4,-3), (2,1) 8) (0,2), (5,-2) O O O Glencoe Course 3 Workbook Page 20 of 23
Lesson5 Golden Problem Golden Problem A juice box has a base of 6 cm by 8 cm and a height of 12 cm. A straw is inserted into a hole in the center of the top. The straw must stick out 2 cm so you can drink from it. If the straw must be long enough to touch each bottom corner of the box, what is the minimum length the straw must be? (Assume the diameter of the straw is 0 for the mathematical model.) You must show all your work and state a clear explanation. Include a sketch of the juice box labeling the dimensions. The Illustrative Mathematics Project: http://illustrativemathematics.org/standards/k8 Page 21 of 23
Golden Problem Rubric: 3-Point Response The student uses the Pythagorean Theorem to successfully find the minimum length the straw must be AND The student sketch is 100% accurate. 2-Point Response The student shows correct work but does not provide the correct answer. OR The student commits a significant error but provides a correct response based on their incorrect work with clear explanations. OR The student provides the correct response and shows correct work but fails to provide clear explanations for each part. 1-Point Response The student only begins to provide a solution 0-Point Response The response demonstrates insufficient understanding of the problem s essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the required solutions, or the explanation may not be understandable. How decisions were made may not be readily understandable. OR The student shows no work or justification. Page 22 of 23
New Vocabulary Coordinate plane A coordinate plane is formed by the intersection of a horizontal number line called the x-axis and a vertical number line called the y-axis Irrational Number An irrational number is a number that cannot be written as the ratio of two integers. Ordered Pair An ordered pair identifies the locations of a point. The x-coordinate shows a point s position left or right from the origin. The y coordinate shows a points position up or down from the origin Perfect Square A perfect square is a number that is the square of an integer Pythagorean Theorem In any right triangle, the sum of the squares of the lengths of the legs (a and b) is square to the square of the length of the hypotenuse. Quadrants The x- and y- axes divide the coordinate plane into four regions called quadrants Square roots The square root of a number is a number that when multiplied by itself is equal to the original number. Page 23 of 23