Sixth Grade Mars 2007 Task Descriptions Overview of Exam Core Idea Task Score Rational Numbers Division This task asks students to analyze the answer to a division problem in a variety of contexts. Students need to think about how to interpret the decimal and round it appropriately to answer the question for each situation. Students were also asked to write a word problem that would fit a given calculation and pick the appropriate answer to fit their own problem. Successful students could determine how to write the correct interpretation of the answer to each of the contexts from the same numerical answer and understood how to round money. Probability Card Game This task asks students to use probability in the context of picking cards, numbered 1 to 10. Students needed to think about a changing sample space. Some cards were no longer available after they had been drawn. Successful students could give a numerical value for each probability. Number Properties Factors This task asks students to list factors of numbers and think about types of numbers with an odd number of factors or an equal amount of even and odd factors. Successful students used the factor list given to help them find and describe patterns in the numbers with different factor characteristics. Data Analysis Household Statistics This task asks students to use a bar graph about number of households and number of children per household and record the information in a table. Students were also asked to interpret information from the graph and calculate percentages. Successful students could think about total number of households, total number of children, or mean (or average) and match it to the correct computation. Geometry and Building Blocks Measurement This task asks students to calculate volume of rectangular prisms, given a picture of the object and its dimensions. Successful students could also calculate the surface area. 6 th grade 2007 1
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Division This problem gives you the chance to: relate a given division calculation to appropriate practical situations When you calculate 100 6 using a calculator, the result is 16.6666667. This result can be used to give a sensible answer to all the following questions except one. 1. Write down the sensible answers and find the question that cannot be answered using this result. a. How much does each person pay when 6 people share the cost of a meal costing $100? b. 100 children each need a pencil. Pencils are sold in packs of 6. How many packs are needed? c. What is the cost per gram of shampoo costing $6 for 100 grams? d. How many CDs costing $6 each can be bought for $100? e. What is the average distance per day, to the nearest mile, travelled by a hiker on the Appalachian Trail, who covers 100 miles in 6 days? 2. Write another question, together with its sensible answer, that can be answered using 100 6. 7 Copyright 2007 by Mathematics Assessment Page 4 Card Game Test 6
Task 1: Division Rubric The core elements of performance required by this task are: relate a given division calculation to appropriate practical situations Based on these, credit for specific aspects of performance should be assigned as follows points section points 1a. Gives correct answer: $16.67 Accept $16.70, S16.75, $17.00 1 1b. Gives correct answer: 17 1 1c. Gives correct answer such as: cannot be done using this result 1 1d. Gives correct answer: 16 1 1e. Gives correct answer: 17 1 5 2 Writes an appropriate question. Writes a sensible answer. 1 1 2 Total Points 7 Copyright 2007 by Mathematics Assessment Page 5 Card Game Test 6
Division Work the task. Look at the rubric. What are the big mathematical ideas involved in solving this task? Look at student work for part 1. How many of your students put: 1a 1b $16.67 16.66 16.6 16.6666667 $16 Other 17 16.6666667 16 16.67 16.66 Other 1c Can t be done 16.6666667 17 cents 16 cents $16.67 16 6/100= $.06 Nothing 600 Other 1d 1e 16 16.6666667 16.67 17 16.66 Other 17 16.6666667 16.67 16.7 16 Other Why do you think it was difficult for students to make sense of the answers? Did you see evidence of students still working the problem each time, rather than using the answer given at the top of the page? What type of experiences do students have to help them make sense of answers and relevant digits? Rounding in context? Using technology? Now look at student work for designing their division question. How many of the students posed a division question for the original problem? How many posed a different division problem (like 6/100)? How many students wrote a multiplication question? What surprised you in reading student work? 6 th grade 2007 6
Looking at Student Work for Division Student A is able to look at the information about the calculation and interpret each division situation individually to find an answer. The student is able to write an original story problem that uses the calculation and answers the question. Student A 6 th grade 2007 7
Student B goes that extra step and explains how the answer was chosen or why it fits each context. The student doesn t understand that part c cannot be solved with the original calculation. Student B Student C solves all of 1 and writes a typical money problem. The student even uses appropriate language for division problems (to the nearest dollar) and has rounded the answer. What is wrong with the answer? Student C 6 th grade 2007 8
