# Performance Assessment Task Parking Cars Grade 3. Common Core State Standards Math - Content Standards

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2 Parking Cars This problem gives you the chance to: interpret a bar graph This bar chart shows the number of cars parked in the multi-level parking garage at a shopping center today. Each floor holds up to 50 cars. 50 Number of cars Floor number 1. On which floor are the most cars parked? How many cars are there on this floor? 2. On which two floors are the same number of cars parked? Floors and 3. How many more cars are parked on Floor 1 than on Floor 8? Copyright 2007 by Mathematics Assessment Page 4 Parking Cars Test 3

3 4. How many cars, in all, are parked on the parking garage? Show how you figured this out. 5. Fifteen more cars come into the parking garage. Show these cars on the graph, parking them in the lowest floors. Explain why you parked these cars in this way. 7 Copyright 2007 by Mathematics Assessment 33 Parking Cars Test 3

4 Task 2: Parking Cars Rubric The core elements of performance required by this task are: interpret a bar chart Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answers: Floor 1 and Gives correct answer: 3and Gives correct answer: Gives correct answer: 150 Shows work such as: = Accept I counted them if student has given 150. Draws 15 more cars on the block graph. 10 cars on Floor number 2 and 5 cars on Floor number 3. Gives a correct reason for their positioning Partial credit Places 15 cars in garage. (1) 2 Total Points 7 Copyright 2007 by Mathematics Assessment 34 Parking Cars Test 3

6 Looking at Student Work on Parking Cars Student A shows a thorough understanding of reading and interpreting a graph by identifying the most, the same, and using comparison subtraction. Notice the clear use of labels to explain thinking with the computations. The student seems very comfortable with using scale. The explanation of parking the 15 cars is very complete. Student A Copyright 2007 by Noyce Foundation 36

7 Student A, part 2 Student B also has a good explanation for how the cars were parked. Student B Copyright 2007 by Noyce Foundation 37

8 Student C has a very thorough understanding of scale and of unit. However the scorer did not understand the strategy in part 4. Can you think why this strategy is not just a convenient number sentence but actually uses the idea of scale? What understanding underlies this strategy? Student C Issues of Scale and Issues of Mathematical Literacy Student D has a good understanding of scale, as evidenced by the work in part 4. The student even explains how 15 cars would be marked with a bar. The student struggles with a mathematically literacy issue, understanding the difference between the contextual meaning of adding 15 more cars to the total already represented in the graph and having a bar with a height representing 15 cars. (adding 15, an operation, or height 15, absolute value) Student D Copyright 2007 by Noyce Foundation 38

9 Student D, part 2 When learning a new idea, an understanding may drift in and out of consciousness. It doesn t hold for all occasions. Look at the work of Student E. The student seems to understand the scale in answering part 1 and in the readings to find the total in part 4. However when making the comparison between cars in floor 1 and 8, the student reverts back to counting squares rather than using a subtraction of the values represented. The student seems to think of floors 3 and 4 as one single bar instead to two bars with the same height. There is literacy issue. The student, instead of adding 15 more cars to the graph, simply recreates the original graph. Instead of explaining how to put the additional cars on the graph, the student explains again how she figured out the total number of cars. How is the level of detail needed for reading a math problem different from the level of detail for reading a story? Student E Copyright 2007 by Noyce Foundation 39

10 Student E, part 2 Copyright 2007 by Noyce Foundation 40

11 Student F again illustrates the idea about an inconstant understanding of scale. In questions one and two the student looks at the value of the columns height without connection to the context of floors. Is this a literacy issue of just reading numbers off a graph versus thinking about context? The students seems to understand the idea a square =10 cars except when doing comparison, again the idea of thinking for counting comparison versus comparison subtraction is well-illustrated by the child s drawing to the right of part 1. While the student knows that a whole square is 10, the student doesn t know what a half square represents. Do you think the student may have only worked with scales of one and 2? Student F Copyright 2007 by Noyce Foundation 41

12 Student G only understands the square as representing one unit. The student does have the sophistication to combine the two half squares to make a whole in part 4. Why might the student have put floor 2 and 8 together? What type of experience can help the student move to thinking about a scale larger than 1? Do you think making a graph from collected data, where there isn t enough room would help? Do you think a class discussion of several answers to part 4 would help the student rethink her position? What would be your next move? Student G Copyright 2007 by Noyce Foundation 42

13 Student H makes several errors around understanding scale. On the original for part 1, 50 is erased and replaced with 15. How many different issues of scale can you find? What are the different ways the student makes sense of the half-squares? How do you think the student got an answer of 42 for part 3? Student H Copyright 2007 by Noyce Foundation 43

14 Student I may be having trouble with mathematical literacy. In part 3 the student seems to be thinking about how many cars versus how many more. Each action describes a different operation. Can you think of two reasons why the student may have added in part 4? Student I Copyright 2007 by Noyce Foundation 44

15 Student J is interesting, because the student has all 5 points preceding part 5 of the task. Then the student reverts to drawing an array of 15 instead of adding 15 cars to the graph. Student J Student K does not distinguish between the numbers representing frequency or scale (number of cars parked) and numbers as a category (floor 1, floor 2). Notice that the student adds to the bars at 50 and 40 to get a total of 15 squares for those two rows. There is some logic, although incorrect, in most of the student s responses. Why might the student think floors 10 and 0 have the same number of cars? Where might the 3 have come from in part 3? Student K Copyright 2007 by Noyce Foundation 45

16 Student K, part 2 Copyright 2007 by Noyce Foundation 46

18 The maximum score available on this task is 7 points. The minimum score for a level 3 response is 3 points. Most students, 87%, could identify the floors with the most number of cars and the same number of cars. Many student, 73%, could also do comparison subtraction with numbers from the graph. More than half, 60%, could also add 15 cars to the graph, but not necessarily in the desired location. 12% could meet all the demands of the task including finding the total number of cars represented on the graph and putting 15 cars on the lowest floors with an appropriate explanation. 5% scored no points on this task. All the students in the sample with this score attempted the task. Copyright 2007 by Noyce Foundation 48

21 Ideas for Action Research Assessing Understanding of Scale Think about an interesting graphing task from your textbook or some other source that has a scale other than 1. How could you redesign the task to look at different layers of cognitive demand, such as using scale for computation and making a scale to fit the numbers? Then try out different versions of the task with different groups of students to see how the thinking and misunderstandings vary between the groups. Take for example this task from a third grade book: Copyright 2007 by Noyce Foundation 51

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