Intent In this activity, students begin using variables to represent unknown values and equations to represent and answer questions. Through context-driven, informal methods, they begin to learn that symbolic representation and manipulation have meaning.

Mathematics This activity demonstrates some of the ways that meaning for symbolic algebra is developed in the IMP curriculum. Students start their work using variables and equations to understand numeric conditions. The focus is on the meaning of equations and the use of variables in context, and the approach to equations is intuitive. For instance, students are asked to explain what it means to represent one individual’s age by C and then represent another individual’s age by C + 20. The activity also provides an opportunity for continuing the development of students’ understanding of the distributive property.

Progression This activity is completed individually, followed by a teacher-led discussion in which students share approaches, ideas, and unresolved questions.

Approximate Time 10 minutes for introduction 20 minutes for activity (at home or in class) 20 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Doing the Activity To introduce the activity, you may want to have students work on Question 1 and then bring the class together to share approaches and ideas. Students should understand how to use variables and constants to represent the ages of people in different generations.

Discussing and Debriefing the Activity Bring the class together to share conclusions and to discuss unresolved problems. Here are some suggestions to focus students’ thinking. Take a solution that one person in your group came up with for Question 1a, and write an equation for it similar to the one shown in Question 1c. Was it possible to come up with a household with two grandparents in Question 2c? Why or why not? Calculate the average age of family members for each solution ou found for Question 1. What do you notice? Can anyone explain why that happens? What about Question 2? Ensure that the discussion makes use of the term equation. Also review the term algebraic expression and introduce the term solution to an equation in discussing the meaning of variables and equations. Ask, How can you use the equation to find the ages? As students share various techniques, discuss the meanings they have placed on the variable C or the expressions C + 20 and C + 40. This might be elicited with the question, What does the 20 in this expression tell us? Ask for replies to Question 1c: What does C mean in this equation? Record replies on the board to ensure that students understand the need to state this meaning of C as a number—in particular, the age of the child. Bring out the connection between the equation and the situation. Emphasize that in each instance, the same number must replace the variable. Use the phrase “solving an equation” to refer to the idea of finding a number to substitute into the equation that will yield a true statement—that is, make the two sides of the equation come out numerically equal. Ask for ideas about an equation to represent Question 3, which might have been challenging for some students. One solution might be as follows. T + T + T + ( T + 25) + ( T + 25) + ( T + 25) + ( T + 25) + ( T + 45) = 201 Select one of the equations volunteered—assuming the main difference between replies is the constant term—and determine the ages of the quadruplets. Then ask students to represent the sum of the ages of the quadruplets with an equation. (T + 25) + ( T + 25) + ( T + 25) + ( T + 25) = 128 To encourage an alternative expression, equivalent to the left side above by the distributive property, ask, Can you find another way to write the sum of the ages of the quadruplets? Verify that all given expressions are equivalent, reminding students that any value of T entered should result in the same value. For example, when T = 5: (T + 25) + (T + 25) + (T + 25) + (T + 25) = 120 4(T + 25) = 120 T + T + T + T + 25 + 25 + 25 + 25 = 120 4T + 100 = 120 Students should recognize that adding the four ages is equivalent to multiplying the age of one quadruplet by 4, and that the two processes are recorded as (T + 25) + (T + 25) + (T + 25) + (T + 25) and 4(T + 25) and mean the same thing.

Key Questions Can you write an equation for someone’s solution to Question 1a? Calculate the average age in the family for different solutions to Question 1. How can you use the equation to find the ages? What does C mean in the equation in Question 1c ?