# MiddleSchoolPortal/Data Analysis: As Real As It Gets

<keywords content="middle school mathematics real-world problems local student curiosity mathematics relevant

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<metadescription content="This free, standards-based, online publication, developed for middle school mathematics teachers, supports teaching data analysis by linking to and describing inquiry-based lessons and activities." />

### Data Analysis: As Real World As It Gets - Introduction

In Data Analysis: As Real World As It Gets, we feature resources for teaching about data and statistics as supported by the NCTM Standards (NCTM, 2000). Data collection and analysis can be an avenue into the meaningful mathematics and problem-solving skills needed by students in the twenty-first century. And an answer to the student question, Why do we have to study math? can be found when teaching mathematics with a real-world statistics approach.

## Contents

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Educator Sheila Tobias, author of Overcoming Math Anxiety, states, "The teaching of math and science suffers from being all scales and not enough music. The emphasis on skills, facts memorization, and mastery very often kills curiosity and interest, even though it is necessary. Better to vary problem solving with discussion about applications that show the power of mathematics and science to provide analysis of complex situations" (Delisio, 2002).

Lovers of mathematics can immediately appreciate Tobias's emphasis on the importance of student interest and curiosity and the value of mathematics when analyzing the complex situations found in daily life. The Teacher Background section features online resources to strengthen teacher content knowledge, as well as an inspirational story about a mathematics lesson taught in context. To begin, we suggest looking to your community for local issues to investigate; see the "Principles" resource in Teacher Background for a stunning example of using mathematics to examine a local question.

The Lessons and Interdisciplinary Projects were selected to promote student interest by focusing on real-world situations and developing skills for using the power of mathematics to form important conclusions relevant to life. Challenge students to do original research with the spirit of adventure—where no one knows what will be discovered. Significantly, when using a problem-solving approach with data analysis, students develop an understanding of the team process and teachers avoid teaching skills out of context. Students learn that working with data offers insights into society's problems and issues. As we seek to develop the passion of scientific inquiry, the teacher may need outside expertise to understand the real-world topic—don't be afraid to ask!

If tempted to embark on an interdisciplinary teaching adventure, be assured that there are many pluses beyond having the benefit of shared teacher expertise on the issue under investigation. When teaching as part of an interdisciplinary team, blocks of time can be assembled for an in-depth investigation, and there are opportunities for professional mutual support in carrying out what may seem like an overwhelming project. In particular, a math/science duo would work well for many of the resources mentioned.

Finally, giving students a context for the mathematics skills they are learning increases their chances of successful engagement and learning. Teaching with a contextual mathematics focus offers additional opportunities for students to apply their reading, writing, and analytical skills to issues or problems they care about in their community or even in the larger world.

### Teacher Background

The first step you take away from the textbook approach to a mathematics topic is often uncomfortable. Are you intimidated by the idea of using real data in your classroom? Do you ever wonder how to teach everything that is required in middle school mathematics? You are not alone! Researcher Robert Reys tells us: Regardless of the textbook, the teachers were most likely to omit data analysis and probability (Cavanaugh and Diegmueller, 2005).

Let's take a minute to think about the content knowledge that teachers should know and understand to teach about data and statistics. The NCTM Data Analysis and Probability standard for grades 6-8 includes topics such as formulating questions; designing studies and collecting data for one or two populations and characteristics; making graphical representations such as histograms, box plots, and scatter plots; finding and using measures of center and spread; using samples to make conjectures and raise additional questions; and drawing conclusions about populations based on samples and approximate lines of fit (NCTM, 2000, p. 248). The resources below can help you prepare and get necessary background to teach standards-based statistics concepts in the middle school classroom.

