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1 Percent Cards This problem gives you the chance to: relate fractions, decimals and percents Mrs. Lopez makes sets of cards for her math class. All the cards in a set have the same value. Set A %! Simple fraction! 1. Complete these sets of cards. Decimal! Percent fraction! Percent Set B %! Set C 0.65!! 65% Set D! 3 8!! Copyright 2008 by Mathematics Assessment Resource Service 47

2 2. Show 2, 65% and on the number line below. 0!! 100% 3 4! 7 Copyright 2008 by Mathematics Assessment Resource Service 48

3 Percent Cards Rubric The core elements of performance required by this task are: relate fractions, decimals and percents Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answers: Set B 0.40 Set C 13/20, 65/100 Set D 0.375, 37.5/100 (accept 375/1000), 37.5% Partial credit One error 2. Values correctly indicated on number line. points 1 2 x 1 2 (1) section points % Partial credit One error % 3 4 (1) 2 Total Points 7 2 Copyright 2008 by Mathematics Assessment Resource Service 49

4 Percent Cards Work the task and look at the rubric. What are the big mathematical ideas being assessed in this task? What strategies might you expect students to use to find the simple fraction in Set C? Look at student work on C. How many students gave a correct answer of 13/20? Make a list of other answers. How might students have gotten these answers? Look at student work for set D. How many of your students put: /100 or 375/ %.38 38/100 38% /100 37% /100 30% /100 26% Other What types of strategies did successful students use? What misconceptions did you see as you looked at student work? Did your students show what they were thinking by writing calculations? What does it mean for something to be all the same value? Do students get opportunities to have discussions about when to round and when rounding is not appropriate? If the instructions had told students not to round off, what percentage of the students who did round do you think would be able to make sense of the number in the thousands place and make a correct representation for it? What evidence did you consider in their work? 50

5 Now look at the work on the number line. How many of your students: Could correctly place all the numbers on the number line? Added values on the line to show the scale? Redid the number conversions? How many of your students had values for 3/8 between: 0.375% Other values between 30-39% 20-29% 70-79% 80-89% % Other How many of your students had a relative order of: o 2/5, 3/8/ 65%? o 2/5, 65%, 3/8? Have your students worked with the number line this year? How can this mathematical representation be used to help students solve problems or check their work? How can this representation be used to show the meaning of operations with rational numbers? Have you ever used the double number line to help make sense of percents? What are some other uses of the number line that help students develop a deeper understanding of number? 51

6 Looking at Student Work on Percent Cards Student A uses division to convert the simple fractions to decimals. The student reduces the percent fraction in C to find the simple fraction. The student is able to place all the values on the number line. Student A 52

7 Student B uses a unit fraction approach to thinking about the conversions. If one fifth equals 2/10ths, then 2/5 equals 4/10ths. If 1/20 equals 5/100ths, then 13/20ths equals 65/100ths. Student C makes a common error by reversing the numbers in the division. The student divides the top number into the bottom number. Notice that the student changes the 3 to a 30 so that the answer will be a decimal. How do we help students master a procedure but also learn enough understanding of why it works to help them correct errors as they occur? What kind of experiences could push students thinking about the whys of the procedure? 53

8 Look at the work of Student D. The student seems to know that is equal to 2/3. Can you figure out what the student was doing for the 3/8? Student E is able to complete the division for the 3/8, but doesn t know how to work with the decimal to convert to a percent fraction or a percent? Beyond procedural knowledge, what does a student need to understand about place value and the number system to make these conversions? 54

9 Student F uses two approaches to the conversion of fractions to decimals. In Set B the student finds equivalent fractions. In Set D the student uses division. The student then shows the rounding to go from three to two decimal places. On the second page of the task the student doesn t necessarily connect the values to those on the previous page and reconverts the fractions to decimals and percents. The student can place the fractions correctly on the number line, but misplaces 65%. What might have confused the student? Student F 55

10 Student F, part 2 56

11 Student G adds marks to the number line to scale the values between 30 and 40%. Student G Student H numbers all the lines on the number line to show a scale in percents. The student uses an incorrect value from page 1 of the task for 3/8, but places that value correctly on the line. Student H 57

