# Modeling Situations with Linear Equations

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1 Modeling Situations with Linear Equations MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students use algebra in context, and in particular, how well students: Explore relationships between variables in everyday situations. Find unknown values from known values. Find relationships between pairs of unknowns, and express these as tables and graphs. Find general relationships between several variables, and express these in different ways by rearranging formulae. COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: 8.F Construct a function to model a linear relationship between two quantities. This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics: 2. Reason abstractly and quantitatively. 4. Model with mathematics. INTRODUCTION This activity links several aspects of algebra. A situation is presented, and relationships between the variables are explored in depth. Letters are used to represent unknowns, generalized numbers, and variables. The problem-solving context gives you a chance to assess how well students are able to combine and apply different aspects of their algebra knowledge and skills. Before the lesson, students work individually on the assessment task The Guitar Class. You then review their work and create questions for students to answer at the end of the lesson, to help them to improve their solutions. During the lesson, students translate between words, algebraic formulae, tables, and graphs in an interactive whole-class discussion. The intention is that you focus students on making sense of the context using algebra, rather than just the routine use of techniques and skills. Students then work in pairs to graph the relationship between two of the variables. In a final whole-class discussion, students identify general formulae showing the relationships between all the variables. At the end of the lesson, students review and improve their individual work. MATERIALS REQUIRED Each student will need two copies of the assessment task The Guitar Class, either the lesson task sheet Making and Selling Candles, or the lesson task sheet Rescue Helicopter, depending on the outcome of the assessment task, a mini-whiteboard, a pen, and an eraser. Each pair of students will need a sheet of graph paper. There are some projector resources to support whole-class discussion. TIME NEEDED 15 minutes before the lesson and a one-hour lesson. All timings are approximate. Exact timings will depend on the needs of the class. Teacher guide Modeling Situations with Linear Equations T-1

3 Common issues: Student writes no equation, or just a few numbers and letters (Q1) Student writes an equation with a particular value of n (Q1) For example: The student substitutes n = 30. Student uses incorrect operation in equation (Q1) For example: The student divides the cost by the number of students, and adds rather than subtracts c to get p = 70n+ c. Student draws incorrect graph (Q2) For example: The graph has negative slope, or is not a straight line. Student does not explain or misinterprets the significance of the x-intercept (Q2) For example: The student does not link the answer back to the context. She just writes p = 5.5, and does not mention that this is the point at which the teacher begins to make a profit. Student uses incorrect operations in formulas in Q3, Q4 For example: The student writes p = fn + c or f = pn c. Student answers Q1-5 correctly Suggested questions and prompts: What do you know from the question? Suppose you have 10 students. How do you figure out how much profit the teacher would make? Does the calculation method change as you vary n? How can you write the calculation for any value of n? How much money does each student pay? How much money do the students pay altogether? Is the amount of money you have calculated before paying costs more or less than the profit? What do you think happens to the amount of profit as the number of students increases? What kind of function links n and p? What graph would you expect from this equation? What does n stand for? Does your answer make sense? Reread the first part of the sheet. What does p > 0 mean? What does profit mean? What does f stand for? Explain how you would calculate the profit in words. Suppose you can make 100 candles from a kit costing \$70. Use the profit you would expect to make to calculate the selling price. Graph this relationship. Teacher guide Modeling Situations with Linear Equations T-3

6 Question 4 Erase two numbers, n and p: The cost of buying the kit: (This includes the molds, wax and wicks.) \$ k n The number of candles that can be made with the kit: candles s The price at which he sells each candle: \$ per candle p Total profit made if all candles are sold: \$ What numbers could n and p be? Is this the only solution? Give me another solution and another and another This time there are two numbers that I don t know, n and p. What is different from not knowing one number? Write a table of possibilities and draw me a graph to show how p depends on n. Working in pairs (10 minutes) Hand out graph paper and allow time for students to work on the problem in pairs. As they do this, go round and prompt them to try different pairs of numbers, using the structure of the situation: If n were 20, what would this equation mean? How could you calculate the profit? How can you write this method in words or symbols? n p What equation fits this table? What would the graph look like? What does the graph show? During the paired work you have two tasks: to note student difficulties and to support them in their thinking about the graphing activity. Note student difficulties Students have not been given a table to complete, nor is there a set of axes or scales on the axes. This allows you to find out whether students are able to use their knowledge in context. Is students use of algebraic notation accurate? Conventional? Do they choose a sensible range for the number of students when drawing the graph, including values of n from 0 to 60? Do students calculate an appropriate scale for each axis to fit all sensible values of n and p? Teacher guide Modeling Situations with Linear Equations T-6

7 Do they label axes accurately and use equal increments? Do they know how to draw and complete an accurate table of values? Do they plot the points accurately? Do they make a point plot, or join the points to make a linear graph? Do students use features such as intercepts and linearity to check the accuracy of their plots? Support students thinking If students are stuck or making errors, try to support their thinking rather than solve the problems for them. If a pair of students is stuck on a problem, suggest that they seek help from another pair who have dealt with that problem successfully. You have chosen values of n from x to y. Why did you choose those values? How do you know that your plot is accurate? What does the x-intercept show? Encourage students to continue to link their mathematical representations with the context. What happens to the profit if the kit only makes five candles? At this point, n = 3.5. Does that make sense? If students complete this quickly, set one of the following questions: Is the graph best drawn as a point graph or as a linear graph? Explain your answer. Explain the relevance of the x- and y-intercepts. Erase two different variables [s and p, or n and s] and express the relationship between them if all other variables are kept constant. Whole-class discussion (10 minutes) Ask students to try to answer Question 5 in pairs. Although students have already done part of this question, they need to recognize this for themselves. Q5: Write a general formula for showing the relationships between the variables. What do you think you are being asked to do? Suppose you know s, k, and n and you want to calculate p. What method are you going to use? Does the method change if you change the value of n? What s the method for calculating p, whatever the values of s, k, and n? If students are really stuck, refer them back to their earlier work: Look back at Q2. What did you do to calculate s when you knew p, k, n? What would happen to the calculation method if n were any other value? Tell me in words how to calculate the total amount of money you collect. How would you write that using algebra? How do you calculate the selling price from that? After a few minutes, ask each pair to show you the four general formulas using their miniwhiteboards: p = ns - k k = ns - p n = p + k s s = p + k n Teacher guide Modeling Situations with Linear Equations T-7

9 3. For example: s = p + k or s =. n 60 As in Question 2, the method stays the same whatever values are substituted. 4. Look for a table of values with an appropriate range of values for n, including cases when p is negative. Check that the graph shows values in equal increments along the axes, is plotted accurately, shows the intercepts, and that the points fit a straight line. The x-intercept is (12.5,0): since you cannot sell half a candle (n is a discrete variable) this means that at least 13 candles must be sold in order to make a profit. You could discuss whether or not it is correct to draw a line joining the points. 5. p = ns - k k = ns - p s = p + k n n = p + k. s Rescue Helicopter The formulas are: t = d s + w d = s(t - w) s = d t - w Stress that t is the total duration, not the time taken to cover the distance. w = t - d s. Teacher guide Modeling Situations with Linear Equations T-9

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