MODULE. MATHEMATICS Areas & Volumes

Transcription

1 MODULE 1 MATHEMATICS Areas & Volumes

2 National Math + Science Initiative 8350 North Central Expressway, Suite M-2200, Dallas, TX Office phone: Fax:

3 Mathematics MATHEMATICS Module 1 Areas & Volumes 2014 EDITION i

4 Mathematics Copyright 2014 National Math + Science Initiative. All rights reserved. No part of this publication may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher. Student activity pages may be photocopied for classroom use only. Printed and bound in the United States of America. ISBN Grateful acknowledgment is given authors, publishers, and agents for permission to reprint copyrighted material. Every effort has been made to determine copyright owners. In case of any omission, the publisher will be pleased to make suitable acknowledgments in future editions. Published by: National Math + Science Initiative 8350 North Central Expressway Suite M-2200 Dallas, TX ii

5 Mathematics Areas & Volumes CONTENTS Belief Statements... i Middle Grades Learner Outcomes.... ii High School Learner Outcomes....iii Lessons & Assessments... 1 Area & Volume Content Progression Chart Area Concept Development Chart Volume Concept Development Chart Surface Area and Volume Unit Dog Solids of Revolution Solving Systems of Linear Equations Working With Formulas and Function Notation Volumes of Revolution Systems of Linear Inequalities Cone Exploration and Optimization Cone Exploration and Optimization Revisited Sample Quiz Questions Free Response Questions Appendix.... A1 Standards for Mathematical Practice... A3 Additional Graphs and Materials.... A11 Graphical Organizer.... A13 Cardstock Copy.... A15 iii

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7 Mathematics National Math + Science Belief Statements Accomplished, dynamic teachers are knowledgeable in their content and confident in their abilities to prepare students for higher education. They create classrooms in which students: engage intellectually to develop conceptual understanding generate their own ideas, questions, and propositions interact collegially with one another to solve problems employ appropriate resources for inquiry-based learning Our teacher training program offers meaningful support to teachers as they construct these effective classrooms. Through tested content materials and research-based instructional strategies, our program enables and encourages them to: choose significant and worthwhile content and connect it to other knowledge use appropriate questioning strategies to develop conceptual understanding clarify to students the importance of abstract concepts and big questions use formative assessments to improve instruction and achieve higher goals guarantee equitable access for all students to information and achievement i

8 Mathematics Areas & Volumes Middle Grades Learner Outcomes MODULE DESCRIPTION Middle school teachers examine how concepts involving areas and volumes progress from sixth grade to calculus. Training begins with manipulative-rich student lessons that explore the surface area and volume of three-dimensional solids. As the lessons progress through the vertical strand, participants will come to realize the necessity of teaching both area and volume on the coordinate plane. They work selected questions from and discuss teaching strategies for model lessons for middle grades in which students plot coordinate points to create planar figures, calculate the areas of those figures, revolve the figures about a horizontal or vertical line to create three-dimensional solids, and calculate the volumes of those solids. In addition, they will work through and discuss lessons from Algebra 1 and Geometry or Math 1 and Math 2 in which students begin by graphing linear equations to create these models. LEARNER OUTCOMES Participants will: Compare expectations for students from sixth grade math through pre-calculus on the topics of areas and volumes to increase vertical alignment. Apply deeper content-based knowledge to increase instructional rigor in order to prepare students for high school math courses leading to college-level calculus in an AP class or university setting. Describe the effect of scale changes on surface area and volume. Represent three-dimensional figures with nets and use the nets to calculate surface area. Model three-dimensional solids formed when various regions are revolved about horizontal or vertical lines. Calculate the surface area and volume of solids of revolution. Identify instructional strategies that they can use to assist students in developing the habits of mind that are required for college and career readiness. ii

9 Mathematics Areas & Volumes High School Learner Outcomes MODULE DESCRIPTION High school teachers examine how concepts involving areas and volumes progress from sixth grade to calculus. Participants begin by reviewing how these concepts are introduced at the middle school level. As the lessons progress through the vertical strand, participants come to realize the necessity of teaching both area and volume on the coordinate plane. They work selected questions from and discuss teaching strategies for high school model lessons. In these activities, teachers experience how students graph linear equations and/or inequalities to create planar figures, calculate the areas of those figures, revolve the figures about a horizontal or vertical line to create three-dimensional solids, and calculate the volumes of those solids. In addition, teachers engage in an activity to explore maximizing the volume of a cone. LEARNER OUTCOMES Participants will: Compare expectations for students from sixth grade math through pre-calculus on the topics of areas and volumes to increase vertical alignment. Apply deeper content-based knowledge to increase instructional rigor in order to prepare students for college-level calculus in an AP class or university setting. Express area and volume formulas in terms of specific variables. Model three-dimensional solids formed when various regions are revolved about horizontal or vertical lines. Calculate the surface area and volume of solids of revolution. Identify instructional strategies that teachers can use to assist students in developing the habits of mind that are required for college and career readiness. iii

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11 Mathematics Areas & Volumes LESSONS & ASSESSMENTS 1

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13 3rd Grade Skills/ Write the formula for the area of a rectangle using a variable for the missing length or width. In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. 4th Grade Skills/ Isolate the variable for length or width in the formula for area of a rectangle. In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) Area & Volume Content Progression Chart 5th Grade Skills/ Isolate the variable for length or width in the formulas for area and perimeter of a rectangle. In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) 6th Grade Skills/ Solve literal equations (perimeter, area, and volume). In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) 7th Grade Skills/ Solve literal equations (perimeter, area, and volume). Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Algebra 1 Skills/ Solve literal equations (perimeter, area, and volume). Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Geometry Skills/ Solve literal equations (perimeter, area, and volume). Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Algebra 2 Skills/ Solve literal equations (perimeter, area, and volume). Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and non-overlapping right rectangular prisms with whole number edge lengths. In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and nonoverlapping right rectangular prisms with fractional and decimal edge lengths and use scale factors to determine the volume of similar solids. Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. Given the equations of lines (at least one of which is horizontal or vertical) that bound a triangular, rectangular, or trapezoidal region, calculate the surface area and/or volume of the solid formed by revolving the region about the line that is horizontal or vertical. Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Pre-Calculus Skills/ Solve literal equations (perimeter, area, and volume). Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. AP Calculus Skills/ Solve and use literal equations in real life and mathematical applications. Calculate the area between curves. Calculate the volume of a solid formed by revolving the region between two curves about a horizontal or vertical line. 3

14 Area Concept Development Chart Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Show three different methods to calculate the area of the figure using only two rectangles. The area of the outer square is 36 square units. A rectangle whose area is 10 square units and whose width is 2 units is removed from the bottom corner of the square. Fill in the missing lengths in the diagram provided. s w x Show three different methods to calculate the area of the figure using only two rectangles. What is the area of a triangle with vertices (1, 2) (7, 2) and (3, 6)? What is the area of a triangle that is formed by connecting the points and? s l Algebra 1 Geometry Algebra 2 Pre-Calculus AP-Calculus A triangle is formed by the intersection of the lines and. What is the area of the triangle? A triangle with an area of 4 square units is formed by the intersection of three lines. One of the bases is formed by the y-axis whose equation is, and one of the sides is formed by the line. An obtuse triangle with an area of 12 square units is formed by the intersection of three lines. The equations of two of the lines are and. Determine the area of a triangle bounded by the lines, x = c, and where,, and. (2013 AB Calculus Q06a; noncalculator) Let and. Let R be the region bounded by the graphs of f and g, as shown in the figure. Which of the following could not be an equation of a line containing the third side of the triangle? Which of the following could be an equation of a line containing a third side that forms an obtuse triangle? A. y = 2 B. y = 5 2 x + 4 C. x + y = 2 D. y = 1 2 x 4 Find the area of R. E. y = 3 2 x 4 4

15 Volume Concept Development Chart Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 A box is made by stacking two rectangular prisms as shown. Box A has in., in., and in. Box B is placed on top of Box A and must be at least 1 inch from any edge of the top of Box A. The volume of Box B is 1/5 the volume of Box A. A rectangular region is formed by the line segments connecting the points (0, 0), (5, 0), (5, 3), and (0, 3) and forms the base of a rectangular prism whose height is 5 1 units. 2 A right triangular region is formed by the line segments connecting the points (0, 0), (3, 0), and (0, 4). Revolving the region about the x-axis creates a cone. List all possible whole number dimensions,, that meet these criteria. What is the volume of a similar rectangular prism whose edges are multiplied by a scale factor of 2? What is the ratio of the volume of this similar prism to the original prism? What is the volume of the cone? Algebra 1 Geometry Algebra 2 Pre-Calculus AP-Calculus A rectangular region is bounded by the lines,, y = 0, and. When the rectangular region is revolved about the x-axis, a cylinder is created. A rectangular region is bounded by the lines,,, and. A cylinder with volume cubic units is created by revolving a rectangle about the y-axis. The rectangle is formed by the intersection of four lines. The equations of three of the lines are,, and. The volume of a cone that has been created by revolving a triangle about the line is cubic units. The equations of two of the lines are x = 2 and y = 2. (2013 AB Calculus Q06b; noncalculator) Let and. Let R be the region bounded by the graphs of f and g, as shown in the figure. What is the volume of the cylinder? What is the volume of the solid formed when the rectangle is revolved about the x-axis? Which of the following could be an equation of a line containing the third side of the triangle? Which of the following could be an equation of a line containing the fourth side? I. y = 3 8 II. y = 8 3 III. y = 3 8 (x + 2)+1 (x + 2)+ 6 (x + 2) 5 Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line. 5

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17 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Surface Area and Volume LEVEL Grade 6 or Grade 7 in a unit on surface area and volume MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. ABOUT THIS LESSON In this lesson, students use linking cubes to build cubes with a variety of side lengths and to develop a concrete understanding of how changes in dimensions affect the surface area and volume of three-dimensional figures. Students apply this conceptual understanding as they answer a variety of application questions at the end of the activity. OBJECTIVES Students will determine the surface area and volume of cubes. develop the formulas for surface area and volume of cubes based on patterns in a table. write expressions using exponents. describe the effect on surface area and volume when the dimensions of a cube are changed proportionally. T E A C H E R P A G E S G N P A V P Physical V Verbal A Analytical N Numerical G Graphical 7

18 Mathematics Surface Area and Volume T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. Targeted Standards 7.G.6: Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. See table and questions RP.3: Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. See questions 1-15 Reinforced/Applied Standards 6.EE.6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. See last row of table and questions 1d, 2d, 3c, 4d, 5c COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.5: Use appropriate tools strategically. Students investigate using linking cubes and a table. MP.8: Look for and express regularity in repeated reasoning. Students connect numbers and symbols with a concrete model and generalize patterns relating the length of the edge to the surface area and volume and then extend their understanding to the effects of scale factor changes on length, surface area, and volume. 8

19 Mathematics Surface Area and Volume FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Calculate surface area and volume of cubes Write and simplify ratios ASSESSMENTS The following types of formative assessments are embedded in this lesson: Students summarize a process or procedure. Students engage in independent practice. Students apply knowledge to a new situation. The following additional assessments are located on our website: Areas and Volumes 6th Grade Free Response Questions Areas and Volumes 6th Grade Multiple Choice Questions Areas and Volumes 7th Grade Free Response Questions Areas and Volumes 7th Grade Multiple Choice Questions MATERIALS AND RESOURCES Student Activity pages Scientific or graphing calculators Linking cubes Interactive applet that fills a rectangular prism with cubes and shows the volume: illuminations.nctm.org/activitydetail.aspx?id=6 T E A C H E R P A G E S 9

20 Mathematics Surface Area and Volume T E A C H E R P A G E S TEACHING SUGGESTIONS This lesson is most appropriately done in a group setting where students explore and discuss multiple approaches to filling in the table and answering the questions. As groups work through this lesson, the teacher should monitor to be sure that all levels of students understand the process columns, that students discover the formulas for surface area and volume, and that all are included in the small group discussions. The teacher should ask leading questions about the table as appropriate to point students toward the essential conclusions. All students should have a 1 unit cube, should build a cube with dimensions of 2 units, and should participate in a group discussion of the surface area and volume of these two cubes, with the goal that each group member can use the cube to explain how to determine surface area and how to determine volume. Have the groups continue to build the cubes in order to complete the table. Discuss as a group the patterns in the table to calculate the surface area and volume for 10 units, 12 units, and n units. Discuss question 5 with the entire group. Students should come to the understanding that if the ratio of the lengths of any two corresponding sides of two similar solids is x, then the ratio of their surface y 3 x areas is x2, and the ratio of their volumes is 2 y y. 3 Questions 6 and 7 can be completed without using the concepts presented in the lesson; however, reinforce the idea of using the relationship between scale factors and volumes by asking students how this question connects to question 4d. It is very important that students realize that question 7 is asking for the effect of the changes rather than the actual volume and surface area. Students tend to think that the answers for volume and surface area in this question are the same. Take this opportunity to be sure students understand how they are the different, that one is measured in square units and the other in cubic units. Be sure that students have a concrete understanding of the difference between these two units of measure. You may wish to support this activity with TI- Nspire technology. See Working with Fractions and Decimals and Calculations Using Special Keys in the NMSI TI-Nspire Skill Builders. Suggested modifications for additional scaffolding include the following: 3, 5 Provide more numerical examples, possibly in a chart, prior to requiring the answer in terms of k Highlight or notate whether the scale factor will affect the area or the volume Eliminate two of the answer choices. 10

