<p>The lesson focuses on the concepts of correlation and scatter plot representations. Students will:&nbsp;<br>- create and explore scatter plots.&nbsp;<br>- use the correlation coefficient to determine the strength of the relationship.&nbsp;<br>- plot and evaluate a real-world sequence, then make&nbsp;additional conclusions/connections.&nbsp;<br>- create a research using two variables.&nbsp;<br>&nbsp;</p>

<p>- What does it mean to analyze or estimate numerical quantities ?&nbsp;<br>- What qualifies a tool or approach as suitable for a certain task?&nbsp;<br>- How may data be arranged and portrayed to reveal the link between quantities?&nbsp;<br>- What impact does the type of data have on the display option?&nbsp;<br>- How can predictions be made using data analysis and probability?&nbsp;<br>&nbsp;</p>

<p>- Clustering: When many data points on a scatter plot are grouped closely together.&nbsp;<br>- Linear Association: When the relationship between two variables shows a linear trend. On a scatter plot, data points that have a linear association can clearly be modeled by a line of best fit.&nbsp;<br>- Line of Best Fit: The line that most closely approximates the data in a scatter plot (provided the data demonstrates a linear association).&nbsp;<br>- Negative Correlation: Describes a relationship between two variables such that as the values of one variable increase, the values of the other variable decrease.&nbsp;<br>- Nonlinear Association: When the relationship between two variables does not show a linear trend. On a scatter plot, data points that have a nonlinear association cannot be modeled by a line of best fit, as there is no obvious linear pattern.&nbsp;<br>- Outlier: A data point that diverges greatly from the overall pattern of the data.&nbsp;<br>- Positive Correlation: Describes a relationship between two variables such that as the values of one variable increase, the values of the other also increase.&nbsp;<br>- Scatter Plot: A graph with points plotted to show a relationship between two variables.</p>

<p>- graphing calculator&nbsp;<br>- GeoGebra (free geometric software)&nbsp;<br>- Lesson 2 Exit Ticket (M-8-7-2_Lesson 2 Exit Ticket)</p>

<p>- Utilize the findings from the applet exploration to assess the extent of students' comprehension.<br>- Evaluate the caliber of students' academic pursuits to measure their understanding. Evaluate students' delineation of the problem, methodology for data collection, and choice of analysis. The data presentation should be compatible with the kind of data found in the findings.<br>- Assess student understanding of general correlation ideas by utilizing the Lesson 2 Exit Ticket (M-8-7-2_Lesson 2 Exit Ticket).<br>- The quiz available on the website provided can be utilized to assess students' comprehensive understanding of scatter plots. The website address is: <a href="http://www.regentsprep.org/Regents/math/ALGEBRA/AD4/PracPlot.htm"><span style="color:#1155cc;"><u>http://www.regentsprep.org/Regents/math/ALGEBRA/AD4/PracPlot.htm</u></span></a>&nbsp;<br>&nbsp;</p>

<p>Modeling, Explicit Instruction, and Active Engagement&nbsp;<br>W: The lesson explores the relationships between data as they are uncovered through analysis, conclusions,&nbsp;and predictions. With the tools provided, students can investigate the impact of outliers on correlation and the relationship between the variables, build and analyze scatter plots, and learn about correlation. Students must be able to classify data and distinguish between correlation and causation to&nbsp;be able to develop meaningful judgments of the degree to which data correlate with one another.&nbsp;<br>H: Students will use one of NCTM's applets to plot points and calculate correlation coefficients, connecting the plotted data's appearance to the quantity's relationship description. Additionally, students have the option to estimate the correlation coefficient and then view the correlation coefficient's actual value, or r-value. The exploration is intriguing and advances rapidly, capturing and retaining student interest.&nbsp;<br>E: Students investigate the relationships between the quantities they generate in real-world contexts and key concepts by employing the NCTM applet (or graphing calculator), which enables them to generate fictional data sets and evaluate the relationships between their fictional variables.&nbsp;<br>R: Students can develop their unique ideas into a cohesive strategy, work with peers, learn from failures, and assess their own and group preparations through small-group presentations to the class (4–5 students).&nbsp;<br>E: Students can utilize their creativity to blend lesson skills and knowledge into presentations that can be evaluated by both the class and the teacher.&nbsp;<br>T: Use the Extension section to customize the lesson to match the needs of the students. Throughout the year, the Routine section offers suggestions for reviewing the concepts covered in the lessons. The Small Group section offers the&nbsp;opportunity for relearning and practice. The Expansion area offers an opportunity for students who are willing to surpass the standard standards.&nbsp;<br>O: This lesson takes a hands-on, exploratory approach. Students are encouraged to solve problems, make deductions, brainstorm, and apply their knowledge. All of the explorations covered in the class are connected to the final exercise.&nbsp;</p>

