Overall Frequency Distribution by Total Score Grade 8 Mean=18.07; S.D.=11.01 300 200 Frequency 100 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Frequency Eighth Grade 2003 pg. 1
Level Frequency Distribution Chart and Frequency Distribution 2003 - Numbers of students Grade 8: 8178 tested: Grade 8 2000-2001 Level % at ('00) % at least ('00) % at ('01) % at least ('01) 1 19% 100% 19% 100% 2 47% 81% 42% 81% 3 26% 34% 31% 39% 4 8% 8% 8% 8% Grade 8 2002-2003 Level % at ('02) % at least ('02) % at ('03) % at least ('03) 1 27% 100% 31% 100% 2 37% 73% 26% 69% 3 23% 36% 19% 43% 4 13% 13% 24% 24% 3000 2500 2000 Frequency 1500 1000 500 0 0-10 1 Minimal Success 11-19 2 Below Standard 20-27 3 At Standard 28-40 4 High Standard Frequency 2509 2158 1556 1955 Eighth Grade 2003 pg. 2
8 th grade Task 1 Pete s Numbers Student Task Core Idea 2 Mathematical Reasoning Core Idea 3 Algebra and Functions Use mathematical reasoning to solve number 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 Verify and interpret results of a problem. Invoke problem-solving strategies Use mathematical language and representations including numerical tales and equations, simple algebraic equations, formulas, and graphs to make complex situations easier to understand Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Recognize and generate equivalent forms of simple algebraic expressions and solve linear equations Eighth Grade 2003 pg. 3
Eighth Grade 2003 pg. 4
Eighth Grade 2003 pg. 5
Looking at Student Work Pete s Numbers Students did very well on Pete s Numbers, with more than 1/3 getting perfect scores. Looking carefully at the methods or strategies these students were using can give us insight into how students are progressing in their use and understanding of symbolic algebra, algebra as problem-solving tool, and understanding of variable. Student A successfully translates the problem statements into equations and uses symbol manipulation to solve systems of equations. Student A Eighth Grade 2003 pg. 6
Student B has a clear understanding of constraints and how to systematically check that the demands of each constraint have been met. In part 3 Student B uses number sentences that mirror the language from the prompt. The student uses guess and check and substitution to find a correct solution to fit both constraints. Student B Eighth Grade 2003 pg. 7
Student C appears to understand how to use symbolic algebra to write down the constraints of the problem in the form of equations. Examining the student s thinking, there is no evidence of using the algebra to solve the equations. Instead Student C uses a systematic guess and check approach for checking the equations. What questions might a teacher ask to help the student connect his strategy to the concept of a variable? Student C Eighth Grade 2003 pg. 8
Student D shows an understanding of how to write equations appropriate for the constraints of the problem. Student D can successfully solve equations if there is the possibility for substitution (see part 2), but struggles or doesn t have the skills for correctly manipulating the variables when simple substitution does not work. Student D Eighth Grade 2003 pg. 9
Student E is still struggling with the idea of constraints. The student is manipulating numbers, but unclear as to what the answers represent. In part 2 Student E solves correctly for both constraints, but confuses the numbers (variables) for the partial answers in the second equation. In part 3 the student only solves for one of the two constraints and therefore does not reach the unique solution defined in the problem. Student E Eighth Grade 2003 pg. 10
Sometimes students with low scores demonstrate a strong foundation that will assist them in solving problems as they gain new skills. Student F shows the ability to use algebraic notation to write the appropriate equations for part 2 and 3. The student is still in the acquiring skills stage of working with symbol manipulation and does not solve the equations accurately. Student F Eighth Grade 2003 pg. 11
Other students do not understand to how look at constraints and figure out all the demands presented. In the work of Student G only one of the two constraints is addressed in each part of the task. The student does not connect the two sentences in the problem statement. Student G Teacher Notes: Eighth Grade 2003 pg. 12
Frequency Distribution for Each Task Grade 8 Grade 8 Pete s Numbers Pete's Numbers Mean:4.49, S.D.: 2.99 3000 2500 Frequency 2000 1500 1000 500 0 0 1 2 3 4 5 6 7 8 Frequency 1310 283 1343 320 340 1410 293 387 2492 Score Score: 0 1 2 3 4 5 6 7 8 % < = 16.0% 19.5% 35.9% 39.8% 44.0% 61.2% 64.8% 69.5% 100.0% % > = 100.0% 84.0% 80.5% 64.1% 60.2% 56.0% 38.8% 35.2% 30.5% The maximum score available for this task is 8 points. The cut score for a level 3 response is 5 points. Most students (about 84%) could find the numbers using the clues in part 1 of the task and about 80% could show the proof for picking those numbers. Many students (over 60%) could find the numbers and prove how they met the conditions for part 1 and find the numbers to fit the clues in part 2. About 30% of the students could use the clues to find the missing numbers in all 3 parts of the task and show how the numbers fit the clues. About 16% of the students scored no points on this task. 91% of the students with a score of zero attempted the task. Eighth Grade 2003 pg. 13