21 Mathematics Surface Area and Volume NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. 4th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) 5th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and non-overlapping right rectangular prisms with whole number edge lengths. 6th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and nonoverlapping right rectangular prisms with fractional and decimal edge lengths and use scale factors to determine the volume of similar solids. 7th Grade Skills/ Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. Algebra 1 Skills/ Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Given the equations of lines (at least one of which is horizontal or vertical) that bound a triangular, rectangular, or trapezoidal region, calculate the surface area and/or volume of the solid formed by revolving the region about the line that is horizontal or vertical. Geometry Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Algebra 2 Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Pre-Calculus Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. AP Calculus Skills/ Calculate the area between curves. Calculate the volume of a solid formed by revolving the region between two curves about a horizontal or vertical line. T E A C H E R P A G E S 11

22 Mathematics Surface Area and Volume T E A C H E R P A G E S 12

23 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Surface Area and Volume Use linking cubes to construct five larger cubes that have edges of 2, 3, 4, 5, and 6 units, respectively. Use these cubes to complete the table. Length of Edge Process Column to determine Surface Area Surface Area Process Column to determine Volume Volume 1 unit 6 1unit 1unit 6square units 1unit 1unit 1unit 1cubic unit 2 units 3 units 4 units 5 units 6 units 10 units 12 units n 1. There are several comparisons and relationships that one can make between the lengths of the edges of similar cubes and their surface areas. Use the table to examine the surface area of a cube with a length of 6 units and the surface area of a cube with a length of 3 units. a. What is the ratio of the length of a 6-unit cube to the length of a 3-unit cube? b. What is the ratio of the surface area of a 6-unit cube to the surface area of a 3-unit cube? c. From the numerical list in the table, list three other pairs of cubes for which the length of one cube is twice the length of the other cube. For the three pairs, give the ratio of the surface area of the larger cube to the surface area of the smaller cube. 13

24 Mathematics Surface Area and Volume d. The length of the cube is doubled. Determine the area of one face of each cube. { n 2n n n n { 2n A = A = Determine the total surface area of each cube. SA = SA = When the length of the edge of the cube is doubled, the surface area of the new cube is times the surface area of the original cube. 14

25 Mathematics Surface Area and Volume 2. Use the table to compare the surface area of a cube with a length of 6 units to the surface area of a cube with a length of 2 units. a. What is the ratio of the lengths of the edges of a 6-unit cube to a 2-unit cube? b. What is the ratio of the surface area of a 6-unit cube to the surface area of a 2-unit cube? c. List two other pairs of cubes for which the length of one cube is three times the length of the other cube. For the two other pairs, give the ratio of the surface area of the larger cube to the surface area of the smaller cube. 15

26 Mathematics Surface Area and Volume d. The length of the cube is tripled. 3n Determine the area of one face of each cube. 3n { 3n n n n A = n n n { 3n A = Determine the total surface area of each cube. 3n SA = 3n SA = When the length of the edge of the cube is tripled, the surface area of the new cube is times the surface area of the original cube. 16

27 Mathematics Surface Area and Volume 3. Fill in the blanks. a. When the length of a cube is increased by a factor of 2, the surface area increases by a factor of = ( ) 2. b. When the length of a cube is increased by a factor of 3, the surface area increases by a factor of = ( ) 2. c. When the length of a cube is increased by a factor of k, the surface area increases by a factor of. 4. Use the table to examine the volume of a cube with a length of 3 units and the volume of a cube with a length of 6 units. a. What is the ratio of the length of a 6-unit cube to the length of a 3-unit cube? b. What is the ratio of the volume of a 6-unit cube to the volume of a 3-unit cube? c. List three other pairs of cubes for which the length of one cube is twice the length of the other cube. For the three pairs, list the ratio of the volume of the larger cube to the volume of the smaller cube. d. When the length of the cube is doubled, the volume of the new cube is times the volume of the original cube. 17

28 Mathematics Surface Area and Volume 5. Fill in the blanks to summarize what you have learned. a. When the length of a cube is increased by a factor of 2, the volume increases by a factor of. b. When the length of a cube is increased by a factor of 3, the volume increases by a factor of. c. When the length of a cube is increased by a factor of k, the volume increases by a factor of. 6. Krista loves growing flowers. Last spring she planted begonias in a planter that measured 7 inches by 7 inches by 7 inches. She had such great luck with those plants that she decided to buy a larger planter this year. The new planter measures 14 inches by 14 inches by 14 inches. Compare the amount of potting soil needed to fill those two planters. 7. The city recreational department is planning to construct a new pool in the city park. In the original plans, the pool was to be rectangular in shape, 75 feet wide, 150 feet long, and 6 feet deep. Due to budget cuts, the plans now call for the pool to be 1 3 as long and 1 3 as wide as in the original plans. The depth will remain the same. Describe the effect that these changes will have on the pool cover and on the amount of water that will be needed to fill the pool. 18

29 Mathematics Surface Area and Volume 8. These flower pots are similar. If the smaller pot has a volume of 100 cubic inches, what is the volume of the larger pot? 9. Myrna wants to construct a box that will hold 64 times as much as a similar box. By what factor should she multiply the dimensions of the smaller box in order to determine the dimensions of her new, larger box? 10. Quincy works in the customer service department of a store and is responsible for ordering supplies for the gift wrap department. The smallest box requires 88 square inches of wrapping paper. How much wrapping paper would be needed for a box having dimensions that are three times those of the smallest box? 19

30 Mathematics Surface Area and Volume 11. A delivery service charges a fee based upon the volume of the box to be delivered. If they charge $3 for a 6 inch by 6 inch by 6 inch box, then how much would you expect them to charge for a box that is 1 foot by 1 foot by 1 foot? 12. An architect is working on a scale model home for a client. The linear dimensions of the scaled model will be 1 the size of the linear dimensions of the actual house. If he uses 2 square feet of wallpaper for 20 the kitchen in the scale model, how much would he need for the kitchen in the actual house? 13. A shipping company sells two types of cartons that are shaped like rectangular prisms. The larger carton has a volume of 720 cubic inches. The smaller carton has dimensions that are half the size of the larger carton. What is the volume, in cubic inches, of the smaller carton? A. 90 in. 3 B. 120 in. 3 C. 240 in. 3 D. 360 in. 3 20

31 Mathematics Surface Area and Volume 14. An ice-cream carton has a volume of 64 fluid ounces. A second ice-cream carton has dimensions that are ¾ the size of the larger carton. Which is closest to the volume of the smaller carton? A. 20 fl oz B. 27 fl oz C. 36 fl oz D. 48 fl oz 15. The radius of the larger sphere was multiplied by a factor of 1 2 to produce the smaller sphere. Radius = r Radius = 1 2 r How does the surface area of the smaller sphere compare to the surface area of the larger sphere? A. The surface area of the smaller sphere is 1 2 as large. B. The surface area of the smaller sphere is 1 π as large. C. The surface area of the smaller sphere is 1 4 as large. D. The surface area of the smaller sphere is 1 8 as large. 21

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33 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Unit Dog LEVEL Grade 6 or Grade 7 in a unit on surface area and volume MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. G N P A V P Physical V Verbal A Analytical N Numerical G Graphical ABOUT THIS LESSON In this lesson, students use linking cubes and nets to explore how the surface area and volume of an irregularly shaped object, unit dog, change as the dimensions are scaled. The linking cubes are used to construct the dog and determine the surface area and volume by counting. Students create nets for the legs, head, and torso of the dog. Scale factors are applied to the unit dog and students construct the larger dogs using nets and cardstock models. Students use ratios to determine how the surface area and volume are affected by the scale factor and create an expression that can be applied to any scale factor to calculate the surface area and volume for this irregular shape. This activity provides an engaging setting for students to practice and apply their skills with nets, surface area, volume, and scale factors. This lesson can be used as a follow up activity to Surface Area and Volume where students explore the effects of scale factors on the surface area and volume of a cube using ratios. OBJECTIVES Students will determine the surface area and volume of an irregularly shaped object. draw front, side, and top views of the object. create nets for various parts of the object. apply scale factors to the nets. build models with unit cubes and with cardstock nets. investigate the resulting effects on surface area and volume when dimensions of a shape change proportionally. T E A C H E R P A G E S 23

34 Mathematics Unit Dog T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. 7.G.6: 6.G.4: Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. See questions 2-3, 8-9 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. See questions 5a-b, 6-7 Reinforced/Applied Standards 6.RP.3d: Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. See questions 7-9 COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.3: Construct viable arguments and critique the reasoning of others. Students compare a variety of student generated nets and then discuss how different nets can create the same 3-dimensional figure. MP.5: Use appropriate tools strategically. Students investigate using linking cubes, nets, and a table. MP.8: Look for and express regularity in repeated reasoning. Students use repeated calculations to develop general rules for determining surface area and volume using scale factors. 6.EE.6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. See question 8 s last row 24

35 Mathematics Unit Dog FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Calculate surface area and volume of cubes and rectangular prisms Sketch nets of cubes and rectangular prisms ASSESSMENTS The following types of formative assessments are embedded in this lesson: Student cooperative learning groups create appropriately scaled models. Students engage in independent practice. Students summarize a process or procedure. The following additional assessments are located on our website: Areas and Volumes 6th Grade Free Response Questions Areas and Volumes 6th Grade Multiple Choice Questions Areas and Volumes 7th Grade Free Response Questions Areas and Volumes 7th Grade Multiple Choice Questions The following additional assessments are located on our website. Assessments by Course 6th Grade Assessments by Course 7th Grade MATERIALS AND RESOURCES Student Activity pages 13 linking cubes for each student Copies of 1 inch square grid paper printed on cardstock Tape Scissors Sample unit dog made from linking cubes Sample unit dog made from grid paper Large copy of the chart provided in question 10 Interactive applet that shows how the surface area and volume of a cube change as the lengths of the sides change: mathopenref.com/cubearea.html and mathopenref.com/cubevolume.html Interactive applet that shows front, back, side, and three-dimensional views of a rectangular prims and shows how the surface area and volume of a rectangular prism change as the lengths of the sides change: org/interactivate/activities/surfaceareaandvolume/ Interactive applet that shows nets for a cube: and Interactive applet that allows the virtual construction of solids with unit cubes that can be rotated to examine front, back, side, and three-dimensional views: nav/frames_asid_195_g_1_t_3.html?open=activities&f rom=topic_t_3.html T E A C H E R P A G E S 25

