Total Student Count: Grade pg. 2

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1 Grade pg. 1

2 Total Student Count: 3170 Grade pg. 2

3 8 th grade Task 1 Pen Pal Student Task Core Idea 3 Algebra and Functions Core Idea 2 Mathematical Reasoning Convert cake baking temperatures between Celsius and Fahrenheit. Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Express mathematical relationships using expressions and equations Use symbolic algebra to represent situations to solve problems Recognize and generate equivalent forms of simple algebraic expressions and solve linear equations Employ forms of mathematical reasoning and justification appropriately to the solution of a problem. Invoke problem-solving strategies Use mathematical language to make complex situations easier to understand Grade pg. 3

4 Grade pg. 4

5 Pen Pal Rubric The core elements of performance required by this task are: use a formula Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: 338 Shows correct work such as: 9/5 x = 2. Gives correct answer: (Accept 176 to 177) Shows correct work such as: = 9 / 5 C and 318 x Gives correct answer: C = 5 / 9 (F - 32) or equivalent Partial credit C = 5 / 9 F (1) 2 Total Points 6 2 Grade pg. 5

6 Looking at Student Work on Pen Pal: Student A shows a very organized approach to solving parts one and two, starting with substitution and then using order of operations to solve each stage of the computations. Student A is able to use inverse operations to change the equation from solving for Fahrenheit to solving for Celsius, including proper notation to show that the 5/9 is multiplied by both the F and the 32. Student A Grade pg. 6

7 Student B also has an organized way of showing the steps in the solution process, including the conversion of the formula. Student B shows the distribution of the 5/9 over both terms as well as demonstrating a good command of working with fractions. Student B Grade pg. 7

8 Some students, like Student C, convert the fraction in the equation to a decimal equivalent to simplify the calculations. Student C Grade pg. 8

9 Student D has the number sense to solve the equation in part 1, converting Celsius to Fahrenheit, and to use inverse operations to change Fahrenheit to Celsius in part 2. The student knows the order to do the computations, but does not follow order of operations for the number sentence written for part 2. The number sentence is an expression of what the student did in the order the calculations were made. It does not follow mathematical convention. In part three the student does not use parenthesis around the (F-32) to indicate what should be done first. The student still has an idea but inverse operations, but not the mathematical tools to express it. The incorrect parentheses in part 3 were added by a scorer. Student D Grade pg. 9

10 Student E solves the notational issues by using a verbal description of how to convert from Fahrenheit to Celsius. Also notice how this student shows a separate problem for each computational process, unlike the work of Students A and B. Student E Grade pg. 10

11 Order of operations is crucial to working with inverse operations. Student F divides first and then subtracts, instead of the reverse. This is shown both in the solution to part 2 and the algebraic equation in part 3. Student F Grade pg. 11

12 For some students, this task exposes misconceptions about working with fractions. Student G knows the rule about invert and multiply, but uses it inappropriately in the first part of the task. However the rule does work successfully for part 2. Student G does not understand how to solve for an unknown and just exchanges the variables in the formula in part 3. Student G Student H seems to understand the process of working backwards in part 2 by first subtracting the 32 from both sides of the equation and then attempting to divide both sides by 9/5. However the student does not know how to do the computation. The student cannot solve the formula for C in the final portion of the task. Student H Grade pg. 12

13 Student I also struggles with how to compute with fractions. In part 1 the student subtracts the 32 from 170, not even recognizing implied operation of multiplication and ignoring the fraction. In part 3 the student just exchanges the variables, rather than trying to solve the equation for C. Student I Student J also struggles with fractions and the idea of substitution. Instead of multiplying the 170 by 9/5, the student adds the 9/5 to 32 and then adds that to the 170. Student J Teacher Notes: Grade pg. 13

14 Score Student Count %<= 44.5% 52.1% 71.1% 79.1% 86.5% 91.5% 100.0% %>=44.5% 100.0% 55.5% 47.9% 38.9% 20.9% 13.5% 8.5% The maximum score available for this task is 6 points. The minimum score for a level 3 response, meeting standards, is 2 points. A little less than half, about 48%, the students could substitute a value into a formula and do calculations involving multiplying by an improper fraction. Some students, about 21%, could use inverse operations to solve an equation working from the independent variable to the dependent variable in part 2 of the task. Almost 9% of the students could meet all the demands of the task, including converting the formula from solving for Fahrenheit to solving for Celsius. Almost 45% of the students scored no points on this task. 78% of the students with this score attempted the task. Grade pg. 14

