measures of center lesson plans, mathematical tasks, and solutions

Grade and Context Area: 6th Grade Mathematics Content Standards Addressed:*Note: Engaging in this task does not mean that students completely meet these standards. The following standards are listed because students will likely complete work that matches all or specific parts of the standard.

CCSS.MATH.CONTENT.6.SPA.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

CCSS.MATH.CONTENT.6.SPA.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

CCSS.MATH.CONTENT.6.SPA.5C Summarize numerical data sets in relation to their context, such as by: Giving quantities measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

CCSS.MATH.CONTENT.6.SPA.5.D Summarize numerical data sets in relation to their context, such as by: Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Information taken from the Common Core State Standards: http://www.corestandards.org/Math/Content/6/SP/

Assumed Student Prior Knowledge: Students have used TinkerPlots before and understand the following functions of

the program:o How to create a plot and pull attributes from the cards onto the x-axis and

y-axis o How to pull different plot points in order to spread out the data or put it

into separate bins. o How to stack data vertically and horizontallyo How to add/remove a vertical and/or horizontal reference/movable lineo How to add/remove the measures of center (mean, median, and mode)o How to have the mouse hover over a measure (mean, median, mode,

aspects of box plots, etc.) and look in the bottom left hand corner to see the associated value(s)

o How to create a box ploto How to show/hide outliers on a box plot

Students will know the definition of and will have had practice finding the following terms. This means they will know what these terms represent (i.e. the mode is the data point that occurs most frequently in a data set) but may not yet be comfortable deciding which measure to use in a given context.

o Meano Mediano Modeo Range

Students will have used dot plots to plot uni-variate data. Students will have had practice using a dot plot to analyze data sets and draw conclusions about those data sets.

Students will have used box plots to analyze uni-variate data. Students will be comfortable using the different measures in box plots, including quartiles and interquartile range, to draw conclusions about the data.

Mathematical Goals: Content Goals:

o Students will be able to describe the measures of center (mean, median, and mode).

o Students will be able to demonstrate that they understand certain measures of center apply better to certain situations than other measures of center by identifying specific measures of center for given situations.

o Students will be able to correctly and appropriately use dot and box plots to analyze data and draw conclusions from it.

Mathematical Thinking Goal:o Students will develop skills that help them think about mathematical

concepts in real life contexts. Mathematical Practices Goals:

o Students will engage in the 1st Mathematical Practice, Make Sense of Problems and Persevere in Solving Them, by attempting to analyze the data in multiple ways, asking questions when needed, and pushing through the problem until it is complete even if it gets difficult.

o Students will engage in the 3rd Mathematical Practice, Construct Viable Arguments and Critique the Reasoning of Others, by justifying their choice of measure of center and plot given different situations. Students will possibly engage in conversations with peers about these choices where they may have to defend their reasoning or ask questions of others.

o Students will engage in the 5th Mathematical Practice, Use Appropriate Tools Strategically, by effectively using TinkerPlots, an exploratory data analysis program, and its various functions to draw conclusions about a data set.

Objective Written in Student-Friend Language:o Students will be able to collect and organize data, make predictions about

its measure of center, manipulate data, and draw conclusions by using TinkerPlots.

Lesson Outline:Day One, Getting Prepared:

Students will engage in an experiment to collect data because it is easier for students, especially 6th graders, to think about mathematical situations if they are somehow meaningful to them.

o I will pass out a bag of M&Ms to each student in the class and provide a worksheet to prompt students to complete the following tasks and keep their information organized.

o Students will use scales to weigh their bag of M&Ms. o Students will open the bags and count the total number of M&Ms. o Students will divide the weight of the bag by the number of candies inside

it in order to determine the unit weight for each M&Ms. o Students will count the number of each color of M&Ms. o Students will record their individual data on a piece of paper and transfer

it to a whole-class excel spreadsheet. Once all students are finished experimenting and all information is put into the

spreadsheet, I will use it to create cards and a table in TinkerPlots. I will project the table for students to see and I will lead students in a discussion

focused on making predictions about the data set. We will write our predictions on the board and check them after we explore the data set.

o This will engage students with the data prior to manipulating it. o This will be helpful because students will gain practice with making

predictions and thinking about what logically makes sense for a given set of data in a specific context.

After the class makes predictions, I will front load important vocabulary by asking students to complete an activity where they will review the terms, mean, median, mode, and range. In this way, students will be prepared to engage with these concepts the next day.

After school I will look at the data the students collected and compiled into one spreadsheet. I will plot it in TinkerPlots and explore different aspects of it. I will find specific examples I wish to point out to students and have them explore. I plan to use the following task with students but if it does not make sense with the data they collected, I will change it accordingly.

o For an example I have used the Sample M&Ms data found in TinkerPlots.