36 Mathematics Unit Dog T E A C H E R P A G E S TEACHING SUGGESTIONS The Unit Dog lesson should be teacher-led. Organize the students into small groups and lead the students through the following steps. Question 1: Demonstrate how to make a unit dog from linking cubes and then have each student build their own dog, using their 13 linking cubes. Some students may require a step-by-step process along with the demonstration to successfully build the dog. This part of the lesson provides an opportunity to review the vocabulary associated with a cube such as edge and face along with a review of how to calculate surface area and volume. Note: Larger dogs should not be built with linking cubes. They are heavy and very unstable. Question 2: Have students determine the volume of their unit dogs. Question 3: Generally students do not correctly determine surface area if they try to count. Give them time to think about a good procedure. Once students begin to have the correct answer, have them share their process with the rest of the class. Question 4: Have each student draw front, side, and top views of the unit dog. Question 5: Show the students how to draw the nets for the leg/head and torso. Use the linking cubes as a model to help students make the transition from a 3-dimensional object to a 2-dimensional net. Students may need to draw a net for a cube before creating the more complex nets for the dog. Discuss the difference between the unit cube and the leg/head and torso to help them see a pattern and make the connection between the two shapes. Note: The lesson Nets for Cubes on the NSMI website provides pictures of 11 different ways to construct a net for a cube. Question 6: Students create different nets and compare these to ones drawn by classmates. Some students may need to cut out and fold the net to confirm whether or not the net is correct. Question 7: Have students sketch the nets using a scale factor of two. Use the linking cubes to help students visualize how the scale factor affects the leg/head and torso. To help students make this connection, explain that each square of the net in question 5 is 2 by 2 instead of 1 by 1. Working with the individual squares of the net help students understand how the net changes based on the scale factor. Question 8 9: To ensure the scaled nets will fit on the grid provide, display a net for all students to use. The nets provided in the answers for questions 5a and 5b will fit on the provide grids. Question 10: Before class begins, construct a scaled unit dog to use as a model. In order to create a variety of different-sized dogs, assign groups to build particular dogs from grid paper nets. Scaled net drawings from questions 7 9 should be used to construct the paper scaled nets. Help the students realize they are to construct one net for the dog s torso and that the other 5 nets (head and legs) are the same net. Each of the different body parts can be assigned to different group members. Note: Some students might construct their dogs but cutting separate pieces for each part of the body instead of building a net. Providing a sample of a net of the leg/head for a larger dog will help students from making this mistake. Each group should calculate surface area and volume for their particular dog. If there are two groups for each size dog, then the groups can verify the answers of the other group. Once all of the dogs are built, have students post their measurements on the large 26

37 Mathematics Unit Dog copy of the chart from question 10. As the students fill in the factors by which surface area and volume have increased, ask them how they determined the factors and how they used the factors in the calculations. Students can use ratios to calculate these factors. As a check for understanding, have the students predict the surface areas and volumes for dogs with dimensions six, seven, and ten times those of the unit dog before moving to the last row of the table. Showing the process in the third and last column of the table will help students see the patterns. Display all of the dogs. Question 11: Students use the expressions from the last row of the table to calculate the surface area and volume with very large scale factors. You may wish to support this activity with TI- Nspire technology. See Working with Fractions and Decimals and Calculations Using Special Keys in the NMSI TI-Nspire Skill Builders. Suggested modifications for additional scaffolding include the following: 4 Provide a sketch of one of the requested views. 5 Provide a sketch of the net in part (a). 7 Provide a sketch of one of the required nets. T E A C H E R P A G E S 27

38 Mathematics Unit Dog T E A C H E R P A G E S NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. 4th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) 5th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and non-overlapping right rectangular prisms with whole number edge lengths. 6th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and nonoverlapping right rectangular prisms with fractional and decimal edge lengths and use scale factors to determine the volume of similar solids. 7th Grade Skills/ Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. Algebra 1 Skills/ Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Geometry Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Algebra 2 Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Pre-Calculus Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. AP Calculus Skills/ Calculate the area between curves. Given the Given the Given the Given the Calculate the equations of equations of lines equations of lines equations of lines volume of a lines (at least or circles that or circles or a set or circles or a set solid formed by one of which bound a triangular, of inequalities that of inequalities that revolving the is horizontal or rectangular, bound a triangular, bound a triangular, region between vertical) that trapezoidal, or rectangular, rectangular, two curves about bound a triangular, circular region, trapezoidal, or trapezoidal, or a horizontal or rectangular, calculate the circular region, circular region, vertical line. or trapezoidal volume and/or calculate the calculate the region, calculate surface area of volume and/or volume and/or the surface area the solid formed surface area of surface area of and/or volume of by revolving the solid formed the solid formed the solid formed the region about by revolving by revolving by revolving a horizontal or the region about the region about the region about vertical line. a horizontal or a horizontal or the line that is vertical line. vertical line. horizontal or vertical. 28

39 T E A C H E R P A G E S Mathematics Unit Dog 29

40 Mathematics Unit Dog T E A C H E R P A G E S 30

41 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Unit Dog 1. As you built the original unit dog, how many linking cubes did it take to construct the entire dog? 2. What is the volume of the original unit dog? Explain how to calculate the volume. 3. What is the surface area of the original unit dog? Explain how to calculate the surface area. 4. Draw the front, top, and side views of the original unit dog. Label each appropriately. 5. a. The head and legs of the original unit dog can be constructed from the same net. Sketch a net of the original unit dog s head or leg on the grid. b. The original unit dog s torso can be constructed from a different net. Sketch a net of the original unit dog s torso on the grid. 31

42 Mathematics Unit Dog 6. Compare your net of the original unit dog s head/leg to the nets of other class members. Discuss why the nets create the same 3-dimensional figure. Sketch at least two different correct nets on the grid. 7. Using a scale factor of two, sketch a net of the scaled dog s head/leg and torso on the grid. 8. Using a scale factor of three, sketch a net of the scaled dog s head/leg and torso on the grid. 32

43 Mathematics Unit Dog 9. Using a scale factor of four, sketch a net of the scaled dog s head/leg and torso on the grid. 33

44 Mathematics Unit Dog 10. Complete the row for your assigned unit dog. Add the data from the other unit dogs created by your class. Dimensions scaled by a factor of... Surface area of the dog Area scaled by a factor of... Volume of dog Volume scaled by a factor of... 1 (the unit dog) n 11. Use the information from the table to determine the surface area and volume for a dog when the dimensions are scaled with a factor of 50. What is the surface area and volume for a dog with a scale factor of 100? 34

45 Cardstock Copy : Unit Dog 1-inch Grid (7 x 10) 35

46 36

47 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solids of Revolution LEVEL Grade 8 as part of a unit on volumes of cylinders and cones MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. ABOUT THIS LESSON In this lesson, students plot ordered pairs in the first quadrant to create and determine the areas of two-dimensional figures. The planar regions are revolved about a horizontal or vertical line to create three-dimensional figures, and students calculate the volume for these solids. Students transfer the dimensions of the planar region to the solid as they identify the dimensions needed to determine the volume. As students compare the volumes when the same planar region is revolved about different axes, they develop a deeper conceptual understanding of how the change in the radius and height impacts the overall volume of a solid. The activity provides an engaging setting for students to practice and apply their skills with plotting points on a coordinate plane and calculating areas and volumes. OBJECTIVES Students will plot the points for a plane figure on a coordinate plane. determine the area of that plane figure. draw and describe the solid formed by revolving the plane figure about a vertical or horizontal line. calculate the volume of the solid. compare the volumes determined when revolving about different axes. T E A C H E R P A G E S G N P A V P Physical V Verbal A Analytical N Numerical G Graphical 37

48 Mathematics Solids of Revolution T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. Targeted Standards 8.G.9: Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. See questions 1d, 1f, 2d, 2f, 3d, 3f, 4d, 4f, 5d, 6d, 6f, 6h Reinforced/Applied Standards 6.G.3: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. See questions G.1: Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems. See questions 1b, 2b, 3b, 4b, 5b, 6b COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.1: Make sense of problems and persevere in solving them. Students plot and connect points to create 2-dimensional figures on a coordinate plane, determine the area of the planar region, revolve the figure to form a 3-dimensional solid, identify the solid, determine its dimensions, and calculate the volume. MP.3: Construct viable arguments and critique the reasoning of others. Students explain why the solids formed from the same planar figure may have different volumes. MP.5: Use appropriate tools strategically. Students create 3-dimensional models using paper and pencil. Other visualization tools can include physical or computer-generated models. MP.7: Look for and make use of structure. Students associate the dimensions of the planar figure to the radius and height of the solid of revolution. 38

49 Mathematics Solids of Revolution FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Plot ordered pairs in the coordinate plane Calculate areas of rectangles and triangles Calculate volumes of cylinders and cones ASSESSMENTS The following type of formative assessment is embedded in this lesson: Students engage in independent practice. The following additional assessments are located on our website: Areas and Volumes 7th Grade Free Response Questions Areas and Volumes 7th Grade Multiple Choice Questions MATERIALS AND RESOURCES Student Activity pages Cut-out of planar region Pencil, straw, or skewer Tape Scientific or graphing calculators Graph paper Mathematica demonstration simulating a planar region revolved around either the x or y axis to generate a 3-dimensional figure located on the NMSI website with resources for this lesson (a free download of Mathematica Player is required). T E A C H E R P A G E S 39

50 Mathematics Solids of Revolution T E A C H E R P A G E S TEACHING SUGGESTIONS Almost all students have difficulty in visualizing 3-dimensional solids. Before beginning the lesson, model the revolutions around different axes by taping a cut-out of one of the planar regions provided in the activity to a pencil, straw, or skewer, and then rolling the pencil between your hands to simulate the sweeping out of the solid. Provide students with different shapes and have them remove various tabs so that the solids produced will have different radii and heights. Ask students to describe the 3-dimensional shape that is produced by revolving the figure along with the radius and height of the solid. If students are having difficulty identifying the solid when revolving the shape, turn the manipulative upside down on the table and twirl. An applet is listed in the resources which demonstrates the solid formed when a planar shape is revolved about either a horizontal or vertical line. The questions in this lesson build from cylinders and cones to solids that are a combination of these two solids. Carefully lead students through the steps for drawing a solid of revolution. On the first grid, plot the ordered pairs. Draw the boundary lines by joining the points and determine the area. On the second grid, redraw and shade the planar region. Reflect the shape about the indicated axis. Connect the vertices and any other significant points and their reflections with ellipses. Refer students back to either their manipulative or applet if they are having trouble identifying the solid. Questions 1 and 2 can be used as classroom examples and require teacher guidance. Question 3 can be used as a formative assessment. These three questions ask students to compare the volume of the solids that have been revolved about both axes. Even though the planar regions are the same, the solids have different radii and different heights. Students are asked to explain why these volumes are different. Using the formulas for the two solids side-by-side may help them deepen their understanding. Question 4 asks students to revolve the planar region about a line instead of one of the axes. Both questions 5 and 6 increase the level of rigor by revolving two geometric figures about either an axis or a line. The instructions for calculating the volume of the solids ask students to leave their answers in terms of and also as a decimal answer rounded to three decimal places. If a decimal approximation is used, students should use the on their calculator instead of using 3.14 for. You may wish to support this activity with TI- Nspire technology. See Calculations Using Special Keys in the NMSI TI-Nspire Skill Builders. Suggested modifications for additional scaffolding include the following: 1 Provide the sketches of the regions and solids for parts (a), (c), and (e). 40

51 Mathematics Solids of Revolution NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. 4th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) 5th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and non-overlapping right rectangular prisms with whole number edge lengths. 6th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and nonoverlapping right rectangular prisms with fractional and decimal edge lengths and use scale factors to determine the volume of similar solids. 7th Grade Skills/ Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. Algebra 1 Skills/ Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Given the equations of lines (at least one of which is horizontal or vertical) that bound a triangular, rectangular, or trapezoidal region, calculate the surface area and/or volume of the solid formed by revolving the region about the line that is horizontal or vertical. Geometry Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Algebra 2 Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Pre-Calculus Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. AP Calculus Skills/ Calculate the area between curves. Calculate the volume of a solid formed by revolving the region between two curves about a horizontal or vertical line. T E A C H E R P A G E S 41

52 Mathematics Solids of Revolution T E A C H E R P A G E S 42

53 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solids of Revolution General instructions: When calculating volumes of cylinders and cones, give your answer both in terms of and also as a decimal accurate to three decimal places. Use the key on your calculator and then round your answer as the last step. 1. a. Draw line segments joining the points (0, 0), (0, 2), (3, 2), and (3, 0). y a) b. Calculate the area of the region formed. x c. Draw and describe the solid formed by revolving the region about the x-axis. c) y d. Calculate the volume of the solid formed. x e. Draw and describe the solid formed by revolving the region about the y-axis. e) y x f. Calculate the volume of the resulting solid. g. Compare the volumes in parts (d) and (f). Explain why these volumes are different. 43

54 Mathematics Solids of Revolution 2. a. Draw line segments joining the points (0, 0), (0, 3), and (2, 0). b. Calculate the area of the region formed. a) y x c. Draw and describe the solid formed by revolving the region about the x-axis. c) y d. Calculate the volume of the solid formed. x e. Draw and describe the solid formed by revolving the region about the y-axis. e) y f. Calculate the volume of the resulting solid. x g. Compare the volumes in parts (d) and (f). Explain why these volumes are different. 44

55 Mathematics Solids of Revolution 3. a. Draw line segments joining the points (0, 0), (0, 1), and (5, 0). b. Calculate the area of the region formed. a) y x c. Draw and describe the solid formed by revolving the region about the x-axis. c) y d. Calculate the volume of the solid formed. x e. Draw and describe the solid formed by revolving the region about the y-axis. e) y f. Calculate the volume of the resulting solid. x g. Compare the volumes in parts (d) and (f). Explain why these volumes are different. 45

