Intent

This activity extends students’ work with writing expressions for mathematical situations.

Mathematics

In this activity, students are asked to create equations to represent the relationship between future and present populations. They are encouraged to use numeric examples to clarify their use of variables. The equivalent expressions that emerge offer another opportunity to review the distributive property.

Progression

Students work individually and in their groups before coming together as a class to share their thinking about converting from written problems to algebraic equations.

Approximate Time

10 minutes for introduction

20 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, then small groups, followed by whole-class discussion

Materials

Information about each group’s four families

Doing the Activity

Read the passage from Life of George Bent aloud with students before they start their work. On the map, point out the part of the trail to which the activity refers—from Fort Kearny, Nebraska, to Fort Laramie, Wyoming, along the North Platte River.

If the activity is assigned for homework, it may be worth having students spend time in their groups to do at least one numeric example for Question 1 to ensure that everyone realizes that the process will involve multiplying the population by the decimal equivalent of the percentage of loss. Remind students of the difference between an expression and an equation as necessary.

This activity might be difficult for many students, primarily because they may be unfamiliar with what it means for a population to decrease by a given percentage. The suggestion in the student book to begin with a particular number to represent the starting population is useful, but students may still need encouragement to complete the necessary mathematics. Once a student has selected a population for 1492, you might ask, How much is 90% of that population?

Although the computations in the activity are simple (involving multiplication and subtraction), the process of using variables and equations to represent information about real-world situations can be challenging. As this is a fundamental idea in mathematics, allow ample time for questions and discussion. Students may also need help identifying the variables in the situations.

Students will need to know the total number of adults and total number of children on the class wagon train. They should have this information from their work on Overland Trail Families.

Discussing and Debriefing the Activity

Have students share ideas in their groups as you assess what they have been able to accomplish with the task.

Ask the students presenting Question 1 to describe exactly what they did, especially the process by which they arrived at an equation. Most students will use one variable (for example, B) for the population at the beginning of the time period and another (for example, E) for the population at the end of the time period and write an equation such as E = 0.1B or E = B – 0.09B to express the relationship between them. (Estimates of the Native American population in 1492 vary widely, from 800,000 to 30,000,000. The 1900 census reported the Native American population as 237,000.)

Next, have students share ideas about creating a rule for the revised wagon train size in Question 2. This is more difficult than Question 1 because it involves two input variables.

Students may have had trouble finding a rule for their tables in Question 2c. If so, have them describe what they did to get their numeric results in Questions 2a and 2b, which can help them to see an arithmetic pattern and produce an equation like N = 0935A + 0.9C to describe the situation. Be aware that some students may not yet be completely comfortable with such algebraic symbolism.

Someone may point out that it doesn’t make sense for the output from this table to be anything other than a whole number. You can let the class come up with a way to resolve this dilemma, such as rounding off to the next highest integer.

Key Questions

What did you do to come up with an equation for Question 1?

How did you get the numeric results in Questions 2a and 2b?

Supplemental Activities

Painting the General Cube (extension) is a challenging activity that asks students to create equations to describe a geometric situation.

Integers Only (extension) introduces the greatest integer function. Interested students might try to use this function to devise a formula for Question 2c of If I Could See This Thing.