Students will learn about various types of data displays, including tally charts, tables, pictographs, bar graphs, and line graphs. Students will: 
- display data using a variety of data displays. 
- transfer the data from one display to another. 
- interpret data from various data displays.
 

- What does it mean to analyze and estimate numerical quantities? 
- What makes a tool and/or strategy suitable for a certain task?
- How may data be organized and represented to reveal the relationship between quantities? 
- How does the type of data effect the display method? 
- How can probability and data analysis be used to make predictions? 

- 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.

- graph paper for each student 
- one display copy (or projection-ready copy) of the Full Ticket and Half Ticket (M-5-4-3_Full and Half Ticket) 
- Lesson 3 Exit Ticket (M-5-4-3_Lesson 3 Exit Ticket and KEY) 
- Lesson 3 Optional Writing Assignment (M-5-4-3_Lesson 3 Optional Writing Assignment)

- Examine the graphs students submitted, looking for errors and missing information (labels, etc.) to measure their level of comprehension of displaying data. 
- Use the Lesson 3 Exit Ticket (M-5-4-3_Lesson 3 Exit Ticket and KEY) to assess students' proficiency with various types of data displays. 
- Use the Lesson 3 Optional Writing Assignment (M-5-4-3_Lesson 3 Optional Writing Assignment) to see if students comprehend the differences and benefits of each type of data display. This may be used as a homework assignment.

Scaffolding, Active Engagement, Modeling, and Explicit Instruction 
W: The class will discuss both familiar and unfamiliar data displays. Differences and similarities between the displays will be identified. Students will learn how to analyze and construct different types of data displays. 
H: Introduce a hypothetical situation where students create a website to rate movies. Encourage them to think about how they would generate their own data, what movies they would grade, and so on. The data supplied is directly related to their experiences. 
E: Students will first follow teacher-guided instruction, but eventually make educated estimates about future graph construction based on similarities between their current graphs. Students will create graphs on their own and in small groups, and then examine the data by answering questions regarding the data being displayed. 
R: The similarities between different types of graph allow students to practice and improve their graph-making skills. Because students create various graphs during the activity, they have multiple opportunities to correct errors and try again. 
E: Students will evaluate their work by comparing it to that of their peers and receiving feedback from you. 
T: Use the Extension section to customize the lesson to match the needs of the students. The Routine section can be utilized throughout the year to review and reinforce lesson concepts. The Small Group section offers additional learning chances for students who could benefit from more time with the idea. The Expansion section contains optional suggestions for students who are ready for a challenge that goes beyond the criteria of the standard. 
O: The class starts with teacher-guided instruction, followed by having students work with partner or in small groups work before finally constructing their own line graph utilizing information and techniques learned in the previous activities. 

Activity 1

"What options do we have for displaying data in a graph or chart? Don't answer aloud; instead, think about some of the various options for displaying data. We shall talk about them in a few minutes. Imagine you have a website where you can rate movies you've seen. Each movie can be rated on a scale of 0 to 5, with 0 representing a poor movie and 5 representing your favorite movie. Assume you create a tally chart to indicate how many movies you gave each rating after rating a bunch of movies." Display the tally chart displayed below.



"How many movies did you rate as a 5?" (2)

"How many movies did you rate as a 3?" (5) When reading tally charts, make sure students understand that each group of five is represented by the fifth mark drawn diagonally. 

"How many movies did you rate in total?" (19) 

Ask students if they can also display this information in another way. Suggest a bar graph and ask students why this data is suited for a bar graph. (One possible answer is that there are six distinct categories, with each movie belonging to only one category.) 

Give each student a piece of graph paper.

"First, we need to draw a horizontal and vertical line to make the edges of our graph." Show this on a piece of graph paper.

"On the horizontal line, we're going to make space for each rating we could give a particular movie." Arrange the numbers 0, 1, 2, 3, 4, and 5 equally along the horizontal axis, then have students do the same. 

"Now, the vertical line will help display the amount of movies we gave each rating. We must begin at the bottom with 0 and go upward. How high should the numbers on our vertical line go?" Make sure students understand that the vertical axis must begin at 0 and extend to the highest possible number in our data collection.