56 Mathematics Solids of Revolution 4. a. Draw line segments joining the points (0, 0), (0, 3), (5, 3), and (5, 0). b. Calculate the area of the region formed. a) y x c. Draw and describe the solid formed by revolving the region about the x-axis. c) y d. Calculate the volume of the solid formed. x e. Draw and describe the solid formed by revolving the region about the vertical line x = 5. e) y f. Calculate the volume of the resulting solid. x 46

57 Mathematics Solids of Revolution 5. a. Draw line segments joining the points (0, 0), (0, 5), (2, 3), and (2, 0). b. Calculate the area of the region formed. a) y x c. Draw and describe the solid formed by revolving the region about the y-axis. c) y d. Calculate the volume of the solid formed. x 47

58 Mathematics Solids of Revolution 6. a. Draw line segments joining the points (0, 0), (2, 4), and (2, 0). b. Calculate the area of the region formed. a) y x c. Draw and describe the solid formed by revolving the region about the x-axis. c) y d. Calculate the volume of the solid formed. x e. Draw and describe the solid formed by revolving the region about the vertical line x = 2. e) y f. Calculate the volume of the resulting solid. x g. Draw and describe the solid formed by revolving the region about the y-axis. g) y h. Calculate the volume of the solid formed. x 48

59 Mathematics Solids of Revolution 7. a. List three ordered pairs to create a triangular region that will produce the same volume when revolved about the x- and y-axes. y x b. List three ordered pairs to create a triangular region that will produce different volumes when revolved about the x- and y-axes. y x c. Compare these figures with a classmate. Write a general statement about the characteristics of the triangles that produce the same volume when revolved about the x-and y-axes and the characteristics of the triangles that produce different volumes when revolved about x- and y-axes. 49

60 Mathematics Solids of Revolution 50

61 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solving Systems of Linear Equations LEVEL Grade 8, Algebra 1, or Math 1 in a unit on solving systems of equations MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. ABOUT THIS LESSON This lesson gives students practice in graphing linear equations; however, the primary purpose is to define regions in a coordinate plane formed by the intersection of linear equations and to determine the area of a bounded region. Students will build on the work they did in middle grades plotting points to determine bounded regions as they graph linear equations on restricted domains and identify the intersections of systems of linear equations to determine areas. This lesson focuses on solving systems of linear equations. At the same time the lesson reinforces the use of area formulas and enhances student understanding of this standard by developing coherence and connections among a variety of mathematical concepts, skills, and practices. OBJECTIVES Students will solve systems of linear equations graphically. determine the area of a polygon formed by the intersection of linear equations. T E A C H E R P A G E S G N P A V P Physical V Verbal A Analytical N Numerical G Graphical 51

62 Mathematics Solving Systems of Linear Equations T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Targeted Standards (if used in Grade 8) 8.EE.8a-b: Analyze and solve pairs of simultaneous linear equations. (a) Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. (b) Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. See questions 1-9 Reinforced/Applied Standards (if used in Grade 8) 6.G.1: Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. See questions G.3: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems scales. See question 9 Targeted Standards (if used in Algebra 1) A-REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. See questions 1-9 Reinforced/Applied Standards (if used in Algebra 1) G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. See questions 1-8 A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. See question 9 52

63 Mathematics Solving Systems of Linear Equations COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.1: Make sense of problems and persevere in solving them. Students graph equations, identify the planar region formed, use a variety of algebraic techniques to determine the dimensions necessary to calculate the area of the region, and determine the area. MP.7: Look for and make use of structure. In question 5, students draw an auxiliary line to determine the height of the triangle. FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Graph linear equations Calculate areas of rectangles, triangles, and trapezoids ASSESSMENTS The following formative assessment is embedded in this lesson: Students engage in independent practice. The following additional assessments are located on our website: Areas and Volumes Algebra 1 Free Response Questions Areas and Volumes Algebra 1 Multiple Choice Questions MATERIALS AND RESOURCES Student Activity pages Graph paper Straight edges Applet to draw regions and calculate area between curves: T E A C H E R P A G E S 53

64 Mathematics Solving Systems of Linear Equations T E A C H E R P A G E S TEACHING SUGGESTIONS This lesson is richer if class time is allowed for a discussion of the various techniques that students use to determine the areas. Emphasize to students that they cannot just guess the intercepts or intersection points. Questions 1, 3, and 7 require students to calculate the intercepts to obtain the dimensions of the triangles. For questions 2, 4, and 8, students may calculate the area of the rectangle in which each trapezoid is enclosed and then subtract the non-shaded triangle. Question 6 requires determining the intersection of two oblique lines. Question 5 has a variety of solutions. This is an excellent opportunity to let students investigate mathematics by asking them to find as many different methods to determine the area as possible. Some students see a large triangle and subtract out a trapezoid and a right triangle. Others subtract each of three non-shaded triangles from the 6 by 9 rectangle which encloses the shaded triangle, that is, 54 ( ). Consider challenging students to calculate the shaded area using the smallest number of geometric figures. A solution using only two figures involves subtracting the area of the obtuse triangle with vertices (0,0),(0,6),and(2,7) from the obtuse triangle with vertices (0,0),(0,6),and (6, 9). Question 9 can be solved algebraically by writing the equations of the lines and finding their point of intersection or by using similar triangles to set up a proportion to solve for the height of the triangular region. You may wish to support this activity with TI- Nspire technology. See Graphing a Function and Displaying a Table, Graphing Piecewise Functions, and Adjusting the Window in the NMSI TI-Nspire Skill Builders. Suggested modifications for additional scaffolding include the following: 2, 4, 8 Turn the paper a quarter turn to the right so that the trapezoid is sitting on one of its bases. 6 Turn the paper a quarter turn to the left in order consider the base of the triangle to be on the y-axis. 54

65 Mathematics Solving Systems of Linear Equations NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. 4th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) 5th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) 6th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) 7th Grade Skills/ Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Algebra 1 Skills/ Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Geometry Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Algebra 2 Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Pre-Calculus Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. AP Calculus Skills/ Calculate the area between curves. T E A C H E R P A G E S 55

66 Mathematics Solving Systems of Linear Equations T E A C H E R P A G E S 56

67 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solving Systems of Linear Equations For questions 1 8, graph and shade the region then calculate the area of the region. Show the work that leads to your answers in questions What is the area of the region in the first quadrant that is below the graph of f( x) = 2( x 3)? 2. What is the area of the region enclosed by the graphs of y = 0, x = 0, x = 6, and 1 y = ( x 3) + 4? 3 3. What is the area of the region in the first quadrant that is below the graph of y = 5( x 1) + 4? 57

68 Mathematics Solving Systems of Linear Equations 4. Let R be the region in the first quadrant under the graph of y = 2x for 4 x 9. What is the area of R? 5. What is the area of the region bounded by the graphs of y = x+ 6, y = x, and y = x? What is the area of the region enclosed by the graphs 1 2 of x = 0, y = x+ 1, and y = ( x 3) + 6? What is the area of the region in the first quadrant under the graph of 2x + 4y = 25? 58

69 Mathematics Solving Systems of Linear Equations 8. What is the area of the region enclosed by the graphs 2 of y = 0, y = x+ 9 for 0 x 5? 3 9. What is the area of the region R bounded by line m, line p, and the x-axis as shown in the graph? 59

70 Mathematics Solving Systems of Linear Equations 60

71 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Working With Formulas and Function Notation LEVEL Geometry or Math 2 in a unit on areas and volumes MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. ABOUT THIS LESSON In this lesson, students review geometric formulas and reinforce algebraic skills as they manipulate familiar formulas and/or given information to create specific new functions, such as the area of a circle in terms of its circumference or the volume of a cone in terms of only its radius. Students isolate a variable in a familiar equation and then use that variable to substitute into a different known formula. This lesson focuses on building familiarity with geometric formulas for perimeter, circumference, area and volume. At the same time, the lesson reinforces the skill of solving literal equations and enhances student understanding of these standards by developing coherence and connections among a variety of mathematical concepts, skills, and practices. OBJECTIVES Students will recall and apply geometric formulas for perimeter, circumference, area, and volume. solve literal equations by isolating a specific variable. T E A C H E R P A G E S G N P A V P Physical V Verbal A Analytical N Numerical G Graphical 61

72 Mathematics Working With Formulas and Function Notation T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Targeted Standards A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. See questions 1-10 Reinforced/Applied Standards G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. See questions 3-5, 8, 10 COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.1: Make sense of problems and persevere in solving them. Students identify primary and secondary equations, isolate the variable for substitution, manipulate the equation, and re-express the function in terms of a single variable. MP.2: Reason abstractly and quantitatively. Students translate a verbal description to a symbolic representation, manipulate the symbols to compose a new function in terms of a single variable, and relate the new function to the verbal description. 62

73 Mathematics Working With Formulas and Function Notation FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Recall basic geometric formulas Isolate a variable in an algebraic equation MATERIALS AND RESOURCES Student Activity pages ASSESSMENTS The following types of formative assessments are embedded in this lesson: Students engage in independent practice. The following additional assessments are located on our website: Areas and Volumes Geometry Free Response Questions Areas and Volumes Geometry Multiple Choice Questions T E A C H E R P A G E S 63

74 Mathematics Working With Formulas and Function Notation T E A C H E R P A G E S TEACHING SUGGESTIONS The skill of manipulating literal equations to solve for a particular variable is essential in mathematics and science. When the equation involves several variables, students often struggle to isolate the required variable. Experience and practice with this skill will benefit students at all levels. This lesson could be used as independent practice. If students are struggling, the entire group could work questions 1 and 2. Then students would have a model for the procedure needed to work other questions. The teacher may need to model sketching the figure in questions 3 6. In questions 8 and 9, students use rational exponents. After students have completed the activity, the teacher might ask partners to read and interpret the use of the function notation in each question. Then students could share out their interpretations. For example, in question 1, A(C) means that the area of the circle is a function of its circumference. In other words, the value of the area depends upon the value of the circumference. Suggested modifications for additional scaffolding include the following: 1-2 Provide a drawing of the figures to which the question refers. 3-5 Provide a drawing of the three dimensional figure with pertinent dimensions labeled Provide the necessary formulas or a formula chart. 64

75 Mathematics Working With Formulas and Function Notation NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ Write the formula for the area of a rectangle using a variable for the missing length or width. 4th Grade Skills/ Isolate the variable for length or width in the formula for area of a rectangle. 5th Grade Skills/ Isolate the variable for length or width in the formulas for area and perimeter of a rectangle. 6th Grade Skills/ Solve literal equations (perimeter, area, and volume). 7th Grade Skills/ Solve literal equations (perimeter, area, and volume). Algebra 1 Skills/ Solve literal equations (perimeter, area, and volume). Geometry Skills/ Solve literal equations (perimeter, area, and volume). Algebra 2 Skills/ Solve literal equations (perimeter, area, and volume). Pre-Calculus Skills/ Solve literal equations (perimeter, area, and volume). AP Calculus Skills/ Solve and use literal equations in real life and mathematical applications. T E A C H E R P A G E S 65

76 Mathematics Working With Formulas and Function Notation T E A C H E R P A G E S 66

77 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Working With Formulas and Function Notation 1. What is an equation for the area of a circle, A, as a function of its circumference, C? 2. What is an equation for the perimeter of a square, P, as a function of its area, A? 3. If the volume of a right cylinder is 400 π cubic units, what is an equation for its lateral surface area, S, as a function of its radius, r? 4. A conical tank is built with its vertex pointed down. It has a height of 12 feet and a base diameter of 8 feet. Water is stored in the tank to a height of h. What is an equation for the volume of water in the conical tank, V, as a function of h? 5. Water is being stored in a cylindrical tank that has a base with area of 900 π square feet. What is an equation for the volume of water in the cylindrical tank, V, as a function of h, the height of the water in the tank? 67

78 Mathematics Working With Formulas and Function Notation 6. What is an equation for the area of an equilateral triangle, A, as a function of the length of one of its sides, s? 7. What is an equation for the area of a circle, A, as a function of its diameter, d? 8. What is an equation for the surface area of a sphere, S, as a function of its volume, V? 9. What is an equation for the volume of a cube, V, as a function of its surface area, S? 10. If the area of the base of a cone is 25 π square units, what is the volume of the cone, V, as a function of its height, h? 68

