Overall Frequency Distribution by Total Score

Overall Frequency Distribution by Total Score Grade 6 Mean=17.08; S.D.=9.24 400 300 Frequency 200 100 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Frequency Sixth Grade 2003 pg. 1

Level Frequency Distribution Chart and Frequency Distribution 2003 - Numbers of students Grade 6: tested: 9865 Grade 6 2000-2001 Level % at ('00) % at least ('00) % at ('01) % at least ('01) 1 26% 100% 16% 100% 2 47% 74% 43% 84% 3 17% 27% 28% 41% 4 10% 10% 13% 13% Grade 6 2002-2003 Level % at ('02) % at least ('02) % at ('03) % at least ('03) 1 21% 100% 21% 100% 2 25% 79% 28% 79% 3 36% 55% 33% 50% 4 19% 19% 18% 18% 3500 3000 2500 Frequency 2000 1500 1000 500 0 0-8 1 Minimal Success 9-16 2 Below Standard 17-26 3 At Standard 27-40 4 High Standard Frequency 2095 2796 3207 1767 Sixth Grade 2003 pg. 2

6 th grade Task 1 Baseball Players Student Task Core Idea 5 Statistics Determine measures of center and spread for two baseball teams. Select and use appropriate statistical methods to display, analyze, compare and interpret different data sets. Use measures of center and spread (mean, median, and range) and understand what each does and does not indicate about the data set. (5 th grade) Sixth Grade 2003 pg. 3

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Looking at Student Work Baseball Players Student A shows a clear understanding of the relationships of individual weights to mean and can express them clearly in mathematical expressions. The student shows no hesitation about the part/whole relationships being discussed in the problem. Student A Sixth Grade 2003 pg. 6

While Student B arrives at the same answers, there is evidence of considerable confusion in thought processes. In part 1 Student B first tries use the algorithm to divide to find the average. Perhaps a familiarity with weight and estimation helps the student to self correct and find the total weight. In part 2 the student tries several things before reaching an answer. The student tries to divide 188 by 3 and gets an answer that is too small. The student then multiplies by the 12 players to find total weight and then divides by 4 to find the weight of 3 players. It is unclear how the student finally reached the correct answer. Student B Sixth Grade 2003 pg. 7

Student C recognizes the need of doing and undoing to find the total number of players, the connection between multiplication and division. However in part 2 Student C does not seem to think about how adding more people affects the average. The student relies on procedures and not number sense to find one-third of a ball player. Student C Sixth Grade 2003 pg. 8

More than 20% of all students found the total average weight of the three new players. They did not think that to raise the average of the other seven, these new players had to weigh more than the 188 pounds. They also lacked the number sense to think about 563 pounds as too large a weight for one player. See the work of Student D. Student D Sixth Grade 2003 pg. 9

Almost 10% of the students thought the total weight of 9 players would be 19.6 pounds, even though this does not make sense as a weight for one player. Had the arithmetic been correct Student E would have arrived back at the 177 pounds she started with. In part 2 the student finds the difference in weight between original and new players, again an answer found by almost 10% of the students. Student E then seems convinced that mean needs to be division and uses that operation to attempt solving the problem. There is not evidence of understanding of meaning of average as an attempt to even out a group of numbers. Yet the student knows and can apply procedures in straightforward ways to find median and range. Student E Sixth Grade 2003 pg. 10

Frequency Distribution for each Task Grade 6 Grade 6 Baseball Players Baseball Players Mean: 2.41, S.D.: 1.83 3000 2500 2000 Frequency 1500 1000 500 0 0 1 2 3 4 5 6 7 Frequency 1924 1206 2472 1465 1744 448 221 385 Score Score: 0 1 2 3 4 5 6 7 % < = 19.5% 31.7% 56.8% 71.6% 89.3% 93.9% 96.1% 100.0% % > = 100.0% 80.5% 68.3% 43.2% 28.4% 10.7% 6.1% 3.9% The maximum score available for this task is 7 points. The cut score for a level 3 response is 3 points. Most students (about 80%) could find the range or median of a set of numbers. A little less than half (about 40%) could find the total weight of players given the average weight and find the range or median. 6% of the students could meet all the demands of the task. Almost 20% of the students scored no points on this task. 90% of those students attempted the problem. Sixth Grade 2003 pg. 11

