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Developing Algebraic Thinking By Using A Spreadsheet Approach To Algebra Word Problems or Using Guess and Check As A Viable Problem Solving Strategy
Why Use a Spreadsheet? When students use a spreadsheet they become immersed in the language of the spreadsheet. A spreadsheet may act as a bridge which students can use to cross over into the world of mathematics. They can use the more intuitive problem solving method of making a table, since the computer can generate tables so easily. The following problems contained in this article have always been solved by most of my students by using guess and check, despite my best efforts to teach other methods. A computer spreadsheet proves to be a powerful tool for students because it allows us to follow our intuition by using a guess and check strategy. Problems that can be attacked by finding patterns, guessing and checking, and making tables naturally fit within the realm of computer spreadsheets. In fact any problem which requires finding a numerical value where two quantities are equal lends itself well to the "don’t spend as much time guessing and just check all possible values" strategy of a computer spreadsheet. The great advantage of using a computer is that we can check every relevant possibility for an answer. Setting up the spreadsheet to solve the problems forces students to break down the problem into "smaller pieces." Modeling a Problem Using the Spreadsheet The first problem is a typical Algebra word problem. There were 166 paid admissions to a game. The price was $2.10 for adults and $.75 each for children. The amount taken in was $293.25. How many adults and children attended? Figure 1 shows one possible method for setting up this problem using a spreadsheet. Figure 2 shows the spreadsheet and the underlying formulas. Most students will understand how to create a table. The number of adults that could have attended ranges from 0 to 166. So far our explanation for setting up the problem is not any different from a traditional algebraic approach. However when we set up the spreadsheet to solve this problem, there are subtle yet important differences. When setting up the column for the number of adults, we are going to list every possible number of adults that could have attended. In essence we are listing every possible domain value for the independent variable. We can see that the number of adults can be an integer from 1 to 166, but have not yet incorporated the level of abstraction that would result from introducing a variable such as "x" into the problem. To set up the columns for the number of children, we reason that since there were 166 total people, the number of children that could have attended is therefore 166-the number of adults. For example, if 3 adults attended then 166-3 or 163 children must have attended. If your students have some skill with spreadsheets, they will already know the correct formula to enter into the column for the number of children. If the students are lacking in spreadsheet experience, this is an excellent example with which to develop the process of using a formula. Students can easily understand that when we use notation such as "=166-A2", that we are telling the spreadsheet to subtract the quantity contained in, stored in, or represented by, cell A2. The concept of variable is developed and still the student can see all specific instances, together with the generalization on the same page. We next set up the column for the money that would be taken in from adults paying $2.10 for a ticket. The spreadsheet can be used to develop the formulas, such as "=a2*2.10". Similarly, the money from the number of children attending will be a formula such as "=b2*.75". The formula for setting up the last column is obviously the money from the adult tickets plus the money from the student tickets—"=d2+e2". A solution to the original question can now be found. We look in the column headed "Total Money," for a total of $293.25 and see how many adults and children attended. This is found for 125 adults and 41 children.
Another Example—The Coin Problem Another typical problem that arises in the same section of an Algebra text is the coin problem. On a table are 20 coins consisting of quarters and dimes. Their combined value is $3.05. How many of each kind of coin are there? This problem follows the same format and is solved using a spreadsheet, in the same fashion as the ticket problem (See Fig. 3). It would be a reasonable assumption that more students would get the coin problem correct than the ticket problem using only pencil and paper. The reason for this being that students are more likely to be successful guessing and checking, and have a better understanding of money than tickets. Using a spreadsheet to solve these problems, involves a variation of the guess and check strategy, we simply guess and check all possibilities, and in the process, begin to develop the language and abstract reasoning skills necessary for understanding algebra.
