MINIATURE GOLF TARGET COMMON CORE STATE STANDARD(S) IN MATHEMATICS G.SRT.8


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1 This task was developed by high school and postsecondary mathematics and design/preconstruction educators, and validated by content experts in the Common Core State Standards in mathematics and the National Career Clusters Knowledge & Skills Statements. It was developed with the purpose of demonstrating how the Common Core and CTE Knowledge & Skills Statements can be integrated into classroom learning and to provide classroom teachers with a truly authentic task for either mathematics or CTE courses. TASK: MINIATURE GOLF TARGET COMMON CORE STATE STANDARD(S) IN MATHEMATICS: G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* 6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 7.NS.3 Solve realworld and mathematical problems involving the four operations with rational numbers. 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. 7.G.6 Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. TARGET STANDARDS FOR MATHEMATICAL PRACTICES MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. TARGET COMMON CORE STATE STANDARD(S) IN ELA/LITERACY: RST Cite specific textual evidence to support analysis of science and technical texts, attending to the precise details of explanations or descriptions. RST Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. RST Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. WHST Write arguments focused on disciplinespecific content. WHST Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. TARGET CAREER AND TECHNICAL EDUCATION (CTE) KNOWLEDGE & SKILLS STATEMENTS: ACC Use appropriate formulas to determine percentages /decimals. ACC Plan, organize, schedule and manage a project/job to optimize workflow and outcome. ACC Estimate resources/materials required for a specific project or problem. ACC Determine alternative solutions for a specific project/problem. 1
2 ACC Understand the context of the projects. ACC Interpret drawings used in project planning. ACPA Deliver a presentation that explains a concept of design or preconstruction. ACPA Identify client requirements. ACPA Build models using referenced drawings and sketches. ACPA Utilize computer technology when communicating concepts and designs. RECOMMENDED COURSE: Geometry; Integrated Math II; Applications in Design & PreConstruction ADDITIONAL INSTRUCTIONS: This task should be completed over an extended period of time, including some time for research of designs and general golf specifications and for writing the report. Some extensions might be completed with small groups or partners. * Modeling standards appear throughout the CCSS high school standards and are indicated by a star symbol (*). About the Common Core State Standards in Mathematics The Common Core State Standards (CCSS) for Mathematics are organized by grade level in grades K 8. At the high school level, the standards are organized by conceptual category (number and quantity, algebra, functions, geometry, and probability and statistics), showing the body of knowledge students should learn in each category to be college and career ready, and to be prepared to study more advanced mathematics. The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. About the Common Core State Standards in English Language Arts/Literacy The Common Core State Standards (CCSS) for ELA/Literacy are organized by grade level in grades K 8. At the high school level, the standards are organized by 910 and grade bands. Across K12 there are four major strands: Reading, Writing, Speaking and Listening, and Language. The CCSS also include Standards for Literacy in History/Social Studies, Science, and Technical Subjects, with contentspecific (Reading and Writing) literacy standards provided for grades 68, 910, and 1112, to demonstrate that literacy needs to be taught and nurtured across all subjects. About the Career Cluster Knowledge and Skill Statements As an organizing tool for curriculum design and instruction, Career Clusters provide the essential knowledge and skills for the 16 Career Clusters and their Career Pathways. It also functions as a useful guide in developing programs of study bridging secondary and postsecondary curriculum and for creating individual student plans of study for a complete range of career options. As such, it helps students discover their interests and their passions, and empowers them to choose the educational pathway that can lead to success in high school, college and career. Although not included in this template, all Clusters and Pathways have Foundational Academic Expectations and Essential Knowledge & Skills Statements, which, in some cases, overlap with the Common Core State Standards. Golf hole Cup Bank shot Bank angle, angle of approach Tee box KEY TERMS 2
