a. How many cups of peas does she need? b. How many cups of milk does she need?

Transcription

1 Pea Soup This problem gives you the chance to: use proportional reasoning This is a recipe for Pea Soup. Pea Soup For 2 people 1 cup of peas 2 cups of milk 1 onion Seasoning 1. Carla makes pea soup for 6 people. a. How many cups of peas does she need? b. How many cups of milk does she need? 2. The graph on the opposite page shows how many cups of peas Carla needs to make the soup for different numbers of people. a. Explain how to use the graph to find how many cups of peas Carla needs for 10 people. b. Complete this table to show the numbers of cups of milk Carla needs for different numbers of people. Number of people Number of cups of milk c. Mark the points in the table on the graph and join the points with a straight line. 51 Copyright 2008 by Mathematics Assessment Resource Service

2 d. What is the difference between your line and the Number of cups of peas line? e. Explain why the line for the number of onions Carla needs is the same as the line for the number of cups of peas Quantity needed Number of cups of peas Number of people Copyright 2008 by Mathematics Assessment Resource Service

3 Pea Soup Rubric The core elements of performance required by this task are: use proportional reasoning Based on these, credit for specific aspects of performance should be assigned as follows points section points 1a. Gives correct answer: 3 1 1b. Gives correct answer: a. Gives correct explanation such as: Draw vertical and horizontal lines from point to axes. Partial credit For a partially correct answer. 2b. Table correctly completed. 2 (1) 2 2c. Partial credit For 2-3 correct numbers in table. Points plotted and line drawn correctly. Partial credit For 1 error. 2d. Makes a correct statement such as: My line is steeper. 1 2e. Gives a correct explanation such as: The number of onions needed is always the same as the number of cups of peas. (1) 2ft (1) Total Points Copyright 2008 by Mathematics Assessment Resource Service 53

4 Pea Soup Work the task. Look at the rubric. What are the big mathematical ideas of this task? Look at student work for part 1 of the task. For the amount of peas, how many of your students put: Other For the amount of milk, how many of your students put: Other What is some of the logic behind these errors? What were students thinking? What didn t they understand? Now look at the student work on explaining how to use the graph. How many of your students could: Give an explanation on how to use the graph? Gave an explanation about how to find the answer another way? What other errors or misconceptions did you see? How many of your students did not plot the graph? Made errors in plotting the graph? What opportunities do students have to make their own graphs about functions? Make a list of the types of words students used to describe the difference in the lines on the graph. Which words describe the steepness of the slope? Which words did students use that don t give enough information, like the numbers are different numbers? Which words talk about nonmathematical ideas, like flavor of the soup? How often do students in your class practice making meaningful comparisons? 54

5 Looking at Student Work on Pea Soup Student A shows the thinking for increasing the recipe for part 1 by identify the scale factor. The student alludes to slope in part 2, although doesn t really compare the lines globally. The explanation in part 5 is concise and right on the mathematical point. Student A 55

6 Student A, part 2 56

7 Student B dismisses the graph in part 2 and gives a different method. Unfortunately even the alternative method is not relating people to peas. In part 2e, the student talks about the size of an onion compared to a cup rather than thinking about the ration 1 for every 2 people. When doing part 2e, the student changes the labels on the graph from milk to onions. Student B 57

8 Student C is interesting because she appears to learn as she moves through the parts of the task. In part 1 she answers as if the original recipe is for one person. This thinking continues in part 2. However by 2e the student understands that the ingredients are 1 for two people. Some researchers suggest that learning occurs during an activity. What do you think? Student C 58

9 Student D pays attention to the wrong details in the recipe. In part 1 the student seems to think of the milk as 1 more cup than the amount of peas, instead of thinking across measures: 2 cups of milk for 2 people. The student is able to make the graph correctly, but does not understand the information being conveyed. In describing how to use the graph the student only talks about how the number of people grows across, leading to an incorrect statement in 2. Part 2e is scored incorrectly. The explanation for the number of onions should be because the ratio is the same 1 for 2, not because of flavor. Although, maintaining the ratio will contribute to the flavor staying the same. Student D 59

10 Student E has an incomplete explanation of using the graph in part 2. While the student was able to make a correct table of value for milk, the student does not enter the values correctly on the graph. Why might the student have made the line parallel? Student E 60

