Volume UNIT 9 CONTENTS. 1117A Unit 9

Transcription

1 UNIT 9 Volume CONTENTS COMMON CORE G-GMD.A.1 G-GMD.A.1 G-GMD.A.1 G-GMD.A. G-GMD.A. MODULE 1 Volume Formulas Lesson 1.1 Volume of Prisms and Cylinders Lesson 1. Volume of Pyramids Lesson 1. Volume of Cones Lesson 1.4 Volume of Spheres Lesson 1.5 Scale Factor A Unit 9

2 Unit Pacing Guide UNIT 9 45-Minute Classes Module 1 DAY 1 DAY DAY DAY 4 DAY 5 Lesson 1.1 Lesson 1. Lesson 1. Lesson 1.4 Lesson 1.5 DAY 6 DAY 7 Module Review and Assessment Readiness Unit Review and Assessment Readiness 90-Minute Classes Module 1 DAY 1 DAY DAY Lesson 1.1 Lesson 1. Module Review and Assessment Readiness Lesson 1. Lesson 1. Lesson 1.4 Lesson 1.5 Unit Review and Assessment Readiness Unit B

3 COMMON CORE COMMON CORE INMNLESE8976_U8ML COMMON CORE G-CO.C.10 Prove theorems about triangles. An isosceles triangle is a triangle with at least two congruent sides. The congruent sides are called the legs of the triangle. The angle formed by the legs is the vertex angle. The side opposite the vertex angle is the base. The angles that have the base as a side are the base angles. Label your angle A, as shown in the figure. A Resource Locker Vertex angle Legs Base Base angles Check students construtions. Resource Locker 4/19/141:10 1:10 PM PM INMNLESE8976_U8ML /19/141:10 1:10 PM PM Program Resources PLAN HMH Teacher App Access a full suite of teacher resources online and offline on a variety of devices. Plan present, and manage classes, assignments, and activities. ENGAGE AND EXPLORE Real-World Videos Engage students with interesting and relevant applications of the mathematical content of each module. eplanner Easily plan your classes, create and view assignments, and access all program resources with your online, customizable planning tool. Explore Activities Students interactively explore new concepts using a variety of tools and approaches. Professional Development Videos Authors Juli Dixon and Matt Larson model successful teaching practices and strategies in actual classroom settings. QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources. DO NOT EDIT--Changes must be made through "File info" CorrectionKeyNL-A;CA-A DO NOT EDIT--Changes must be made through "File info" CorrectionKeyNL-A;CA-A DO NOT EDIT--Changes must be be made through "File info" CorrectionKeyNL-A;CA-A LESSON. Isosceles and Equilateral Triangles Common Core Math Standards The student is expected to: G-CO.C.10 Prove theorems about triangles. Mathematical Practices MP. Logic Language Objective Explain to a partner what you can deduce about a triangle if it has two sides with the same length. ENGAGE Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? In an isosceles triangle, the angles opposite the congruent sides are congruent. In an equilateral triangle, all the sides and angles are congruent, and the measure of each angle is 60. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo, explaining that the instrument is a sextant and that long ago it was used to measure the elevation of the sun and stars, allowing one s position on Earth s surface to be calculated. Then preview the Lesson Performance Task. Teacher s Edition Support students with point-of-use Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more. Name Class Date. Isosceles and Equilateral Triangles Essential Question: What are are the the special relationships among angles and sides in in isosceles and equilateral triangles? Explore Investigating Isosceles Triangles An An isosceles triangle is is a a triangle with at at least two two congruent sides. Vertex angle Legs The congruent sides are are called the the legs of of the the triangle. The angle formed by by the the legs legs is is the the vertex angle. The side opposite the the vertex angle is is the the base. Base Base angles The angles that have the the base as as a a side are are the the base angles. In In this this activity, you you will will construct isosceles triangles and and investigate other potential characteristics/properties of of these special triangles. A Do Do your work in in the the space provided. Use a a straightedge to to draw an an angle. Label your angle A, as as shown in in the the figure. AA B Using a a compass, place the the point on on the the vertex and draw an an arc arc that intersects the the sides of of the the angle. Label the the points B B and C. C. A A B B C C Check students construtions. Module 1097 Lesson DO NOT EDIT--Changes must be made through "File info" CorrectionKeyNL-A;CA-A Name Class Date. Isosceles and Equilateral Triangles Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? Explore Investigating Isosceles Triangles In this activity, you will construct isosceles triangles and investigate other potential characteristics/properties of these special triangles. Do your work in the space provided. Use a straightedge to draw an angle. HARDCOVER PAGES Watch for the hardcover student edition page numbers for this lesson. DO NOT EDIT--Changes must be made through "File info" CorrectionKeyNL-A;CA-A DO NOT EDIT--Changes must be be made through "File info" CorrectionKeyNL-A;CA-A C Use the the straightedge to to draw line segment BC BC _.. AA BB D Use a a protractor to to measure each angle. Record the the measures in in the the table under the the column for for Triangle Triangle 11 Triangle Triangle Triangle 44 m A m B m C Possible answer for for Triangle 1: 1: m A 70 ; m B 55 ; m C 55. E Repeat steps A D at at least two more times and record the the results in in the the table. Make sure A A is is a a different size each time. Reflect How do do you know the the triangles you constructed are are isosceles triangles? The compass marks equal lengths on on both sides of of A; therefore, AB AB AC AC.... Make a a Conjecture Looking at at your results, what conjecture can can be be made about the the base angles, B B and C? The base angles are congruent. Explain 1 Proving the Isosceles Triangle Theorem and Its Converse In In the the Explore, you you made a a conjecture that the the base angles of of an an isosceles triangle are are congruent. This conjecture can can be be proven so so it it can can be be stated as as a a theorem. Isosceles Triangle Theorem If If two two sides of of a a triangle are are congruent, then the the two two angles opposite the the sides are are congruent. This theorem is is sometimes called the the Base Angles Theorem and and can can also be be stated as as Base angles of of an an isosceles triangle are are congruent. CC Module 1098 Lesson PROFESSIONAL DEVELOPMENT Learning Progressions In In this lesson, students add to to their prior knowledge of of isosceles and equilateral DO NOT EDIT--Changes must be be made through "File info" CorrectionKeyNL-A;CA-A EXPLORE Investigating Isosceles Triangles INTEGRATE TECHNOLOGY Students have the option of completing the isosceles triangle activity either in the book or online. QUESTIONING STRATEGIES What must be true about the triangles you construct in order for them to be isosceles triangles? They must have two congruent sides. How could you draw isosceles triangles without using a compass? Possible answer: Draw A and plot point B on one side of A. Then use a ruler to measure AB _ and plot point C on the other side of A so that AC AB. EXPLAIN 1 Proving the Isosceles Triangle Theorem and Its Converse CONNECT VOCABULARY Ask a volunteer to define isosceles triangle and have students give real-world examples of them. If possible, show the class a baseball pennant or other flag in the shape of an isosceles triangle. Tell students they will be proving theorems about isosceles triangles and investigating their properties in this lesson. ing Company Name Class Date. Isosceles and Equilateral Triangles Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? COMMON CORE G-CO.C.10 Prove theorems about triangles. Explore Investigating Isosceles Triangles An isosceles triangle is a triangle with at least two congruent sides. The congruent sides are called the legs of the triangle. The angle formed by the legs is the vertex angle. The side opposite the vertex angle is the base. The angles that have the base as a side are the base angles. In this activity, you will construct isosceles triangles and investigate other potential characteristics/properties of these special triangles. Do your work in the space provided. Use a straightedge to draw an angle. Label your angle A, as shown in the figure. A Vertex angle Legs Base Base angles Resource Locker

