Using Regression Analysis to Develop Algebraic Understanding
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The document discusses using regression analysis to develop algebraic understanding. It describes how regression analysis can be used to identify and describe patterns in data using functions, flexibly represent functions to model ideas and solve problems, and make connections to develop algebraic reasoning. Examples are provided of using regression analysis with different data sets to determine relationships and ask questions.
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Using Regression Analysis to Develop Algebraic Understanding
- 1. Using Regression Analysis to Develop Algebraic Understanding KCTM Fall 2009 Conference Roland O’Daniel
- 2. Standards for the Presentation Participants will understand how to create learning experiences that: • Make connections and develop a framework for algebraic reasoning • Identify and describe patterns in data using functions that approximate the data. • Flexibly move between multiple representations of functions to model mathematical ideas, solve problems, and communicate understandings. The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 3. Where do you see algebra outside of your classroom? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 4. The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 5. When looking for a data source, what are you looking for? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 6. GRE Score Concentrations in 28 Fields of PhD Study The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 7. Cost/ in US $ month Electricity Gas The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 8. U.S. Origin Patent Data 100 90 Patents in Thousands 80 70 60 50 40 30 0 5 10 15 20 25 30 Years Since 1980 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 9. Height of teachers in school 210 200 190 y = 2.5687x - 1.7889 Height in cms 180 170 160 150 60 65 70 75 80 Height in inches The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 10. Finding a Source of Data • Gather your own! – Measure/CBR/Go Motion • Swivel – http://www.swivel.com/graphs • The Numbers – http://www.the-numbers.com/charts/thisweek.php • ZIPskinny – http://zipskinny.com/ The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 11. Example 1 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 12. Determining a Relationship • Choose the tool you wish to use – TI graphing calculator – Excel spreadsheet • Create a representation of the data set • What are the kinds of questions that we can ask students regarding the representation of data? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 13. Titanic W.B.O. Gross 12/97- 3/98 Date Weekend B.O. Gross 12/19/1997 1 $28,638,131 12/26/1997 2 $35,455,673 1/2/1998 3 $33,315,278 1/9/1998 4 $28,716,310 1/16/1998 5 $36,014,544 1/23/1998 6 $25,238,720 1/30/1998 7 $25,907,172 2/6/1998 8 $23,027,838 2/13/1998 9 $32,876,424 2/20/1998 10 $21,036,343 2/27/1998 11 $19,633,056 3/6/1998 12 $17,605,849 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 14. Titanic Weekend Box Office Gross December ‘97 thru March ‘98 What questions do you have about the graphs? What information, is necessary for the graphs to be more understandable? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 15. Researching a Data Set: Titanic WBO Gross 12/97-3/98 $40 Why do these points behave differently? $35 1/16/98 $30 2/13/98 Millions 12/19/97 $25 $20 y = -1E+06x + 4E+07 R² = 0.5483 $15 1 3 5 7 9 11 13 15 17 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 16. Creating a Regression Analysis • Identify regression equation • Display trend line and data set • Interpolate/Extrapolate – Trace function – Point of intersection • What is interesting to you? What is connects to what you have already studied/taught? • What questions can we ask? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 17. Kinds of Tasks/Questions to Ask • Describe what the values in the regression equation mean in terms of this problem. • What does ( , ) mean in terms of this data? • What would happen to the model if the data were changed by ….? • Compare trend lines… The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 18. What did we find? How can we use it? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 19. Example 2 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 20. y= -3,107,000 x + 16,063,000 r2 = 0.9442 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 21. $20,000,000 $18,000,000 y = 19,160,000∙0.646x $16,000,000 $14,000,000 y = 19,160,000∙e-0.436x $12,000,000 R² = 0.9837 $10,000,000 Gross $8,000,000 Predicted Gross $6,000,000 Expon. (Gross) $4,000,000 $2,000,000 $0 0 1 2 3 4 5 6 7 Weekend Since March 20, 2009 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 22. Another Example of a System Olympic Men 400 Meter Free Style Swimming Winning Times Since 1924 y1 = -1.3x + 305 y2 = -0.31x + 248 What kind of questions could your students ask about this system? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 23. Men 310 300 290 280 270 260 250 240 230 220 1920 1940 1960 1980 2000 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 24. Questions? Next Steps? Link to Presentation Materials: http://ctlonline.pbworks.com/ Regression-Analysis The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 25. Photo Acknowledgements Lush ...-Wink-... (taking a break) Traffic Jam ...-Wink-... (taking a break) Hey, I'm all Bundled up ...-Wink-... (taking a break) Don’t Bother Me ...-Wink-... (taking a break) Todavía hay esperanza Piulet (Daniel) Brooklyn Bridge Nfalsey motion study experiment Monkeyc.net stopped in motion JarkkoS Ladder of Knowledge Degreezero2000 Steps to Knowledge Sanjibm The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 26. What algebra do you see in this graph? The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 27. Number of Patents Granted in US by Origin Year Year from 1980 U.S. Origin Foreign Origin 2008 28 92000 93244 2005 25 82586 75155 2000 20 97011 79072 1995 15 64510 49445 1990 10 52977 46243 1985 5 43393 33880 1980 0 40764 25455 Full Data Set from Swivel.com The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 28. Notes The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 29. Titanic Data w/ Trend Line $40.00 $35.00 $30.00 $25.00 $20.00 $15.00 y = -1.2839x + 35.634 $10.00 r² = 0.5483 0 2 4 6 8 10 12 14 16 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
- 30. Patent Data with Trend Line 100 90 Patents in Thousands 80 70 60 U.S. Origin U.S. Origin 50 40 y = 2069.x + 37148 30 R² = 0.857 -10 0 10 20 30 40 Years Since 1980 The Collaborative for Teaching and Learning Regression Analysis Roland O’Daniel September 2009
Editor's Notes
- Entry board: What roles does regression analysis play in your classroom?What is challenging about teaching regression analysis?As teachers enter, they capture their thoughts about these two questions on post its and add them to the chart paper
- Standards of the day.Without have internet access download several data sets, their graphs, and distribute to tables. Make some version of this data available via links with TIs, make directions for linking/sharing data via TI link cables
- This really is about the nature of algebra being a branch of mathematics concerning the study of structure, relation, and quantity. Data analysis is one way we understand relationships in algebra more effectively.The idea of creating opportunities to “see” (literally) algebra and fostering mathematical creativity in students. 1) Discuss the first question, whole group, quickly2) At tables look through the pictures to identify where they can see algebra outside of their classroom- Goal of this discussion is to:Help students identify variables & relationshipsIdentify/discuss “relationships” between variables
- What are the key characteristics when representing data?Want to create an interaction here that will have teachers brainstorming important information or information needed to understand a data setAlso, need to think about what I want participants doing during this activity:- Does it make sense that the data be represented continuously? Discussing independent/dependent data Discussing causal versus correlational relationships and no relationship Different characteristics of functions and does the data display those characterisitics and can you think about extending the data to identify key characteristics (i.e. a negative linear relationship will cross the x- axis, does it make sense if the trend continues that it would cross the x- axis? Or would an exponential decay function better represent the data) Domain/Range values Labeling of axis Manipulation of variables for representation purposes (i.e. 3/23/08 becomes 0 as a starting point, $34,456,009 becomes 34.5 million)
- A totally random sampling of data, there does not seem to be a relationship between quantitative and verbal scores in PhD studies from this data
- Does this data display any mathematical relationship?
