Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight

Transcription

1 Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight Contents by Grade Level: Overview of Exam Grade Level Results Cut Score and Grade History Reflection of the Whole Grade Level By Task Task Rubric Questions for Reflection Student Work Core Ideas, Strengths and Weaknesses Graph and Analysis of the Data Understandings and Misunderstandings Implications for Instruction Ideas for Action Research Grade Level Analysis Examining the Ramp

2 Balanced Assessment Test Eighth Grade 2009 Core Idea Task Data Averages This task asks students to work with weighted averages and percents. Successful students could find and correct an error in calculating the average test scores. Algebra and Functions Square Patterns This task asks students to work with extending geometric patterns and calculating percentages. Successful students could count the number of black and white tiles in a pattern and calculate the percentage of the pattern containing black tiles. Probability Marble Game This task asks students to calculate compound probability for pulling marbles from two bags, to compare experimental and theoretical probabilities, and find the probability for a simple spinner. Successful students were able to find probabilities in different situations, but often struggled with making comparisons between probabilities or applying probabilities to context. Algebra and Vincent s Graph Mathematical Reasoning This task asks students to read and interpret graphs in a context. Students needed to interpret the slope of graphs in terms of the action of the context. Successful students could interpret a graph about measures and time and make some correct work in drawing their own graph to match a given story. Geometry and Photos Measurement This task asks students to work with equivalent ratios and show understanding of spatial relationships by finding sizes of photos on a page. Successful students could identify equivalent rations, find the size of photos for a given number per page, and find the number of pages needed for making a given number of copies. 1

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5 Averages This problem gives you the chance to: identify an error about means calculate means and solve a problem involving means 1. In a test, the 10 students in Group 1 get a mean score of 43%. The 15 students in Group 2 get a mean score of 57%. Hank says, The mean score for all 25 students is 50%. a. What mistake has Hank made? b. What is the correct mean score for all 25 students? Show how you figured it out. 2. In a different test, the mean score of 50 students is 54%. The students are split into two groups, Group A and Group B. The mean score for Group A is 60% and the mean score for Group B is 50%. How many students are in Group A and how many are in Group B? Group A Group B Show how you figured it out. 9 Copyright 2009 by Mathematics Assessment Resource Service. Averages 4

6 Averages Rubric The core elements of performance required by this task are: identify an error about means calculate means and solve a problem involving means Based on these, credit for specific aspects of performance should be assigned as follows points section points 1.a. b. Gives correct explanation such as: Hank has found the mean of 43% and 57%. He has not taken into account the sizes of the groups.. Gives correct answers: " " 57 Mean = = ! 2. Gives correct answers: 20 and 30 Shows work such as: If number in group A is X and number in Group B is Y, X + Y = 50 60X + 50Y = 50 x 54 Solution of the above to give 20 in group A and 30 in group B or Guess and check 2 or 2 4 Total Points 9 2 Copyright 2009 by Mathematics Assessment Resource Service. 5

7 Averages Work the task and look at the rubric. What are the key mathematical ideas being assessed in this task? What does a student need to understand to be successful on this task? Look at student work on part one, finding Hank s mistake. How many of your students put: Hank had not considered the number of students? Hank made a division error? Forgot to divide? Should have multiplied? An addition error? Did the math wrong? Didn t make a mistake? The average should be higher? How are these misconceptions different? How do we help students learn a concept well enough to be able to work backwards or use it in unexpected ways? How is understanding this concept different from knowing the procedures? Now look at the work in 1b, finding the mean for all 25 students. How many of your students put: 51.4% 100% 12.5% 40% 25% 50% 81% Other What misconceptions did students have who picked 100% or 40%? What don t they understand about averages? What experiences might these have? Now look at student work on part 2 finding the mean of two new groups. How many of your students put: 20/30 30/20 35/20 25/25 No answer Total >50 Total <50 Look at student operations. What types of procedures did they do? What do they not understand about operation? How do we help students learn to think deeper about concepts at higher grade levels? What activities and discussions help students to clarify and push their understandings about big mathematical ideas? 6

8 Looking at Student Work on Averages Student A meets all the demands of the task. He has identified what Hank calculated and produced the correct solution for the weighted average. The student sets up a very good number sentence to solve for the number of students in part 2. Student A 7

9 Student B is able to complete all of part 1. However in part 2 the student confuses the test scores with the percentage of students. What prompt might help this student rethink his process in 2? Student B Student C is also able to solve all of part 1 correctly. In part 2 the student understands that the weighted average should be connected to the mean times the total students. However the student tries a variety of strategies, but can t reason about the next step in the process. This student has a lot of correct knowledge to work with. What question might help the student think through the final steps? Student C 8

10 Student D has some understanding of the weighted average. The student doesn t appear to connect repeated addition with multiplication in part 1b. In part 2 the thinking breaks down and the student divides the total students by 2 instead of dividing the total scores. Student D Student E is able to solve part 1 but struggles with part 2. The student attempts to analyze the differences, which could lead to a correct solution. However the student then just tries to use all the numbers in the problem. Student E 9

11 Student F has conflicting answers within his work. In part 2 the student tries to find a score for each of the 50 students that will add to 54%, rather than thinking of the process of averaging. Student F 10

12 Student G has some knowledge about averages. She can find the correct average for part 1 and discuss one possible error Hank could have made. In part 2 she tries to use the percentages to make sense of the numbers in each group instead of thinking of the percentages as test scores. Notice that the total students adds to more than 50. Student G 11

13 Student H is able to reason out what Hank has calculated, but cannot find a method to correctly calculate the mean in part 1. In part 2 the student tries to reason about the effect of the test scores on the average, but can t find a process for making a weighted average. Student H 12

14 Student I identifies Hank s error and then repeats it in part b. The student also tries to divide into the total students instead of the total test scores in part 2. Student I Student J divides the number of students into two equal groups instead of finding the mean of the test scores. The student has some concept of dividing to find a mean but consistently uses the wrong number to divide up. Notice that the total in part 2 is more than 50. Student J 13

15 Student K tries to reason about the size of the scores to determine why Hank is wrong in part 1. The student finds the total of the test scores instead of the average in part one. The student seems to be comfortable only with the operations of addition and subtraction. At no point does the student connect mean with division. What question might you ask of the student to find out what he knows about mean? What experiences might this student need? Student K 14

16 Student L confuses percents with an absolute value rather than a relative value in part 1. In part 2 the student attempts to use a double number line to find the solution. Knowing alternative strategies is useful, but only if they make sense to you. There is no advantage to learning another rote procedure. Student L 15

17 8 th Grade Task 1 Averages Student Task Core Idea 5 Data Analysis Identify an error about mean. Calculate means and solve problems about means. Formulate questions that can be addressed with data and collect, organize, analyze, and display relevant data to answer them. Find, use, and interpret measures of center and spread, including interquartile range. Mathematics of the task: Analyzing a common error in finding weighted averages Calculating a weighted average Working backwards from an average to the number of students in each group Understanding what category the total should be for finding average Reasoning about the connection between the size of the scores and average on the size of the groups Based on teacher observation, this is what eighth graders know and are able to do: Understand that finding mean implies using a total and division Reason that higher scores raise the average Identify Hank s error Areas of difficulty for eighth graders: Using the wrong total to find the mean Not understanding weighted average Not looking at constraints, such as: the total students in 2 should add to 50 16