4 Evaluate Lesson 19. Precision and Accuracy PROFESSIONAL DEVELOPMENT TEACH ASSESSMENT AND INTERVENTION C1 teacher Support Lesson XX.X Comparing Linear, Exponential, and Quadratic Models Lesson 19. Precision and Accuracy 1 EXPLAIN Concept 1 Explain Concept ComPLEtINg the SquArE with EXPrES- SIoNS Determining Precision Avoid Common Errors Some students may not pay attention to whether b is positive or negative, since c is positive regardless of the sign of b. Have As you have seen, measurements are given to a certain precision. Therefore, student change the sign of b in some problems and compare the factored forms of both the value reported does not necessarily represent the actual value of the expressions. measurement. For example, a measurement of 5 centimeters, which is questioning Strategies In a perfect square trinomial, is the last term given to the nearest whole unit, can actually range from 0.5 units below the always positive? Explain. es, a perfect square trinomial can be reported value, 4.5 centimeters, up to, but not including, 0.5 units above either (a + b) or (a b) which can be fac- it, 5.5 centimeters. The actual length, l, is within a range of possible values: tored as (a + b) a + ab b and (a b) a + ab b. In both cases the last term is centimeters. Similarly, a length given to the nearest tenth can actually range positive. reflect from 0.05 units below the reported value up to, but not including, 0.05 units. The sign of b has no effect on the sign of above it. So a length reported as 4.5 cm could actually be as low as 4.45 cm or c because c ( b ) and a nonzero number squared is always positive. Thus, c is as high as nearly 4.55 cm. always positive. c ( b ) and a nonzero number c ( b ) and a nonzero number Name Date Class LESSON Precision and Significant Digits 1-1 Success for English Learners Minimum length cm and maximum length < cm The precision of a measurement is determined Find by the the range smallest of values unit or for the actual length and width of the rectangle. fraction of a unit used. Problem 1 Minimum Area Minimum width Minimum length Choose the more precise measurement. 7.5 cm cm 4. g is to the nearest tenth. Math On the Spot video tutorials, featuring program authors Dr. Edward Burger and Martha Sandoval-Martinez, accompany every example in the textbook and give students step-by-step instructions and explanations of key math concepts. Interactive Teacher Edition Customize and present course materials with collaborative activities and integrated formative assessment. Differentiated Instruction Resources Calculate the minimum and maximum possible areas. Round your answer to Support all learners with Differentiated the nearest square centimeters. Instruction Resources, The width and length of a rectangle including are 8 cm and 19.5 cm, respectively. Leveled Practice and Problem Solving Find the range of values for the actual length and width of the rectangle. Reading Strategies Minimum width 7.5 cm and maximum width < 8.5 cm Success for English Learners My answer Challenge Find the range of values for the actual length and width of the rectangle. 4.7 g is to the nearest hundredth. 4. g or 4.7 g Because a hundredth of a gram is smaller than a tenth of a gram, 4.7 g is more precise. Problem Choose the more precise measurement: 6 inches or feet. Reflect In the above exercise, the location of the uncertainty in the linear measurements results in different amounts of uncertainty in the calculated measurement. Explain how to fix this problem. Name Date Class LESSON 6-1 Linear Functions Reteach The graph of a linear function is a straight line. Ax + By + C 0 is the standard form for the equation of a linear function. A, B, and C are real numbers. A and B are not both zero. The variables x and y have exponents of 1 are not multiplied together are not in denominators, exponents or radical signs. Examples These are NOT linear functions: no variable x 9 exponent on x 1 xy 8 x and y multiplied together 6 x in denominator x y 8 y in exponent y 5 y in a square root Tell whether each function is linear or not. The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or Common Core standards. Practice With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, step-by-step solutions, and video tutorials. Assessments Choose from course assignments or customize your own based on course content, Common Core standards, difficulty levels, and more. Homework Students can complete online homework with a wide variety of problem types, including the ability to enter expressions, equations, and graphs. Let the system automatically grade homework, so you can focus where your students need help the most! Intervention Let the Personal Math Trainer automatically prescribe a targeted, personalized intervention path for your students. Elaborate Question of 17 Save & Close Solve the quadratic equation by factoring. 7x + 44x 7x 10 x, Personal Math Trainer View Step by Step Focus on Higher Order Thinking Raise the bar with homework and practice that incorporates higher-order thinking and mathematical practices in every lesson. Assessment Readiness Prepare students for success on high stakes tests for Integrated Mathematics with practice at every module and unit Assessment Resources Tailor assessments and response to intervention to meet the needs of all your classes and students, including Leveled Module Quizzes Leveled Unit Tests Unit Performance Tasks Placement, Diagnostic, and Quarterly Benchmark Tests Tier 1, Tier, and Tier Resources Video Tutor?! Textbook X Check Animated Math Turn It In Look Back x. xy x 1 x The graph of y C is always a horizontal line. The graph of x C is always a vertical line. 1. When deciding which measurement is more precise, what should you Formula consider? Examples Send to Notebook Unit D. An object is weighed on three different scales. The results are shown in the table. Which scale is the Explore most precise? Explain your answer. Your Turn y 1 x y x Scale Measurement T