Pete s Numbers Points Understandings Misunderstandings 0 90% of the students with this score attempted the problem. 2 Students with this score could find the missing numbers in part one of the task and show how they checked the numbers with the clues. 4 Students could find the missing numbers for 2 of the 3 parts of the task and show the work to prove that one of those sets was correct. 5 Students with this score could find the numbers using the clues in parts one and two and show how the numbers fit the clues. Students usually found numbers that worked for only one of the two clues in each part of the problem. They did not make a connection between the two clues. While many students can use guess and test and do mental computations for finding the numbers, they have not formed a habit of checking numbers with the clues to prove that the numbers meet all the demands of the task. They do not see the need to justify their answers. Students do not consistently check their answers against constraints. In some cases they would check one, but not both constraints or they could write algebraic equations, but students lacked the skills to solve them. About 1/3 of the students made some attempt to use algebra. Students with this score could not find the missing numbers for part 3 or show the work to prove why they were correct. 7 Students did not justify either part 1,2, or 3. 8 Students could use number clues to find missing numbers. Students may have used algebra, substitution in number sentences, or guess and check to show why the numbers were correct. Teacher Notes: Eighth Grade 2003 pg. 14
Based on teacher observations, this is what eighth grade students seemed to know and be able to do: Find numbers to fit number clues Use guess and check strategies to find missing numbers Write symbolic expressions or number sentences to fit problem statements Areas of difficulty for eighth grade students, eighth grade students struggled with: Solving two equations for two unknowns Justifying solutions Questions for Reflection: Look carefully at the work of students with scores above 5. How many of these students: Used guess and check Forgot part of a justification Wrote number sentences and/or used substitution Wrote appropriate equations Used rules of algebra to correctly solve equations How often do students in your classrooms have the opportunity to make justifications or get credit for showing their work? What types of experiences do students have identifying constraints within a problem? Are they asked how they know they have met all the constraints? What activities or experiences have students had with the concept of variable? In what ways to do you help students make the transition or connections from guess and test to algebra? What might you like to do or remember for next year? Do students in your algebra classes see the connections between the procedures they are learning to their usefulness in solving problems or are they still more comfortable with more familiar strategies? What might be holding them back? Implications for Instruction: Students at this grade level should be comfortable with the idea of constraints and finding numbers that meet multiple conditions. They should also have experience with the need to justify how a solution fits the conditions in the problem. Students should ask themselves have I checked for each condition in the problem? Just as students at lower grades need help transitioning from the concrete to numeric and symbolic representations, students in eighth grade need help transitioning from guess and check to the use of algebra as a problem-solving tool. What questions or experiences can help them make that transition? How can we help them connect the multiple guesses to the concept of a variable? How can we get them to see the connection between the guesses and a unique solution in equations with multiple variables? Could graphing help them to see some of these connections or relationships? Perhaps their experiences in Pre-Algebra need to be more focused on some of these big ideas around variables and justification. Eighth Grade 2003 pg. 15
8 th grade Task 2 Squares and Rectangles Student Task Core Idea 4 Geometry and Measurement Core Idea 2 Mathematical Reasoning Use the properties of shapes to find similar shapes. Analyze characteristics and properties of two- and threedimensional geometric shapes; develop mathematical arguments about geometric relationships; apply transformations and use symmetry to analyze mathematical situations; and apply appropriate techniques tools, and formulas to determine measurements. Understand relationships among the angles, side lengths, perimeter, and area of similar objects Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling. Employ forms of mathematical reasoning and justification appropriately to the solution of a problem. Formulate conjectures and test them for validity Eighth Grade 2003 pg. 16
Eighth Grade 2003 pg. 17
Eighth Grade 2003 pg. 18
Eighth Grade 2003 pg. 19
