##
* *Reflection: Developing a Conceptual Understanding
Central Tendency - Section 2: Class Notes

Central tendency is a complex concept when it comes to understanding ** what** it describes about data. In 7

^{th}grade, students also learn about variability. As educators we must keep in mind the Common Core shifts of depth of knowledge. Before this curriculum existed, “mean” was taught along with median, mode

**range. Some of my students this year learned it this way. By grouping mean, median, mode and range this way, it becomes difficult for some students to separate range as a measure of something different. It is no longer expected to solely teach the process by which mean is derived; students must also understand how this measure can be limited so that they are ready to understand the role that measures of variability can play when analyzing data. By changing the way we discuss mean in this grade we can prepare our students for higher levels of statistics, rather than giving them a limited understanding which only ends up confusing them later in life and sometimes discouraging them from seeking careers in this field.**

*and*One way to change these notes and encourage rigorous discussions around ** the limitations** of the “mean” is to give students 2 sets of data with the same averages (or close to), but different variability. The first page of this Common Core lesson includes a great example elated to temperature. Some questions that could push this rigorous conversation include:

** “Find the mean temperature for each city” **– this question will allow students to practice the process of finding the average

** “Are the averages similar? Equivalent? What does this mean?”** – this question can push students to gradually consider

**mean is measuring. It may require some scaffolds, such as “**

*what*

*what does the ___ degrees represent?”, “what does it tell us about the city’s temperature?”, “since they are similar averages, does it mean that both cities have the same kind of climate? Why or why not?”*I would NOT introduce any measures of variability in this lesson, I would still keep that separate and investigate in the next lesson. The push during these notes and discussion should be to get students to recognize that while the two cities have very similar averages, they do not share the same kind of climate. Identifying ** what** is different about those climates (the variability) is a great way to provide extension for the higher kids. This will make them experts the following day when they compare variability of data sets. Additionally, the constant questioning of “

**” throughout this unit is necessary to push this complex topic.**

*what is the mean? What does it represent?**A More Rigorous Discussion*

*Developing a Conceptual Understanding: A More Rigorous Discussion*

# Central Tendency

Lesson 1 of 11

## Objective: SWBAT calculate mean, median, mode, and compare them as measures of central tendency.

#### Do Now

*10 min*

Students enter silently according to the Daily Entrance Routine. The Do Now includes percent problems students must review as evidenced by the data from the Unit Test. Students will also be provided with calculators since my goal is to focus on the procedures for solving problems like these. When I looked closely at students’ work on the Unit Test, I noticed many of them had no work and seemed to be guessing. I can’t be sure of the problem (is it conceptual or skill based?), thus I am utilizing the Do Now to assess.

As I walk around the room I am looking at students’ work, or lack of work, and forming strategic groups. Some students seem to understand the steps needed to solve. ** These students will be leading the review of the answers in the last five minutes of this section**. I make sure to flag at least 5 different students. I also notice the students who seem to struggle to complete the problems or answer using the wrong strategies; these students will be sitting with me when it is time to review.

**It is important to also select the students who are using the strategies taught in class (i.e. the formula for percent change and percent error).**

*Students selected to lead the review in each group will be given a review sheet to defend their correct answer.*After 5 minutes, I ask those 5 students I flagged to sit at booths and/or other empty tables in the room. Then I ask all students to form groups of 4 – 5 with these targeted students. I also announce that I will be pulling some students to work with me. While students are moving around the room, I tap the struggling students and ask them to sit with me.

** A note about negative reactions:** some students do not like to be told “you’re going work with me”. Any time that a student reacts negatively to me pulling them into my group, they will usually get the following response from me: “Please do not react negatively, I am trying to help you. If you do not want the help, please find a more appropriate way to let me know that. Do you think you can do that now?” If the student is so upset that they cannot, I ask them to sit out of this activity. If they DO find an appropriate way to respond, then I explain, “ok, you can go work with another group, however, I will be check back in with you at the end of class to make sure you understand how to do these problems. If you still do not understand, we will need to find a time to meet so that you can get the information you need”. This is an example of a situation where I stop to address character. Students are expected to communicate in an appropriate way and are held accountable for their choices. I usually come back to these students and we reflect on whether the choice was good or bad based on the data I pull from resulting work or exit tickets. This way, students are coached through making tough decisions which require prioritizing learning over friends.

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#### Class Notes

*15 min*

Students return to their seats and get ready for class notes. They receive the Cornell Notes style handout and are asked to fill out the heading, aim, and to read the first paragraph in the notes.

After about 2-3 minutes I ask for a student reader and we review the class notes, filling in the blanks and finding the mean, median and mode of the numbers. I do not talk about range because it is a measure of variation. Many students group mean, median, mode, AND range as measures of center, making it difficult to correctly identify these measures in word problems. Keeping these topics apart will help with the distinction between measures of center and variation.

Many students know how to calculate the mean but do not understand its meaning as a measure of center. The question at the end of the notes aims to get students to think about the purpose of an average in a real world problem. The big idea is that measures of center describe the middle set of the data, the “average joe”. This information can be used to compare two or more sets of data.

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#### Task

*20 min*

After reviewing the notes and answering questions, students receive their task. The questions on this task mainly aim to have students practice calculating measures of center. There are two open response questions which ask students to analyze the meaning of a measure of center. Asking the question, “which of these measures best describes her monthly spending habits?” is the same as asking students to consider the logic behind deviations. The measure which best describes the data will be the value which is closest to all of the distinct data points. This kind of connection is high on Webb’s wheel , thus many students will need guidance in understanding what the question is asking about and how to answer/think about the answer. This is definitely a question worth reviewing whole class at the end.

In the last 5 minutes of this section, I will be asking students to write their answers on the board. We will be reviewing the answers at the end of class.

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#### Closing

*10 min*

Students who were able to put their work up on the board ** will be asked to explain the logic behind his steps. **He/she may also choose to have me review the work for them.

Since students have calculators throughout this lesson, this is a great time to *emphasize showing work neatly, linearly showing the numbers used for each step, as shown in the example below.*

After reviewing each problem, students pack up and go to the next class.

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##### Similar Lessons

Environment: Suburban

- UNIT 1: Integers
- UNIT 2: Operations with Rational Numbers
- UNIT 3: Expressions and Equations - The Basics
- UNIT 4: Multi-step Equations, Inequalities, and Factoring
- UNIT 5: Ratios and Proportional Relationships
- UNIT 6: Percent Applications
- UNIT 7: Statistics and Probability
- UNIT 8: Test Prep
- UNIT 9: Geometry

- LESSON 1: Central Tendency
- LESSON 2: Comparing Distributions
- LESSON 3: Line plot & Stem-and-Leaf Plot
- LESSON 4: Comparing Distributions Part II
- LESSON 5: Variability
- LESSON 6: Measures of Variation - Range and IQR
- LESSON 7: Box and Whisker Plots
- LESSON 8: Mean Absolute Deviation
- LESSON 9: Quiz + The Language of Probability
- LESSON 10: Theoretical vs Experimental Probabilities
- LESSON 11: Compound Probability