Baseball Players Points Understandings Misunderstandings 0 90% of the students with this score attempted the problem. 2 Students could see the connection between average weight and total weight for all the players. 3 Students could find total weight and median or range. 4 Students could find total weight, median, and range. 7 Students could think about average weight and understand how changes in average must be compensated for by the additional data. They could correctly calculate median and range. 10% of all students divided to find an average weight of 19.6 in part 1. Answers in part 1 ranged from 1.59 to 1791, showing that students are not making sense of the situation and trying random operations on the given numbers. Many students took the average weight of 177 and added the 9 players to get a total of 186. They don t think about unit analysis and that adding pounds and players gives??? Students did not know how to find median and range. A common error for median was to forget the two numbers in the first line of the problem or forget to rearrange the data by size. Range was slightly more difficult for students. Fewer students attempted to answer this part of the task. Some students put the highest number. Answers ranged from 3 to 1999. Students could not think about how the new players raised the average and therefore needed to each weigh more than the new average. More than 20% of all students simply multiplied 188 by 3. 10% of all students divided 188 by 3. About 5% got a weight of 11 for the new player and another 5% multiplied 12 x 188 to get 2256 pounds as the weight of the new player. Sixth Grade 2003 pg. 12

Based on teacher observations, this is what sixth grade students seemed to know and be able to do: Find the total weight of players if they knew the average weight Find the median Areas of difficulty for sixth graders, sixth grade students struggled with: Finding the range Understanding the relationship between individual weights and average weights Thinking about how additional data affects the average, how new weights must be higher than the average to raise the total average Questions for Reflection on Baseball Players: What types of experiences or questions do you ask students to get them to reflect on the meaning or purpose of average? Do you think students in your class could draw a picture or model to show what average means? Do they see it as making everything even or the same? What opportunities do students have to think about the relationship between individual pieces of data to the whole or how changes in the average will be reflected in the additional data? What other kinds of questions might promote or check for sense making with your students? Look at your student work to part 2. How many of them put: 221 lbs. 564 lbs. 188 x3 62.6 lbs. 188/3 11 lbs. 188-177 2256 lbs. 188 x 12 No response other Could your students find the mean and range? What further questions would you want to ask them to probe their understanding of these concepts to see if they know the purpose for these measures? Teacher Notes: Instructional Implications: Students need more understanding of the meaning and application of mean (average) in a real situation. While many students can calculate the mean, they don t understand the relationship of that answer to the problem. They can t work backwards from the average to the total. This makes it difficult for them to reason about how changes in the situation will affect the average. Students need more Sixth Grade 2003 pg. 13

experiences understanding what the mean reflects about the data and how changes in data will affect the average. The purpose of statistics is to give a picture about the data. Students need to be able to use other measures of center, like median and range, to help them make sense of the situation. 6 th grade Task 2 Gym Student Task Core Idea 3 Algebra and Functions Core Idea 1 Number and Operation Analyze gym membership costs to solve a practical money problem. Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Model and solve contextualized problems using various equations Understand number systems, the meanings of operations, and ways of representing numbers, relationships, and number systems Understand and use proportional reasoning to represent quantitative relationships Sixth Grade 2003 pg. 14

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Looking at Student Work Gym Students seem to have a difficult time analyzing the effects of operations on units (dimensional analysis) and making comparisons. Student A does a good job of tracking the units as he works through different parts of the task. The work is organized in a manner that makes the logic very is easy to follow. Part 3 starts with the cost of visits per month to visits per year, then monthly fee to yearly fee, and finally total cost. Student A Sixth Grade 2003 pg. 18

Student A Sixth Grade 2003 pg. 19

Student B makes the most common error in part 3. The student finds the yearly cost of gym visits and yearly total for monthly fees, but forgets to add in the cost of joining the gym. Student B Sixth Grade 2003 pg. 20