5 Math Background Volume COMMON CORE LESSONS 1.1 to 1.4 B G-GMD.A. The formula for the volume of a rectangular prism (V Bh) is the starting point for developing the volume formulas for other three-dimensional figures. Another important ingredient in developing volume formulas is Cavalieri s Principle. This principle says that if two three-dimensional figures have the same height and the same cross-sectional area at every level, then they have the same volume. To illustrate the use of Cavalieri s Principle, consider the following informal argument for the volume formula for a cylinder with height h and base area B. Construct a rectangular prism of height h so that each rectangular cross-section has area B. As shown in the following figure, the prism and cylinder can be positioned so that the area of any cross-section of the cylinder created by a plane parallel to the base has the same area as the corresponding cross-section of the prism. By Cavalieri s Principle, the volume of the cylinder is equal to that of the prism. That is, V Bh. Note that the above argument works for all cylinders, oblique or right, and regardless of the shape of the base. In the case of a cylinder with a circular base, the volume formula may be written as V π r h. B h Students can use a stack of pennies to understand Cavalieri s principle. Have students arrange the pennies to form a right cylinder and ask them to estimate the volume. Then, have students push the stack to form an oblique cylinder. Students should see that the volume of the stack does not change. This is supported by Cavalieri s principle because the cross-sectional area at each level (that is, the area of the face of a penny) is unchanged when the stack is pushed to formed the oblique cylinder. Students may have worked primarily with two-dimensional figures. Here the focus shifts to three-dimensional figures. Students have probably already explored three-dimensional figures in earlier grades, especially in the context of volume problems. Students will learn and use the essential volume formulas. The formula for the volume, V, of a prism and the volume, V, of a right or oblique cylinder is the same, V Bh, where B is the area of the base and h is the height. Review the definition of pyramid and the associated vocabulary (lateral face, vertex, base). To sketch a pyramid, start by drawing a polygonal base and plotting a point for the vertex. Then draw straight lines from the vertex to each vertex of the polygonal base. Use dashed lines for edges that are hidden when the pyramid is viewed from the front. A key postulate about pyramids states that pyramids with equal base areas and equal heights have equal volumes. A wedge pyramid is a pyramid in which the base is a triangle and a perpendicular segment from the pyramid s vertex to the base intersects the base at a vertex of the triangle. Any pyramid can be divided into wedge pyramids. For pyramids, cones, and spheres, the situation is somewhat more complex. Students can fill models of solids with sand or water to gain an intuitive sense of how the volumes of pyramids and cones are related to the volumes of prisms and cylinders, respectively. However, rigorous justifications of the volume formulas for pyramids, cones, and spheres all rely on clever applications of Cavalieri s Principle. 1117E Unit 9

6 PROFESSIONAL DEVELOPMENT The following hands-on activity can help students develop the formula for the volume of a pyramid from an inductivereasoning perspective. Have students make nets for a square-based pyramid and a square-based prism that has the same height as the pyramid. Then, have students cut out, fold, and tape the nets to form the three-dimensional figures. Students can model the volume of the pyramid by filling it with uncooked rice, sand, or another granular material. Ask students to pour the rice from the pyramid into the prism as many times as necessary to see how the volumes of the figures are related. Students will discover that it takes three batches of rice from the pyramid to fill the prism. That is, the volume of the pyramid is one-third the volume of the associated prism. One approach to finding a formula for the volume of a cone is similar to a method for finding a formula for the circumference of a circle. This can be done by using inscribed regular polygons and an informal limit argument to show that the circumference, C, of a circle with radius r is given by C πr. You can inscribe a sequence of pyramids in a given cone and use similar reasoning to show that the volume, V, of the cone is given by V _ Bh, where B is the base area and h is the cone s height. One way to develop a formula for the volume of a sphere offers a surprising application of Cavalieri s principle. It is surprising because the argument is based on showing that two seemingly unrelated solids have the same volume. The solids a hemisphere and a cylinder from which a cone has been removed are shown to have the same crosssectional area at every level and therefore must have the same volume. The formula for the volume, V, of a sphere with radius r is V 4 πr. A bit of algebra shows that the volume of a sphere is equal to the volume of its circumscribed cylinder. This result, which has been known since ancient times, was of such importance to the Greek mathematician Archimedes that he requested that a drawing of a sphere and cylinder be placed on his tomb. Scale Factor LESSON 1.5 COMMON CORE G-GMD.A. When all dimensions of a figure are multiplied by a nonzero constant k, the perimeter or circumference changes by a factor of k and the area changes by a factor of k. This principle can be proved for various categories of figures by using established formulas. Consider the case of a triangle, ABC, with sides of length a, b, and c. When all dimensions are multiplied by k (k 0), the resulting triangle has sides of length ka, kb, and kc. The perimeter of the new triangle is therefore ka + kb + kc or k (a + b + c), which is k times the perimeter of ABC. The three-dimensional analogue of this principle says that, when all the dimensions of a three-dimensional figure are multiplied by a nonzero constant k, the surface area changes by a factor of k and the volume changes by a factor of k. Unit F

