Mean as Fair Share and Balance Point VMSTC November 8, 2010

Mean: Fair Share and Balance Point

Mean: Fair Share 2009 5.16: The student will a) describe mean, median, and mode as measures of center; b) describe mean as fair share; c) find the mean, median, mode, and range of a set of data; and d) describe the range of a set of data as a measure of variation. Understanding the Standard: “Mean represents a fair share concept of the data. Dividing the data constitutes a fair share. This is done by equally dividing the data points. This should be demonstrated visually and with manipulatives.” The blue text represents additional changes in the Standard. The red text represents the change that we are addressing today. *This convention is also used on Slide 76 for SOL 2009.6.15. Understanding the Standard is copied directly from the 2009 Curriculum Framework.

Understanding the Mean Each person at the table should: Grab one handful of snap cubes. Count them and write the number on a sticky note. Snap the cubes together to form a train. We will be asking participants to engage in two separate activities – this one uses snap cubes to demonstrate mean as fair share, and the sticky note activity demonstrates mean as balance point. Michael has suggested that we should take the group up through Slide 79 (the example where the mean doesn’t work out neatly to a whole number) BEFORE we conduct the second activity. The instructions for that activity are in the notes for Slide 79.

Understanding the Mean Work together at your table to answer the following question: If you redistributed all of the cubes from your handfuls so that everyone had the same amount (so that they were “shared fairly”), how many cubes would each person receive? “Teach” this lesson similarly to the way you would with students.

Understanding the Mean What was your answer? - How did you handle “leftovers”? - Add up all of the numbers from the original handfuls and divide the sum by the number of people at the table. - Did you get the same result? Obviously, the answer represents the mean – but make sure they understand it represents a “fair share” as the underlying concept of the mean. - What does your answer represent?

Understanding the Mean Take your sticky note and place it on the wall, so they are ordered… Horizontally: Low to high, left to right; leave one space if there is a missing number. Vertically: If your number is already on the wall, place your sticky note in the next open space above that number. Building a line plot for the activity at the end of Slide 79.

Understanding the Mean How did we display our data? From the Curriculum Framework. 2009 3.17c

Understanding the Mean Looking at our line plot, how can we describe our data set? How can we use our line plot to: - Find the range? - Find the mode? - Find the median? “Teach” again to make sure they can find the range, mode, and median before moving to the next slide about using a line plot to find the mean (the balance point). - Find the mean?

Mean: Balance Point 2009 6.15: The student will a) describe mean as balance point; and b) decide which measure of center is appropriate for a given purpose. Understanding the Standard: “Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of all the points above the mean is equal to the sum of the distances of all the data points below the mean.” Understanding the Standard and Essential Knowledge & Skills info is taken directly from the Curriculum Framework. *For the record, the last bullet on this slide is not the only bullet in the Framework: • Find the mean for a set of data. • Describe the three measures of center and a situation in which each would best represent a set of data. • Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data. Essential Knowledge & Skills: Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data.

Where is the balance point for this data set? X X X X X X We can manipulate the data points to help us see where the balance point would be. Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 11

Where is the balance point for this data set? X X X X X X We can manipulate the data points to help us see where the balance point would be. Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 12

Where is the balance point for this data set? X X X X X X We can manipulate the data points to help us see where the balance point would be. Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 13

Where is the balance point for this data set? X X X X X X We can manipulate the data points to help us see where the balance point would be. Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 14

Where is the balance point for this data set? X X X X X X We can manipulate the data points to help us see where the balance point would be. Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 15

Where is the balance point for this data set? MEAN Sum of the distances below the mean 1+1+1+2 = 5 Sum of the distances above the mean 2 + 3 = 5 X X X X X X Now refer back to the original line plot. We know that 3 is the balance point or mean. Talk about the sum of the distances above the mean being the same as the sum of distances from the mean below the mean. 16

Where is the balance point for this data set? Move 2 Steps Move 2 Steps Move 2 Steps Move 2 Steps We can manipulate the data points to help us see where the balance point would be. Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean. *Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance. And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 4 is the Balance Point 17

The Mean is the Balance Point We can confirm this by calculating: 2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36 36 ÷ 9 = 4 The Mean is the Balance Point 18

Where is the balance point for this data set? If we could “zoom in” on the space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point. Move 1 Step The Balance Point is between 10 and 11 (closer to 10). Move 2 Steps Move 1 Step Move 2 Steps Sticky Note Activity: Work with the whole group to use this strategy to find the mean number of cubes in one handful based on our data set. If it doesn’t work out to be whole number, discuss how we could find the exact decimal value of the mean if we could “zoom in” and how we could estimate the mean based on the modified line plot. *You may want to have a calculator handy to find the actual mean of your whole group data set. 19

Mean: Balance Point When demonstrating finding the balance point: CHOOSE YOUR DEMONSTRATION DATA SETS INTENTIONALLY. Use a line plot to represent the data set. Begin with the extreme data points. Balance the moves, moving one data point from each side an equal number of steps toward the center. Continue until the data is distributed symmetrically or until there are only two values left on the line plot. By “intentionally” we mean that teachers should begin with data sets that have whole number means, only progressing to rational number means once students understand the concept and process. When introducing decimal means, teachers should begin with “neat” data sets. - For example: {2, 3, 4, 5, 6, 7} would have a mean of 4.5 (27/6 = 4.5), which would be easy to see once the line plot was transformed so that there were three 4’s and three 5’s.

Assessing Higher-Level Thinking Key Points for 2009 5.16 & 6.15: Students still need to be able to calculate the mean by summing up and dividing, but they also need to understand: - why it’s calculated this way (“fair share”); - how the mean compares to the median and the mode for describing the center of a data set; and - when each measure of center might be used to represent a data set. Emphasize the shift to higher-order thinking from the 2001 to 2009 Standards.

Mean: Fair Share & Balance Point “Students need to understand that the mean ‘evens out’ or ‘balances’ a set of data and that the median identifies the ‘middle’ of a data set. They should compare the utility of the mean and the median as measures of center for different data sets. …students often fail to apprehend many subtle aspects of the mean as a measure of center. Thus, the teacher has an important role in providing experiences that help students construct a solid understanding of the mean and its relation to other measures of center.” - NCTM Principles & Standards for School Mathematics, p. 250 Connection between the new SOL and the NCTM Standards emphasizing that the teacher must provide learning experiences that will promote students’ conceptual understanding. The days of “just add ‘em up and divide” are behind us :o)