Student D does not accept the given calculations, but needs to recalculate the problem each time to help make sense of the situation. Notice that the student goes an extra step to correctly solve part c. The student does not round appropriately in part a and e. What about these situations might be confusing to a student? Student D 6 th grade 2007 9
Student E has learned some rules about rounding off answers with decimals and applies these rules equally to all the situations in the problem. The student has a procedure without asking questions about what fits for this situation. The student is not making sense of how the situations vary. This difficulty with context is shown in the problem written in part 2 ( which is scored incorrectly). What would be the correct number sentence for this problem? Student E Student F has also written a problem that cannot be solved by 100/6. What would be the correct number sentence for this problem? Many countries have students write their own problems on a regular basis as a way of assessing their understanding of operation. What does this process reveal about students misconceptions? Do your students have this kind of opportunity to construct their own problems on a regular basis? Student F 6 th grade 2007 10
Student G writes down the problem and then shows the rounding for each problem. Notice that the student uses multiplication to help make sense of part d. The student makes an interesting error of only using the cents part of the answer in part c. Notice that in part e the student decides to change the calculations because for average the rule is to add numbers and then divide by the number of addends. How could you use this piece of work to promote a discussion about the meaning of average in this situation? The student writes a good problem in part 2 that fits a context similar to textbook questions. Why would this problem be difficult to solve with the given information? What would the student have to do to find the solution? Student G 6 th grade 2007 11
Student H does not appear comfortable with the process of division. The student needs to work each situation as a multiplication problem and to find the appropriate answers. Could this strategy help the student solve each problem? What questions would the student need to ask herself to make sense of each multiplication problem? What might be next steps for this student? Student H 6 th grade 2007 12
Student I uses the same rounded answer for every situation. The student has procedural knowledge, but has not learned the logic of division or an understanding of division in context. What types of activities or discussions help children to learn to differentiate different types of division situations and how to interpret their answers? How do they develop the ability to identify significant digits for varying contexts? Look at sentence one of the student s problem. This seems to suggest multiplication rather than division. Student I 6 th grade 2007 13
Student J seems to understand rounding as a procedure and can do it appropriately when asked in part 3. However the student does not understand significant digits in the context of a problem or does not pull out the tool of rounding on his own. What opportunities do students in your class have to interpret answers from calculations? Student J 6 th grade 2007 14
Student K understands that answers need to fit the context. The student needs to keep reworking the problem to check the calculations or make sense of the situation. It is almost like there is an issue about conservation of number. The student tries to look for significant digits, but does not interpret the situations correctly. Notice that in part d the student doesn t realize that the answer is not dollars. This difficulty with labels is again apparent in part 2. The student gives information about kilometers per hour, but asks a question about kilometers per day. Does Mr. Koti drive for 24 hours straight? How could you rewrite this problem to better fit the situation? Student K 6 th grade 2007 15
Finally look at the work of Student L. What does this student understand about the operation of multiplication and division? Why doesn t the word problem make sense? What information is needed to make this problem work? Student L 6 th grade 2007 16
6 th Grade Task 1 Division Student Task Core Idea 1 Number and Operation Relate a given division calculation to appropriate practical situations. Interpret a remainder in a problem context. Understand number systems, the meanings of operation, and ways of represent numbers, relationships, and number systems. Based on teacher observation, this is what sixth graders know and are able to do: Write a division problem that can be solved with 100/6. Find the number of cd s that can be bought for $100. Round money appropriately. Areas of difficulty for sixth graders: Identifying the problem that couldn t be solved. Using rounding in context Interpreting a decimal answer in context. 6 th grade 2007 17