7 UNIT 9 Volume MATH IN CAREERS Unit Activity Preview After completing this unit, students will complete a Math in Careers task by investigating the volume of a piece of jewelry. Critical skills include modeling real-world situations and applying volume formulas. For more information about careers in mathematics as well as various mathematics appreciation topics, visit The American Mathematical Society at UNIT 9 Volume MODULE 1 Volume Formulas MATH IN CAREERS Image Credits: Hero Images/Corbis Jewelry Maker A jewelry maker designs and creates jewelry. Jewelry makers often employ geometric designs and shapes in their work, and so they need a good understanding of geometry. For example, they must calculate volume and surface area to determine the amount of materials needed. They can also use computer designing programs to help them with their design specifications. Jewelry makers often need to calculate costs of materials and labor to determine production costs for their designs. If you are interested in a career as a jewelry maker, you should study these mathematical subjects: Algebra Geometry Business Math Research other careers that require knowing the geometry of threedimensional objects. Check out the career activity at the end of the unit to find out how jewelry makers use math. Unit TRACKING YOUR LEARNING PROGRESSION IN_MNLESE89847_U9UO /19/14 9:10 AM Before In this Unit After Students understand: central and inscribed angles chords, secants, tangent lines, and arcs inscribed quadrilaterals segment lengths in circles formulas for circumference, area, and equation of a circle arc lengths, concentric circles, radian measure area of a sector 1117 Unit 9 Students will learn about: formulas for the volume of a prism, cylinder, pyramid, cone, and sphere scale factor Students will study: probability permutations and combinations conditional probability independent and dependent events making and analyzing decisions using probability

8 Reading Start -Up Visualize Vocabulary Use the words and draw examples to complete the chart. Object Example cone cylinder pyramid Vocabulary Review Words area (área) composite figure (figura compuesta) cone (cono) cylinder (cilindro) pyramid (pirámide) sphere (esfera) volume (volume) Preview Words apothem (apotema) oblique cylinder (cilindro oblicuo) oblique prism (prisma oblicuo) regular pyramid (pirámide regular) right cone (cono recto) right cylinder (cilindro recto) right prism (prisma recto) Reading Start Up Have students complete the activities on this page by working alone or with others. VISUALIZE VOCABULARY The example chart helps students review vocabulary associated with three-dimensional figures. If time allows, discuss the characteristics that helped students identify each figure. UNDERSTAND VOCABULARY Use the following explanations to help students learn the preview words. A regular pyramid has a base that is a regular polygon and lateral faces that are congruent isosceles triangles. sphere Understand Vocabulary Complete the sentences using the preview words. 1. A cone whose axis is perpendicular to its base is called a(n) right cone.. A prism that has at least one nonrectangular lateral face is called a(n) oblique prism. Active Reading Pyramid Create a Pyramid and organize the adjectives used to describe different objects right, regular, oblique on each of its faces. When listening to descriptions of objects, look for these words and associate them with the object that follows. A cylinder whose axis is perpendicular to its bases is a right cylinder. A cylinder in which this is not the case is an oblique cylinder. ACTIVE READING Students can use these reading and note-taking strategies to help them organize and understand the new concepts and vocabulary. Encourage students to make connections among the figures and their properties by using descriptive vocabulary when constructing their vocabulary pyramids. Remind students to keep asking questions about any vocabulary that they find confusing or unclear. Unit 9 IN_MNLESE89847_U9UO /19/14 9:10 AM ADDITIONAL RESOURCES Differentiated Instruction Reading Strategies Unit

9 MODULE This version is for Algebra 1 and Geometry only 1 Volume Formulas ESSENTIAL QUESTION: Answer: For one example, volume formulas are useful when you want to find how much liquid something can hold, such as a cup or a swimming pool. PROFESSIONAL DEVELOPMENT VIDEO Professional Development Video Author Juli Dixon models successful teaching practices in an actual high-school classroom. MODULE Volume Formulas 1 LESSON 1.1 Essential Question: How can you use volume formulas to solve real-world problems? Volume of Prisms and Cylinders LESSON 1. Volume of Pyramids LESSON 1. Volume of Cones LESSON 1.4 Volume of Spheres LESSON 1.5 Scale Factor Professional Development my.hrw.com Image Credits: Stringer/ Reuters/Corbis REAL WORLD VIDEO Check out how volume formulas can be used to find the volumes of real-world objects, including sinkholes. MODULE PERFORMANCE TASK PREVIEW How Big Is That Sinkhole? In 010, a giant sinkhole opened up in a neighborhood in Guatemala and swallowed up the three-story building that stood above it. In this module, you will choose and apply an appropriate formula to determine the volume of this giant sinkhole. Module DIGITAL IN_MNLESE89847_U9M1MO TEACHER EDITION 1119 Access a full suite of teaching resources when and where you need them: Access content online or offline Customize lessons to share with your class Communicate with your students in real-time View student grades and data instantly to target your instruction where it is needed most PERSONAL MATH TRAINER Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, Common Core-aligned practice tests. 4/19/14 8:57 AM 1119 Module 1