12 Student I has placed the values in the correct relative order, but does not connect the values to the line. Student I Student J has misplaced the fractions on the number line. The student had correctly identified 2/5=0.40 for Set B and had a value of 3/8 =0.22 for Set B. Notice that neither of these values account for the positions on the number line. What conjecture can you make for the location of the fractions made by the student? Student J Student K has made the common measurement mistake of counting using the 0 as one. All the locations are off exactly 10%. Student K 58

13 6 th Grade Task 4 Percent Cards Student Task Core Idea 1 Number and Operation Relate fractions, decimals and percents. Use number line to locate and compare percents, fractions, and decimals. Understand number systems, the meanings of operations, and ways of representing numbers, relationships, and number systems. Understand fractions, decimals, and percents as parts of unit wholes and as parts of a collection. Recognize and generate equivalent forms of commonly used fractions, decimals, and percents. Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line. The mathematics of this task: Understanding decimal place value and working with values in the thousandths place Converting between representations using fractions, decimals and percents Understanding equivalency Using scale to place values on a number line Based on teacher observations, this is what sixth graders know and are able to do: Convert decimals to percents or percents to decimals Convert decimals to percent fractions and percents Understand the idea of a percent fraction Areas of difficulty for sixth graders: Locating fractions on a number line Reducing 65/100 to a simple fraction Converting fractions to decimals for values smaller than 100ths Understanding equivalency, same value Understanding the relationship between percents on the number line and the value of the fractions Strategies used by successful students: Using calculations from part 1 Marking some other benchmark numbers on the number line, like 1/2, 50%, or 10% 59

14 The maximum score available on this task is 7 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Most students, 94%, could change 2/5 into a decimal and change 0.65 into a percent fraction. Many students, 69%, could also give a simple fraction for 65%. About half the students could find the decimal for 2/5, the fraction and decimal fraction for 65%, and place both these values on the number line. Almost 14% of the students could meet all the demands of the task including changing 3/8 to equivalent decimal and percent values and place 3/8 on the number line. About 3% of the students scored no points on the examine. All the students in the sample with that score did not attempt the task. 60

15 Percent Cards Points Understandings Misunderstandings 0 All the students in the sample with this score did not attempt the task. 2 Students could change 2/5 into a decimal and change 0.65 into a percent fraction. 4 Students could find the decimal for 2/5, the fraction and decimal fraction for 65%, and place both these values on the number line. 7 Students could comfortably convert between simple fractions, decimals, percent fractions and percents. Students could work with values in the 1000ths place. Students could place decimals and fractions on a number line scaled in percents. Students had difficulty changing the 65/100 to a simple fraction. About10% of the students put 2/5 between 20 and 30% on the number line. About 7.5% put 2/5 between 40 and 50%. About 7% put 2/5 between 10 and 20%. Students struggled with 3/8. Almost 9% did not finish the division or rounded to 0.37%. About 7.5% rounded up to 38%. 5% of the students divided 3 into 8 to get 26%. 5% were able to give the decimal of 0.375, but then used 37/100 for the percent fraction. Students also struggled with how to place 3/8 on the number line. Almost 18% had a relative order of 2/5, 65%, 3/8. 9% had a relative order of 2/5, 3/8, 65%. 13% of the students placed 3/8 between 80 and 90%. 10% of the students placed 3/8 between 20 and 30%. 61

17 Area and Perimeter This problem gives you the chance to: work with area and perimeter of rectangles 1. The perimeter of this rectangle is 2(5 + 2) = 14 inches. The area of this rectangle is 2 x 5 = 10 square inches. 2 inches a. Draw a diagram of a rectangle with the same perimeter, but a larger area. Write down the area of your rectangle. 5 inches b. Draw a diagram of a rectangle with the same perimeter, but a smaller area. Write down the area of your rectangle. 2. The perimeter of this rectangle is 22 inches. The area of this rectangle is 24 square inches a. Is it possible to draw a rectangle with the same area as the one on the right, but a larger perimeter? Explain your reasoning. 3 inches 8 inches b. Is it possible to draw a rectangle with the same area, but a smaller perimeter? Explain your reasoning. Copyright 2008 by Mathematics Assessment Resource Service 10 63

