Balanced Assessment Test Fifth Grade 2008 Core Idea Task Score Measurement Shopping Bags This task asks students to work with pounds and ounces to find weight of items in a shopping bag. Successful students knew how to convert ounces to pounds and could decide which items to put into the shopping bag so that it wouldn t break. Number Operations Breakfast Time This task asks students to calculate costs for groups of people eating at a café and find change. Given the size of the bill, successful students could use multiplication or division with decimals to find the number of people served. Rational Numbers Fruity Fractions This task asks students to use equivalence to change improper fractions to whole numbers. Successful students could create improper fractions to match whole number values. Algebra Pea Soup This task gives students a chance to use proportional reasoning to think about expanding a recipe. Students need to be able to find the amount of the ingredients for different numbers of peoples, record the information in a table, and graph the data from the table. Successful students could explain how to read the graph, and they could compare the slopes of two lines on the graph. Data Bar Charts This task asks students to interpret and construct bar charts. Students needed to think about how to make all the bars total to a given number. Then students needed to change one bar on the graph to keep the mode the same. Successful students were able to reason about combining information on the graph to calculate number of total children given the number of families and the children per family. 1
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Shopping Bags This problem gives you the chance to: work with standard units in the customary system The supermarket s shopping bags can only carry 6 lbs of goods. Here is Yusef s shopping list. Item Weight laundry detergent 2lbs 10oz oranges 2lbs 11oz toothpaste 5oz pineapple 3lbs 6oz liquid soap 1lb 1oz book 7oz paper tissues 8oz. 16oz = 1lb Yusef has two shopping bags. He has already put his laundry detergent in one bag and his oranges in the other. 1. Show what other items Yusef can put in the bags so that the bags don t break. Bag #1 Bag #2 laundry detergent oranges 2. What is the weight of the items in Bag #1? Show how you figured this out. 3. What is the weight of the items in Bag #2? Show how you figured this out. 6 Copyright 2008 by Mathematics Assessment Resource Service 4
Shopping Bags Rubric The core elements of performance required by this task are: work with standard units in the customary system. Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: Bag #1 pineapple Gives correct answer: Bag #2 toothpaste, liquid soap, book, paper tissues 2. Gives correct answer: 6 lbs or 96oz Shows correct work for items in their bag, such as: 2lbs 10 oz + 3lbs 6oz. 3. Gives correct answer: 5 lbs or 80oz Shows correct work for items in their bag, such as: 2 lb 11 oz + 5 oz + 1 lb 1 oz + 7 oz + 8 oz 1 1 1ft 1ft 1ft 1ft 2 Total Points 6 2 2 Item Weight Ounces laundry detergent 2lbs 10oz 42 oz oranges 2lbs 11oz 43 oz toothpaste 5oz 5 oz pineapple 3lbs 6oz 54 oz liquid soap 1lb 1oz 17 oz book 7oz 7 oz paper tissues 8oz. 8 oz Copyright 2008 by Mathematics Assessment Resource Service 5
Shopping Bags Work the task. Look at the rubric. What are some of the key mathematical ideas in this task? What does a student need to understand in order to be successful? Look at the work in the shopping bags. How many of your students picked: A correct solution Used an item more than once Didn t use all the items Made a shopping bag with more than 6 lbs. Didn t put anything in the bags Now look at work for adding pounds and ounces. How many of your students: Had a system for adding pounds and ounces and making conversions Only added the pounds Only added the ounces Tried to use decimals What experiences have your students had with making conversions? When working with conversions, do you try to help students generalize about a process that will work for all measurements or do you just focus on the units for a particular problem or problem set? What is the same about changing any units? What is different in changing different types of units? What were some of the strategies used by successful students? What were some of the strategies used by unsuccessful students? 6
Looking at Student Work on Shopping Bags Students had a variety of strategies for combining weights and making conversions. Student A searches for number combinations to total 16, because 16 oz. equals one pound. Notice that the student experiments with the numbers near the sides of the bags. Then the student lays out the complete process, step-by-step, with labels. Student A 7
Student B also looks for combinations to make 16 oz. Pounds and ounces are treated as separate number problems. Student B Student C has correct work for each shopping bag, but puts it in the wrong space. The student was given benefit of the doubt because there is clear evidence of understanding. The student avoids many of the complications of dealing with mixed units by converting everything to ounces. The student starts each calculation chain with finding the maximum a bag will hold, 96 ounces. Then the student subtracts out the number of ounces until the bag is full or maximized. 8