10 Are YOU Ready? Complete these exercises to review skills you will need for this module. Area of a Circle Example 1 Find the area of a circle with radius equal to 5. A πr Write the equation for the area of a circle of radius r. A π (5) Substitute the radius. A 5π Simplify. Find each area. 1. A circle with radius 4 16π. A circle with radius 5. A circle with radius π 9 π 4. A circle with radius π _ Volume Properties Online Homework Hints and Help Extra Practice Example Find the number of cubes that are 1 cm in size that fit into a cube of size 1 m. Notice that the base has a length and width of 1 m or 100 cm, so its area is 1 m or 10,000 cm. The 1 m cube is 1 m or 100 cm high, so multiply the area of the base by the height to find the volume of 1,000,000 cm. Find the volume. 5. The volume of a 1 km piece of land in m 1,000,000,000 m 6. The volume of a 1 ft box in in. 178 in 5π 4 π Are You Ready? ASSESS READINESS Use the assessment on this page to determine if students need strategic or intensive intervention for the module s prerequisite skills. ASSESSMENT AND INTERVENTION 1 TIER 1, TIER, TIER SKILLS Personal Math Trainer will automatically create a standards-based, personalized intervention assignment for your students, targeting each student s individual needs! Volume of Rectangular Prisms Example Find each volume. Find the area of a rectangular prism with height 4 cm, length cm, and width 5 cm. V Bh Write the equation for the volume of a rectangular prism. V ()(5)(4) The volume of a rectangular prism is the area of the base times the height. V 60 cm Simplify. 7. A rectangular prism with length m, width 4 m, and height 7 m 84 m ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: Tier Skill Pre-Tests for each Module Tier Skill Post-Tests for each skill 8. A rectangular prism with length cm, width 5 cm, and height 1 cm 10 cm Module IN_MNLESE89847_U9M1MO 110 Response to Intervention Differentiated Instruction 4/19/14 8:57 AM Tier 1 Lesson Intervention Worksheets Tier Strategic Intervention Skills Intervention Worksheets Tier Intensive Intervention Worksheets available online Reteach 1.1 Reteach 1. Reteach 1. Reteach 1.4 Reteach Surface Area 44 Volume Building Block Skills 9, 10, 11, 14, 0, 1, 77, 8, 101, 106 Challenge worksheets Extend the Math Lesson Activities in TE Module 1 110

11 COMMON CORE Locker LESSON Common Core Math Standards The student is expected to: COMMON CORE G-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Also G-GMD.A., G-GMD.A., G-MG.A.1, G-MG.A. Mathematical Practices COMMON CORE 1.1 Volume of Prisms and Cylinders MP.4 Modeling Language Objective Explain to a partner how to apply the formulas for the volume of a prism and a cylinder. ENGAGE Essential Question: How do the formulas for the volume of a prism and a cylinder relate to area formulas that you already know? The formula for the volume of a prism involves the formula for the area of a rectangle, and the formula for the volume of a cylinder involves the formula for the area of a circle. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photograph. Ask students to suggest possible connections between the photo and the subject of this lesson, the volume of prisms and cylinders. Then preview the Lesson Performance Task. Name Class Date 1.1 Volume of Prisms and Cylinders Essential Question: How do the formulas for the volume of a prism and cylinder relate to area formulas that you already know? Explore Developing a Basic Volume Formula The volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure. This prism is made up of 8 cubes, each with a volume of 1 cubic centimeter, so it has a volume of 8 cubic centimeters. You can use this idea to develop volume formulas. In this activity you ll explore how to develop a volume formula for a right prism and a right cylinder. A right prism has lateral edges that are perpendicular to the bases, with faces that are all rectangles. A B IN_MNLESE89847_U9M1L1.indd 111 right prism axis right right cylinder prism On a sheet of paper draw a quadrilateral shape. Make sure the sides aren t parallel. Assume the figure has an area of B square units. Use it as the base for a prism. Take a block of Styrofoam and cut to the shape of the base. Assume the prism has a height of 1 unit. How would changing the area of the base change the volume of the prism? An increase in the area of the base would increase the volume. A decrease in the area of the base would decrease the volume. axis right cylinder Resource Locker Volume 1 cubic unit A right cylinder has bases that are perpendicular to it center axis. area is B square units height is 1 unit Module Lesson 1 DO NOT EDIT--Changes must be made through File info CorrectionKeyNL-A;CA-A Name Class Date 1. 1 Volume of Prisms and Cylinders Essential Question: How do the formulas for the volume of a prism and cylinder relate to area formulas that you already know? G-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Also G-GMD.A., G-GMD.A., G-MG.A.1, G-MG.A. Explore Developing a Basic Volume Formula The volume of a three-dimensional figure is the number of nonoverlapping cubic units contained in the interior of the figure. This prism is made up of 8 cubes, each with a volume of 1 cubic centimeter, so it has a volume of 8 cubic centimeters. Volume 1 cubic unit You can use this idea to develop volume formulas. In this activity you ll explore how to develop a volume formula for a right prism and a right cylinder. A right prism has lateral edges that are perpendicular A right cylinder has bases that are perpendicular to it to the bases, with faces that are all rectangles. center axis. right prism axis right right cylinder prism On a sheet of paper draw a quadrilateral shape. Make sure the sides aren t parallel. Assume the figure has an area of B square units. axis right cylinder Resource area is B square units HARDCOVER PAGES Watch for the hardcover student edition page numbers for this lesson. 4/19/14 7:58 AM Use it as the base for a prism. Take a block of Styrofoam and cut to the shape of the base. Assume the prism has a height of 1 unit. How would changing the area of the base change the volume of the prism? An increase in the area of the base would increase the volume. A decrease in the area of the base would decrease the volume. height is 1 unit Module Lesson 1 IN_MNLESE89847_U9M1L1.indd 111 4/19/14 8:00 AM 111 Lesson 1.1

12 If the base has an area of B square units, how many cubic units does the prism contain? B cubic units Now use the base to build a prism with a height of h units. EXPLORE Developing a Basic Volume Formula height is h units INTEGRATE TECHNOLOGY Students have the option of doing the Explore activity either in the book or online. B How much greater is the volume of this prism compared to the one with a height of 1? The volume of the prism is B h, which is h times the volume of the smaller prism. Reflect 1. Suppose the base of the prism was a rectangle of sides l and w. Write a formula for the volume of the prism using l, w, and h. V lwh. A cylinder has a circular base. Use the results of the Explore to write a formula for the volume of a cylinder. Explain what you did. V π r h; I multiplied the area of a circle, π r, by the height of the cylinder. Explain 1 h Finding the Volume of a Prism The general formula for the volume of a prism is V B h. With certain prisms the volume formula can include the formula for the area of the base. Volume of a Prism The formula for the volume of a right rectangular prism with length l, width w, and height h is V lwh. W B l h WS The formula for the volume of a cube with edge length s is V s. l S S h S S S QUESTIONING STRATEGIES How are the units used to measure volume related to the units used to measure length? The units used to measure volume are cubes of the units used to measure length. How can you estimate the volume of a cylinder whose radius and height are both 1 cm? πr h π (1) 1.14 cm INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP. Begin by briefly reviewing the definitions of prism and cylinder. Be sure that students recognize the similarities in these three-dimensional figures and that they can identify the bases of a given prism or cylinder. EXPLAIN 1 Finding the Volume of a Prism Module 1 11 Lesson 1 PROFESSIONAL DEVELOPMENT IN_MNLESE89847_U9M1L1.indd 11 Learning Progressions Previously, students saw that the formula for the area of a rectangle is the starting point for developing the area formulas for other polygons. In much the same way, the formula for the volume of a rectangular prism (V Bh) is the starting point for developing the volume formulas for other three-dimensional figures. Another important idea in developing volume formulas is Cavalieri s Principle, which states that if two threedimensional figures have the same height and the same cross-sectional area at every level, then they have the same volume. As students progress through more advanced courses, such as calculus, they will apply Cavalieri s Principle to more complex solid figures. 4/19/14 7:58 AM AVOID COMMON ERRORS Students may have difficulty finding the volume of a prism if they need to use conversion factors. Review how to use unit analysis and conversion factors to find volume. Volume of Prisms and Cylinders 11