18 The maximum score available for this task is 9 points. The minimum score for a level 3 response, meeting standards, is 4 points. Some students, 28%, could identify Hank s error of averaging the test scores instead of compensating for the number of students in each group. 6% of the students could also calculate the weighted average in part 1. Less than one percent of the students could meet all the demands of the task, including identifying Hank s error, calculating a weighed average, and working backwards from a weighted average to find the size of the two groups. 71% of the students scored no points on this task. 80% of the students with this score attempted the task. 17

19 Averages Points Understandings Misunderstandings 0 80% of the students with this score attempted the task. Students couldn t identify the error in Hank s thinking. 9% thought he had calculated wrong. An additional 6% thought he added wrong and another 4% thought he divided wrong. 5% did not 2 Students noticed that Hank had averaged the two scores without considering the number of students in each group. 4 Students could identify Hank s error and calculate the weighted average. 5 Students could identify Hank s error, calculate the weighted average, and show how they figured it out. 9 Students could identify Hank s error, calculate the weighted average, and show how they figured it out. Students could also work backwards from the weighted average to the number of students in each group. think Hank had made a mistake. Students had difficulty calculating the weighted average in part 1. 18% thought the average was 100%(added the two test scores). 7% thought the mean was 12.5% (dividing the total students by 2). About 6% each picked 40%, 25%, or 50%. They didn t show their work for the calculations. Students struggled with working backwards from the weighted average to the number of students in each group. 13% thought there would 25 in each group. 14% thought there would be 30 in group A and 20 in group B. This was usually because their process was incorrect and not just a transcription error. 13% of the students did not attempt this part of the task. Almost 20% of the students gave answers totaling more than

20 Implications for Instruction Students need more work with understanding averages and how they are effected by the number of people. Students who struggled often followed procedures without considering frequency of scores. Students should have opportunities to think about the meaning of averages apart from the procedures. One activity is to give everyone in the room different sizes of columns of linker cubes. Then have students move around the room comparing columns. If the columns are different heights the students should share cubes to make equal size columns, when everyone can no longer make exchanges have students discuss the average. Students can then develop a working definition of averaging as leveling off, making everything equal. This concrete activity helps them connect physical reality with the procedure. Students should now be able to think about 2 different numbers and discuss which number will be closer to the mean or average. Try to design a lesson based on discussion without necessarily using any calculations. Another useful Balanced Assessment Task to use on this topic is 2003 Baseball Players. Ideas for Action Research - Re-engagement Confronting misconceptions, providing feedback on thinking, going deeper into the mathematics. (See overview at beginning of toolkit). 1. Start with a simple problem to bring all the students along. This allows students to clarify and articulate the mathematical ideas. 2. Make sense of another person s strategy. Try on a strategy. Compare strategies. 3. Have students analyze misconceptions and discuss why they don t make sense. In the process students can let go of misconceptions and clarify their thinking about the big ideas. 4. Find out how a strategy could be modified to get the right answer. Find the seeds of mathematical thinking in student work. In this task students had difficulty understanding just the idea of weighted averages. 71% of the students scored no points on this task. So students need to use the re-engagement lesson to develop some strategies that they can use in future work and make sense of the underlying mathematical ideas. Before starting the re-engagement lesson, it might be good to give students some visual ideas of the concept of average. A nice, quick task is to give everyone a length of attached linker cubes (make sure the lengths vary). Then have everyone get up and check their links with those of classmates. If the lengths are the same do nothing. If the lengths are different then the person with the longer chain gives up cubes until the two are equal. Students keep moving around the room until no trades are possible. Here students develop a visual for thinking about average as leveling off or evening the amounts. What questions might you pose at the end of the activity to help push at this idea? 19

21 Now have students look at the picture: How does this show the idea of average? How could we use number calculations to check to see if the average is the same? Now have students look at the second picture. Can you use the picture to find the average? Can you use numbers to find the average? To begin any re-engagement lesson it is essential to spend time on or almost overemphasize the basic facts to allow students access to higher thinking that is the focus of the lesson. Because of the number of students who scored no points, I am choosing to look at partially correct work rather than going to misconceptions with this class. I want them to be able to adopt some successful strategies. Now I would have students look at some of the work samples from the task and try and make sense of what is happening. I think it is easier to start with the calculations in 1b before trying to make the generalizations in 1a. 20

22 I might ask, When looking at the work of Julie, I was confused. What do you think she is doing? Where do the numbers come from? I would give students time to think alone, then share in pairs before starting a class discussion. I want to maximize the amount of student talking and hold all students accountable for having an opinion. I might want them to predict if the average is going to be closer to 57 or 43, by having them give a thumb signal on the chest or using a white board. Again, I want all students to commit before having the class discussion. Then I would ask them to give reasons to support their choice. Finally I might ask, What do you think Julie did next? Why? Give students individual think time, then pair/share, and finally have a class discussion. I might also show the work of Joshua. 43 x 10 = 430, and 57 x 15 = 845 What is Joshua doing? Why? How does it help him solve the problem? How is it different from what Julie did? How is it the same? I want students to make the connection between addition of equal groups and multiplication. I think this is important for students functioning at this level. Next I would want students to think about the generalization in part 1a. I might start by saying that on Kevin s paper I saw = 100, 100 divided by 2 = 50. What was Kevin thinking? What is wrong with that? I want to know if students can now connect the thinking about averages with the misconception. Now look at student samples from your class or from the toolkits. How might you use student work to promote understanding on part 2? Discuss with colleagues how to continue the lesson? What are the big mathematical ideas you want students to get from the discussion? What kind of writing prompt might you give at the end of the lesson to check for understanding? What are some similar problems that you might give students a few days later to see if they understand the idea of weighted averages? 21

23 Square Patterns This problem gives you the chance to: work out percentages In Prague some sidewalks are made from small square tiles. The blocks are made from black and white tiles. This is one of the patterns. 1. How many black and white tiles are there in this pattern? black white 2. What percent of the tiles is black? Give your answer to one decimal place. Show how you figured it out. % Copyright 2009 by Mathematics Assessment Resource Service. Square Patterns 22

24 On the sidewalks the patterns are separated by areas of white tiles. 3. On one sidewalk there are 10 patterns separated by areas of 13 by 13 white tiles, with a 13 by 13 area of white tiles at each end. What percent of the tiles on the sidewalk is black? Explain how you figured it out. % 7 Copyright 2009 by Mathematics Assessment Resource Service. Square Patterns 23

25 2009 Rubrics Averages Rubric The core elements of performance required by this task are: identify an error about means calculate means and solve a problem involving means Based on these, credit for specific aspects of performance should be assigned as follows points section points 1.a. b. Gives correct explanation such as: Hank has found the mean of 43% and 57%. He has not taken into account the sizes of the groups.. Gives correct answers: " " 57 Mean = = ! 2. Gives correct answers: 20 and 30 Shows work such as: If number in group A is X and number in Group B is Y, X + Y = 50 60X + 50Y = 50 x 54 Solution of the above to give 20 in group A and 30 in group B or Guess and check 2 or 2 4 Total Points 9 2 Copyright 2009 by Mathematics Assessment Resource Service. 24

