Tubes are Fun. By: Douglas A. Ruby Date: 6/9/2003 Class: Geometry or Trigonometry Grades: 9-12 INSTRUCTIONAL OBJECTIVES:

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1 Tubes are Fun B: Douglas A. Rub Date: 6/9/2003 Class: Geometr or Trigonometr Grades: 9-2 INSTRUCTIONAL OBJECTIVES: Using a view tube students will conduct an eperiment involving variation of the viewing distance from a surface compared with the height of the viewable area. Students will analze collected data to discern patterns. At the end of this activit students will be able to:. Devise an eperimental technique that produces reproducible results. 2. Propose a conjecture that models the data collected. 3. Suggest a geometric model that would eplain the observed phenomenon. 4. Analze the geometric model to derive an analtic equation for the observed phenomenon. 5. Use one of a variet of techniques (plotting, graphing, tables, etc.) to test the fit of the analtic model with the observed data. Relevant Massachusetts Curriculum Framework 8.G.2 Classif figures in terms of congruence and similarit, and appl these relationships to the solution of problems. 8.G.3 Demonstrate an understanding of the relationships of angles formed b intersecting lines, including parallel lines cut b a transversal. 8.G.4 Use a straightedge, compass, or other tools to formulate and test conjectures, and to draw geometric figures. 8.M. Given the formulas, convert from one sstem of measurement to another. Use technolog as appropriate. 8.M.4 Use ratio and proportion (including scale factors) in the solution of problems, including problems involving similar plane figures and indirect measurement. 8.P. Etend, represent, analze, and generalize a variet of patterns with tables, graphs, words, and, when possible, smbolic epressions. Include arithmetic and geometric progressions, e.g., compounding. 8.P.2 Create and use smbolic epressions and relate them to verbal, tabular, and graphical representations. 8.P.3 Identif the slope of a line as a measure of its steepness and as a constant rate of change from its table of values, equation, or graph. Appl the concept of slope to the solution of problems. 8.P.4 Set up and solve linear equations and inequalities with one or two variables, using algebraic methods, models, and/or graphs. 8.P.5 Use linear equations to model and analze problems involving proportional relationships. Use technolog as appropriate. 8.P.6 Use tables and graphs to represent and compare linear growth patterns. In particular, compare rates of change and - and -intercepts of different linear patterns. Page

2 Tubes are Fun Mr. Rub MENTAL MATH (5 Minutes) Prior Knowledge Inventor:. Modeling of linear and quadratic equations from data sets 2. Geometric properties of triangles 3. Properties of congruence of triangles 4. Pthagorean theorem 5. Properties of proportionalit 6. (Optional: Trigonometric functions including tangent) Mental Math Eercises: Solve the following: ft.3 in. ft.7 in (solve for ) Page 2

3 Tubes are Fun Mr. Rub CLASS ACTIVITIES Note: This lesson is intended to be given over a several da period (at least 2-3 das). Homework will consist of working on and completing the analsis of the data suggested in the attached worksheet. ) Materials (per team) To perform the data collection portion of the eperiment, we need: a) Two ardsticks b) One View tube c) A piece of paper to record measurements d) A pencil to write measurements e) A brain (or two, or three) to think with 2) Team Composition There are three people per team. Each team is divided into: a) A Viewer Looks through view tube and indicates the field of view seen against the target ardstick. b) A Measurer Measures the distance between viewer and the target ardstick. Also assists viewer in sizing the field of view. c) A Recorder Record resultant observations in / pairs. Works with Viewer and Measurer to analze patterns in data. 3) Objectives: a) Objective: Understand the mathematical relationship between distance and field of view when looking through a view tube b analzing observed data b) Objective: Tr to develop an analtical model that describes relationship between field of view and distance from viewing plane when looking through a view tube. Page 3

4 Tubes are Fun Mr. Rub 4) Procedure a) Measure field of view when looking at measuring device through view tube b) Measure distance of ee from measuring device c) Do this for at least 5 different distances and record results X Y d) Hints: i) Viewer Make sure the end of tube is round and the same shape ever time ii) Measurer Use a pen or pencil to help point out and record field of view iii) Recorder create a table of values for data using attached worksheet iv) All Tr to be accurate to /6 Page 4

5 Tubes are Fun Mr. Rub 5) Data Analsis Using the data ou have collected in our eperiment, perform the following data analsis: a) Analze the relationship between the independent variable (distance of viewer s ee from the viewing target) and dependent variable (field of view) b) What kind of technique will ou use to do the analsis? c) What kind of mathematical relationship does this data suggest? d) Can ou show a graph, plot, chart, or other technique that illustrates our conjecture? e) How well does our model fit our conjecture? f) Are there other factors that influence the measurements? 6) Analtic Model a) Show a geometric diagram that accounts for all of the factors affecting the outcome of our measurements. i) Your diagram should account for and ii) Are there other variables that affect the field of view and distance? b) Can ou create an equation from our analtic model that predicts given an? i) Show our equation and how it relates to the geometric model. ii) Test this analtic model to see if our data support it. iii) Does our model work for an size of tube at an distance from the viewing target? 7) Conclusions i) What can ou conclude about the relationship between the viewing tube and the field of view at a given distance? What kind of relationship is this? Did ou graph this? ii) Wh did our actual measurements differ(match) the predictions from our analtic model? Account for errors in measurement as much as ou can. Page 5