An extensive tutorial is found in this college-level online mathematics course designed to teach data analysis, statistics, and probability to elementary and middle school teachers. The site focuses on statistics as a problem-solving process, and it engages teachers through video lessons, activities, and online demonstrations. Topics addressed include minimizing measurement biases, organizing data in a stem-and-leaf plot, measuring variation, and fitting lines to data. Estimation, five-number summaries, comparative studies, and binomial probabilities are also covered. The course consists of 10 sessions, each rich with video lessons, activities, and homework problems. The final session presents case studies that teachers use to examine students' perspectives on statistical problem-solving processes. Solutions are included for the activities and homework problems.

Are you looking for ideas to make data analysis real? This 11-minute video offers an inspiring look at how a teacher, with help from a local college, engages his 5/6 classroom in an in-depth study to explore the safety of well water in their community. The students become immersed in the project as they take samples from 63 wells to test for trace metals contamination. They use spreadsheets to analyze the data and share the surprising results with their community.

### Interactive Online Activities

Here are digital teaching resources that demonstrate how data and statistics are a vital part of learning mathematics in a meaningful context. The resource activities are often interdisciplinary, which makes them time-consuming to prepare, as additional expertise is often needed. But the payoffs can be huge: student engagement, in-depth learning, and a real-world context for learning mathematics.

The starting point for developing an interdisciplinary activity can be a teacher's personal interests or a student's passion. Are you a birder or pool player? Check out the ideas in Classroom FeederWatch, Backyard Birding--Research Project, or Analyzing Numeric and Geometric Patterns of Paper Pool below. Or if you want to begin with a simpler data collection lesson, take a look at Junk Mail below or Backpack Project.

Another approach is to look at situations in your community or larger world issues and have the students frame questions to investigate. Students may develop a passion for scientific inquiry when a topic can be analyzed with numbers. Requiring quality work and including a component about sharing results with the community will add value to an interdisciplinary contextual learning experience. Teachers may want to enlist a science teacher or community person to provide additional expertise. Whether thinking small activity or big project, be ready to be surprised at what the data analysis reveals!

Individual Lessons

Look out, pool sharks! Begin the study of data and statistics with this super student exploration where data are collected and analyzed while students apply mathematical topics studied in grades 6 and 7: factors, multiples, rectangles, and the meaning of being relatively prime. In the Paper Pool applet, a ball is hit from the lower left-hand corner of a grid-lined pool table at a 45-degree angle. Students modify the size of the rectangular pool table and observe how the ball always travels on diagonals of the grid squares. After gathering and organizing data, students look for patterns to predict the corner pocket into which a ball will fall and the number of side hits the ball makes as it moves on the table to a corner pocket. The goal is to determine how the number of hits, final pocket, and number of squares crossed depend upon the relative lengths of the sides of the pool table. Sounds like fun, doesn't it?

No one is immune from receiving junk mail, but just how much of it is really finding its way to your address? In this simple activity, data collection and analysis are a key part of a project to learn about the importance of recycling. For one week, students count and record the number of pieces of junk mail received in their homes. The display and organization of the data can be modified to address the data and statistics topics the class is working on.

Just how on-target are those weather forecasters we watch and listen to? In this webquest, students work in groups to track online weather reports for several locations over the course of three days and determine the accuracy of forecasts. Students develop an understanding of how weather can be described by measurable quantities, such as temperature, wind, and precipitation as they find and compare weather data found on the Internet, chart and graph data, and present their conclusions about forecasting. This straight-forward activity is suitable for students who are just beginning their work with data and statistics.

Projects

With this free online collaborative project, students measure the temperature and record the minutes of sunlight for one week. Data are collected on the web site, and average daily temperatures and amount of sunlight are compared. Students draw conclusions about how the distance from the equator influences temperature. If you like this collaborative project, be sure to check out Down the Drain: How Much Water Do You Use?, another collaborative data project from the Center for Innovation in Engineering and Science Education (CIESE).

If ocean travel is your passion, this site offers a way to spend time at sea without ever leaving your classroom. Here is a science project that uses actual data to help students investigate the science and history of the Gulf Stream. Math students can greatly benefit from the opportunity to collect data and draw conclusions based on the data. In the lesson called Current Now, students use real-time data and satellite images to determine how the Gulf Stream moves in the course of a year. In another activity, students use data about water temperature obtained from ships and buoys to determine the course of the Gulf Stream.