Student D attempts this strategy by finding the maximum weight per bag, but doesn t understand that this needs to be used in connection with the weights put into the bag. Notice that the student doesn t understand the idea of distributing all the items between the two bags. The student puts the same items in both. Student D Student E uses fractions to add the ounces. Student E 9
Student F misinterprets the task and tries to make each bag exactly 6 lb. The student simplifies the mathematics by using multiple amounts of some items (paper tissues), which conveniently make even amounts of pounds. Then the student uses the item in the bag and other items needed to make the target weight. Student G doesn t understand that all the items need to be used. The student adds the pounds and ounces separately and then converts correctly the extra ounces into pounds. 10
Student H creates a table to separate pounds and ounces. This makes a nice visual organizer. Student I uses decimals to create the same type of visual organizer. However the mathematics is not correct. The value for toothpaste is not 5/100 of a pound. While the student makes the correct conversions from ounces to pounds to get the answer, will this notational error create problems in later mathematical work? Student J also attempts to use decimals as an organizer. The student is not even consistent with place value, but reaches a correct solution by searching for combinations that make 16. What are your concerns about the decimal approach? What do you think they understand about measurement? What do you think they understand about decimals? Student H 11
Student I Student J 12
Student K understands that if the ounces are more than 16 then they need to be changed to pounds. However the student just goes up 1 pound regardless of the number of extra ounces. So 32 ounces goes to 1 additional pound and 25 ounces also goes to 1 additional pound. Extra ounces are just dismissed. What questions might you ask this student to push his thinking? Student K Student L only adds the ounces for each item. Then converts the ounces to pounds. The student ignored or forgot to use the complete weight for each item. Why might a student have made this error? What doesn t the student understand? Student L 13
Student M adds together the ounces for all the items and uses an alternative algorithm for division to find the total pounds that come from ounces. The student then adds together all the pounds to find the total weight of the purchases, 11 lbs. The student doesn t seem to understand the distributing of the items into the two bags. The student just gives the weight in pounds of the items already placed in the bag. Student M While Students D, K, L, and M had no points, there was some evidence of understanding units and working on the mathematics of the task. However, because there were so many students who scored zero points, it is important to examine some other examples of zeroes. 14
Student N only adds the ounces. Then the student makes no attempt to convert to pounds and ounces. The student puts the same item in both bags, rather than trying to distribute all the items. Student N Student O just tried to find an item that has a value close to 6. The student ignores the fact that there are different units at play. Again the student does not try to distribute all the items. The student ignores the item already in the bag. Student 0 15
Student P adds ounces to pounds. The student does not distribute all the items. The student s answers go over the limit of 6. The student ignores the items already in the bag. What might be you next step with this student? Student P 16
5 th Grade Task 1 Shopping Bags Student Task Core Idea 4 Geometry and Measurement Core Idea 2 Number and Operation Divide up a shopping list of items into two bags, neither of which can hold more than 6 lbs. Convert between ounces and pounds when adding weights. Apply appropriate techniques, tools, and formulas to determine measurements. Select appropriate type of unit for measuring an attribute, like weight. Reason about and solve problem situations that involve more than one operation in multi-step problems. The mathematics of this task: Understanding multiple constraints: all the items must be used, the total weight of any bag cannot exceed 6 lbs., items can t be used more than once Converting between pounds and ounces and using mixed measures of weight Estimation to help narrow the choices for items in the first bag Based on teacher observation, this is what fifth graders know and are able to do: Add the sums of items in their bags Convert ounces to pounds Areas of difficulties for fifth graders: Knowing to convert between ounces and pounds Omitting items Using items more than once Adding only the ounces, only the pounds, or adding pounds to ounces Including the weights of items already in the bag Incorrect use of decimals 17
The maximum score available on this task is 6 points. The minimum score needed for a level 3 response, meeting standards, is 3 points. More than half the students, 58%, could put the correct item into shopping bag number 1. Almost half the students could put the right item in one shopping bag and add the total for the bag. Some students, 24%, could meet all the demands of the tasks including distributing all the items between the two bags so that no bag exceeded a weight of 6 lb., and could add the totals for each bag. 41% of the students scored no points on this task. 97% of the students with this score attempted the task. 18