79 NATIONAL MATH + SCIENCE INITIATIVE Mathematics LEVEL Geometry or Math 2 as part of a unit on volume MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. G N P A V P Physical V Verbal A Analytical N Numerical G Graphical Volumes of Revolution ABOUT THIS LESSON In this lesson, students calculate the perimeter and area of a planar region defined by a system of equations and then determine the volume and surface area of the solid generated by revolving the region around the x-axis or the y-axis. The lesson provides students with a physical method to visualize 3-dimensional solids and a specific procedure to sketch a solid of revolution. This lesson focuses on computing perimeters, areas, and volumes of geometric figures presented from an algebraic perspective, while reinforcing the skills of graphing linear equations and solving linear systems. The lesson enhances student understanding of these standards by developing coherence and connections among a variety of mathematical concepts, skills, and practices. OBJECTIVES Students will graph equations that define a plane figure on a coordinate plane. determine the perimeter and area of that plane figure. draw and describe the solid formed by revolving the plane figure about a vertical or horizontal line. calculate the volume and surface area of the solid. compare the volumes determined when revolving about different axes. T E A C H E R P A G E S 69

80 Mathematics Volumes of Revolution T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Targeted Standards G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. See questions 1f, 2f, 3f G-GMD.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects. See questions 1e, 2e, 3e G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. See questions 1b-c, 2b-c Reinforced/Applied Standards A-REI.6: A-REI.7: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. See question 2a Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x 2 + y 2 = 3. See question 4a COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.1: Make sense of problems and persevere in solving them. Students graph equations to create a two-dimensional figure on a coordinate plane, revolve the figure to form a three-dimensional solid, identify the solid, determine its dimensions, and calculate the volume and surface area of the solid. MP.3: Construct viable arguments and critique the reasoning of others. Students explain the differences in the solids formed when a two-dimensional figure is revolved about a vertical line versus a horizontal line. MP.5: Use appropriate tools strategically. Students create three-dimensional models using paper and pencil. Other visualization tools can include physical or computergenerated models. MP.7: Look for and make use of structure. Students associate the dimensions of the planar figure to the radius and height of the solid of revolution. 70

81 Mathematics Volumes of Revolution FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Graph linear equations Calculate perimeter and area of planar figures and volumes of cylinders and cones ASSESSMENTS The following formative assessment is embedded in this lesson: Students engage in independent practice. The following additional assessments are located on our website: Areas and Volumes Geometry Free Response Questions Areas and Volumes Geometry Multiple Choice Questions MATERIALS AND RESOURCES Student Activity pages Cut-outs of planar regions Pencils, straws, or skewers Tape Straight edges Applets for creating volumes of revolution: volumewashers.html T E A C H E R P A G E S applets/revolution.html html 71

82 Mathematics Volumes of Revolution T E A C H E R P A G E S TEACHING SUGGESTIONS The concept of revolving a region about an axis is fundamental to integral calculus. Students will experience creating boundaries of a planar region to calculate area and volume and then determine the volume and surface area generated by revolving the region around the x-axis or the y-axis. Make sure that students are correctly identifying the radius and the height for each generated solid. Ask students to explain how changing the axis of revolution changes the dimensions of the solid that is formed. Discuss with students known real life examples for the solids of revolution. For example, question 1d models a toilet paper roll. To help students visualize the solid generated by revolving the figure about an axis, have them glue or tape a right triangle or a rectangle onto a pencil, straw or skewer and then revolve the triangle horizontally and vertically by rolling the stick between their hands so the students can see the 3-dimensional figure that will be generated. As an extension of the lesson, have students bring three-dimensional objects to class and then have the students draw cross-sections. Being able to visualize solids and their cross sections is a valuable tool for calculus. When drawing a sketch of a solid revolution, use the following procedure: Draw the boundaries. Shade the region to be revolved. Draw the reflection (mirror image) of the region about the axis of revolution. Connect significant points and their reflections with ellipses (oval shapes) to illustrate the solid nature of the figure. Suggested modifications for additional scaffolding include the following: 1a Label the points of intersection on the graph and the distance between each of the points of the rectangle for calculating the perimeter and area of the rectangle. 1d Label the outer radius, R, of the cylinder and the inner radius, r, for the smaller cylinder that is removed from the center. 1g, 2h, Provide a graph so the student can draw a 2j, 3h sketch of the given scenario to visualize the problem situation before proving their answer symbolically. You may wish to support this activity with TI- Nspire technology. See Working with Fractions and Decimals in the NMSI TI-Nspire Skill Builders. 72

83 Mathematics Volumes of Revolution NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. 4th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) 5th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and non-overlapping right rectangular prisms with whole number edge lengths. 6th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and nonoverlapping right rectangular prisms with fractional and decimal edge lengths and use scale factors to determine the volume of similar solids. 7th Grade Skills/ Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. Algebra 1 Skills/ Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Given the equations of lines (at least one of which is horizontal or vertical) that bound a triangular, rectangular, or trapezoidal region, calculate the surface area and/or volume of the solid formed by revolving the region about the line that is horizontal or vertical. Geometry Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Algebra 2 Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Pre-Calculus Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. AP Calculus Skills/ Calculate the area between curves. Calculate the volume of a solid formed by revolving the region between two curves about a horizontal or vertical line. T E A C H E R P A G E S 73

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85 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Volumes of Revolution When drawing a sketch of a solid revolution, use the following procedure: Draw the boundaries. Shade the region to be revolved. Draw the reflection (mirror image) of the region across the axis of revolution. Connect significant points and their reflections with ellipses (oval shapes) to illustrate the solid nature of the figure. 1. A geometric figure is created when bounded by the lines y = 3, y = 1, x = 1, and x = 6. a. Sketch and shade the region. b. Determine the perimeter of the region c. Determine the area of the region d. Draw a picture of the region being revolved about the x-axis. e. Describe the geometric solid formed by revolving the region about the x-axis f. Determine the volume of the geometric solid

86 Mathematics Volumes of Revolution g. Julia was asked to change the constraint to. Jeremiah was asked to change the constraint y = 3 to y = Verify that Julia s and Jeremiah s new regions have areas that are one-half of the area of the original figure. Both students assumed that, since the change of their constraints cut the area of the figure in half, the volume of the revolved figure about the x-axis would also be half of the volume of the original geometric solid from part (f). Demonstrate graphically and algebraically whether each student s assumption is correct or incorrect. 76

87 Mathematics Volumes of Revolution 2. A geometric figure is created when bounded by the lines y = 3 x 3, y = 0, and x = 0. 4 a. Sketch and shade the region. b. Determine the perimeter of the region. c. Determine the area of the region. d. Draw a picture of the region being revolved about the x-axis e. What geometric figure is formed by revolving the region about the x-axis? f. Determine the volume of the geometric solid g. Determine the surface area of the geometric solid h. If the region were revolved about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer. Compare the surface areas. i. Name another region that could be revolved about the x-axis to create exactly the same geometric solid defined in part (d). 77

88 Mathematics Volumes of Revolution j. In answering question (i): Billy used a figure bounded by x = 4,, and, revolved about. Jennifer used the same constraints as Billy, but revolved her figure about Both students claim that their figure has the same area, perimeter, and volume as the figure in part (a) and part (e). Determine which student is correct and justify your answer. 78

89 Mathematics Volumes of Revolution 3. A geometric figure is formed by the curve y = and the line y = 0. a. Sketch and shade the region. b. Determine the perimeter of the region. c. Determine the area of the region. d. Draw a picture of the region being revolved about the x-axis e. What geometric figure is formed by revolving the region about the x-axis? f. Determine the volume of the geometric solid g. Determine the surface area of the geometric solid h. If the region were revolved about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer. 79

90 Mathematics Volumes of Revolution 4. A region is bounded by the graphs of a. Sketch and shade the region. 5 y x b. Draw a picture of the region revolved around the x-axis. 5 y x c. Draw a picture of the region revolved around the y-axis. 5 y x 80

91 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Systems of Linear Inequalities LEVEL Algebra 1 or Math 1 in a unit on systems of linear inequalities Algebra 2 or Math 3 as a review of systems of linear inequalities before graphing non-linear inequalities MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. ABOUT THIS LESSON This lesson provides students with an opportunity to practice graphing systems of linear inequalities and to identify the points of intersection creating the bounded region formed by the inequalities. Students are asked to determine the perimeter and area of the bounded region and revolve the region about the y-axis. The student then describes the solid formed and calculates the volume and surface area. This lesson focuses on graphing systems of linear inequalities in two variables as a half plane, determining the points of intersection of the system, and identifying the solution by shading the region defined by the system. At the same time, the lesson reviews using formulas to calculate perimeter, area, surface area, and volume of geometric figures to reinforce student understanding of these standards. OBJECTIVES Students will graph and calculate the perimeter and area of the region formed by a system of linear inequalities. determine the volume and surface area of the solid formed by revolving this region about the y-axis. T E A C H E R P A G E S G N P A V P Physical V Verbal A Analytical N Numerical G Graphical 81

92 Mathematics Systems of Linear Inequalities T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Targeted Standards A-REI.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. See questions 1a, 2a, 3a Reinforced/Applied Standards G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. See questions 1c-d, 2c-d, 3c-d G-GMD.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects. See questions 1e, 2e, 3e COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.1: Make sense of problems and persevere in solving them. Students graph inequalities to create 2-dimensional figures on a coordinate plane, revolve the figure to form a 3-dimensional solid, identify the solid, determine its dimensions, and calculate the volume and surface area. MP.5: Use appropriate tools strategically. Students create 3-dimensional models using paper and pencil. Other visualization tools can include physical or computer-generated models. MP.7: Look for and make use of structure. Students associate the dimensions of the planar figure with the radius and height of the solid of revolution. G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. See questions 1f, 2f, 3f 82

93 Mathematics Systems of Linear Inequalities FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Graph linear equations and inequalities Calculate perimeter and area of triangles and trapezoids Calculate volumes of cylinders and cones ASSESSMENTS The following types of formative assessments are embedded in this lesson: Students engage in independent practice. MATERIALS AND RESOURCES Student Activity pages Cardstock Tape Scissors Applets for creating volumes of revolution: volumewashers.html SolidsOfRevolution/ The following additional assessments are located on our website: Areas and Volumes Algebra 1 Free Response Questions Areas and Volumes Algebra 1 Multiple Choice Questions Areas and Volumes Algebra 2 Free Response Questions Areas and Volumes Algebra 2 Multiple Choice Questions T E A C H E R P A G E S 83

94 Mathematics Systems of Linear Inequalities TEACHING SUGGESTIONS To help students visualize the solid generated by revolving the figure about an axis, glue or tape a right triangle onto a stick or dowel and then rotate the triangle horizontally and vertically by rolling the stick between your hands so that students can see the cone that will be generated. Another option is for students to create a model using the following directions: From cardstock, cut out about 20 copies of the region formed by the linear inequality in the example. See figure 1. Suggested modifications for additional scaffolding include the following: 1a, 2a, 3a Provide the equations in slope-intercept or point-slope form. 1e Graph the solid for the first question. Lay 2 copies of the region together and tape them together. See figure 2. T E A C H E R P A G E S Fold the taped regions together then lay another cardstock region beside the folded region and tape together. Continue to fold and tape another card stock region to the folded, taped cardstock regions until all of the cardstock regions are taped together. Connect the first and last cardstock regions together, like an accordion, to illustrate revolving the region about the y-axis. You may wish to support this activity with TI- Nspire technology. See Graphing Functions and Equations and Graphing Inequalities in the NMSI TI-Nspire Skill Builders. 84

95 Mathematics Systems of Linear Inequalities NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. 4th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) 5th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and non-overlapping right rectangular prisms with whole number edge lengths. 6th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) In a problemsolving situation with a real-world application, determine the volume of rectangular prisms and nonoverlapping right rectangular prisms with fractional and decimal edge lengths and use scale factors to determine the volume of similar solids. 7th Grade Skills/ Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Given 3 or 4 coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or cylinder formed by revolving the bounded region about either of the lines. Algebra 1 Skills/ Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Given the equations of lines (at least one of which is horizontal or vertical) that bound a triangular, rectangular, or trapezoidal region, calculate the surface area and/or volume of the solid formed by revolving the region about the line that is horizontal or vertical. Geometry Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Algebra 2 Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. Pre-Calculus Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed by revolving the region about a horizontal or vertical line. AP Calculus Skills/ Calculate the area between curves. Calculate the volume of a solid formed by revolving the region between two curves about a horizontal or vertical line. T E A C H E R P A G E S 85