11 Student E, part 2 61

12 Student F also has difficulty with graphing. The student answers all the questions correctly, including filling out the table of values, but then draws vertical lines for the graph. What might the student have been thinking about? Student F 62

13 Student G does not understand the recipe in part 1, but is still able to complete the table in part 2. In part 2a the student is able to use the correct relationship to find the number of peas, but doesn t go back to fix the answers above. The student appears not to read for details carefully, because in part 2d the student seems to think that the table is for number of peas. The student never notices that the numbers in the table are different from the numbers in the graph. What might help this student?(the student shows no work on the graph.) Student G 63

14 Look at the work for Student H. Do you think the student has identified the correct 2 in the table? Why or why not? What patterns do you see in the table? Can you find a logic for how the student made the graph? What do you think this student understands? Student H 64

15 Student H, part 2 65

16 Look at the work of Student I. Notice that the student has two patterns in the table; one pattern for the first 3 columns and a different pattern for the second 3 columns. What do you think the student doing? Where would you go next with this student? Student I 66

17 Student I, part 2 67

18 5 th Grade Task 4 Pea Soup Student Task Core Idea 3 Patterns, Functions and Algebra Use proportional reasoning to increase the size of a recipe. Explain how to read a line graph and graph data from a table. Compare slopes of lines on a graph. Understand patterns and use mathematical models such as algebraic symbols and graphs to represent and understand quantitative relationships. Express mathematical relationships by graphing them on a coordinate grid. Investigate how a change in one variable relates to a change in a second variable. The mathematics of this task: Reasoning about a proportional relationship to increase a recipe Being able to explain how to read information from a graph Graphing coordinate pairs Comparing slope of lines on a graph Comparing ratios of two ingredients Based on teacher observation, this is what fifth graders know and are able to do: Plot points on a graph Use a ratio to increase a recipe Areas of difficulty for fifth graders: Explaining how to read a graph; many gave alternative strategies to find the same information Writing a comparison about the two lines on the graph, missing the vocabulary about steeper or larger Using comparison language, instead giving information about one or the other Understanding how to compare onions to peas by ratio rather than volume 68

19 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, 93%, could complete the table for milk. Many students, 72%, could increase the recipe for peas, fill in the table, and graph the information on the table. More than half the students could also increase the recipe for milk in part 1. Almost half, 43%, could also explain how to read the graph. Almost 18% of the students could meet all the demands of the task including explaining how the ratio for onions and peas was the same and compare the slopes of the two lines on the graph. About 4% of the students scored no points on this task. 66% of the students with this score attempted the task. 69

20 Pea Soup Points Understandings Misunderstandings 0 66% of the students with this Students could not extend the pattern in the score attempted the task. table. 5% put 2,3,4,5. 5% put 6,8,10,12. 2 Students could fill in the table for milk. 4 Students could fill in the table and extend the recipe in part 1. 5 Students could extend the recipe, fill in a table and explain how to read a graph. 6 Students could extend the recipe, fill in a tabl, and explain how to read a graph. Students could also explain that both onions and peas and 5% put 8, 12, 16, 20. Students had difficulty extending the recipe in part 1. Almost 22% of the students thought there would be 6 cups of peas and 12 cups of milk. 8% thought there would be 2 cups of milk. 6% thought there would be 3 cups of milk. And 5% thought there would be 4 cups of milk. Students struggled with explaining how to read the graph. Many tried to explain some other method for finding the number of peas. Students struggled with explaining that the ratio of onions to people is the same as the ratio of peas to people (1 to 2). Students didn t know how to make the graph. 17% of the students did not attempt to plot the points. are 1 for 2 people. 8 Students struggled with comparing the two lines on the graph. They didn t have the vocabulary of steeper or larger. They talked about one line or the other, without using comparative language. 10 Students could extend the recipe, fill in a table and explain how to read a graph. Students could also explain that both onions and peas are 1 for 2 people. They could compare slopes of lines on a graph and the ratios of onions and peas. 70