Student C finds the cost per month of visits and fees and compares them to the cost per month of the Regular deal gym. This would have worked if the student had additionally found the monthly cost of the initiation fee ($25) and added that to the Superfit total. The student would still have needed to multiply the difference by 12 to find the yearly difference. Student C Sixth Grade 2003 pg. 21

Student D makes two common errors. In part 2 of the task the student confuses the increase in number of visits with the total number of visits. In part 2 the student finds the cost of 20 visits per month, but forgets to multiply that number by 12 to find the cost for visits per year. Student D Sixth Grade 2003 pg. 22

Student E forgets the monthly cost for regular deal and finds the number of visits that will equal all-in-one price. Students have difficulty keeping track of the different constraints or features involved in the different pricing schemes. Student E Teacher Notes: Sixth Grade 2003 pg. 23

Grade 6 Gym Gym Mean: 3.30, S.D.: 2.63 2500 2000 Frequency 1500 1000 500 0 0 1 2 3 4 5 6 7 8 Frequency 2299 961 655 1567 618 1793 659 308 1005 Score Score: 0 1 2 3 4 5 6 7 8 % < = 23.3% 33.0% 39.7% 55.6% 61.8% 80.0% 86.7% 89.8% 100.0% % > = 100.0% 76.7% 67.0% 60.3% 44.4% 38.2% 20.0% 13.3% 10.2% The maximum score available for this task is 8 points. The cut score for a level 3 response is 3 points. Most students (about 80%) could find the cost of the Pay as you go plan and the Allin-one plan. A little more than half of the students (about 67%) could correctly calculate the costs of all three options and compare to find the better deal. Almost 44% could compare the 3 options and find the number of visits that would make Allin-one and regular deal the same. About 10% of the students could meet all the demands of the task, including comparing the yearly costs and savings of two different gym plans. Almost 20% of the students scored no points on this task. 90% of those students attempted the task. Sixth Grade 2003 pg. 24

Gym Points Understandings 0 Almost 20% of the students scored no points on this task. 90% of the students with this score attempted the problem. 1 Students with this score could generally find the monthly cost for Pay as you go and All-In- One deal. 3 Students could find the monthly cost of all the gym plans and compare to find the best value. 5 Students could find the monthly costs of all the gym plans and find the number of visits needed at the Regular Price to equal the cost of the All-in-one plan. Misunderstandings Students had difficulty calculating the costs for Regular deal. Many students put $50 (just the monthly fee),$100 (50 x 2), $52 (50+2), 1000 (20 x 50), or 2000 (40x 50). These answers make up 19% of the responses. They could not reason clearly about how the how visits, costs per visit, and monthly costs fit together. They are testing out operations without thinking through how those operations contribute to the solution. Students had trouble finding the number of visits to equalize the cost for regular plan and All-In-One. About 10% of all the students did not attempt this part of the task. Another 10% thought there would need to be 50 visits, forgetting to include the monthly fee in their calculations. About 12% thought the number of monthly visits would be 2 or 5. Students had difficulty dealing with all the constraints for comparing the yearly costs of Superfit and Regular Deal. Almost 10% of the students forgot to add in the initiation fee for Superfit. Another 10% forgot to multiply the cost of monthly visits by 12, so they got a total cost of $520 instead of $960. 6 Students with this score could not find the number of visits needed in part two of the task or could not complete the comparison in part 3 even though they correctly found the yearly cost for Superfit. 8 Students could calculate and compare different options for gym memberships in dollars and number of visits. Sixth Grade 2003 pg. 25