26 Square Patterns Work the task and look at the rubric. What are the big mathematical ideas being assessed? Look at student work for part 1. How many of your students could not count the correct number of black tiles? white? Did students show evidence of finding tiles in sections and adding them up? Did they appear to rely on counting every one? In part two, finding the percent of black tiles, how many of your students put: 50.3% 85% 51% 52% 8.5% No work Other Thinking about their answers, what did students not understand about the idea of percents? Did they reverse the division (169/85)? Did they divide by 100? Did they divide by the whites? Did they think 85 was a percent (percent as an absolute value rather than a relative value)? Did they use estimation? What experiences do these students need to let go of these misconceptions? How is teaching to a student with partially learned or misinformation different from working with a clean slate? What knowledge that they do have can you tap into to help them make sense of their errors? Now look at work in part 3, extending the pattern. How many of your students only used the tiles in the diagram? How many used only the two patterns and the center white space? How many attempted to use all 10 pattern pieces? Were your students able to correctly account for the part of the pattern they were looking at? Now look at how they set up the percents: Were they using a correct amount of blacks for their pattern? Did they divide by all the tiles? Just the whites? Did they omit some of the whites in the pattern and only count the whites in the space blocks? Did they try to make the black into a percent without using the whites at all? Did students seem to have a system for organizing their thinking about the bigger pattern, such as using labels or showing calculations for different parts of the pattern? Thinking about their work as a whole, what would be your next steps for students? 25

27 Looking at Student Work on Square Pattern Student A shows thinking about the sections of the pattern to find the number of black and white tiles. In part 3 the student is able to generalize that all the columns are 13 and multiplies by the exact number of columns in the diagram. The student is comfortable with the process of finding percents. However the student does not consider how to replicate the larger pattern to 10 pieces with separators. Student A 26

28 Student A, continued In part 2, Student B understands that percents are made by dividing. The student can set up the fraction correctly but is unclear whether to divide the top into the bottom or the bottom into the top. The student tries both and is able to reason by the size of the fraction which one makes most sense. In part 3, Student B is also able to find the number of tiles in the diagram and sets up the correct fraction for making the percent. However the student does the calculation for total tiles divided by number of black. 27

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30 Student C is able to solve parts 1 and 2 of the task. In part 3 the student only considers the 2 patterns shown and the 1 white separator, ignoring the two end columns of white. While the student knows the number of whites and the number of blacks the student does not find a total number of tiles. The student tries to find the percentage by dividing the blacks by the whites. Student C 29

31 Student D is able to do parts 1 and 2 successfully. Student D finds the number of tiles in the two patterns and 1 separator section. The student understands that the bottom of the fraction should represent the total tiles. However the student is unclear what the numerator should represent. The student first thinks about the ten patterns in the prompt and then reverts to the black tiles from part 1. The student doesn t consider that there are two pattern pieces, doubling the number of black tiles. Student D 30

32 Student E is able to successfully complete parts 1 and 2. Student E is the first example of a student attempting to replicate the pattern 10 times. The student makes the assumption that if the diagram shows two sections of the patterns, then duplicating the diagram 5 times will produce 10 patterns. Why doesn t this work? What about this won t work? The student has a clear sense of organizing and labeling work but does not calculate the number of whites correctly. What error has the student made? The student makes a transcription error in finding the number of blacks. Student E 31

33 Student F has some understanding of proportionality and percents. In part 2 the student compares the sizes of the two parts and makes a reasonable estimate of the percent. The student also attempts to estimate the percent in the second part, but without calculating the total whites and blacks the estimate is not very close. What would be your next step with this student? Student F 32

34 Student F, continued Student G has some partially learned procedural information about percents and moving decimals points. The student has no understanding of percents as a comparison of part to whole. Student G 33

35 Student H does not know how to organize his thinking. In part 1 the student relies on counting every tile, which is laborious and leads to errors. The student counts 88 black and 81 while. The student divides the black into the total tiles to find the percentage. In part 3 the student shows numbers larger than 108, but is not concerned with inconsistencies. To find the percent the student uses language to suggest again dividing the total by part, but in actuality the student does the reverse. The student lacks the language of division to help organize thinking and is unclear about the meaning of the division expressed in the fractional notation. There is a conjecture that we can only think about things we have words for. How do we help students develop mathematical language, both in words and in notation? What would be your next steps with this student? Student H 34

36 Student H, continued Student I is able to look at sections of the diagram and reason about the size. The student has some partially learned procedural knowledge about using percents to multiply. The student looks at the two patterns and one separator section in the diagram in part 3 and understands how to find the number of white in the patterns, black in the patterns, and white in the middle section. But the student can t reason about the meaning of operation to find the total tiles and forgets that the number of blacks is 170. So the student does a series of unconnected operations of addition and subtraction. The student knows the percents are used in multiplication and so multiplies by the 10 pattern pieces. How could you pose a series of questions to get the student to think about labels and meaning of operations? 35

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38 Student I, continued 37

39 Student J also has some partially learned procedural information about percents being part of 100 In part 2 the student shows division by 100 notationally, but in fact divides the 84 into 100. The student makes a similar error in part 3. How do we work with students with this partial information or misinformation? What types of discussions or experiences help them let go of these ideas and make the effort to learn new ones? What understandings do they show that we might build upon? Student J 38

40 8 th Grade Task 2 Square Patterns Student Task Core Idea 4 Geometry & Measurement Core Idea 1 Number and Operations Core Idea 2 Mathematical Reasoning Recognize and extend a pattern. Calculate percentages in a geometric pattern. Apply appropriate techniques, tools and formulas to determine measurements. Understand number systems, the meanings of operations, and ways of representing numbers, relationships, and number systems. Work flexibly with fractions, decimals, and percents to solve problems. Understand and use the inverse relationships to simplify computations and solve problems. Employ forms of mathematical reasoning and justification appropriately to the solution of a problem. Extract pertinent information from situations and determine what additional information is needed. Formulate conjectures and test them for validity. Use mathematical language and representations to make complex situations easier to understand. Mathematics of this task: Composing and decomposing a figure to find the number of tiles of each color Understanding percents as a comparison of part to whole Calculating percents and rounding Reasoning about extending a pattern, identifying the units to be replicated Understanding that the white separators will be one more than the number of pattern pieces Organizing calculations, showing what is known and what needs to be calculated Understanding division, the language of division, and divisional notation Based on teacher observation, this is what eighth graders know and are able to do: Count the number of black and white tiles in the pattern Students could find the total number of tiles Areas of difficulty for eighth graders: Calculating a percentage Organizing information Extending the pattern beyond the diagram, understanding what it means to have 10 patterns Understanding the language of division and divisional notation 39

41 The maximum score available for this task is 7 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Many students, 85%, could count the number of black or white tiles. Almost 70% could count both accurately. Some students, 28% could find the percentage of black tiles in the pattern. 16% could show how they calculated the percentage. About 7% of the students could reason about the tiles shown in the diagram to find a percentage in part 3. About 1% of the students could meet all the demands of the task, including extending the pattern to 10 steps. Almost 15% scored no points on this task. All of the students in the sample with this score attempted the task. 40

42 Square Patterns Points Understandings Misunderstandings 0 All the students in the sample with this score attempted the task. Students did not break the diagram into parts to find the number of tiles. They attempted to count every square. 2 Students could find the total number of black tiles and white tiles. 3 Students could find the number of black and white tiles and calculate the percentage of black tiles in the pattern. 4 Students could find the number of black and white tiles and calculate the percentage of black tiles in the pattern and show their work. 5 Students could also reason about the black and whites in a new diagram and find the percentage of blacks. 7 Students could find the total number of black and white tiles in a pattern and calculate the percentage of blacks. They could reason about extending the pattern to 10 sections and determine the number of white spacers needed and calculate the percentage. They decomposed their shapes into parts and used multiplication to find the number of tiles in the section. They had an systematic approach to organizing calculations. Students did not know how to find a percentage. 13% just moved decimal points to get 85%. 14% showed no work for this part. Students were confused about which numbers to divide into, often dividing the part into the whole. Students could not think beyond the diagram to extend the pattern to 10 pieces. Therefore they never grappled with how many large white squares would be needed. 41