18 Area and Perimeter Rubric The core elements of performance required by this task are: work with area and perimeter of rectangles Based on these, credit for specific aspects of performance should be assigned as follows point s sectio n points 1.a. Draws a rectangle with sides such as: 3 inches x 4 inches area = 12 square inches 1 1 b. Draws a rectangle with sides such as: 1 inch x 6 inches area = 6 square inches a. Gives correct answer: Yes and Gives correct explanation such as: Area 2 x 12 = 24, Perimeter 2(2 + 12) = 28 inches or Area 1 x 24 = 24, Perimeter 2(1 + 24) = 50 inches 3 or 3 Partial credit Allow partial credit for a partially correct answer. (2) b. Gives correct answer: Yes and Gives correct explanation such as: Area 4 x 6 = 24, Perimeter 2(4 + 6) = 20 inches 3 Partial credit Allow partial credit for a partially correct answer. (2) 6 Total Points 10 Copyright 2008 by Mathematics Assessment Resource Service 64

19 Percents Work the task and look at the rubric. What does a student need to understand to be successful on this task? Look at student work for part 1. How many students gave answers, such as: A complete response with rectangles of correct dimensions that fit the constraints of the same perimeter and a larger or smaller perimeter and quantifified the area for the new shape? Gave rectangles with correct dimensions but didn t give a value for area? Gave rectangles that had larger or smaller areas, but also changed the perimeter? Used the same rectangle of 2 x 5? Gave shapes that were not rectangles? What misconceptions did you see in student work? Now look at the work for part 2a and 2b. How many students gave answers, such as: Used correct dimensions, showed that area is the same, and calculated the new perimeters? Used correct dimensions but didn t quantify why or how they fit the constraints? Thought one was possible and the other was not possible? Thought changing the area would change the dimensions? What other types of errors did you see in students thinking? How do you communicate values around justifying an answer in your classroom? Do students know that quantifying is an important mathematical value? How often do students have the opportunity to do investigations in your classroom? What misunderstandings or habits of mind prevented students from making attempts to find solutions in part 2? Although incomplete, what pieces of thinking showed that students had some understanding of the concepts? What would need to be done to these answers to get a complete solution? How could you use this idea to develop a class discussion? 64

20 Looking at Student Work on Area and Perimeter Student A is able to find rectangles to meet the constraints. The student is able to put numbers into a formula to calculate perimeter and also calculates the area. Notice that the student uses greater than or less than to compare new calculations to originals to complete the proof. Student A 65

21 Student B is able to meet the demands of the task for part 1. In part 2 the student shows an understanding of how to change the shape and restrictions about what types of numbers can be used for the dimensions, but doesn t use specific examples to complete the argument that it is possible to change perimeter without changing the area. Student B 66

22 Student C is able to find rectangles to justify part 2. However the student does not notice the constraint of keeping the perimeter the same, when solving part 1. How do we help students develop the habit of mind to look for all the constraints and check that they have been met? Student C 67

23 Student D is able to find correct dimensions for rectangles to meet the criteria for part 1. In part 2 the student uses different logic for part a and b. In part a the student has the common misconception that a larger perimeter must have a larger area. In part b the student knows that the dimensions need to be factors of 24, but only tests one case which doesn t work. What might be next steps for this student? Student D 68

24 Student E makes nonrectangular shapes in part 1, but still uses the formula for area of a rectangle. How do students develop the connection between shape and possible dimensions? What types of experiences does this student need? In part 2 the student can do part 2b, but doesn t think the dimensions can be changed in 2a. Why do you think students have these kinds of inconsistencies? Student E 69