Based on teacher observations, this is what sixth grade students seemed to know and be able to do: Calculate the costs of different options for gym memberships for one month Compare costs of the different options Find the number of visits needed in one plan to equal the cost with another plan Areas of difficulty for sixth graders, sixth grade students struggled with: Identifying and using multiple constraints within the different gym options Analyzing the affects of operations on units (e.g. adding visits to dollars) Converting all constraints to the same measure to allow a comparison (e.g. forgetting to converting cost of visits per month to cost of visits per year) Questions for Reflection on Gym: Look at the errors in student work for finding the monthly cost of regular deal. What types of error patterns do you see? What were students thinking about and not thinking about? Were the errors caused by ignoring constraints? Not understanding the units? How did your students make sense of part 2? How many put: 50 5 2 8 95 4000 What is the logic behind each wrong answer? Is the logic consistent for each wrong answer? What are the implications for further instruction? Look at the work for students in part 3. How many of your students put: 660 600 520 300 180 240 What is the logic behind each wrong answer? Is the logic errors related to the types of errors you noticed in part 2 or different? In what way? How would you summarize student thinking? What are the implications for further instruction? Do students need more work with units? With the logic of comparisons? Problems dealing with multiple constraints? Models to help them understand the effects of operations on numbers? Organizational tools? What is your evidence? Teacher Notes: Sixth Grade 2003 pg. 26

Instructional Implications: Students need more experiences making comparisons. They need to calculate the full costs of both options before making a decision. It is not enough to just calculate the cost for the favored choice. Students might consider different choices about family entertainment options or cell phone plans to determine the best price. Students lack experience analyzing the effects of operations on units or systems for keeping track of units as they work through steps in a problem. Dimensional analysis is a topic that many teachers need to deal with more explicitly. Students also have trouble identifying the various constraints of within each option and seeing how they interconnect. Students need to work with more problems in context instead of working isolated number calculation skill sets. Sixth Grade 2003 pg. 27

6 th grade Task 3 Square Elk Student Task Core Idea 4 Geometry and Measurement Find the area and perimeter of letter shapes on a square grid. Apply the appropriate techniques, tools, and formulas to determine measurements. Select and apply techniques and tools to accurately find length and area to appropriate levels of precision Investigate, describe, and reason about he results of subdividing, combining, and transforming shapes Sixth Grade 2003 pg. 28

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Looking at Student Work Square Elk Unlike many problems where looking at strategies of good students gives us many insights into student thinking, students who did well on Square Elk seemed to just know it and their work reads like a rubric. However examining the error patterns reveals insights into the logic load of this task. Looking first at the area part of the task, students were generally not able to use a formula, but instead needed to match parts to make wholes. Student A shows an understanding of the idea that two half squares make a whole. However, the numbering system is off slightly when counting shape K. In part 3 of the task, Student A does use the formula for area of a triangle to find the area of the right triangles. Student A Sixth Grade 2003 pg. 32

Student B tries to look at the whole shape and think of ways to use the area formula. For the letter E, the student realizes that if a line is drawn at the ends of the E it will make a 3 x 4 rectangle. Then Student B subtracts the two whole squares. However he neglects the 2 half-squares by the middle prong of the E. In finding the area of the N the student again tries to find the area rectangle, but uses the dimensions 1 1/2 x 4 instead of 2 1/2 by 4. Student B also does not understand how to interpret the key for perimeter. He apparently thinks that all sides are 1.4, not differentiating between the sides and hypotenuse of the triangle. Sixth Grade 2003 pg. 33

Student C also has trouble interpreting the sides of the key for perimeter. In finding the perimeter for K, the student incorrectly uses the 1.4 as the distance for the top and for the diagonal on K In finding the perimeter for N, the student does not count the leg of the right triangle. It is unclear if the student thinks the 2.34 goes to the whole triangle or just neglects the leg. Sixth Grade 2003 pg. 34

Student D has trouble keeping track of what has been counted and what has not. In letter E the student incorrectly counts the middle prong as 5 instead of 4, but forgets to count one of the 2-1 combinations for the top and bottom prongs of the E. In finding the perimeter of the K the student counts the straight and diagonal sides of the triangle as 1.4 and neglects to count the top and bottom of the diagonal part of the K. Student E does not recognize that the hypotenuse for the right triangle goes across two squares instead of one. Sixth Grade 2003 pg. 35