43 Implications for Instruction Students need to have explicit help developing the vocabulary and notation for divisional thinking. Students need to understand the line in the fraction as division. They need to be fluent with many ways that division is represented and which number goes into something else. Students need to have more than procedural knowledge about percents. They need to think of it as a comparison between part and whole. This information helps them to understand how the division is set up. Many students have partially learned or misinformation that needs to be confronted and discussed so they can see the errors in their logic and let them go. Students at this grade level should work with a variety of patterns, identifying the unit to be replicated. This idea is critical to thinking about the idea of a variable and algebraic thinking. Most of the students in the sample are working at a very concrete level, not being able to move beyond the diagrams given in the problem. They need to be challenged to think about what the pattern would look like with ten patterns. What does that mean? How can you draw it? Is there a simpler way to find the total without drawing every tile. Some students could not even decompose the given concrete diagram into pieces to assist themselves in finding the number of tiles. They still relied on counting every tile. How can you devise an activity to push students beyond counting? Ideas for Action Research Re-engagement Confronting misconceptions, providing feedback on thinking, going deeper into the mathematics. (See overview at beginning of toolkit). 1. Start with a simple problem to bring all the students along. This allows students to clarify and articulate the mathematical ideas. 2. Make sense of another person s strategy. Try on a strategy. Compare strategies. 3. Have students analyze misconceptions and discuss why they don t make sense. In the process students can let go of misconceptions and clarify their thinking about the big ideas. 4. Find out how a strategy could be modified to get the right answer. Find the seeds of mathematical thinking in student work. Part of the story of this task is that students just had difficulty counting the tiles or visualizing the pattern. To start the lesson, it is important for students to know the correct number of white and black tiles. Also at this grade level, it is important for students to start connecting multiplication with equal size groups and to find short cuts instead of relying on counting every tile. The numbers are too large for that to be a reliable strategy. The beginning of the lesson might be to have a copy of the tile or sidewalk piece on the overhead. The teacher might say that one student had the following work on their paper: 4 x 12 = What do you think the student was looking at? What was the student trying to figure out? 42

44 After some discussion, the teacher might say that on another paper she saw: 12 x 7 = What do you think the student was looking at? What was the student trying to figure out? The idea is for students to think of different ways of decomposing the shape into smaller parts and be able to link number sentences to the geometric shapes. By giving students more than one set of number sentences for the white squares, students with different ways of viewing the shape are validated for their thinking. Now the teacher might ask students to find a number sentence for finding the black tiles. What do you think students might come up with? If a student said: 13 x =. What do you think they were looking at? How were they seeing the shape? Students had a very difficult time calculating percentages. Looking through the toolkit or your own student work, what pieces might you use to pose some questions for the class? How could you use just parts of the work so everyone has to rethink the problem? What is the mathematics you want students to walk away with? What misconceptions do you want students to confront and talk about? Why is it important for students to discuss misconceptions and decide why they don t make sense? In the final part of the pattern, students had a difficult time just visualizing the whole pattern. Using some Cuisenaire rods, use the unit rod for the sidewalk pieces and one of the intermediate rods, (maybe a size 3) to represent the white spacers. Have students build a sidewalk pattern for 5 sidewalk tiles. Ask them how many spacers would they need? How many sidewalk tiles? How could they find the total number of whites? How could they find the total number of blacks? How could they make a rule for finding the total for any number of sidewalk tiles in words? In algebraic expressions? Now have students work individually or in pairs to find the solution to part 3 of the task. How would you ask some questions to get closure on the lesson? What are some of the big mathematical ideas you hope students walk away with at the end of the lesson? 43

45 Marble Game This problem gives you the chance to: use probability in an everyday situation Linda has designed a marble game. Bag A Bag B 1. Bag A contains 3 marbles one red, one blue and one green. Bag B contains 2 marbles one red and one blue. G B R R B To play this game, a player draws one marble from each bag without looking. If the two marbles match (are the same color), the player wins a prize. What is the theoretical probability of winning a prize at a single try? Show your work. 2. Here are the results for the first 30 games. How do the results in this table and the theoretical probability you found compare? Win (Match) No Win (No Match) Explain any differences. Copyright 2009 by Mathematics Assessment Resource Service. Marble Game 44

46 3. Linda has designed a second game. The spinner has nine equal sections. To play the game, a player spins the spinner. Blue Red Gold If the spinner lands on a Gold section, the player wins a prize. Does the player have a better chance of winning with the bag game or the spinner game? Gold Red Gold Red Red Gold Explain your reasoning. 7 Copyright 2009 by Mathematics Assessment Resource Service. Marble Game 45

47 Marble Game Rubric The core elements of performance required by this task are: listed here Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: 2 6 = 1 3 Lists all possibilities: RR, RB, BR, BB, GR, GB p(rr or BB) = 2/6 or 1/3 or! Shows work such as: Probability R " R = 1 3 # 1 2 Probability B " B = 1 3 # or 1 1! Probability both same color =! 2. These results are quite close; ! but the number of trials is not large enough to give an accurate estimate. or Explains that from these results the experimental probability 9 = 30 = 3 10 = 0.3 The theoretical probability = 0.33 recurring! 3. Gives correct answer: the spinner game Shows work such as: 1 the probability of winning on the spinner game is 4/9 = 0.44 recurring 2 Total Points 7 Copyright 2009 by Mathematics Assessment Resource Service. 46

48 Marble Game Work the task and look at the rubric. What are the key mathematical ideas being assessed in this task? What activities or experiences have students had with probability this year to help prepare them for the High School Exit Exam? How do you help students draw or define sample spaces? What models do they know? Look at student work on part 1. How many of your students put: 1/3 2/3 4/5 2/5 1/5 Red Other Make a list of models students used to figure out probability: Organized list Formula Now look at student work of part 2. How many students: Knew the probability was 9/30 or 30%? Did not compare to the two probabilities? Thought that winning was unlikely (nonnumerical)? Thought there were less wins than losses? Thought the probability was 9/21 or 3/7? What are some of the big ideas about quantifying probability that your students did not understand? Now look at student work on part 3, finding the probability of the spinner. How many of your students thought the spinner was better? the bag game was better? they were equal? How many of your students did not attempt to quantify the probability for winning the spinner? 47

49 Looking at Student Work on Marble Game Student A is able to make an organized list to show the sample space in part 1 and quantify the probability. The student does not how to convert the frequency table into a probability but recognizes that the total is important to the fraction. The student is able to quantify the probability on the spinner and compare it correctly to the bag game. Student A 48

50 Student B is able to get the correct probability, but from drawing the sample space incorrectly. What error has the student made? The student is able to calculate the experimental probability from the frequency table and understands that the denominators need to be the same to compare the two probabilities. However the student makes the wrong choice when comparing the two. The student has a complete explanation for part 3. Student B 49

51 Student C is able to draw a complete sample space and identify the winning choices, the student even gives the numerical probability. The student does not see the importance of the numerical probability with the idea of likely, unlikely, and certain, etc. dominating the thinking. Notice that in part 2 the student doesn t quantify the experimental probability. The student does give the correct numerical probability in part 3. Student C 50

