How Do We Get From Here to There? Why Measurement Matter Authors: Beth Dichter, Carolyn Gardner, Rachel Stavely-Hale, Antigoni Tzoumakas District: Northampton School District Unit or Course: Algebra 1 (focus on measurement) Lesson Duration: 255 minutes Prep Time: 90 minutes Topics or Keywords: measurement, unit analysis, conversion, linear equations, quadradic equations, cubic equations, polynomials, area and volume, perimeter Lesson Overview: Students learn about skills related to measurement of familiar shapes and solids through the use of videos from Discovery Education as well as java applets and other Web 2.0 tools. Aligned Standards This lesson is aligned to the follow standards: Mathematics – May 2004, Measurement Strand 10.M.1. Calculate perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles. Mathematics – May 2004, Measurement Strand 10.M.2. Given the formula, find the lateral area, surface area, and volume of prisms, pyramids, spheres, cylinders, and cones, e.g., find the volume of a sphere with a specified surface area. Mathematics – May 2004, Measurement Strand 1 10.M.3. Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume. Mathematics – May 2004, Measurement Strand 10.M.4. Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Mathematics – May 2004, Geometry Strand 10.G.6. Use the properties of special triangles (e.g., isosceles, equilateral, 30°-60°-90°, 45°-45°-90°) to solve problems. (10.G.6) 10.G.9 Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems. Mathematics – May 2004, Geometry Strand 10.G.5 Solve simple triangle problems using the triangle angle sum property, and/or the Pythagorean theorem. Mathematics – May 2004, Geometry Strand G.G.10. Apply the triangle inequality and other inequalities associated with triangles (e.g., the longest side is opposite the greatest angle) to prove theorems and solve problems. Mathematics – May 2004, Geometry Strand G.G.9. Define the sine, cosine and tangent of an acute angle. Apply to the solutions of problems. Mathematics – May 2004, Measurement Strand 12.M.2. use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. Technology Literacy Standards and Expectations – April 20008, Use of Computers and Applications Strand 1.19. Enter formulas and functions; use the auto-fill feature in a spreadsheet application. Technology Literacy Standards and Expectations – April 20008, Use of Computers and Applications Strand 1.20. Explain and use advanced formatting features of a spreadsheet application (e.g., reposition columns and rows, add and name worksheets). Technology Literacy Standards and Expectations – April 20008, Use of Computers and Applications Strand 1.21. Differentiate between formulas with absolute and relative cell references. 2 Technology Literacy Standards and Expectations – April 20008, Ethics and Safety Strand 2.1. Demonstrate compliance with the school's Acceptable Use Policy. Research, Problem Solving and Communication Strand Technology Literacy Standards and Expectations – April 20008, Problem Solving and Communication Strand 3.8. Use online communication tools to collaborate with peers, community members, and field experts as appropriate (e.g., bulletin boards, discussion forums, listservs, Web conferencing). Assessment Instructions Formative Assessments: Prior knowledge assessment during warm-up using clickers or Post-It histograms that will prove for misconceptions Questions and student discourse throughout lesson Assessment of understanding through practice with MCAS questions and real life application (in this lesson plan this was done through the Park Project Design) Summative Assessments: Class presentation of scale model projects (virtual or actual) When this class was taught the students rapidly became engaged in the creation of their garden plot. Two examples of plot plans are included at the end of the lesson plan as well as descriptions of how they came to their conclusions (one description is a part of the Sketch-Up diagram and the other was written in a word processor). In addition some of the students explored circles and cubes. They were asked to write down a hypothesis as to what would happen if a cube was increased, doubling or tripling in size. They then created a cube in Sketch-Up and resized it to determine if their hypothesis was correct. Other students did a similar activity exploring the radius of a circle. An example of each of these activities is included at the end of this lesson plan. Each group presented their findings to the class and discussion followed where students were highly engaged with sharing their findings and exploring the findings of the other students. 3 Products and Performances: Sketch-Up Measurement Project: 1) Design: Pretend you have a plot of land that is 30 ft by 30 ft and wish to make a mini park. Create two different basic designs of your mini park using the following guidelines. a. Use the entire plot area to design your park. b. Construct 4 ft wide walk paths around the perimeter. c. Ensure that visitors to the park have access 4 ft wide walk paths providing access to water fountain in the center by constructing two 4 ft wide walk path with seating areas at the end of the paths. d. Your design should allow for four separate grass and/or flower beds. e. Be creative. 