# Change and Central Tendency

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## Objective

SWBAT determine how a change in a data set impacts the measure of center.

#### Big Idea

We can make predictions as to the effect additional values will have on the measures of central tendency by having an understanding of numbers and their influence on the measures of central tendency.

## Think About It

7 minutes

Students work in pairs on the Think About It problem.  Because of the work from the Data Distribution Unit, students are able to quickly find the mean of the data set in this problem.

For the second question in this problem, I found some students were a bit vague in their responses.  Many wrote 'The average will change,' without making a prediction about how the average would change.  We had a conversation around what would happen to the numbers, and I had students articulate why they thought the average would increase (some students also decided to complete the 3rd part first, and then answer with something along the lines of 'The average will increase to 82.5.')

I ask the students why the overall average only increased by 2 points, if the quiz score increased by 8 points.  I want students to think through how we find the mean, and what an average really tells us.

You can see student thinking in this Think About It sample.

## Intro to New Material

20 minutes

The new piece of learning in this lesson is around what the effect of adding additional values (or changing values) to the data set will have on the measures of central tendency.  The Intro to New Material section of this lesson provides many opportunities for students to practice their fluency with finding the mean, median, mode, and range of a data set. They can do this work without my support.

I have students read Example 1, and then jot down their response on the line.  I ask a student what they think will happen to the median when the chilly day is added.  I then ask another student to justify the response - why will the mean decrease?  I go through the same process for the question around the mode of the data set.

We then move on to Example 2.  Much like the first problem, I have students write down their predictions.  During this time, I am watching to be sure that kids aren't trying to calculate the actual measures of central tendency.  I want them to make predictions, based on the magnitude of the numbers.  It's okay if students are wrong with their predictions - they will solidify their conceptual understanding as we move through the 'Prove It' sections.

Students help me fill in the blanks for the Key Points.  I have students whisper the words they think should go in each blank.

## Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.  As they work, I circulate around the room and check in with each group.  I am looking for:

• Are students making accurate predictions as to the impact values will have on a data set
• Are students looking at data and extracting information in order to calculate measures of central tendency?
• Are students correctly calculating measures of central tendency?
• Are students correctly justifying why a change happened to a measure of central tendency given the addition of another data point?

I'm asking:

• How did you know the mean/median/mode/range would increase/decrease?
• How can you prove this answer?
• What impact would the value _______ have on this data set? How do you know?
• What would happen to the median if we added ____ (the number that is the median)?  Why?
• Is there a number that we could add to this data set that would not change the mean/median/mode?  Why?

After 10 minutes of partner work time, students complete the Check for Understanding problem independently.  They then turn back to their partners and share out their thinking for each of the two problems.

## Independent Practice

15 minutes

Students work on the Independent Practice problem set.  Students have access to calculators throughout this lesson.  As I circulate, I am watching to be sure that students are making predictions, and not finding the actual measures of central tendency first.

In this lesson, my students struggled with the problems that asked them to add to the data sets.  The next time I teach this lesson, I will spend more time discussing the questions at the end of the problem set.

After independent work time, I have students turn and talk with their partners about problem 3, and have them share out their 'sometimes, always, never' responses with one another.  These problems give students the opportunity to give examples and counter examples, and gives the opportunity for students to make a claim and justify it with an example.

## Closing and Exit Ticket

8 minutes

After student pairs have discussed Problem 3 from the Independent Practice set, I bring the class back together for a conversation about Problem 8.  If my students didn't have a chance to get to this problem, I'll give 2 minutes of silent work time before we discuss.  I really like how this problem requires students to reason about range and how data might impact it.

Students then work on the Exit Ticket to close the lesson.  As this exit ticket sample shows, part of the last question was difficult for students.  The data from the exit tickets let me know that I did not do enough to ensure students mastered the idea of adding a number to the data set that would keep the measure of central tendency the same.  Many students responded that a 0 would keep the median at 32.5.