52 Student D is not very explanatory and relies on numbers. In part 1 the student thinks about a variety of choices, but lands on the correct probability. In part 2 the student quantifies the probability, but makes no attempt to compare the probabilities. In part 3 the student finds the numerical probability. Student D 51

53 Student E is able to define the sample space and give a definition for finding probability. The student can t turn the definition into a numerical expression. Instead the student quantifies the number of wins. In part 2 the student is also concerned with winning or losing rather than a numerical expression. The student subtracts wins from losses with the frequency table, rather than comparing the probability in 1 with the probability in 2. The student could find the numerical probability for the spinner. How do we help students to see the relevance for the numerical probability when their focus is on the importance of winning? What would make this important to them? Student E 52

54 Student F gets the correct probability, but for the wrong reasons. The student seems to think about 2 bags, so 50% and 50% to get 100. Then the student divides by the 3 marbles in the first bag. The student does calculate correctly the experimental probability. In part 3 the student incorrectly counts the number of sections on the spinner. The student doesn t combine the winning choices. Student F 53

55 Student G is able to make a list for the sample space, but doesn t know how to use it to write a correct probability. The student does not use the total in either part 1 or part 2. The student doesn t understand how to set up the probability for the spinner, and instead seems to compare the reds from the two games. Student G 54

56 Student H attempts to draw the sample space, but uses the first bag twice. The student seems to think that games should be fair, have a 50% chance of winning in part 2. The student is able to identify the winning spaces on the spinner, but then compares chances to chances rather than probability. Student H 55

57 Student I identifies the winning combinations in part 1, but sets up the probability with total marbles instead of total possibilities. What experiences help students to understand the distinction between the two? In part 2 the student seems to compare the winning to not winning rather than set up a probability. In part 3 the student doesn t quantify the fraction for winning on the spinner. Student I 56

58 8 th Grade Task 3 Marble Game Student Task Use probability in an everyday situation. Core Idea 2 Probability Apply and deepen understanding of theoretical and empirical probability. Determine theoretical and experimental probabilities and use these to make predictions about events. Represent probabilities as ratios, proportions, decimals or percents. Represent the sample space for a given event in an organized way. Mathematics of this task: Define a sample space for a game Identify winning combinations Express theoretical probability numerically Convert data from a frequency table to an experimental probability Compare probabilities by finding commonality (percents, common denominator) Find simple probability for a spinner Based on teacher observation, this is what eighth graders know and are able to do: Determine the game with the best probability of winning Identify the probability on a spinner Areas of difficulty for eighth graders: Understanding the difference between marbles and the total possibilities in setting up the probabilities Defining a sample space for a compound event Still thinking about likely/ unlikely or wining/losing instead of in terms of numerical probabilities Comparing colors in the bag game and spinner instead of the probabilities Using numbers to quantify the probability in the frequency table and spinner 57

59 The maximum score available for this task is 7 points. The minimum score for a level 3 response, meeting standards, is 4 points. More than half the students, 60%, could identify the winning game. Some students, 28%, could also give the numerical probability for winning the spinner game. About 7% of the students could find the probability for winning the bag game (their reasoning may not have been correct). Less than 1% of the students could meet all the demands of the task including defining the sample space for the bag game and giving the theoretical and experimental probability and noticing that they were close in value. 40% of the students scored no points on this task. 92% of the students in the sample with this score attempted the task. 58

60 Marble Game Points Understandings Misunderstandings 0 40% of the students in the sample got this score. 92% of them attempted the task. 18% of the students thought the chances of winning the bag game were better. 10% thought the chances were the same for both games. 20% made no attempt to quantify the probability on the spinner game. Some students subtracted winning outcomes (4-1 Students could select the game with the highest probability of winning. 2 Students could give the probability of winning on the spinner and knew it was a higher probability than the bag game. 3 Students could work with the spinner game and find the probability for the bag game. 4 Students could find the correct probability for the bag game with correct reasoning, give the correct probability for the spinner game, and compare the two values. 7 Students could reason about a compound event, define the sample space, and write a probability. Students could convert a frequency table into an experimental probability. Students could compare probabilities by converting them to decimals, percents or fractions with common denominators. Students could give a probability for a simple event like a spinner. 2). Students could not support their reasoning by quantifying the probability on the spinner. Students could not find the probability for winning the bag game. 8% thought the probability was 4/5. 8% thought the probability was 2/3. 8% thought the probability was 2/5. Students could not give convincing or correct justification for their probability. Many students arrived at a probability of 1/3 using incorrect logic. Students could not write the experimental probability or compare it to the probability for the bag game. 13% used the word unlikely instead of giving a numerical value. 13% did not attempt this part of the task. 8% of the students compared winning to losing in the frequency table instead of comparing the frequency table to the theoretical probability in part 1. 59

61 Implications for Instruction Students need more exposure to probability. While in elementary school, students learn about probability in terms of likelihood, in middles school students should start to make progress on quantifying probability. Students need to see the difference between odds (favorable outcomes to unfavorable outcomes) and probability (favorable outcomes to total possible outcomes). Before any of this can be determined, students need to be able to develop sample space. The spinner is easy because all of the spaces were equal size and could be added together. Students understand generally how to determine probability on a single number cube. They have trouble with compound events: rolling a die and then rolling a second die or spinning a spinner; drawing from two different bags. Students need to practice with strategies like making an organized list or drawing a tree diagram to help them develop all the possible outcomes. Some students still have trouble distinguishing between the actual objects used in the event (marbles in the bag) and the outcome of drawing the red marble. Some students need to understand that probability is the chance of something happening rather than winning. Many students were only focused on winning the game. Students need to think about the difference between theoretical probability and experimental probability. While many students could quantify the size of the experimental probability, there was little evidence about the difference between chance and expected values. Ideas for Action Research With 40% of the students scoring no points on this task, students need to do some serious work with the idea of probability. While this isn t an eighth grade standard, it is tested on the High School Exit exam. Probability is also an interesting way to help students review ideas about fractions and decimals, making them feel like they are studying a new topic. Before trying to formalize the information, students just need to build up some experiential knowledge. They need lots of experiences with games of chance. Some interesting tasks can be found in Chance Encounters by Education Development Center, Inc. One group of activities involves describing probabilities as fractions, percents, decimals and ratios. It is called Designing Mystery Spinner Games. See a part of the activity on the next page. A different set of activities is to decide if games are Fair. See examples on following pages. A problem of the month on Fair Games is also available on the SVMI website: 60

62 How does this activity press students to practice skills such as using fractions, decimals, and percentages? How does it help students use justification and mathematical reasoning? 61

63 62

64 In this game, students are asked to help Terry decide which is the better game to play. How does this help students practice basic skills? What strategies do you think students might use to rank the games? 63

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66 Vincent s Graphs This problem gives you the chance to: interpret graphs draw a graph Vincent is eating a packet of raisins. This graph shows the changes in the mass of raisins in the packet as time passes. M ass i n g r ams Ti me i n m i n u te s 1.a. What is Vincent doing when there is a vertical line on the graph? b. Why are the vertical lines of different lengths? c. Did Vincent eat all the raisins? Explain how you know. 2. Ellie is drinking with a straw from a box of fruit juice. The graph shows the volume of juice in the box as time passes. Volume in milliliters Time in minutes a. What is happening when the line on the graph is horizontal? b. Why do the lines going downwards on this graph go at an angle? Copyright 2009 by Mathematics Assessment Resource Service. Vincent s Graph 65