Guide students through the process of selecting an appropriate scale, one that will clearly indicate the differences in the number of movies given specific ratings while leaving enough space to graph all of the data. Explain to students that the scale must be constant. 

After students have labeled the vertical axis, ask them, "How high should the first bar be, the one that represents the number of movies rated 0?" Students should understand that it should be 1 "unit" high. Students should draw a bar for that data and then complete the process of representing the data in the bar graph.

After students have done, ask them, "Is that all we need to do to make a bar graph?" If students are unsure what is missing, ask them what the numbers at the bottom of the graph (along the horizontal axis) signify. Guide them to the conclusion that both axes need labels, and the graph needs a title. The completed graph should look like as follows:



Remind students that for each table, graph, or plot they create, they must name both axes as well as offer a title. Without this information, the graph has no meaning and no one can interpret it correctly.

Activity 2

Tell students another option to display the data about our movie ratings is in a pictograph. Discuss the parts of the word and what they mean. Students should be able to make guesses regarding both picto- (referring to photos) and -graph (officially, "written," but for the purposes of this lesson, students should just admit that it is a graph). Ultimately, students should understand that a pictograph is essentially a graph with pictures.

On the board, create a vertical axis labeled "Movie Ratings" and a row for 0, 1, 2, 3, 4, and 5. "When creating a pictograph, we need a vertical axis. Our pictograph is going to be kind of like the bar graph we made but turned sideways.” Show students a copy of the complete ticket drawing (M-5-4-3_Full and Half Ticket) and explain that each movie ticket in this pictograph will represent one movie. 

"Now you know that each movie ticket is going to represent one movie, but we need to tell someone who might not know that, so we'll make a key."

Write the word “Key” at the bottom of the graph. Tape up a copy of the Full Ticket next to the word "Key". To the right of the ticket, write, "= 1 movie."

"The key is an essential part of a pictograph. It tells the reader of the graph what each symbol represents. Why might we use a key other than '1 ticket equals 1 movie'?" Guide students through a discussion of data sets with a large number of responses, when drawing tens or hundreds of tickets may not be the best option. Tell students that an example of such a pictogram is coming up, but first we'll finish the pictogram we've started.

"So, if one ticket represents one movie, how many tickets should we put in the row representing movies rated 0?" (Students can use the tally chart or bar graphs to see how many movies received each rating.) Place a single ticket in the 0 row. Have the class lead you through the creation of the rest of the pictograph. 

"Are we finished now that we have all the rated movies represented?" Students should take note that the pictogram does not yet have a title. Write a title for it on the board.

Now, tell students to draw an axis for their pictogram on graph paper. Tell them they'll recreate the pictograph from the board with one change: one ticket should symbolize two movies rather than just one. Students should work together in pairs to create a new pictograph representing the data. If students are wondering what to do with odd-numbered data, ask them, "If a whole movie ticket represents two movies, what part of a movie ticket represents one movie?"

After each group has completed their pictograph, have them compare it to their neighbors' graphs to ensure they completed it correctly. Then, have one group come forward and adjust the pictograph on the board (while also adjusting the key) with Full Tickets and Half Tickets to recreate the proper pictograph.

"What if we changed the key so that each movie ticket represented 10 movies? Is that a good choice?" Students ought to recognize that, given the data, this would not be a good key since they would need to distinguish between, say, \(1 \over 10\) of a ticket and \(2 \over 10\) of a ticket. "So, selecting a key that makes sense is an essential step in creating a pictogram. Here, a key of 1 or 2 is usually the best option."

"Now, suppose that in another year, after you've seen and rated more movies, we create a pictograph to represent your ratings." Create a new pictograph with the Full and Half Tickets, changing the key so that 1 ticket equals 10 movies. Ask students to interpret the data on the new pictograph, using questions like: 

"How many movies were rated a 1?" (30) 

"How many more movies were rated a 3 than a 2?" (10) 

"How many movies were rated in total?" (190)

Activity 3

"We have one more type of graph to investigate. This is a line graph. When we build up our graph, it will seem similar to a bar graph. We'll have a scale up the side of the graph and values along the bottom. Of course, we'll include labels and a title so that people understand what our graph is about. This table contains the data that will be used to create our line graph." Display this table:

Daily Temperature

Give students another sheet of graph paper and ask them to draw a horizontal and vertical axis. "Label the lines on the horizontal axis as Monday, Tuesday, and so on. As with a bar graph, keep an equal gap between the days." Point out that in a line graph, typically labels mark individual lines, whereas in a bar graph, labels may be located between vertical lines. After students have finished that stage, ask them to indicate the vertical axis.

