Playing with Measures of Central Tendency

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want students to analyze a line graph in order to answer questions. Scholastic News  has a 5^th-6^th grade edition that typically includes a graph on its back page.

I ask for students to share their thinking.  I am interested to hear about the strategies students used to determine the answer to question 3.  I ask students what we should get when we add up all parts of a circle graph.  I want students to understand that the entire circle graph represents 100%.  I also want students to realize that this circle graph is not accurate because the sum of all of the pieces is 99%.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

I have students work in partners to complete this review from the previous lesson.  The most common mistakes I see are that students confuse the median, mean, and mode.  I want to make sure students understand what each term means and how to find it. Another common mistake is that students divide the new sum by 5 rather than 6.

Problem two gets to the big idea of this lesson.  It is important that before calculating the changes, students make predictions.  I ask students to share out their findings for problem 1 and 2.  I ask students:

• Why did the mode and the median stay the same?
• Why did the mean change?

I want students to understand that the mode stayed the same, since $3 is still the value that occurs most often.  The median stayed the same, since $3 is still in the middle.  The mean increased because we added a game that was more money.  Our new sum was $30, and we divide it by 6 since there are now 6 games.  Some students may not see this pattern yet, and that is okay.  I want students to start to recognize and anticipate how changes will affect the mode, median, and mean of a data set during their group work.

Notes:

• Before this lesson, I use the exit tickets from the previous lesson to Create Homogeneous groups of 3-4 students. 
• I copy sheets A-E and place them in the back of the room.
• Each group also gets a Group Work Rubric.
• I create and Post a Key.

Students move into groups and I pass out materials.  We work together to calculate the mean, median, and mode of the six games.  I have a volunteer read the directions for the groups.  I make sure that students understand that with each new problem, they are starting with the original set of six games and making a change.

Students work in their groups while I walk around to monitor student progress and behavior.  Students are engaging in MP2: Reason abstractly and quantitatively, MP6: Attend to precision, and MP8: Look for and express regularity in repeated reasoning.

If students are struggling, I may ask them one or more of the following questions:

• How do you calculate the mode/median/mean?
• What do you think will happen with this new piece of information?  Why?
• How did your prediction compare with the new mode, median, mean?  Why is this?

If students are on track and need some extension I may ask them one or more of the following questions?

• What will happen if you add another game that is more than the original mean? Prove it.
•  What will happen if you add another game that is less than the original mean? Prove it.
•  What will happen if you add another game that is the same price as the original mean? Prove it.

When students finish a page, I quickly check in with them.  If they are on track, I let them use the key to check their work and move on.  If groups complete all of this work, they can work on the challenge problems.

I ask students to share what patterns they noticed when they added a new game to the original six games.  Students participate in a Think Pair Share.  I want students to be able to anticipate how a new piece of information will affect the mode, median, and mean of a data set.   Students are participating in MP3: Construct viable arguments and critique the reasoning of others and MP8: Look for and express regularity in repeated reasoning.

If I have time I ask students, “What if we add one game to the original set and the new game causes the mean to change from $7 to $8.  How much did the new game cost?”  Students participate in a Think Pair Share.   I am interested to see the strategies students use.  Students should be able to understand that the new game must be more than the original mean of $7.  Some students may guess and check.  Other students may work backwards and realize that the sum of the 7 games must equal $56, since 56 divided by 7 equals $8.

I pass out the Ticket to Go and the Homework.