The purpose of this Warm-up is to review students’ prior knowledge about representation of numerical data. Students may be familiar with line plots from previous grades but unfamiliar with the term dot plot, which is what will be used in this unit and beyond. Students learn that both terms are commonly used for the same type of diagram.
Students examine a dot plot of data and consider which contexts may make sense for the data shown. Then students invent their own context for the data and interpret what additional data in that set would mean in their context.
This is an opportunity to see how students make sense of a data representation in the context of situations (MP2).
During the Activity Synthesis, introduce the terms numerical data and categorical data to refer to the type of responses we would expect for different questions. The question about the number of children per class would give numerical data, which are responses that are measurements or quantities. The question about the favorite subject in school would give categorical data, which are responses that are not quantities but can be sorted into categories.
Launch
Display the image and data for all to see.
0
0.25
0.5
0.5
0.75
0.75
0.75
1
1
1.5
1.5
1.5
1.5
1.75
1.75
1.75
2
2.25
2.25
2.5
2.5
Ask students what they notice and wonder about the image. Students may notice:
There are 21 dots over the number line.
The data match the dots over the number line.
Some of the dots are over fractional values.
Some values have no dots.
Students may wonder:
What do the dots and data represent?
Why are there no dots over 1.25 or after 2.5?
Why are there so many dots over 1.5?
Is this a line plot?
If not mentioned by students, remind them that the display is interpreted like a line plot. Tell students that this is called a dot plot, and that line plots use Xs and dot plots use dots, but the meaning is very similar.
Give students 2 minutes of quiet work time to complete the task, followed by a whole-class discussion.
Activity
None
Here is a dot plot for a data set.
Determine if each of these would be an appropriate label to represent the data in the dot plot? Be prepared to explain your reasoning.
Number of children per class.
Distance between home and school, in miles.
Favorite subject in school.
Weight of elephants, in pounds.
Points received on a homework assignment.
Think of another label that can be used with the dot plot.
Write it below the scale of the dot plot. Be sure to include the unit of measurement.
In your scenario, what does one dot represent?
In your scenario, what would a data point of 0 mean? What would a data point of mean?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of the discussion is for students to understand how to read a dot plot, including what dots represent in context.
Invite students to share their responses to each scenario presented in the first question, and ask a few students to explain their reasoning for each.
Invite a few students to share another label that they think could be used with the set of data and what each dot would represent. Based on each response, ask the class these questions:
“What would a data point of 0 mean in the context mentioned?”
“What would a data point of mean in the context mentioned?”
In this activity, students reason abstractly and quantitatively (MP2) about numerical data sets to match them with questions that are likely to produce the data. They see that some survey questions lead to responses that are expected to vary when posed to different people, but others produce responses that are likely to be the same. Students categorize data sets based on whether more than one different value is present and make use of this structure (MP7) to define variability.
As students match questions with data sets, look out for different plausible explanations for their choices. Their matches are reasonable if they can explain why the given data could be responses to a question. Identify one student to share the response to each question. Also notice students who offer different but equally reasonable explanations for the same data set, and invite them to share later.
Adjust the timing of this activity to 10 minutes. To move more quickly through the activity, consider telling students to be prepared to share their reasoning with their partner but that they do not need to write down their reasoning.
During the Activity Synthesis, explain that a statistical question is a question that can be answered by using data in which variability is expected. Ask students which questions in this activity were statistical questions (1, 2, and 4) and which ones were not (3 and 5).
Launch
Tell students that they will be looking at numerical data sets and thinking about what question could produce the responses in each data set. Emphasize that they need to be able to support their matching decisions with reasonable explanations.
If necessary, guide students to understand how to read the table by asking:
“What values are included in Data Set A?” (0, 1, 1, 3, 0, 0, 0, 2, 1, 1)
“To which data set does the number 8 belong?” (Data Set C)
“How many people answered '1' to the question that produced the data for Data Set A?” (4)
Keep students in groups of 2. Give them 5 minutes of quiet work time and 1–2 minutes to share their responses with their partner.