Shopping Bags Points Understandings Misunderstandings 0 97% of the students with this score attempted the task. Students didn t understand the constraints. They may have used an item more than once or omitted some items. They may 1 Students put the correct item in bag #1. 3 Students put the correct item in bag # one and found the correct total for the items they placed in bag #2. 6 Students could distribute all the items between the two bags so that no bag exceeded a weight of 6 lb. and could add the totals for each bag using proper units. have put nothing in the bags. Students couldn t total the weights. They may have added only the ounces or only the pounds. They may have added pounds and ounces together, e.g. 1lb + 6 oz. = 16 lb. Students ignored or didn t understand the constraints of the problem. They showed some understanding of adding weights with the proper units. 19
Implications for Instruction Students need more practice with rich problem-solving tasks dealing with multiple constraints. Students had difficulty attending to all the details needed to complete the tasks, such as using all the items and not using multiple quantities of an item than you purchased. In working these types of tasks, students need to learn organizational skills to keep track of what they have calculated, such as using labels. Students need more practice with measurement. Students need a variety of ways for thinking about how to move from smaller to larger units. Some students did not understand that decimals are not appropriate for units that don t come in groups of 10 s or 100 s. While they may have reached a correct solution, the underlying misunderstanding may cause problems in multiplicative problems or in future learning. They need to think about other ways of visually organizing the numbers. Decimal place value is a critical understanding for this grade level. Ideas for Action Research Re-engagement One useful strategy when student work does meet your expectations is to use sample work to promote deeper thinking about the mathematical issues in the task. In planning for re-engagement it is important to think about what is the story of the task, what are the common errors and what are the mathematical ideas that students need to think about more deeply. Then look through student work to pick key pieces of student work to use to pose questions for class discussion. Often students will need to have time to rework part of the task or engage in a pair/share discussion before they are ready to discuss the issue with the whole class. This reworking of the mathematics with a new eye or new perspective is the key to this strategy. To plan a follow-up lesson using this task, pick some interesting pieces of student work that will help students confront and grapple with some of the major misconceptions. Make the misconceptions explicit and up for public debate. During the discussion, it is important for students to notice and point out the errors in thinking. I might start the lesson, by confronting the issue of adding units appropriately. I might start by posing the question: I noticed a student adding the items in the shopping bag #2 like this: 2 + 1 + 1 + 1 = 5 11 + 5 = 16 1 + 7 + 8 = 16 What do you think the student was doing? Where do the numbers come from? What items did the student put in the bag? Why did he group the numbers this way? What do you think the students final answer was? This is to push students to think about finding groups of 16 ounces to make a whole pound. 20
A different student did this work: What is this student doing? Whose work do you think is easier to understand? Why? This prompt is to help students think about the purpose of labels and ways of organizing their work. There were other choices that might also have been useful. I noticed this work on a different paper for the second shopping bag. 2 lb. 11 oz. 3 lb. 6 oz. 5 lb. 25 oz. = 6 lb. What do you think this student was thinking? Does this answer make sense? Why or why not? Here I want students to think about what to do with leftover ounces. I want them to grapple with numbers that don t come out evenly to 16. One student put the following items in a shopping bag: Liquid soap, Book, Paper tissues, Pineapple, and Laundry detergent Here is the work: 10 + 6 = 16= 1 lb. 8 + 7 = 15 15 + 1 = 16 = 1 lb the answer is 2 lb. What was the student doing? Where do the numbers come from? Do you think the answer is correct? Why or Why not? I want students to think about the need to consider both measures. I might now pose some follow up question to see if students are making sense of the strategies. Suppose Sophia and David went to the store. They bought the following items: Popcorn 2 lb. 5 oz. Cookies 3 lb. 9 oz. Ice cream 5 lb.12 oz. Chips 3 lb. 8 oz. Ignoring the fact that they will probably get a tummy ache, what is the total weight of items? How did you figure it out? 21
