1. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved Chapter Sixteen Collecting, Organizing, and Interpreting Data Test Grades Stem Leaf 4 2 5 6 7 7 3 8 8 2 4 6 9 1 4 9

2. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-2 Elementary school research model 1. Brainstorm for questions that children would like answered. 2. Choose one of the questions or problems. 3. Predict what the outcome will be. 4. Develop a plan to test the predicted outcome. 5. Carry out the plan. 6. Analyze the data. Is the hypothesis supported? 7. Look back. Answer the question. Should the information be shared? With whom? How could it be shared? (Bohan, Irby, & Vogel, 1995)

3. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-3 The level of graphing experience is not determined by the chosen topic but the way the data are presented. Concrete Stage Concrete – Pictorial Stage Pictorial – Abstract Stage Abstract Stage

4. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-4 Early graphing experiences should involve constructing graphs with concrete materials. www.icanteachmychild.com/2010/ 07/early-graphing-activity/

5. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-5 Involves using both pictorial representations of objects and concrete materials. Children are able to compare more than two events. A one-to-one correspondence between the object or picture and what is being graphed is still present. www.kindergarten- lessons.com/graphing- activities.html

6. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-6 Bar Graphs Pictures or objects are used to represent an item on the graph. www.athens.edu www.education.com

7. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-7 Children continue making bar graphs with pictures but are now beginning to use abstract objects to form the graph. Pictographs Used to read, interpret, and discuss results through questioning www.brainpopjr.com/ Trees planted on Earth Day http://www.livebinders.com/play/play?i d=69022

8. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-8 One-to-one correspondence of objects has been replaced with a one-to-many correspondence. Rectangular bars can replace colored squares in a bar graph Pictographs can be used with the objects representing many different things crayola.com www.eduplace.com

9. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-9 One-to-one correspondence of objects has been replaced with a one-to-many correspondence. Line graphs can be introduced Data points are joined with a line http://vickimartinez.c om/math_gc_unit3.ht ml#Line

10. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-10

11. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-11 Shows how the whole is broken into parts. Steps to help children construct a circle graph: 1. Collect the data and calculate the total. 2. Calculate the fractional part each data piece is of the total. 3. Express each fraction as a percent. 4. Calculate the number of degrees out of 360° that each fractional part represents. 5. Draw the graph, using the degrees from step 4 to determine the size of each sector. 6. Label the graph and each sector. www.ck12.org/ www.themathlab.com

12. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-12

13. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-13 Histograms – a graphical representation of the frequency with which scores occur

14. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-14 Line Plots – uses a symbol, such as an X, to indicate the frequency of each data point http://www.eduplace.com/

15. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-15 Stem-and-Leaf Plots Method of showing the frequency that data occurs Stem – part of the data that shows the beginning value Leaf – next digit to the right of the stem Complete a stem-and-leaf plot for the following list of grades on a recent test: 73, 42, 67, 78, 99, 84, 91, 82, 86, 94 I'll use the tens digits as the stem values and the ones digits as the leaves. 42, 67, 73, 78, 82, 84, 86, 91, 94, 99 Since I know where these data points came from ("a recent test"), I'll use a title. Then my plot looks like this: Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved Test Grades Stem Leaf 4 2 5 6 7 7 3 8 8 2 4 6 9 1 4 9

16. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-16 Five reasons to include statistics and probability in the school mathematics program: 1. They provide meaningful applications of mathematics at all levels. 2. They provide methods for dealing with uncertainty. 3. They give us some understanding of the statistical arguments, good and bad, with which we are continually bombarded. 4. They help consumers distinguish sound use of statistical procedures from unsound or deceptive uses. 5. They are inherently interesting, exciting, and motivating topics for most children. (Shulte & Smith, 1981, p. ix)

17. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-17 Descriptive statistics refers to the collection, organization, presentation, and interpretation of data. Frequency Measures of Central Tendency Mode – most frequently occurring value in a set of data Prices of 10 Ice Cream Specialties sold at Ben & Jerry’s Mode = $1.20 STEMLEAF 9 0 10 0 0 5 11 0 5 12 0 0 0 5 13

18. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-18 Measures of Central Tendency Median – the middle number in a set of data 100, 100, 99, 98, 92, 91, 91, 90, 88, 87, 87, 85, 85, 85,80, 79, 76,72, 67,66, 45 Median is 87 Mean – the average of the set of data American History Test Marco 90.....Adriane85 Linda 75.....Christy 99 Chantelle 88.....Jay 45 Ralph 68.....Marcus97 Chi Bo 92.....Donnie 85 Class mean is 82.4

19. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-19 A plot that shows the summary of data and provides a visual representation of the variability within the data. Requires five values Median = 80 Upper quartile = 95 Lower quartile = 66 Upper extreme = 100 Lower extreme = 45 Averages of Math class: 45, 55, 66, 66, 70, 80, 88, 90, 95, 98, 100 50 55 60 65 70 75 80 85 90 95 100 45 66 80 95 100

20. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-20 Probability is the analysis of the chance of something occurring. P(event) = Number of Desired Favorable Outcomes Number of Total Possible Outcomes

21. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-21 1. Probability is a measure of the likelihood of future events. 2. A sample space is a set of all possible outcomes and their associated probabilities of occurring. 3. Probability helps predict outcomes of simple experiments. 4. Students can make estimates of probability by using data from experiments. (Tarr, 2002)

22. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-22 Informal and nonnumeric probability activities should help children think about concepts such as: Certain – Class of 30 girls. Teacher picks 1 child. Certain to pick a girl. Impossible – Spinner has no purple. Impossible to spin and land on purple. Equally likely – Spinner has 4 colors. Equally likely to land on any color. More likely – More likely to land on red or blue Less likely – Less likely to land on yellow or green Activities should include Prediction Experimentation

23. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-23 Children should begin Using conventional probability terms Assigning probability values to some events Experimental probability - based on the outcomes of an experiment Theoretical probability – based on mathematical formula P(event) = Number of Desired Favorable Outcomes Number of Total Possible Outcomes Engaging in simple simulations

24. Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk ©2011 Pearson Education, Inc. All Rights Reserved 16-24 Probability activities at this level should include: An emphasis on determining theoretical values Experimental probability Advanced work with simulating events The Game of SKUNK http://illuminations.nctm.org/LessonDetail.aspx?id=L248