Grade 6 Square Elk Square Elk Mean: 3.27, S.D.: 2.36 2500 2000 Frequency 1500 1000 500 0 0 1 2 3 4 5 6 7 8 Frequency 2017 742 868 1479 1677 1172 925 608 377 Score Score: 0 1 2 3 4 5 6 7 8 % < = 20.4% 28.0% 36.8% 51.8% 68.8% 80.6% 90.0% 96.2% 100.0% % > = 100.0% 79.6% 72.0% 63.2% 48.2% 31.2% 19.4% 10.0% 3.8% The maximum score available for this task is 8 points. The cut score for a level 3 response is 4 points. Most students (about 80%) could find one or more of the areas in elk. A little less than half the students (about 49%) could find all the areas in part 1 and 3. Only about 4% of the students could meet all the demands of the task. About 20% of the students scored no points on the task. Most of those attempted the problem. Sixth Grade 2003 pg. 36

Square Elk Points Understandings Misunderstandings 0 Most students attempted the task. 2 Students with this score could find the area of E and L. 3 Students could kind all the areas in part one. 4 Students could find areas of all the shapes. 5 Students could find the areas of all shapes and the perimeter of L. Some students had difficulty with area. The most common areas given for E and L were both 12. Students had difficulty matching the half squares in the K to find area. The most common errors for k were 8, 10, and 6. Students had difficulty with area in part 3 for the letter N. Common areas were 11, 12, and answers ending in.68. Students confused the length of the diagonal with the area of the right triangle when looking at the key. Counting perimeters for irregular shapes was difficult for students. Most students showed some understanding of perimeter going around the outside, but they don t have the spatial skills to see the rotation of the diagonals within the letters or mark and count all the subparts to the shape. Students often forgot to count the half square lengths next to the middle prong of the E. Students had trouble with diagonals and legs of the right triangles. Students thought they were both same or did the length of the hypotenuse count for two sides of the triangle. 7 Students with this score could not find the perimeter of N. 8 Students could find the area of shapes by counting squares and finding areas of triangles. They could interpret a key for the length of a hypotenuse and recognize it in rotated positions to find perimeter of irregular shapes. Sixth Grade 2003 pg. 37

Based on teacher observations, this is what sixth graders seemed to know and be able to do: Count squares to find area Understand the relationship between right triangles and rectangles to find area Understand that the distance around a shape was equal to the perimeter Areas of difficulty for sixth graders, sixth grade students struggled with: Using a key to find perimeter of shapes representing the hypotenuse of a right triangle Recognizing rotations of a shape Systematically identifying and counting all the parts of a perimeter Questions for Reflection on Square Elk: Were your students successful finding the areas of all the shapes? Were there any common wrong answers for area? What do you think was the logic that led to those errors? What experiences have your students had using keys? What experiences have your students had with identifying attributes? What experiences have your students had that would help them understand the difference in size of a diagonal versus a leg of a right triangle? Do you notice other relationships that may have prevented students from solving the problems for perimeter? Teacher Notes: Implications for Instruction: Students who had difficulty with this problem could not break down the shape into simpler parts. When calculating the area, they could not find the fractional parts. When finding perimeter, they did not recognize the connection between the scale model and the diagonals on the diagram. They need more practice with spatial visualization and rotations. Students at this grade level need to be able to work with more complex shapes. When working with detailed figures, they need to develop strategies for keeping track of their calculations, what s been done and what still needs to be calculated. Sixth Grade 2003 pg. 38

6 th grade Task 4 Spinners Student Task Core Idea 2 Probability Work with probabilities for two different spinners. Identify the sample space and the likelihood for certain events. Demonstrate understanding and use of probability in problem situations. Determine theoretical and experimental probabilities and use these to make predictions about events. Represent the sample space for a given event in an organized way (e.g., table, diagram, organized list, and tree diagram) Sixth Grade 2003 pg. 39

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Looking at Student Work Spinners Student A struggled with likeliness when the possibilities for making 2 or 3 were not contiguous. The student did not combine the sections to compare areas. However Student A has a very convincing mathematical argument for why the combined scores for the spinners will be less than 10. Student A uses an organized list to show all the different scores Tasha can get in part 3. Student A Sixth Grade 2003 pg. 43