21 Implications for Instruction Students need to work with simple proportional relationships and extend the pattern. For example if 1 cup of peas is needed for 2 people, then 3 cups is needed for 6 people. Students should be exposed to tools, such as ratio tables and equivalent fractions to help them reason about increasing or decreasing ratios. Some students are paying attention to the wrong information in the ratio, for example, noticing that the milk is one larger than the peas. Students need to look across ratios to see that there are 2 cups of milk for every two people. Modeling good vocabulary for describing these multiplicative relationships help students to develop academic vocabulary and focuses their attention on the essential mathematical relationships. Sometimes we want students to notice a variety of patterns, the more the better. However, in this case, the attention needs to be across the measures. Students at this grade level should have many opportunities to plot data on graphs. Students should be comfortable explaining how to read a graph, using a graph to find solutions, extending a graph to find solutions, and discussing the shape or slope of the graph in general terms. Students should work with a variety of patterns to eliminate false generalizations, such as all graphs go through zero or all graphs go up in a one to one pattern. Ideas for Action Research Investigating Graphs Exploring Multiple Representations Students at the grade level would benefit from investigating graphs and seeing a purpose for using them to solve problems. While it is not important for them to know the academic language of slope, they should be able to notice that some graphs are steeper or flatter than others. Think about some situations, like buying t-shirts where they can combine proportional reasoning and graphing. For example, say Mack s sells t-shirts 2 for $7 and Norwalk s sell the same shirts at 3 for $10. After they compare the two graphs have them look at a third store: Murve s is having a sale on t-shirts: Buy one for $4, each addition is $3. Ask them to notice what is different about this graph. Now think about using graphs to make predictions. Which one do they think would be cheaper for buying 10 shirts or $20 shirts? Have them try to base their arguments by looking at the graph? If they didn t want to make the calculations, how could they find the cost for each store of buying the 10 shirts. The choice of numbers was purposeful to show steeper lines, to show lines that cross or have the same value for some amount of shirts, and to show that not all lines would continue to 0. For this situation a 0 shirt doesn t make sense. It would be good to think of situations where extending the line to zero would make sense. The problem was also designed to help students see that graphs could help them solve problems. They should also notice when one choice is cheaper than another choice. The MARS task 2007 Candies is also a proportional reasoning problem. After you have done some work with graphing, you might have the class work this task, then use graphing to check their results. Functions could also be used to point out some of these ideas without using context. Students don t need to know the rules, but should play with how the numbers effect the shape of the graph. 71

22 Bar Charts This problem gives you the chance to: interpret and construct bar charts Number of families Here is a bar chart about the number of children in families. The bar to show 2 children per family is missing. 1. Draw the missing bar so that the total number of families is 50. Show how you figured it out Number of children per family Number of families Draw the missing bar so that the mode of the number of children per family is 1. Explain why there are several different possible answers Number of children per family 72 Copyright 2008 by Mathematics Assessment Resource Service

23 Draw the missing bar so that the total number of children is 104. Show how you figured it out Number of children per family 73 Copyright 2008 by Mathematics Assessment Resource Service 8

24 Bar Charts Rubric The core elements of performance required by this task are: interpret and construct bar charts Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Shows correct answer: Bar of height 18 units Shows work such as: = = 18 Partial credit For 1 error 2. Shows correct answer: Bar of height < 12 units Gives correct explanation such as: Because the bar can be any height less than 12 families. 3.. Gives correct answer: Bar of height 19 units Shows correct work such as: 0 x x x x 6 = = Partial credit For 1 error 1 2 (1) (1) 3 Total Points Copyright 2008 by Mathematics Assessment Resource Service

25 Bar Charts Work the task. Look at the rubric. What are some of the big mathematical concepts that a student needs to understand to be successful on this task? Look at student work for part 1. How many of your students put: Added to more than 1 bar Other What were some of the student misconceptions? Now look at student work for part 2. How many of your students put: bar< 12 No bar Added to more than 1 bar Other Now look at some of the reasons for why there is more than one answer. How many of their answers were: Misunderstanding of mode (e.g. every bar needs to be a different height or things can be divided in half): About life rather than math (e.g. babies are born all the time or different families have different numbers of children): Almost right but forgot scale so omitted 11: Other reasons Make a list of some of the errors: What surprised you about their reasons? Why do you think this was so difficult? Now look at work for part 3. How many students put: 19 No bar Added to all bars 72 Other What types of misconceptions led to these errors? 75

26 Looking at Student Work on Bar Charts Student A is able to find the number of families needed to make the total families equal to 50 in part 1. In part 2 the student lists all the possible answers for the bar height. This is possible because there can t be fractional families. In part 3 the student shows the multiplication of the families times number of children to find the children needed to make 104 and then divides that by 2 because each family in the missing bar has 2 children. Student A 76

27 Student A, part 2 Student B uses the idea that the mode is the highest bar. 77

28 Student C explains the mode by showing the frequency for each value on the bar chart. Notice that the student writes a working definition of mode to go with his work. Also notice the use of labels in part 1. Student C 78

29 Student D has correct work on all parts of the task except the explanation in 2. The student forgets about the scale and doesn t consider that 11 is also a possibility for 2 families. Student D Student E concentrates on the word average and doesn t think about the frequency of each number. How do we help students to distinguish between number as a category, like children in a family, and number as a frequency, like number of this type of family? Student E 79

30 Student F talks about the fact that there are many answers that equal 1. Can you figure out what the student is doing in the work? Student F attempts to calculate the missing number of children but makes arithmetic errors. The student also forgets to divide by 2, but doesn t seem disturbed that her answer is larger than the available room on the graph. Student F 80

31 Student G does not think about the meaning of mode. The student sees that 1 could appear in different places on the graph, because there is more than one bar, so the student is thinking about a frequency of 1, not a mode of 1. Notice how the student copes with the short scale in part 3 for his solution. Student G 81

32 Student H adds more 1 s to the graph to make it the mode. What is the conceptual misunderstanding of this student? Do we overemphasize a short version of the definition? Notice that the student counts the families with 0 children as children. The student does not consider how to apply the scale for the number of children to the height of the bar in part 3. What is the student thinking when subtracting all the 32 s? Student H 82

33 Student I thinks there are multiple answers because the scale allows you to draw bars of several different heights. The student is thinking about scale instead of mode. Notice that the student redrew the graph to conform to his answer. Student I 83

34 Student J seems to have an invented algorithm of just dividing by 2 until the answer will fit on the graph. Again, the explanation in part 2 has nothing to do with mode, but is about the ability to divide and get different answers. Student J 84

35 Student K doesn t count the families with 0 children when adding up the total families. Again the explanation in part 2 has nothing to do with mode. In part 3 the student experiments with several computations until she gets an answer that seems suitable to fit in with the other bars. Can you see in sense-making in the work? Student K 85

36 Student K, part 2 86

37 Student L increases the size of the graphs to fit his interpretations of the questions. In part two the student expresses confusion on mode; maybe it means this or maybe it means something else. Student L 87

38 Student L, part 2 88

39 Student M does not know what the question is asking and answers his own unique question in part 1. What might the student be thinking in part 2?Notice that the student adds to all the bars to try and get 104 families rather than 104 children. What might be the next steps for this student? Where would you start? Student M 89

40 5 th Grade Task 5 Bar Charts Student Task Core Idea 5 Data Analysis Interpret and construct bar charts. Use statistical measures, such as mode, in a problem situation. Display, analyze, compare and interpret different data sets. Use measures of center (mean, median, and mode) and understand what each does and does not indicate about the data set. Represent data using bar graphs. Interpret data to answer questions about a situation. The mathematics of this task: Reading and interpreting a scale that is not going up by ones Understanding that the vertical height is a frequency Understanding the double meaning of numbers on the horizontal scale, both as a label (type of family) and as a value or scale factor (each family represents a number of children per family) Understanding mode in the context of a graph, understanding mode as the most meaning highest bar rather than the number labels on the bottom or side scale Calculating total families and total children from a graph Based on teacher observation, this is what fifth graders knew and were able to do: Create a bar to represent information Create a bar that is less than the modal bar Basic computations Calculate the number of families Areas of difficulty for fifth graders: Explaining mode or most in the context of a graph Understanding that the height of the bar for families with 0 children wouldn t be used to find the number of children represented on the graph Applying the scale for children to the frequency to find the total children on the graph On the final graph remembering that the number of children remaining needed to be divided by 2 because each family on that bar had two children 90

41 The maximum score available for this task is 8 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Most students, 84%, could draw a bar lower than the mode in part 2. More than half the students could find the total number of families and find and graph the remaining families needed to make the graph represent a total of 50 families. Almost half the students, 46%, could both draw the lower bar in part 2 and do all of part one, finding and graphing the families to make a total of 50 on the graph. About 15% of the students could reason about the number of children on the graph combining frequency and scale to make the graph represent 104 children in part 3. 10% of the students could meet all the demands of the task including explaining why there could be bars of different sizes and still maintain a mode of 1 on graph 2. 16% of the students scored no points on this task. 81% of the students with this score attempted the task. 91

