Lesson 3

Pockets Graph

Est. Class Sessions: 1–2

Developing the Lesson

X

Allow adequate time to complete the data collection in one class session. The data won't be the same with a change of clothing.

Gather Data. Begin the lesson by commenting about yesterday's homework asking everyone to wear clothing with pockets today and how it made you wonder about the number of pockets the class would have, as well as other questions.

Ask:

X
  • What do you think is the greatest number of pockets someone is wearing today? What do you think is the smallest number?
  • What do you think is the total number of pockets in our class today?
  • Would 20 pockets be a good estimate of the total? How about an estimate of 300?
  • How can we find answers to these questions?

Briefly discuss how students determined if 20 and 300 are reasonable estimates. Congratulate the suggestion of using a data table as in Lesson 1, Favorite Colors, in response to the last prompt. Remind students that using efficient strategies and tools is a good math practice. Then introduce the graph as another tool for organizing and keeping track of the data the class will collect to answer these questions.

Suggest using connecting cubes to help keep track of their pockets. Demonstrate by placing one cube in each of your pockets. Then remove all your cubes and connect them to make a cube train. Ask students to complete this process. Ask students to count the number of cubes in their trains.

X

Connecting cubes help those students still working on
one-to-one correspondence keep track and count pockets and also serve as a visual representation helping bridge concrete (pockets) to symbolic (number) representations of the number of pockets. If the students in your class do not need the aid of the cubes you can simply have each student use the self-adhesive note to write his or her name and the number of pockets worn below it.

X

To get an accurate survey, ask students to set aside any extra cubes that are not used.

Ask all students with a given number of pockets (e.g., three pockets) to line up in front of the class and show their cube trains. Check that students have the right number of cubes in their trains by counting and comparing to see that all are the same length. Repeat this procedure with other numbers of pockets.

Graph the Data. Call attention to the graph you prepared before class, reading the title and labels including abbreviations on both the horizontal and the vertical axes. Explain or elicit from students what the numbers on each axis represent. Tell students that the graph will help us remember and organize the data about the number of pockets each student is wearing today. Explain that the graph will give us a picture of the data and help us find answers to the questions posed at the start of class.

Instruct each student to write his or her name on a self-adhesive note. As you name a value from 0 to 12, students with that number of pockets (and length of cube train) bring their notes to the graph. Students should place their own self-adhesive notes on the graph to develop a concrete understanding of what a graph represents.

Make sure students place their notes on the lines above the numbers and not in the spaces between. This is a scientific convention that is used throughout the curriculum. A sample graph is shown in Figure 3.

X

Have a student who has no pockets place the self-adhesive note on the vertical axis—the zero line.

Ask students with two pockets to stand. Count the number of students standing. Write the number on the board. Count the number of self-adhesive notes above the number two on the bar graph.

Ask the following questions:

X
  • Should the number of students standing and the number of self-adhesive notes be the same? Why? (Yes, because each student put a note on the graph. The graph shows everybody here today.)
  • If I read the names on the notes above the number two, should they be the names of the people standing? Why? (Yes, because they are the people who had two pockets. They stood up for two pockets and they put their name on the graph for two pockets.)

Repeat this process with other numbers of pockets.

X

If your students wear uniforms there may not be much variety in the data. You may choose to have students collect the data anyway and talk about why all the students seem to have a similar number of pockets. Show students the data in Figure 3 and ask whether they think these students are wearing similar clothing or whether their outfits have more variety.

Rather than asking students to wear an outfit with pockets, you may choose to have students draw or take a picture of themselves wearing an outfit with pockets. Be sure students show the pockets on the front and back of their outfits. See the drawing below.

X

Representations Support Strategy Development. Observe students as they work on questions requiring addition. Students can use fingers as a support for counting on, but watch for students who use counting all as the strategy they use repeatedly. Students who count from one repeatedly need to be encouraged to start with the larger addend, add the number of counters or fingers for the second addend, then count on to determine the answer more efficiently. Look for opportunities to help students use 5 and 10 as benchmarks to help them develop the strategy of counting on. Ten frames provide a visual representation of these benchmarks.

Gather Questions about the Data. Begin the analysis of the data represented in the bar graph by asking students to think about what the graph is showing. Encourage them to think of questions about the data. As students share their questions aloud write each question on the front of one of the question cards you prepared. See Materials Preparation. Some sample questions are listed here and are based on the data represented in Figure 3. If students do not think of the questions listed, introduce them into the discussion using your class data.

Students will work in pairs to answer the questions later, so record at least one question for each pair of students and two additional questions for class discussion.