Day Two, Implementing the Task: I will set up today’s task by having students review the terminology we discussed

yesterday (, mean, median, mode, and range) with peers around them. I will also have students review the predictions we made yesterday and will set the stage for the lesson by explaining that we will explore the data we collected in order to determine if our predictions were correct.

Before handing out the task, I will share my goals with students. I will use the following language so the students can best understand what is expected of them and will then post the goals in a visible area for students to refer to throughout the lesson as needed. I will provide time for students to ask questions about the goals they are attempting to meet before passing out the mathematical task.

Mathematical Content Goals in Student-Friendly Language:o Describe the three measures of center (mean, median, and mode).o Be able to answer the question “does context matter when I choose which

measures of center to use?”o Use dot and box plots to learn about data.o Be able to determine which type of plot to use for certain sets of data.

I will pass out a piece of paper with the task printed on it. Students are to complete the task in a word document and include screenshots of plots when they

feel the plots are appropriate/support their answer. Instructions for how to type their answers and include screenshots of their work will be provided, as well.

While students engage in the task, I will walk around the classroom and interact with individuals or small groups of students. As I check in with students I will ask myself,

o Do students understand what the task is asking?o How are students responding to questions in the task?o What measures of center are students using?o Are students explaining their thought process/justifying their mathematical

choices? Have students work independently but allow them to complete the following math

task focused on exploring uni-variate data.

The Task:1. Create a plot in TinkerPlots and pull the data Number_Candies attribute onto the

x-axis. Spread the data out as far as it can go so it is not in separated bins. Remember the different tools in TinkerPlots (stacking, box plot, measures of center, etc.) can be used to help you at any point in this investigation.

2. What is the range for the number of candies per bag? Do you think this range is big or small? Why do you think it is like that?

3. Change your plot so the Weight attribute is now on the x-axis. The M&M company claims that each bag contains 47.5 grams of candy. How does this compare to your bag?

4. How does the company’s claim of 47.5 grams compare to what the entire class found? Use evidence from your plot to support your answer.

5. What measures of center did you use to find your answer for Questions 3 and 4? Why? What type of plot did you use? Would using a different measure of center change your answer? If so, how would your answer change and if not, why not?

6. Pretend the M&M company no longer wants to label their bags by weight. The company instead wants to label them by number of candies per bag. What would you expect the company to label the bag with? Why? Use evidence from your plot to support your answer.

7. What measures of center did you use to find your answer for Questions 6? Why? What type of plot would you use? Would using a different measure of center change your answer? If so, how would your answer have changed?

8. Based on this information and your knowledge of companies and producing goods (think back to your social studies class!), what do you think is the best way to label the bags? Defend your answer and use evidence from TinkerPlots as needed.

Day Two, Implementing the Task Continued: Once students complete this task, which may take multiple days, I will bring them

all back for a discussion on the following concepts:o I will have students share answers and responses to Question 3 and talk

about why everyone’s answers differed.

o I will have students make predictions about the class’ responses to Question 4. Will everyone’s answers be different again? Will they be the same? Why or why not?

o As a class, we will review responses to Questions 4 and 5 by first having students compare their answers to the answers of peers around them. Then students will share what they found in the whole class setting.

If students all used the same measure of center (not very likely), their answers may be similar.

If students all used different measures of center, their answers may differ.

o As a class, we will review responses to Questions 6 and 7 by first having students compare their answers to the answers of peers around them. Then students will share what they found in the whole class setting.

If students need support I will have them list their answers on the board along with the measure of center they used to find their solution. This will hopefully help students begin to see the connections between measure of center used and solution.

Students will likely see the connection between different measures of center and different answers.

o I will strategically call on students who I know can explain their thinking well to peers. I will ask them to explain why they chose the measure of center they did.

o I will then summarize what those students said and hint at the idea that context (the problem situation) played a role in how they decided what measure of center to use.

o As a class, we will review answers to Question 8. I will ask students to raise their hands if they supported labeling the bags by weight and then by number of candies. I will ask students from each side to defend their answer(s).

After the whole-group discussion, I will pass out the following Exit Ticket for students to complete by writing answers on the handout before leaving for the day:

Exit Ticket:1. Define briefly in your own words the following terms:

a. Meanb. Medianc. Mode

2. Is one of the 4 measures of center we used today always the best choice for every situation? Why or why not? Feel free to use evidence from today’s task to support your answer.

3. Describe the differences between dot and box plots. a. Describe a situation where using a dot plot to analyze data would be

helpful. Why would a dot plot be the best choice in your situation? b. Describe a situation where using a box plot to analyze data would be

helpful. Why would a box plot be the best choice in your situation?