13 QUESTIONING STRATEGIES How would you find the volume of the contents of a right prism that is 4 full? Multiply the volume found by using the volume formula by 4. Example 1 Use volume formulas to solve real world problems. A shark and ray tank at the aquarium has the dimensions shown. Estimate the volume of water in gallons. Use the conversion 1 gallon 0.14 f t. Step 1 Find the volume of the aquarium in cubic feet. V lwh (10) (60) (8) 57,600 f t Step Use the conversion factor _ 1 gallon to estimate 0.14 ft the volume of the aquarium gallons. 8 ft 10 ft 60 ft 1 gallon 57,600 ft _ 0.14 ft 49,851 gallons _ 1 gallon 0.14 ft 1 1 gallon Step Use the conversion factor _ to estimate the weight of the water. 8. pounds 8. pounds 49,851 gallons,580,6549 pounds 1 gallon 8. pounds 1 1 gallon The aquarium holds about 49,851 gallons. The water in the aquarium weighs about,580,659 pounds. Chemistry Ice takes up more volume than water. This cubic container is filled to the brim with ice. Estimate the volume of water once the ice melts. ge AB Density of ice: g/c m Density of water: 1 g/ cm Step 1 Find the volume of the cube of ice. V s 7 c m Step Convert the volume to mass using the conversion factor g_ cm. cm 7 c m g_ c m 4.8 g cm Step Use the mass of ice to find the volume of water. Use the conversion factor 1 _ g. Reflect 4.8 g cm 1 g 4.8 c m. The general formula for the volume of a prism is V B h. Suppose the base of a prism is a parallelogram of length l and altitude h. Use H as the variable to represent the height of the prism. Write a volume formula for this prism. V l h H Module 1 11 Lesson 1 IN_MNLESE89847_U9M1L1.indd 11 COLLABORATIVE LEARNING Small Group Activity Have students work in groups to make cylinders out of modeling clay. Then have them use string to slice the cylinders into eight congruent sectors and arrange the sectors to approximate a rectangular prism. Have them use this shape to explain the volume formula for a cylinder. 4/19/14 7:58 AM 11 Lesson 1.1

14 Your Turn 4. Find the volume of the figure. 5. Find the volume of the figure. EXPLAIN Finding the Volume of a Cylinder Volume 8 cubic units 88 cubic units 7 k cubic units Explain Finding the Volume of a Cylinder You can also find the volume of prisms and cylinders whose edges are not perpendicular to the base. Oblique Prism An oblique prism is a prism that has at least one non-rectangular lateral face. Oblique Cylinder Each cube has a side of k. An oblique cylinder is a cylinder whose axis is not perpendicular to the bases. QUESTIONING STRATEGIES How is an oblique cylinder similar to a right cylinder? The bases are circles. The bases are connected by a curved lateral surface. The same volume formula works for both types of cylinders. How is an oblique cylinder different from a right cylinder? In an oblique cylinder, the axis is not perpendicular to the base. h h h h Cavalieri s Principle If two solids have the same height and the same cross-sectional area at every level, then the two solids have the same volume. h h h h Module Lesson 1 DIFFERENTIATE INSTRUCTION Kinesthetic Experience IN_MNLESE89847_U9M1L1.indd 114 Kinesthetic learners can use a stack of pennies to understand Cavalieri s Principle. Have students arrange the pennies to form a right cylinder and ask them to estimate the volume. Then have students push the stack to form an oblique cylinder. Students should see that the volume of the stack does not change, because the number and size of the pennies have not changed. This is supported by Cavalieri s Principle because the cross-sectional area at each level (that is, the area of the face of a penny) is unchanged when the stack is pushed to form the oblique cylinder. 4/19/14 7:58 AM Volume of Prisms and Cylinders 114

15 AVOID COMMON ERRORS Students may make careless errors if they do not read the problem carefully and use the given information correctly. For example, if the diameter of a cylinder is given, they must divide by to find the radius before using the formula. Also, the base area may be given instead of the radius. Example To find the volume of an oblique cylinder or oblique prism, use Cavalieri s Principle to find the volume of a comparable right cylinder or prism. The height of this oblique cylinder is three times that of its radius. What is the volume of this cylinder? Round to the nearest tenth. Use Cavalieri s Principle to find the volume of a comparable right cylinder. Represent the height of the oblique cylinder: h r Use the area of the base to find r: π r 81π cm Calculate the height: h r 7 cm Calculate the volume: V Bh (81π) The volume is about cubic centimeters. B 81π cm The height of this oblique square-based prism is four times that of side length of the base. What is the volume of this cylinder? Round to the nearest tenth. Calculate the height of the oblique cylinder: h 4 s, where s is the length of the square base. Use the area of the base to find s. s 75 cm s _ 75 cm Calculate the height. h 4s 4 75 cm Calculate the volume. V Bh (75 c m )( 4 _ 75 cm) c m B 75 cm Your Turn Find the volume r 1 in. h 45 in. 4x cm 5x cm h (x + ) cm V π r h π (1 in.) (4 in.) 6480 in. V (4x) (5x) (x + ) 0x (x + ) Module Lesson 1 IN_MNLESE89847_U9M1L1.indd 115 LANGUAGE SUPPORT Connect Vocabulary To help students remember the vocabulary in the lesson, including right prism, right cylinder, oblique prism, and oblique cylinder, have students make a small poster showing examples of each solid figure. Then have them use colored pencils to mark the dimensions of the base of each in one color, the heights in another color, and then list the formulas for volume in different colors. Have them label the figures with the vocabulary words, and display the posters in the classroom. Invite students to present their posters to the class. 4/19/14 7:58 AM 115 Lesson 1.1

