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* Ken Schmarge Spring, 2015 * Teacher of Math at Ramsey High School * Geometry, Calculus, Computer Science * Infuse technology to coursework to enhance student understanding * * Geometry Honors Course (10th grade) * Perimeter & area of a regular polygon * Develop formulas * Use a spreadsheet to implement formulas * Use graphing capabilities of the software to recognize a pattern * * 1. Apply trigonometry to a new problem 2. Apply knowledge of circles and triangles to a new problem 3. Solve a complex problem by breaking it into bite-sized pieces 4. Apply statistical analysis techniques to repeated calculations * CCSS.MATH.CONTENT.HSG.GMD.A.1 (Give an informal argument for the formulas for the circumference of a circle, area of a circle…) CCSS.MATH.CONTENT.HSG.SRT.C.6 (Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles) CCSS.MATH.CONTENT.HSG.SRT.C.8 (Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. * • • 2 days total • 1 day in the computer lab to complete excel work 1 day for students to complete pre-work *Pre-work The “look” of a regular polygon as the # of sides increase Introductory problem – find the area of an isosceles triangle *Excel piece Set up columns Enter formulas Populate the chart Build the graphs Percent error analysis Find the area of regular 3-gon, 4-gon, & 5-gon Develop needed formulas for general case * Data in rows & columns Column widths and text alignment customized * Color utilized to highlight perimeters and areas that are within a specified tolerance * * Formulas created by using cell references on same sheet – ex: central angle = 360 / n * Formulas created by using cell references across different sheets – ex: calculation results from “Perimeter and Area” sheet called in “Percent Error” analyis * Formulas created using Excel’s built=in functions – ex: PI() * Formula created by an absolute reference – ex: Circumference = 2*PI() and Area = PI() * Conditional formatting – ex: shading cells within a certain tolerance * * Fill Series feature utilized – ex: number of sides in the polygon auto-filled from 3-50 * Headers and print titles used – ex: both sheets have a title header and, due to the length of the sheet, print titles * * Graph of polygon perimeter * * Graph of polygon area * * Dropdown menu created for user to pick a tolerance * * Students will connect the concepts of trigonometry, area, statistics, and graphical analysis to demonstrate that the perimeter and area of regular polygon inscribed in a circle approach the circumference and area of the circle. * Students can pick a tolerance to visually identify how many sides are needed for the approximation to be “close enough” to the known answer – this is a building block of calculus (concept of limit introduced). * Students will develop a concrete association with the irrational number pi through table and graphical analysis. * Category 0 1-3 4-5 Pre-work Work is not done, or shows little to no effort Work is partially complete and contains mathematical and conceptual errors. Work is completed with little or no mathematical or conceptual errors Excel Sheet 1 Sheet is not completed Sheet is completed, but formulas either have errors or have not been entered. Sheet is completed with all formulas correct and properly expanded to fill all cells. Sheet has been appropriately named. Excel Sheet 2 Sheet is not completed Sheet is completed, but formulas either have errors or have not been entered. Sheet is completed with all formulas correct and properly expanded to fill all cells. Sheet has been appropriately named. Graphs Graphs are not completed Graphs are constructed, but have no title, axis labels, may have incorrect data selected, and may not be properly located Graphs are correctly constructed with titles, labels, and correctly located on their own page. Sheets have been appropriately named. * * Concrete Sequential learners will benefit from the programmed instructions and application of known topics to a new area * Concrete Random learners are curious and follow their instincts. They will benefit from the problem-solving aspect to this lesson and the ability to work independently through the guide. * * • • • • Target graphic: blog.7gees.com • Sousa, David. How the Brain Learns, 4th. Thousand Oaks: Corwin Press, 2011. 1-120. Kindle Edition. • Diagrams created in Geometer Sketchpad Common Core graphic: johnstonpublicschools.org Clock image: metro.co.uk Common Core Standards: www.corestandards.org/Math/Content/HSG/introduction/