2) Calculate: Use area formulas and show your process for calculating the following: a. The total area of the park. b. The total area of the grass and/or flower beds. c. The total area of the walk paths (include the water fountain and seating areas). d. Using the “Entity Info” window in SketchUp, how do your calculations compare to the SketchUp values? 3) Discuss: Base on your scale model project investigations, discuss how the calculated areas in part (2) will be altered if the land you had was twice the size (60 ft by 60 ft). Please note - samples of work from this assignment are included in this handout. Please look at page 9. MCAS Questions – these may be pulled from the DESE website if interested in using them. Key Concepts/Essential Questions Essential Question How does math help us understand our world? Unit Questions: In what way are measurements used in the real world that impact our daily lives? Content Questions: • How do you determine the perimeter, circumference and area of a: parallelogram, trapezoid, circle and triangle? 4 • • How do you find the lateral area, surface area, and volume of a prism, pyramid, sphere, cylinder and cone? How do you relate changes in the measurement of one attribute of an object to changes in another attribute, e.g. how does changing the radius or height of a cylinder affect the surface area or volume? Specific Skills Students will need to be introduced to Sketch-Up. Activities and Procedures: Opening Activity (Please note that TA = Teacher Activity and SA = Student Activity) TA – introduce the lesson plan, using white board configuration (date, objective, agenda for the day). SA – quick warm-up for assessment of prior knowledge using the clickers or a show of hands. Motivational Activity View Discovery video on area and surface area with students and then have students select an object of interest in the classroom to measure dimensions and find surface area. Follow up with discussion of their challenges in carrying out the activity, what they discovered during the activity, why accurate measurement matters, and how these sorts of measurements are relevant not just for achieving proficiency on the MCAS but also in a broad range of applications in the real world. Citation: Discovering Math: Measurement (Grades 9-12). Discovery Education. 2007. Discovery Education. 25 March 2009 http://streaming.discoveryeducation.com/ The specific segments we use are from the segment called Surface Area and Volume. The four videos include Introduction: Geometric Quantities and Fantastic Animation; Example 1: Surface Area – Boxes and Cans; Example 2: Volume – Pools and Cans; and Example 3: Surface Area and Volume – Cheese. The total time for the clips is 10 minutes 47 seconds and they do not all need to be shown in one sitting. TA - begin discussion of unit analysis, note that equations produced when finding surface area were quadratic and show second Discovery video to introduce what happens when you consider a third dimension, such as when calculating volume. SA - calculate volume of object measured in motivational activity. 5 TA - give students a selection of MCAS measurement questions. Be sure to select "real world" problems of the sort that students might realistically encounter in their lives, if possible. First let students attempt the problems without the MCAS reference sheet, then provide them with the sheets, then provide them with the enhanced sheets. To accommodate students who may have difficulty determining what are their own areas of weakness, the Special Education Teacher may facilitate individual conversations with students to gauge understanding of the various area and volume formulas. Individual or small-group conversations may help assess and engage students who otherwise may perform inadequately on MCAS questions due to test anxiety, reading disabilities, or lack of motivation. The Special Education Teacher will then report back to the General Education Algebra Teacher results of oral assessments. Follow up with discussion of what students perceive as their areas of weakness and how/if the reference sheet and enhanced sheet were of help. SA - assess for themselves during the activity what their knowledge is of the various area and volume formulas and how to apply them. TA - give students a copy of the MCAS scoring rubric and the response to one question, then review it in a whole-class discussion. SA - take turns scoring their own responses to the MCAS questions in group. The group could then hand the question off to a different group (or student) and ask them what they would do to improve the response. TA - show Java applets demonstrating how altering one dimension affects area/volume/surface area. The applets we are utilizing include: • http://illuminations.nctm.org/ActivityDetail.aspx?ID=176 • http://www.shodor.org/interactivate/activities/surfaceareaandvolume/?version=1.6.0_11&browser=Mozilla&vendor=Sun_Mic rosystems_Inc. • http://mste.uiuc.edu/pavel/java/cylinder/ SA - return to the object measured in the motivational activity