Data+and+Probability

**Learning about data affords students the ** **opportunity to interact with (collecting,** **analyzing and representing) data in a much** **more personalized way. Students have to formulate questions about the data and use critical thinking skills to answer the questions they pose. In general, this strand is the closest to real life situations that students undergo in math class. Whenever they use data, it is something they or someone else has collected about a situation that occurs outside of the math classroom. This outlet gives students the opportunity to see how math can be used in everyday life. Students can also see the various ways data can be manipulated and discuss best practices of reflecting on and representing data. In addition to Microsoft Excel and Google Spreadsheets, try these data and probabilty interactives to help motivate your students to analyze and understand data and probabilities.**

Data and Probabilty Evaluation #1:

 **//An interactive lesson in statistical analysis //** || **Data Analysis and Probability ** probability, graphing and graph interpretation, measures of central tendency, and variance || * || //Rationale //: This applet is designed to allow students to investigate how mathematics can help them analyze a simulated forest fire. Students will be able to work with the concept of probability and manipulate different probabilities within the simulation. Students are also introduced to the concept of chaos and variance as they organize and analyze the data. Students then graph and analyze the data using a line plot and calculate the mean of various data points to see how they change. || Instrumental understanding Relational understanding || * || //Rationale //<span style="font-family: 'Times New Roman','serif';">: Although each concept is defined and reviewed in the “learner” section of the applet, the simulation serves as a way for students to develop relationships between probability, variances and the mean of data and apply this to the real-world simulation of a forest fire. Within this activity, students collect data using the simulator and then graph the data to see if a pattern can be determined. They then determine if this model accurately predicts the forest burned for different sets of probabilities. Students should come to the conclusion it would be more accurate by testing various data points for different burn probabilities in approximately equal intervals. || · <span style="color: black; font-family: 'Times New Roman','serif';">learning through exploration   || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Although there are some direct instruction in the “learner” tab, students learn through exploration by observing what happens in several different trials and organizing the information to make predictions and conjectures about how a fire could destroy a forest and at what variance total destruction is expected to happen. Through guided practice, students answer critical thinking questions based on the interactive applet and how their manipulations affect specific outcomes. ||
 * <span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> **//<span style="font-family: 'Times New Roman','serif'; font-size: 18pt;">[|Fire, Probability and Chaos] //**
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Which standard? **
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|formulate questions] that can be addressed with data and collect, organize, and display relevant data to answer them;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|select and use] appropriate statistical methods to analyze data;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|develop and evaluate] inferences and predictions that are based on data;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|understand and apply] basic concepts of probability. || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: This applet simulates a forest fire and focuses on using data and statistics to make informed decisions about preventing forest destruction. Students use the simulator to test probabilities of how many tress will burn by setting different percentages in the “probabilities section. Upon several trials, they should be able to make conjectures about the relationship between probabilities and the amount of trees burned. Keeping track of the data, students can create a graph by using the Simple Plot activity. Using the graph, students can make predictions based on that data and apply it to new simulations in the fire activity. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What mathematical content is being learned (or intended to be learned)? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">Is the focus on instrumental or relational understanding? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What role does technology play? ** || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Response //<span style="font-family: 'Times New Roman','serif';">: Technology provides students an opportunity to visualize an actual forest fire and manipulate the outcome in several different ways. Students gather information through the simulation and discover the relationship between probability and variance. This technology gives students this access to virtually seeing a forest fire and its devastation quickly and easily. Students can also create several trials within seconds to confirm their predictions and conjectures as opposed to hand written graphs and simulations. This technology also represents knowledge and thinking in a different way that teachers could present the material without the technology. Simulating the fire allows students to make necessary connections between the real world and probability of events occurring. ||
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What instructional function(s) does the resource serve? **
 * **<span style="color: green; font-family: 'Times New Roman','serif';">What kinds of representations of the mathematics are used? **

<span style="font-family: 'Times New Roman','serif'; line-height: normal;">symbolic <span style="font-family: 'Times New Roman','serif'; line-height: normal;">graphical <span style="font-family: 'Times New Roman','serif'; line-height: normal;">visual/spatial <span style="font-family: 'Times New Roman','serif'; line-height: normal;">concrete or real-world objects <span style="font-family: 'Times New Roman','serif'; line-height: normal;">dynamic || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Mathematics is constructed and represented in several different ways within this applet. Numerical representations are presented symbolically using numbers and mathematic symbols such as percentages and decimals for probability. The forest fire is represented on a grid that “burns down” trees in various squares. Students are able to visualize the forest destruction, or lack thereof within each trial and manipulate the symbols to change how the fire burns (and how much). Within the applet, pictures of pine trees represent the tress found in a forest and most students are familiar with the concept of fire and forest destruction. It is dynamic in each trial as tress are wiped out through flames and smoke and students can reset the interactive to try multiple trials of forest destruction to make their inferences and estimations. ||
 * <span style="color: #403152; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Data and Probability Evaluation #2: **