4. What was your biggest take-away from today’s task and exploration?5. What questions do you have? The Exit Ticket will provide me with information about my students’

understanding of the content goals I set out originally because I wrote the questions with the mathematical goals in mind. Each question on the Exit Ticket provides me with information about where a student is in relation to understanding and meeting the mathematical content goals. I believe that the questions on my Exit Ticket also align with the task students completed. I plan to gain information about how students meet the mathematical thinking and mathematical practices goals through observations of and discussions with students because I feel these goals are more individualized and subjective.

Day Three, Extending:

Because examining bi-variate data is complex and can be considered advanced for 6th grade students, I created a task that will help students transition into this type of data analysis as an extension activity.

The extension is meant to introduce students to thinking about bi-variate data and help them determine what the plot they create actually represents. I have assumed they have prior knowledge (likely from previous math or science courses) about the x-axis and y-axis as well as independent and dependent variables. The questions for this extension scaffold students to start thinking about what the points on the graph, and what the graph as a whole, actually mean. This extension was created not only to introduce students to bi-variate data sets, but also to help them figure out how to think about them compared to uni-variate data. The questions were also written to help students understand that it is still important to think about the context of the problem when analyzing bi-variate data sets.

Mathematical Content Goals for Day Three, Extending:o Students will be able to demonstrate their understanding that people look

at uni-variate and bi-variate data sets in different ways. o Students will be able to list things to consider when looking at uni-variate

data sets. o Students will be able to list things to consider when looking at bi-variate

data sets. This extension includes a task made up of eight questions. Questions 1-5 prompt

students to explore independently, Questions 6 and 7 are challenge questions, and Question 8 will be used for whole-class discussion, reflection, or an Exit Ticket.

The Extension Task: 1. Create a plot with the Number_Candies attribute on the x-axis and the Weight

attribute on the y-axis. In this situation, which attribute is the dependent variable? Which attribute is the independent variable? Does this make sense in the context of the problem? Why or why not?

2. Based on your plot, what can you say about the relationship between the weight of the bag of M&Ms and the number of candies per bag? Does this make sense in the context of the problem? Why or why not?

3. Do you think a similar relationship to the one you found in Question 2 exists if we looked at other candies, such as skittles? Why or why not?

4. Make a prediction about the Unit Weight attribute and its relationship to the number of candies per bag. Make a prediction about the Unit Weight attribute and its relationship to the weight of each bag. How would you use TinkerPlots to test your predictions?

5. Test your prediction using the methods you just identified in Question 4. Was your prediction correct? Why or why not?

6. Do you think that the color of an M&M matter in determining its weight? Put differently, do you think that certain colored M&Ms weigh more than others? How might you use TinkerPlots to discover relationships between the color of M&Ms and the weight of the bag?

7. Make a prediction about the relationship between the color of M&Ms and the weight of the bag. Then use TinkerPlots to test your prediction. Were you correct? Why or why not?

8. Based on the questions in this extension and the plots you created, compare methods for analyzing uni-variate versus bi-variate data.

a. What should be considered when analyzing uni-variate data? b. What should be considered when analyzing bi-variate data?

How Each Aspect of the Original Mathematical Task Relates to the Common Core State Standards and Mathematical Goals:I will address this question by listing each individual question in the task and explaining its purpose as well as how it aligns with the standards and goals I have laid out.

1. Create a plot in TinkerPlots and pull the data Number_Candies attribute onto the x-axis. Spread the data out as far as it can go so it is not in separated bins. Remember the different tools in TinkerPlots (stacking, box plot, measures of center, etc.) can be used to help you at any point in this investigation.

The purpose of this question is to help students set up a plot that will be beneficial for them as they complete this task.

This question aligns the standard CCSS.MATH.CONTENT.6.SPA.4 because students use TinkerPlots to display numerical data in plots. Although students follow directions to do so, they still independently engage with this standard. Other questions found in this task encourage students to engage more independently with this standard.

This question does not specifically align to a certain mathematical goal because its purpose is to help prepare students to complete the rest of the task.


The purpose of this question is to help students start thinking about all of the data in a way that is familiar to them. Because students have likely had much practice with the idea of range before 6th grade, this question should not be too difficult for them to answer. This question asks students to think about all of the data in the data set. Getting in this mindset (rather than just thinking about individual data points) will be beneficial as students complete this task.

This question aligns with the standard CCSS.MATH.CONTENT.6.SPA.5.C because students need to think about the data as well as the context. In order to answer this question, students must interpret the size of the range of cadies per bag and think about why that might be. Likely students will relate this to some aspect of the context of the problem such as the production process.

The mathematical thinking goal associated with this question is Students Will Develop Skills that Help Them Think About Mathematical Concepts in Real Life Contexts because students must think about the idea of range in the context of the problem situation.


The purpose of this question is to help students set up a plot that will be beneficial for them as they complete this aspect of the task. Another purpose of this question is to act as the first part in a two-step scaffold (see Question 4). By asking students to look at one specific data point, students will begin to analyze the data set in a way familiar to them.

This question does not necessarily align with a specific standard, rather, it helps prepare students to answer questions that align with standards later in the task.

The mathematical think goal associated with this question is Students Will Develop Skills that Help Them Think About Mathematical Concepts in Real Life Contexts because students must think about their bag of candy in relation to the M&M company’s estimate of 47.5 grams per bag. This helps students begin to think about mathematical concepts in real life contexts.

4. How does the company’s claim of 47.5 grams compare to what the whole class found? Use evidence from your plot to support your answer.

The purpose of this question is to act as the second part in a two-step scaffold (see Question 3). By asking students to look at one measure compared to a large set of data, students must begin to think about the data as an entire class rather than think about individual data points.

This question aligns with the standard CCSS.MATH.CONTENT.6.SPA.2 because students must think about the distribution of the data set. Although they are not necessarily asked to describe the distribution using measures of center, they will likely have to think this way to answer to question. They are prompted to explain which measures of center they used to answer this question in Question 5.

This question aligns with two mathematical goals. The first mathematical goal it aligns with is the thinking goal Students Will Develop Skills that Help Them Think About Mathematical Concepts in Real Life Contexts because students are asked to think about the class’ data in relation to the M&M company’s estimate meaning students will think about mathematical ideas in real life contexts. The second mathematical goal it aligns with is the content goal Students Will Be Able to Describe The Measures of Center (Mean, Median, and Mode) because students will likely use a measure of center to find their solution to this question.

5. What measures of center did you use to find your answer for Questions 3 and 4? Why? What type of plot did you use? Would using a different measure of center change your answer? If so, how would your answer change and if not, why not?

The purpose of this question is to have students explain the thinking they experienced answering Questions 3 and 4. Students are asked to think about how they decided to measure the spread of the data. It is important for them to have an idea of why they chose which measure at this stage of the task so they can answer other, similar questions later in the task. Students are also asked to think abstractly about the impacts of using different measures of center to interpret the data. This helps further or solidify students’ knowledge of the measures of center.

This question aligns with standard CCSS.MATH.CONTENT.6.SPA.2 because students must think about the distribution of the data set. In this question, unlike in Question 4, students are asked to describe which measure of center they chose to use to describe the data set and why they chose it.

This question aligns with standard CCSS.MATH.CONTENT.6.SPA.5.D because students must explain why their choice of measure of center makes sense in the context of the problem situation.

This question aligns with two mathematical goals. The first is the mathematical thinking goal Students Will Develop Skills that Help Them Think About Mathematical Concepts in Real Life Contexts because students must explain why the measures of center they used make sense and will likely cite the context of the problem situation. The second is the mathematical content goal Students Will Be Able to Demonstrate That They Understand Certain Measures of Center Apply Better to Certain Situations than Other Measures of Center by Identifying Specific Measures of Center for Given Situations because in order to answer this question they must identify and justify the choice of one measure of center over others.


The purpose of this question is to first help students determine which attribute(s) to plot in order to best find a solution for this question and second to get students to think about and analyze a whole set of data, rather than individual plot points, without as much scaffolding like in Questions 3-5.

This question aligns with standard CCSS.MATH.CONTENT.6.SPA.4 because students use TinkerPlots to display numerical data in plots. Unlike in Question 1, students must figure out independently which attributes(s) to plot in order to help them answer this question.

This question aligns with standard CCSS.MATH.CONTENT.6.SPA.5.C because students are asked to analyze the data set and then interpret it. In this question, they are asked to think about the data and their interpretation in the context of the problem situation.

This question aligns with three mathematical goals. The first is the mathematical thinking goal Students Will Develop Skills that Help Them Think About Mathematical Concepts in Real Life Contexts because students must take into consideration the demands of the problem situation (i.e. what the company wants to do) when answering this question. The second is the mathematical content goal Students Will Be Able to Demonstrate That They Understand Certain Measures of Center Apply Better to Certain Situations than Other Measures of Center by

Identifying Specific Measures of Center for Given Situations because students will likely need to choose a specific measure of center to use in order to answer the question. The third is the mathematical content goal Students Will Be Able to Correctly and Appropriately Use Dot and Box Plots to Analyze Data and Draw Conclusions From It because students will have to decide which plot best helps them answer the question.

7. What measures of center did you use to find your answer for Questions 6? Why? What type of plot would you use? Would using a different measure of center change your answer? If so, how would your answer have changed?