16 Explain Finding the Volume of a Composite Figure Recall that a composite figure is made up of simple shapes that combine to create a more complex shape. A composite three-dimensional figure is formed from prisms and cylinders. You can find the volume of each separate figure and then add the volumes together to find the volume of the composite figure. Example Find the volume of each composite figure. 5 ft EXPLAIN Finding the Volume of a Composite Figure Find the volume of the composite figure, which is an oblique cylinder on a cubic base. Round to the nearest tenth. The base area of the cylinder is B π r π (5) 5π ft. The cube has side lengths equal to the diameter of the cylinder s circular base: s 10. The height of the cylinder is h ft. The volume of the cube is V s f t. The volume of the cylinder is V Bh (5π ft ) (1 ft) 94.5 f t. The total volume of the composite figure is the sum of the individual volumes. V 1000 f t f t f t This periscope is made up of two congruent cylinders and two congruent triangular prisms, each of which is a cube cut in half along one of its diagonals. The height of each cylinder is 6 times the length of the radius. Use the measurements provided to estimate the volume of this composite figure. Round to the nearest tenth. Use the area of the base to find the radius. B π r π r 6 π, so r 6 in. Calculate the height of the cylinder: h 6r in. The faces of the triangular prism that intersect the cylinders are congruent squares. The length of the sides of each square are the same as the diameter of the circle. s d 6 1 in. The two triangular prisms form a cube. What is the volume of this cube? V s in Find the volume of the two cylinders: V 6π 6 4π in The total volume of the composite figure is the sum of the individual volumes. V 178 in + 4π in in h ft B 6π in. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students should be able to recognize the solid figures that make up a composite figure. Ask them to make an organized list of all figures comprising the composite figures and then give the formulas for their volumes. Emphasize that while the volume of each cylinder or prism has the formula V Bh, where B is the area of the base and h is the height, finding B changes from one solid figure to the next, depending on the figure. After organizing how to find the volume of the composite figure, have students substitute for the variables in each formula, find the volume of each solid, and then add the volumes to get the total. QUESTIONING STRATEGIES How can you find the volume of a composite figure? You can divide a composite figure into component figures, use the volume formula for each component figure, then add the individual volumes. Module Lesson 1 IN_MNLESE89847_U9M1L1.indd 116 4/19/14 7:58 AM Volume of Prisms and Cylinders 116

17 ELABORATE QUESTIONING STRATEGIES How do you find the volume of a prism or cylinder? Multiply the area of the base of the figure by the height. Reflect 8. A pipe consists of two concentric cylinders, with the inner cylinder hollowed out. Describe how you could calculate the volume of the solid pipe. Write a formula for the volume. Find the volume of the large cylinder and subtract the volume of the smaller cylinder. If r 1 is the radius of the larger cylinder and r is the radius of the smaller cylinder and h is the common height, this is the volume of the solid pipe. V πr 1 h - πr h πh (r 1 - r ) r 1 r h SUMMARIZE THE LESSON How is the formula for the volume of a prism similar to the formula for the volume of a cylinder? How are the formulas different? Both of the formulas may be written as V Bh, where B is the area of the base and h is the height, but in the formula for the volume of a cylinder, B represents a circular area, while in the formula for the volume of a prism, B represents the area of a polygon. \ Your Turn 9. This robotic arm is made up of two cylinders with equal volume and two triangular prism hands. The volume of each hand is _ r _ r r, where r is the radius of the cylinder s base. What fraction of the total volume do the robotic hands take up? V Arms _ r V Total (π r h) + _ r π r h + _ r r ( πh + r ) V Arms _ V Total _ r r ( πh + r ) r_ ( πh + r ) r_ πh + r Elaborate 10. If an oblique cylinder and a right cylinder have the same height but not the same volume, what can you conclude about the cylinders? They have different radii. 11. A right square prism and a right cylinder have the same height and volume. What can you conclude about the radius of the cylinder and side lengths of the square base? Let V 1 be the volume of the prism. Let V be the volume of the cylinder. V 1 V s h π r h s π r s π r 1. Essential Question Check-In How does the formula for the area of a circle relate to the formula for the volume of a cylinder? Sample answer: I can multiply the expression for the area of a circle by the height of the cylinder to get an expression for the volume of the cylinder. Module Lesson 1 IN_MNLESE89847_U9M1L1.indd 117 4/19/14 7:58 AM 117 Lesson 1.1