3 MINIATURE GOLF The Task The local miniature golf course would like to increase their profits by increasing the difficulty and number of holes in the course. The owners believe players will like being able to play more holes for their money and then may spend more at the concession stand. In order to find designs for three new holes, they have decided to hold a design contest. Each design will be judged on the most creative use of materials, the least likelihood that someone will score a holeinone, and the best use of the available space. The contest requires documentation of your design, including a scale drawing of the design with dimensions, all of your calculations, and a report to the owners that explains your design. Tasks: 1. Sketch a miniature golf hole design including any obstacles or general items required. The designed hole must fit within a rectangle of no more than 128 square feet of area. Draw your design to scale, using appropriate tools, and include all measurements. Make sure your dimensions will allow for a player to comfortably approach the ball within the golf hole area. Identify the cup, the tee box, obstacles, and all angles of approach. For purposes of computing area, your design should be composed of twodimensional shapes for which you can find the area. 2. Show that your design will fit within a rectangle of dimensions that meet the 128 sq ft specification. Be sure to use as much of the area as possible, while still maintaining a creative design. 3. Calculate the total area of your golf hole design and the percentage of this total area you have used. Show all your work. (Remember: You will be judged partially on the best use of space.) 4. Research and describe the regulation size for a golf cup for consideration and placement in your design. 5. Identify and sketch two possible paths from tee to cup, with at least one path being a bank shot. Explain the possible approaches and show and calculate all angles associated with your bank shot(s). 6. Write a report to submit with your design to explain why yours should be selected. Include information regarding the creativity of your design, the level of difficulty, and the best use of available space. 3
4 MINIATURE GOLF Possible Solution(s) 1. The design: (Scale: one unit = 6 inches) 2. The total design fits inside a rectangle that is 11 x 11 6, for a total of square feet: 11 x 11 6 = 11 x 11.5 = square feet This is less than the maximum by 1.5 square feet. 3. Calculate the area: The total used area can be most easily calculated by subtracting the unused area from the total: Total rectangle area: square feet 4
5 The unused area includes three right triangles and one rectangle (shaded in this version of the design). The rectangle is 2 6 by 5, two triangles have legs of 3 and 5, and the third triangle has legs of 3 and 6. Total unused area: 2.5 x 5 + 2[0.5(3 x 5 )] + 0.5(3 x 6 ) = = 36.5 square feet of unused space = 90 square feet of used space = This is approximately 71% of the total space. 4. According to the USGA the regulation golf cup measures 4.25 inches in diameter. Since this design drawing has one unit equal to 6 inches, the cup will fit easily into one square on the grid. 5. There are two possible shots shown on this sample version of the design. In #1 (dark green line) the goal will be to hit the ball with the best possible speed to stop as close to the center of point 1 as possible. This plan is not a bank shot and may be accomplished in as few as two shots total. In #2 (blue line) the ball will bank off the far wall of the hole and enter the hole. This shot is daring and hopes for a holeinone. This can happen only if the bank angles are the same measure, ma = mb in this version of the design. ma can be found by using the right triangle formed with the distance from the bank point to the hole as the hypotenuse. Using the scale in the design of one unit = 6 inches, the triangle legs will measure 36 (six full units) and 93 (15.5 units). The base angle, angle a, will have a tangent of 93/36. So ma = or approximately 69. Since both a and b have measures of 69, the angle at the bank point (2) would be 180 2(69) = Reports will vary based on the individual designs. 5
6 MINIATURE GOLF Possible Extensions The extensions below represent potential ways in which mathematics and/or CTE teachers can build on the task above. All of the extensions are optional and can be used in the classroom, as homework assignments, and/or as longterm interdisciplinary projects. 1. Consider possible placement options for the cup and propose two alternative locations, providing three total cup sites. Discuss the implications that each cup position has on the level of difficulty and recommend one cup site. 2. Focus Group Study: Select the favorite three designs of the class and present them to a random sample of people to collect data on the preferred design. Compile, graph, analyze, and compare the data and summary statistics based demographics, such as gender, age, and golf experience. 3. Create a project plan to complete the hole. Include local pricing for materials and total cost. 4. Create a threedimensional sketch of the hole (e.g., using Google Sketchup). Create (and draw) components of the hole that reflect its theme that you have determined with art students from your school. 6
7 MINIATURE GOLF Appendix: Alignment Ratings The rating system used in the following charts is as follows: 3 EXCELLENT ALIGNMENT: The content/performance of the task is clearly consistent with the content/performance of the Common Core State Standard. 2 GOOD ALIGNMENT: The task is consistent with important elements of the content/performance of the CCSS statement, but part of the CCSS is not addressed. 1 WEAK ALIGNMENT: There is a partial alignment between the task and the CCSS, however important elements of the CCSS are not addressed in the task. N/A: For Mathematical Practices a content rating does not apply. In the charts C = Content Rating and P = Performance Rating COLOR KEY Black = Part of CCSS/K&S Statement aligned to task Gray = Part of CCSS/K&S Statement not aligned to task 7
