Students must match a portion of a circle graph with data in a table. This multiple-choice question is a sample item used in the 2005 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. The test item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 OGT for Mathematics, Annotated Item 22. (author/sw)

Students are asked to interpret data shown in a box-and-whiskers plot. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. The test item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 2. (author/sw)

Students must determine the interval of greatest increase on a scatterplot. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. The test item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 34. (author/sw)

Students are asked to construct a circle graph using percentage data in a table. They have the option of using a calculator. This constructed-response question is a sample test item used in grade 8 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-8M9, No. 16. (sw)

Students are asked to interpret a bar graph and determine the approximate number of votes a candidate received. This multiple-choice question is a sample item used in the 2008 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OGT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2008 OGT for Mathematics, Annotated Item 18. (author/sw)

Students are asked to interpret the smallest value displayed on a stem-and-leaf plot. This multiple-choice question is a sample test item used in grade 8 in the 2003 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 2003-8M10, No. 8. (sw)

Students must interpret box-and-whiskers plots to find a difference of medians. This multiple-choice question is a sample item used in the 2005 Ohio Grade 8 Achievement Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 Ohio Grade 8 Achievement Test for Mathematics, Annotated Item 34. (author/sw)

Students must read a temperature versus time graph and find the difference in temperatures at two different times. This multiple-choice question is a sample test item used in grade 12 in the 1996 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1996-12M13, No. 1. (sw)

Students must interpret a pictograph in which each icon represents four students. They have the option of using a calculator. This multiple-choice question is a sample test item used in grade 12 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-12M9, No. 6. (sw)

Students write their interpretation of a bicycle trip based on a line graph of the speed over an 80-minute period of time. This constructed-response question is a sample test item used in grade 8 in the 2003 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 2003-8M10, No. 19. (sw)

Students are asked to estimate the number of radios represented by a 37° sector of a circle. They have the option of using a calculator. This multiple-choice question is a sample test item used in grades 8 and 12 in the 1992 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1992-8M12, No. 4. (sw)

Students are asked to solve a multi-step problem involving average hourly wages using data in a table. They have the option of using a calculator. This constructed-response question is a sample test item used in grade 12 in the 1996 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1996-12M12, No. 2. (sw)

Students are asked to determine the average of data given in a frequency bar graph. They have the option of using a calculator. This constructed-response question is a sample test item used in grade 12 in the 1992 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1992-12M7, No. 12. (sw)

Students are asked to explain why a pictograph is misleading. This constructed-response question is a sample test item used in grade 8 in the 1992 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1992-8M15, No. 17. (sw)

This lesson is most meaningful to students in states like Ohio, where turnpike tolls are proportional to distance. Students use a toll ticket or a state website like The Ohio Turnpike to collect data on distance traveled versus the toll collected, and they create a scatterplot using spreadsheet software or a graphing calculator. They find the line of best fit and discuss the interpretation. Activity sheets, discussion questions, suggestions for assessment, lesson extensions, and questions for teacher reflection are included. (sw)

In this lesson, students model linear data collected in a variety of settings that range from car repair to sports to medicine. Students can work alone or in small groups to construct scatterplots, interpret data points and trends, and investigate the line of best fit. Activity sheets and guiding questions are provided for four applied data contexts. (author/sw)

Students are asked to find the average birth month. Analysis of the data and examination of just what “average” means in the case of months leads to the mode being selected as the best way to find an "average" with categorical data. Students create appropriate graphs to illustrate what the mode is. Next, students are asked if this information is representative of the entire population, and why or why not. Questions about sampling arise in this portion of the lesson. Finally, students are asked to create a method to determine the average day of the week for birthdays. Students explore a question that engages them even as it leads to deeper understanding of basic statistical concepts. This mathematically rich problem was prepared by the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools developed by the Ohio Department of Education. (author/sw)