"Just like a bar graph, we'll begin with 0 at the bottom. How high should our labels go?" (65) "To count by 1s, we'll need to have 65 squares. Is there a better number to count by?" Students should recommend 5s. Discuss any further suggestions as they make them. For example, counting by twos would still take up a lot of space on the graph, whereas counting by 10s would make the graph slightly smaller, forcing us to fit numbers like 45° between 40° and 50°. Ask students to label their graphs in 5s.

"Now we just plot the points. For example, follow the line that symbolizes Monday until you reach the horizontal line that represents 45°. Put a dot there to indicate that the temperature was 45° on Monday at noon. Then move on to the next day, Tuesday, and place a dot where the lines for Tuesday and 50° meet." Students should finish charting the five points that will make up the base of the line graph, and then connect the data points using straight lines.

"One significant difference between line graphs and other plots we've created is that after plotting our separate points, we link them with a line. Instead of a graph with 5 different points, we have a graph that covers the entire time from Monday at noon (when it was 50°) to Friday at noon (when it was 65°). Remember that every point reflects the temperature at noon." 

Again, ask students if they are finished with their graphs. They should note that they require labels and a title; have them add those.

Ask students questions like, "Which two days saw the highest increase in noontime temperature?" (Thursday and Friday) Explain to students that one advantage of line graphs is that they make it simpler to notice differences between consecutive numbers by observing how steep the line connecting the two values is.

"Now, based on your graph, what was the temperature on Monday just before midnight?" Students may notice that this point is missing on their graph. Discuss how 11:59 on Monday (to avoid confusion about whether midnight on Monday is at the beginning or conclusion of the day) is pretty much halfway between noon on Monday and noon on Tuesday. "So, to determine the temperature based on the graph, we just look at where our line is halfway between the point for Monday and the point for Tuesday." The temperature should be around 47°. 

Lead a discussion among students on whether 47° is a reliable predictor by examining what happens to temperatures after the sun goes down (i.e., they drop dramatically and then rise again in the morning).

"Increasing the number of points on the graph would improve our prediction. For example, if we graphed a point every hour on Monday and Tuesday, we'd get a far more accurate picture of how the temperature is changing. So, line graphs are useful for making predictions, but the quality or accuracy of the predictions is determined on the number of data points available in the first place." 

Students should turn in their bar graphs, pictographs, and line graphs at the end of this activity.

The lesson encourages students to consider different methods to express data. It also encourages students to be creative when designing their graphs, highlighting that no scale is absolutely correct. Students can be creative in the extension activities, such as collecting their own data or creating their own surveys. 

Extension: 

Use the strategies listed below to adjust the lesson to your students' needs throughout the year.

Routine: Newspapers like as USA Today frequently use "infographics," which take the form of bar graphs, pictographs, or line graphs. These can be brought in on a regular basis throughout the year, and students should discuss the findings presented in the display. They can also transfer the data to another type of display than the one that was originally published.

Small Group: Students that require additional help may be divided into small groups and assigned the following task. Give each small group a chart, table, or tally chart containing data that can be displayed using bar graphs, pictographs, and line graphs. Encourage each member of each small group to design a different type of graph. After everyone in a group has completed their data displays, the members of the group should compare and contrast them. Have each group create a list of the similarities and differences between their graphs.

Expansion: This task may be assigned to students who are prepared for a challenge that exceeds the requirements of the standard. Line graphs can include negative numbers (primarily y-values such as negative temperatures, elevations, etc.) Allow students to investigate double and/or stacked bar graphs. Furthermore, with any sort of graph, students can design and conduct their own survey or gather their own data (such as actual temperature data) and make a brief presentation on their survey/data, utilizing a data display to support up their conclusions.