<p>The lesson explores scatter plots, specifically how to draw a line of best fit and utilize it to answer contextual questions about the data in the plot. Students are going to:<br>- identify a line of best fit by hand (estimation) or with a calculator.&nbsp;<br>- use the equation for the line of best fit to investigate the meaning of the <i>y</i>-intercept and slope.</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>- Bivariate Data: Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team.&nbsp;<br>- 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>- Scatter Plot worksheets (M-8-7-3_Scatter Plot) for each student and one for the overhead projector&nbsp;<br>- graphing calculator or access to spreadsheet program (Excel, etc.)</p>

<p>- Evaluate each group's ability to generate bivariate data and understand the slope and <i>y</i>-intercept.<br>&nbsp;</p>

<p>Active Engagement, Metacognition, Modeling&nbsp;<br>W: Students will use given data points to forecast unknown values. They will also discover the limitations of those predictions.&nbsp;<br>H: Students will be given freely accessible data on height based on age and encouraged to make "strange" predictions, such as forecasting a 40-year-old's height using a line of best fit with a result of 12 feet.&nbsp;<br>E: Students will explore and consider&nbsp;non-mathematical themes such as growth rates and growth restrictions. Students will develop their own lines of best fit and predictions.&nbsp;<br>R: Students will alter their predictions depending on the line of best fit they identified. They will also have the opportunity to design and update their lines using better procedures (for example, "eyeballing" the line, drawing it with a straightedge, and computing the line's real equation).&nbsp;<br>In Activity 3, students evaluate their comprehension of the idea by producing a scatter plot, estimating and computing the equation of the line of best fit, and interpreting the equation in context of the problem. In addition, a PowerPoint assignment is included to help evaluate student mastery.&nbsp;<br>T: Students are provided numerous approaches to draw their line of greatest fit and urged to weigh the advantages and cons (easy vs. accuracy). The Extension section can be utilized to adapt the lesson to the needs of the students. Use the Routine section for ideas on how to review course content throughout the year. The Small Group portion is suitable for those who require further practice. The Expansion section includes suggestions for students who are ready for a higher challenge.&nbsp;<br>O: The lesson focuses on graphing lines to approximate data, but students can also explore real-world themes like age vs. height and create their own. They have the option to provide "realistic" data and present a&nbsp;rationale to explain their findings.&nbsp;</p>

<p><strong>Activity 1</strong><br><br>Give students a Scatter Plot worksheet (M-8-7-3_Scatter Plot), and then project the following scatter plot for them to view (the plot below is the&nbsp;same as the one on the worksheet):&nbsp;</p><figure class="image"><img style="aspect-ratio:382/366;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_15.png" width="382" height="366"></figure><p><strong>"This scatter plot displays the heights of 24 different children, ranging in age from 1 to 7 years."</strong> To check that students are accurately interpreting a scatter plot, ask them to estimate the heights of the three 1-year-olds, the 5-year-olds, and so on.&nbsp;<br><br><strong>"Can the scatter plot provide any information regarding the heights of any 8-year-olds?"</strong> <i>(No)</i> <strong>"Is it possible to make an informed estimation of the height of an 8-year-old based on the data from the scatter plot?"</strong> <i>(Yes)</i><br><br>Have students write down their extrapolated guesses for an 8-year-old's height in inches based on the data in the plot. Then, ask how many students predicted 40 inches. Most likely, no students will have guessed 40 inches; if they did, simply record the number of guesses on the board. Otherwise, ask,<strong> "Why did nobody guess 40 inches?" </strong>Students should point out that 40 inches appears to be an average height for a 5-year-old and that children's heights increase year after year.<br><br><strong>"Is it possible for an 8-year-old to be 40 inches tall?" </strong>Help students understand that any prediction they make is just that—a prediction. It is not a guarantee; it is only a hypothesis about what might happen in the future or with a group that we have not yet examined.<br><br>Continue to ask about predictions, increasing by one or two each time (for example, next ask if anyone projected 42 inches, and so on). Identify the most popular prediction. Ask the students who made the prediction to explain how they got at it. Ask other students who made the same prediction how they came up with it. Collect as many methods of prediction as possible.<br><br>Direct the conversation toward using a ruler, the edge of a sheet of notebook paper, or another straightedge to assist in making a prediction. If no students used a straightedge to make their prediction, demonstrate how they could have done so.<br><br><strong>"What makes using a straightedge easy with this particular set of data?" </strong>Students should understand that the data is essentially on a straight line. <strong>"So we can take that straight line and imagine extending it to the right to see how tall an 8-year-old would be. Would everyone draw the&nbsp;same line if they were all to approximate the data with a straight line?"