Ask:

X
  • Which number of pockets is most common?
    (8 pockets. It is the tallest bar on the graph.)
  • Which is the least common? (3 pockets. Only one person had three pockets and it is the shortest bar on the graph.)
  • Who has the least number of pockets? (Sue and Bob had zero pockets.)
  • Who has the most pockets? (Amy and Robin had
    9 pockets; it's the biggest number.)
  • If a new student walked in, how many pockets do you think he or she might have? (The tall bars mean more people had that number of pockets. So 8 is the highest, but 7 and 6 pockets are pretty close to 8, so maybe the graph means the new student would have
    6 or 7 or 8 pockets.)
  • How many students have clothes with five pockets? (3 students)
  • What is the total number of students who have more than seven pockets? (It's 13 students, because
    8 pockets is 11 students and 9 pockets is 2 students.
    11 + 2 = 13.)
  • What is the total number of students who have four, five, or six pockets? (Four pockets shows 5 students, five pockets is 3, six pockets is 7;
    5 + 3 + 7 = 15 students.)
  • How many more students have 8 pockets than
    6 pockets?     (4 more students; I compared the columns and counted on.)
  • How many fewer students have 5 pockets than
    7 pockets?     (5 fewer students; I compared the columns and counted on.)
  • Are students more likely to have 5 pockets or
    7 pockets?     (7 pockets because more students have
    7 pockets.)
  • I want a group of 5 students. There are 3 students with 6 pockets. Which other group of students can join them to make a group of 5 students? (There are 2 students with zero pockets and 2 students with
    9 pockets. Either group of students could join the
    3 students with 5 pockets to make a group of
    5 students.)
  • I want a group of 10 students. There are 7 students with 6 pockets. Which other groups of students can join them to make a group of 10 students? (There are several ways to add 3 students to this group. The
    2 students with 0 pockets and the 1 student with
    3 pockets could join them, the 2 students with 9 pockets and the 1 student with 3 pockets could join them, or the 3 students with 5 pockets could join this group.)
  • How many students have 5 pockets? (3 students have 5 pockets.) More students joined the class.
    Now 5 children have 5 pockets. How many students joined the class?     (2 students joined the class because there were 3 students and now there are 5 students with 5 pockets.)
  • How many students have 6 pockets? (7 students have 6 pockets.) More students joined the class.
    Now 10 students have 6 pockets. How many students joined the class?     (3 students joined the class because there were 7 students with 6 pockets and now there are 10.)
  • How many students have 8 pockets? (11 students have 8 pockets. I counted the tally marks.) Some students left. There are now 10 students with
    8 pockets. How many students left the room?
    (1 student left the room. There were 11 students with
    8 pockets and one less is 10 students with 8 pockets.)

These kinds of questions and the related discussion help students to reflect on the "story" of the data represented in the graph.

Analyze the Data. Display the Math Practices page in the Student Activity Book Reference section and discuss Math Practices Expectation 5, Show my work.

Pose a question about the graph such as:

X
  • How many [more, fewer] students have [number] pockets than [number] pockets? Tell how you know.

Record the question on one of the question cards you prepared. Ask students to show or tell how to use the graph to answer the question. Students should be able to show which information from the graph they used. For example, if you asked how many more students have 7 pockets than 3 pockets, students might show that they compared the 7-pockets column and the 3-pockets column and then counted on to figure out how many more pockets they need to get to 7. Other students might determine the number of students for each number of pockets and then use their fingers or counters to solve 8 − 1.

After hearing the different strategies from students, record the answer to the question on the inside of the question card. See Figure 2. You may want to represent the answer in a variety of ways to show students that there are many ways to show their answer: drawing, number symbol, and/or words.

Tell students they will now have a chance to answer some of the questions they generated. Remind students to record their answer on the inside of the card and, while referring to Math Practices Expectation 5, that they should be prepared to show or tell how they answered the question. As you distribute the question cards to each pair of students, read the question aloud once or twice.

Give students a chance to interpret their assigned question. Some students may need their question read again. Some students may also need to work more closely with the displayed graph. Circulate among the pairs supporting students' thinking and their interpretation of the question. Allow students some flexibility in communicating their answer. Students could draw a picture or write symbols or words. It is important that they can read and share their answer and be ready to show others how they came to that answer. See the Meeting Individual Needs Box for additional instructional strategies to help students access the question and share their solution strategies.

X

Observe students while they analyze the data using their assigned question to assess students' abilities to read a bar graph [E9] and to explain how they arrived at their answer so someone else can understand their thinking [MPE5].

X

  • Encourage students to use invented spelling and drawings to first interpret their assigned questions and then share their answers and solution strategies.
  • Distribute fewer questions to the class at a time and have students devise a solution and answer orally with some writing. For example, ask half the students to focus on one question and the other half of the class to focus on another question. Then have one or two students from each group share their solution strategies.
  • Recording devices may help students listen to the questions written on the question cards and share their strategies for analyzing the graphs. Students can listen to the question repeatedly, and these devices can help capture student thinking. Voice recognition software may also be used to translate students' oral language into text.
  • Use the inherent variety in the questions to meet the individual needs of students.

After students have had adequate time to record answers to their assigned questions, pose each question aloud to the class providing time for students to silently determine the answer. Then display the written answer and have the authors explain to the class how they arrived at it using the graph or using the data to make predictions.

Adventure Book:
Student Activity Book: R-10
X
SG_Mini
+
Master:
Question cards for analyzing data
X
+
Pockets Bar Graph
X
+