# Cal s Dinner Card Deals

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1 Cal s Dinner Card Deals Overview: In this lesson students compare three linear functions in the context of Dinner Card Deals. Students are required to interpret a graph for each Dinner Card Deal to help them decide which is the better deal and why. Goals: Students will solve the problem using an equation, a graph, and a table. Students will interpret a linear equation in terms of the problem. Students will justify their solutions to the problem. Students will interpret a graph of the linear function in terms of the problem. Students will identify and interpret the x- and y-intercepts of the graph of the linear function in terms of the problem. Algebra Standards: 1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results: 1.1 Write and solve one-step linear equation in one variable. 1.3 Apply algebraic order of operations and the commutative, associate, and distributive properties to evaluate expressions; and justify each step in the process. 1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator. 2.0 Students analyze and use tables, graphs, and rules to solve problems involving rate and proportions. 2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. Mathematical Reasoning Standard: 1.0 Students make decisions about how to approach problems: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. 1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. LEARNING RESEARCH AND DEVELOPMENT CENTER 2005 University of Pittsburgh 1

2 1.3 Determine when and how to break a problem into simpler parts. 2.0 Students use strategies, skills and concepts in finding solutions: 2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.5 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solution with evidence in both verbal and symbolic work. 2.7 Make precise calculation and check the validity of the results form the context of the problem. 3.0 Students move beyond a particular problem by generalizing to other situations: 3.1 Evaluate the reasonableness of the solution in the context of the original situation. 3.3 Develop generalization of the results obtained and the strategies used and apply them in new problem situations. Building on Prior Knowledge: Students graph and interpret linear and some nonlinear functions. Students solve simple linear equations and inequalities over the rational numbers. Materials: Cal s Dinner Card Deals Task handout and transparency, calculators, chart paper and graph paper. LEARNING RESEARCH AND DEVELOPMENT CENTER 2005 University of Pittsburgh 2

3 CAL s DINNER CARD DEALS The graph below shows data for three dinner plans. Make observations about each of the graphs. What is the formula for determining the cost of each dinner plan? Decide which plan is the best and explain your reasoning. Cal's Dinner Card Deals Regular Price Plan A Plan B Number of Dinners Purchased (N) With permission from Smith, Silver and Stein (2005) Using cases to transform mathematics teaching and learning, Vol. 2: Algebra. New York: Teachers College Press. The COMET Project is funded by the National Science Foundation (ESI ). The project is co-directed by Margaret Smith, Edward Silver, and Mary Kay Stein and is housed at the Learning Research and Development Center at the University of Pittsburgh. LEARNING RESEARCH AND DEVELOPMENT CENTER 2005 University of Pittsburgh 3

6 make notes of important information in the task. Teacher may record for the student depending of the English-language level of the students. What misconceptions might students have? Students might connect lines on the graph interpreting the data as continuous rather than discrete. Ask: What does the line mean? How much of a meal is between two points? USING A TABLE - Students only make a table for one plan. Ask the student: - Which Dinner Card Deal is a better deal? How do you know this f from the table? How do you know this from the graph? - What patterns do you notice from your table? - (Point to the y-intercept on the graph) Where is this amount in your table? What does this amount mean? - How is the cost of the meals changing from one meal to the next? - How can you solve for 100 dinners in the regular plan? For Plan A? For Plan B? What misconceptions might students have? Misconceptions are common. Students may have learned the information incorrectly or they may generalize ideas prematurely. - Press students for the meaning of the numbers within the context of the problem. Consistently asking about the context helps students to make sense of the problem and appreciate the benefit when problem solving, especially if it helps them make sense of the problem. USING A TABLE A table can be created from the data points shown on the graph. By looking at the difference in cost between consecutive numbers of dinners purchased, a pattern can be found for each dinner plan. For the Regular Plan, there is a +10 pattern (the cost increases by 10 between consecutive rows). For Plan A, there is a +8 pattern (the cost increases by 8 between consecutive rows). For Plan B, there is a +6 pattern (the cost increases by 6 between consecutive rows). After identifying this pattern, it is possible, but not necessary, to fill in the costs for 4, 6, and 7 dinners purchased. Possible Solution Paths LEARNING RESEARCH AND DEVELOPMENT CENTER 2005 University of Pittsburgh 6

7 Phase Action Comments Which Dinner Card Deal is a better deal? How do you know this from the table? How do you know this from the graph? The Regular Plan is the best deal if the diner only uses the card to LEARNING RESEARCH AND DEVELOPMENT CENTER 2005 University of Pittsburgh 7

11 like? intercepts because this is where the number of meals (the x value) is equal - What is the slope of the Regular Plan, Plan A, and Plan B? to zero and where the graphs touch or cross the y-axis. Mathematical Reasoning: The graph could be used to count out the rate of change between data points for each dinner plan to determine the cost per meal. Students would then need to make sense of the y-intercept as the cost for 0 meals purchased or the initial fee of the dinner card. The coefficient - When you ask students why they multiplied 8(5) or 6(5) they should say that every meal that you buy you are charged \$8 or \$6 per meal so if you buy 20 meals this is 20 times the 8 or 6. You can refer to the 8 or 6 as the coefficient. y-intercept or Initial Fee Students should state that the y-intercepts are the costs of the plans for buying zero meals. They may call the amount the base fee. The constant - Students should explain that the base fee is added because it is a one-time fee, not a per-meal fee. Will the lines be connected on the graph? If the data are treated as continuous rather than discrete, the points on the graph can be connected to form lines for each dinner plan. Then the y-intercept (b) and slope (m) of each line can be identified. Using the slope-intercept form of a linear equation, y = mx + b, the formulas presented in the solutions A and B can be written for the cost (y) of any number of dinners (x) purchased on each plan. HOMEWORK Create two other plans, one that is a better deal than Plan B if you really like to each out a lot. Make a table and show your new plan on the graph. Explain why your plan is a better plan than the three plans from Edith Hart s classroom. ELL: Make sure you give any extensions or homework to your students in writing. Your students may be up to the challenge, but if the problem is given verbally they may miss the question entirely LEARNING RESEARCH AND DEVELOPMENT CENTER 2005 University of Pittsburgh 11

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