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 i
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. ii
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 iii
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. iv
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/ Objectives Write the formula for the area of a rectangle using a variable for the missing length or width. 4th Grade Skills/ Objectives Isolate the variable for length or width in the formula for area of a rectangle. 5th Grade Skills/ Objectives Isolate the variable for length or width in the formulas for area and perimeter of a rectangle. 6th Grade Skills/ Objectives Solve literal equations (perimeter, area, and volume). 7th Grade Skills/ Objectives Solve literal equations (perimeter, area, and volume). Algebra 1 Skills/ Objectives Solve literal equations (perimeter, area, and volume). Geometry Skills/ Objectives Solve literal equations (perimeter, area, and volume). Algebra 2 Skills/ Objectives Solve literal equations (perimeter, area, and volume). Pre-Calculus Skills/Objectives Solve literal equations (perimeter, area, and volume). AP Calculus Skills/ Objectives Solve and use literal equations in real life and mathematical applications. T E A C H E R P A G E S v
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NATIONAL MATH + SCIENCE INITIATIVE Mathematics Cone Exploration and Optimization Answers 1. The circumference of the base of the cone decreases from cm to 0 cm. The slant height does not change. The slant height of the cone is the radius of the original circle (length Q). 2. a. The radius decreases from Q cm to 0 cm. b. The height increases from 0 cm to Q cm. c. The area of the base of the cone decreases from to 0 cm 2. d. The lateral surface area decreases from to 0 cm 2. 3. a. The radius increases from 0 cm to Q cm. b. The height decreases from Q cm to 0 cm. c. The area of the base of the cone increases from 0 cm 2 to d. The lateral surface area increases. 0 cm 2 to. 4. a. If a degenerate cone is considered, the lateral surface area is at its greatest when the height of the cone is 0. The radius of the cone is the same length as the radius of the original circle. The volume of the cone will be 0. b. If a degenerate cone is considered, the lateral surface area is at its smallest when the height of the cone is equal to Q, the radius of the original circle. The radius of the cone is 0. The volume of the cone will be 0. c. As the radius of the cone changes between its maximum of Q and its minimum of 0, the volume must increase and then decrease. T E A C H E R P A G E S 5. a. 0 cm h 10 cm if degenerate cone permitted or 0 cm < h < 10 cm if not permitted b. 0 cm r 10 cm if degenerate cone permitted or 0 cm < r < 10 cm if not permitted 6. Accept any answers. A common (but incorrect) conjecture is that the slant height forms a 45 degree angle with the base so that. Another common (but incorrect) conjecture is that. 7. 8. From a strictly algebraic perspective, the domain of this function is determined by the radicand. so and. In the context of the situation, this function represents the volume of a cone. This function is always non-negative so there are no restrictions to assure that the volume is greater than or equal to 0. Since r represents the radius of the cone, the domain must eliminate any negative values for r. The situational domain is then either 0 r 10 if we allow the radius and the volume to take on values of 0 or 0 < r < 10 if we want to hold r and V to positive values. vii
9. See graph: 10. V 450 400 350 300 250 200 150 100 50 r 1 2 3 4 5 6 7 8 9 10 T E A C H E R P A G E S 11. 12. 13. cm, cm. Comparisons with student conjectures will vary. 14. cm; cm. 15. 11.530 = θ (2 10 π ) = 66.061 360 16. Students compare their measurements to the dimensions in question 13. viii
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 1
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? 2
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 450 400 350 300 9. Graph the volume of the cone with respect to the radius of the cone. 250 200 150 100 50 r 0 1 2 3 4 5 6 7 8 9 10 3
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. 4
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