Breakfast Time This problem gives you the chance to: calculate costs and charges for a group 1. Linda had breakfast in a café. It cost $12.40. She paid with a $20 bill. How much change did Linda get? $ Show how you figured it out. 2. Basic Continental Breakfast $6.40 each A group of nine people had the basic continental breakfast. How much did they pay in all? $ Show your work. 3. A different group of people had the basic continental breakfast. They paid $32 in all. How many people were in the group? Show how you figured it out. 6 Copyright 2008 by Mathematics Assessment Resource Service 22
Breakfast Time Rubric The core elements of performance required by this task are: calculate costs and charges for a group Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answer: $7.60 Shows work such as: 20.00 12.40 2. Gives correct answer: $57.60 Shows work such as: 6.40 x 9 3. Gives correct answer: 5 Shows work such as: 32 6.40 or repeated subtraction points 1 1 1 1 section points 1 2 Total Points 6 1 2 2 Copyright 2008 by Mathematics Assessment Resource Service 23
Breakfast Time Work the task. Look at the rubric. What are the important mathematical ideas in this task? Look at student work in part 1. How many of your students: Chose subtraction and could do the calculations correctly: Chose subtraction, set up the problem correctly, but made a regrouping error $8.40: Tried to subtract $20 from $12.40: Used multiplication Other Look at work in part 2. How many students used multiplication? How many students used repeated addition? How many students chose a different operation? How many students used extra numbers in their calculations? Did students seem to be able to use place value appropriately in this part? Finally look at part 3, finding the number of people if the tab was $32. How many students correctly used division? How many students correctly used multiplication (or guess and check)? How many students had an incorrect answer of 4? How many students had an incorrect answer of 20? What might these students been thinking? What are the misconceptions they have about this situation? What concerns you about their understanding of place value and computations with decimals? What concerns you about students ability to chose the correct operation? What structures do you have in classroom to help students who are struggling? What structures are in place at your school to help students who are struggling before they fall further behind and need intervention in middle school? 24
Looking at Student Work on Breakfast Time Student A is able to choose the proper operations and work efficiently with decimals, including division with decimals. Notice how the student uses labels to make sense of the operations and what is being calculated. Student A 25
Student B uses a guess and check strategy, but is thoughtful about choosing the guess. How do we help students become more effective in their number choices? How do we help them apply things they already know when using this strategy? Student B 26
Student C is able to work part 2 and 3 of the task correctly. The student solves part 3 by using guess and check and needs 3 guesses before arriving at the solution. The student is confused about subtraction with decimals. What experiences would help this student? Student C Student D doesn t understand some basics about dividing by decimals. The student seems to have an internal misconception that when dividing, the shorter number goes on the outside. The student was successful on part 1 and 2 of the task. What contextual clues should have alerted the student that 20 people was an unreasonable answer? Student D Student E solves part 1 and 2 of the task, but struggles with the meaning of part 3. The student inserts a new number into the problem to make the division easier. Student E 27
Student F has full marks for part 1. The student tries to invent her own algorithm for multiplying decimals by decomposing the number into dollars and cents. However the student doesn t understand how place value comes into play between the two groups. Thinking of digits separately from their place value leads to many computational errors. Students need to think about the value of the digit in all their mathematical conversations. Student F chooses an incorrect operation in part 3. The student also attempts to use the 9 from part 2 above. How could the student have finished the problem successfully using the subtraction problem? What would she need to do next? Student F 28
Student G has a strategy for finding the number of people who spent $32, but doesn t know how to interpret his answer. How could this work have led to a correct solution? What does the student need to think about to interpret this information? Student G 29
Student H does not show much of her work. In part 1 the student has the answer typical of someone who doesn t regroup when subtracting. In part 2 the student multiplies the dollars by 9 but keeps the cents the same (no multiplication). In part 3 the student attempts to use estimation to solve the problem. If 1 breakfast is about $6, the 2 breakfasts would be about $12. While there is sense-making of the context, why doesn t the strategy work? Student H 30
Student I subtracts incorrectly in part 1. The student doesn t know to add zeroes to the end of the $20. In part 2 the student tries to multiply, treating the dollars and cents as separate units. The student did understand how to add the cents back to the dollars, but only added 6 groups of 40 cents instead of 9 groups. In the process of compartmentalizing the numbers, the student lost some of the facts of the problem. The student had an idea about division in part 3 but clearly didn t understand how to do the computation. Student I 31
