Mean as a Balance Point

So, if you teach in Virginia you would that the state (and subsequently my county) have once again changed our pacing guide! We have new standards and objectives that we have to teach...and most days it feels like less time...

BUT, one of the new concepts that we have to teach is "mean as a balance point".  This seems like a pretty simple concept...in fact I applaud you whoever came up with this because for once it is a developmentally appropriate concept for 6th graders...

BUT in 5th grade, students are supposed to learn mean/average as a fair share concept...let's face it most of us do not do this.  We just teach the students to do the algorithm by adding them all up and dividing.

BUT, what does this mean??? Why do we do it??  Why does adding them all up and dividing work? I want my students to be able to communicate what mean is. I want them to show me how they can prove it without using the formula, and let's face it I want them to pass the new standardized test that asks them to use the balance point method.

So, while they may understand that mean is another word for average, which can be found by using the "fair share" or "balance point" methods, it is difficult for students to UNLEARN an algorithm   You would think I asked them to solve a 10th grade math problem!  They freak out when I do not allow calculations of any kind.

What I did allow was manipulatives.

Using a word problem from "Navigating through Data Analysis and Probability in Grades 3-5" I asked the students to create a line plot that shows me seven possible scores from a soccer game.  The students have some discussion about what is probable and what they have background knowledge of.  I then ask the students to use the algorithm to find their mean.  The majority of students have different scores.  I then ask them, what if my mean is 2, what could my seven scores be then?  The students continue to use the algorithm to try and solve my problem.  Most of them can find these scores.  We then take a gallery walk to see all the other line plots.  The students start to notice that line plots differ, that the data can be different even if the mean is the same.  I think do my mini-lesson on balance.  I connect it to a balance beam and absolute value (the distance from 0).  From there I ask them to try to find two more line plots that have a balance point of 2.  This took the rest of my 1 hr. long class.  The next day, we used the same line plots and manipulatives to find a line plot that has a balance point of 2:
A. only 1 game had 2 goals
B. Exactly two games had 4 goals
C. One game had an amazing 7 goals!
D. The median of the data set is 3.

This 1 activity took two days, but I am confident and have two tests (1 multiple choice and 1 open ended) to prove their understanding of the topic.  They are able to communicate what "fair share" is, what a "balance point" is, and what does average mean.


I also tied this math concept into our social studies unit about the House of Representatives.  We took a census of how many people live in your household and created a bar graph, table, and line plot from this data. The students had to find the balance point from this data.