25 6 th Grade Task 5 Area and Perimeter Student Task Core Idea 4 Geometry and Measurement Work with area and perimeter of rectangles. Investigate how change linear measures effects area and perimeter. Analyze characteristics and properties of two-dimensional geometric shapes. Apply appropriate techniques, tools, and formulas to determine measurements. Develop, understand, and use formulas to determine area of quadrilaterals. Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes. The mathematics of this task: Willingness to investigate different cases to see if something is possible Identifying constraints See relationships between dimensions and the measurements of area and perimeter Quantify measurements in order to make comparisons Understanding which dimensional measurements will and will not make a rectangle and why Understanding that rectangles have two sets of parallel sides with the same side length for each set Calculating area and perimeter Understanding the logic of justification Understanding and tracking units Based on teacher observations, this is what sixth graders knew and were able to do: Rectangles have four sides Know formulas for area and perimeter and how to use them to calculate Draw rectangles and give appropriate dimensions Areas of difficulty for sixth graders: Confusing perimeter with diameter, so they kept one dimension the same and changed to height to get the new areas Conducting an investigation. Some students did not believe it was possible to keep the perimeter the same and change the area, but they did not try numbers to test their conjectures. Applying knowledge to a complex problem Understanding that opposite sides of a rectangle needed to be the same size Making connections between area and factors to develop possible side lengths Using all the constraints in the problem Changing the wrong measurement Thought more square-like shapes were no longer rectangles Thought that making changes in one of the dimensions would change both measurements Strategy used by successful students: Thinking about factors to determine possible dimensions for the rectangle 70

26 The maximum score available for this task is 10 points. The minimum score needed for a level 3 response, meeting standards, is 5 points. About half the students could give the dimensions for rectangles with the same perimeter and larger or smaller areas. Some students, about 39% could also calculate the area of the new rectangles. A few students, 30%, could give the dimensions for rectangles with the same perimeter and larger and smaller areas, calculate the areas for the new rectangles, and give the dimensions for a rectangle with the same area and either a larger or smaller perimeter. Almost 14% of the students could meet all the demands of the task including giving the dimensions for a rectangle with the same area and larger and smaller perimeters and justify the dimensions by giving the new perimeters. Almost half the students scored no points on this task. 81% of the students with this score attempted the task. 71

27 Area and Perimeter Points Understandings Misunderstandings 0 81% of the students with this score attempted the task. Students could not give dimensions for rectangles with the same perimeter. Students gave dimensions for shapes that were not rectangular. Students gave shapes 2 Students could give dimensions for rectangles with the same perimeter but larger and smaller areas. 4 Students could give dimensions for rectangles with the same perimeter but larger and smaller areas and justify their solution by calculating the new areas. 5 Students could give dimensions of new rectangles, for parts 1 and 2, but in general did not calculate the area or perimeter for the new shapes. with dimensions that did not add to 14. Students did not justify their solution by calculating the new areas. Students did not understand how to change the perimeter while maintaining the same area. 14% said that if the area is larger, then the perimeter must be larger. Another 8% said that if the area stays the same, the perimeter will stay the same. 10% thought it was possible, but did not attempt to give the new dimensions or test their conjectures. Students don t understand the logic of justification. Once they have a dimension they are done with the task. 8 Students did not calculate the perimeters in part 2 of the task. 10 Students could investigate hypotheses, such as maintaining perimeter and changing the area, and then calculate the values for area and perimeter of the new shape to verify the hypothesis. 72

29 Several students use the same dimensions in part one as the sample rectangle. Why might they do this? What aren t they grasping about the prompt? Some of the student reasoning is based on beliefs about the effects of operations on numbers. What is correct and incorrect in their understanding? After examining and discussing all the student work, how might you plan a small mini-unit to dig into the misconceptions and help students explore area and perimeter to develop a deeper level of understanding? Which misconceptions should be dealt with explicitly? Which misconceptions should be dealt with through investigation and direct experience? 74

30 Student 1 75

31 Student 2 Student 3 76

32 Student 4 Student 5 77

33 Student 6 Student 7 78

34 Student 8 Can you figure out what the student is calculating in part 1. In 2b the student gives the perimeter for the correct rectangle, but no dimensions. What do you think the student understands? 79

35 Student 9 Student 10 80

36 Student 11 Student 12 81

37 Ideas for Action Research 2 Problem of the Month One interesting task to help students stretch their thinking about 3-dimensional shapes is the problem of the month: Surrounded and Covered, from the Noyce Website: Ask students to work individually or in teams to solve the problem. Have them make posters of calculations they made, their conclusions, and graphics or visuals to support their thinking. The poster might also include other ideas they want to explore or conjectures they haven t had time to test. The purpose is to give them some complex mathematical thinking, that requires persistence, willingness to make mistakes, edit and revise, and is worth understanding the thinking of others. Now give students a chance to investigate part D of this task. 82