67 3. Ralph is eating cherries from a bag. After eating a cherry he puts the stone back into the bag before taking out the next cherry. On the grid draw a graph to show the changes in the mass of the bag of cherries as time passes. Mass in grams Time in minutes 8 Copyright 2009 by Mathematics Assessment Resource Service. Vincent s Graph 66

68 Vincent s Graphs Rubric The core elements of performance required by this task are: interpreting graphs drawing graphs Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. a. Gives correct answer such as: he is taking raisins out of the packet. b. Gives correct answer such as: he takes different numbers of raisins from the packet. c. Gives correct answer: No and a correct explanation such as: the line does not reach the x axis. 2. a. Gives correct answer such as: Ellie is not drinking. b. Shows correct work such as: the volume decreases steadily as the juice is sucked out. 3. Draws a correct graph: First a short horizontal line Followed by a short line downwards. A short horizontal line followed by a short line upwards. The line upwards should be shorter than the first line downwards. 3 Total Points Copyright 2009 by Mathematics Assessment Resource Service. 67

69 Vincent s Graph Work the task and look at the rubric. What are the key mathematical ideas being assessed in the task? Look at student work on part 1. How many of your students: Showed an understanding that taking raisins out of the box is the cause of the change in weight? Thought the vertical line was because of eating the raisins? Said Vincent had stopped eating? Made comments about the labels rather than referring to the context? Thought the vertical line was about time? Couldn t attempt an explanation? Look at student work on part 2. How many students thought the horizontal line was: Ellie not drinking? Ellie drinking? The speed of Ellie drinking? Something moving up and down or high and low? Couldn t attempt an explanation? Now look at student graphs. How many of your students: Drew a correct graph? Drew a graph showing a series of horizontal, vertical lines making a downward staircase? Drew graphs with an increasing height? Drew graphs going up and down, but making a horizontal pattern never going lower on the height? What other things did you notice about student graphs? Why do you think students had so much difficulty relating the graph in context? Do students need more experiences with scale and measuring? What kinds of experiences and discussions have students had with time graphs? How do you help them connect the action of the story to shape of the graph? Do they have exposure to graphs where the action of the story is the reverse of the graph? 68

70 Looking at Student Work on Vincent s Graphs For the most part Student A is able to correlate parts of the graph to the context of the story. The student doesn t understand that removing the raisins from the box is what changes the weight. The student is able to make some progress on the graph, but does not return the cherry pits to the bag. Student A 69

71 Student B makes the common mistake of taking the action of the graph as eating rather than removing raisins from the box. While the actions are connected, it is the removing the raisins that causes the instantaneous change in weight. Student B is one of the few students able to make a correct graph in part 3. Student B 70

72 Student C also confuses eating in part 1a and 1b with the action of removing raisins. The student has good explanations for the other parts. The student doesn t understand the instantaneous action of removing from and adding objects to a scale and therefore has slanted versus vertical lines on the graph. The student allows no time for eating the cherry between removal and return of the pit. Student C 71

73 Student D is starting to focus on the scale of the graphs rather than trying to match the action of the story to the context of the graph. In part 1c the student does not give a full explanation. What does it mean for the line to stop for a while? In part 2 the student makes the common mistake of the thinking the vertical line is drinking or time to drink. Again both answers in 2 are more about scales (in this case the wrong scales) than the context. How do we help students to connect the shape of graph to the real life action being represented? What does the shape have to do with the story? Student D 72

74 Look at the work of Student E. What do you think the student really understands about the situations? What questions would you like to ask the student? Student E 73

75 Student F is an example of a student with very little ability to make sense of the graph. The student thinks about the context and what actions could or could not be happening, but does not relate them to scale or slope. How do we help students make these connections? Student F 74

76 Student G is focused on the labels of the graph. How is this different from relating the graph to the context? Student G 75

77 Student H appears to be reading the graph from bottom to top, rather than top to bottom. The student talks about the mass growing, mass gained. The student does not notice the downward trend of the graph. Student I shows an example of students reversing the slope of the graph. Student H Student I 76

78 Student J is not thinking of the labels or the context of the action. The student only notices the up or downward trend of the graphs. What experiences does this student need? Student J 77

79 8 th Grade Task 4 Vincent s Graphs Student Task Core Idea 3 Algebra and Function Core Idea 2 Mathematical Reasoning Draw and interpret graphs. Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope. Employ forms of mathematical reasoning and justification appropriately to the solution of a problem. Verify and interpret results of a problem. Use mathematical language and representations to make complex situations easier to understand. Mathematics of the task: Make connections between the shape of the graph and the context Understand that adding or taking away objects on a scale causes an instantaneous change in weight Think about slope and the action of a story or context Make a graph to match a verbal description of an action, thinking about the slopes and the different discrete actions taking place Based on teacher observations, this is what eighth graders know and are able to do: Understand the downward motion of removing cherries from a bag and graph that action Understand allowing time to pass before removing more cherries and graphing that action Explain how they know that Vincent did not eat all the raisins Understand the meaning of the horizontal line on Ellie s graph, drinking juice Areas of difficulty for eighth graders: Understanding how to graph returning the cherry pit to the bag Distinguishing instantaneous weight change, removing raisins from the bag, from eating raisins Understanding that a long horizontal line does not mean that all the raisins are gone, just that no action is occurring Confusing labels on the graph with the action of the story; not relating graph to context 78

80 The maximum score available for this task is 8 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Many students, 67%, could understand the downward motion of removing cherries from a bag and allow time to pass between removing cherries and the next action and graph those actions. Some students, about 27%, could explain why the graph shows Vincent did not eat all the raisins, explain the horizontal line on Ellie s graph in terms of the context and graph 2 of the actions about the cherries. About 14% could also explain the downward slanted lines on Ellie s graph as the slow action of sipping. About 3% could meet all the demands of the graph including recognizing the difference between removing objects from a box, instantaneous, and eating raisins, which doesn t effect the weight of the box. Almost 19% of the students scored no points on the task. 83% of the students with this score attempted the task. 79

81 Vincent s Graph Points Understandings Misunderstandings 0 83% of the students with this score Students did not know how to make a attempted the task. time graph. Some drew bar graphs, 2 Students could graph the vertical and horizontal lines on the graph. 4 Students could graph the vertical and horizontal lines on the graph. Students could talk about why Vincent did not eat all the raisins and knew that Ellie had stopped drinking when the graph showed a horizontal line. 5 Students could graph 2 of the 3 parts of the graph, explain that Vincent ate all the raisins, and interpret both parts of Ellie s graph in the context of the story. 8 Students could interpret time graphs in terms of the action they represent and make a time graph of their own. some made lines with an upward trend. Students did not understand the returning the pit to the bag. 52% made just a downward staircase, with no upward parts. 9% thought Vincent had eaten all the raisins because the line was flat or stopped going down. 9% thought he had not eaten all the raisins because the line never stops. 18% of the students thought Ellie was drinking during the horizontal lines on the graph. 6% thought that was the speed of her drinking, faster or slower. 11% of the students did not attempt to explain the slanted lines on Ellie s graph. Many talked about scale rather than context, how much juice is left or drinking as time goes by. Many thought it had to do with the volume being consumed. 80