The purpose of this question is to have students explain the thinking they experienced answering Question 6. Students are asked to think about how they decided to measure the spread of the data and explain why their choice was appropriate for the data set.

This question aligns with standard CCSS.MATH.CONTENT.6.SPA.2 because students must think about the distribution of the data set. In this question, students are asked to describe which measure of center they chose to use to describe the data set as well as explain why they chose it.

This question aligns with standard CCSS.MATH.CONTENT.6.SPA.5.D because students must explain why their choice of measure of center makes sense in the context of the problem situation.

This question aligns with three mathematical goals. The first is the mathematical thinking goal Students Will Develop Skills that Help Them Think About Mathematical Concepts in Real Life Contexts because they must choose which measure of center to use and describe what it tells them about the data set. The second is the mathematical content goal Students Will Be Able to Demonstrate That They Understand Certain Measures of Center Apply Better to Certain Situations than Other Measures of Center by Identifying Specific Measures of Center for Given Situations because student must explain why the measure of center they chose matches the context of the problem situation. The third is the mathematical content goal Students Will Be Able to Correctly and Appropriately Use Dot and Box Plots to Analyze Data and Draw Conclusions From It because students will have to chose and explain why they used specific plot forms.


The purpose of this question is to have students interpret the context of the situation and apply it to the mathematical information they have gained from this task in order to make a statement related to their learning.

This question relates to the standard CCSS.MATH.CONTET.6.SPA.5.C because students are asked to analyze the information they found in the text and interpret it based on the context of the problem situation.

This question aligns with the mathematical thinking goal Students Will Develop Skills that Help Them Think About Mathematical Concepts in Real Life Contexts because students must respond to the question with reference to the context of the problem situation.

How The Task Helps Engage Students in Higher-Level Thinking and Cognitive Demand:

This task includes multiple opportunities for students to engage in higher-level thinking. Although some questions can be categorized as lower-level cognitive demand, those are necessary to help students achieve well at the higher-level cognitive demand questions. For example, I believe Questions 1, 2, and 3 can be categorized as Procedures without Connections questions because students are following directions to produce specific plots. They then look at simple attributes and plot points to answer questions. Here, student focus on producing correct answers rather than building mathematical understanding. These questions are important, however, to students’ overall success with the task. These lower-level cognitive demand questions serve as scaffolds. Without them, students would likely struggle to answer the higher-level cognitive demand questions.

Question 4 can be considered a higher-level cognitive demand question categorized as Procedures with Connections. To answer this question, students must put forth cognitive effort. They need to engage with the ideas that make up different measures of center to effectively answer the question. However, the question provides specific procedures for them to follow in that it explicitly asks them to look at certain aspects of the plot. It is important to remember that for this question, answers can, and likely will, vary and be open to interpretation.

Questions 5-8 can be considered higher-level cognitive demand questions categorized as Doing Mathematics. These questions require students to think in complex and possibly new ways about the data. Because there is no “right” answer, there is no algorithmic approach so students must think about them in a more abstract way in order to come up with an answer. These questions require students think about relevant knowledge (different measures of center, the specific question being asked, and the context of the problem situation) in order to work through them appropriately. Students will use considerable amounts of cognitive effort to think through, answer, and defend their solutions because they are subjective and can be interpreted in many ways.

Why TinkerPlots in Necessary for This Task: TinkerPlots is very important for students’ success with this task. It would be

difficult for students to achieve the mathematical goals of this task without the use of TinkerPlots technology. Below is a summary of why the technology is necessary for success with each mathematical content goal for this task:

Students will be able to describe the measures of center (mean, median, and mode).

o Although students could likely define these terms prior to working with TinkerPlots, I believe the use of TinkerPlots and the engagement with this task will help students develop deeper understandings of these measures of center. Because TinkerPlots can quickly calculate and plot the measures of center, students can manipulate the graph and watch the measures of centers change. They can also change attributes or measures of center in question and see immediate effects. This is helpful because students can think about the meaning of each measure of center without having to compute it for each situation they wish to analyze.

Students will be able to demonstrate that they understand certain measures of center apply better to certain situations than other measures of center by identifying specific measures of center for given situations.

o Because TinkerPlots can quickly calculate and plot measures of center, students can look at different aspects easily. This gives students the ability to easily compare what different measures of center represent and how those measures of center may affect the interpretation of the data.

o Because TinkerPlots is visual, it allows students to see different measures of center at once and because the measures of center adjust as students manipulate the plot, students will be able to better compare the different measures of center for given contexts.

