7.4.2 Circle Graphs and Histograms
Use reasoning with proportions to display and interpret data in circle graphs (pie charts) and histograms. Choose the appropriate data display and know how to create the display using a spreadsheet or other graphing technology.
Overview
Standard 7.4.2 Essential Understandings
Students in 7th grade have been using different displays to show data they have collected. They have used bar graphs, line plots, line graphs and double bar graphs. They will move into circle graphs and histograms in 7th grade, finding a part of a circle (sector) given the percent or other data for the section. They should be able to determine missing values of a circle graph. They will transfer the data into displays to show the data. They will use the information gathered or make a frequency table to help them in the display of the data. They will also deepen their understanding of proportional reasoning to display and interpret the data.
All Standard Benchmarks
7.4.2.1
Use reasoning with proportions to display and interpret data in circle graphs (pie charts) and histograms. Choose the appropriate data display and know how to create the display using a spreadsheet or other graphing technology.
7.4.2 Circle Graphs and Histograms
7.4.2.1
Use reasoning with proportions to display and interpret data in circle graphs (pie charts) and histograms. Choose the appropriate data display and know how to create the display using a spreadsheet or other graphing technology.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Find degrees in a part of a circle (sector) given the percent or other data for the section;
- Be able to determine the percent of a sector of a circle;
- Use given data to determine missing values of a circle graph;
- Transfer the data from a table to the data in a circle graph;
- Use given data to determine missing values.
- Display and interpret data with a histogram;
- Create circle graphs and histograms using technology (spreadsheet or other graphing technology).
Work from previous grades that supports this new learning includes:
- Multiply;
- Find a percent of a number;
- Find and write fractions;
- Scale up fractions (find equivalent fractions);
- Work with circles;
- Round correctly;
- Find equivalent fractions and ratios;
- Use frequency tables;
- Use/make bar graphs, picture graphs, line plots, tables, timelines, Venn diagrams and double graphs.
NCTM Standards
Formulate questions that can be addressed with data and collect, organize and display relevant data to answer them.
- Select, create and use appropriate graphical representations of data, including histograms, box plots and scatterplots.
Common Core State Standards (CCSS)
[None]
Misconceptions
Student Misconceptions and Common Errors
- Students use the percent of a circle and graph for degrees instead of finding the number of degrees (by taking percent x 360).
- Students will sometimes choose to make the wrong type of graph when a different graph type might have been a better choice for a given data set.
- Students have confusion about the difference between a histogram and a bar graph.
- As students make histograms, sometimes they do not use intervals that are equal on the horizontal axis.
- When comparing two histograms, students sometimes will not pay attention to the scale used.
- Students will sometimes make calculation errors when preparing pie charts, resulting in their percentages of the "pie sections" not adding up to 100%.
Vignette
In the Classroom
In this vignette, students use information from a budget scenario to make circle graphs.
Teacher: What types of graphs have you learned to make?
Student: Line graphs, bar graphs and histograms.
Teacher: What type of graph do you think you would use in this scenario? You have a certain amount of money you can spend each month, and you want to record how much you are spending in different categories.
Student: Um, I think a histogram could work for that.
Teacher: It could, but there is another type of graph that would work too: a pie graph.
Student: How do you know when to use a pie graph and when to use a histogram?
Teacher: A pie graph works well for data with categories and for data that has a total, like 24 hours in a day, or a budget like I mentioned. The advantage of a pie graph is it displays the data in a "part-to-whole" representation. The entire circle is always the whole and the slices of the pie, or sectors of the circle, represent the proportional parts.
Student: Why is a pie chart better for that?
Teacher: Let's try using the example of a budget to make a pie chart, and I think you will see why. What are some categories you might spend money on?
Student: Clothes, food, video games, gifts, savings and music.
Teacher: OK, let's say your parents give you $100 a month for allowance. How much money would you spend in each of those categories?
Student: $20 on clothes, $15 on food, $30 on video games, $10 on gifts, $10 for savings and $15 on music.
Teacher: OK. Now we can use the circle graph activity to create a pie chart based on your data. Enter your data into the box below the circle and then click "Update Chart."
Students view their results.
Student: Maybe I should cut back on video games!
Teacher: Maybe you should! Let's see what happens if you decrease your spending in that category.
Student: The other sections got bigger!
Teacher: Yes, the sections got bigger because in a pie chart each category is proportional to the other categories. Pie charts provide a very clear image of how categories relate not only the parts to each other, but also compare them to the whole circle.
Student: Oh! Is that what makes it different from a histogram?