Student J chooses the incorrect operation in part 2 and 3. Notice that the student changes the $32 to $3.20 to make it the right size for the operation. What activities help students learn to choose the correct operations? Student J 32
5 th Grade Task 2 Breakfast Time Student Task Core Idea 2 Number and Operation Core Idea 1 Number Properties Calculate costs and charges using decimals for a group buying breakfast. Understand the meanings of operations and how they relate to each other, make reasonable estimates, and compute fluently. Develop and use strategies to solve problems involving number operations with fractions and decimals relevant to students experience. Reason about and solve problem situations that involve more than one operation in multi-step problems. Understand the place-value structure of the base-ten number system including being able to represent and compare rational numbers. The mathematics of this task: Choosing operations Understanding decimal place value Calculating with decimals Based on teacher observations, this is what fifth graders know and are able to do: Use correct money notation Choose appropriate operation Use multiplication to help solve a division problem Multiply decimals Understand the concept of change Areas of difficulty for fifth graders: Division with decimals Subtraction with regrouping or adding in the extra zeroes before subtracting from the $20 Where to put the numbers in a division problem (some students want the shorter number on the outside) 33
The maximum score available for this task is 6 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Most students, about 96%, could set up the subtraction of decimals to find change in part 1. Most students, 82%, could also multiply with decimals to find the cost of breakfast for 5 people. More than half the students could meet all the demands of the task including using division or multiplication to find the number of people who spent $32. Less than 4% of the students scored no points on this task. All the students in the sample with this score attempted the task. 34
Breakfast Time Points Understandings Misunderstandings 0 All the students in the sample with this score attempted the task. ($8.40) 2 Students could subtract decimals to calculate change. 4 Students could subtract and multiply with decimals. Students did not know how to subtract with decimals. 4% of the students didn t regroup Students had difficulty with multiplying decimals. Some students decomposed the number into dollars and cents and didn t know how to put them back together. Other students made arithmetic errors. Students did not know how to divide with decimals. Many students were able to use multiplication and guess and check to solve the problem. 7% of the students reversed the order of the numbers for division and had an answer of 20 people. 7% had an answer of 4 people. 5 About half the students with this score made a calculation error in part 1. The other half made a calculation error in the division in part 3. 6 Students could reason with decimals in context to solve subtraction, multiplication, and division problems. 35
Implications for Instruction Students need more practice with problems in context. They need to be able to recognize the operation needed to solve the problem. Students, who are still struggling with operation, might benefit from learning how to use bar models to mirror the action of word problems and thus help them choose the correct operation. Although students have been working with money notation since early grades, some students are still struggling with the idea of place value. There is evidence of students thinking about digits in isolation with no place value attached. I multiplied 6 and 4 and got 24. The new level of understanding for this grade level is to start understanding decimals as representing fractional parts of a whole, not as a convenient way of segregating numbers or units. One effective way to help students learn academic vocabulary is have the teacher try to weave in definitions when using the mathematical vocabulary in communicating with students and during instruction. Students might benefit from some number talks with decimals, so that the issue of place value can be brought up with clarifying questions. Students should be comfortable using decimals in money for addition, subtraction, multiplication, and division. Often students have a logic for their misconceptions based on a faulty generalization from earlier work. There is evidence that some students think that the $32 needs to go on the outside (32 into 6.40) because the 32 is shorter than 6.40. To let go of these misconceptions, students need to confront them explicitly and see why they don t make sense. Ideas for Action Research- Exploring Place Value Students need opportunities to play with numbers and explore how they operate. One idea is to do an investigation around exact sums and differences. (from Teaching Student- Centered Mathematics by John Van de Walle). Give students a sum such as 83.53 + 7.4 + 0.649. First ask students to estimate the answer and explain how they made the estimate. Second, have students