8 MINIATURE GOLF TasktoMathematical Practice Alignment Recording Sheet Task Name Aligned CCSS Mathematical Practice Standards MP.1 Make sense of problems and persevere in solving them. C P Alignment Comments N/A 3 MP.2 Reason abstractly and quantitatively. N/A 3 MP.3 Construct viable arguments and critique the reasoning of others. N/A 2 MP.4 Model with mathematics. N/A 3 MP.5 Use appropriate tools strategically. N/A 3 MP.6 Attend to precision. N/A 3 For this task students analyze givens, constraints, relationships, and goals. They must make conjectures about the form and meaning of the solution and plan a solution pathway. They must check the reasonableness of their solution, continually asking themselves, Does this make sense? This task involves quantitative relationships. It requires that students make sense of quantities and their relationships in the problem situation. They must attend to the meaning of the quantities and pay attention to units as they represent the quantities and measures in the design, work, and their report. Students must present their design with a written report describing their findings. There is no requirement to critique the reasoning of others This task is an application from everyday life requiring that the student create a geometric representation (model) that can replace the situation described in the prompt. Students must identify important quantities in the practical situation and use them to answer the questions posed. Students are required to explain their decisions and assumptions. Students are directed to use appropriate tools in their design. They may use drafting tools, CAD programs, or other software. This task requires that students communicate precisely, organizing their information, as they perform their calculations and show their mathematical thinking. Task Comments (Strengths, weaknesses, possible improvements, effectiveness, etc) This is a multistage problem with real life applications and considerations. Students must identify measurements and lengths to make sure their design meets the specifications. They must use both abstract reasoning and quantitative calculations. 8
9 MINIATURE GOLF TasktoCommon Core State Standards Alignment Recording Sheet Task Name Aligned CCSS Content Standards C P Alignment Comments Task Comments G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* Students must use trigonometric ratios to find the angles associated with the bank shots. This task is a design problem that involves specifications and constraints. 6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities NS.3 Solve realworld and mathematical problems involving the four operations with rational numbers G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure As students form their design and calculate the area, they will need to use and convert units of length and area. Students must calculate the lengths and areas associated with their design. This task requires that students are able to create their design to scale. They are not required to reproduce a drawing at a different scale. Students use facts about supplementary and adjacent angles only. They must determine the measure of the angle between the two congruent bank angles using the sum of the three angles equals 180, which involves an equation. This task combines planning, design, calculating, comparing, and presentation with writing and using communication skills. 7.G.6 Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 2 3 This task focuses on 2dimensional design composed of polygons, for which the student must calculate area only. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. 3 3 The task requires that students find the percent their design uses of the total. * Modeling standards appear throughout the CCSS high school standards and are indicated by a star symbol (*). 9
10 MINIATURE GOLF TasktoNational Career Cluster Knowledge & Skills Statements Alignment Recording Sheet Task Name Aligned National Career Cluster Knowledge & Skills Statements C P Alignment Comments ACC Use appropriate formulas to determine percentages /decimals. 3 3 Task requires use of percentages and decimals. ACC Plan, organize, schedule and manage a project/job to optimize workflow and outcome. 1 1 Task requires design concepts not project management. ACC Estimate resources/materials required for a specific project or problem. 1 1 Task does not require pricing calculation. ACC Determine alternative solutions for a specific project/problem. 3 3 Task requires several solutions to determine best outcome for the design of the project. ACC Understand the context of the projects. 3 3 Task requires research for scale design. ACC Interpret drawings used in project planning. 1 3 ACPA Deliver a presentation that explains a concept of design or preconstruction. 3 2 ACPA Identify client requirements. 2 1 Task requires contest applicants to produce original designs for the contest. Task requires a written report that fully explains the design project. Task provides basic client requirements but requires students to act as contest applicants not contractors. ACPA Build models using referenced drawings and sketches. 3 2 Task requires drawings to scale. Models are not built. ACPA Utilize computer technology when communicating concepts and designs. 2 1 Task does not identify use of specific computer technology to be competitive in the competition. 10
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