Teacher: Both histograms and pie charts show the relationships between categories, but in different ways. However, what makes a pie chart different from a histogram is that you can visually see the categories making up portions of a whole. Pie charts make it very clear how one category compares to the total. In other words, pie charts make it easier to see how much of the whole each category represents.
Student: I understand! Using the pie chart, it was easy for me to see just how much of my total money I would spend on video games and how much I would spend on food.
Source: http://www.shodor.org/interactivate/discussions/PieChart/
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- In preparation for interpreting circle graphs, have students review the percent of a circle equivalent to sectors with various angle measures. Ask them to identify the angle measures equivalent to 20%, 25%, 50% and 75%. Answers: 72°, 90°, 180°, 270°
- Students can find the number of degrees in a sector in multiple ways. Ensure they know why they are doing what they are doing instead of just using a proportion to solve it.
- To find degrees in a sector given the percent of the sector, students can set up a proportion to find missing value. For example, $\frac{x}{360}=\frac{34}{100}$; the percent of the sector of the circle to find the degrees in is 34%.
- Have students try finding the number of degrees in a sector of a circle by finding percent or by using a proportion to solve.
- Help students make the connection between histograms and stem-and-leaf plots.
- Encourage students to make a frequency table before they start developing a histogram. First determine what intervals would be best for the data and then make the frequency table (possibly by using tally marks to begin with). For example:
- Students should be reminded that the percents in a pie chart should add to 100%.
- Brainstorm with the students about situations where circle graphs would be a better choice than a histogram or other graph. Some examples might include:
1. when the parts make a meaningful whole (total budget, total number of votes);
2. when you want to compare the parts to the whole (total) and not to each other;
3. when there aren't too many "slices" of the pie; too many slices make a pie graph hard to read.
- Based on the circle graph given below, answer the following questions.
1. What type of information is being presented on this graph? 2. If the total spending is $50,000, how much money was spent on highways? 3. Approximately how many times the amount of spending on highways is spent on education? 4. Approximately what fraction of the total expenditures are spent on highways and public welfare combined? |
Expenditures for State and Local Governments
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Answers:
- The graph title is Expenditures for State and Local Governments. The circle is cut into four sectors, each representing one category of expenditures, with the percentage of the total for each category.
- Money spent on highways was $3,500.
- Education spent approximately five times more on education than on highways.
- Approximately 1/5 of the total is spent on highways and public welfare combined.
Source: http://cstl.syr.edu/fipse/tabbar/ReadCirc/REVCIRCL.HTM
This tool can be used to create a histogram for analyzing the distribution of a data set using data that you enter or using pre-loaded data that you select.
This website allows students to graph data sets in bar graphs. The color, thickness and scale of the graph are adjustable. Students can input their own data or use or alter pre-made data sets.
The activities at this level focus on studying a series of circle graphs that report the population of the United States and of selected states-Massachusetts, New York and Pennsylvania-in fifty-year intervals from 1800 to 1950. The students are asked to discuss and describe the information and to explain why the percent of the total population changed for the three states. They are asked to write a justification for their explanation.
Using data from the Internet, students summarize information about party affiliation and ages at inauguration of Presidents of the United States in frequency tables and graphs. This leads to a discussion about categorical data (party affiliations) vs. numerical data (inauguration ages), and histograms vs. bar graphs.
Additional Instructional Resources
- Histogram vs. bar graph
This website prompts an example discussion about the difference between a histogram and a bar graph.
- Pie charts and histograms
This website has links to activities involving pie charts and histograms.
- Circle Graph
This activity allows the user to graph data on a circle graph (pie graph). Students can use predefined data sets or enter their own data. It also includes additional resources, such as related activities or worksheets.
- Circle Graphs, Instructions
This website lets students create labels for each category and enter the number of items in each category that they want to display on a circle graph.
- Constructing Circle Graphs
This website offers step-by-step instructions and could be used by all levels of students. It covers the construction of a circle graph from a table of information and how to critically read circle graphs, particularly comparing parts to a whole.
This website offers a great review of circle graphs, vocabulary, how to construct them, what they show and many other valuable resources for help with circle graphs.
circle graph: a graph in the form of a circle that is divided into sectors, with each sector representing a part of a set of data; also called pie chart or a pie graph. The circle represents the whole and each sector of the circle proportionately represents a part of the whole.
histogram: a histogram is a type of bar graph in which the bars are used to represent the frequency of numerical data that have been organized into equal intervals.
Example:
frequency table: a frequency table is a table that lists items and uses tally marks to record and show the number of times they occur.
Example:
sector: the part of a circle enclosed by two radii of a circle and their intercepted arc; a pie-shaped part of a circle. A sector is a part of the whole circle graph, which proportionally represents part of the data that the graph represents.