Students investigate the average earnings of a movie in a given week using data they collect from web or print resources. In this investigation, students use the measures of center and different graphical displays of the data to make an argument that each of the measures of center may be appropriate, providing an explanation for their choice of mean, median, or mode. They identify outliers and create box plots (both modified and non-modified), histograms, and stem-and-leaf plots. Students also analyze spread and distribution of the data as part of their investigation. To conclude the exploration, students prepare a poster and give a report to the class. As students analyze actual data about a topic that engages them, they apply mathematics to a real-world situation and can see the value of the statistical concepts they are studying. Evaluation rubrics are included. This mathematically rich problem was developed for the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools proposed by the Ohio Department of Education. (author/sw)

Using simulated data, students determine the probability that a customer calling to order a shirt gets a busy signal. They then estimate the amount of money the company might lose when customers get a busy signal. Activity sheets take students step by step through the basic problem. A second activity considers the situation if the company installs a second phone line. A 31-page background manual for the teacher explains the mathematical ideas behind the lesson and the numerous real-life applications of these ideas. Blackline masters and Internet extensions are included. (sw)

This site offers graphical images that portray data accumulated from a range of sources (historical events, spread of disease, distribution of resources, etc.). The purpose of the collection is to give examples of the "best and worst of statistical graphics." The author contrasts the differences between the best and worst by showing how some images communicate data clearly and truthfully, while others misrepresent, lie, or totally fail to "say something." Among the images are Florence Nightengale's coxcomb graphic of deaths among soldiers and a cartogram showing the 2004 U.S. election results, an unusual map of the country in which the size of each red or blue state corresponds to its population rather than its land area. Teachers looking for examples of innovative representations of data or examples of misrepresentation will find this resource helpful. (author/th)

Students must determine the minimum test score that will yield a given mean. This multiple-choice question is a sample item used in the 2005 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 OGT for Mathematics, Annotated Item 19. (author/sw) 

Students must determine the minimum test score that will yield a given mean. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 31. (author/sw)

Students must find the range of 10 students' scores on a final examination. This multiple-choice question is a sample test item used in grade 12 in the 1992 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1992-12M5, No. 9. (sw)

Students are asked to solve a multi-step problem involving average hourly wages using data in a table. They have the option of using a calculator. This constructed-response question is a sample test item used in grade 12 in the 1996 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1996-12M12, No. 2. (sw)

Students are asked to determine the average of data given in a frequency bar graph. They have the option of using a calculator. This constructed-response question is a sample test item used in grade 12 in the 1992 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1992-12M7, No. 12. (sw)

Students are asked which statistic--mean or median--they would use to describe typical daily attendance at a movie theater, given five data points, one of which is an outlier. They must explain their reasoning in detail. They have the option of using a calculator. This constructed-response question is a sample test item used in grade 12 in the 1996 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1996-12M12, No. 10. (sw)

Students must compare measures of center and spread for two sets of data. This multiple-choice question is a sample item used in the 2005 Ohio Grade 8 Achievement Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 Ohio Grade 8 Achievement Test for Mathematics, Annotated Item 41. (author/sw)

Students identify combinations of numbers that have a given mean. This multiple-choice question is a sample test item used in grade 8 in the 1992 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1992-8M7, No. 12. (sw)

Students are asked to use data in a table to determine the mean weekly salary at a video store. This multiple-choice question is a sample item used in the 2008 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OGT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2008 OGT for Mathematics, Annotated Item 15. (author/sw)

Students are asked to find the median of a data set from a scatterplot. They have the option of using a calculator. This multiple-choice question is a sample test item used in grades 8 and 12 in the 1992 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1992-8M14, No. 3. (sw)

Students are asked to determine the combined weight of 50 tomatoes when they are given the average weight of a single tomato. This multiple-choice question is a sample test item used in grades 8 and 12 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-8M7, No. 5. (sw)

Students must interpret box-and-whiskers plots to find a difference of medians. This multiple-choice question is a sample item used in the 2005 Ohio Grade 8 Achievement Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 Ohio Grade 8 Achievement Test for Mathematics, Annotated Item 34. (author/sw)

