Students will learn how to make and understand line plots, particularly ones using fractions. Students will
- develop line plots with rational data.
- interpret and solve problems using line plots that represent rational data.
- In what ways are the mathematical attributes of objects or processes measured, calculated, and/or interpreted?
- What makes a tool and/or strategy suitable for a certain task?
- Line Plot: A method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot.
- Pictograph: A way of representing statistical data using symbols to match the frequencies of different kinds of data.
- Tally Chart: A way of representing statistical data using tallies to match the frequencies of different kinds of data.
- one copy of the Number Line worksheet (M-5-4-2_Number Line) per student
- one copy of the Tally Chart worksheet (M-5-4-2_Tally Chart) for every two students
- one copy of the Match the Line Plots exercise (M-5-4-2_Match the Line Plots) per student. (optional)
- one copy of the Line Plot Quick Quiz (M-5-4-2_Line Plot Quick Quiz and KEY) per student. (optional)
- As an exit ticket, each pair of students should submit a copy of their line plot that shows a completed line plot, including titles, that shows the data on the bar graph and the data represented on the Tally Chart.
- The Match the Line Plots activity can be used as a quick way to ensure that students understand how to match line plots to data sets.
- The line plot A Quick Quiz can be used to assess how well students can read, understand, and interpret information in a line plot in order to answer questions regarding the data given.
Scaffolding, Active Engagement, Modeling, and Formative Assessment.
W: Students will learn to design, understand, and use line plots to address data-related questions.
H: Students analyze a bar graph on a relevant topic and they immediately start analyzing the data and constructing their own line plot from the data.
E: Teachers lead students through the process of creating a line plot. They are presented with values that do not "fit" on their line plot and are given the opportunity to experiment with how to include those values into their line plot. They are also free in Activity 3 to come up with their own methods for calculating the overall number of books read.
R: Students will improve their grasp of a line plot as they progress from representing only integer values to values containing \(1 \over 2\), \(1 \over 4\), or \(3 \over 4\). They will collaborate with a partner to improve their skills in labeling a number line with a consistent scale and creating an accurate line plot.
E: Students evaluate their work by comparing it to others' line plots. They also evaluate their work in Activity 3 by comparing their responses to those of other pairs and working with other pairs to either help them comprehend or identify where any of their own misunderstandings or mistakes occurred.
T: Use the Extension section to personalize the lesson to students' specific requirements. The Routine section offers opportunity for concept assessment throughout the year. The Small Group section provides additional learning or practice opportunities for students who might benefit from more time with lesson concept. The Expansion section proposes additional demanding exercises for students who are ready to go above and beyond the requirements of the standard.
O: The class starts with large group instruction and Socratic questioning, gradually increasing student responsibility for interpreting and understanding the content. The lesson concludes with students helping one another understand the content.
Activity 1
"Today we will discuss about graphs. Our main focus will be on something known as a line plot. Some of you may be familiar with that term. We'll build a line plot together in a minute. But first, I'd want to show you a bar graph. I'll need your help determining what the bar graph represents."
Present the following bar graph to students (either on individual worksheets or projected so that all students can see it):
"This bar graph shows the results of a survey."
To ensure that students understand how to interpret the bar graph, ask (and provide logic to support their responses) questions like:
"What do you think the survey question was?" (How many books did you read last month?)
"How many people were surveyed?" (26)
"What is the most number of books any of the surveyed people read?" (4)
"How many people read 1 book last month?" (6)
"How many people read at least 3 books last month?" (6)
"What is the most common number of books people read last month?" (2)
"What is the total number of books read by all 26 people surveyed last month?" (48)
"There is another way to display this data besides a bar graph. We can visualize the data using a line plot. The first step in creating a line plot is to draw a number line." Give each student a copy of the Number Line worksheet (M-5-4-2_Number Line). "Notice that the number line is unmarked. Let's give it some labels. Begin on the left side of the number line with the smallest number of books that respondents in our survey could have read and identify the mark beneath the number line." Make sure that students label one of the marks on the left end of the number line with a 0.