96 Mathematics Systems of Linear Inequalities T E A C H E R P A G E S 86

97 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Systems of Linear Inequalities When drawing a sketch of a solid revolution in part (e), use the following procedure: Draw the boundaries. Shade the region to be revolved. Draw the reflection (mirror image) of the region across the axis of revolution. Connect significant points and their reflections with ellipses (oval shapes) to illustrate the solid nature of the figure. 1. Use the following system of inequalities to answer parts (a) (g). Write all of the answers in terms of and in simplified radical form. 8 6 a. Graph the solution to the system of inequalities. b. Determine the vertices of the region formed by the solution c. Determine the area of the region. d. Determine the perimeter of the region. e. Sketch the solid formed by revolving the region about the y-axis on the graph in part (a) and describe the solid formed. f. Determine the volume of the solid. g. Determine the surface area of the solid. 87

98 Mathematics Systems of Linear Inequalities 2. Use the following system of inequalities to answer parts (a) (g). Write all of the answers in terms of and in simplified radical form a. Graph the solution to the system of inequalities. b. Determine the vertices of the region formed by the solution c. Determine the area of the region d. Determine the perimeter of the region. e. Sketch the solid formed by revolving the region about the y-axis on the graph in part (a) and describe the solid formed. f. Determine the volume of the solid. g. Determine the surface area of the solid. 88

99 Mathematics Systems of Linear Inequalities 3. Use the following system of inequalities to answer parts (a) (g). Write all of the answers in decimal form. Decimal answers should be accurate to 3 decimal places. To avoid round off errors, use exact answers in all of your work. 8 a. Graph the solution to the system of inequalities. 6 b. Determine the vertices of the region formed by the solution. 4 2 c. Determine the area of the region d. Determine the perimeter of the region. e. Sketch the solid formed by revolving the region about the y-axis on the graph in part (a) and describe the solid formed. f. Determine the volume of the solid. g. Determine the surface area of the solid. 89

100 Mathematics Systems of Linear Inequalities 90

101 NATIONAL MATH + SCIENCE INITIATIVE Mathematics G F E D C Cone Exploration and Optimization I H J K L M LEVEL Algebra 2, Math 3, Pre-Calculus, or Math 4 in a unit on polynomials MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. N Q O B P A ABOUT THIS LESSON In this lesson, students manipulate a paper cone by cutting along the radius of a circle and overlapping the edges. They explore the manner in which the cone s dimensions change as the amount of overlap increases or decreases. Students conclude the lesson by determining the dimensions of the cone with greatest volume. Students consider domain issues while exploring the limits of the cone construction. This lesson focuses on understanding and creating a function that models a relationship, while reinforcing the skills of determining a reasonable domain, graphing a non-linear function, and calculating volumes. Students will graph functions and use the table feature on their graphing calculator. The lesson enhances student understanding of these standards by developing coherence and connections among a variety of mathematical concepts, skills, and practices. OBJECTIVES Students will explore the effect of changing the dimensions of a cone. determine the cone of maximum volume. T E A C H E R P A G E S G N P A V P Physical V Verbal A Analytical N Numerical G Graphical 91

102 Mathematics Cone Exploration and Optimization T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Targeted Standards F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. See questions 5-15 Reinforced/Applied Standards F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. See questions 5-6, 8 A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. See questions 7, 9 G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. See question 7 G-SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. See question 7 COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.1: Make sense of problems and persevere in solving them. Students cut along a radius of a circle and overlap the edges to create cones of varying dimensions, explore the manner in which the cone s dimensions change as the amount of overlap increases or decreases, write an equation to determine the volume, and determine the dimensions of the cone with maximum volume. MP.2: Reason abstractly and quantitatively. Students create equations involving the dimensions of a cone based on the manipulation of a concrete model of a circle and manipulate the symbols to compose a volume function in terms of a single variable. MP.5: Use appropriate tools strategically. Students cut along a radius of a circle and overlap the edges to create cones of varying dimensions. Students use graphing calculators to determine the maximum volume of the cone. 92

103 Mathematics Cone Exploration and Optimization FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Identify features of a cone (radius, height, slant height, circumference, surface area, volume) Calculate the maximum value of a function using a graphing calculator MATERIALS AND RESOURCES Student Activity pages Scissors Skewers Protractors Rulers Tape ASSESSMENTS The following types of formative assessments are embedded in this lesson: Students engage in independent practice. Students apply knowledge to a new situation. The following additional assessments are located on our website: Areas and Volumes Algebra 2 Free Response Questions Areas and Volumes Algebra 2 Multiple Choice Questions Areas and Volumes Pre-Calculus Free Response Questions Areas and Volumes Pre-Calculus Multiple Choice Questions T E A C H E R P A G E S 93

104 Mathematics Cone Exploration and Optimization T E A C H E R P A G E S TEACHING SUGGESTIONS This lesson is most powerful when students work in groups as they manipulate their circles to form various cones and discuss their observations. Listen to their discussions and provide guidance as needed. Since the radius, height, and surface area all change monotonically, students sometimes mistakenly conclude that the volume of the cone always increases or always decreases. Observing the very small volume of the cone at the extremes of the domain helps students draw the correct conclusion. In calculus, the definition of a cone is extended to permit degenerate cones with either a height of zero or a radius of zero. For calculus students, this extension makes the identification of an absolute maximum volume easier to justify. The answers provided for this lesson consider the degenerate figures. You may wish to support this activity with TI- Nspire technology. See Graphing Functions and Equations and Finding Points of Interest in the NMSI TI-Nspire Skill Builders. Suggested modifications for additional scaffolding include the following: 1-4 Provide sketches of what the cone would look like at various points of overlap. Ask the student to label the dimensions of the cone before attempting to draw conclusions. 5 Refer to the sketches in questions 1 4 to help the student visualize the limited nature of the domains for radius and height. 7 Ask the student to begin with the familiar formula for the volume of a cone,. Provide a sketch of the right triangular cross section of the cone, with the legs labeled r and h and the hypotenuse fixed at 10 cm. Lead the student to recognize that the Pythagorean Theorem provides a relationship between r and h that can be solved for h and substituted into the volume formula. 94

105 Mathematics Cone Exploration and Optimization NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ Write the formula for the area of a rectangle using a variable for the missing length or width. 4th Grade Skills/ Isolate the variable for length or width in the formula for area of a rectangle. 5th Grade Skills/ Isolate the variable for length or width in the formulas for area and perimeter of a rectangle. 6th Grade Skills/ Solve literal equations (perimeter, area, and volume). 7th Grade Skills/ Solve literal equations (perimeter, area, and volume). Algebra 1 Skills/ Solve literal equations (perimeter, area, and volume). Geometry Skills/ Solve literal equations (perimeter, area, and volume). Algebra 2 Skills/ Solve literal equations (perimeter, area, and volume). Pre-Calculus Skills/ Solve literal equations (perimeter, area, and volume). AP Calculus Skills/ Solve and use literal equations in real life and mathematical applications. T E A C H E R P A G E S 95

106 Mathematics Cone Exploration and Optimization T E A C H E R P A G E S 96

107 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Cone Exploration and Optimization For the following activities, cut out the circle and cut along Q. F E D C G B H I Q A P J O K L M N 97

108 Mathematics Cone Exploration and Optimization Consider that the radius of the circle is length Q. Overlap the sides of the cut in the circle, moving point A counterclockwise to various points on the circumference of the circle to form a cone. 1. What happens to the circumference of the base of the cone as point A moves over the circle through points B, C, D, and continuing around the circle until it reaches its original position? What happens to the slant height of the cone? What is the relationship between the radius of the original circle and the slant height of the cone? 2. When the circumference of the cone decreases, what happens to a. the radius of the base of the cone? b. the height of the cone? c. the area of the base? d. the lateral surface area of the cone? 3. When the circumference is its smallest length, point A then moves clockwise around the circle back to its starting point. As the circumference of the cone increases, what happens to a. the radius of the base of the cone? b. the height of the cone? c. the area of the base? d. the lateral surface area of the cone? 4. a. At what height is the cone s lateral surface area the greatest? What is the radius of the cone when the lateral surface area is the greatest? What is the volume of the cone when the lateral surface area is the greatest? 98

109 Mathematics Cone Exploration and Optimization b. At what height is the cone s lateral surface area the least? What is the radius of the cone when the lateral surface area is the least? What is the volume of the cone when the lateral surface area is the least? c. What happens to the volume of the cone between the instances when the radius of the cone is Q and the radius of the cone is 0? For questions 5 12, assume that the radius of the original circle is 10 cm. 5. a. Determine the domain for the height of the cone formed. b. Determine the domain for the radius of the cone formed. 6. Make a conjecture as to what the cone s radius should be to maximize the volume of the cone. 7. Determine the equation for the volume of the cone as a function of r, the radius of the base of the cone. 8. What is the domain of the function determined in question 7? Explain the domain algebraically, based on the equation. V Graph the volume of the cone with respect to the radius of the cone r

110 Mathematics Cone Exploration and Optimization 10. Verify that the volume of the cone is 0 when r is 10 cm. 11. What is the volume of the cone when r is 5 cm? 12. Using a graphing calculator, determine the maximum volume of the cone created from a circle with a radius of 10 cm. 13. Determine the radius and the height of the cone with maximum volume. How do these results compare to your conjecture in question 6? 14. What is the circumference of the base of the cone with maximum volume? What is the length of the arc that should be cut from the circumference of the original circle in order to form the cone with the maximum volume? 15. What is the measure of the arc, in degrees, that is cut from the circumference of the circle? 16. Based on the answers to questions 14 and 15, remove the appropriate sector from the circle and tape the cut edges together to form the cone. Measure the radius and the height of the cone and compare their lengths to the calculated dimensions. 100

111 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Cone Exploration and Optimization Revisited LEVEL Algebra 2, Math 3, Pre-Calculus or Math 4 in a unit on polynomials MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. MODALITY NMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. ABOUT THIS LESSON In this lesson, students investigate the maximum volume of a cone, given various values for the slant height of the cone, as well as the relationship between the height and the radius of the cone with maximum volume. Students generalize the formula for the volume of a cone using the central angle of the circle as the independent variable. To simplify the algebra, the angle is calculated in radians instead of degrees. This lesson focuses on rearranging formulas to highlight a quantity of interest while it extends students ability to solve equations. It enhances student understanding of these and other standards by developing coherence and connections among a variety of mathematical concepts, skills, and practices. OBJECTIVES Students will explore the effect of changing the dimensions of a cone. determine the cone of maximum volume. T E A C H E R P A G E S G N P A V P Physical V Verbal A Analytical N Numerical G Graphical 101

112 Mathematics Cone Exploration and Optimization Revisited T E A C H E R P A G E S COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol ( ) at the end of a specific standard indicates that the high school standard is connected to modeling. Targeted Standards A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. See questions 1, 8-12 Reinforced/Applied Standards F-BF.1: Write a function that describes a relationship between two quantities. See questions 1, 3, 8-12 G-SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. See questions 1, 9b-c, 10, 12 A-SSE.2: Interpret expressions that represent a quantity in terms of its context. Interpret the structure of expressions. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 ) (x 2 + y 2 ). See question 12 G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. See questions 1, 3, 6b-c, 10, COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice. MP.1: Make sense of problems and persevere in solving them. Students remove a sector of given arc length from a circle and use the remaining sector to form a cone, determine the maximum volume of the cone, collaborate with other students to compare cones with different slant heights, write an equation to determine the volume in terms of θ, and determine the dimensions of the cone with maximum volume. MP.2: Reason abstractly and quantitatively. Students create equations involving the dimensions of a cone based on the manipulation of a concrete model of a circle and manipulate the symbols to compose a volume function in terms of a single variable. MP.5: Use appropriate tools strategically. Students use a graphing calculator to create a scatterplot and to determine the maximum volume of the cone. MP.8: Look for and express regularity in repeated reasoning. Students are assigned a circle with a unique radius and, after collecting the class results, determine that the ratio of the radius to the height is constant for a cone of maximum volume. 102

113 Mathematics Cone Exploration and Optimization Revisited FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Rewrite equations and formulas to isolate a specific variable Calculate the maximum value of a function using a graphing calculator MATERIALS AND RESOURCES Student Activity pages Scissors Conical paper cups ASSESSMENTS The following types of formative assessments are embedded in this lesson: Students engage in independent practice. Students apply knowledge to a new situation. The following additional assessments are located on our website: Areas and Volumes Algebra 2 Free Response Questions Areas and Volumes Algebra 2 Multiple Choice Questions Areas and Volumes Pre-Calculus Free Response Questions Areas and Volumes Pre-Calculus Multiple Choice Questions T E A C H E R P A G E S 103

