Using the Spreadsheet to Promote Algebraic Thinking

4 downloads 0 Views 562KB Size Report
dents enter $1 in cell C2 and “=2*C2” in cell. C3 and scroll down until age 18. Then using the summation notation under each column to get the total, students ...
Using the Spreadsheet to Promote Algebraic Thinking by Meryle Hirotsu, Leeward Community College, Pearl City, HI [email protected]

H

ow does pattern recognition help algebraic reasoning? And how can the use of the spreadsheet help students to make sense of patterns? Pattern recognition is a teaching strategy that helps students develop algebraic reasoning. It is a way for students to understand how to solve algebraic problems and develop algebraic thinking. Through group collaboration and discussion, students can solve problems by noticing patterns and making generalizations or rules to solve the problems. This strategy promotes successful problem solving in Algebra. The spreadsheet is a technological tool that teachers and students can use to help facilitate the process of solving algebraic problems since it simplifies tedious computations and promotes deeper learning. Technological tools such as the spreadsheet enhance mathematical learning.



➤ Plan A: She will give $1,000 each year from age one to age 18.

➤ Plan B: She will give $1 at age one, $2 at age 2, and $4 at age 3, doubling it each year until age 18. Which plan would you choose and why? How much would each plan give at age 18?

By using the spreadsheet and entering $1,000 in row 2 under Plan A and copying down, students will generate amounts for each year. Under column C and Plan B, students enter $1 in cell C2 and “=2*C2” in cell C3 and scroll down until age 18. Then using the summation notation under each column to get the total, students would be able to find the totals for each plan. They will notice that a series of answers can be generated, saving time in making tedious calculations.

Instructional Approach

I teach remedial Algebra classes in a two-year college. The majority of the students express negative emotions when asked about their experiences with mathematics. They are in remedial mathematics classes because of their results on the admissions test. Since the students have the use of computers in a Math Lab, the use of spreadsheets to solve real world application problems offers them a different perspective on algebraic thinking. The following examples illustrate how students can find meaningful ways to use spreadsheets, rather than paper and pencil, to connect to algebraic reasoning through a technological tool. These examples are also appropriate for middle school grade levels through grade 9 Algebra in high school. Example 1: Which Plan Is Better? A rich aunt offers to pay for your college expenses. Page  20





CMC ComMuniCator

Figure 1 Volume 37, Number 2

Looking at the Totals for Plan A and Plan B, students can immediately see that Plan B will give much more money than Plan A. By looking at the pattern generated, they can formulate an algebraic equation for each plan. The formula for Plan A is y = 1000x, where x is the age in years and the independent variable. The formula for Plan B is y = 2x – 1, where x is the age in years and also the independent variable. By making a graph of both plans, students should notice that Plan B is not linear, but exponential. By scrolling down the entire chart and clicking on the Charts menu, students can create a graph that shows that Plan B generates much more money than Plan A in 18 years.

Figure 3

Examples 1 and 2 compare a linear function to an exponential function and illustrate how money can grow exponentially over time. They are good examples of how students can utilize spreadsheets to solve real-world problems.

Example 3: Which job pays better?

Figure 2

Example 2: Penny Doubling

A similar problem, the “Penny Doubling” problem states: You have the choice of receiving $1,000 per day for a month or a penny the first day, two pennies the second day, four the third day, and so on. Which is more? By using spreadsheets, students learn how to generate patterns, create a running total, and create a pattern using a formula (Ploger, Klinger, and Rooney 1997). By solving this problem with the use of a spreadsheet, students should see that Plan A would give $31,000 and Plan B would total $2,147,483,647 or 214,748,364,700 pennies. The screenshot of this problem (Figure 3) shows that Plan B is an exponential function y = 2n – 1, where n is the number of the year. If n = 1 or year one, y = 21 – 1 = 20, so y = 1; if n = 2 or year two, then y = 22 – 1 = 21 and y = 2. Thus plan B would generate a series “1, 2, 4, 8, . . ., 1,073,741,824 pennies. December 2012

A student is trying to get a part-time job and has two offers: Plan A: $50 to start and $15 per hour Plan B: $35 to start and $20 per hour How many hours does the student need to work to earn $200? Which offer is better for working for a prolonged period of time? Why? Both plans take 3 hours to earn $95. However, since after 3 hours Plan B earns the student more money, Plan B would be a better paying job for more than 3 hours.