</strong> <i>(No) </i><strong>"Why not?"</strong>&nbsp;<br><br>Help students understand that our line is only an approximation. We're attempting to draw it as close as possible to all of the points. However, with only our estimation skills, we cannot design the perfect line that reduces the distance from all places at once. Draw a line like this on the scatter plot:</p><figure class="image"><img style="aspect-ratio:441/396;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_16.png" width="441" height="396"></figure><p><strong>"Let's use this as an estimate. Your line might be somewhat different, which is fine. We'll refer to this as the </strong><i><strong>line of best fit</strong></i><strong>."</strong> Make a note of this term somewhere visible to students.<strong> "It's the line that best fits our data, at least according to our guesses. Again, it may not be perfect. Based on this line of best fit , what do you estimate an 8-year-old's height might be?"</strong><i><strong> </strong>(50 inches)</i><br><br><strong>"The average height for 8-year-old boys is about 51 inches, and for girls, it is about 50.75 inches, depending on the source. So, our guess is reasonably accurate. What do you think a newborn infant's height should be based on the line of best fit?"</strong><i><strong> </strong>(25 inches)</i><br><br><strong>"Again, depending on the source, the average height for newborns is 20 inches for boys and 19 inches for girls. So, our estimate was somewhat close, it was not as close as our estimate for 8-year-olds. Why could our estimate not be as accurate?"</strong><br><br>(Possible explanations include: the heights shown on our scatter plot were slightly higher than normal; infants grow at a faster pace from years 0 to 1 than during other intervals; children do not grow at a steady rate throughout childhood, and so on.) Use this chance to discuss the veracity of predictions based on lines of best fit.<br><br><strong>Activity 2</strong><br><br>Ask students to review the second section of their worksheet and to post the following information in a visible location for the class as a whole:</p><figure class="image"><img style="aspect-ratio:148/681;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_20.png" width="148" height="681"></figure><figure class="image"><img style="aspect-ratio:151/764;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_21.png" width="151" height="764"></figure><p><strong>"This is the same data that was used to create the scatter plot we've been discussing. Using&nbsp;real statistics, we can determine the line of best fit. The ones we drew were just guesses, and it isn’t possible to determine points on our line very accurately just by looking at it. For instance, was an 8-year-old's estimated height precisely 50 inches? Was it 51? Or 50.5? However, if we have the correct equation, we may use it to find the extract point on our line."</strong><br><br>If students use graphing calculators, instruct them to enter the data into the list section of their calculator (for TI calculators), and then direct them to have the calculator generate a line of best fit (using the stat menu). Students may need help comprehending the output of the calculator's linreg feature, which normally returns a value for <i>a</i> and a value for <i>b</i> that must be factored into the equation<i> y = ax + b</i>.<br><br><strong>"The equation of the line of best fit can also be found by hand, although it takes a lot of time,&nbsp;or with the aid of a spreadsheet application like Microsoft Excel or any other program. The equation for our data's best-fit line is </strong><i><strong>y</strong></i><strong> = 2.96</strong><i><strong>x</strong></i><strong> + 25.14."</strong> Write this equation on the board as a reference.<br><br><strong>"In our equation, </strong><i><strong>x</strong></i><strong> represents the age in years, as shown by the </strong><i><strong>x</strong></i><strong>-axis on the scatter plot and </strong><i><strong>y</strong></i><strong> represents the height in inches. So, use this equation to determine the predicted height of an 8-year-old."</strong> Allow students to work with partners if necessary, and provide supplementary teaching based on how they are performing.<br><br>Ask students to offer their responses. If there are many responses, work through the procedure with the class to determine the correct answer and clear up any mistakes that other students may have made. The correct height is 48.82.<br><br><strong>"What is the predicted height of a newborn according to our equation?" </strong>Assist students in understanding that when 0 is replaced with <i>x</i>, we are left with the predicted value of 25.14. On the scatter plot, add a point to symbolize this.<br><br><strong>"What do we call a point like (0, 25.14)?" </strong><i>(The y-intercept)</i> <strong>"When we look at our equation, we already have the </strong><i><strong>y</strong></i><strong>-intercept; all we need to do is add the number to the </strong><i><strong>x</strong></i><strong> term. How about 2.96? What does that mean?" </strong>If students forget, remind them that the equation <i>y = mx + b</i> is known as the "slope-intercept" version. Students should find the slope of the line is 2.96.