- In the late 90s, the US patent office went online creating a spike in patent submissions/
- Data allows a representation of an algebraic relationship. It is what allows us to identify/understand/visualize the relationship, understand characteristics of the relationship, show predicted outcomes based on the relationship, etc.What’s the difference between a formula and an algebraic relationship? (the purpose of this question is to open the conversation about gathering your own data to test/represent formulary relationships, (i.e. height in inches/height in cms, measurement error, build in error (make a measurement that intentionally includes ¾ of an inch of error))Activity steps:(with internet connectiviy)Look through the siteFind a data set that has some characteristic that you want to shareCopy and paste the URL of your dataset into the chat window(without internet connectivity) Don’t explore the sites, but discuss the question aboveThis is a discussion of what we find in our exploration of Swivel and The Numbers.comFocus on the key characteristics of the graphs, how we can manipulate those graphs, how we can quantify those relationships, how we can generalize based on those observationsHaven’t discussed but want to pay attention to:Independent vs Dependent (the whole idea of causal relationship)Discussion of Domain and range (especially in labeling those and how we can label them differently, how we can move data from a 1st quadrant relationship to a second/third quadrant relationship based on some subjective description of 0.
- Guide participants through the first activity, creating a scatter plot of data.Focus on the fact that actually creating the scatter plot is not as important as being able to use the scatter plot, which is why we do the question:What questions can we ask about data? Before we do this activity.
- This is the actual activity in which teachers will be creating a scatter plot of data using the TI 84 or Excel Spreadsheet capabilitiesImportant component of this next section is the
- Create the scatter plot of data, Show/explain how to manipulate the window, add grids, Develop/determine questions that can be asked.
- Second aspect of the regression activity, described above!
- Gets at asking the students different questions, but need to scaffold what this looks like, use the modality routinely to make sure students learn the process as well as the content. Once a student understands the process of what a single/simple linear relationship is then it becomes a tool that they can use to explore concepts for the rest of their mathematical career. C/d relationship in geometry to determine pi experimentallyMore advanced relationships (next slide)
- This is the place where participants will create, capture the regression they have already doneExtend the analysis of the data Interpolate/Extrapolate information Analyze the information from the regression equation
- Simple relationship but allow students to explore this data set before they know what an exponential function is, why?Allowing students to extend the regression equation and seeing that the linear model moves into negative money is a great discussion topic. Does a linear relationship really model what we are seeing? Does it make sense when we extend the graph? What do we think is going to happen? What would that look like on the graph?
- Weekend Box Office Gross for "I Love You, Man"Revisiting the same data set later with an advanced group, why?Representing the exponential function using the natural base, and percent decay function, finding both of those functions using ExcelCreating meaning of the numbers as they relate to this problem, The regression created by Excel is a natural base exponential function, great opportunity for students to discuss the differences between the equations, the similarities, etc. i.e. the base of the non-natural function 0.64 means? (loss of 36% of the income each week, etc)Also, note that the red (maroon), and regression equations predict a week into the future, questions about what does the point (6, 1,400,000) mean in terms of this problem. Remember it’s the weekend box office gross for the movie I Love You Man. When you get new information in it is easy for students to re-evaluate the regression analysis, identify characteristics that changed, propose reasons for those changes, predict future changes and provide rationale, etc.
- I like this system because it doesn’t translate quite as nicely as most systems. It really is an example of a piecewise function, but is in many ways a system. I just throw it in here because it’s a great example of just letting students look at all kinds of data and figure out that something is happening, try and figure out why it’s happening, can we use that information to our advantage? Can we predict something about the future? How well can we predict? If we are looking at the Olympics in 2009 are there events that we think will have an impact on the Olympic records (i.e. I swimming the introduction of the Speedo suits was predicted to have a huge impact on the record book and it did)Not included in this presentation, but a fantastic example of real-world and complex regression analysis is the Usain Bolt blog post that I did on the Ning. I think that idea has great potential and would be glad to expand that thinking with anyone.
- Very complex graph, only for use with advanced students, but how do I think about what I see here and present it to the beginning algebra students. Will be sharing the document here so the different aspects of the data can be maximized and utilized.Make the data less complicatedOnly look at one set of dependent variablesEven compare US vs Foreign and graph, I haven’t done that so maybe we could do that together?
- Links to slide 24 for full data set
- What caused the data point from 2000 to differ from the other data points to radically? What would happen if we dropped it from our data set? Is that acceptable?