Lesson 2

This is the first Notice and Wonder activity in the course. Students are shown three statistical displays representing the same data set. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to write down things they notice and things they wonder. After students have had a chance to write down their responses, ask several students to share things they noticed and things they wondered; record these for all to see. Often, the goal is to steer the conversation to wondering about something mathematical that the class is about to focus on. The purpose is to make a mathematical task accessible to all students with these two approachable questions. By thinking about them and responding, students gain entry into the context and might get their curiosity piqued.

The purpose of this warm-up is to elicit the idea that the same data can be displayed in different ways, which will be useful when students create different data displays in a later activity. While students may notice and wonder many things about these images, the comparison of the three representations and interpreting the information in each representation are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). Specifically, they might use the structure of the three representations, particularly the structure of the horizontal number line, to find mathematically important similarities in how the same set of data is represented.

Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. 

The goal is to help students recall different ways to represent distributions of data. Highlight the similarities between the dot plot and the histogram. Tell students that the tallest bar in the histogram is created from the two data values at 5 and the six data values at 5.5 in the dot plot, and that the final bar is created from the two data values at 6 and the two data values at 6.5 in the dot plot. If time permits, discuss questions such as . . .

• “Which representation(s) shows all the data values?” (The dot plot shows all the data values.)
• “How do you create a box plot?” (You calculate the values for the five-number summary and then graph them on a number line. The first quartile, the median, and the third quartile are used for the box, and the minimum value and maximum value are used for the whiskers.)

The mathematical purpose of this activity is to represent and analyze data with histograms. Students will create two different histograms from the same data set by organizing data into different intervals.

Arrange students in groups of 2.

Students may struggle to know how to place numbers that lie on the boundary between intervals. For example, students may not know if a value like 60 should be included in the interval 50–60 or 60–70. Explain to students that the lower boundary value is included in the interval, and the upper boundary value is not. For example, the interval 60–70 includes all the values that are greater than or equal to 60 and less than 70.

The purpose of this discussion is to make sure that students know how to create and begin to interpret histograms. Here are some questions for discussion.

• “Where did you put the 60? In the 50–60 interval or 60–70 interval?” (Tell students that we use the convention of including the 60 in the 60–70 interval. The interval 60–70 means all the values greater than or equal to 60, but less than 70. The interval 50–60 means all values greater than or equal to 50 but less than 60.)
• “What information is easily seen in the histogram?” (The shape of the distribution as well as estimates for the measure of center and measure of variability.)
• “According to each histogram, what appears to be the typical number of days it takes a tomato plant to produce tomatoes?” (Using the first histogram, it appears that the typical number of days is somewhere between 60 and 80. Looking at the second histogram, it appears that the typical number of days could be between 75 and 80.)
• “What information is not seen in the histogram?” (You are not able to see the actual values. You only know the number of values within an interval rather than the values themselves.)

The mathematical purpose of this activity is to represent the distribution of data on the real number line with a box plot and help students think informally about the median as a measure of center. Students calculate the values for the five-number summary and create a box plot. The median, quartiles, and extreme values split the data set into four intervals with approximately the same number of data values in each. Students engage in MP2 when they interpret these values in the given context. Although these intervals are often called “quartiles,” the term “quarters” is used in these materials to avoid confusion with the quartile values Q1 and Q3.

Keep students in groups of 2. Give students 5 minutes to work the questions. Ask them to compare their answers with their partner after each question.

For students who have difficulty calculating the median, remind them that the median is the middle of a sequential data set. For students who have difficulty finding Q1 and Q3, ask them how many groups should we have if we are splitting the data into “quarters.” The data should be divided into four equal groups and the median of the lower half of the values is Q1 and the median of the upper half of the values is Q3.

The goal is to make sure students understand the five-number summary and to help them think informally about the median as a measure of center. Here are some questions for discussion.

• “What information is easily seen in the box plot?” (The minimum value, quartiles including the median, and the maximum value. This also highlights the interquartile range and the range.)
• “According to the box plot, what is the typical number of days it takes a tomato plant to produce fruit?” (The typical number of days is 71 because the median of the data is 71 days.)