"Now, skip 3 marks and label the 4th with a 1, indicating the quantity of books participants in our survey could have read. "You should now have a 0, three blank spots, and then a 1." Explain to students that they should always make sure they have enough space to graph all of the data they need. "For example, if we were surveying someone who read 30 books last month, we probably shouldn't skip too many spaces since we'll run out of space on our number line before we get to 30. However, because we only need to get up to 4 for this line plot (the most books read by anyone in our survey), we can skip 3 marks between each full number."
"Where should we put the 2 on our number line for the spot where we'll show how many people read 2 books?" Engage students in a discussion about maintaining a constant scale on the number line, which involves placing three empty marks between whole numbers. Students should finish labeling their number lines and then compare them with a classmate to ensure they match. (Remind students that they may have started in slightly different areas. The crucial factor is that the distances between the complete numbers are equal.
"Now we're ready to putting in statistics on how many people read books last month (0, 1, 2, 3, or 4). We're going to use the same data as the bar graph. How many people read 0 books last month?" (3)
"So, above the spot marked 0 on your number line, put 3 Xs, stacked on top of each other like you're building a tower." If necessary, illustrate how to draw the Xs on the board. "That's it. Each X indicates one person; in this case, each X represents someone who did not read any books last month."
"How many people read 1 book last month, based on the data in the bar graph?" (6) "So how many Xs should we put above the spot marked 1 on the number line?" (6)
Students should finish their line plots and compare their work to that of their neighbors. When all students have finished transferring data from the bar graph to the line plot, ask, "Are we done making our line plot?" If students say yes, ask them: "If you handed this to someone who wasn't in our class and hadn't seen the bar graph, would that person know what the data represented?" Encourage students to recognize that they must still offer a title (which can be the same as the title of the bar graph) and label the number line (which can be the same as the label on the horizontal axis of the bar graph). "Remember, titles and labels are important; without them, nobody will be able to tell what the line plot is about."
"That's it—that's all there is to making a line plot." Ask students to list some of the most significant factors to consider while creating a line plot. Considerations should include:
accurate counting of the number of Xs labels
equal spacing between values on the number line
Activity 2
Remove the projected bar graph; it will not be used for the remainder of the lesson. Only the line plot will be updated to include new, extra data.
"But now we've got a problem. I decided to survey more people. The first person I asked stated she read half a book last month but didn't finish it. What should we do with this information?" If students suggest rounding up to 1, ask, "What if we have two persons who each read half a book? How many books did the two of them read?" (One) "However, if we round each of them up to one, it appears that they read two books, which is twice as many. There are many cases where rounding is appropriate, but I don't think it is what we want to do in this situation."
If students do not want to simply plot the data on their line plot, ask them what the spaces on the number line between 0 and 1 represent, particularly the space halfway between 0 and 1. After some discussion, ask students to label the space halfway between 0 and 1 as \(1 \over 2\).
"Now, we have a spot on our line plot for people who read \(1 \over 2\) of a book last month, so let's represent our new person with an X in that spot."
"The second person I surveyed read \(1 {1 \over 2} \) books. Where should we put him?" Encourage students to talk to one another before adding them to the line plot. Also, have students label the places on the number line that correspond to \(2 {1 \over 2} \) and 3 books.
"The third person I surveyed, however, is causing more complications. She has read \(3 \over 4\) of a book. She just didn't have time to finish everything. How should we represent her on our line plot?" Help students understand that \(3 \over 4\) is halfway between \(1 \over 2\) and 1. Students should label the relevant mark on their line plot, then add an X to indicate the new person.
Have students work in pairs. Give each pair a copy of the Tally Chart worksheet (M-5-4-2_Tally Chart). "This tally chart covers ALL of the extra people I surveyed. This covers the person who read \(1 \over 2\) a book, the person who reads \(1 {1 \over 2} \) books, and the person who reads about \(3 \over 4\) of a book. Your task is to make sure that all of the people shown in the tally chart are appropriately represented on the line plot. Remember that you have previously represented three of the people on the tally chart."