114 Mathematics Cone Exploration and Optimization Revisited T E A C H E R P A G E S TEACHING SUGGESTIONS This lesson continues the students exploration of the cone that was introduced in the lesson, Cone Exploration and Optimization and can be used as independent practice. After determining the volume formula for the cone formed with the assigned R value, students will use a graphing calculator in question 6 to consider the maximum volume and other features of that cone. In question 7, students will use the table features of the graphing calculator to plot a scatterplot of all the data generated by the class. In questions 8 12, students practice rewriting formulas in terms of a given variable. Question 12b asks students to rewrite a fairly complicated equation and simplify it to a particular form. This will be a challenge for many students but a good exercise in manipulating the algebra into a desired form. Question 13 explores the cone with respect to the central angle removed. You may wish to support this activity with TI- Nspire technology. See Finding Points of Interest and Displaying Data in a Scatter Plot in the NMSI TI-Nspire Skill Builders. Suggested modifications for additional scaffolding include the following: 6 Provide a class copy of the table shown in the answers, with several rows completed. Ask the student to first confirm one or more of the rows before calculating the entries for one of the blank rows. 12b Omit the algebraic manipulation, or provide some of the steps. 104

115 Mathematics Cone Exploration and Optimization Revisited NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ Write the formula for the area of a rectangle using a variable for the missing length or width. 4th Grade Skills/ Isolate the variable for length or width in the formula for area of a rectangle. 5th Grade Skills/ Isolate the variable for length or width in the formulas for area and perimeter of a rectangle. 6th Grade Skills/ Solve literal equations (perimeter, area, and volume). 7th Grade Skills/ Solve literal equations (perimeter, area, and volume). Algebra 1 Skills/ Solve literal equations (perimeter, area, and volume). Geometry Skills/ Solve literal equations (perimeter, area, and volume). Algebra 2 Skills/ Solve literal equations (perimeter, area, and volume). Pre-Calculus Skills/ Solve literal equations (perimeter, area, and volume). AP Calculus Skills/ Solve and use literal equations in real life and mathematical applications. T E A C H E R P A G E S 105

116 Mathematics Cone Exploration and Optimization Revisited T E A C H E R P A G E S 106

117 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Cone Exploration and Optimization Revisited A cone of height h and radius r is constructed from a flat circular disk of radius R by removing a sector of arc length x and connecting the edges. x Θ R h R r 1. Calculate r 2 in terms of R and h, and then write a formula for the volume of the cone in terms of h and R. 2. Each student in the classroom will be assigned a different R. Ask your teacher for your specific value of R. 3. Determine the formula for the volume of your cone in terms of h, the height of the cone, and your particular value of R. 4. How many cones can be formed using your R? Using your value for R, what additional piece of information is required to actually make a specific cone? 5. What is the domain of the volume function using your given value of R? 107

118 Mathematics Cone Exploration and Optimization Revisited 6. Use a graphing calculator and your particular value of R to answer the following: a. What is the maximum volume of the cone? b. What is the height of the cone with maximum volume? c. What is the radius of the cone with maximum volume? d. What is the circumference of the base of the cone with maximum volume? e. What is the ratio of r to h when the volume is a maximum? f. What is the circumference of the original circle with your given R value? g. Using your value of R and the value of r you determined in part (c), determine the length of the arc, x, that should be removed from the original circle to create the cone with maximum volume. h. What is the measure of, in degrees, required to obtain the arc used to form the cone of maximum volume? 7. Create a scatterplot of the ordered pairs (h, r), using the data from all of your classmates. What do you notice? 8. What is the circumference of the base of the cone in terms of R and x? 9. a. What is the radius of the base of the cone in terms of R and x? b. What is a formula that relates the height, the radius, and the slant height of the cone? c. What is an expression for h, in terms of R and x? 108

119 Mathematics Cone Exploration and Optimization Revisited 10. Write a formula, in terms of R and x, for the volume of the cone. 11. The formula for the length of an arc of a circle comes from the ratio: Let x be the length of the arc, let R be the radius of the circle, and let radians. Solve the ratio for x. be the central angle measured in 12. a. Rewrite the formula for volume from question 10 as a function of in radians, instead of as a function of x. b. Show the algebra that transforms the equation from part (a) into. c. State the domain for the function of. d. If a sector with a central angle of is removed from a circle with a radius of 10 cm, what is the volume of the cone that can be formed from the remaining portion of the circle? 109

120 Mathematics Cone Exploration and Optimization Revisited 13. a. Using the formula from question 12b, calculate the maximum volume of a cone that can be constructed by cutting a sector with a central angle of from a circle with a radius of 10 cm. b. What angle measured in degrees gives the maximum volume? c. Does the angle required to produce a cone of maximum volume depend on the radius of the original circle? Use a graphing calculator to investigate the function for various values of R and explain why the value of R does or does not matter. 14. Cut open a conical paper cup to form a circle with a sector removed. Measure the slant height of the cone and the central angle of the sector that has been removed. Using the measurement information, determine if this cup has the maximum volume that could have been made from the circle. If not, suggest some possible reasons for the choice of these dimensions. 110

121 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Introduction to the NMSI Mathematics Multiple Choice Quizzes The National Math and Science Initiative multiple choice questions are modeled after multiple choice questions on the AP* Calculus and Statistics exams. The questions are assigned a course-level designation based on an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The grade-level multiple choice quizzes for sixth grade through pre-calculus assess the skills and concepts introduced in each module. These quizzes reflect the module s Content Progression Chart, which outlines the mathematics imbedded in the activities for each grade level, and the Concept Development Chart, which provides examples of how those concepts or skills might be assessed. Additionally, the quizzes are directly linked to the NMSI posttests for each grade level. Once students have completed the activities, teachers may use the quiz questions to determine student understanding and to prepare students for the level of rigor on the posttests. When scoring the multiple choice questions, teachers should remember that the quizzes are intended to model the rigor of questioning on AP exams. A suggested scoring guideline, which is also included with the rationales for each quiz, is: Percent Correct Grade All of these materials lessons and activities with answer keys, grade-level quizzes with rationales, and free response questions with scoring rubrics and student samples are available for each module on the NMSI website. 111

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123 Mathematics Sample Quiz Questions 1. 6 th Grade Module 1 Question 4 A right triangular region is formed by line segments connecting the points (0, 0), (3, 0), and (0, 4). Revolving the region about the y-axis creates a cone. What is the volume of the cone? A. 4π cubic units B. 8π cubic units C. 12π cubic units D. 18π cubic units E. 36π cubic units 2. 6 th Grade Module 1 Question 6 Sally has a chocolate candy bar that she is going to share with her friend Sam. The candy bar is a rectangular prism with the following dimensions: length =10 cm ; width = 2.5 cm ; and height =2.25 cm. She cuts off a piece that is 2 cm long from the length of the candy bar and gives the shorter piece to Sam. Which phrase accurately describes the volume of the remaining candy bar in relation to the volume of the original candy bar? A. The volume is 32 of the original volume. 225 B. The volume is 1 of the original volume. 5 C. The volume is 4 of the original volume. 225 D. The volume is 64 of the original volume. 125 E. The volume is 4 of the original volume

124 Mathematics Sample Quiz Questions 3. 7 th Grade Module 1 Question 3 A rectangular region is formed by the line segments connecting the points (0, 0), (5, 0), (5, 3), and (0, 3). Revolving the rectangular region about the x-axis creates a cylinder. What is the volume of the cylinder? A. 15π cubic units B. 30π cubic units C. 45π cubic units D. 75π cubic units E. 225π cubic units 4. 7 th Grade Module 1 Question 6 If the radius of a cone is multiplied by 6, how does the volume of the new cone compare to the original cone? A. The new volume will be 6 times the original volume. B. The new volume will be 12 times the original volume. C. The new volume will be 36 times the original volume. D. The new volume will be 72 times the original volume. E. The new volume will be 216 times the original volume. 114

125 Mathematics Sample Quiz Questions 5. Algebra 1 Module 1 Question 3 A triangular region is bounded by the lines y = 3, y = x + 4, and y = 2(x 6)+1. What is the area of the triangular region? A. 12 square units B square units C. 21 square units D square units E square units 115

126 Mathematics Sample Quiz Questions 6. Algebra 1 Module 1 Question 6 A triangular region is bounded by the lines x = 0, y = 0, and 3x + 6y =18. What is the volume of the cone that is formed when the triangular region is revolved about the y-axis? A. 18π cubic units B. 27π cubic units C. 36π cubic units D. 54π cubic units E. 108π cubic units 116

127 Mathematics Sample Quiz Questions 7. Geometry Module 1 Question 4 The volume of a cylinder that is created by revolving a rectangle about the y-axis is 54π cubic units. The rectangle is formed by the intersection of four lines. The equations of three of the lines are: x = 0, y = 0, and y = 6. Which of the following is the area of the rectangle that is revolved to form the cylinder? A. 3 square units B. 6 square units C. 9 square units D. 18 square units E. 27 square units 8. Geometry Module 1 Question 7 In the cone provided, what is the volume, V, in terms of the slant height, k? A. V = πk 3 8 B. V = πk 3 4 C. V = 3πk 3 8 D. V = πk E. V = πk

128 Mathematics Sample Quiz Questions 9. Algebra 2 Module 1 Question 3 The region bounded by the lines x = 1, y =1, y = 5, and y = 4 ( 3 x 5 )+1 is revolved about the line y =1. What is the volume of the solid? A. 48π cubic units B. 64π cubic units C. 80π cubic units D. 84π cubic units E. 96π cubic units 118

129 Mathematics Sample Quiz Questions 10. Algebra 2 Module 1 Question 6 The volume of a cone that has been created by revolving a triangle about the x-axis is 16π cubic units. The triangle is formed by the intersection of three lines. The equations of two of the lines are x = 0 and y = 0. Which of the following could be the equation of a line containing the third side? I. y = 4 3 x 4 II. y = 4 3 x + 4 III. y = 4 ( 3 x 3 ) A. I only B. II only C. III only D. I and II only E. I, II, and III 119

130 Mathematics Sample Quiz Questions 11. Pre-Calculus Module 1 Question 2 A triangle with an area of 9 square units is formed by the intersection of three lines. The equations of two of the lines are x = 0 and x + y = 0. Which of the following could not be an equation of a line containing the third side? A. y = 3(x 2) B. y = 3(x + 2) C. y = x + 6 D. y = 6 x E. y = x 6 120

131 Mathematics Sample Quiz Questions 12. Pre-Calculus Module 1 Question 5 The volume of a cylinder that has been created by revolving a rectangle about the line x = 1 is 18π cubic units. The rectangle is formed by the intersection of four lines. The equations of three of these lines are x = 1, y =1, and y = 3. Which of the following could be the equation of the line containing the fourth side? A. x = 2 B. x = 3 C. x = 3.5 D. x = 4.5 E. x = 9 121

132 Mathematics Sample Quiz Questions Selected Rationales 1. 6 th Grade Module 1 Question 4 A. Student does not square the radius. B. Student doubles the radius rather than squaring it. C. Correct. Student uses 3 units for the radius and 4 units for the height. D. Student uses 1 2 instead of 1 3 in the formula. E. Student does not use one-third in the formula th Grade Module 1 Question 6 A. Student uses the scale factor given in the question. B. Student doubles the scale factor or squares the scale factor and divides by 3. C. Correct. Student squares the scale factor since only the radius is affected. D. Student cubes the scale factor then divides by 3. E. Student cubes the scale factor. 6. Algebra 1 Module 1 Question 6 A. Student revolves the figure about the x-axis. B. Student uses 6 units for the height and 3 units for the radius, or student revolves the figure about the x-axis, and then divides by 2 rather than 3. C. Correct. Student uses a radius of 6 units and a height of 3 units to determine the volume of the cone. D. Student divides by 2 rather than 3 in the formula for the volume of a cone. Alternatively, the student revolves the figure about the x-axis or uses 3 units for the radius and 6 units for the height and does not divide by 3. E. Student does not divide by

133 Mathematics Sample Quiz Questions 10. Algebra 2 Module 1 Question 6 I. True. The line segment in Quadrant IV creates a triangle with a horizontal leg of 3 units and a vertical leg of 4 units that revolves about the x-axis to create a cone whose volume is π 3 42 II. True. The line segment in Quadrant II creates a triangle with a horizontal leg of 3 units and a vertical leg of 4 units. III. True. The line segment in Quadrant I creates a triangle with a horizontal leg of 3 units and a vertical leg of 4 units. A. Student graphs the triangle in Quadrant IV but does not consider that other such triangles can be created by different lines in other quadrants. B. Student graphs the triangle in Quadrant II but does not consider that other such triangles can be created by different lines in other quadrants. C. Student graphs the triangle in Quadrant I but does not consider that other such triangles can be created by different lines in other quadrants. D. Student selects lines with positive slopes only or selects lines in slope-intercept form only. E. Correct. Student realizes that the cones created by each line will have the same volume. ( ) 3 ( ). 123