Given a frequency table of scores on a test, students are asked to find the average score. They have the option of using a calculator. This constructed-response question is a sample test item used in grades 8 and 12 in the 2003 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 2003-8M7, No. 13. (sw)

Students collect data on their study and TV viewing time over a period of several days. They use a graphing calculator, with step by step instructions, to find measures of central tendency, construct a box and whiskers graph, construct a scatter plot of class data, and look for a line of best fit. Activity sheets and suggestions for assessment are included. (author/sw)

Working in groups, students use a graphing calculator and the instructional activity sheets to gather data, calculate statistics, construct graphs, and make inferences. The students communicate their findings in a summary letter. The lesson is listed for grades 8-10 but reviewers felt the emphasis on measures of center made it better suited to grades 5-8. Activity sheets, interpretation questions, and extensions of the lesson are included. (author/sw)

Students investigate the average earnings of a movie in a given week using data they collect from web or print resources. In this investigation, students use the measures of center and different graphical displays of the data to make an argument that each of the measures of center may be appropriate, providing an explanation for their choice of mean, median, or mode. They identify outliers and create box plots (both modified and non-modified), histograms, and stem-and-leaf plots. Students also analyze spread and distribution of the data as part of their investigation. To conclude the exploration, students prepare a poster and give a report to the class. As students analyze actual data about a topic that engages them, they apply mathematics to a real-world situation and can see the value of the statistical concepts they are studying. Evaluation rubrics are included. This mathematically rich problem was developed for the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools proposed by the Ohio Department of Education. (author/sw)

For students who have already had some experience calculating mean, median, and mode, this applet provides a handy mechanism for exploring how the length of the interval affects the shape of a histogram and how outliers do or do not affect the measures of center. (author/sw)

Students are asked how many different outfits can be made from 4 tops, 3 shorts, and 2 pairs of shoes. This multiple-choice question is a sample item used in the 2005 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 OGT for Mathematics, Annotated Item 17. (author/sw)

Students must determine how many different meals consisting of a main dish, vegetable, and dessert can be ordered from a menu of choices. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 43. (author/sw)

Students must list all possible permutations of three distinct objects. This constructed-response question is a sample test item used in grade 12 in the 1996 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1996-12M13, No. 6. (sw)

Students find the number of arrangements, or permutations, of 5 different houses being built in a row. They have the option of using a calculator. This multiple-choice question is a sample test item used in grade 12 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-12M9, No. 7. (sw)

Students must find the number of permutations of five colored hats. This multiple-choice question is a sample item used in the 2005 Ohio Grade 8 Achievement Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 Ohio Grade 8 Achievement Test for Mathematics, Annotated Item 7. (author/sw)

Students are asked to determine how many different combinations of features are available in a jacket.  This multiple-choice question is a sample item used in the 2008 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OGT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2008 OGT for Mathematics, Annotated Item 08. (author/sw)

Students are asked to determine the probability that 2 heads and 1 tail will come up, in any order, if a fair coin is tossed three times. Students have the option to use a calculator. This constructed-response question is a sample test item used in grade 12 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-12M9, No. 17. (sw)

Students are asked to list all possible results of drawing two chips, with replacement, from a bag of three chips. This constructed-response question is a sample test item used in grades 4, 8, and 12 in the 2003 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 2003-8M6, No. 25. (sw)

Students learn about various voting methods, ways in which these methods can be manipulated to achieve certain outcomes, and the impossibility of fair elections when more than two alternatives are available. Plurality voting is the most familiar method of voting. In this method, each voter is given one vote and the option that receives the most votes wins. When a candidate receives more than 50 percent of the vote, including situations in which only two candidates are being considered, then plurality does produce a preferred candidate. However, in many situations, plurality may not produce a clear preference. The second half of the first activity sheet introduces students to three voting methods: the Hare system, Borda count, and sequential pairwise voting. Although the sheet gives instructions for each voting method, some student groups may need help following the directions. A second activity sheet introduces students to strategic voting, and a third sheet engages them in constructing tournament digraphs and the analysis of Condorcet winners. This unit on voting can be engaged at many different levels, beginning about 8th grade and including right up through college mathematics majors, who will be challenged by the proof of Arrow's Impossibility Theorem that there is no guaranteed fair way of electing one from among three or more candidates. (author/sw)