Students should work in pairs to complete one copy of their line plot. (Students should write both their names on the line plot that will be used for evaluation.) After each couple has revised their line plot, bring the class back together.
Activity 3
"Now that each pair has a complete, correct line plot, let's look at the new data. According to your line plot, how many people were surveyed in total?" (43)
"How many people read 2 books or more?" (23)
Ask questions related to the line plot as a whole, such as:
"What is one number of books that no body read?" (\(1 \over 5\))
"Which fraction or mixed number of books was the most popular survey response?" (\(1 {1 \over 2} \))
"How many people read more than 1 book but less than 2 books?" (5)
And finish with "How many books were read in total by all 43 people surveyed?" (\(77 {1 \over 4} \))
This question should be followed by an explanation of "strategy," or how to address the problem. Point out that all of the whole-number answers matched the ones on the bar graph, and the class had already decided that those replies equaled a total of 48 books. Now, have students concentrate on the fractional numbers.
Discuss several strategies, with a focus on "pairing up" numbers to make wholes, such as pairing up a \(1 \over 4\) and \(3 \over 4\) to make a whole book. Recommend that students find appropriate pairs and cross off or otherwise mark the Xs on their line plot to indicate that they have counted the person.
Point out that values involving \(1 \over 2\) can be paired up with themselves (or other values involving). Depending on the lesson, a quick review of adding mixed numbers may be helpful. (For this activity, students should be able to add mixed numbers conceptually, i.e., without using common denominators, etc.)
Have each couple collaborate to come up with a grand total (including the 48 shown on the bar graph). After each couple has a grand total, collect all of the different responses and write them on the board. (Hopefully there is only one different answer, but there could be more.) If there are many responses, create a tally chart to show how many groups provided each one.
The correct answer is \(77 {1 \over 4} \). Choose a group that provided the correct answer and ask them how they calculated the fractional amounts from their line plot. (If there are multiple groups with the proper answer and multiple groups with the incorrect answer, have the group with the correct answer collaborate with one or two of the "incorrect" groups to explain their reasons.)
Each pair of students should turn in their Number Line worksheet (the completed line plot) to be assessed for comprehension. You can use this as an exit ticket or check them at another time.
The Match the Line Plots exercise (M-5-4-2_Match the Line Plots) can be used as an alternative way to assess students' comprehension of how data sets relate to line plots. It will take only a minute or two for students to draw a line connecting each line plot to its data source. There are no labels or titles for students to utilize as clues; instead, they must match the labels and results in the line plot to the data in the tables and tally charts.
The Line Plot Quick Quiz (M-5-4-2_Line Plot Quick Quiz and KEY) is one final option that you can use if time allows. This quick quiz assesses students' ability to read, analyze, and understand data from line plots, as well as answer questions based on the data in the plot.
Extension:
Use the strategies listed below to adjust the lesson to your students' needs throughout the year.
Routine: As students learn about different types of data displays, they can transfer data from those displays to line plots. Students may also be asked questions related to the data (whether in line plots or not) that require them to perform computations with rational numbers.
Students can also gain further practice improving their skills for working with mixed numbers by setting up problems that require them to analyze line plots and work with more "difficult" fractions.
Small Group: Students might work in small groups to come up with survey questions that may receive fractional responses. Have each group generate data and a data display that is not a line plot, as well as write questions (similar to those you asked during this lesson) to produce a worksheet for another group. The other group should start finishing the worksheet by drawing a line plot to represent the data.
Expansion: The concepts presented in this lesson can be explored by having students perform actual surveys and create line plots and worksheets to reflect the results. The lesson can easily be modified to incorporate additional "difficult" fractions, requiring students to use common denominators and practice mixed number computation.