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135 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Introduction to the NMSI Mathematics Free Response Questions The National Math and Science Initiative free response questions are modeled after free response questions on the AP* Calculus and Statistics exams. The questions are assigned a course-level designation based on an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content. The free response rubric is a guide to assist the reader, not a detailed solution to the question. Sometimes a method is outlined in the rubric, but another more efficient method may work as well. A student s correct solution may earn all of the points to be awarded for a particular part of the question, even though the approach does not match the one shown in the rubric. The rubric shows a way to work the problem, not the way to work the problem. When scoring the free response questions, teachers should practice reading with a student s error. This statement means that the student is penalized for the error when it first occurs, but the reader then follows the student s process for full credit in subsequent parts of the question, even when the student continues to use the results of the earlier error. For the free response, the reader should be in the mindset of awarding points, not taking them away. Students start at 0 and can earn up to 9 points rather than starting at 9 points and losing points. Student directions for the free response questions include the following: All work for a given part of a question must be shown in the space provided. Answers do not need to be simplified completely; however, when calculating approximate answers, do not round intermediate values. Your final answers should be accurate to three places after the decimal point. Questions that contain units require units in the answers. The setup for all mathematical computations and equations is required using mathematical notation rather than calculator syntax. Intermediate calculations do not have to be shown when determining: the answer to basic arithmetic computations; the zeroes of a function; the maximum/minimum of a function; the intersection point between two functions; a regression equation. Part A and B are given equal weight, but parts of a particular question are not necessarily given equal weight. During the timed portion for Part A, you may work only on Part A. A calculator may be used on Part A only. During the timed portion for Part B, in addition to working on the question in Part B, you may continue to work on Part A without a calculator. 125

136 Mathematics Free Response Questions Grade Free Response Question - Calculator Not Allowed Region R is a rectangle formed by connecting the points (0, 6), (0, 9), (9, 9), and (9, 6). Region S is a triangle formed by connecting the points (9, 6), (9, 9), and. Identify the points that form the vertices of each region on the grid provided by labeling their coordinate points. (a) What is the area of the rectangle represented by region R? What is the area of the triangle represented by region S? Show the work that leads to the answer. (b) When region R is revolved about the line, a cylinder is formed. (, where B is the area of the base and h is the height of the cylinder.) Sketch the cylinder on the grid by reflecting the rectangle across the line. Connect the corresponding outside vertices of the combined regions with oval-like shapes. What is the volume of the cylinder? Show the work that leads to the answer. Leave in the answer. 126

137 Mathematics Free Response Questions (c) When region S is revolved about the line, a cone is formed. (, where B is the area of the base and h is the height of the cone.) What is the volume of the cone? Show the work that leads to the answer. Leave in the answer. (d) When revolved about the line, what is the combined volume of the solids formed by regions R and S? Show the work that leads to the answer. Leave in the answer. (e) Twenty students calculated the volume of a wooden model of the solid described in part (d) by measuring the outer dimensions. Their volumes, when the answers are divided by, are shown in the stemplot. Create a histogram based on their data. Volume of Solid Divided by Key: 8 4 means 84 cubic units What is the numerical difference between the median of the data multiplied by volume calculated in part (d)? Show the work that leads to the answer. Leave and the actual in the answer. 127

138 Mathematics Free Response Questions Algebra Free Response Question - Calculator Not Allowed In the figure provided: Region R is a rectangle bounded by the lines y = 6, x = 0, y = 9, and x = 9. Region S is a triangle bounded by the lines x = 9, y = 9, and a line, h(x), containing the points (0, 0) and (9, 6). Region T is a triangle bounded by y = 6,x = 0, and the line, h(x). Region Z is formed by grouping regions R, S, and T. (a) What is the equation of the line, h(x), containing the points (0, 0) and (9, 6)? What is the value of x when? Show the work that leads to the answer. Using this information, identify the point on region S by labeling the coordinates. Identify the other two vertices of region S by labeling their coordinates. What is the area of region S? Show the work that leads to the answer. (b) When region R is revolved about the line, a cylinder is formed. Sketch the cylinder on the graph by reflecting the rectangle across the line. Connect the corresponding outside vertices of the combined regions with oval-like shapes. What is the volume of the cylinder? Show the work that leads to the answer. Leave in the answer. 128

139 Mathematics Free Response Questions (c) When region S is revolved about the line, a cone is formed. On the graph provided on the previous page, sketch the cone in the correct position. What is the volume of the cone in terms of? Show the work that leads to the answer. (, where B is the area of the base and h is the height of the cone) (d) When region Z is revolved about the line, a cone with a radius of 9 is formed. What is the volume of the cone in terms of? Show the work that leads to the answer. (e) When region T is revolved about the line, a frustum with a cylinder removed is formed that has a volume of 270 cubic units. Two hundred students measured the dimensions of a wooden model of the solid and calculated the volume. They divided their calculations by and collected the data. The five-number summary for the data is {min = 249, Q1 = 261, med = 268, Q3 = 279, and max = 292} Create a boxplot of the data. Is the statement: Based on the boxplot, at least one student calculated a volume between the median and the actual volume correct? Justify the answer. 129

140 Mathematics Free Response Questions Pre-Calculus 2009 Free Response Question - Calculator Allowed The shaded figure below is a sector of a circle whose radius is 8 cm. A cone can be constructed from the sector by connecting the edges AO and CO. Arc, whose length is x cm, becomes the circumference of the base of the cone. The radius of the circle, which is 8 cm, forms the slant height of the cone. (a) If the arc length, x, of the shaded sector is 12 cm, what is the radius of the base of the cone? What is the height of the cone? What is the volume of the cone? (The volume V of a cone with radius r and height h is.) 130

141 Mathematics Free Response Questions (b) If the arc length of the shaded sector is x cm, what is the radius of the base of the cone in terms of x? What is the height of the cone in terms of x? What is the volume of the cone in terms of x? (c) What is the domain for the arc length, x? (d) What is the value of x when the volume of the cone is maximized? Use your graphing calculator to determine the answer. Note: When dividing by 2 on the calculator, parentheses must be placed around the. (e) What is the maximum volume of the cone? 131

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145 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Standards for Mathematical Practice MP.1 - Make sense of problems and persevere in solving them. Mathematically proficient students: start by explaining to themselves the meaning of a problem and looking for entry points to its solution. analyze givens, constraints, relationships, and goals. make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need, explain correspondences between equations, verbal descriptions, tables, and graphs, draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might: rely on using concrete objects or pictures to help conceptualize and solve a problem. check their answers to problems using a different method, and they continually ask themselves, Does this make sense? understand the approaches of others to solving complex problems and identify correspondences between different approaches. In assessments, the question: is designed to take a typical student a long time to solve. leads to a more difficult problem. requires a large number of routine and fairly easy steps. contains several givens. the statement of the problem itself is designed not to allow for jumping in and working the problem immediately. posed using abstract statements that must be parsed carefully before they make sense. require students to construct their own solution pathway rather than to follow a provided one. may be unscaffolded so that a multi-step strategy must be autonomously devised by the student. involve ideas that are currently at the forefront of the student s developing mathematical knowledge in a word problem. A3

146 Mathematics Standards for Mathematical Practice MP.2 - Reason abstractly and quantitatively. Mathematically proficient students: make sense of quantities and their relationships in problem situations. bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. use quantitative reasoning that entails habits of creating a coherent representation of the problem at hand: considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. In assessment, the question is designed to: be contextual so that the student can gain insight into the problem by relating the algebraic form of an answer or intermediate step to the given context. require the use symbolic calculations to generalize a situation and draw conclusions from those calculations. A4

147 Mathematics Standards for Mathematical Practice MP.3 - Construct viable arguments and critique the reasoning of others. Mathematically proficient students: understand and use stated assumptions, definitions, and previously established results in constructing arguments. make conjectures and build a logical progression of statements to explore the truth of their conjectures. analyze situations by breaking them into cases, and can recognize and use counterexamples. justify their conclusions, communicate them to others, and respond to the arguments of others. reason inductively about data, making plausible arguments that take into account the context from which the data arose. compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. In assessments, require students to: base explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student). construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. determine conditions under which an argument does and does not apply. distinguish correct explanations/reasoning from that which is flawed, and if there is a flaw in the argument explain what it is. provide informal justifications. use of diagrams, words, and/or equations to solve. reason about key grade-level mathematics. apply rigorous deductive proof based on clearly stated axioms. state logical assumptions being used. test propositions or conjectures with specific examples. apply a series of logical and well-motivated steps with precise language and terms. A5

148 Mathematics Standards for Mathematical Practice MP.4 - Model with mathematics. Mathematically proficient students: apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. analyze those relationships mathematically to draw conclusions. routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. In assessments, require students to: apply a known technique from pure mathematics to a real-world situation in which the technique yields valuable results even though it is not obviously applicable in a strict mathematical sense. execute some or all of the modeling cycle: formulate, compute, interpret, validate, and report. select from a data source, analyze the data and draw reasonable conclusions from it, often resulting in an evaluation or recommendation. use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity. make assumptions and simplifications. select from the data at hand or estimate data that are missing. use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity. A6

149 Mathematics Standards for Mathematical Practice MP.5 - Use appropriate tools strategically. Mathematically proficient students: consider the available tools when solving a mathematical problem (these tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software). are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. High school students: analyze graphs of functions and solutions generated using a graphing calculator. detect possible errors by strategically using estimation and other mathematical knowledge. when making mathematical models, know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. use technological tools to explore and deepen their understanding of concepts. In assessments, questions involve making the coordinate plane essential for solving the problem, yet no direction is given to the student to use coordinates. creating circumstances for poor use or misuse of tools. posing questions that are fairly easy to solve or to answer correctly if a diagram is drawn first, but very hard to solve or to answer correctly if a diagram is not drawn, yet no direction is given to draw a diagram. using formulas or conversions where there is no prompting to use them. data sets of numbers. using a calculator to test conjectures with many specific cases. substituting messy numerical values into a complicated expression and find the numerical result. A7

150 Mathematics Standards for Mathematical Practice MP.6 - Attend to precision. Mathematically proficient students: try to communicate precisely to others. try to use clear definitions in discussion with others and in their own reasoning. state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem. calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school, students have learned to examine claims and make explicit use of definitions. In assessments, require students to: use reasoned solving of equations, such as those in which extraneous solutions are likely to be found and must be discarded. solve algebraic word problems in which success depends on carefully defining variables. present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately. A8

151 Mathematics Standards for Mathematical Practice MP.7 - Look for and make use of structure. Mathematically proficient students: look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well-remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. step back for an overview and shift perspective. see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. In assessments, questions: can be solved by analyzing parts of figures in relation to one another. can be solved by introducing auxiliary lines into a figure. reward seeing structure in an algebraic expression and using the structure to rewrite it for a purpose. reward or require deferring calculation steps until one sees the overall structure. assess how aware students are of how concepts link together and why mathematical procedures work in the way that they do. A9

152 Mathematics Standards for Mathematical Practice MP.8 - Look for and express regularity in repeated reasoning. Mathematically proficient students: notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. maintain oversight of the process, while attending to the details. continually evaluate the reasonableness of their intermediate results. In assessments, questions require: repeating calculations to lead to the articulation of a conjecture. working repetitively with numerical examples leading without prompting to the writing of equations or functions that describe modeling situations. recognizing that tedious and repetitive calculation can be made shorter by observing regularity in the repeated steps. answers like multiplying by any number and then dividing by the same number gets you back to where you started. using recursive definitions of functions. using patterns to shed light on the addition table, the times table, the properties of operations, the relationship between addition and subtraction or multiplication and division, and the place value system. A10

153 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Additional Graphs and Materials Cone Exploration and Optimization F E D C G B H I Q A P J O K L M N A11

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155 NATIONAL MATH + SCIENCE INITIATIVE Mathematics NMSI Lessons New Ideas Technology Tips Areas & Volumes Teaching Strategies Graphical Organizer A13

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168 MODULE 1 NMSI s model lessons and instructional resources can be integrated into any existing curriculum to raise the level of instructional rigor for all students. MATHEMATICS Areas & Volumes Visit us on the web at nms.org