Activity
None
Ten sixth-grade students at a school were each asked five survey questions. Their answers to each question are shown here.
Data Set A
0
1
1
3
0
0
0
2
1
1
Data Set B
12
12
12
12
12
12
12
12
12
12
Data Set C
6
5
7
6
4
5
3
4
6
8
Data Set D
6
6
6
6
6
6
6
6
6
6
Data Set E
3
7
9
11
6
4
2
16
6
10
Here are the five survey questions. Match each question to a data set that could represent the students’ answers. Explain your reasoning.
Question 1: Flip a coin 10 times. How many heads did you get?
Question 2: How many books did you read in the last year?
Question 3: What grade are you in?
Question 4: How many dogs and cats do you have?
Question 5: How many inches are in 1 foot?
How are survey Questions 3 and 5 different from the other questions?
Activity Synthesis
The purpose of this discussion is for students to define variability and recognize when it is present.
Select previously identified students to share their choices and explanations. Briefly poll the class after each explanation to see if others made the same choice for the same reason. If not, invite students with different explanations to share.
Discuss how the question about grade level and the one about number of inches in a foot are different from the others. If not mentioned by students, highlight the idea of variability. Explain that we use the term variability to describe data sets in which not every data value is the same. Data sets B and D are unlike the other sets because they show no variability.
1.3
Activity
Standards Alignment
Building On
Addressing
6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
In this activity, students create a dot plot on their own. The work of drawing dot plots is not new, but students are asked to describe their analysis of the data broadly. They also learn to use the terms frequency and distribution.
Launch
Keep students in groups of 3–4. Display the dot plot for all to see.
Ask students how many cats have 1 spot. (4 cats) Explain that the term frequency associated with a particular item represents the number of times an item occurs in the data set. The value 1 in this data set has a frequency of 4. Ask,
“What is the frequency of cats with 2 spots?” (3)
“What is the frequency of cats with 3 spots?” (0)
Then ask students to describe how the values are arranged in the dot plot. (There are a lot of cats with only a couple of spots, and then 2 with 4 or more spots.) Explain that the distribution refers to a description of how the data are arranged (or distributed) in the dot plot.
Give groups 5–6 minutes to work together to organize the given data and draw their dot plots. Then, give them 2–3 minutes of quiet time to analyze the dot plots and answer the last two questions, and time for a brief small-group discussion about their responses. Reconvene as a whole class afterward.
As students work and discuss, identify those who are able to describe the distribution of the data clearly and succinctly, as well as students who can articulate why a dot plot is an appropriate representation. Ask them to share later.
Representation: Develop Language and Symbols. Maintain a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of distributions. Terms may include "spread," "distribution." and "frequency." Supports accessibility for: Conceptual Processing, Language
Activity
None
Priya wants to know if basketball players on 2 teams have had prior experience in international competitions. She gathers data on the number of times the players were on a team before 2016.
Team 1
3
0
0
0
0
1
0
0
0
0
0
0
Team 2
2
3
3
1
0
2
0
1
1
0
3
1
Did Priya collect categorical or numerical data?
Organize the information on the two basketball teams into these tables.
Team 1
number of prior
competitions
frequency
(number)
0
1
2
3
4
Team 2
number of prior
competitions
frequency
(number)
0
1
2
3
4
Make a dot plot for each table.
Team 1
Team 2
Study your dot plots. What do they tell you about the competition participation of:
The players on team 1?
The players on team 2?
Explain why a dot plot is an appropriate representation for Priya’s data.
Student Response
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Building on Student Thinking
When drawing dot plots, some students might use dots of different sizes or neglect to stack the dots in a straight column. Remind students to use uniform dots and to stack them vertically, using a straightedge as a guide, if needed. For some students, the use of graph paper may be helpful.
Activity Synthesis
Select previously identified students to share their comments on the competition participation of the two teams and why a dot plot is an appropriate representation. Then, invite them to compare the features and merits of the representations that they used in this lesson. Discuss:
“In what ways might dot plots be more useful than lists and tables?” (Dot plots give us a visual snapshot of the data so that we can see patterns or anything unusual.)
