For this lesson students will work in partnerships to share ideas and critique each other's thinking around several of the questions that are posed during the direct instruction portion of the lesson. I will use these slides to focus students' attention: Measures_of_Center
Slide #1: The title of this slide says it all...this should be an investigatory opening activity. Students are experimenting with how changing the data set can have an effect on the mean (MP2). I have students make a table in their notebooks to keep track of the "original average" 82.5% then "new score" either 95%, 60%, and 83% and the "new average" either 86.7%, 75%, or 82.7%. Once students find the new averages I ask them to work with their partners to write a summary of what they have discovered (MP3) as a result of their work. We summarize by having different groups share out on their summaries.
Instructional Note: While most students will make the observation that a score higher than the original average makes the average go up, I push them to look at the amount of the change. This is an important understanding to see that while a score of 95% (for example) is significantly higher than 82.5% it only moves the average about 4.2%. When questioned, students can usually come up with the fact that it is because the 82.5% is a representation of two scores not one. If it was a representation of even more scores you would get even less "movement" of the average with a higher score. I find that often students have a misunderstanding of this concept.
Slide 3: Some students really struggle to come up with one set of four numbers that will work for this question. Others come up with the numbers relatively easily. For those students, I challenge them to come up with as many ways as then can in the few minutes that they are working on this task. By coming up with multiple solutions students begin to see the pattern that any four numbers that sum to 24 will be a viable solution. Being able to fabricate a set of numbers with a given mean helps them to push their understanding further.
Slide 4: This is a typical "exam" style question that I ask student to solve purely by "systematic" trial and error. Once they calculate that the average of the four test scores so far is 85%, they know that they must have a score that his higher than 85% in order to pull the average up to an 87% (refer back to the activity in slide 2). By listening to student conversations and looking at their trials you can learn A LOT. Students who start with 86 as their first guess to not have the understanding that the new score must be much higher than 85% because the 85% is a representation of four scores. Students who have that understanding will usually start with a score in the 90's and adjust from there. Once all students have successfully completed the problem by trial and error, we discuss an algebraic solution by cross multiplication. While the algebraic solution is obviously the preferred and most efficient method, an understanding of how the number work together to come up with the desired mean goes a long way to judging the reasonableness of a solution.
Slide 5 and Slide 6: Both of these slides help students begin to uncover an important idea surrounding measures of center, namely, that the mean of a data set can be effected by small or large members of that set. In slide 5, we are asking if the median or the mean is a better representation of the data. Most students will say the median because a majority of the data are the numbers 3 or below making is a better representation of the set as a whole. Because students can now have a solid understanding of the effects of data on the mean, they instantly see what the 6, 8, and 10 in this data set will do. Slide 6 brings up a similar concept and when students calculate that the average of these 4 scores is a 77% they begin to question the idea of using averages to represent understanding. The vocabulary term outlier is also addressed here.
This Independenent Practice is differentiated based on need. Students who still need work simply calculating the measures of central tendency are given the opportunity to do so while those that have a fluent understanding can challenge themselves with problems that have an evaluative aspect.
Before I return the ticket out the door from yesterday's class, I put a green dot on the students who should be doing the first two pages and a red dot on the students who will be working on pages three and four.
On Page seven of the Measures_of_Center PowerPoint there is a short writing prompt. The student answers to this question will give you a good idea regarding which students have gained an understanding of the lesson. Again, many students will be able to calculate the measures of center but having an understanding about how the structure of the data set effects those measures is the goal of the lesson.
Instructional Note: Students who have made progress with the goals of this lesson will be given access to the more complex questions.