Looking at Student Work Squares and Rectangles Student A gives a completely acceptable response. In part one the student gives a good explanation for rectangle. The student can find the square on the diagonal in part 3 and correctly name the coordinates. The student gives an answer to part 7 based on the assumption that the vertices of the rectangle need to be on an intersection of the grid lines, and even includes the squares as member of the set of rectangles. Student A Eighth Grade 2003 pg. 20
Student A Eighth Grade 2003 pg. 21
Part 7 was a more open question allowing students to make more than one interpretation. Student B makes that assumption that the vertices do not need to be on the intersections of grid, but can appear anywhere along the appropriate parallel lines. Student B Eighth Grade 2003 pg. 22
Student C assumes the rectangle must stay in the ratio of 2/3.Student C does not consider the possibility that the horizontal dimension could be the height. Student D also assumes the ratio must stay 2:3, but considers the possibility of fractional sides with a size of 2 2/3. Eighth Grade 2003 pg. 23
Many students make some false assumptions. Students E and F struggle with the idea that squares are not rectangles. Student E Student F Eighth Grade 2003 pg. 24
Some students with low scores show quite a bit of understanding. Student G gives a good definition for rectangle. Student G can draw and correctly name the coordinates for two of the squares. Student G also demonstrates misconceptions or difficulties with measurements. When trying to draw the rectangle in part 5, the student counts the width as 3 instead of 4, giving a common mistake of 6 for the area. The student s explanation in part 7 is unclear. Student G Eighth Grade 2003 pg. 25
Student G, part 2 Eighth Grade 2003 pg. 26
A few students did not understand that the lengths of diagonal lines are not the same as perpendicular lines. Some students made a diagonal rectangle for part 5. See the work of Student H. Student H also demonstrates a common misconception that rectangles have 2 short sides and 2 long sides. Student H reverses the coordinates in part 4. Student H Eighth Grade 2003 pg. 27
Student H, part 2 Teacher Notes: Eighth Grade 2003 pg. 28
Grade 8 Squares and Rectangles Squares and Rectangles Mean: 5.24, S.D.: 3.56 1000 800 Frequency 600 400 200 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Frequency 875 611 692 826 659 911 572 740 527 525 510 232 498 Score Score: 0 1 2 3 4 5 6 7 8 9 10 11 12 % < = 10.7% 18.2% 26.6% 36.7% 44.8% 55.9% 62.9% 72.0% 78.4% 84.8% 91.1% 93.9% 100.0% % > = 100.0% 89.3% 81.8% 73.4% 63.3% 55.2% 44.1% 37.1% 28.0% 28.0% 44.1% 8.9% 6.1% The maximum score available for this task is 12 points. The cut score for a level 3 response is 6 points. Most students (more than 80%) could define a square and find 1 or 2 squares in part 3 of the task. More than have the students could define a square, draw 2 squares on the grid, and correctly give the coordinates for the vertices. 20% of the students could define a square, find and give the coordinates for 2 of the squares, draw a rectangle with sides in a given ratio and find the area of the rectangle. 6% of the students met all the demands of the task. 11% of the students scored no points on this task. Of those students, almost 94% attempted the task. Eighth Grade 2003 pg. 29
Squares and Rectangles Points Understandings Misunderstandings 0 Almost 94% of the students with this score attempted the problem. Students with this score often counted all the little squares in the grid and gave definitions, like rectangles have 3 1 Students could give a definition for square. 3 Students could give a definition for square and find 2 of the squares in the grid. 5 Students could give the definition for a square, find 2 squares on the grid and name the appropriate coordinates. 6 Students could give the definition for a square, find squares on the gird, give coordinates for some of the squares, find the rectangle on the grid given a ratio of sides. 8 Students could define a square, find the non-diagonal squares, give their coordinates, draw the rectangle with the proper ratio, give the area of the rectangle, and find the total number of rectangles that could be made in the grid. sides. 18% of the students thought that rectangles have 2 long and 2 short sides (with about half stating the opposite sides should be parallel). 13% stated that 2 sides were equal and the other 2 sides were equal. 9% just stated rectangles have 4 sides. Many students had difficulty giving the coordinates correctly, often reversing the x and y coordinates. Students had difficulty measuring the sides for the rectangle in part 5. They may have counted the two given coordinates as 3 apart instead of 4. They may have attempted a diagonal rectangle, not realizing that diagonal lines are longer than horizontal or vertical lines between grid points. They could not find the square on the diagonal, had difficulty defining rectangle, and finding the total number of rectangles on the grid. Student assumptions about question 7 varied. Almost 10% gave an answer of 13, 14% gave an answer of 12, 9% gave an answer of 3, and only 3% gave an answer of infinity. Another 14% gave a response of 1, followed by incorrect responses like 2,8, and 6. 10 Students had trouble with the definition for rectangle or giving the number of rectangles in part 7 and some other small error. 12 Students could define rectangle and square, locate squares on a grid, give coordinates of vertices, find a rectangle with sides in a 2:3 ration find the area, and find the total number of rectangles. Many students with this score did not think that the vertices could go in between the intersections of the grids. Eighth Grade 2003 pg. 30