<span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> //<span style="color: #000000; font-family: 'Times New Roman','serif'; font-size: 18pt;">An interactive lesson in representing data // || <span style="color: green; font-family: 'Times New Roman','serif'; line-height: normal; margin: 9.35pt 0in;">Data Analysis and Probability <span style="color: black; font-family: 'Times New Roman','serif'; line-height: 13.5pt; margin: 0in 0in 0pt;">Learning about different types of graphs and applying them to various situations. || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Using this interactive, students will be able to analyze data including body fat percentages in men, gas mileage, and SAT scores, to see how each set of data can be represent in various ways. Students then describe the shape and important features of that data set, with an emphasis on how the data are distributed over time. Using any of the scenarios, students formulate questions, construct data tables, and display the relevant data in a graph of their choice. Students must justify their conclusions and design further studies based on their information. || <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt;">Relational understanding || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: While students are given explicit instructions on how to work the applet, the learning that takes place is based on the relationships students find through their individual analysis of the data they choose to use. In the Exploration portion of the applet students are given the data they must analyze and import that data into the applet. Students then analyze the data and choose an appropriate graph to represent their data. Students must critically think about their choice of representation and justify why that graph works best. || <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">Practice <span style="color: #000000; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in; text-indent: -0.25in;"> · <span style="color: black; font-family: 'Times New Roman','serif';">learning through exploration  || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* ||  //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: Students practice investigating different types of graphs and how those graphs represent data. Students are able to see that a scatter plot might work best for statistical data and histograms might work best for categorical data. Through their exploration, students can see a variety of different data represented in using different plots. <span style="color: black; font-family: 'Times New Roman','serif';">Within each data set, students choose the best graph for a set of data, depending on the question being asked of the data. Then they create a graph using appropriate scale, intervals, and titles using the interactive. Students the interpret each graph to determine if it is showing any patterns or relationships among the data. ||
 * <span style="display: block; line-height: normal; margin: 0in 0in 0pt; mso-element-anchor-horizontal: margin; mso-element-anchor-vertical: page; mso-element-frame-hspace: 9.0pt; mso-element-top: 44.3pt; mso-element-wrap: around; mso-element: frame; mso-height-rule: exactly; text-align: center;"> //<span style="font-family: 'Times New Roman','serif'; font-size: 18pt;">[|Advanced Data Grapher] //
 * <span style="color: green; font-family: 'Times New Roman','serif'; line-height: normal; margin: 9.35pt 0in;">Which standard?
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|formulate questions] that can be addressed with data and collect, organize, and display relevant data to answer them;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|select and use] appropriate statistical methods to analyze data;
 * <span style="font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 10pt; tabstops: list .5in;">[|develop and evaluate] inferences and predictions that are based on data; || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: In this activity, students can use a given set of data, or import a data source of their own to represent data using a stem-and leaf plot, a box and whisker plot, a scatter plot or a histogram. Within the interactive, students conduct activities in which they can use data given in the applet or other data available on the Web to examine questions about that set of data. Working on such activities, students also formulate their own questions and use the mathematics they are studying to address these questions. They can propose and justify conclusions that are based on data and design further studies on the basis of conclusions or predictions.  ||
 * What mathematical content is being learned (or intended to be learned)?
 * <span style="color: green; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">Is the focus on instrumental or relational understanding?
 * What role does technology play? || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Response //<span style="font-family: 'Times New Roman','serif';">: Technology allows students to see the same data set represented using various types of graphs in real time. Students can then compare and contrast each graph to choose a graph that best represents the data. Students are given three “ready-made” sets of data, but they can also go out to Web and find their own data to analyze and investigate. Within the applet, students need to examine data sets and choose different graphs to best represent the data. Being able to see these graphs and compare them easily allows students to focus on the appropriateness of the graph and not just on creating the graph. ||
 * <span style="color: green; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">What instructional function(s) does the resource serve?
 * <span style="color: green; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">What kinds of representations of the mathematics are used?

<span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">symbolic <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">graphical <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">visual/spatial <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">concrete or real-world objects <span style="color: #000000; font-family: 'Times New Roman','serif'; line-height: normal;">dynamic || <span style="color: white; font-family: 'Times New Roman','serif'; line-height: normal; margin: 0in 0in 0pt;">* || //<span style="font-family: 'Times New Roman','serif';">Rationale //<span style="font-family: 'Times New Roman','serif';">: While using this applet, students represent mathematics in many different ways. Symbolically, students import data sets typing in numbers and appropriate symbols. Data given is represented numerically in tables and graphs. Students compare different graphs to analyze the data using real world constraints (i.e. sports statistics, population density, etc.). This interactive is also dynamic as it changes the data into different graphical representations and allows the user to import data from the internet. ||


 * <span style="color: #403152; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Data and Probability Annotated Links: **


 * ** Screen Shot ** ||  ** Name and Description **  ||
 * [[image:State_Data_Map.jpg width="321" height="213" align="center" link="http://illuminations.nctm.org/activitydetail.aspx?id=151"]] || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">The [|State Data Map] is an interactive way for students to collect, analyze, represent, and make conjectures about either given data, such as land area, population density, gasoline use and many more seen in the picture to the left, or students can collect their own data from another source and type it in the applet to begin to represent their data. Students can choose from any of the buttons at the top of the interactive to analyze the data and see how the box-and-whisker plot changes as they manipulate data. Within the extention part of the applet, several critical thinking activities about mean, median, mode, and population density challenge students to make predictions and conjectures about the data they are given. ||
 * [[image:Stema_and_Leaf_Plot.jpg width="333" height="210" align="center" link="http://www.shodor.org/interactivate/activities/StemAndLeafPlotter/"]] || <span style="font-family: 'Times New Roman','serif'; font-size: 12pt; line-height: normal; margin: 0in 0in 0pt;">In this [|Stem-and Leaf Plot] activity, students can use given data such as those in the "Stem-and-Leaf Plots" Lesson Companion, or enter their own data to create stem and leaf plots quickly and easily. Students can visually see the stem-and leaf plot and make conjectures about outliers and best fit data. In doing the activity, students can use the automated calculator to find mean, median and mode, or they can use the “guess” features that allows then to input these calculations on their own. Students can access the procedures for finding measures of central tendency and creating stem-and-leaf plots on the “Learner” tab within the interactive. Students can also print their plots along with their documents at the end of the activity for assessment. ||