15 Pen Pal Points Understandings Misunderstandings 0 About 78% of the students attempted the task. Students had difficulty working with fractions and using substitution. Some students tried to invert and multiply when the situation did not call for division. Some students did not know that a number preceding a variable 2 Students could substitute a value into a formula and do calculations involving multiplying by an improper fraction. 3 Students could substitute into the formula and convert from Celsius to Fahrenheit. They could also get partial credit on solving the formula for Fahrenheit. 4 Students could use a formula for converting Celsius to Fahrenheit and use inverse operations to solve for Celsius given the temperature in degrees Fahrenheit. 6 Students could convert between temperature scales, multiplying and dividing with fractions. Students could convert a formula for finding Celsius to a formula for finding Fahrenheit. meant multiplication. Students had difficulty calculating with fractions. 5% of the students used the 350º from part 2 of the task as their answer to part 1. Some students added or subtracted Answers ranged from 6 1/2 to 888. Students either did not understand that the 5/9 needed to be distributed over both the F and the +32 or they did not know how to use parentheses to make the order of the desired operations clear. Many students, about 7%, multiplied before subtracting giving them 162º. About 6% just repeated the original fact that the temperature was 170º. 3% continued to use the original formula and treated the 350º as a Celsius temperature. 17% of the students did not attempt part 2 of the task. Many students, 14%, just traded the positions of the variables, so C=9/5 F+32. 7% of the students understood something about inverse operations writing a formula with 32 instead of +32, C=9/5 F-32. Some students have trouble with division notation and wrote 9/5 divided by F % of the students did not attempt to do part 3 of the task. Grade pg. 15

16 Based on teacher observation, this is what eighth graders know and are able to do: Substitute numbers into a formula Areas of difficulty for eighth graders: Calculations with fractions Order of operations and use of parentheses Inverse operations Symbolic manipulation to solve an equation for a variable Strategies used by successful students: Converting fractions to decimals Working through the solution one step at a time Questions for Reflection on Pen Pal: To work this task students need to have basic computational skills with fractions. Students may try to use invert and multiply for multiplication as well as division. They may use inverse operations for whole numbers, but not for fractions. They may be able to show that the expression in part 2 needs to be divided by 9/5, but be unable to complete the calculation. Some students had trouble with algebraic or symbolic notation. They did not understand that a number before a variable implies multiplication. Did you see evidence of this error in student work? Students also had trouble with use of parentheses. They may have understood order of operation in order to solve part 2, but did not use parentheses in part 3. Some of this is caused because they do not have a clear idea about order of operations and/or distributive property. Some students had difficult manipulating a variable to solve an algebraic equation. They don t realize that variables can be operated on in the same way numbers are operated on to solve expressions. For example, they may simply exchange one variable for the other. Some students have basic misconceptions about variables representing numbers or values and how they work within an algebraic expression. When writing their own formulas, they may put both the independent and dependent variable on the same side of the equations ( C = C + F) or they may try to eliminate the variableness by writing a specific value instead of a formula ( C = 37.5) Other students do not distinguish the two types of variables writing formulas like C=9/5 divided by C. When looking through student work, see if you can identify which type of misunderstanding contributed to each particular error. Look at student work in part 1. What evidence do you have that students were able to multiply 9/5 x 170? How many students changed the 9/5 into a decimal? Did students think about solving the equation as a whole or chop it into several individual calculations? Did any of your students try to invert and multiply? Grade pg. 16

17 338º 350º used # from part 2 138º 170º º Adding before multiplying More than 370º Less than 300º How do these errors relate to the big ideas listed above? Now look at student work in part 2. How many of your students gave answers of: 176º 162º 170º 318º or 316º 662º No answer Other Finally look at the algebraic formulas for part three. How many of your students: C= 5/9(F-32) C=5/9 F 32 C=F 32/9/5 (Partial credit) C=9/5 F +32 C= 9/5 /F 32 C=9/5 F 32 Other No response Make a list of other formulas to see what they might reveal about student thinking. Teacher Notes: Implications for Instruction: Students at this grade level should be proficient with fractions. If uncomfortable with fractions, they should also be familiar with converting fractions to decimals to help them think about the operations and strategies needed to solve problems. Often students get confused when working with fractions because they have some basic holes in their understanding of number operations with whole numbers. Students should be exposed to a variety of types of multiplication and division problems with whole numbers, fractions, and mixed numbers (measurement, partitive, and factor/ product). Models, like the bar model, can help students to picture the actions occurring in the different types of problems. Students should also explore and start to make generalizations about how and why different categories of numbers effect solutions when multiplying and dividing. Students need to confront such previous misconceptions as multiplying makes numbers larger and dividing makes numbers smaller. Students need to start articulating what equal groups are represented in the problem. This will help them make sense of multiplication and division, as well as increasing their understanding of proportional situations. Grade pg. 17

18 Students at this grade level should also be preparing the foundation needed for understanding algebra. Part of this foundation should be the idea of doing and undoing (or working backwards). While students are working on computational proficiency, they should be exposed to a variety of problems, which work with this notion of inverse operations. While learning to find percent of a number, questions should also be posed like, If 30% of a quantity is 42, what is the original number? or If 2/3 of a number is 26, what was the number? Students in earlier grades have become comfortable with guess and check as a strategy to solve problems. Now is the time to help them transition to writing number sentences to describe the situation and working backwards to find the answer. This idea can be promoted by changing the problem to include fractions and decimals. The teacher might first describe a problem situation with whole numbers. Then change the numbers involved and have students discuss which strategies are most useful for each case. This will allow students to see the need for a more formal approach and work on computational fluency. Emphasizing problems which require students to work backwards also helps to sharpen students understanding of order of operations. By posing the problems in context, students can check their solutions for reasonableness and through classroom discussions and probing questions find a need for learning the importance of notation such as the use of parentheses. In working with simple algebraic equations, students need explicit help understanding that the variables work the same way as numbers. They need to learn to think about the computational steps they would use if the variable were replaced with a number and then express those number operations symbolically. Students need to learn how to manipulate or do operations on the variable. Teacher Notes: Grade pg. 18

19 8 th grade Task 2 Picking Apples Student Task Core Idea 3 Algebra and Functions Core Idea 2 Mathematical Reasoning Determine the cost of apples from the rates given. Solve to find the number of pounds of apples that could be purchased for $30. Compare the two pricing structures for apples. Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Model and solve contextualized problems involving inequalities Use graphs to analyze the nature of changes on quantities in linear relationships Employ forms of mathematical reasoning and justification appropriately to the solution of a problem. Invoke problem-solving strategies Use mathematical language to make complex situations easier to understand Grade pg. 19

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22 Picking Apples Rubric The core elements of performance required by this task are: work out costs from given rules Based on these, credit for specific aspects of performance should be assigned as follows 1. a. Gives correct answer: $50 Shows correct work such as: 10 x $ x $1 b. Gives correct answer: $47.50 Shows correct work such as: $ x $ x $ a. Gives correct answer: 20 pounds Gives a correct explanation such as: The first 10 pounds of apples cost $20. The remaining $10 buys 10 pounds. Altogether = 20 pounds. b. Gives correct answer: 16 2 / 3 pounds (accept 16) Gives a correct explanation such as: The entry fee is $10. The first 10 pounds of apples cost $15. The remaining $5 buys 6.6 (accept 6) pounds. Altogether = 16.6 pounds (accept 16) 3. Gives correct answer: more than 30 pounds (Accept 31) points section points 4 4 Shows work such as: David s: 10 x $ x $1 = $40 Pam s: $ x $ x $0.75 = $40 or Draws a correct graph 1 or 1 2 Total Points 10 Grade pg. 22

23 Looking at Student Work on Picking Apples Student A shows a clear understanding of the proportional relations described in per pound by showing the multiplication for the different amounts of apples. The student uses labels clearly to define what each computation represents. To solve for part 3 the student makes an organized list to show where David s cost is less than Pam s, at what point the costs are the same, and the where Pam s becomes less expensive. Student A Grade pg. 23

24 Student A, part 2 Student B is able to find the break-even point, where the cost for each person is the same. Then the student shows that at any point after 30 lbs. Pam s cost would be less. Student B has that sense than apples can be bought in pounds or fractions of a pound. Student B Grade pg. 24

25 Student C approaches the problem from a different perspective by identifying the cause for Pam s initial higher costs, i.e. the entrance fee. The student then shows how many pounds must be purchased to compensate for that initial cost. Student C Student D tries to use the organized list to find where Pam s cost is less than David s. The student knows that the change occurs between 30 lb. and 35 lb. The student s logic breaks down by forgetting that he is looking for the smallest amount where Pam s is cheaper rather than any amount where Pam s is cheaper. When looking at the table of values for 30 lb. and 35 lb., the student should have jumped to 31lb. Student D Grade pg. 25

26 Student E and Student F don t appear to try to narrow to the smallest amount of apples, where picking at Pam s is a better deal. They seem to be content with any value that will yield the desired results. Student E Student F Grade pg. 26

27 Many students, including almost half the students with scores of 8/10, left part 3 blank. They did not have any strategies for how to approach the problem. Student G has managed to make the amount of money the same, but has different amounts of apples. The key to the question is to find the point where costs are the same for the same weight of apples. Student G Student H forgets to use all the constraints: different costs for the first 10 lbs. and entry fee. Therefore the solution is incorrect. Student H Grade pg. 27

28 Student I has difficult understanding the proportionality of $2 per pound in David s Orchard, making errors in both 1a and 2a in finding the costs. However the student is able to understand and use it appropriately to find the costs of Pam s orchard in 1b. In part 2b, the student is on the right track for finding the number of apples Pam can buy with $30, however the student forgets to label the + 6 as pounds of apples and adds 6 lbs. to $25 instead of 6 lb. and 10 lbs. Many students had similar difficulty of adding items with different labels. Student I Grade pg. 28

29 Student J also has trouble with the dimensionality of the numbers. The student adds 10 lbs. and $20 to get 30 lbs. in 2a. Student J Grade pg. 29

30 Student K has trouble understanding the constraints of the task. The student does not understand the proportional language of per pound. In part 1 the student knows to multiply the additional pounds by $1, but treats the first 10 lb. as 1 unit costing $2, rather than $2 per pound. In part 1b, the student again understands the second quantity should be multiplied but ignores the entrance fee and cost of the first 10 pounds. In part 2 the student reasons backwards from his answer in part 1. Student K Grade pg. 30

31 Student L has trouble with the idea of per pound, thinking in terms of groups of tens instead of groups of 1. Notice also that the student does not change the cost after the first ten pounds. Student L Teacher Notes: Grade pg. 31

32 Score Count %<= 32% % 46.1% 55.0% 60.3% 71.4% 78.8% % 100% % % %>= 100% 68% 64.9% 57.8% 53.9% 45% 39.7% 28.6% 21.2 % 6.9% 3.8% The maximum score available for this task is 10 points. The minimum score for a level 3 response, meeting standards, is 5 points. Many students, about 64% could find the cost of 40 lbs. of apples picked at David s orchard and show how they figured it out. About half the students could find the cost of 40 lb. of apples at David s orchard or work backwards from an amount of money to the number of pounds picked at David s. About 45% could also make some sense of picking 40 lb. of apples at Pam s orchard by either getting a correct solution or a correct process. Some students, about 21% could find the cost of 40 lb. of apples at either orchard and the number of pounds that could be purchased for $30 at either orchard. 3.8% of the students could meet all the demands of the task, including finding the lowest point where Pam s orchard is less expensive than David s including some justification to back up that answer. 32% of the students scored no points on this task. 92% of the students with this score attempted the task. Grade pg. 32

33 Picking Apples Points Understandings Misunderstandings 0 92% of the students with this score attempted the task. Some students had difficulty interpreting the language of proportionality, per pound. 8% of all students had answers of $32 for apples at David s, because they didn t multiply the first 10 lbs. by $2. 8% of the students thought the cost of 40 lbs. was $40, just taking $1 for every pound and ignoring the difference in price for the first 2 Students could interpret the meaning of per pound and calculate the cost of 40 lbs. of apples at David s orchard. 4 Students could calculate the cost of buying 40 lbs. of apples at David s and find the number of pounds that could be purchased for $30, showing appropriate calculations for each. 6 Students could find the cost of 40 lbs. at both orchards. They could work backwards from a cost of $30 to the amount of apples purchased at David s, but not at Pam s. 8 Students could find the cost of 40 lbs. or the number of pounds that could be purchased for $30 at both orchards, showing appropriate calculations. 10 Students could find the cost of 40 lbs. or the number of pounds that could be purchased for $30 at both orchards, showing appropriate calculations. Students could also find the point where Pam s orchards were cheaper than David s. 10 lbs. Students had difficulty interpreting the 3 constraints for Pam s orchard. 11% of the students ignored the entry fee. 4% only added $1.50 for the first 10 lbs. getting answer of $34. 5% did not use monetary notation giving an answer of $ % of the students, who missed part 2a, ignored the change in cost for the first 10 lb. They thought $30 would get 30 lbs. 6% thought that if 40 lbs. = $40 and the first 10 lb. = $2, then $30 would buy 38 lbs. Other common answers for 2a were 15, 150, and 40 pounds. Some students had difficulty working backwards from a cost of $30 to the number of pounds at Pam s. Some calculated it as if all apples cost $.75, giving them an answer of 40 lb. Some students did not realize you can by fractional amounts of pounds, so they picked answers that would use most of the money like 13 or 15 pounds. Many students who were successful at all other parts of the task did not attempt the final part of the task. They did not know how to attack the problem. 65% of all students did not attempt this part of the task. Students might try guess and check to find some value where Pam s orchard was cheaper, but not narrow it down to the lowest amount giving answers as high as 100 or 187 lbs. Grade pg. 33

34 Based on teacher observations, this is what eighth graders knew and were able to do: They could multiply to find the costs of additional pounds Areas of difficulty for eighth graders: Interpreting the language of proportions, per pound Identifying and using all the constraints in a problem Labeling answers to understand what had been calculated Adding together inappropriate items like pounds and dollars The concept of a break-even point, setting up an equality to find out when two alternatives will yield the same value Standard monetary notation Questions for Reflection on Picking Apples What kinds of language do your students use to make sense of rates or proportional situations? Do you think they understand terms like per pound, per hour, per box? Do you think they see these terms as sets of equal size groups? What types of activities do students do to help them make sense of the meaning behind rates or proportions? Do they associate multiplication/ division with these ideas? In what ways? How are labels used in the classroom when solving problems? Do you provide explicit instruction to help students deal with dimensional analysis or how operations effect or change labels? Looking at student work in part 1 and 2, were students thinking in terms of function or doing several individual calculations? Look at student work for part 1a. How many of your students put: $50 $40 $32 $8 $30 $20 Other Can you follow the reasoning chain that led to each particular error pattern? What does this show you about student misunderstandings? Look at student work for 1b. How many of your students put: $47.50 $47.5 or $47.05 $55 $34 $37.50 Other What additional misconceptions contributed to the problems in this part of the task? What types of problems do students work requiring them to work backwards or do inverse operations? Do students work with this idea with computational procedures or just with problem-solving tasks? Are students comfortable with order of operations and how that works when undoing a procedure? Grade pg. 34

35 How comfortable are students with symbolic notation like parentheses, division symbols? When looking at student work in part 2, check student thinking to see if they combined inappropriate terms like $ and pounds. Did they lose track of the meaning behind their computations? Look at student work on 2a. How many students put: Other Look at student work on 2b. How many students put: 16 2/3 or No response Other What misconceptions led to these error patterns? What made this part more difficult for students? Now look at work in the final section. Many students did not know how to approach this part of the task. They did not have the sense of finding the point where the number of pounds and the cost were the same for both orchards. How many of your students did not attempt this part of the task? How many of your students guessed a large value (34 to 187) that made Pam s cheaper? How many lost track of some of the constraints (like entrance fee) when making their calculations for this section? Were successful students able to set up an equation to solve the problem or did they use guess and check to solve the problem? Do you have any problems in your text dealing with the idea of break-even point? Have you students worked with problems graphing two equations to find where they intersect? What strategies would you have expected or wanted your students to be able to use? Implications for Instruction: Students at this grade level should be comfortable with identifying and using constraints to solve problems. Students should be starting to do operations with labels to keep track of how the calculations change the labels. Students should also start to use equations to express the multiple constraints, rather than using a string of calculations. A big idea for middle grades is the ability to use proportional reasoning or understand multiplicative relationships. Students should be comfortable with the language of proportions or rates, like per pound, per hour, per person. Students need explicit instruction of help them connect cost per pound or miles per hour as representing equal groups that can be multiplied or divided. Students also need help seeing how these operations change the units. Grade pg. 35

36 Students at this grade level should be preparing for the transition to algebra. They need many opportunities to work problems involving inverse operations. They need to be confronted with situations with multiple steps, where order of operations makes a difference. So when students share solutions, it is important for them to be asked why they have different answers. The teacher might pose questions, such as, If two students both subtracted and both divided, how is it possible for them to get two different solutions? How can we determine which one is correct? Having students grapple with these issues helps them see the logic behind the rules or algorithms in a way that direct instruction alone doesn t. Another big idea to help then prepare for algebraic thinking is the idea of equality. In order for students to think about the idea of when Pam s orchard is cheaper, it is helpful to ask the question, At what point are the two orchard s the same? As more than half the students had no idea how to even start this part of the task, giving them this as a discussion point when returning the papers would be a good classroom activity. Teacher Notes: Grade pg. 36

37 8 th grade Task 3 Fractions of a Square Student Task Core Idea 4 Geometry and Measurement Core Idea 1 Number and Operation Core Idea 2 Mathematical Reasoning Calculate the areas and name the fractional pieces of 6 regions of a square. Analyze characteristics and properties of two-dimensional geometric shapes, develop mathematical arguments about geometric relationships, and apply appropriate techniques, tools, and formulas to determine measurements. Develop strategies to determine area Create and critique arguments concerning geometric ideas and relationships Understand relationships among the angels, side lengths, perimeters and area of shapes Understand number systems, the meanings of operations, and ways of representing numbers, relationships, and number systems. Understand and use the inverse relationships of squaring and finding square roots 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 Invoke problem-solving strategies Verify and interpret results of a problem Use mathematical language to make complex situations easier to understand Grade pg. 37

38 Grade pg. 38

39 Grade pg. 39

40 Fractions of a Square Rubric The core elements of performance required by this task are: calculate areas and fractional regions of a square Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answer: Piece A = 1/4 accept equivalent fractions 2. Gives correct answer: Piece B = 1/16 accept equivalent fractions 3. Gives correct answer: Piece C = 1/32 accept equivalent fractions points sectio n points 1 1 Shows correct work such as: The area of Piece C is 1/2 cm 2 or 1/2 /16 4. Gives correct answer: Piece D = 5/32 accept equivalent fractions Shows correct work such as: The area of Piece D is 2 1/2 cm 2 or 21/2 /16 5. Gives correct answer: Piece E = 6/16 or 3/8 accept equivalent fractions Shows correct work such as: The area of Piece E is 6 cm 2 or 6/16 6. Gives correct answer: Piece F = 1/8 accept equivalent fractions Shows correct work such as: Adds all their values A through E, then subtracts them from 1. 1 / / / / / 8 = 28 / 32 = 7 / 8 Total Points Grade pg. 40

41 Looking at Student Work on Fractions of a Square: Student A solves this problem by thinking about the area of the whole (4 x 4) and area of the part. For piece A the student calculates the two measurements and tries to make a fraction, then realizes that the part should be the numerator. In trying to find the area of piece D, the student breaks the piece into two parts, a small square representing 1/16 and a triangle. The student tries to find the area of the triangle and has to try a couple of approaches before finding a successful strategy. Again in calculating piece F, the student tries to use Pythagorean theorem before using the cumbersome process of totaling all the other parts. This task was definitely a problem-solving situation rather than an exercise for the student. But good number sense and checking of solutions for sense making, allowed the student to sort through the work and arrive at correct final solutions. Student A Grade pg. 41

42 Student A, part 2 Grade pg. 42

43 Student B also reasons about the relationship of part to whole by first calculating the areas and then reducing the ratios. Notice that the student is able to simplify the fraction of 1/2/16 to 1/32 in part 2 and that the formula for area of a triangle in clearly present. Notice that the sketches help to clarify the areas being calculated. In calculating piece F, Student B incorrectly uses 4/32 for part A instead of 8/32. The strategy could have led to a correct solution. Student B Grade pg. 43

44 Student B, part 2 Grade pg. 44

45 Student C has a clear approach to solving the problem by comparing all pieces to a common denominator. It is unclear if the student knew how to simplify the complex fractions in parts C and D. Notice that by using a common denominator for all the pieces, that adding all the areas was much easier in finding the value of piece F. Student C Grade pg. 45

46 Student D uses diagrams to make equal size parts, then counts the parts to find the whole. This might be considered a measurement model. In thinking about piece C, the student draws a useful diagram but makes some errors in trying to count all the pieces. It is unclear how the student calculated the size of piece F. Grade pg. 46

47 Student E tries to find the number of pieces that will fit into the whole. The student attempts to make each piece into a unit fraction. Spatial visualization without considering measurements causes an error in estimating piece D. In trying to fit three of piece E into the larger square, the student discovers there is not enough room for the final piece. So if E is a little smaller than 1/3, maybe it is 1/4. The student is relying on visual estimation. Student E Grade pg. 47

48 Student F labels the dimensions on the diagram and calculates the area of 3 squares to use as building blocks for thinking about the area of the different pieces. Student F struggles with the mathematical notation. For example in calculations for piece B and C, the student shows multiplying by 4 and by 2 when the student is actually multiplying by 1/4 and by 1/2 (or dividing by 4 and dividing by 2). In calculating the size of piece D, the student seems to be trying to follow a pattern rather than reasoning about the shapes. It is unclear how the student reasoned about piece E. Student F Grade pg. 48

49 Student F, part 2 Grade pg. 49

50 Some students did not appear to have even visual estimation to help them reason about the size of the pieces. Student G has some idea about fourths, but reasons that part A is 1/2, which is visually incorrect. In reasoning about piece B the student may have lost track of the whole and thought that B is 1/4 the size of A. Just looking at the answers, the student should have been able to think that C is smaller than B, so if B=1/4 then C cannot be 3/4. Student G Grade pg. 50

51 Some students thought about area and lost the idea of finding the fractional part represented by each piece. Student H calculates a correct area for piece A and has an area for the whole (although the number sentence is incorrect). Student A is unable to even follow some of the simple calculations given in the number sentences. For example 1 x 1 2. Student H Grade pg. 51

52 Student H, part 2 Teacher Notes: Grade pg. 52

53 Score Count %< 43.7% 55.9% 63.7% 73.6% 83.2% 87.1% 92.6% 95% 97.2% 98.3% 100% %>= 100% 56.3% 44.1% 36.3% 26.4% 16.8% 12.9% 7.4% 5% 2.8% 1.7% The maximum score available for this task is 10 points. The minimum score for a level 3 response, meeting standards, is 4 points. About half the students could reason that piece A was 1/4 of the whole square. Some students, about 44%, could reason the A represented 1/4, and that piece B was 1/16 of the whole square. About 26% of the students could also calculate that piece C was half the size of piece B or 1/32 and show either diagrams or area calculations to back up their solution. Less than 10% of the students could reason about the complex shapes of D and F. Less than 2% of the students met all the demands of the task. Almost 44% of the students scored no points on this task. 58% of the students with this score attempted the task. Grade pg. 53

54 Fractions of a Square Points Understandings Misunderstandings 0 58% of the students with this score attempted the task. 6% of all students gave whole number answers rather than fractions. 5% of the students thought piece A was 1/2. 5% thought piece A was 3/4, maybe thinking about the remaining part instead of the 2 Students were able to use diagrams or area to find the fractional parts for pieces A and B. 4 Students were able to find the fractional parts for pieces A,B, and C and give some justification for C. 6 Students were able to find the fractional parts for pieces A,B,C, and E. 8 Students were able to find the fractional parts for pieces A,B,C,D, and E. 10 Students could use diagrams, calculate areas of rectangles and triangles, and reason about part/whole relationships to find the fractions represented by all of the pieces. actual piece. 7% of the students though that piece B represented 1/8 instead of 1/16. 3% thought piece B was 1/4, losing track of what represented the whole. 6% of the students did not change the complex fraction.5/16 to 1/32. 6% thought C was equal to 1/8, losing track of the whole. 4% thought C was equal to 3/16. Some students tried to use visual estimation to find the size of piece E, getting answers like 1/3. Some students gave whole number answers. Common answers included 1/2, 1/6, and 3/4. Students used visual estimation to think about how many D s would fit in the whole square. 10% of all students thought D = 1/4. 6% of the students thought 1/5 and another 6% thought 1/6. Some students, 3%, did not reduce the complex fraction 2.5/16. Of the students who attempted the task, 23% did not attempt to find the fraction for piece F. Many students just guessed a size by thinking about relative sizes, giving answers like 1/6 or 1/4. Other students gave whole number answers. Grade pg. 54

55 Based on teacher observations, this is what eighth graders know and are able to do: Decompose the diagram into familiar shapes, like rectangles and triangles, or into equal size parts Calculate areas for rectangles Find the fractional parts of small squares within the larger square Areas of difficulty for eighth graders: Explaining their thinking or showing their work Giving areas instead of fractional parts Decomposing more complex shapes and finding the area of triangles Trying to estimate rather than having a strategy to find the exact size of the piece Remembering what shape represented the whole Questions for Reflection What opportunities have students in your class had with composing and decomposing shapes? How might this help them make sense of this task? When working with area, are students expected to memorize formulas and/or are they asked to make sense of the procedure and give justifications for the operations involved? Do you think your students could explain why the formula for an area of a triangle has a divide by 2? Do you think students could explain why the bases are added in finding the area of a trapezoid? How many of your students tried to solve the task by: o Thinking how many pieces of each shape would fit into the whole o Comparing area of the part to area of the whole o Subdividing the whole shape into equal size small shapes and counting the number of small pieces that would fit into the larger piece o Estimation o Other What strategies would you have expected students to use in solving this task? What surprised or disappointed you about their work? Look at student work for piece C. How many of your students thought the piece represented: 1/32 1/8.5/16 3/16 1/16 No response Other Look at student work for piece D. How many of your students thought the piece represented: 5/32 1/4 1/5 1/8 2.5/16 Integer No response Other How is the logic involved different for the different error patterns? Grade pg. 55

56 Implications for Instruction: Students at this grade level should be fluent in using formulas to find area of rectangles and triangles, particularly right triangles where the height is one of the dimensions. They should be able to decompose complex shapes into familiar shapes to calculate areas. The ability to interpret diagrams, find or derive appropriate dimensions, is an important skill that needs to be fostered through much practice and discussion. Successful students had two strategies for thinking about the fractional relationships involved in this task. The first was to calculate the area of the whole and the area of the part in square cm. and use this to make a comparison in the form of a fraction or ratio. The second was to think of the definition of fractions, dividing a shape into congruent parts, and then comparing the number of parts of the small shape that can fit into the total number of parts. Students using this second approach, relied on visual thinking and diagrams, rather than calculating area. Students should be encouraged to develop fluency with both approaches. In sharing student solutions during classroom discussions, teachers should try to get students to notice the similarities and differences in the two approaches and help provide them with language to think about or classify the strategies. Students at this grade level should understand that mathematics is about quantifying relationships whenever possible. This task was not about estimating that a particular piece looks to be about one-fifth or one-ninth. The dimensions were given so that a more precise relationship could be calculated. Students should have frequent discussions about when estimations are acceptable and when exact calculations are called for. On the other hand, students should be encouraged to develop the habit of mind of checking their work for reasonableness. Some students gave answers for small pieces that were larger than their answers for larger pieces. For example, students might have given an answer of 2/3 for piece D or 1/2 for piece E, when those pieces are both clearly less than 1/2. In these cases, the teacher should do further checking to find out if students have a clear understanding of the relative sizes of common benchmark fractions and find out what they know about fraction size. For these students, experiences estimating relative size of part to whole may be very appropriate to develop understanding of meaning before they do further work on precise calculations. Some students are not clear that fractions are a way of describing a relationship between two things. Like students at lower grade levels, they confuse finding the area of a part with the description of a part-to-whole relationship. These students need more work with sense-making and developing a working definition for fractions, before continuing with work on algorithms and procedures. Teacher Notes: Grade pg. 56

57 8 th grade Task 4 Sports Student Task Core Idea 1 Number and Operation Core Idea 5 Data Analysis Core Idea 2 Mathematical Reasoning Work with percentages to create and interpret favorite sports information in circle graphs. Communicate mathematical understanding of percentages. 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 Students deepen their understanding of statistical methods used to display, analyze, compare and interpret data sets. Represent and analyze data in the form of graphs including circle graphs Discuss and understand the correspondence between data sets and their graphical representations 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 Invoke problem-solving strategies Verify and interpret results of a problem Use mathematical language to make complex situations easier to understand Grade pg. 57

58 Grade pg. 58

59 Grade pg. 59

60 Sports Rubric The core elements of performance required by this task are: work with percentages use circle graphs Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives three correct percentages: 30, 45, 25 Partial credit 1 value correct and total 100% or 2 values correct Shows a correct method for at least one calculation such as: 78/260 x Draws and labels one sector correctly. Draws and labels the other two sectors correctly. 3. Gives a correct answer: 180 points 2 (1) (1) 1 1ft 1ft 1 section points 3 2 Shows correct work such as: 72/40 x Gives a correct explanation such as: We only know about the percentages, not the actual numbers. or 1 There might be fewer 8 th grade boys in Appleton than in Lake City. 1 Total Points Grade pg. 60

61 Looking at Student Work on Sports Student A clearly labels each computation, including finding the total number of girls surveyed. The student is able to set up an equation and solve it to find the total in part 3. Student A Grade pg. 61

62 Student A, part 2 Student B has the habit of mind to check that all the percents equal 100. The student also verifies the solution to part 3 in two different ways. Student B Grade pg. 62

63 Student B, part 2 Successful students had a variety of strategies for finding the total number of boys in part 3. Student C cannot use inverse operations, so he uses guess and check to find the solution. This could be cumbersome for problems with larger numbers or an answer that was not an integer. Student C Grade pg. 63

64 Student D applies composing and decomposing percents to arrive at 100% (100%=40% +40% + 20%, 20% = 40% /2) Student D Student E also uses knowledge of percents to find 100%. (20% = 1/2 x 40%, and 20% x 5 = 100%) Student E Grade pg. 64

65 Students F and G think about decomposing 40% into groups of 10%. Then they use 10 groups of 10% to find the total. Student F Student G Grade pg. 65

66 Student H is able to use proportional reasoning to find the total. 2 1/2 x 40% = 100%, so 2.5 x 72 = total students surveyed. Student H Student I treats the 72 like a total instead of the number of boys liking basketball. The student takes 35% of 72 and 25% of 72 in an attempt to find the boys liking baseball and soccer. The student is not thinking about the 72 as a percentage of some number. Student I Grade pg. 66

67 Student J also treats the 72 as a total. Student J Student K confuses the rule that percents should total 100% with the total number of students surveyed. If 72 out of 100 people like basketball, there are 28 people left to divide between baseball and soccer. Student K Grade pg. 67

68 Student L is not able to calculate percents. The student appears to make a guess based on the relative size of girls surveyed. Student L has made up an algorithm, that students equal the percent This was used by 5% of all students. Student L Grade pg. 68

69 Student M appears to know some procedure connected with percents about moving the decimal 2 places. For soccer and tennis this makes the percent the same as the number of students surveyed. For basketball, the process is confusing, because the value would be less that the other two categories when the number surveyed is more. So the student solves this discrepancy by only using the numbers after the decimal. The student does not notice or check to see if the percents add to 100. Student M Teacher Notes: Grade pg. 69

70 Score Student Count %<= 36.6% 44.5% 50.6% 55.8% 62.0% 73.8% 80.9% 88.4% 100% %>= 100% 63.4% 55.5% 49.4% 44.2% 38% 26.2% 19.1% 11.6% The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 5 points. Many students, about 63%, could reason that the sample size for Appleton and Lake City was unknown so the percents wouldn t necessarily be the same as the relative number of boys represented. Almost half the students could calculate some of the percents in part 1 of the task and relate at least 1 percent to an area on a circle graph. Some students, about 38%, could calculate percents of a number and plot percents in a circle graph. Almost 12% of the students could meet all the demands of the task, including working backwards from a percent of a number is 72 to find the missing total number and reason about why there might not be more boys preferring baseball in Appleton than Lake City. About 36% of the students scored no points on this task. 74% of the students with this score attempted the task. Grade pg. 70

71 Sports Points Understandings Misunderstandings 0 74% of the students attempted the task. 2 Students could calculate some of the percents of girls preferring different sports or they could convert incorrect percentages into a correct representation on the circle graph. 4 Students could calculate percentages of girls preferring a sport and represent one of the sports correctly on a circle graph. 5 Students could calculate percentages and make the appropriate representation on a circle graph. 6 Students could calculate percentages and make the appropriate representation on a circle graph. Students could reason that comparing students by percents in not necessarily an accurate predictor of total students preferring the sport. 8 Students could calculate percents and work with inverse operations, graph percents in a circle graph, and reason about making comparisons with percents. Many students had values totaling more than 100%. A few students had percents that totaled less than 50%. Students tried to move decimals to go from number of people surveyed to percentages. Students are trying to remember procedures without thinking about the meaning of percents and how they reflect part/whole relationships. Percents for soccer ranged from 3% to 48%, with 35% as the most popular incorrect answer. For basketball answers ranged from 13% to 117%, with 40% and 46% as the most common. For tennis, answers ranged from 5% to 65%. 24% and 65% were tied for the most popular choices. Students had trouble with the scale on the circle graph. They did not see that each section on the graph represented 5%. Students had difficulty explaining why 45% of the boys from Appleton might not be more than 35% of the boys from Lake City. 24% of the students did not attempt part 4 of the task. Students could not reason with percents. Incorrect responses included ideas like 55% preferred something else, there s the same number on each team so it doesn t matter, maybe the players on the other team are better. Students struggled with inverse operations, working from a percent of a number is 40% to finding the original total. 18% of the students did not attempt this part of the task. 5% thought the answer was 196 instead of % thought the total for all three sports was less than the number who liked basketball. 4% thought the total would be 100 students. Grade pg. 71