PoW-TER Problem Packet A Phone-y Deal? (Author: Peggy McCloskey) 1. The Problem: A Phone-y Deal? [Problem #3280] With cell phones being so common these days, the phone companies are all competing to earn business by offering various calling plans. One of them, Horizon, offers 700 minutes of calls per month for $45.99, and additional minutes are charged at 6 cents per minute. Another company, Stingular, offers 700 minutes for $29.99 per month, and additional minutes are 35 cents each. 1. For each plan, write an expression that represents the total cost per month in terms of the number of additional minutes used. 2. For how many total minutes of calls per month is Horizon's plan a better deal? Extra: A third company, Dash, offers a plan that costs $49 for 500 minutes, and extra minutes are 2 cents each. For how many total minutes of calls per month is the Dash plan the best deal of the three? Note: This problem, A Phone-yDeal?, is one of many from the Math Forum @ Drexel's Problems of the Week Library. Are you interested in having access to more and also the many teacher resources that the Math Forum provides? View information about the different levels of membership including a free Trial Account from this page: http://mathforum.org/pow/productinfo.html 2. About the Problem A Phone-y Deal can be found in the Math Forum Problems of the Week Library. Many students today possess a cell phone and know about cell phone usage. The problem is slightly outdated since now all sorts of plans exist for cellular phones, individual and family plans along with the fees for texting, downloads, and web usage. In the middle school classroom we start with the basics and then we can assign others independent research on the cost today for using the phones and the services they enjoy. There are a variety of places where one can insert the challenge posed by this problem into the middle school and algebra 1 curriculum. In the 6 th grade curriculum students write stories for graphs and draw graphs for stories. Eliminate the two questions and have students draw three graphs to represent all three companies. These stories can lead to a discussion on the horizontal segment and why we call that a constant function. Although slope is not introduced in your curriculum, a discussion on which segment is steeper and A Phone-y Deal? 1
why, can lead to some critical thinking among students. It is a good exercise then to work with the labels on the x- and y-axis and put the first two companies on a single axis. Compare and contrast discussions can begin. The children could be introduced to the questions to write the expression for the pricing of each phone plan and to find the number of minutes for Horizon to be the better plan. I would not expect them to solve algebraically but the students are capable of evaluating the expressions and creating in/out tables. In Pre-Algebra or Algebra class the problem can be used in the curriculum when writing and evaluating expressions, equations and inequalities. This problem demonstrates the effect of different slopes. In an Algebra class one might insert this problem into the lessons on systems of equations or inequalities and look for an algebraic solution to this system of equations. The primary goal for introducing this problem can be aligned to the algebra standard for 6 th -8th grade students published by NCTM: Represent situations using algebraic symbols; Analyze situations using algebraic symbols. You would hope that the translation of this problem would lead to expressions for the cost of each company as asked for in question number 1. Question number 2 is asking to compare these expressions to find when one company offers the better deal. The extra includes a third company with different constraints. You would hope younger students could construct tables to solve the problem and that pre-algebra and algebra students would write equations or inequalities to solve a system. The NTCM algebra standard is expanded for 9 th -12 th grade to include fluent solving of systems of equations or inequalities. Asking students to submit a written explanation to the Math Forum allows them to work on the NCTM communication standard. Working in groups, and or class room discussion will also increase the student s ability to communicate with peers and teachers. NCTM standard of representation and problem solving can be aligned as this problem will lead to extensions in representing the given functions using graphs and or spreadsheet models. Or as in the case of 6 th graders the story and graph can lead to the representation in algebraic symbols. Recently, the Math Forum has encouraged children to solve this problem using technology by listing the problem in the tpows (technology problems). Published Geometer s Sketch pad activities has a similar problem listed in a unit on matching graphs with a story. The NCTM curriculum standard on problem solving encourages the use of problems that will lead to building math knowledge. Another extension building on this problem would be to discuss domain intervals and piece wise functions. A Phone-y Deal? 2