Be part of an annual event: Enroll your class in this free Internet-based collaborative project. Students discover which factors--room temperature, elevation, volume of water, or heating device--have the greatest influence on boiling point. Students boil water, record their data, and send it via email to be included in the site's database of results. Student activities focus on analyzing the compiled data to find answers to questions about how and why water boils.

Birds are everywhere, and here are ideas for creating a data collection project. Work with a science teacher and, possibly, an industrial tech teacher to expand this multiweek activity into a cross-curricular project to help students see how data analysis can support an understanding of nature.

These nine online lesson/activities investigate population growth and its impacts. Students use archived census and demographic data from the U.S. Census Bureau to model population growth and examine how population change affects the environment. Teachers will want to carefully review this resource to choose the activities most appropriate for their students' mathematics background. Linear, quadratic, and exponential functions are used in some lessons.

Lesson and Project Collections

The Center for Innovation in Engineering and Science Education at the Stevens Institute of Technology develops free standards-based projects that offer real-time data found on the Internet and online collaborative projects linking classrooms from around the world. There are opportunities for cross-curricular activities and for rich experiences with data collection, data display, and for drawing conclusions or inferences based on the data. The Global Sun Temperature Project and the Gulf Stream Voyage highlighted above are two examples of this site's materials.

What better way to make data and statistics lessons real than to use the daily newspaper? Here are interdisciplinary lesson materials based on New York Times articles. The stories offer ways to draw on real-world issues and statistics to develop lessons in mathematics. For example, in one lesson, "students convert statistics about gun injuries into visual presentations, then use these as the basis for a poster campaign to teach children about the dangers of guns in home," while another lesson idea involves analyzing the job market for older workers.

This unit contains seven lessons with hands-on learning activities to help students explore statistics and probability. The lessons, designed for mentoring situations, may also be helpful to teachers new to teaching data and statistics. Each lesson features an overview of the mathematics, preparation guidelines, teaching tips, and suggestions for how to use each activity to develop specific mathematics concepts. Statistics activities focus on posing questions, gathering data and identifying bias, understanding measures of central tendency, representing data with graphs, and interpreting data.

### 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.

### SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related math-focused virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities.

### Careers

The FunWorks Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

### NCTM Data Analysis and Probability Standard

The NCTM Principles and Standards for School Mathematics document states that "Prior to the middle grades, students should have had experiences collecting, organizing, and representing sets of data. They should be facile both with representational tools (such as tables, line plots, bar graphs, and line graphs) and with measures of center and spread (such as median, mode, and range). They should have had experience in using some methods of analyzing information and answering questions, typically about a single population" (NCTM, 2000, p. 249).

The NCTM document further states that middle grades students should be challenged to "Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them…. Select and use appropriate statistical methods to analyze data… Develop and evaluate inferences and predictions that are based on data" (NCTM, 2000, pp. 249-250).

The resources highlighted in Data Analysis: As Real World as it Gets can be used to facilitate student activities focused on designing experiments and collecting data in the context of important issues and concerns. The Data Analysis and Probability standard calls for students to learn to select and apply statistical ideas such as mean, medium, and mode to analyze data and for students to have experience with making and evaluating inferences and predictions based on data. As students formulate questions, analyze data, and make predictions using the highlighted resources, teachers will also find excellent opportunities to incorporate ideas from the Connections, Problem Solving, and Communication standards. Two other valuable resources from NCTM related to standards-based teaching include the book, Navigating Through Data Analysis in Grades 6-8, and the Illuminations web site featuring teaching materials and interactive activities.

We hope the contextual approach to teaching the topics found in the Data Analysis and Probability standard will help students build mathematics knowledge that is perceived as useful and connected to life.

## Author and Copyright

Judy Spicer is the mathematics education resource specialist for digital library projects at Ohio State University. She has taught mathematics in grades 9-14.