82 Implication for Instruction Students need more work with interpreting graphs in context. Students at this grade should develop an understanding of slope and the relationship to what activities are occurring over time. For example, the weight of items on a scale goes down instantaneously when items are removed. Ideas for Action Research Making Graphs for Stories Having students make their own graphs about situations helps them to understand the logic of the graph and see how the lines do not represent the action of the story. Consider giving your students some story situations and have them make a graph of the general situation, not necessarily dealing with the issues of an exact scale. For example: A factory cafeteria contains a vending machine selling drinks. On a typical day: - The machine starts half full. - No drinks are sold before 9 a.m. or after 5 p.m. - Drinks are sold at a slow rate throughout the day, except during the morning and lunch breads (10:30-11 am and 102 pm) when there is a greater demand. - The machine is filled up just before the lunch break. (It takes about 10 minutes to fill). Make a sketch to the graph to show how the number of drinks in the machine might vary from 8 am to 6 pm. What does the student have to understand about graphing to do this task? What do you think the graph might look like? What errors do you anticipate students might make? How does this help you think about how to process this activity? What are the mathematics you want to bring out or highlight as students discuss their work? 81

83 Now consider another task from the Shell Centre book, The Language of Functions and Graphs. 82

84 83

85 How did this activity add to student understanding of line graphs? What did the student have to think about to be successful? What evidence of understanding did you see in student work? 84

86 Photos This problem gives you the chance to: show understanding of spatial relationships The aspect ratio of a picture is the ratio of its width to its height. 8in 1. Which of the following are correct expressions for the aspect ratio for this 8 inch by 12 inch baby photo? Draw rings around all of the correct ratios. 8:12 12: 8 4:6 2:3 12in 3:2 1:1.5 16:24 24:36 8in 2. When the length and width of the photo are halved, four photos fit onto the 8 inch by 12 inch page. The aspect ratio is still the same. a. What are the new measurements of the photo? 12in b. Jane wants 75 copies of this size of the photo for Christmas cards. How many 8 inch by 12 inch pages does she need? Show your work. Copyright 2009 by Mathematics Assessment Resource Service. Photos 85

87 3. For use on key chains, the photo is reduced to a width of 1 inch, keeping the aspect ratio the same. a. What is the height of the photo? b. How many of the photos will fit onto one 8 inch by 12 inch page? 4. Explain how to calculate the number of photos that will fit on one page if you know the width of the photo. 9 Copyright 2009 by Mathematics Assessment Resource Service. Photos 86

88 Photos Rubric The core elements of performance required by this task are: show understanding of spatial relationships Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answers: 8:12 4:6 2:3 1:1.5 16:24 24:36 with no extras 2.a. b. 3.a. b. Partial credit 4 or 5 correct with no extras or 6 correct and 1 extra 3 or 2 correct with no extras or 4/5 correct and 1 extra Gives correct answer: 4 inches by 6 inches Gives correct answer: 19 Gives correct answer: 1.5 inches Gives correct answer: 64 3 (2) (1) Gives correct explanation such as: Find how many times the width goes into 8 inches and square that number. Or Find how many times the width goes into 8 inches and multiply by how 2 or many times the height of the photo goes into Total Points 9 Copyright 2009 by Mathematics Assessment Resource Service. 87

89 Photos Work the task and look at the rubric. What are the key mathematical ideas being assessed? Think about the work you have done with ratios this year. What were some of the key concepts that you hope students walked away with? Now look at student work on part 1. How many of your students: 8:12 4:6 2:3 1:1.5 16:24 24:36 12:8 3:2 Omitted/ added How many only marked the 8:12? What made some of these ratios more difficult for students? Now look at student work for part 2a, finding the new measurements of the photo. How many of your students put: 4:6 2:3 8:12 18:12 12:8 Other Why doesn t reducing the ratio work for this situation? How do we help students understand the difference between the ratio and the measurement? Now look at work for part 2b, finding the number of papers needed to print the pictures. How many of your students put: or 18? 75 Answer 100 No answer Other How often do students work problems in context, where they need to interpret a remainder? How did students get the answer of 75? What was their misconception? What other misconceptions did you see? :11 No answer Other What might the student be thinking who put these answers? Now look at 3b, finding the number of photos on a page. How many of your students put: No answer Other Can you find some of the thinking behind the incorrect thinking? What were students misconceptions about the problem? 88

90 Finally look at the explanations for finding the number of photos on a page given the width of the formula. How many of your students: Made a generalized rule for finding the number of photos? Made a rule for a specific case(one of the widths in the problem)? Understood that the width of the paper needed to be divided by the width of the photo, but couldn t finish the rule? Gave an area formula( l x w)? Gave a rule with multiplication? Gave a rule with addition or subtraction? Tried to use a counting strategy? Did not attempt to write a rule? How do we help students move from concrete, specific examples to making generalizations? What opportunities have students in your class had this year to help them learn to make generalizations? How might you use student work with area formula to bridge to making generalization? What were the seeds of mathematical understanding that you have to work with? How do we help students at this grade level develop an understanding of operation? Why was it so difficult for students to choose a correct operation? 89

91 Looking at Student Work on Photos Student A is able to identify all but one of the equivalent ratios in part 1. The student can find the measurement of the new photos and make sense of rounding in context. The student struggled with the idea of making a generalization. The student shows a clear process for finding the solution of a specific case. How would you help this student move to the next level? Student A 90

92 Student B is able to do all of part 1 and 2 correctly. Notice the student brainstorms with multiplication before settling on division to find the number of pages. In part 3 the student can t use the same procedure to find the number of photos on the page. What about this part is different from part 2? The student does attempt a generalization, but can t complete the process. What step is missing? What would the student need to visualize to think about the final step? What question might you pose to push the student s thinking? Student B 91

93 Student C tries to use the measurements as a ratio to solve for the number of pages needed. However the student is able to solve for the number of pages and explain clearly how the calculations were made. How could you help this student move from the specific example to the generalization in part 4? Student C 92

94 Student D makes a typical error of only picking one of the ratios in part 1. In part 2 the student is able to find the size of the new photo and calculate the number of pages. To find the number of pages in part 3 the student attempts to make a drawing. Why doesn t the drawing work? What error does the student make? Again, the student gives the process for a specific case in part 4 rather than reaching for a generalization. Student D 93

95 Look at the process Student E used for finding the number of pages. What does the student not understand about the meaning of operations? In part 3 the student is unable to visualize the relationship between the height and width, even though the student could identify the ratio in part 1. Notice the struggle with visualization by examining the drawing in part 4. What might be your next steps with this student? Student E 94

96 Student F is able to identify most of the equivalent ratios. The student calculates the size of the new photo, but then reduces the answer. Why isn t this appropriate? Notice that the student needs to multiply to do the division in part 2b. The student isn t able to find the measurements in part 3, but does attempt to make a generalization in part 4. What is missing from the explanation to make it complete? Student F 95

97 Student G uses a ratio table to check and compare ratios in part 1. The student solves part 2a correctly, but can t use that information to find the number of pages. Can you make sense of the work in part 2b? In part 3 the student is able to find the correct dimension. The student makes a diagram that could have helped to find the number of photos on the page, but can t use it correctly. The student does not attempt a generalization in part 4. Student G 96

98 Student F, continued 97

99 Student G does not have a firm understanding of the equivalent ratios. Notice that in 2a the student cannot find the correct measurements even with the drawing. The student uses multiplication to solve part 2b. The student does not attempt to use mathematics to find the number of photos on a page, but tries to talk about objects like photo boxes. Student G 98

100 Student H only identifies one ratio in part 1. The student can find the measurements in part 2a. In part 2b the student doesn t connect the question to the previous answer or the picture. The student is very literal in the interpretation. The student lack of number sense is shown in the explanation in part 4. If you halve the dimensions, the student wants to double the number of pages. What might be next steps for this student? Student H 99

101 Student H, continued 100

102 8 th Grade Task 5 Photos Student Task Core Idea 4 Geometry & Measurement Show understanding of spatial relationships. Analyze characteristics and properties of two-dimensional geometric shapes; develop mathematical arguments about geometric relationship; and apply appropriate techniques, tools, and formulas to determine measurements. Understand relationships among the angles, side lengths, perimeter, areas, and volumes of similar objects. Solve problems involving similarity and scale factors, using proportional reasoning. Mathematics of this task: Identify equivalent ratios Find dimensions of similar rectangles and reason about the number that will fit on a page Write a generalization to find the number of photos on a page given the width Based on teacher observation, this is what eighth graders know and are able to do: Find the new dimensions of the photo in 2a Identify 2 or 3 of the equivalent ratios Find the height of the photo in part 3 Areas of difficulty for eighth graders: Finding all the equivalent ratios Finding the number of photos on a page Making a generalization or even understanding that a generalization is being asked for Identifying equivalent ratios: 8% only marked 8:12 41% omitted 4:6 45% omitted 2:3 62% omitted 1:1.5 51% omitted 16:24 52% omitted 24:36 29% added in 12:8 5% added in 3:2 101

103 The maximum score available for this task is 9 points. The minimum score for a level 3 response, meeting standards, is 5 points. More than half the students, 65%, could find the measurements of the new photo in part 2a. 39% could also find 2 or 3 equivalent ratios in part 1. 14% could identify 3 to 5 ratios in part 1, find either the new measurements or number of photos in part 2, and solve all of part 3(finding measurements and number of photos per page without a drawing). 8% could find 4or 5 ratios in part 1, find the new measurements in part 2, and solve all of part 3. Less than 1% of the students could meet all the demands of the task including making a generalization. 35% of the students scored no points on this task. 87% of the students with this score attempted the task. 102

104 Photos Points Understandings Misunderstandings 0 87% of the students with this score attempted the task. Students struggled with the size of the new photo in part 2a. 6% of the students thought the dimensions would be 12:8. 5% thought the answer was 8:12. 4% thought the new dimensions 1 Students could find the dimensions of the new photo. They struggled with identifying the correct ratios in part 1. 2 Students could find the dimensions in 2 and identify 2 or 3 equivalent ratios. 4 Students could identify 2 or 3 equivalent ratios, find either the dimensions of the photos in 2 or the number of pages needed and solve all of part 3. 5 Students could find 4 or more equivalent ratios, find the dimensions of the new photo in 2 and 3, and find either the number of pages in 2 or the number of photos per page in 3. 8 Students could identify equivalent ratios, find dimensions of new photos, reason about the number of photos on a page and find the number of pages needed for a given number of photos. Students could make a generalization about how to find the number of photos on a page given the width. were 18:12. 8% only marked the given ratio, 8:12. See chart. Students struggled with the height in part 3. 8% thought the height would be 11 in. 8% thought the height would remain 12 in. 6% thought the height would be 2 in. 8% did not attempt this part. 7% of the students thought the number of pages in 2b would be 75. 6% thought the number of pages would be % gave 18 or some other fraction with 18. In finding the number of photos on a page in 3b, 24% of the students thought 4 photos would fit per page. 10% thought 8 photos would fit. Other popular answers were 16, 48, and 96. Students did not interpret question 4 as asking for a generalization. Most students attempted to give a process for a specific case. Some students gave rules with addition or subtraction (6%). 7% gave the area formula. 14% had other rules involving multiplication. 103

105 Implications for Instruction Students need more experience with ratios and using them in context. Students need to have experience with spatial relationships, being able to visualize how items fit on a page, and using those relationships to make calculations. Students need more experiences with problems in context involving multiple steps to find a solution. This helps them develop the ability to organize their calculations and make sense of what they know and what they need to find. Students at this grade level need frequent opportunities to make generalizations or rules for how to calculate photos on a page, how to express a geometric formula, or how to find any term in an algebraic pattern. The grade level expectation for eighth grade is to move from thinking about specific cases and specific events to thinking about generalizations, what is true for all or for a larger set. Ideas for Action Research Many students had difficulty finding equal ratios. One way to help them see connections might be to play an adaptation of Silent Board Games. Usually, the teacher puts up a variety of inputs, and students try to guess outputs. For example, the teacher might start with: x ?? y All students are given a chance to think about the rule. If they think they know an answer, they come down and write and answer in a blank space. If they are correct, the answer stays in the table. If an answer is incorrect, the teacher erases the answer. The game continues for a while and then the teacher checks to see if people know the rule. Students should have individual think time and pair share time before sharing the results with the class. In the modified version, the teacher would use ratios: Students should try to find the missing number in each ratio. Then students should try to find a rule to prove that all the ratios are equivalent. Students also had difficulty defining a rule for how many pictures fit on a page. Students may need some practical experience with fitting something on a page. Maybe have them work in pairs with 2 or 3 sizes of post-its. Have them try to write instructions to read over the phone for someone trying to find the number of post its to fit on a page. Then ask them if they can write a number sentence to describe their verbal rule: first for a specific case and then try to write a generalized rules. 104

106 After some generalized, hands on experience you might have the class critique some of the rules from the toolkit or from your own student work. For example look at the work of Student G. I might show students this piece of the work: What do you think this student is thinking? How could you write it as a number sentence? Is this a rule for any number? How would the number sentence change? Next I might show them the work of Student F. How might this answer be improved? What do you think the student is unclear about? If a final example is needed, I might ask: Jesse gives a rule for a specific size photo. What suggestion or hint could you give Jesse to help her write a rule for any size photo? Are there any other big mathematical ideas you think students should discuss regarding this lesson? What pieces of student work could you use to help pose questions to get students to examine these ideas? Is there a similar task that you can use with students to see if they can apply what they have learned? 105

107 Reflecting on the Results for Eighth 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 were 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? 106

108 Five areas stood out for the Collaborative as a whole: 1. Spatial Visualization In Square Patterns, many students lacked the ability to take the tile apart into sections to help find the number of black and white tiles. They relied on counting every tile. In the last part of the task, almost all students only considered the tiles or some part of the tiles shown in the diagram. Students did not know how to think about or expand the pattern beyond the diagram to find what ten patterns would look like. In Photos, students could not reason about how the photos would fit on the page. 2. Applying Procedural Knowledge In Averages students could not apply procedures about averages to reason about a weighted average. Students knew it was something about adding and divided, but used the incorrect numbers in the procedure. Students struggled in Square Patterns with how to do the division in finding the percents. They often divided the top number into the bottom number. Their language for talking about division was weak. 3. Comparisons Students did not understand how to set up percents as a comparison of part to whole. Students often compared part to part or whole to part. In making probabilities, students often compared favourable to unfavourable outcomes rather than favourable to possible outcomes. In Photos students did not recognize equivalent ratios and had difficulty using ratios in context. 4. Generalizations Students could not generalize about pattern growth in Square Patterns. Students seldom recognized that the final question in Photos was even asking for a generalization about how to find the number of photos on a page given any size width. Most students tried to describe a specific case. Some students struggled with just picking the correct operation. 5. Interpreting graphs Students did not understand the idea of instantaneous change. Students did not see that the change in weight was caused by adding or subtracting objects from a scale. Students need more experiences with measurement on scales. Students had trouble identifying the slope or changes in slope with the action of a story over time. Students were often very literal, looking only at the labels on the graph, rather than trying to refer back to the context of the story and describe the actual action taking place. 107