3. Common Misconceptions Assigning and labeling of variables and expressions. This student did a wonderful job in communicating the labels for the variables and correctly simplified the equation labeled cost per month for two companies. The equations did lead to the correct answer since the bill for Horizon was less than Singular when the total minutes was over 756 minutes. The student solved problem #2 by setting up an inequality. The solution is greater than 755.17. The process of rounding up means the student should use this symbol 756. The student knew this since the statement reads 756 minutes per month or more. Horizon is the better deal. To emphasize that the cost given is only when the usage is over 700 minutes wonder what is the cost for Horizon, Singular and Dash if one only used 100 minutes. Two of the equations would yield a negative value. Wonder what the cost would be if the students evaluate the equations using x as 500 minutes in the first two companies. What the student A Phone-y Deal? 3
needed to label y was the cost per month when the total minutes was over 700 minutes for Horizon and Singular. The misconception follows for the extra that is not written correctly. The student also needs to change.2 to.02. I might wonder how the student would now label the cost for Dash. Then I might wonder how one can compare cost and minutes. I wonder what the cost for Dash is for 500 minutes to make sure the equation is correct. Here we need to lead the student and most likely the entire class to discover an equation to compare. Wondering what the cost of Dash is at 700 minutes might help. These wonderings should lead the student to discover a new equation for the cost of Dash when the total usage is over 700 minutes. In the student presentations we will discover the solution to the extra. Interpreting the Solution to answer the question Again labeling m = the total minutes talked, hurt the student when checking the answer. The student was aware when checking the answer that the answer did not correspond. Wonder why she thought there was a typo. Wonder if there is another label for m. If m is labeled minutes over 700 wonder what the total minutes would be. The student did solve the equation correctly. This student found when the two bills would be equal. In solving this problems we will see that some used the logic of finding when the cost were equal an equation; while others looked for the solution when one bill was less than the other bill. Labels and making sure one answers the question are important skills for the middle school child to develop. You can point out the skill of eliminating decimals was used correctly, the expressions for more than 700 are correct and the solution is correct. The reporting of the answer does not match the question. The question was for the total minutes talked not the minutes over 700. A Phone-y Deal? 4
Equations or a Label Here we have a student who used an expression for a label. The expression for cost is 45.99 +.06 x. This is the cost when total minutes is 700 +x. These two expressions are not equal. When solving the student used substitution to make the cost are equal and was able to solve the equation correctly because the two expression in the equation were both cost. The solution does not answer the question since the reported response should be the expression 700 +x. Assuming the companies charge per minute only not part of a minute the answer should be rounded up. This student found the number of minutes over 700 to make the two cost equivalent. A Phone-y Deal? 5
4. Sample Student Solutions Trial and Error: Using minutes over 700 The student answered question 2 correctly and the expressions for Horizon and Singular are correct if labeled total cost for usage over 700 minutes. Labeling would make the solution easier for peers to follow. This student did not show how the solution was derived. There is a magnitude error in the solution. Since the check for 5.51 did not work it appears the student decided to use trial and error method. The solution is discovered in the table when the cost for 56 extra minutes makes Horizon a better deal than Singular. Encourage the student to revise and add labels. Encourage all students to verify their work. The student reports the answer using total minutes which is correct. A Phone-y Deal? 6
Trial and Error: Equations for total minutes. A Phone-y Deal? 7
The solution above uses the total minutes for the variable t and reports the cost at the end of month using the expression (t-700). The labeling is correct and the evaluating at 800, 750, 753, and 755 show that the student was looking for a time when Horizon was less than Singular. Remind students that this equation for C is correct only when the usage is over 700. We need to begin to suggest that for the first 700 minutes there is another equation. Be sure to point out the reasoning why the student did not need to verify that 756 was the solution to the problem. Wonder if everyone follows that line of reasoning. A Phone-y Deal? 8