“Could a dot plot be used to represent the first letter of the last names of the players represented on these teams? Why or why not?” (No, because a dot plot uses a number line. Each point on the number line represents a number, so where a data point is placed matters. A bar graph could be used where the categories could be shown in any order, and the width of the bars is flexible.)
Explain to students that a frequency table, bar graph, and dot plot all tell us about the distribution of a data set: each of them lists or shows all the possible values or categories in a data set and how often each one occurs.
Lesson Synthesis
The purpose of the discussion is to clarify the new vocabulary. Ask students,
“What are some examples of categorical data?” (brand of a product, color, location)
“What are some examples of numerical data?” (how many of something, the length of something, volume of something)
“What does it mean to say that data have variability?” (Not all the data values are the same.)
“When might we expect data to have variability?” (When the question we are trying to answer is about a feature or a characteristic of a group that has different members, with different features or characteristics.)
“When might we expect data to have no variability?” (When the question we are trying to answer is about an individual, or about a feature that all group members have in common.)
“What is the distribution of data?” (Data and how often they occur. This can be seen in a dot plot or bar graph.)
“How do we tell the frequency of a data value from a data set? What about from a frequency table?” (If all the data are listed, we can count how often a value occurs to find the frequency. In a frequency table, the frequency is listed next to the value.)
“How does a dot plot represent data?” (It represents each data value with a point above a number line.)
“How do we tell the frequencies of different data values from a dot plot?” (The number of dots stacked over the value on a number line is the frequency.)
Consider creating a permanent display of the new terms introduced throughout the unit.
Student Lesson Summary
We often collect data to answer questions about something. The data we collect may show variability, which means the data values are not all the same.
Some data sets have more variability than others. Here are two sets of figures.
A
B
Set A has more figures with the same shape, color, or size. Set B shows more figures with different shapes, colors, or sizes, so Set B has greater variability than Set A.
When a question can be answered only by using data, and we expect that data to have variability, we call it a statistical question. Here are some examples.
Who is the most popular musical artist at your school?
When do students in your class typically eat dinner?
Which classroom in your school has the most books?
How much do the dogs at a pet adoption center weigh?
The table contains data about 10 dogs.
The weights of the dogs are an example of numerical data, which are data that are numbers, quantities, or measurements. The weights of the dogs are measurements in kilograms.
The dog breeds are an example of categorical data, which are data containing values that can be sorted into categories. In this case, there are three categories for dog breeds: pug, beagle, and German shepherd.
dog name
weight (kg)
breed
Duke
36
German shepherd
Coco
6
pug
Pierre
7
pug
Ginger
35
German shepherd
Lucky
10
beagle
Daisy
10
beagle
Buster
35
German shepherd
Pepper
7
pug
Rocky
7
beagle
Lady
32
German shepherd
When we analyze data, we are often interested in the distribution, which is information that shows all the data values and how often they occur. We can see the distribution of the dog weights in a table such as this one.
weight in kilograms
frequency
6
1
7
3
10
2
32
1
35
2
36
1
The term frequency refers to the number of times a data value occurs. In this case, we see that there are 3 dogs that weigh 7 kilograms, so 3 is the frequency for the value “7 kilograms.”
Like a frequency table, a dot plot also shows the distribution of a data set. This dot plot shows the distribution of dog weights.
A dot plot for "dog weights in kilograms". The numbers 5 through 40, in increments of 5, are indicated. The data are as follows: 6 kilograms, 1 dot. 7 kilograms, 3 dots. 10 kilograms, 2 dots. 32 kilograms, 1 dot. 35 kilograms, 2 dots. 36 kilograms, 1 dot.
A dot plot uses a horizontal number line. We show the frequency of a value by the number of dots drawn above that value. Here, the two dots above the number 35 tell us that there are two dogs that weigh 35 kilograms.
Some students may have trouble matching questions and data sets because they do not attend carefully to the range of possible solutions. For example, they may not notice that a data set with 11 as a data value cannot be a response to the first question about flipping a coin 10 times. Ask them to study the questions and data values more closely, and to look for values that seem unlikely or impossible for a given context.