get an exact answer (without using a calculator) and explain how they got their answer. Finally ask students to generalize about how to add numbers with decimals with any two numbers. When student pairs have finished all three parts, have them share their strategies with other students and test their strategies with other numbers. This same format can be used with subtracting decimals. A second way to get students to explore numbers is to do a matching activity. Problems in the matching set are designed to bring up common errors. As students work through they activity they need to explain how and why they matched the different representations. During the activity there is nothing to discuss except the mathematics of place value. Matching activities can be set up for looking at place value or computations with decimals. Here is one example from the new web magazine, Educational Design: http://www.educationaldesigner.org/ed/volume1/issue1 36
A final choice of activities around place value is to give students 3 or 4 problems for any operation you want to explore with decimals. Plan the problem so that typical errors will occur. Then give students number lines to check their answers. 37
Fruity Fractions This problem gives you the chance to: use equivalence to write fractions in simplest form 1. Change these improper fractions into whole numbers. a. 1 1 20 5 18 9 25 5 Then use the code to find the mystery fruit. 1 2 3 4 5 6 7 8 9 P A L E R U M G S Write your answers in the boxes. Whole number Letter The name of the fruit is. b. 100 100 24 8 36 6 14 2 Write your answers in the boxes. Whole number Letter The name of the fruit is. 2. Now it is your turn to make a fraction puzzle to which the answer is GRAPES. Use the same code. Write improper fractions in the grid to make the fruit. G R A P E S 10 Grade 5-2008 Copyright 2008 by Mathematics Assessment Resource Service 38
Fruity Fractions Rubric The core elements of performance required by this task are: use equivalence to reduce fractions to simplest form Based on these, credit for specific aspects of performance should be assigned as follows 1.a. b. Gives correct answers: 1 4 2 5 or 1 4 5 2 All correct 3 points Partial credit 3 correct 2 points 2-1 correct 1 point P E A R P E R A Gives correct answer: 1 3 6 7 or 1 6 7 2 P L U M P I N A All correct 3 points Partial credit 3 correct 2 points. 2-1 correct 1 point. 2. Produces a correct grid of improper fractions using the code to give the word GRAPES. or MELON points 3 (2) (1) 3 (2) (1) section points 6 G R A P E S 8 5 2 1 4 9 All 6 correct 4 points Partial credit 5 correct 3 points. 4-3 correct 2 point 2-1 correct 1 point. M E L O N 9 4 3 8 7 All 5 correct 4 points Partial credit 4 correct 3 points. (3) 3 correct 2 points (2) 2-1 correct 1 point. (1) 4 Total Points 10 4 Grade 5-2008 Copyright 2008 by Mathematics Assessment Resource Service 39
Fruity Fractions Work the task. Look at the rubric. What are the important concepts a student needs to understand about fractions to be successful on this task? Look at student work on part. How many of your students: Worked only one or two of the fractions and then tried to make a word, rather than continue with the mathematics? Just put any in 4-letter fruit like kiwi? What other types of errors did you notice? In part 2, how many of your students: Just put down the whole number value for the letters? Rearranged the letters? Didn t use improper fractions? Made some progress on the task, but made computational errors? Do your students get opportunities to work tasks with multiple steps? Do they have frequent opportunities to make sense of problem situations for themselves? What types of struggles did you see with the mathematics? Did your struggling students show any of their computations which might allow you to see where the understanding is breaking down? What is your classroom norm around showing work? Grade 5-2008 40
Looking at Student Work on Fruity Fractions Student A is able to meet all the demands of the task. Notice that the student shows the thinking at the side of each fraction as a strategy to help think through the problem. Student A Grade 5-2008 41
Student B also has the habit of mind of showing thinking. The student actually shows the division involved in finding the whole number associated with each fraction. The student makes a computational error in part two. Student B Grade 5-2008 42
Student C has a score of 8. The student shows work by the fractions. While the student has some ideas about improper fractions, the student makes several errors. Notice that in part 1 the student changes 20/5 to 4 3/5, but because of the code was able to get the correct letter. Student C Grade 5-2008 43
Student D has a total score of 8. The student shows an understanding of improper fractions, but makes some computational errors. Student D It is unclear what the Student E understands. The student misinterprets the demands of part 3 and just writes the whole number code for each letter. This is the typical response of a 6 score. Student E Grade 5-2008 44
Student F appears to find the value of the first improper fraction and then search for 4 letter words that start with that letter. In part 2 the improper fractions have no relationship to the letter values, but just form a pattern. It is unclear what the student knows about improper fractions and what is just decoding skills about what the task is asking. What question would you ask this student? Student F Grade 5-2008 45
