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History of Populations

Data Analysis and Probability
Location: Unknown

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.

Begin the lesson with a discussion on the use and importance of population data. You might want to ask questions such as:

  • How is population data collected? [Through the Census, which is taken every 10 years.]
  • How is population data used? [To determine voting districts, public services, to make predictions for economic and population growth, etc.]
  • What are some ways of displaying population data? [Tables, line graphs, circle graphs, etc.]

Distribute the History of Populations Activity Sheet to the students.

pdficonHistory of Populations Activity Sheet 

Individually, students should describe what they see in the circle graphs. Ask them to give an explanation of the information shown in the graphs. Discuss these explanations as a class.

Prior to using the circle graphs, students should create a title for all four graphs. Students might suggest "Percent of Total U.S. Population by State." Students should record their title at the top of the first page of the activity sheet.

Ask the students to:

  • describe the information in each circle graph, and
  • explain what steps would have been followed to construct the graphs.

Record the following information on the chalkboard or overhead. Then ask the students to individually figure out the population of the three states for the four dates.

Population of the United States
(nearest million)

Year Population 

In pairs, students should compare their solutions and how they arrived at them.


  • New York: 1800 (400,000); 1850 (2,300,000); 1900 (12,160,000); 1950 (15,300,000)
  • Massachusetts: 1800 (400,00); 1850 (690,000); 1900 (3,800,000); 1950 (3,400,000)
  • Pennsylvania: 1800 (440,000); 1850 (2,070.000); 1900 (7,600,000); 1950 (10,200,000)

Individually, students should record their observations about the graphs and share as a class. Some of these observations might include the following:

  • How do the percents for each of the three states and the other states change from one graph to the other?
  • What was happening to the population for the three states throughout the time span covered by the graphs?
  • Did the percents for the three states decrease from one fifty-year period to the next?
  • Why do you think the percent for New York increased between 1800 and 1850?

Have the students work in pairs to complete the activity sheet, questions 1 through 4. Ask the students to share their explanations with their partners, as well as the whole class.

Questions 5 and 6 require other resources, such as web access, in order to find census data. These can be completed in class, or they may be assigned as homework.

Solutions for the History of Populations Activity Sheet 

Question 1. 31%; decreased (in 1850), increased from 850 (in 1900); decreased (in 1950). The percent changes were 22% (in 1850), 31% (in 1900), and 17% (in 1950.)

Question 2. In 1800, Pennsylvania had the largest percent of the population. In 1850, 1900, and 1950, New York had the largest percent of the population.

Question 3. From 1800 to 1950, Massachusett's population percent decreased from 10% in 1800 to 2% in 1950, an overall decline of 8%.

Question 4. One possible explanation for New York in 1900 was the number of immigrants coming through Ellis Island.

Question 5. In 1800, there were 13 other states. In 1850, there were 28. In 1900, there were 42. And in 1950, there were 45. Students may use an almanac (either in book form or online) to help them answer this question.


Irons, Calvin and Irons, Rosemary. The Arithmetic Teacher. October, 1991. vol. 39, no. 2. p 26 - 33.



Assessment Option 

Students can complete the Gathering Graphs Activity Sheet at home and bring it to class the next day. Note that when students are cutting out graphs, instruct them not to cut a graph out of a book. Instead, they should make a photocopy of the graph.


1. Have the students find out population information for other states that existed in 1800 for each of the years shown in the pie graphs on the worksheet: 1800, 1850, 1900, 1950.

  • Ask the students to calculate the percents for the population for each of those years for each state.
  • Then they can study the percents to check if the general trend was similar to the trend shown in the circle graphs.

2. Find out the current population of Massachusetts, New York, Pennsylvania, and the United States.  

  • Use Microsoft Excel or another spreadsheet/graphing program to construct a circle graph which represents the population information.
  • Determine if the general trend shown on the activity sheet continues.
  • Ask the students to predict what a circle graph for the three states and other states will look like for the next census.

3. Students can use their state and neighboring states to construct a circle graph, representing the percent of the total U.S. population as compared to their state, neighboring states, and "other states."

Data Analysis and Probability

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In this lesson, students analyze information represented by pictographs. Students are asked to discuss, describe, read, write, and make predictions about the graphs and the information they contain.
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Name Your Tune

The activities in this lesson focuses on studying information displayed in pictographs that report the sales of records, tapes, and compact discs for one week at a music store. The students are asked to discuss and then describe the information and to explain the numbers sold in each category. They are asked to defend their explanation.

Learning Objectives

Students will:

  • Describe and explain information displayed in a circle graph.
  • Make written predictions from the information.

NCTM Standards and Expectations

  • Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.
  • Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken.