# High School Functions Building Functions Build a function that models a relationship between two quantities.

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2 Coffee This problem gives you the chance to: use a chart to solve simultaneous equations This chart shows the cost, in cents. of different numbers of small and large cups of coffee. 6 5 This means that 2 small cups and 4 large cups cost 680 Number of large cups Number of small cups 1. Explain what the number 500 in the chart means. 2. Use the information in the chart to find the cost of a small cup of coffee and the cost of a large cup of coffee. Show how you figured it out. Small cup of coffee costs cents Large cup of coffee costs cents Coffee Copyright 2009 by Mathematics Assessment Resource Services. 81

3 3. What number should go in the empty box that the arrow is pointing to. Explain your work. Coffee Copyright 2009 by Mathematics Assessment Resource Services. 82 7

4 Coffee Rubric The core elements of performance required by this task are: use a chart to solve simultaneous equations Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer such as: the cost of 3 small cups of coffee and 2 large cups of coffee is Gives correct answer: small 80, large 130 Shows work such as: solving simultaneous equations Partial credit Some correct work x 1 3. Gives correct answer: 340 and Gives correct explanation such as: The cost of one small cup and two large cups. (is half the cost of two small 2ft cups and four large cups.) 2 Total Points 7 2 (1) 4 Copyright 2009 by Mathematics Assessment Resource Services. 83

5 Coffee Work the task and look at the rubric. What are the big mathematical ideas being assessed in the task? Look at student work on part 1, explaining the 500. How many of your students put: Complete explanation, 2 large and 3 small cups cost 500 cents? Cost of the coffees?(ignored the units cents or didn t quantify the cups of coffee)? Misread the scale: o 2 large and 2 small? o 3 large and 2 1/2 small? o 2 large and 1 small? o Other? Now look at student work on part 2, finding the cost of small and large coffees. How many of your students: Could solve simultaneous equations to find the cost? Tried to set up an equality between the 2 equations? Used guess and check or substitution? Gave answers with no work? Did not attempt this part of the task? How often do students get opportunities to apply their algebraic skills in context? Why do you think guess and check is such a prevalent strategy? How can you design a lesson that would help students see the reason to leave behind comfortable arithmetic and make it worthwhile to learn new mathematics? Now look at work in part 3, finding the cost of the empty square. How many students: Could use the correct or incorrect solutions in part 2 were able to read the scale and calculate the cost of 2 large and 1 small coffee? Used proportional reasoning to find the cost? Tried to estimate using one of the values in the table? Only looked at the horizontal or vertical scale, but not both, to compute a value? Why do you think students struggled with part 3? What was confusing for them? What misconceptions did they have? 84

6 Looking at Student Work on Coffee Student A uses equations to find the cost of small and large coffee. The student is competent computing with negative numbers. In part 3 the student writes an equation and uses substitution to find the cost represented by the empty box. Student A 85

7 Student A, part 2 Student B understands the table, shown by the clear labeling next to the arrow. The student does not know how to set up equations in part 2. Student B uses proportional reasoning to find the cost represented by the empty box. 86

8 Student B Student B, part 2 Student C tries to use the 2 equations to set up a new equality. The problem with this strategy is that there are always 2 variables. The student at some point needs to use guess and check to get the values. The student can use the values found from guess and check to find the cost of the empty box. 87

9 Student C 88

10 Student D tries to use a proportion to solve for the cost of large and small coffees in part 2. Like Student C the student has formed an equation that still has two variables and therefore can t be solved. Student D Most students do not understand how to use algebra to find the cost of the coffees. Instead Student E makes an organized list and uses guess and check to help find the cost. The student is then able to use the costs to find the missing value in the table. The student relies on arithmetic skills to by-pass the algebra. 89

11 Student E 90

12 Student F is able to set up ratios to find the missing value in the table. The thinking for this is shown in the section 2. The student attempts to use algebra in part 2 to find the cost of the coffees. However the student doesn t see a second equation. Instead the student tries to add cups of coffee to make a second equation, x + y = 6. Student F 91

13 Student G makes the same assumption that the coffees can be added together to make the second equation. When Student G works out the cost of the coffee in the empty box, the size of the answer is way out of scale with other numbers in the table. Why doesn t the student use this as a trigger that something is wrong? Why do you think students are doing this? What is their misunderstanding? What prompt could you pose to these students to get them to rethink their strategy? Student G 92

14 Student G, part 2 Student H tries to make guesses on scale. The student uses 100 as a unit. What evidence would help the student see the problem with this scale? 93

15 Student H 94

16 Student H, part 2 Student I also tries to find a scale. The student looks at one scale at a time to find equal units to get to the 500. Unfortunately this method ignores the idea that both scales contribute to the 500. In part 3 the student tries to use the same logic to find the value of the empty box, but finally adds the two prices together. This ignores the quantity of large and small coffees. 95

17 Student I 96

18 Student I, part 2 97

19 Student J scores no points on the task. In part 1 the student confuses the scales. The student does not connect the numbers in the box with money in part 2. The student tries to estimate the value of the empty box in part 3 by comparing it with the 680, but ignores the 500. Why should the student know that 580 is too large? Student J 98

20 Task 5 Coffee Student Task Core Idea 3 ic Properties and Representations Use a chart to solve simultaneous equations. Represent and analyze mathematical situations and structures using algebraic symbols. Write equivalent forms of equations, inequalities and systems of equations and solve them. Understand the meaning of equivalent forms of expressions, equations, inequalities, or relations. Mathematics of this task: Read and interpret a diagram Write equations from a graphical representation Solve 2 equations with 2 unknowns to find a unit price Use proportional reasoning or equations to find the cost of an amount not on the table Based on teacher observations, this is what algebra students knew and were able to do: Read and interpret the table to describe the 500 using appropriate units Find the cost in part 2 Use guess and check to find the unit prices in part 2 Areas of difficulty for algebra students: Using the information in the table to write equations Solving simultaneous equations or recognizing the using equations could be a strategy to find the unit prices Trying to impose a unit of scale on the table without considering all the information 99

21 The maximum score available on this task is 7 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Many students, 79%, were able to describe the meaning of the 500 in the table using appropriate units. More than half the students could find the cost of two large and one small coffee using proportional reasoning or values from their work in part 2. Almost half, 41%, could describe the 500 in the table, find the unit prices in part 2, and showed some algebra to find the unit price. 22% of the students met all the demands of the task including solving simultaneous equations to find the unit price. 20% of the students scored no points on this task. 93% of the students with the students with this score attempted the task. 100

24 cover the 200 yen gap. Students figure out that the gap will need 7 of the cheaper cakes to make up the difference. A student in the class then shares a strategy using an inequality. The student first sets the tasty cakes = x. So the amount of the other cake would be (10 x). The inequality would then be 230x + 200(x-10) The class then discusses the solution. The purpose of letting the class solve the problem with their familiar solution strategy of guess and check first is to help them become grounded in the context and meaning of the solution. They are then ready to process other strategies and compare how the methods are similar. The teacher now makes the unusual move, from our cultural perspective, of discussing the good qualities of the student s solution. The teacher points out using guess and check you need to do a lot of calculation, but with this method, using an inequality, the answer comes out quickly. Finally, the teacher poses a follow up question with the prompt to try on the new strategy. This encourages students to make an effort and see a purpose in using the new mathematics. 103

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