Building Bridges

In this lesson, students transition from arithmetical to algebraic thinking by exploring problems that are not limited to single-solution responses. Values organized into tables and graphs are used to move toward symbolic representations. Problem situations involving linear, quadratic, and exponential models are employed. This lesson is based upon the article "Building Bridges to Algebraic Thinking" by Roger Day, which appeared in the February 1997 edition of Mathematics Teaching in the Middle School.

Introductory Comments

This lesson assists teachers in building a bridge between their current instructional goals and new goals that emphasize an earlier introduction to algebraic thinking. In the activities which follow, we illustrate an approach to algebraic thinking that is based on an extension of problem-solving tasks typically investigated by elementary and middle school students.

 

Problem Tasks

Pose the following question to students. They may work individually or in groups, depending on their needs.

The Hamburger Problem

Every Saturday you play basketball in the local community youth club. At the end of the season after a club tournament, the players in the club meet at a fast-food restaurant for a party. If hamburgers cost 59¢, what is the total cost for 15 players to have a hamburger each?

 

Note that the task is completed once the single solution has been determined. The situation does not necessarily motivate further exploration or mathematical thinking.

When a different question is posed, however, problems like the one above can be used as a bridge to further mathematical thinking.

 

Read the following problem to your students. Once again, students may work individually or in groups, according to their needs.
 

A variation of the Hamburger Problem

Every Saturday you play basketball in the local community youth club. At the end of the season after a club tournament, the players in the club meet at a fast-food restaurant for a party. If hamburgers cost 59¢, find a way to determine the total cost of hamburgers when various numbers of players in the club each have a hamburger.

 

As you discuss this new problem with your students, ask them the following questions:

• How is this question similar to the original Hamburger Problem?
• How is this question different from the original Hamburger Problem?
• How did you solve the new problem differently?

The posing of a new question does not guarantee a change in students' thinking. Students may adopt a thinking process that merely generates the hamburger costs for specific numbers of players, treating each number of players or hamburgers as a separate and unrelated situation. Under this scenario, despite the change in task, students' thinking may not have progressed beyond what was expected for the original task. However, by posing the revised Hamburger Problem, teachers give students an opportunity to do more than simply generate specific solutions.

Under this second scenario, students may make a table of values, construct some portion of a graph, or invoke the use of variables. More important, they may identify, describe, and extend a pattern. In other words, they are engaged in what we understand as algebraic thinking.

Students may use algebraic symbols to respond to this new problem. For example, a student might respond with C = $0.59 × p. In this equation, C represents the total cost, and p represents the number of players.

 

Using a Problem Task as a Bridge to Algebraic Thinking

To illustrate the use of a problem situation for middle school students that serves as a bridge to algebraic thinking, consider the problem described below.
 
 

The Valentine-Cards Problem

Each class in your middle school is making valentine cards to sell at affordable prices to elementary school students in your district. The cards are boxed in groups of 12 before they are routed to the elementary schools. Find a way to determine how many cards have been made when various numbers of boxes have been routed to the elementary schools.

 

As students will undoubtedly use different strategies to solve this problem, listed below are different probes you, the teacher, can use to help guide students during the problem solving process.

 

For students finding it difficult to begin the problem, the teacher might pose this series of questions:

• Can you find how many cards were made if 25 boxes were routed to the elementary schools? If 40 boxes were routed?
• What did you do in each case? Write an explanation to describe what you did.

 

For students who have already begun to list the number of cards made for specific numbers of boxes, the teacher might pose these questions:

• Can you organize your information into a table or graph?
• How would you describe your table or graph?

 

For students who have already begun to organize a table or construct a graph, the teacher might use some of the following probes:

• What if the district routed more boxes than those shown in your table or graph?
• Can you see a pattern that goes beyond your table or graph? Write a description of your pattern.

 

For students who have already begun to describe the pattern in words, the teacher might use some of the following probes:

• What is the key aspect of the pattern you have described?
• Can you express the pattern you see using b to stand for the number of boxes and C for the total number of cards made for the elementary schools?

 

The ongoing interactions suggested by these questions enable the teacher to capitalize on students' previous problem-solving experience and, by using a different focus, to extend their thinking beyond specific cases. Students are challenged not only to compute specific numerical values but generalize their thinking beyond specific values to values they have not handled, have not included in a table, or have not shown in a graph. In essence, the intention is to help students recognize and describe in their thinking process, in their table, or in their graph, numerical patterns that will lead to a more general description of the problem solution. The most viable demonstration of this outcome is a representation that expresses the solution process in a symbolic way using variables. When students are able to construct such a symbolic relationship, we believe that algebraic thinking has begun.

 

The approach outlined through this illustration needs to be used repeatedly with students if they are to recognize all the different representations and the connections among them, as well as realize that a representation involving variables is the more informative and concise description of a mathematical pattern. However, it is not enough for students merely to develop algebraic relationships; they need to see that these relationships empower them to solve a range of related problems and to move to a more advanced stage in algebraic thinking.

Extending the Approach to Further Algebraic Thinking

Once students have engaged in algebraic thinking and modeling through a series of problems like the valentine-cards problem, further explorations in algebraic thinking can be generated by using problems that are related to the original problem. The table below illustrates a set of related problems, stemming from the valentine-cards context, that enable students' algebraic thinking to be extended to new algebraic representations; substitution, linear equations and inequalities, and equivalence of algebraic expressions. As did the original problem, the related problems shown in the table should stimulate various solution strategies.

Related Problems, Algebraic Representations, and Potential Solution Strategies
(Stemming from the Valentine-Card Problem and the Algebraic Representation C = 12b)

RELATED PROBLEMS	ALGEBRAIC REPRESENTATIONS	SOLUTION STRATEGIES
(1)  How many cards were made if 600 boxes were routed to elementary schools?	C= 12(600)	Compute; return to the original problem   Read value; use a graphic representation or a table of values   Solve; use formal algebraic thinking
(2)  By the end of the first week in February, 1440 cards had been sent to the elementary schools. How many boxes had been sent?	1440 = 12b
(3)  If more than 9600 cards have been made by the district's middle schools, how many boxes will be routed to the elementary schools?	12b > 9600
(4)  In one shipment of b boxes of cards, water damage destroys two cards in each box. How many cards in the shipment are still usable? Solve this problem in two different ways.	C = 12b-2b, C = 10b,       or 12b - 2b = 10b

 

For example, students might solve problem 2 by returning to the original valentine-cards problem to compute how many 12s are needed to reach 1440 using trial and error or division. However, the availability of the algebraic representation C = 12b, where C represents total number of cards made and b represents the number of boxes of cards, and the fact that 1440 is a specific value for the variable C, provide the momentum for enabling students to generate the algebraic equation 1440 = 12b. Once this equation is represented, students can use a graph or a table or, eventually, more formal strategies to solve the problem.

Students may also use a graphing calculator, or similar technology tools, to generate a graph or table in solving problems like problem 2. With the availability of such technology as the graphing calculator, graphical solutions have taken on a new meaning as a key process in algebraic thinking at all levels. This new emphasis on graphical methods and technology notwithstanding, teachers may want to use this opportunity to guide students toward more formal solution strategies, such as the use of inverse operations, for solving such equations as 1440 =  12b.
 
 

(a)	(b)

           (a) Graph and (b) table for valentine-card problem, generated from the algebraic representation C = 12b

 

Where Do You Go from Here?

Thus far, this lesson has focused on an approach to algebraic thinking, involving linear models, that could easily begin as early as grade 5. This same approach can also be used in the later middle grades to develop algebraic thinking that incorporates nonlinear models arising from quadratic, exponential, or even rational models. The essence of the approach is to identify problems that embody such nonlinear models.

Two such problems, representing quadratic and exponential relationships, are presented in the Building Bridges activity sheet to illustrate the kind of starting points we envisage for exploring algebraic thinking in nonlinear contexts.

Building Bridges Activity Sheet

Once such modeling relationships as N = G(G-1)/2 and F =  2ⁿ have been developed, the relationships can be used to extend algebraic thinking to representations involving equations, inequalities, and equivalent expressions. As with linear relationships, tables and graphs generated with a graphing calculator or similar technology can be used to solve problems and investigate the properties of quadratic and exponential relationships.

Solutions to the Activity Sheet

1. This is a quadratic relationship, N = G(G-1)/2, where G represents the number of girls registered and N is the number of games to be played.)
2. The number of friends invited on the nth day is an exponential relationship, F = 2ⁿ, where n represents the day number and F is the number of friends. The total money raised after n days is also an exponential relationship, T =  (2^{n+ 1}- 2)(0.25), where n represents the day number and T is the total money raised.)