Students will be able to correctly and appropriately use dot and box plots to analyze data and draw conclusions from it.

o Because students will not be focused on how to accurately construct the dot and box plots, they can focus more on interpreting the plots to determine what they mean and what information they provide.

o TinkerPlots allows students to create a stacked dot plot and a box plot at the same time. This allows students to easily compare the usefulness of each type of plot in helping them answer the questions found in the task.

This use of technology is necessary in supporting students as they work to achieve the mathematical content goals. I believe the use of technology in this task acts as a reorganizer because it allows students to shift their focus from simple computations of measures of center to be more analytical about their usefulness in different contexts. Students can easily compare different measures of center to each other to gain a better understanding of what each means.

Possible Solutions for The Original Mathematical Task:1. Create a plot in TinkerPlots and pull the data Number_Candies attribute onto the

x-axis. Spread the data out as far as it can go so it is not in separated bins. Remember the different tools in TinkerPlots (stacking, box plot, measures of center, etc.) can be used to help you at any point in this investigation.

Student responses to this question will likely look one of two ways. First, students may simply pull the Number_Candies attribute onto the x-axis and then leave the plot looking like this:

Students may also create a plot with Number_Candies on the x-axis but they may chose to use the vertical stack tool. If this is the case, their plot will likely look like this:

A misconception students may have when completing this first question is knowing which axis is the x-axis. Students may create the following plot (which also could not be stacked). Although this could work for this task, it would likely be difficult for students to use because they are not used to looking at data represented in this way:


In order to answer this question, students will first define the range for the data set. Because the highest plot point is 61 and the lowest plot point is 50, the range is 11.

A misconception students may have is about the meaning of range. Because I decided to front load vocabulary and because we reviewed the term range prior to starting the task investigation, I do not think students will misunderstand or possibly forget the idea of range as the difference between the upper-most and lower-most limits/bounds in a set of data. Rather, I think students could possibly struggle with how to represent the range. I expect some students to write the range as the whole set of values and others to write the highest and lowest limits/bounds in the data set. If students do this, I will know they are likely on the right track to understanding the concept of range and how to represent it but I will know this is an area of instruction I should focus with them on improving.

It will be interesting to see if students start thinking about the data in the context of the problem situation here. For example, some students may not think 11 is a very big range at all but if their bag of M&Ms had 11 less pieces of candy than say their sibling’s of friend’s bag of M&Ms, they would likely think it is not fair

and become frustrated. In the context of the problem, the 11 M&M range seems like a lot.

To answer the second half of the question, students will likely have to think in the context of the problem situation. Answers to this question may vary but students could respond with answers such as poor machinery at the production plant, variability in weight of M&M, variability of size of M&M, the possibility of broken candies (or pieces) getting into bags, etc.

A misconception students may have with the second half of this question is to not think about the mathematical concept of range with respect of the context of the problem (M&Ms). Instead, students may answer by simply saying that different M&M bags contained different amounts of candy or by explaining the 11 M&M difference as classmates’ counting error.


The first half of this question asks students to rearrange their plot. Students may create a plot that looks like this:

or one that is vertically stacked and looks like this:

For the second half of this question, students are asked to explain how the weight of their bag compares to what the M&M company lists as the weight of the bag. I would expect some students to click on and therefore highlight their case/bag of M&Ms on their plot but this is in no way a necessary step to answering this question.

I anticipate most students to respond to this question by explaining if their bag is heavier or lighter than what the company says and then by stating the difference in weight.

Some students may explain more subtle differences such as the weight of their bag only differed from what the company listed by one or two unit M&M weights. If students answer in this way, that means they are not only comparing the weight of their bag to the company’s estimate of 47.5 grams but they are also looking at other factors that may play a role (i.e. if their bag had just one more M&M than what was actually in it, then the weight of their bag would match the company’s estimate). Although I do not expect students to answer in this way (this type of answer may not even be applicable for some students), I think this shows extra thinking about the context of the problem.

A misconception students may have regarding the first half of this question is that they may make the plot wrong. This will not impact their ability to answer this question but it could possibly impact how they respond to other questions in this task.

A misconception students may have regarding the second half of this question is that they may compare the weight of the bag to the range of values on the plot to see whether it is the range rather than comparing it to the M&M company’s estimated 47.5 grams. This would show that students who answer in this way likely do not understand what the range fully represents because they would not

automatically assume their data point was used to create the plot and therefore included in the range.

4. How does the company’s claim of 47.5 grams compare to what the whole class found? Use evidence from your plot to support your answer.

Students may approach this question in multiple ways. One way students may look at this question is to add a movable line to their plot and place it at 47.5 grams as shown below:

This will help students visualize what the M&M company claims as the normal weight of a bag of M&Ms. Students can see that most of the data falls to the right of the moveable line and therefore shows that in general, the data for the class is heavier than what the company predicted. Although this approach is not incorrect, I believe it shows a basic level of understanding regarding the use of TinkerPlots to help students analyze data because they are simply looking at a clump of data points on a plot instead of analyzing other aspects.