CMC ComMuniCator

Figure 4 Continued on page 22 ➢ ➢ Page  21

Figure 5

A screenshot of the spreadsheet and graph is shown in Figure 5. This problem can be solved so that the initial and increment are used as references. If G7 is the cell of the initial value, enter the absolute formula in G7 as “=$I$4” and enter the formula “=G7+$K$4” in cell H7. G7 is the relative reference and $K$4 is the absolute reference. By doing this, a student could change the initial and increment values to change the graph and data for different formulas. To make a dynamic spreadsheet that can accept different pay situations, we can use absolute references in the example above. By entering “=$A$8*G7+$D$8” in cell G8 and

“=$A$9*G7+$D$9” in cell G9, the spreadsheet can now consider different plans of part-time job payrolls. So suppose the plans changed to include the following payment options: Plan A: $70 to start and $15 an hour Plan B: $25 to start and $20 an hour Which plan would earn more money over a period of time? How long will it take for both plans to pay the same amount of money? The screenshot (Figure 6) of the spreadsheet and graph shows that it will take 9 hours for the plans to pay $205 each. Plan B eventually pays more than Plan A after 9 hours.

Figure 6 Page  22

CMC ComMuniCator

Volume 37, Number 2

Using relative and absolute referencing is a challenge for most students. However, once the students begin to use referencing and understand its applications, other spreadsheet applications become clearer to them.

Reflection

The three problems are examples of how the spreadsheet can help students develop algebraic thinking and solve real world application problems. I would begin with a simple problem and build on it. For example, a problem like “Find the 100th term of the series: 1, 4, 7, 10, . . . would be a good start in showing students how to fill a cell with a formula and use “fill-down” to find the 100th term. Students used to do the problems presented in this article by using paper and pencil. Some of the table calculations were tedious, even when using the TI-84+ graphing calculator. However, when students use a spreadsheet, they see the problem in a different light. Students express awe when they find that the spreadsheet does tedious calculations in a matter of a second and displays clear graphical representations of data with ease. As a next step, students can learn about downloading real world data from the Internet and making charts, analyses, and generalizations of their findings. A helpful website is the End Users Shaping Effective Software (EUSES) website which can be found at eusesconsortium.org/edu/education.php.

Conclusions

The spreadsheet can shift the students’ focus from merely performing operations to reflecting on, analyzing, manipulating, and communicating about sequences of operations. Spreadsheets expose students to a method of symbolically describing numerical procedures in a more algebraic manner (Batista and van Auken Borrow 1998). Spreadsheets “allow students to model in a dynamic, reflective way, facilitating a variety of learning styles that can be characterized by the terms: open-ended, problem-orientated, investigative, active, and student centered” (Calder 2010). Niess (2005) discusses the importance of students using the spreadsheet to help them visualize patterns. Through the use of scafDecember 2012

folding techniques and careful planning, students will be able to learn how to use the spreadsheet as a tool for learning mathematics. Students need to be prepared for the technological demands of the 21st Century. As Drew Polly (2011) states: “Technology has been shown to positively influence student learning when students explore technologyrich tasks that simultaneously require them to use higher-order thinking skills (HOTS), such as analyzing or evaluating information or creating new representations of knowledge.” References

Battista, M.T., and C. van Auken Borrow. 1998 “Using Spreadsheets.” Teaching Children Mathematics 4(8): 470–478. Calder, N. (2010). “Affordances of Spreadsheets in Mathematical Investigation: Potentialities for Learning.” Spreadsheets in Education 3(3): Art. 4. Driscoll, M., and J. Moyer. 2001. “Using Students’ Work as a Lens on Algebraic Thinking.” Mathematics Teaching in the Middle School 6(5): 282–287. Herbert, K., and R. Brown. 1997. “Patterns as Tools for Algebraic Reasoning.” Algebraic Thinking, Grades K–12: Readings from NCTM’s School-Based Journals and Other Publications. Reston, VA: National Council of Teachers of Mathematics: pp.123–28. Niess, M. 2005. “Scaffolding Math Learning with Spreadsheets.” Learning & Teaching with Technology 32(5): 24–26. Ploger, D., L. Klingler, and M. Rooney. 1997. “Spreadsheets, Patterns, and Algebraic Thinking.” Teaching Children Mathematics 3: 330–334. Polly, D. 2011. “Developing Students’ Higher Order Thinking Skills (HOTS) through Technology-rich Tasks.” Educational Technology 51(4): 20–26.

CMC ComMuniCator

Page  23