Student F is an example of a zero score. The student didn t understand the format of the task. The student tried to think of fruits with the given letters. Notice that in part 2 the student uses a mixture of simple fractions and improper fractions. How would you help this student? What structures allow you to give some students extra help? Student F Grade 5-2008 46
5 th Grade Task 3 Fruity Fractions Student Task Core Idea 1 Number Properties Find equivalent improper fractions and convert improper fractions to whole numbers to solve a coded puzzle. Understand the place-value structure of the base-ten number system including being able to represent and compare rational numbers. Use models, benchmarks, and equivalent forms to judge the size of fractions. Recognize and generate equivalent forms of commonly used fractions and decimals. The mathematics of this task: Reducing improper fractions to whole numbers Changing a whole number into an improper fraction Based on teacher observations, this is what fifth graders know and are able to do: How to change improper fractions to whole numbers Multiplication facts Areas of difficulty for fifth graders: Changing whole number to an improper fraction Understanding or interpreting the question in part 2 Grade 5-2008 47
The maximum score available for this task is 10 points. The minimum score needed for a level 3 response, meeting standards, is 6 points. Most students, 96%, were able to decode one or two of the improper fractions in 1a. A large number of students, 90%, were able to simplify all the improper fractions in 1a and find 1 or two of the improper fractions in part b. Many students, 87%, could simplify all the improper fractions in part 1. More than half the students could meet all the demands of the task including writing their own improper fractions for the code numbers in part 2. Less than 5% of the students scored no points on this task. 66% of the students with this score attempted the task. Grade 5-2008 48
Fruity Fractions Points Understandings Misunderstandings 0 66% of the students with this score attempted the task. 1 Students could find the value of 1 or 2 improper fractions in part 1. 4 Students could find all the values of the improper fractions in part 1 and 1 or 2 of the improper fractions in part 4. 6 Students could find all the values of the improper fractions in part 1. 8 Students could simplify improper fractions and invent improper fractions with a given value. Students could use these values to decode a word in the puzzle. Students did not connect the fractions with the problem. They tried to find 4 letter words without referring to the value of the letters or fractions. Many students just put the whole number values for the letters rather than attempting to write an improper fraction. (11% of all students) Grade 5-2008 49
Implications for Instruction While most students understood the task and performed well, a few students struggled with the basic concepts. Teachers need to devise time in their day to work more intensively with these students, so that they don t fall further behind. Research shows that putting a little extra time at the earlier grades as problems first arise, saves time and money needed for remediation at later grades. Some students may need more help than can be provided within the regular classroom structure. Measures should be in place, such as peer tutoring or after school help shops to help these struggling students. Because students didn t need to show their work on this task, some further individual interviews may need to be done to determine the root of their problems. Do students need more opportunities to work on deciphering directions and working tasks without so much detailed instruction? Are students missing some basic computational facts knowledge? Is the problem about understanding fractions? Ideas for Action Research Just in Time Intervention Sit down with your grade level team and talk about the needs of struggling students. What are possible ways of working together to help these students? Is there a time for pulling aside groups from all classes to work intensively on a particular problem? Perhaps having one teacher take the majority of students to work on a big project, freeing up someone else to do the intervention. Is there a way to structure group time to all in-class time to give some students more one-onone time to work through their misunderstandings? What are some of the things you have found effective? Share ideas and strategies. Is there a bring in a grandparent program that could be utilized to help with this problem? Is there a way to set up peer tutoring sessions one day a week after school? How could you provide training for the tutors to make it more effective? Some lesson study groups have experimented with the idea of taking small groups of students after school to preteach new concepts. They have found this to be very effective in helping some disadvantaged or struggling students. What might this look like? Is there a way of setting up an experiment where you all work together to try this idea? What other ideas might you experiment with to help all students succeed? Grade 5-2008 50