XY Chart Here the student uses an XY chart to demonstrate the formula. The student mentions graphing. I would wonder if they graphed and if they used the table function for the figures published above. Writing mathematics and formatting work is hard for the students but the software and tools are improving each year. I might use this solution to review for students how one can insert charts so that there is no fear of someone wondering what, 8 = 46.17 means. Also if students are going to use the technology available to them we can show them how to insert graphs and images to enhance their communication of mathematics. We know the student knows the x is minutes over 700 from his last paragraph. Wonder how his graph would show the cost for a 100 minutes or any value less than 700. This will set the stage for new learning on piecewise functions. A Phone-y Deal? 9
Graphing These two students told us they solved the problem by graphing. Finding the intersection point is the correct process. But even using technology the process of communication needs to improve. The same misconceptions occur. Labeling the variable and the equations are important pieces of communicating thinking. The class now knows that the equations given are for minutes over 700 and over 500 for the third company. We can wonder on the graph the same as we did for the trial and error solutions. What is the cost for 100 minutes of usage? Is there away to combine the two pieces of the billing process in a graph? The first student reports the correct solution whereas the second student report the overage minutes. Finding the intersection is a way to solve systems of equations so it is important to present this method. This is the correct theory but now in our presentation we need to start the questioning about the third company Dash. The reported price increases after 500 minutes. Wonder what the cost of all three companies is at 557 and 575 minutes, at 757 and 775 minutes. If x = the extra minutes would you use the same x for the given minutes for all three companies? We will address this issue after we present the algebraic solutions. Algebraic solutions: x= total minutes, solve an inequality A Phone-y Deal? 10
Algebra x= extra minutes, solve an equality. Here we have two concise simple algebraic solutions. In the first one the student gives the expressions representing the cost in terms of total minutes. We now know it is the cost when usage is over 700 minutes. The student sets up an inequality. The verb to use would be 756 minutes or more is when Horizon is better than Singular. In the second example the student lets x equal the extra minutes and solves an equation. Note that the student uses cents to eliminate the decimals in the original expressions. The student solves an equation looking for the number of minutes where the cost would be equal then reasons that Horizon is cheaper for usage of 756 or more. Both students used the substitution method correctly. Algebraic differences This student uses a different approach. The expressions are given for the first two companies. The student tells us the difference in cost and the difference in the extra cost per minute. I would wonder if he could revise this work and include the sentences showing the differences and quotients mentioned. The quotient of 15.98 and.29 gives the number of minutes for the two plans to be equal. Rounding up he found that at 756 minutes Horizon became a better deal. Although we are not shown the cost of the third A Phone-y Deal? 11
company nor the work this student solved the extra and tell us even more than asked for. Dash was better than Singular when the total is 770 minutes and Dash is a better plan than Horizon when total minutes used in 876. In order to see if the class understands this line of reasoning, I might wonder aloud for the class what was the total cost for Dash, what was the result of Dash Horizon, Dash Singular, Extra cost of Dash extra cost of Horizon, Extra of Dash Extra of Singular. Does a negative value mean anything or are we looking for the absolute value of the difference? The author and the class might need support for finding the expressions that gave 770 minutes and 876 minutes above. Ask students to verify that these values are correct. Wonder what happened with the student who graphed the three expression and found 57 and 75 as the intersection using x as the extra minutes. The first wonderings concerning the extra came up in the graphing example. We raised a few questions. We now know there are two expressions one can use in solving these problems. Let x = extra minutes and the cost of Dash over 500 minutes is.02x +49. Let x = total time and the expression is.02(x-500) + 49. The expressions change depending on the variable label. The extra was solved above using differences but the student did not communicate his ideas clearly enough for the peers. The last presentation creates another equation for dash so that it can be compared with the other two companies using x as the same total. If x = extra minuets 50 extra minutes would be a total of 750 for Horizon and for Singular but for Dash x= 50 is a total of 550; that is why we see conflicting solutions in the extras presented above and the graphing. Had the student written the expressions using x as total minutes the answers would be the same. Here is an approach to the Extra: A Phone-y Deal? 12