Sometimes math tasks provide insight into student understanding that you aren t expecting. Student B does not answer the question about highest score, but gives a clear explanation and understanding about why 3 is equally likely Sixth Grade 2003 pg. 44

Students had a difficult time constructing a mathematical argument for why the score would be less than 10. While Students C and D, know there is nothing that will add to 10, they can t provide any evidence to back up their statements. Student C Student D Student E makes the case by eliminating all the possible combinations that would make 10, so therefore the score is less than 10. Student E Sixth Grade 2003 pg. 45

Student F makes the more common argument about what is the highest possible total, so therefore all totals are less than 10. Student F does a good job of explaining the logic behind the organized list to find the possible scores obtained by multiplying the numbers on the spinners. Student F Sixth Grade 2003 pg. 46

Grade 6 Spinners Spinners Mean: 5.53, S.D.: 2.78 2000 1500 Frequency 1000 500 0 0 1 2 3 4 5 6 7 8 9 Frequency 513 494 775 1019 766 762 1020 1308 1569 1639 Score Score: 0 1 2 3 4 5 6 7 8 9 % < = 5.2% 10.2% 18.1% 28.4% 36.2% 43.9% 54.2% 67.5% 83.4% 100.0% % > = 100.0% 94.8% 89.8% 81.9% 71.6% 63.8% 56.1% 45.8% 56.1% 16.6% The maximum score available for this task is 9 points. The cut score for a level 3 response is 4 points. Most students (about 82%) could determine which spinner was more likely to get a certain number. Many students (about 72%) could also recognize equally likely events. About half the students could recognize more likely and equally likely events and find all the scores when the numbers on the two spinners were multiplied. Almost 17% of the students could meet all the demands of the task including developing a convincing mathematical argument for why scores would be less than 10. About 5% of the students scored no points on this task. All the students in the sample set attempted this task or did not finish any parts on this task or the rest of test. Sixth Grade 2003 pg. 47

Spinners Points Understandings Misunderstandings 0 60% of the students with this score attempted the task. 1 Students could compare the area of two sections on a spinner and determine which one was more likely. 3 Students could combine spaces on a spinner and compare areas on different spinners to determine more likely and equally likely events. 4 Students could find which was more likely for getting a 1 or 2 and do some parts of finding scores when the numbers of the scores were multiplied. Some students with this score could complete all of part 3. 7 Students could make an organized listed of all possible combinations for multiplying the 2 spinners. They could either reason about likeliness or equally likeliness of events on different spinners or construct a convincing argument about why scores on the spinners would be less than 10. 40% of the students with this score did not attempt this task or the final task on the test. Time may have been a factor. They could not combine sections that were not contiguous to make a comparison. They did not understand the idea of equally likely. Students had difficulty listing all the possibilities for multiplying numbers on a spinner to determine highest and lowest score. Students with this score may still have trouble comparing sections that are not contiguous or understanding equally likely events. Students were often more successful with part 3 than some parts of 1. Students who missed parts of 3 were most likely to include extras, forgetting that there was only a five on spinner A or they made only a a partial list of scores. About half the students with this score could not make a convincing mathematical argument for why the scores had to be less than 10. They were not specific or did not provide evidence to back up their statements. The other half still could not complete the likeliness or equally likeliness of sections where some of the parts were not contiguous. They could not combine areas of more than one section. 8 Most students omitted some of the answers needed in part 3. They did not have an organized strategy for listing all the possibilities. 9 Students could determine likelihood of events on a spinner, make mathematical arguments about total scores and find all the outcomes for multiplying numbers from 2 spinners. Sixth Grade 2003 pg. 48