Another way students can approach this question is to plot the median value (49.07grams). This will provide students the following plot:

This will provide the information about the median value. This value shows students the point on the plot where half of the data is below and half of the data is above this point. The median is the middle value of the distribution and can also be thought of as the 50th percentile. By plotting the median value, students can explain that the middle of the class data is greater than the M&M company’s estimate of 47.5 grams by 1.57 grams. Students can use this to explain that the company’s estimate is low compared to the class data.

Students can also approach this problem by plotting the mean value (49.2147 grams). This will provide students the following plot:

This will provide the information about the mean value. This value shows students the point on the plot that is the average. It is found by adding all of the values of the plot points together and dividing that by the total number of plot points. The mean can be thought of as a balancing point for the data set. By plotting the mean value, students can explain that the balancing point of the class data is greater than the M&M company’s estimate of 47.5 grams by 1.7147 grams. Students can use this to explain that the company’s estimate is low compared to the class data.

Students can also approach this problem by plotting the mode for the data set (49.8 grams). This will provide students the following plot:

This will provide the information about the mode for the data set. This shows students the point on the plot that is the most frequent data point. By plotting the mode, students can explain that the M&M company’s estimate is low compared to the data points that occurred the most.

I am also anticipating students will respond to this question by using and analyzing a box plot like in the figure below:

By looking at this box plot, students can explain that more than 75% of the data falls above the M&M company’s estimate of 47.5 grams. If students choose to use a box plot, they may see that 50% of the data falls within 48 and 50 grams per bag. This information may be useful to them later in the task.

One misconception students might have is that they think they can answer the question without using the measures of center (mean, median, and mode) by simply comparing the M&M company’s estimate of 47.5 grams to each individual data point. This may look like the student noting that the 47.5 grams is greater than points and less than others. Although this is technically correct, it shows a misinterpretation of the question. Other misconceptions may surface and those are addressed in Question 5 because these questions are so similar.

5. What measures of center did you use to find your answer for Question 4? Why? What type of plot did you use? Would using a different measure of center change your answer? If so, how would your answer change and if not, why not?

Students’ responses to this question will vary depending on the measure of center they used to answer Question 4. I expect students to reflect on their method from Question 4 and explain why they chose to use that measure. Answers may be simple, such as because the student felt more familiar and comfortable using a particular measure of center, but could also be complex, for example, such as the student knew that comparing the data to the movable line at 47.5 grams would allow them to look at the spread of the data for the whole class rather than to an individual data point. Other complex answers may include students’ justification of using mean, median, or mode to answer the question. I am expecting students to justify these measures by describing what the measure represents and why that is beneficial to helping them answer the question (similar responses to those listed for Question 4).

I am personally partial to two ways to answer Question 4. First, I liked comparing the data as a whole to the movable line placed at 47.5 grams. I thought this provided me with the information I needed to answer the question and allowed me to compare the spread of the data to the specific value listed by the M&M company. Second, I liked using the box plot and the movable line at 47.5 grams because this helped me not only see the spread of the data compared to the M&M company’s estimated weight but it also helped me think about where a large portion of the data laid (the 50% in the box and the greater than 75% right of the movable line). This information made me feel confident in my answer. Although as an individual I am partial to agreeing with students if they answer in this way, as a teacher I must make sure I also listen to the arguments of students who used other measures of center to answer the question. If students provide an answer and back it up with solid information related to the problem, the context of the problem situation, and the meaning of the measure of center they chose, then they likely have responded to this question well.

Students will likely respond to the second part of this question in one of two ways. First, students may explain that they used the dot plot to find their answer. Students could have various reasons for this including the fact that the dot plot was already there and they did not even think of using the box plot or that the dot plot made more sense to them because they could easily look at one (or more)

measure(s) of center. Second, students may explain that they used a box plot to find their answer. This may be because they wanted to see the data broken into quartiles or because they wanted a more visual representation of data measures.

For the last two parts of the question (would using a different measure change your answer and if so, how would your answer change), I expect students to explore (using TinkerPlots) to help them interpret the plot as they apply different measures of center to it. I anticipate students to explain that no, their overall answer did not change (all measures show that the class data is larger than the M&M company’s estimate), but the measures represented different concepts. I anticipate students to explain what these different measures of center represent similar to how they are described in Question 4.

A misconception students may have when answering this question is about the meaning of different measures of center (mean, median, and mode). First, students may simply confuse the definitions of these terms and answer the question incorrectly. Second, students may not be able to describe what each represent and therefore will not accurately be able to attend to the question.