<p><strong>Activity 1</strong><br><br>Inform the class that they will be studying the connections between various data sets. <strong>"Assume you surveyed one hundred children, aged three to eighteen, noting the age and height of each child. What type of correlation do you think one should see between an individual's height and age?"</strong> Students need to acknowledge that as they get&nbsp;older, their height tends to increase.<br><br>Create a simple scatter plot to illustrate fictional&nbsp;data about&nbsp;age and height. The scatter plot should show positive correlation&nbsp;but should&nbsp;not be perfectly linear. Depending on the class, you can ask students if the relationship is linear, for example, <strong>"Does a person grow the same amount each year?"</strong><br><br>Label the scatterplot <i>Positive Correlation</i> and explain&nbsp;to students that a positive correlation can be defined as <strong>"Whenever one of the quantities increases, the other quantity also increases."</strong> Remind them that they do not have to increase by the same amount; all that matters is that they both increase. <strong>"What are some other real-life examples of two things that might show a positive correlation?"</strong> Encourage students to generate examples of positive correlations. Some examples include outdoor temperature vs. air conditioning bill, study time vs. test scores, fat grams vs. calories, and so on.)<br><br>Ask students to work in pairs to identify two or three different pairings of data that show a positive correlation. Remind them that the key&nbsp;is that as one quantity increases, the other one also increases. After each pair has a few ideas, have students write them on the board. Depending on class size, each pair should list one or more ideas. Ask students if any of the combinations on the board are "stronger" than others. In other words, are there any pairs&nbsp;in which when one quantity increases, the other always increases, maybe even in a fix amount? If nothing on the board satisfies this condition, you can suggest a relationship between the number of hours driven and the distance driven by someone driving at 60 miles per hour. Investigate the relationship between the quantities&nbsp;with a significant positive correlation, and create another scatter plot to represent fictional data about these two quantities, ensuring that the data is more closely clumped&nbsp;around an imaginary line with a positive slope.<br><br><strong>"This data clearly shows a positive correlation, but we can also argue it has a </strong><i><strong>strong</strong></i><strong> positive correlation. In other words, this data makes it much obvious that if one quantity increases, the other quantity also increases. For our example of age and height, it is possible that when people age one year, their height scarcely increases, and sometimes&nbsp;it increases by an average amount, or sometimes&nbsp;they grow really quickly. In general, there is still a positive correlation, but not as strong a correlation&nbsp;as in our second example."</strong><br><br>(Note: Students who struggle to comprehend how to interpret correlation frequently overlook the presence of both variables because they tend to think of correlation as a single concept. Emphasize to them that any variable can move in two directions. Positive correlation occurs when the second variable moves in the same positive direction as the first variable. Positive correlation demands that the second variable move in the same negative direction as the first variable. Before using real data, emphasize qualitative rather than quantitative characteristics before using real data.)<br><br><strong>Activity 2</strong><br><br>Repeat this practice with a negative association, beginning with an example like the quantity of money in people's bank accounts and the length of their vacations. <strong>"What happens to the amount of money in their bank account as the duration of their vacation increases?" </strong>Students should be aware that it is decreasing. <strong>"Does it consistently decrease by the same amount?"</strong> Have students suggest reasons for differences in the rate at which the amount of money would decrease (airline flights, changing hotels, eating at pricier restaurants, etc.)<br><br><strong>"There is a </strong><i><strong>negative</strong></i><strong> correlation between these two quantities: the longer their vacation, for example, the lower the other quantity."</strong> Make some fictional data on a scatter plot to illustrate a negative correlation. Make sure the data is not strongly correlated (there should be some scatter). Students should collaborate in pairs to come up with quantities with a negative correlation and then share their ideas like before. If there is no strong negative correlation, suggest the height of a candle and the duration of its burning, or, in the reverse of the strong positive correlation concept, the distance a person is from their destination based on how long they have been driving at a constant rate. Draw some fictional data for the case with a strong negative correlation and explain that this data has a strong negative correlation.<br><br><strong>Activity 3</strong><br><br>Now ask students about the association between a person's height and the number of movies he or she has seen in a theater in the previous year. Ask students to describe the relationship between the two quantities. They may come up with some hypotheses (for example, an extremely tall person may be less likely to go to the theater because he or she does not want to obstruct someone else's view), but they emphasize that there is no real relationship between the statistics for 99% of the population. Create a scatter plot to depict this data and demonstrate that there is no pattern. Make the data cluster around specific values of average height and a suitable number of movies seen in the previous year. Also, include one or two outliers to illustrate extremely tall people who do not go to theater.