Based on teacher observations, this is what sixth grade students seemed to know and be able to do: Compare areas on a spinner to determine which is more likely Multiply numbers from 2 spinners to find possible outcomes Areas of difficulty for sixth grade students, sixth graders struggled with: Combining areas of noncontiguous sections on a spinner Determining equally likely events Constructing a convincing, specific mathematical argument for why some scores are not possible Organizing information or lists in a manner to determine when all possibilities have been given Questions for Reflection on Spinners: How many of your students had difficulty with answering part 1 for scores of 2 and 3? What further questions could you ask to determine whether their confusion was combining sections and working with fractions? Or whether their confusion was on understanding probability and likeliness? What experiences have students had working with angles in a circle and fractional parts? What experiences have students had this year with probability? What went well? What important concepts around probability did not get covered? What measures does your school have in place to help make up these deficiencies in 7 th and 8 th grade? In your probability unit, how do you build up the idea of sample space and comparing areas? What are some of the better problems that help students develop this idea? What types of experiences do your students have in building convincing arguments? Is this a regular part of classroom activities and discourse? How do you use modeling to make explicit what you value in a mathematical argument? Have you worked explicitly with teaching problem solving strategies like make an organized list? What are some of your favorite problems for developing this skill? Teacher Notes: Sixth Grade 2003 pg. 49

Implications for Instruction: Students need more experiences with probability situations. They need experience analyzing areas with different sizes to determine if events are equally likely. They need to be able to visualize how sections can be divided or combined on a spinner and still have the same probability. Students often think the position on the spinner affects the probability, e.g. if a red space appears on both sides of a spinner, then it is more likely than if two red spaces of the same size appear side-by-side. Students also have difficulty combining information from two events like two spinners, two coins, a coin and a spinner. Students need many more opportunities to experiment with probability situations to better their understanding between models and the probability of an event happening. Learning to organize information from the situation by making charts or lists can also help them make sense of the possible outcomes. Teacher Notes: Sixth Grade 2003 pg. 50

6 th grade Task 5 Rabbit Costumes Student Task Core Idea 1 Number and Operations Use proportional reasoning involving fractions to find material needs for making costumes. Understand number systems, the meanings of operations, and ways of representing numbers, relationships, and number systems Understand and use proportional reasoning to represent quantitative relationships Select appropriate methods and tools for computing with fractions, and decimals from among mental computation, estimation, calculators, and paper-and pencil, depending on the situation, and apply selected methods Sixth Grade 2003 pg. 51

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Looking at Student Work Rabbit Costumes Many students had difficulty working with mixed numbers to solve the problems for Rabbit Costumes. Some of the most successful students tended to convert to decimals. Students also struggled with making a mathematical comparison. They did not always check against every constraint to find out precisely which fabric limited the choice. Student A makes a good case of showing that she tested for every constraint in part 3 and worked efficiently with fractions. Student B completes the argument by showing that 6 is less than 7. Student A Sixth Grade 2003 pg. 54

Student B Many students have trouble picking an appropriate operation when solving problems involving fractions. Student C uses division instead of multiplication to find the amount of fabric for 8 costumes. Student C Sixth Grade 2003 pg. 55

Student D converts yards to inches in order to avoid working with the fractions. Student D ignores the blue and pink fabric when trying to make the comparison in part 2. Student D Student E appears to be checking all the fabrics to see how many costumes can be made, but does not clearly describe how the drawings were used in the making the comparison. Student E Sixth Grade 2003 pg. 56

Another common error for working with the mixed numbers is to only use the 1/2 once in 1 1/2 yards of white. For example in part one, students may have thought that 8 x 1 1/2 was 8 1/2. In part 2 student F thinks that there is enough fabric for 9 costumes (9 x1 1/2= 10 1/2). Student F By far the most difficult part of the task was connecting the mathematical answers to the situation and being able to discuss which fabric would run out first. So even if there is evidence of knowing the correct number of costumes, that information is ignored in the explanation. The most common response is to think pink will run out first. Student G Student H Sixth Grade 2003 pg. 57

Student I Many students also think white will be used up first, but not because of their math calculations. Student J Student L Teacher Notes: Sixth Grade 2003 pg. 58