18 Evaluate: Homework and Practice 1. The volume of prisms and cylinders can be represented with Bh, where B represents the area of the base. Identify the type of figure shown and match the prism or cylinder with the appropriate volume formula. Online Homework Hints and Help Extra Practice EVALUATE A. V (π r ) h B. V ( bh ) h C. V lwh rectangular prism triangular prism rectangular prism cylinder cylinder triangular prism C. V lwh Find the volume of each prism or cylinder. Round to the nearest hundredth... A. V (π r ) h B. V ( bh ) h ASSIGNMENT GUIDE Concepts and Skills Explore Developing a Basic Volume Formula Example 1 Finding the Volume of a Prism Example Finding the Volume of a Cylinder Practice Exercise 1 Exercises 6 Exercises cm 8.4 mm 9 cm 5.6 mm.5 mm V (8.4 mm) (.5 mm) (5.6 mm) m m 4. The area of the hexagonal base is ( 54 tan 0 ) m. Its height is 8 m. V Bh ( 54 tan 0 ) m 8 m m V 9 cm 4 cm 6 cm 16 c m 4 cm 15 yd 10 ft 9 yd V ( 9 yd 15 yd ) 1 yd (67.5 yd ) 1 yd 810 y d 1 ft 1 yd 5. The area of the pentagonal base is ( 15 tan 6 ) m. Its height is 15 m. V Bh ( _ 15 tan 6 ) m 15 m m V π (6 ft) 10 ft ft Module Lesson 1 Example Finding the Volume of a Composite Figure Exercises 9 1 INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Some students may benefit from a hands-on approach to finding the volume of a prism. Have students imagine an arrangement of rows of 4 unit cubes each. The cubes form a 4 1 rectangular prism with a volume of 1 cubic units. To find the volume of a 4 prism, imagine another layer of cubes stacked on top of the first one. IN_MNLESE89847_U9M1L1.indd 118 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 4/19/14 7:58 AM Recall of Information MP.5 Using Tools 1 15 Skills/Concepts MP.4 Modeling 16 Skills/Concepts MP.5 Using Tools Skills/Concepts MP.1 Problem Solving Volume of Prisms and Cylinders 118

19 AVOID COMMON ERRORS When finding the volume of composite figures, some students may think they need to subtract the areas of the common surfaces that are shared by the touching cubes, as they would if they were finding the surface area of a composite figure. Using concrete models, show them that the volume of a composite figure made up of stacked blocks is the sum of the volumes of the individual blocks. CONNECT VOCABULARY To remember how the volume of a prism is connected to the volume of familiar figures like cubes and boxes, have students make a graphic organizer listing the volume formulas they know and adding the volume formulas from this module. Sample: Shape Volume Volume Cube V s V Bh Box (prism) V lwh V Bh Cylinder V πr h V Bh 8. Multi-Step A vase in the shape of an oblique cylinder has the dimensions shown. What is the volume of the base? Round to the nearest thundredth. (Hint: Use the right triangle in the cylinder to find its height.) a a Find the volume of each composite figure. Round to the nearest tenth ft in. 4 ft 4 ft 8 cm 1 ft Cylinder V π (4 ft) 4 ft ft 8 cm 14 ft ft ft ft V ft cm 1. The two figures 4 cm 4 cm on each end combine to 6 cm 6 cm form a right cylinder. 6 cm 8 cm V (4 cm) 64 cm V (6 m) 16 cm V (8 cm) a 9 a 9 cm 51 cm V Bh π r h π (7 cm) 9 cm cm Prism V 1 ft 6 ft 14 ft 1008 ft 64 cm + 16 cm + 51 cm 79 cm 17 cm Large cylinder V π (10 in) 15 in in 4 ft 5 in. ft 1 ft 14 cm 15 in. V in in 54.9 in One whole cylinder V π ( ft) 4 ft 50.7 ft Prism V (1 ft) (4 ft) (4 ft) 19 ft _ cm 1 liter ( 1000 cm ) 1.48 liters The volume is 1.5 liters ft + 19 ft 4.7 ft Small cylinder V π (5 in) 15 in in 4 ft ft Module Lesson 1 IN_MNLESE89847_U9M1L1.indd 119 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 4/19/14 7:58 AM 19 Strategic Thinking MP.1 Problem Solving 0 Strategic Thinking MP.4 Modeling 1 Strategic Thinking MP. Reasoning 119 Lesson 1.1

20 1. Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle. If he wants to fill it to a depth of 4 in., how many cubic yards of dirt does he need? Round to the nearest cubic yard. If dirt costs $5 per y d, how much will the project cost? V Bh lwh 9 ft 16 ft ( 4 yd $5 yd $50; yd ; $50 1 ft ) 48 ft ; 48 ft ( 1 yd 14. Persevere in Problem Solving A cylindrical juice container with a in. diameter has a hole for a straw that is 1 in. from the side. Up to 5 in. of a straw can be inserted. a. Find the height h of the container to the nearest tenth. a + b c ; h + 5 ; h + 4 5; h 1; h 1 ; h 4.6 in. b. Find the volume of the container to the nearest tenth. V Bh π r h π (1.5 in) 4.6 in..5 in _ ft ) ( 1 yd ft ) ( 1 yd ft ) 1. _ 7 yd ; 5 in. in. h 1 in. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. When working with the volume of a cylinder formulas, remind students that if they use.14 instead of the π key on a calculator, they are automatically giving an approximation to the volume of a cylinder. Have small groups of students discuss when using the π key on a calculator is appropriate and when it is not necessary. c. How many ounces of juice does the container hold? (Hint: 1 i n 0.55 oz) _.5 in 0.55 oz 17.9 oz 1 in 15. Abigail has a cylindrical candle mold with the dimensions shown. If Abigail has a rectangular block of wax measuring 15 cm by 1 cm by 18 cm, about how many candles can she make after melting the block of wax? Round to the nearest tenth. Wax Mold V Bh V Bh lwh π r h 15 cm 1 cm 18 cm π cm 18 cm 40 cm 18 cm 14.9 Because 0.9 of a candle would not make an entire candle, the answer rounds down to 14 candles. 16. Algebra Find the volume of the three-dimensional figure in terms of x. V Bh π r h π (x + 1) x π ( x + x + 1) x π ( x + x + x) π x + π x + πx V π x + π x + πx x 6.0 cm.4 cm x + 1 INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 If the corresponding linear dimensions of two similar solid figures have a scale factor of k, then their volumes are in the ratio of 1: k. For example, if the dimensions of a cube with an edge length of 4 cm are doubled, then the volumes are in the ratio 1:, or 1:8. This means, the volume goes from 4 64 c m to 8 51 c m. Module Lesson 1 IN_MNLESE89847_U9M1L1.indd 110 4/19/14 7:58 AM Volume of Prisms and Cylinders 110