Example:
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- How can this content be connected to other benchmarks in learning?
- How can students see the connection between making circle graphs and proportions?
- Are students interpreting the graphical data correctly?
- In what areas did students perform best and what weaknesses are evident?
- Were there any misconceptions that were not anticipated?
- Pie chart lesson
- Math resources subscription
This website provides many resources, including a dictionary, lessons, solutions to problems, etc. - Review of circle graphs
Assessment
1.
Answer: d
Source: Minnesota MCA Math Grade 7 Item Sampler
2. Mr. J made a frequency table of the scores his students got on a test.
Score |
Frequency |
Below 75 |
4 |
76 - 80 |
14 |
81 - 85 |
2 |
86 - 90 |
8 |
91 - 95 |
5 |
96 - 100 |
1 |
How many students got a score that was more than 85?
Choices:
A. 14
B. 13
C. 17
D. None of the above
Answer: a
Source: http://www.northstarmath.com/sitemap/FrequencyTable.html
3. Find the angle covered by A, B and C in the circle graph.
Choices:
A. 306°
B. 162°
C. 36°
D. 108°
Answer: a
What percent of the circle is covered by A, B and C in the circle graph above?
A. 46%
B. 306%
C. 85%
D. 54%
Answer: c
Source: http://www.northstarmath.com/SiteMap/CircleGraph.html
4. Given the table below, create an accurate circle graph which represents the same information. Shares of Stock Owned by an Investor
Type of Stock |
Number of Shares |
Coca-Cola |
8 |
Pepsi |
10 |
IBM |
4 |
Exxon |
8 |
General Motors |
20 |
Answer:
Shares of Stock Owned by an Investor
Source: http://cstl.syr.edu/fipse/tabbar/bldcirc/PRAC7A.HTM
5. The discounts offered by a super market are as shown in this table. Which of these histograms is the correct representation of the data?
Amount (in $) |
Discount% |
1,000 - 1,999 |
2 |
2,000 - 2,999 |
4 |
3,000 - 3,999 |
6 |
4,000 - 4,999 |
8 |
5,000 - 5,999 |
10 |
Choices:
A. Figure 1
B. Figure 2
C. Figure 3
D. Figure 4
Answer: a
Source: http://www.icoachmath.com/math_dictionary/histogram.html
6. The circle graph below represents the percentages of people who like different fruits. Find the number of people who like apples, if the total number of people is 1,000.
A. 300
B. 250
C. 400
D. 350
Answer: d
Source: http://www.icoachmath.com/solvedexample/sampleworksheet.aspx?process=/__cstlqvxbefxdxbgefaxkhgmgxdaahejxb&.html
Differentiation
- If the data is not already in a table, help students organize a table and guide them in completing the data table.
- Instead of requiring a compass to draw the circle for making a pie chart, have the students trace around a circular object.
- When making a histogram, help students determine what intervals will work the best for the horizontal axis.
- Encourage students to make a frequency table of a data set before attempting to make a histogram.
- Have students make a table comparing and contrasting bar graphs and histograms.
- Have the students write (or give them) the step-by-step instructions for creating a pie chart.
- Comparing basketball statistics
This website provides links for sports data and statistics, which can be interpreted and used to make charts and graphs.
- Have students find examples of histograms and circle graphs in newspapers and magazines. Ask them to create a poster displaying all of the graphs including information such as title, scales used on histograms, and values for pie charts. Require the students to write a few sentences describing what information the graph is trying to convey.
- Have each students develop a survey question, collect data using that question, and make a pie chart and histogram using the data. This will require students to see if they are asking the right type of question to provide the type of data needed to make these types of graphs.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) |
Teachers are: (descriptive list) |
reading various graphs of data. |
asking students to interpret the data after it is graphed. |
using proportional reasoning to analyze data and create circle graphs. |
requiring students to make both pie charts and histograms. |
interpreting the data in histograms and circle graphs. |
providing students with appropriate materials for making circle graphs and histograms. |
using technology to create circle graphs and histograms. |
allowing students to see the connection these graphs have with proportional reasoning. |
analyzing the data and deciding which graph would be appropriate to represent the data efficiently. |
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Parent Resources
- Circle Graphs
In this activity, students create labels for each category and enter the number of items in each category that they want to display on a circle graph.
- AAAMath
This site features a comprehensive set of interactive arithmetic lessons. Unlimited practice is available on each topic, which allows thorough mastery of the concepts. A wide range of lessons (kindergarten through eighth grade) enables learning or review to occur at each individual's current level