The extra poses a problem since the price changes after 500 minutes. So our student finds the cost of Dash when the total time was 700. Using the expression now of 53+.02m and setting up the equations the solutions tells us when Dash is equal to Horizon and when Dash is equal to Singular. I would wonder with this student if statements could be made when each company is the better deal. I would wonder if the students could revise the compound inequality. The student is correct that Dash is the better deal for 876 minutes or more. Presentation order I would present the student solutions as presented above. Trial and error is a method all should be able to follow and use. I want to stress the two different expressions depending on the label given to the variable. I want to stress the two strategies. Setting up an equation and reasoning from the equality point to report the answer as greater than or equal to the solution is one strategy. The other is setting up the inequality and reporting the answer as such. Trial and Error leads naturally to a XY chart or in out table understandable by younger students. The XY table then leads to graphs. Although the graphing students both used x as the extra minutes and neither produced the graphs, I want to present graphing as a valid method and use this as jumping off point for extending this problem to graphing, piece wise functions and spreadsheet representations. It is important that students see various models of the same data. Algebra students know how to solve a system and substitution is a method of choice. The examples given use the different expressions and one solves equations while the other thinks in terms of inequalities. Finding the differences was a different and accurate way to reason. The extra is a challenge for all. 5. Extensions According to NCTM standard on problem solving good problems lead students to build new mathematical knowledge. The NCTM standard on representation encourages students to select, apply and translate among mathematical representations to solve problems. This problem needs to be modeled by graphing and spreadsheet. This problem can build new knowledge on graphing piece wise functions and discussion on domain intervals. Using the constant price and then using the expressions written by the students for the cost for 500 and 700 minutes or more respectively they can produce this model using a A Phone-y Deal? 13
spreadsheet. From the spreadsheet data students can answers the questions. The yellow shows the better deal total min total min Horizon Singular Dash Horizon Singular Dash 750 48.99 47.49 54.00 870 56.19 89.49 56.40 751 49.05 47.84 54.02 871 56.25 89.84 56.42 752 49.11 48.19 54.04 872 56.31 90.19 56.44 753 49.17 48.54 54.06 873 56.37 90.54 56.46 754 49.23 48.89 54.08 874 56.43 90.89 56.48 755 49.29 49.24 54.10 875 56.49 91.24 56.50 756 49.35 49.59 54.12 876 56.55 91.59 56.52 757 49.41 49.94 54.14 877 56.61 91.94 56.54 758 49.47 50.29 54.16 878 56.67 92.29 56.56 759 49.53 50.64 54.18 879 56.73 92.64 56.58 760 49.59 50.99 54.20 880 56.79 92.99 56.60 The spreadsheet will produce the following model. Even younger students can visualize the constant function and then the change in steepness. Discussion for the younger child can be on what the horizontal segment represents and what does the differences in the steepness signify? Wondering for the algebra students could include wondering about the equation of the constant functions. A Phone-y Deal? 14
Wonder what is the domain for the constant function. What is the domain for the functions that show a constant rate of change. Wonder if they can make any changes to the functions they used for total monthly bill. I would also wonder if this model could be produced by the handheld. Discussion on piece wise functions could be discussed and modeled. Fig. 1 Fig.2 Fig.3 Figures 1 can model the intersection of Horizon and Singular. Figure 2 shows all functions and figure 3 zooms in on the intersection found by graphing. Once students discover reporting total cost with respect to the domain we can graph piece wise functions by adding the interval to the expression. A Geometer s Sketchpad model could be created with the younger students asking them to find the significance of point A, B, and C. A Phone-y Deal? 15
Hopefully one can see the many uses of having students grapple with this Math Forum problem. The primary goals of writing expressions and finding a solution can be achieved by trial and error, graphing, or the substitution method of solving a system. If you re lucky someone might use another line of reasoning as the student who reasoned using the differences in price and the fees for the extra usage. Giving students challenging problems to wonder about, and formulate a written response will increase their mathematical skills, analytic thinking and reasoning ability. Having them produced a written response can enhance the discussion for all on the different approaches to label and solve. The problem leaves open the door to introducing other topics to challenge or enrich and other models to represent the data. Since cell phones are dear to the heart of the pre-teens and teenagers you might even expand the problem to compare the plans offered today. This problem will engage all and all can be encouraged to find a solution through the wonderings of the teachers and peers. A Phone-y Deal? 16