Another misconception students may have is that changing the measure of center changes the answer they will get. It does not. Given this specific data set and question, changing the measure of center merely represents the data in different ways. Students should be able to recognize this, especially when they explain why they chose the measure of center they did and when they answer the last part of this question.


In order to fully answer this question, I expect students to first change their plot so the Number_Candies attribute is once again on the x-axis. Students can either stack or leave random their data plots. One common example may look like this:

Once students have created a plot like this, the next step is to figure out which number best represents the data. Students will use measures of center and possibly a different plot format (box plot) to find their answer.

One possible response to this question can come from students examining the mode of the data set as shown below:

This plot shows the mode at 58. Students could possibly interpret this as the most common number of M&Ms per bag was 58 so it would make sense for the M&M company to label the bags with approximately 58 total M&Ms.

Another possible response to this question can come from students examining the mean value of the data set as shown below:

This plot shows the mean at 57.1. Students could possibly interpret this as the balancing point for the plot. 57.1 here means the number of candies that is closest to all the data points found by the class. Students could explain that it makes sense for the M&M company to label the bags with 57 candies (rounding down for practicality) because it represents, to some extent, something about all of the values we plotted as a class.

A third possible response students could have to answering this question is to examine the value of median as shown below:

In this plot, the median value is 58. Students could interpret this as saying 50% of the data is above 58 candies per bag and 50% of the data is below 58 candies per bag. Students could explain that the M&M company should list 58 number of candies on the bag because it is the “middle” of the data and this would best represent the spread of the data.

Finally, I anticipate some students to respond to this question by looking at the data in the format of a box plot as shown below:

What is interesting about this box plot is that it shows the median value (58) is the same as the 3rd quartile. This box plot shows that 50% of the data collected is represented between the numbers 56 and 58. Although this does not necessarily point to one specific value that could be used on the M&M bags, students could use it to justify a choice they make. For example, students could respond to the question by saying they company should label the bags with 56 candies each. This is because they want to keep their customers happy. It is likely that the amount of candy in each bag will be close to 56 (remembering where 50% of the data lies). Students could explain that choosing 56 candies is a “safe” choice for the company because if bags have more pieces of candy than what is listed, more often than not customers will still be satisfied.

One misconception students may have is that they think the company has to pick a number to label the bags with that is either the maximum or minimum of the range. Students may believe this because they do not think the bag should contain more than what it says (in which case they would pick the maximum) or they do not think the bag should contain less than what is says (in which case they would pick the minimum).

7. What measures of center did you use to find your answer for Question 6? Why? What type of plot would you use? Would using a different measure of center change your answer? If so, how would your answer have changed?

Students’ responses to this question will vary depending on the measure of center they used to answer Question 6. I expect students to reflect on their method from Question 6 and explain why they chose to use that measure. Answers may be simple, such as because the student felt more familiar and comfortable using a particular measure of center or because they used it to answer Question 4.

Responses may also be complex, for example, such as the student knew that certain measures of center represented certain mathematical ideas and they knew which one made the most sense to them in the given context of the problem. Other complex answers may include students’ justification of using mean, median, or mode to answer the question. I am expecting students to justify these measures by describing what the measure represents and why that is beneficial to helping them answer the question (similar responses to those listed for Question 6).

I am personally partial to one way to answer Question 6. I liked using the box plot to determine the value I think the M&M company should list on the bag. This helped me see where the majority of plot points were and, because I was able to visualize them, I was better able to figure out what it meant regarding the data set. I also think using the box plot helped me to best conceptualize the problem in context. The box plot allowed me to think about why the company may want to list the number of candies as lower than a “typical” amount because it may provide as a safeguard and keep their customer base happy. Although as an individual I am partial to agreeing with students if they answer in this way, as a teacher I must make sure I also listen to the arguments of students who used other measures of center to answer the question. If students provide an answer and back it up with solid information related to the problem, the context of the problem situation, and the meaning of the measure of center they chose, then they likely have responded to this question well.

Some misconceptions students may have while answering this question are the same as described in Question 6. Students may also have misconceptions unique to this question. For example, if students chose the measure of center in Question 6 because it was the only one they understood, then they will have misconceptions in Question 7 based on their lack of understanding of the other measures of center.


Students will respond in one of two ways either they will support labeling the bag by weight or they will support labeling the bag by number of candies. Student will defend their position using information and evidence from this task. In addition, I will be looking for students to integrate the context of the problem situation into their response such as the concerns of the producer or consumer of the product. If students provide sufficient, accurate support, they will have responded to this question well.

measures of center lesson plans, mathematical tasks, and solutions

Documents