6 Tubes are Fun Mr. Rub Optional Problem Note: this would be part 2 for a Trigonometr class What would happen if instead of varing the distance of the viewer from the viewing target, we kept the distance constant, and changed the angle of view? Can we develop an analtic model that would predict the observed measurements? What kind of model or function do we see from the data? (See diagram below) Y Reference: Keiser, J. (2003) Variations on a view tube; Mathematics Teacher, March 2003, Vol. 96, #3, p70-76, National Council of Teachers of Math Page 6

7 Tubes are Fun Mr. Rub Grading Rubric Team Members: Element Points Possible Assessment Points. Students complete measurements and record a 20 minimum of 5 data points (more is desirable) 2. Measurements are precise and careful 0 3. Data set analzed for underling pattern using table, 20 chart, or graph 4. Mathematical model is proposed based on analsis of 20 data from step 3. Needs variable descriptions and units of measure 5. Analtic model that takes into account all relevant 0 variables is proposed. Must include diagram, variable descriptions, and relevant measurements. 6. Analtic model is compared with measured data using 20 chart, graph or other means. Differences are accounted for. 7. Supporting evidence is provided in the concluding 20 paragraph to show the measurement and math used to reach the conclusion. 8. Results, assumptions, estimations, and corrections are 0 presented in a precise and well- organized manner. 9. Calculations are accurate. Work is shown Mechanics of English are good. Writing is neat and presentable. 20 Total Possible Points = 60 Team Score: Page 7

8 Tubes are Fun Mr. Rub Data Collection Worksheet Team Members: X VALUE Y VALUE X Y Equation from Measured Data Analtic Model using Geometric Diagram Note: Show view tube, viewer, and viewing target in diagram. Label all angles, vertices, variables, etc. Account for, and an other relevant variable factors. Equation derived from Analtic Model Page 8

9 Tubes are Fun Mr. Rub Analtic Model Worksheet Team Members: PREDICTED X PREDICTED Y ACTUAL X ACTUAL Y Conclusions: Page 9

10 Tubes are Fun Mr. Rub Data Collection Worksheet Team Members: Solution Ke: X VALUE Y VALUE X Y In the above table, actual measurements will var. Students should have observed data from at least 5 measurements, accurate to /6. Of course, these need to be converted to improper fractions so that the and can be calculated. While I do not epect that the will be constant (i.e that each of the measurements were taken at equal distances apart), I would epect that if the data were plotted on the Cartesian plain it would should be nearl linear. What we want the students to look at is ultimatel the ratio of :. This ratio should be nearl constant. Equation from Measured Data If measurements were taken with some degree of accurac, particularl with respect to measuring as the distance from the viewer s ee, not from the front of the view tube, then the ratio of / should be nearl constant. This would impl: f() = = ( / )() Page 0

11 Tubes are Fun Mr. Rub Analtic Model using Geometric Diagram Note: Show view tube, viewer, and viewing target in diagram. Label all angles, vertices, variables, etc. Account for, and an other relevant variable factors. D w F A l B I G C E A = point of viewer s ee = distance in inches from the viewers ee to the target (length of AI) = height in inches of the viewed target area (length of DE) l = length of the view tube in inches (length of FB and GC) w = width of the inner diameter of the view tube in inches (length of BC and FG) Equation derived from Analtic Mode D Y = 2.84 in. F l = 2.73 in. B A m = 0.73 in. X = in. I G C l m = m CAB = 5.6 m DAE = 5.6 X Y = E In this eample created using Geometer s Sketchpad, we see that CAB = DAE (notice the measurements) and that ABC and ADE are similar isosceles triangles formed b the wa the viewer looks at the target. Segment AI (whose length is ) is the median of these two trianbgles. Page

12 Tubes are Fun Mr. Rub Because the triangles are isosceles, it is also a perpendicular bisector of segments BC and DE. Segment DE s length is. Since the two triangles are isocoles and share a common verte angle bisector, we can state that not onl are the sides of the triangles in proportion, but their heights (as measured b l and ) are also in proportion. Thus, we can sa that: l w This is shown above in the two rations whose value is. So for an arbitrar distance from the viewing area, the field of view will be such that: ( ) Notice also, if we were measuring from the front of the view tube instead of from the viewer s ee, then the appropriate proportion would be: l w l Since in our eample the l/w ratio would remain at and l=2.73, we would rewrite our equation as: ( ).7264" Page 2

13 Tubes are Fun Mr. Rub Analtic Model Worksheet Team Members: PREDICTED X PREDICTED Y ACTUAL X ACTUAL Y Conclusions: Here we would epect to see an analsis of the predicted model compared with the actual measured data. The predicted Y should be based completel on the modeled value for. This should be compared with the Actual X/Y measurements taken in the eperiment. Differences should be noted and accounted for. Eamples of error in measurement might be:. The actual aperture (end) of the view tube is not eactl round 2. The abilit to measure the actual distance accuratel is limited b the measuring instrument. 3. The view tube ma not be absolutel perpendicular to the viewing target in all aes, thus making our assumption about the perpendicular bisector/median above somewhat inaccurate. 4. Scaling inaccuracies introduced b the lack of precision (/6 to /8 ) in the measuring instrument (ard stick). These will be most pronounced in the measurements of l and w with shorter viewing tubes or viewing tubes that have a smaller diameter. We should se a graph (here or elsewhere) that compares the theoretical model with the observed data so that we can see how well the model fits the data. Page 3

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