Statistics and Probability Investigate patterns of association in bivariate data.

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2 Course One 2005 pg. 57

3 Course One 2005 pg. 58

4 Scatter Diagram Rubric The core elements of performance required by this task are: discuss and understand a scatter plot of real data Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Point correctly plotted 1 2. Draws a line that best fits the data. Gives a correct statement such as: A line of best fit can be used to estimate a students score in one test if you know their score in the other. 3. Correctly completes the table such as: No. The lowest score on Test A (7) is greater than the lowest score on Test B (3). No. The student with the highest score on Test A (30) does not have the highest score on Test B (28). No. The biggest difference is more than 5 (accept 14). 1 5 Total Points Course One 2005 pg. 59

5 Looking at Student Work on Scatter Diagrams: Student A is able to graph the point and draw in a line of best fit. The student understands that best fit shows the relationship between the two variables and the trend of the graph. The student does not mention that the line can be used to make predictions of one score given the other score. The student is able to analyze the statements in the table, determine whether they are accurate, and make appropriate rewrites where needed using information from the scatterplot. Student A Course One 2005 pg. 60

6 Student A, continued Student B makes the mistake of confusing line of best fit with slope. The student makes the common error of confusing the range of scores with the highest score for test B. The student has trouble with the fourth statement, thinking that we are looking for the highest overall score rather than a comparison of two scores for the same person. Many students realized that the biggest difference in scores was larger than 5, but had difficulty quantifying that value. Course One 2005 pg. 61

7 Student B Student B, continued Course One 2005 pg. 62

8 Student C might be grasping at the idea of trend or correlation, but at the end describes line of best fit as a way of seeing if you did better on one test than the other. Again the student is unable to quantify the range or the exact value of the largest difference between scores on the two tests. Student C Course One 2005 pg. 63

9 Student C, continued Student D thinks the line of best fit determines who is above or below average. Many students, especially in middle school confuse line of best fit with average. Student D Student E shows a line of best fit with a negative slope. Notice that the student connects 1,16 with 19,5; drawing two points instead of one to represent 19,16 or the scores for the two tests. The student confuses the range with the highest value on the graph scale rather than a score represented on the graph. The student realizes the fourth statement is incorrect, but cannot rewrite it. It is unclear what the student is thinking about in his final statement. Maybe he notices that some dots are one apart. Course One 2005 pg. 64

10 Student E Course One 2005 pg. 65

11 Student F tries to connect all the dots. It is unclear what the student understands because, like a few students, all boxes are checked on the second page of the task. Student F Course One 2005 pg. 66

12 Student G recognizes which statements are correct, but cannot figure out how to rewrite the incorrect statements. Like Student G many students did not attempt to rewrite the statements, either because they didn t know how to make the appropriate calculation or were unsure about the meaning of the statement. Student G Teacher Notes: Course One 2005 pg. 67

13 Frequency Distribution for Task 4 Course 1 - Scatter Diagram Scatter Diagram Mean: 3.98 StdDev: 2.05 MARS Task 4 Raw Scores Score: Student Count % < = 6.6% 13.4% 25.5% 39.9% 55.7% 73.0% 89.4% 98.4% 100.0% % > = 100.0% 93.4% 86.6% 74.5% 60.1% 44.3% 27.0% 10.6% 1.6% The maximum score available for this task is 8 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Most students, about 93%, were able to see that the scatterplot showed a positive correlation between scores on the two tests. Many students, about 74%, were able to plot the point and recognize that statements 1 and 3 in the table were true. More than half the students were able to plot the point, recognize that statements 1 and 3 were correct, and either draw a line of best fit or rewrite statement 2 or 4. About 10% of the students could meet most of the demands of the task including plotting a point, drawing a line of best fit, comparing lowest scores on A and B, finding range for test B, comparing highest scores on A and B, and finding the biggest difference. Only about 2% of the students could explain what the line of best fit represents in terms of the context of the problem. 7% of the students scored no points on this task. Only 13% of those students attempted the task. Course One 2005 pg. 68

16 Knowing students ideas about average as the most common misinterpretation of tests scores, the teacher might plan a lesson on returning the papers to help students confront this misconception. The teacher might ask them to talk about what the average is for each test, what the average might mean for one particular student on the graph, say point (24,10) or (19,26). Where would you plot a point to represent each average? If you started to graph all the points, what would you notice about the plots of the averages? Now have students find the number of points and try to design a line dividing the points into equal groups above and below the line? How does this line look different? Which line is the best predictor of someone s score? Test out several cases. Then have the students examine and think about these same questions using some other scatter plots. Students should be exposed to several that have different correlations, positive, negative, no correlation, tight correlation, and loose correlation. Then students should be able to answer why this is a useful tool for researchers. Students should also be given situations that appear to have a correlation, but don t have a cause and effect relationship, where the results are just coincidental. It is in unpacking these types of discrepancies that the mathematics becomes engaging for students and the ideas start to take on meaning. Teacher Notes: Course One 2005 pg. 71

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