Grade 6 Rabbit Costumes Rabbit Costumes Mean: 2.56, S.D.: 2.40 3000 2500 2000 Frequency 1500 1000 500 0 0 1 2 3 4 5 6 7 8 Frequency 2802 1061 1487 1765 561 1031 288 150 720 Score Score: 0 1 2 3 4 5 6 7 8 % < = 28.4% 39.2% 54.2% 72.1% 77.8% 88.3% 91.2% 92.7% 100.0% % > = 100.0% 71.6% 60.8% 45.8% 27.9% 22.2% 11.7% 8.8% 8.8% The maximum score available on this task is 8 points. The cut score for a level 3 response is 3 points. Many students (about 71%) could find the amount of blue fabric needed to make 8 costumes. Almost half the students (45%) could find the amount of fabric needed for each of the 3 colors. A little more than 10% of the students checked all 3 constraints before making a comparison. Less than 10% of the students could meet all the demands of the task. Almost 30% of the students scored no points on this task. About 1/4 of those students did not attempt the task. Sixth Grade 2003 pg. 59

Rabbit Costumes Points Understandings Misunderstandings 0 30% of the students score zero. 3/4 of the students with this score attempted the task. 1 Students could find the amount of blue fabric needed for 8 costumes. 2 Students could find the amount of fabric for blue and pink. It was easier for students to think about a fraction than a mixed number. 3 Students could find the amount of fabric needed for all 3 fabrics. Many successful students converted fractions to decimals. 5 Students could calculate the fabric needed for 8 costumes. They also knew how many costumes could be made with the new amount of fabric. 8 Students could find the amount of fabric needed for a specified number of costumes and use the inverse process to calculate the number of costumes that could be made if the amount of fabric was known. Students could make a comparison, complete with addressing all the different options. Students sometimes copied down the information for one costume as the answer for 8. Others attempted to divide instead of multiply. Students had difficulty working with fractions and mixed numbers. They may have multiplied 8 x 1 1/2 to get 8 1/2 or 8(1 /4) they only multiplied the 1/2 by 8). Students did not know how to make a comparison. Typically they would think pink would run out first because it s a small amount or white would run out first because it s the main color on the costume (put the details on later). More than half the students with this score did not mention any numbers when explaining which fabric would run out first. Almost 25% of all students thought pink would run out first because it s a small amount. More than 10% thought white would be used up first, because rabbits are white or it takes more white. Sixth Grade 2003 pg. 60

Based on teacher observations, this is what sixth grade students seemed to know and be able to do: Find the amount of fabric needed for blue fabric (multiply by whole numbers) Find the amount of fabric needed for pink fabric (multiple by a fraction) Areas of difficulty for sixth graders, sixth grade students struggled with: Multiplying and dividing mixed numbers Checking all constraints before making a comparison Connecting results of calculations to the context of the problem (e.g. being able to calculate how many costumes could be made from each color of fabric, but then using other information to decide which fabric would run out first) Questions for Reflection on Rabbit Costumes: Could most of your students pick the correct operation to solve for part 1? Did your students use fractions, decimals or pictures to help figure out their solutions to part 1? What methods did successful students use? Did your students seem comfortable with fractions? What surprised you or disappointed you about their work with fractions? What do you want to think about more carefully when you prepare your fraction unit next year? Look carefully at the student work on comparisons. Can you find evidence of students: Finding the number of white costumes? Blue Costumes? Blue and Pink Costumes? All 3 fabrics? Incorrect calculations? No attempt to quantify the number of costumes? Did your students use these calculations in making their arguments about which fabric would run out first? What types of problems have students worked on this year to help them develop their skills for making comparisons? Implications for Instruction: Students need more experiences multiplying fractions and mixed numbers. Students also need to show all calculations before making a comparison. When given information on the amount needed for one item they need to determine the amount needed for any number of items. Students also need to work backwards. When they are given the total amount of material available, they should be able to calculate the amount of costumes that can be made. Students had difficulty grasping the difference between having a small amount of something, compared to how many costumes that would make. Sixth Grade 2003 pg. 61