42 Bar Charts Points Understandings Misunderstandings 0 81% of the students with this score attempted the task. Students struggled with the word mode in part 2 and how to draw the graph. 17% of the students did not draw a bar in part 2. 4% drew a bar with a height of 18. 3% added to height of all the bars in part 2. 3% 1 Students could make the height of the new bar in part 2 smaller than the height of the bar for families with 1 child. 3 Students could calculate the number of families on a graph and draw a bar to make the total families equal Students could also draw a bar with a height less than the mode. 7 Students could show the number of children needed to make 104 children on the graph. 8 Students could find the total families, find the total children, and reason about mode in the context of frequency on a graph. Implications for Instruction thought the height should be 12. Students struggled with finding the total families in part 1(32) and then calculating the number of additional families needed to make 50. 5% put a bar of 20, 4% put a bar of 8. 3% put a bar of 22 and 3% put a bar of 16. Some students didn t count the families with no children. Students did not know how to find the number of children. Almost 15% of the students thought the bar should be at 22(highest number on the scale). 3% expanded the graph to 72. 8% added additional squares to all the bars in part 3. Other popular answers were 18, 14, 6, and 12. Students could not explain why there were several answers to make the bar for 2 families and maintain a mode of 1. Students used real life thinking: babies are born all the time. Students thought about doing math problems: there s lots of ways to get an answer of one. Students were confused about what part of the graph needed to be the most : there is more than 1 bar on the graph. Students confused mode with scale: there are other heights that haven t been used 16, 18, 20,

43 Students need more opportunities to think about measures of center in the context of graphing. Students had a very difficult time reasoning about mode. Sometimes students seemed to have a definition that was too short, mode means most ; but they couldn t interpret what about the graph should be the most. In teaching academic language it is important that students get a complete definition, not a short gimmick for quick responses. Oversimplification causes problems in understanding when trying to work with the idea to solve problems. This idea of frequency in the graph is different than frequency in the context of a string of data. It is important for students to be able to connect the two representations. One of the best ways is to give students the opportunities to make their own graphs to see how the two fit together. Making their own graphs also helps students to see that the number on the horizontal scale has two meanings. First the number is a label, the type of family represented by that particular bar, families with 2 children. It is also a scale factor for multiplying enlarging the frequency. Children equal the number of families times the scale. Students had difficulty interpreting the 0 bar on the graph. They might not count the bar when finding the number of families and then add in the bar when calculating the number of children. Students need to be asked a wide variety of questions about graphs to probe their understanding and where see where it breaks down. Students might make graphs of the number of pets per household or the numbers of tables in the cafeteria and children per table to help them think about numbers as labels and as scale. Similar MARS tasks include: 5 th 2000 Chips, 5 th Washington St. and 6 th 2001 Ice Cream. Ideas for Action Research Writing a Re-engagement Lesson One of the effective tools developed by MAC through Action Research is the idea of reengagement, or using student work to plan further lessons. After the initial task the teacher or a group of teachers sits down to examine student work. As teachers read several papers, a story emerges. Sometimes the story is about strategies used by successful students. Sometimes the story is about the different demands in working with a concept in context versus being given the information or data. Other times the story is about a common recurring misconception. Sometimes the story is about the difference in cognitive demand in the task, such as finding derived measurements, i.e. finding the number of families or children represented on a graph or how the mode or frequency seems different than mode in a string of data. Sometimes the story is about recognizing significant digits, understanding remainders, understanding a big mathematical idea, such as what an average does or how to make a comparison. In this part of the tool kit, a larger sample of student work has been provided to give you an opportunity to develop your own re-engagement lesson. Sit down with a group of colleagues to examine the work on Bar Charts or look at your own student work. What are the important mathematical ideas that students are struggling with in this task? What do you think is the story of this task? Now think, what is the minimum that most students in the class can understand and think about a series of 3 to 5 questions to help move student thinking to the more desired goal. What pieces of student work could you use to pose the questions? It helps to personalize the questions by attaching student names (not names from your class, because you don t want to put someone on the spot) or by posing a dilemma to get students intrigued about finding the solution. 93