A round table seats six. Three boys and three girls must be seated around it so that boys and girls alternate. How many ways can this be done? An extension of the problem and a complete discussion of the underlying mathematical ideas are included. This mathematically rich problem was originally developed for the Project Discovery Mathematics by Inquiry institutes for middle grades teachers taught in 1992 - 1994 at Ohio State University. Project Discovery was co-funded by the Ohio Board of Regents and the Statewide Systemic Initiative (SSI) program of the National Science Foundation. (author/sw)

How many license plates can you make if they are each six digits long? This question opens an investigation into permutations. Variations on the question are offered (including solutions) and a final, more complex problem extends the challenge. In a section on mathematics, the author explains the mathematical reasoning underlying the common formula for finding "the number of permutations of n objects taken m at a time." This mathematically rich problem was originally developed for the Project Discovery Mathematics by Inquiry institutes for middle grades teachers taught in 1992 - 1994 at Ohio State University. Project Discovery was co-funded by the Ohio Board of Regents and the Statewide Systemic Initiative (SSI) program of the National Science Foundation. (author/sw)

Students explore a variation of the traveling salesman problem based on cost: What is the cheapest path through a network that will hit all nodes and return to the starting point? Activity sheets guide students through a brute-force approach and then a nearest-neighbor algorithm to find the cheapest route for a college recruiter to follow in visiting several cities. Students discover that the brute-force method always provides the optimal solution but is too inefficient to use in most applications. A 30-page resource for the teacher provides background to the lesson, describes several extensions including adding a city to an already established route, and poses a number of real-life business applications. Blackline masters, complete solutions to problems, and Internet extensions are also included. (sw)

Suppose you have some objects that you want to line up in a row. In how many different ways can you do it? Extensions of the problem and a complete discussion of the underlying mathematical ideas are included. This mathematically rich problem was originally developed for the Project Discovery Mathematics by Inquiry institutes for middle grades teachers taught in 1992 - 1994 at Ohio State University. Project Discovery was co-funded by the Ohio Board of Regents and the Statewide Systemic Initiative (SSI) program of the National Science Foundation. (author/sw)

Students must find the probability of a simple compound event. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 35. (author/sw)

Students must compute the probability of a specified event. This multiple-choice question is a sample item used in the 2004 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2004 OGT for Mathematics, Annotated Item 19. (author/sw)

Students are asked to determine the probability of a simple event and explain their answer. This short-answer question is a sample item used in the 2005 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, a complete solution of the problem, and a scoring rubric. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 OGT for Mathematics, Annotated Item 42. (author/sw)

Students are asked to determine the probability that the arrows on two spinners will both land in the same region. Students must justify their answer, and they have the option of using a calculator. This constructed-response question is a sample test item used in grade 12 in the 1996 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1996-12M12, No. 9. (sw)

Students are asked to determine the probability that 2 heads and 1 tail will come up, in any order, if a fair coin is tossed three times. Students have the option to use a calculator. This constructed-response question is a sample test item used in grade 12 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-12M9, No. 17. (sw)

Students are asked to solve a problem involving sampling without replacement.  This constructed-response question is a sample test item used in grade 8 in the 2005 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring guide, sample student responses, and a discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 2005-8M12, No. 7. (sw)

Students must find the probability of a compound event. This multiple-choice question is a sample item used in the 2005 Ohio Grade 8 Achievement Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OAT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2005 Ohio Grade 8 Achievement Test for Mathematics, Annotated Item 32. (author/sw)