<br><br><strong>"What does 2.96 </strong><i><strong>mean</strong></i><strong> in terms of the problem? We already know the term </strong><i><strong>y</strong></i><strong>-intercept refers to the height at age 0. How about the slope?" </strong>Guide students through a discussion of slope, also known as the <i>rate of change</i>, and how fast one number changes to another. Students might also be reminded of how to compute the slope given two points. The numerator is always the difference of the <i>y</i>-coordinates, and the denominator is always the difference of the <i>x</i>-coordinates.<br><br><strong>"So, when we discuss slope, the numerator represents the </strong><i><strong>y</strong></i><strong>-values. What quantity are our </strong><i><strong>y</strong></i><strong>-values representing? Age or height?"</strong> Students should understand that in this plot, the <i>y</i>-value reflects height. <strong>"So, our slope will represent the change in height each year; a slope of 2.96 indicates that a child's height changes by 2.96 inches every year. Does that imply that his or her height changes by exactly 2.96 each year?"</strong><i><strong> </strong>(No)</i> <strong>"Why not?" </strong>(Answers should mention that children develop at different rates, that our slope is an approximation, and so on.) <strong>"However, we can continue to use it as an estimate. Based on our line of best fit, we already know that an 8-year-old has an expected height of 48.82 inches, with an increase of 2.96 inches per year. What might we expect for a 9-year-old's height based on this data?"</strong> <i>(51.78 inches).</i><br><br>Explain to students that they might have solved by simply adding 2.96 to 48.82 or by entering <i>x</i> = 9 into the equation and obtaining the associated value of <i>y</i>. Depending on the lesson, have students guess heights for 10- and/or 11-year-olds. <i>(54.74 and 57.70 inches, respectively.)</i><br><br><strong>"How tall should a 40-year-old expect to be based on our line of best fit?"</strong> Students must comprehend that they should return to using the equation rather than repeatedly adding 2.96. They should get an answer of 143.54 inches. <strong>"144 inches equals 12 feet tall. So, what went wrong?"</strong><br><br>Help students understand that the line of best fit can only be used to make prediction up to a particular point, depending on the situation. <strong>"Why doesn't our line of best fit work for predicting the height of 40-year-olds?"</strong> Students should understand that at a certain point, people stop growing (and may even begin to shrink), so our line will not be a reliable predictor.<br><br><strong>Activity 3</strong><br><br>Students should collaborate in small groups to brainstorm quantities that they believe have a linear relationship. It doesn't have to be a perfect linear relationship; the two values could even have a negative correlation. Make a list, and then choose one to form a group. All members of the group should should research the relationship to come up with some reasonable values. For example, if they choose shoe size and height, they can use their own shoe sizes and heights as data points. Have students collect 12 to 20 data points and build a scatter plot to demonstrate the relationship. Then, have students estimate or use a graphing calculator to find the line of greatest fit.<br><br>Regroup the class for discussion. One group should define the quantities and indicate which quantity they put on the <i>x</i>-axis and which on the <i>y</i>-axis, and then provide their equation for a line of best fit. If they didn't create an equation, ask them how&nbsp;the y-intercept of their predicted line was determined, as well as the amount by which the <i>y</i>-value changed for each unit of change in the <i>x</i>-direction. Ask other groups about what the slope represent. Their answers should be expressed through the use of phrases such as "the extent to which [quantity] fluctuates for each [quantity]." As an example, "the amount that a person's shoe size increases for each foot s/he grows."<br><br>Examine all of the groups' scatter plots, reviewing the equation of best fit and, in particular, what the slope and y-intercept indicate (in units). This also provides an opportunity to explore the limitations of their line of best fit.<br><br><strong>Extension:</strong><br><br><strong>Routine:</strong> Throughout the year, show students examples of real-world scatter plots with lines of best fit to reinforce the principles of scatter plots and lines of best fit. Students should make predictions regarding the data and discuss the validity and likelihood of those predictions.&nbsp;<br><br><strong>Small Group:</strong> Ask students to collect bivariate data and create a scatter plot. Have them create a line of best fit and use it to make predictions about their data, as well as investigate its validity and limitations. Students can use NCTM's Line of Best Fit: <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>&nbsp;<br><br><strong>Expansion:</strong> Have students investigate non-linearly related data and work with polynomials of best fit of higher orders. Students can compare their predictions based on each model. Students can also use exponential curves of best fit. Even though they haven't seen many (or any) functions of this type, they can comprehend the function for various <i>x</i>-values.<br><br><strong>Technology: </strong>Ask students to&nbsp;use graphing calculators and spreadsheet software to investigate lines of best fit as well as higher-order polynomials of best fit.</p>