44 For example: Julie says, I don t understand how graph number one. There isn t room. The scale only goes to 22. Rhonda says, Oh, I solved that by just making all the bars taller. What advice would you offer to these students? Or Ken says, I am confused about the mode. There is more than 1 way to get an answer of one. Melinda adds, Right, there are several answers because 1 can be in different places on the graph. What do you think these students are talking about? Clara says, It wouldn t work if there was only 1 one. Mode means most. Alex says, I am still confused. One could be the most bar lines have 1 family high or most of the families could have one child. What is each of these children thinking? What does mode mean in a graph? 94

45 Hopefully, class discussion will get students to think about the meaning of mode or frequency when represented on a graph. Students or the teacher may then pose further questions to help clarify the ideas. For example: Bennie suggests that the group should make a list of all the 0 s, 1 s, 2 s, etc. What do you think he means? What would that list look like? When using student work, it is important not to show too much or use labels. The purpose is to make all students in the class have a reason to re-think the mathematics from a slightly different perspective. The purpose of the re-engagement lesson is not to give students another worksheet, but provide a forum for the exchange of ideas so that students develop the logic of making convincing arguments, confront their misconceptions explicitly, learn strategies used by others through the feedback of listening to their peers. Now try writing or designing a series of questions that you might pose for your class to develop the important mathematics of this task. Reflecting on the Results for Fifth 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 ideas 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? 95

46 Four areas that stand out for the Collaborative as a whole are: 1. Converting between units of measure - Students had difficulty converting between units of measure for ounces and pounds. Students sometimes ignored the ounces or ignored the pounds. Some students added ounces to pounds. Students didn t understand the logic of converting between measures. 2. Decimal place value In Shopping Bags, students converted ounces into decimals. There are 16 ounces in a pound, not 100. This gets at understanding the very structure of decimals as a base 10. Students are treating the decimal as just a graphic organizer. In Breakfast Time students had difficulty with decimal place value in operations. Students didn t know to add zeroes to the $20 before subtracting. Students tried to divide the $32 into the $6.40, because the length of the number was shorter or smaller rather than taking the smaller value. Students were not comparing the quantity represented by the numbers. 3. Proportional Reasoning In Pea Soup the students had difficulty thinking about how to compare ratio of onions to ratio of peas or how to compare the ratios when displayed on a graph. In Bar Charts students couldn t work with the scale factor to find the number of children, they didn t apply the scale factor to the frequency. They also didn t understand that they needed to reverse the scale when converting from children to number of families with 2 children. 4. Understanding Mode Students had inadequate definitions of mode as most. They couldn t understand what part of the graph represented frequency. They didn t know which one was the mode. Students need to see the relationship between mode as a string of data and mode on the graph. Students need to be able to relate between multiple representations of the same information. 96

47 Examining the Ramp: Looking at Responses of the Early 4 s (31-33) The ramps for the fifth grade test: Shopping Bags Part 2 Reasoning about bag 2 o Problem solving to find the which items go in the bag o Interpreting the constraints of the task: every item should be used, items could only be used once o Quantifying the total weight of the items by converting between measures of weight Breakfast Time - Part 3 Dividing with decimals o Understanding and comparing size of decimal numbers o Using multiplication or division to find the number of groups of $6.40 in $32 Fruity Fractions - No ramp Pea Soup Part 2a Explaining how to read a graph Part 2d Comparing lines on a graph o Reasoning about the steepness of the slope Bar Charts - Part 2 Reasoning about mode in the context of a graph Part 3 Applying scale to frequency o Using the scale to find the number of children represented on the graph o Understanding that the 0 bar did not contribute to the number of children o Working backwards from the number of children to the height of bar for a family with 2 children 97

48 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 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 understanding from them personally or does it sound more like 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 to 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 understand what they are striving for? Do you think errors were caused by lack of exposure to ideas or misconceptions? 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? 98

49 Mindy, total score 31 points 99

50 Mindy, part 2 100

51 Cameron, total score 32 points 101

52 Cameron, part 2 102

53 Cameron, part 3 Calvin, total score 32/33 103

54 Calvin, part 2 104

55 Calvin, part 3 Sally, total score 33 points 105

56 Sally, part 2 106

57 Martin, total score 32/33 107

58 Martin, part 2 108