The Temperature Problem Find a temperature where degrees Celsius and degrees Fahrenheit are equal. Seventh grade students were given the formulas to convert from degree Celsius to Fahrenheit and vice-versa. They were instructed to set up a spreadsheet as in Figure 4. Students were then instructed to use the spreadsheet and the formula for degrees Celsius—C=5/9(F-32), to find were degrees Celsius was equal to degrees Fahrenheit. (It should be noted that these students already had experience with spreadsheets and using spreadsheet formulas.) When I decided on the domain of the problem and how to label the columns I was essentially solving the problem. When students get enough experience they will be able to choose independent variables, label for their columns, dependent variables, and put in the correct formula. Even though I typed in the formula for degrees Celsius, it is not immediately apparent to all students what to do next. Most of my sixth and seventh grade students began by typing in a formula in cell b2 which looked like: =5/9*(-260-32), (See Figure 5). Some students continued to type in this type of formula for the rest of the class period, and I allowed them to do so. They were checking each of the Celsius formulas, but using the spreadsheet only as a calculator, and doing a lot of typing! Notice that solving the problem in this way offers no advantages over using a calculator or pencil and paper. But it does demonstrate the beginning of some level of abstraction because students demonstrate the ability to apply the formula and replace the variable "F" in the formula with a number. However, it does not take long for most of the students to tire of this type of problem and look for an easier method. After typing in about ten specific instance of the formula, many of the students began to look for ways to use the power of the spreadsheet. I could see the lights turning on in their heads. They were already familiar with the use of formulas and "Filling Down" instead of typing in each cell. They were looking for a way to generalize and use a variable. Through observation of the pattern they had created as in Figure 5, they were able to discern that the only number that was changing (varying) in column B was the Fahrenheit temperature. They next recognized that each of these varying numbers were contained in the column A. They quickly replaced their specific numbers with a more general formula and filled it down (See Figure 6). The solution of -40° was then quickly found (See Figure 7). The Pythagorean Triple Problem Find all integer Pythagorean triples from 1 to 24 inclusive. We will use the spreadsheet to check all possibilities. There are several ways to set up you spreadsheet. Deciding how to label your rows and/or columns, as in many spreadsheet problems, is the key to finding a solution. Figure 8 shows a completed spreadsheet. Figure 9 shows the underlying formulas. The dollar signs in the spreadsheet formulas indicate absolute cell references—they do not vary. The Spreadsheet Bridges the Gap Between the Specific and the General Using the first problem as an example, the difference from a traditional approach is that all 167 cases, 0 adults through 166 adults, have their own unique equation. We are using equations, we are using variables, we are using formulas, but we have not yet jumped immediately to using a single equation to represent the whole problem. Students without training in algebra can setup this problem and solve it. On the spreadsheet page, the students will be able to see every possible situation, so the level of abstraction has initially been reduced from one single equation as in a traditional approach. Using this approach lays the foundation for algebraic reasoning. Students that could set up the problem using a traditional algebra approach, but maybe not solve the equation correctly, now can get a correct solution. Now we can remove the training wheels and lead students to a more general model of the problem. Although each case has its own unique spreadsheet formula, it is a short leap to setting up the equation representing the whole problem, by using the last three columns—a18*2.10+b18*.75=293.25. The substitution property of equality can be employed to replace b18, and is intuitively understood by students, because we can look in cell b18 and see that it contains 166-a18. Our equation would then become "a18*2.10+(166-a18)*.75=293.25. Which is precisely where we may have started the problem without a spreadsheet, only using different notation—"2.10x+.75(166-x)=293.25". Using the spreadsheet can help teachers assist students in reaching the final level of abstraction of this problem. There is evidence to suggest that students, without algebra training, will retain the ability to solve these types of problems even with pencil and paper, using the language of the spreadsheet (Sutherland and Rojano, 1993).
Bibliography and References Esty, W. W. & Teppo, A. R. (1996). Algebraic thinking, language and word problems. In P. Elliot (Eds.), Communication in mathematics, K-12 and beyond (pp. 45-53). Reston, VA: National Council of Teachers of Mathematics. Hoyles, C. (1992). Mathematics teaching and mathematics teachers: A meta-case study. For the Learning of Mathematics, 12 (3), p. 32-44. White Rock, BC, Canada: FLM Publishing Association. Hoyles, C. & Noss, R. (1992). A pedagogy for mathematical microworlds. Educational Studies in Mathematics, 23, 31-57. Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277-289. Sutherland, R. & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. Journal of Mathematical Behavior, 12, 353-383. |