38 Reflecting on the Results for Sixth Grade as a Whole: Think about student work through the collection of tasks and the implications for instruction. What are some of the big misconceptions or difficulties that really hit home for you? If you were to describe one or two big ides to take away and use for planning for next year, what would they be? What are some of the qualities that you saw in good work or strategies used by good students that you would like to help other students develop? Five areas that stand out for the Collaborative as a whole are: 1. Making comparisons Students have difficulty understanding what makes a good comparison mathematically. In Snail s Pace students often made a logical argument or gave a rate for the fastest snail, ignoring the speeds of the other snails. Students might have made a logical comparison between the two top speeds without converting them into equivalent units like inches per hour or inches per 20 minutes. Students did not understand that a good comparison would have used a similar unit for all the snails. In Area and Perimeter students did not think about finding the area or perimeter of the new rectangles so that both could be compared. Many just stated the prompt, such as same area with larger perimeter, with no quantification. Others just gave dimensions with no discussion or back up for how that completed the argument. 2. Understanding academic language of logic and probability Students did not understand the language or logic of a negative statement, such as not black. While this type of statement is not often used in everyday language, it is a typical form of statement in mathematics. 3. Making sense of all the relevant features of a pattern Students had trouble locating all the important features in A Number Pattern. Students might have thought of multiples, if they only looked at row five. Some students didn t notice that every row started and ended with a one. Some students didn t notice that the middle number(s) were the largest. Some students knew to add the above numbers to get the next row, 83

39 but didn t spend enough time noticing where the sum went in the row below. Some students were looking for the simplest possible patterns, like a sequence of even and odd numbers. Students need to develop the habit of mind to test their conjectures against all available evidence. 4. Seeing relationships between different representations of numbers Students did not know what to do with decimals that did not come out evenly at the hundredths place. They didn t understand that rounding gives an approximate answer but not the same or equal value as the original. Students did not know how to locate fractions on the number line. Students were often ordering fractions by the size of the denominator: 3/4, 2/5, then 3/8. 5. Making and testing conjectures Students had difficulty testing conjectures about area and perimeter. Students often neglected some of the constraints. Students didn t calculate the areas and perimeters to compare back to the original rectangles and justify that the conjecture had been fully met. Students didn t test different size rectangles before trying to write a generalization. 84

40 Examining the Ramp: Looking at Responses of the Early 4 s (31-33) The ramps for the sixth grade test: Snail Pace Part 4 Making a comparison using equal rates o Fixing the distance to compare times o Fixing the time to compare distance o Interpreting calculations to find the fastest rate Black and White - Part 2b Understanding a negative probability Number Pattern - Part 3b and 4 explaining patterns in words o Recognizing a doubling pattern o Looking at a pattern growing by increasing consecutive numbers Percent Cards Part D Understanding equivalence o Division with decimals beyond hundredths o Understanding that rounding gives an approximate value not an equivalent value Part 2- Locating fractions on a number line o Seeing relationships between previous work and the values to be placed on the number line o Understanding relative size of fractions Area and Perimeter Part 2 Keeping area the same and changing the perimeter o Conducting an investigation o Quantifying values of perimeter and area to verify conjecture With a group of colleagues look at student work around points. Use the papers provided or pick some from your own students. How are students performing on the ramp? What things impressed you about their performance? What are skills or ideas they still need to work on? Are students relying on previous arithmetic skills rather than moving up to more grade level appropriate strategies? What was missing that you would hope to see from students working at this level? When you read their words, do you have a sense of them understanding ideas personally or does it sound more like they are parroting things they ve heard the teacher say? How do you help students at this level step up their performance or see a standard to aim for in explaining their thinking? Are our expectations high enough for these students? For each response, can you think of some way that it could be improved? How do we provide models to help these students see how their work can be improved or what they are striving for? Do you think errors were caused by lack of exposure to ideas or misconceptions? Grade

41 Student 1 What would a student need to fix or correct their errors? What is missing to make it a top-notch response? What concerns you about their work? What strategies did you see that might be useful to show to the whole class? Grade

42 Student 1, part 2 Student 2 Grade

43 Student 2, part 2 Grade

45 Student 3, part 2 Student 4 Grade

46 Student 4, part 2 Grade

47 Student 4, part 3 Student 5 Grade

48 Student 5, part 2 Grade

49 Student 5, part 3 Grade

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