and discuss how the surface area and volume would change when one dimension is increased or decreased. TA - show introductory Google SketchUp video and demonstrate the use of Google SketchUp. Google SketchUP and the video are both located at http://sketchup.google.com/product/gsu.html. Please note that there is a free version of Google SketchUp and that you must make sure that your IT Department is willing to download it and make it available for student use SA - explore Google SketchUp. TA - assign scale model project, to be completed in groups SA - create a scale model, virtual or actual, and present scale model project to class. Post at regular intervals to Moodle discussion boards to update on progress and discuss challenges. TA – assign Project Park Design to each student as a real life application SA – see attached document that explains this project in detail 6 Closure Student presentations may include actual scale models students have built by themselves in their groups or virtual scale models designed using Google SketchUp or other virtual 3D design software. Presentations should be followed by a question-and-answer period. Of particular interest will be discussion of the constraints on the students' projects. This will be followed up by a teacher led discussion using new measurement questions as a wrap-up for this lesson. This provides an additional opportunity to assess students understanding (and address misconceptions if necessary). It also provides a guide for future lessons. A teacher may ask: Do I need to repeat this? Is it appropriate to move on? A key component of this section is the presentation of the real life application design where students were able to self-assess and teacher was able to assess the students comprehension levels of scaling up the model. (For more information refer to attached plan.) Extensions and Modifications Pre-teaching Activity To accommodate learners that have working memory issues, visual processing disorders, or significant weaknesses in math, the Special Education Teacher will lead a pre-teaching activity. For example, H = height, which is the perpendicular distance from the base of a shape (triangle or rectangle) to its topmost point. By limiting pre-teaching instruction to the formulas for measuring triangles and rectangles, students will not be overwhelmed by the amount of information to which they are attending. Pre-teaching should occur one to three days before the Motivational Activity of the main lesson. Materials or Resources Needed for Lesson Computer Lab with High Speed Internet Access Software used: • Sketch-Up, an online software through Google that allows students to create 3D models. There is a free version for education. • GoogleDocs or Buzzword, online tools that allow students to write and share documents with others in their group as well as the teacher • Moodle, an open source content management system that provides forums for that allow students to share and discuss their work outside of the classroom Discovery Education videos, specifically Introduction: Geometric Quantities and Fantastic Animation; Example 1: Surface Area – Boxes and Cans; Example 2: Volume – Pools and Cans; and Example 3: Surface Area and Volume – Cheese. Available at http://streaming.discoveryeducation.com/ 7 Websites Used • • • • http://illuminations.nctm.org/ActivityDetail.aspx?ID=176 http://www.shodor.org/interactivate/activities/surfaceareaandvolume/?version=1.6.0_11&browser=Mozilla&vendor=Sun_Mic rosystems_Inc. http://mste.uiuc.edu/pavel/java/cylinder/ http://sketchup.google.com/product/gsu.html References: Lesson plan developed by: Beth Dichter, Technology Integration Specialist, [email protected], Northampton High School, Northampton Public Schools Carolyn Gardner, Math Teacher, [email protected], Northampton High School, Northampton Public Schools Rachel Stavely-Hale, Math Teacher, [email protected], Northampton High School, Northampton Public Schools Antigoni Tzoumakas, Special Education Teacher, [email protected], Northampton High School, Northampton Public Schools Students should make use of the MCAS reference sheet used at Northampton High School. Each school may use their own reference sheet. The Geometry textbook is also an excellent resource. The following pages include samples of student work using SketchUp. 8 Cube Investigation 9 Park Radius 10 Park Design #1 11 Park Design Calculation #1 12 Park Design #2 13 Park Design Calculation #2 14 CALCULATIONS A) Area of Park 30 ft ∗ 30 ft = 900 ft 2 B) Area of Grass and Flower beds The edge length of the park subtract three times the width of the walk path each 4 ft gives the edge length of one of the green square. Squaring the edge length gives the area of one green square and then multiplying by 4 gives the total area of the grass and flower beds. ⎛ (30 − 12) ft ⎞ ⎛ 18 ft ⎞ 2 2 ⎟ = (9 ft ) = 81 ft ⎟ =⎜ ⎜ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 Area of one grass area: 2 Total area of grass areas: 81 ft 2 ∗ 4 = 324 ft 2 C) Area of Walk Paths Area of Park – Area of Grass and Flower beds 900 ft 2 − 324 ft 2 = 576 ft 2 My calculated values correspond to Sketchup values. 15