<strong> "Because one quantity does not seem to increase or decrease based on the other quantity, we say this data has </strong><i><strong>no correlation</strong></i><strong>."</strong> Circle the cluster of data and tell students, <strong>"This data also appears to cluster around a specific area, indicating that the majority of the persons represented by this scatter plot are about the same height and attended the movies about the same number of times. This is an example of </strong><i><strong>clustering</strong></i><strong>, when a large number of data points are grouped&nbsp;in the same area. How about the other data points?"</strong> Indicate the outliers. <strong>"What do they indicate?"</strong><br><br>Even if students do not use the term "outliers," they should realize that they "lie" on the outer borders of the data indicated by the scatter plot and are separated from the cluster of data.<br><br><strong>Activity 4</strong><br><br><strong>"In mathematics, we often want to use numbers to quantify or explain relationships. It's fine to remark that a set of data has a positive correlation, for example, but we'd like to be able to quantify that correlation and assign a numerical value to it."</strong><br><br>Allow students to work in groups with the applet at <a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=146."><span style="color:#1155cc;"><u>http://illuminations.nctm.org/ActivityDetail.aspx?ID=146.</u></span></a><br><br>(This activity can also be completed with a graphing calculator, but the applet makes it much easier to plot and move points (by selecting "Move Points") while watching the results change in real time.)<br><br>Some groups should display data with a positive correlation, some with a negative correlation, and others with no association at all. As students plot the data, have them select the "Computer Fit" check box and record the value of <i>r</i> that the Website produces.<br><br><strong>"As your positive correlation increases in strength, what is the impact on the value of </strong><i><strong>r</strong></i><strong>?"</strong> <i>(The value of r increases.)</i> <strong>"What is the greatest value of </strong><i><strong>r</strong></i><strong> you can achieve?"</strong> Students should be able to easily reach 0.9, but it is unlikely that they will reach exactly 1.0. Tell students,<strong> "The greatest possible correlation is 1. What do you think the data looks like when the correlation is 1?" </strong>Based on the fact that the website applet also creates a line of best fit, direct students to recognize that data that all fall on the same line correlates 1.<br><br>Repeat the same questions for negative correlations, noting that the minimum possible correlation is -1.<br><br><strong>"What do you think the value of r is for data that is not correlated at all?"</strong> Students should assume it's 0.<br><br><strong>"The </strong><i><strong>correlation coefficient</strong></i><strong>, or the value of </strong><i><strong>r</strong></i><strong>, is a numerical measure of our data's correlation. It's difficult to calculate by hand, but calculators and spreadsheet tools like Excel can do so quickly (as can this website)."</strong><br><br>Students should collaborate in small groups to create research in which they identify two quantities to measure, construct a strategy, collect, analyze, and interpret data, and make predictions. Students will share their discoveries and visual representations with the class. A discussion of the variable-correlation coefficient relationship must be presented. Students must connect the findings to the study's context and draw real-world conclusions.<br><br>The following website offers an optional scatter plot quiz. The quiz can be used to check students' understanding of scatter plots:<br><br><a href="http://www.regentsprep.org/Regents/math/ALGEBRA/AD4/PracPlot.htm"><span style="color:#1155cc;"><u>http://www.regentsprep.org/Regents/math/ALGEBRA/AD4/PracPlot.htm</u></span></a>&nbsp;</p><h3><strong>Extension:</strong></h3><p><strong>Routine:</strong> Asking students to create and develop their own data based on their specific interests is a motivating force. Topics such as current movie box office sales, top-ten video game lists, possible music downloads, and sports records are all examples of big data troves that are reasonably simple to search. Partner grouping can help students grasp the relationship between variables, the line of best fit, and the correlation coefficient.<br><br>Another option to assess students' comprehension of correlations in scatter plots is to administer the online quiz at the following website:<br><br><a href="http://www.mathopolis.com/questions/q.php?id=3072&amp;site=1&amp;ref=/data/scatter-xy-plots.html&amp;qs=3072_3073_3074_3075_3772_3773_3774_3775_3776"><span style="color:#1155cc;"><u>http://www.mathopolis.com/questions/q.php?id=3072&amp;site=1&amp;ref=/data/scatter-xy-plots.html&amp;qs=3072_3073_3074_3075_3772_3773_3774_3775_3776</u></span></a>&nbsp;<br><br><strong>Small Group:</strong> Assign groups to create and compose lists of data categories that they believe will show strong correlations. Consider student grade level, average shoe size, or daily high temperature and ice cream sales. Assign writing lists with negative correlations to students who are more skilled at representing correlations. For example, average daily temperatures and winter clothing sales.<br><br><strong>Expansion:</strong> Have students conduct further experiments with the applet at<br><a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=146"><span style="color:#1155cc;"><u>http://illuminations.nctm.org/ActivityDetail.aspx?ID=146</u></span></a>. Next, have students draw broad conclusions about data that are weakly and highly correlated.</p>