Students are asked to determine the probability of one of Drake’s tickets being selected. This multiple-choice question is a sample item used in the 2008 Ohio Graduation Test (see Overview of Ohio's Assessment System). The URL link (above) takes the user directly to the OGT test item (PDF), with access to performance data, complexity level of the item, and discussion of incorrect responses. This OGT item is also available in Microsoft® Word. The Ohio Department of Education Instructional Management System website allows visitors to search for test items by subject and grade band and build a printable database of questions using the Add to Your Backpack function. ODE Reference Information: 2008 OGT for Mathematics, Annotated Item 43. (author/sw)

Students are asked to find the probability that a randomly selected student will be a boy, given the number of boys and girls in a class. This multiple-choice question is a sample test item used in grade 8 in the 1990 National Assessment of Educational Progress (see About NAEP). The URL link (above) takes the user directly to the NAEP test item, with access to performance data by various subgroups of students, a scoring key, and discussion of the content on which the item is based. The NAEP website allows users to build their own printable database of test items by clicking on Add Question in the upper right hand corner of the screen. NAEP Reference Number: 1990-8M7, No. 18. (sw)

This lesson presents a classic game-show scenario: You pick one of three doors in hopes of winning a prize. The host opens one of the two remaining doors which reveals no prize, then asks if you want to stick with your original choice or switch to the other unopened door. Which strategy gives you the best chance of winning? Approaches to the solution run from guesses to experiments to computer simulations (links to applets are provided) to theoretical models. The lesson also includes links to appropriate Internet extensions. This lesson plan first appeared in the April 1991 issue of Mathematics Teacher. (author/sw)

This lesson is a step-by-step introduction to tree diagrams for computing probabilities of simple compound events. A Racing Game applet is included, along with discussion questions and suggestions for guided and independent practice. (sw)

Most students understand that the probability of an event occurring can be influenced by another event that has already occurred. However, many students cannot understand that the probability of an event occurring can actually be dependent on an event that occurred later. Having information about the outcome of a later event can be used to revise probabilities of the occurrence of a previous event. This lesson on the time-axis fallacy will help students understand this important, but counterintuitive, idea by developing an understanding of Bayes' Theorem. The lesson plan includes a brief discussion of the mathematical topics, a list of needed materials, a suggested teaching procedure that includes several interesting problem scenarios, and a complete solution guide. (author/pk)

First, students review Pascal's Triangle by completing and discussing the entries of the first eight rows. They then determine the total number of possible girl/boy combinations in a five-child family. This is accomplished by having students investigate the make-up of different five-child families that could be born in a town called Smithville. A coin is used to simulate the births of different children. If the coin shows a head, the child is a girl and if it shows a tail, the child is a boy. The different combinations are presented in an organized manner so that students can discover patterns that will enable them to identify all possibilities. Students are encouraged to look for patterns that will assist them in generating the numbers in subsequent rows of Pascal's Triangle. Finally, students work collaboratively to address and analyze questions regarding the theoretical probabilities of other multiple-child families using Pascal's Triangle. In addition to the lesson plan, the site includes ideas for teacher discussion, extensions of the lesson, and additional resources. The lesson plan is accompanied by video clips illustrating lesson procedures. The user should first locate the Smithville Families lesson and then access the appropriate video clips at the PBS TeacherSource website. The video player necessary to view the video clips can be downloaded for free from the site. (author/sw)

Students investigate the number of people necessary to have a match of the day of the week of birth, month of birth, and date of birth by collecting data from different sizes of groups. A random number generator allows students to simulate this process to estimate the experimental probability of two birthdays matching. Using complementary and independent events, students calculate the theoretical probability of each event. This mathematically rich problem was prepared by the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools developed by the Ohio Department of Education. (author/sw)

This site describes three creative games that give students an opportunity to apply concepts related to probability. Students are asked to determine whether the games are fair, and if unfair, how they might be changed. For many students, exploring situations involving chance can be a challenging and enjoyable mathematical experience. Students can be quite fascinated when their intuitive guesses turn out to be wrong. Complete solutions to the problems are included, and Internet extensions are provided. This resource is adapted from an article that appeared in the April 1992 issue of Mathematics Teacher. (author/sw)