The maximum score available for this task is 7 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, about 87%, could write a story problem that would use the calculation 100/6. Many students, about 70%, could also find the number of cd s that could be purchased with $100. They could round down to an even 16. About 38% of the students could write a question, find the amount of money owed by each person in a restaurant, the number of pencil packs that were needed for 100 children, and find the appropriate number of cd s. About 11% could also answer their own division question and find the average distance traveled per day. About 4% could identify the problem that couldn t be solved with the calculation given. Almost 13% of the students scored no points on this task. 70% of the students with this score attempted the task. 6 th grade 2007 18
Division Points Understandings Misunderstandings 0 70% of the students with this score attempted the task. 1 Students could write a story problem for 100/6. 2 Students could write a story problem and find the number of cd s that could be purchased with $100. 4 Students could write a story problem and find the number of cd s that could be purchased with $100. Students could also think about money in a restaurant and buying pencils in a pack. Students had difficulty writing a division task. Many students wrote a problem that would be solved by reversing the numbers, e.g. 6/100. Others wrote problems that were actually multiplication problems. Others wrote problems without enough information to be solved. Students had difficulty finding the correct number of cd s that could be purchased. 14% of the students thought they could buy 17. 7% used the entire answer, 16.6666667 cd s. Almost 4% rounded the answer inappropriately to 16.67 cd s. Students had difficulty finding the amount of money each person would pay for a meal at a restaurant. 15% thought each person would pay $16.66. Almost 8% thought each person would pay $16. 6% used the given answer of 16.6666667. 4% thought the answer was 16.6 with a bar over the final 6, ignoring money notation. Students also had difficulty with Finding the number of pencil packs. Almost 11% thought only 16 packs needed to be purchased. 5% thought 16.6666667 packs should be purchased. Students had difficulty finding the average distance traveled. Almost 18% thought the average was 16 miles. 8% thought the average was 16.6666667 miles. 4% thought the answer was 16.6 with a bar over the final 6. Almost 3% thought this problem couldn t be solved. Students also had difficulty with answering their own questions. 42% gave no answer to their question. 12% wrote questions that couldn t be answered with the original answer. 7% did not round their answers appropriately for the context. 6 Students could not identify the problem that couldn t be solved with 100/6. 13% of the students thought the answer to part 1c was 16. 6% thought the answer was 16.6666667 cents. Another 6% left it blank. 5% thought the answer was $.16. 4% thought the answer was $16.67 and 4% thought the answer was 17. 7 Students could interpret a decimal answer to a division problem for a variety of situations and write and answer their own division problem for 100/6. 6 th grade 2007 19
Implications for Instruction Students need more experience thinking about the meaning of numbers. Students were asked to think about the calculation 100 divided by 6 = 16.66667 given on a calculator. In different situations, the answer to a question might be 16, 17, or $16.67. The question is asking students to make sense of decimal numbers in a division context. When is it appropriate to round up or round down? What is the level of accuracy that makes sense in a given situation? In a world driven by technology, students need to prepared to think about significant digits or results from computations by calculators. Some students do not understand the operation of division. Their questions they posed are about multiplication or imply reversing the order of the division. Models, such as the bar model or number line, can help students to make sense of the actions of operations. The models help students to identify what is known and what is being asked. Action Research Ideas The Role of Models and Context in Understanding Division Some students at this grade level are still having trouble understanding the operation of division. These students need more experience with the types of actions that call for division action. With your colleagues, study the diagram below. What types of division problem-types do you think that students understand? What types of division problem-types do students need more help with? Can you design a series of problems or classroom lessons to examine different division situations? from Children s Mathematics, Cognitively Guided Instruction by Heinemann Press 6 th grade 2007 20
Models are also useful in helping students visualize the process or action of division. Study the models below. How do they help clarify division? Model #1 Model #2 6 th grade 2007 21
Model #3 from Japanese series: Mathematics for Elementary School available from Global Education Resources in Madison, New Jersey Now think about the actions in the task Division. What models could you make to illustrate each situation? 6 th grade 2007 22