#### Data Analysis, As Real World as It Gets Applets and Activities

When working with applets, also known as virtual manipulatives, students see and interact with mathematics in ways not possible before the development of personal computers. The interactive, dynamic capacity of these online learning tools enables the student to immediately see the result when changing a data point on a graph or checking an answer to a calculation. Here we suggest excellent online resources for visualizing statistical concepts. Each suggested resource is from online collections featuring outstanding teacher support and materials for teaching standards-based topics in grades 6-8.

#### Selected Activities

Students can use this virtual manipulative to construct box plots to summarize data. They enter data into a table, and the manipulative displays the minimum and maximum data values, the lower and upper quartiles, and the median. The number in the data set, the average, and the standard deviation are also shown. In addition, the manipulative allows users to order the data. Instructions for using the manipulative, suggestions for the classroom, and a link to the standard for data analysis and probability, established by the National Council of Teachers of Mathematics (NCTM), are included.

Did you ever wonder how fish and wildlife experts determine how many of which fish to stock in a lake or pond? They use a method called capture-recapture, a statistical tool that lets them estimate the size of populations. This activity challenges students to estimate the total number of fish in a pond, given the numbers of fish initially tagged and released, the tagged fish recaptured, and the total number of recaptured fish.

This applet, with associated lesson plans and statistics background information, offers a great way to dynamically examine the relationship between data points and a best fit function. After entering data points, the student uses a slider bar to select the constants in a linear, quadratic, exponential, or trigonometric function to determine the line or curve that best approximates the data. Whether using linear or higher order equations, the applet offers an excellent visual for understanding the best fit function and making predictions based on data.

Students can use this manipulative to construct histograms by entering data into a table. The program notes minimum and maximum data values, median and average, number in the data set, and standard deviation. Instructions for using the manipulative, suggestions for the classroom, and a link to the standard for data analysis and probability, established by the National Council of Teachers of Mathematics (NCTM), are included.

Visualizing linear regression is easy to do with this colorful and simple-to-use applet. The student enters data points by clicking on a green graph showing the first quadrant. As blue data points appear, the applet displays a red line passing through the point for the average x and average y values. A scroll bar enables the student to rotate the slope of the red line to approximate a line of best fit. The applet can add a calculated blue line of best fit line when a button is clicked. The equations for the red approximate line and the calculated blue line of best fit are displayed along with a thermometer illustrating how well the approximate red line fits the data points. This applet may sound complicated, but be assured it offers a great visual for demonstrating how the equation of a line can serve as a data summary.

This virtual manipulative enables students to construct scatterplots illustrating the relationship between two variables. Students click on points on a grid to enter data. The manipulative displays a red regression line that approximates the set of indicated points. A yellow dot indicates the mean value for the data. The number of points, the average x and y values, the correlation coefficient, and the equation for the regression line appear on the screen. Instructions for using the manipulative, suggestions for the classroom, and a link to the standard for data analysis and probability, established by the National Council of Teachers of Mathematics (NCTM), are included.

Students compare the median salaries of men and women over a period of years and try to determine if women's salaries will ever be equal to those of men. The activity explains how data interpretation is important in such industries as advertising, athletics, and insurance. It suggests that students approach the problem by using fractions or difference to compare the salaries and then looking for trends. The exploration also has activities to investigate percent increase and extrapolation. Questions are posed to encourage students to think about factors that might affect trends in salaries, as well as factors that might cause women's salaries to be lower than men's.

#### Collections

The 80 mathematical challenges found on this site encourage problem solving with students in grades 6 to 8. The index page organizes the activities by math topic. The math challenges focus on concepts and objects found in everyday life, such as how fast your heart beats, what shape container holds the most popcorn, and how much of a person shows in a mirror. Each challenge contains an initial problem with a solution hint, a complete explanation of the answer, and additional problems related to the same challenge. Resources for further investigations are suggested as well. From the Printing the Challenges link on the homepage, pdf files are available for all 80 challenges in English, the first 15 challenges in Spanish, and the family resource materials in English and Spanish.

This site features more than 60 virtual manipulatives designed to function as concept tutorials. The virtual manipulatives encourage active student exploration with numbers and can help students visualize mathematical relationships. Student activities and lesson plans accompany many of the virtual manipulatives, each of which is linked to an explanation of the relevant NCTM Standards. The resources are organized into five categories based on the Standards: number and operations; algebra; geometry; measurement; and data analysis and probability.

These interactive activities, or applets, are designed for individual or small group learning. With each applet there are links to the activity's purpose, a description of functionality, and information about how the mathematics fits into the curriculum. The teacher background information found here is particularly useful. Lesson plans, handouts, and student discussion questions are usually provided. Organized by four content standards found in the NCTM Principles and Standards for School Mathematics and the NCEE New Standards Performance Standards: Middle School, the site's four sections are number and operations; algebra; geometry; measurement; and data analysis and probability.

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