Based on teacher observations, this is what eighth grade students seemed to know and be able to do: Define a square. Locate squares on a grid, whose sides were located on grid lines. Give coordinates of points on a grid. Find area of a rectangle. Areas of difficulty for eighth graders, eighth grade students struggled with: Defining a rectangle. Finding squares whose sides were not parallel to the sides of the grid. Finding the dimensions of a shape when given the ratio of the sides. Finding the total number of rectangles that would fit on a grid. Questions for Reflection on Squares and Rectangles: What types of logic problems have students worked on? How might these help students to be more precise in making their geometric definitions? What experiences have students had with classifying and sorting? Why do you think students have such difficulty understanding that squares are rectangles? What other shapes might give them similar confusion? Look at your student answers for the definition of rectangle. How many of them gave answers such as: 4 right angles Parallel sides, 2 long 2 short 2 sets of parallel lines 4 sides 2 equal sides, 2 other equal sides 3 sides 2 short/ 2 long sides All sides equal When presenting shapes to the class, how often do they look at irregular shapes? Shapes with sides not parallel to the sides of the paper? Have your 8 th graders worked with Pythagorean theorem? Did you have students who did not know how to find the length of diagonal lines? While not needing to measure precisely, do you think most of you students have an intuitive notion that diagonals are longer than the sides of rectangle? What other types of measurement errors did you see in your students work? Students also had trouble defining their assumptions in part 7 of the task. How many of your students gave answers, such as: Infinite 13 12 3 1 4 2 What did students need to think about to get those answers? What would you like to clear up about their thinking around rectangles? Eighth Grade 2003 pg. 31
Many students who worked the parts of the problem (almost 15%) were uncomfortable with giving definitions or explaining their thinking in part one and 7 of the task. How frequently are students in you class asked to give justifications for their answers? Are students in your class comfortable with the demands of making a mathematical argument? Teacher Notes: Implications for Instruction: Students at this level need to know mathematical definitions of common geometric shapes like squares and rectangles and be able to analyze the properties of twodimensional figures. More experiences with classifying and sorting, working with attributes, or logic problems might help them become more precise in the use of definitions. They should be able to find and to draw those shapes on a coordinate graph, including those shapes with sides not parallel to the sides of the grid. They also need to be able to use coordinates to locate points a graph. Students should be comfortable with the idea that not all points in a graph are located on the grid intersections. Points can lie anywhere on the continuum and represent fractional distances. Students at this grade level should also be comfortable working with simple ratios to find different sides. Their thinking should be flexible enough to see that the horizontal axis could be the width or height of a shape. Students should work with shapes in a variety of orientations, including those with sides not parallel to the side of the paper. Eighth Grade 2003 pg. 32
8 th grade Task 3 Sport Injuries Student Task Core Idea 5 Data Analysis Core Idea 2 Mathematical Reasoning Core Idea 1 Number and Operation Use real data to interpret a circle graph regarding sports injuries. Formulate questions that can be addressed with data and collect, organize, analyze, and display relevant data to answer them. Use graphical representations of data 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 Work flexibly with fractions, decimals, and percents to solve problems. Eighth Grade 2003 pg. 33
Eighth Grade 2003 pg. 34
Eighth Grade 2003 pg. 35
Eighth Grade 2003 pg. 36
Looking at Student work Sports Injuries Sports Injuries was bimodal. Students either did very well or scored zero. Student A uses ratios to solve the problems and understands how relationships change when going from percent to degrees or degrees to percent. Student A Eighth Grade 2003 pg. 37
Eighth Grade 2003 pg. 38
Student B uses a different, but accurate approach for solving the problem. Student B shows a good understanding for converting between percents and decimals, using a slightly unconventional choice of numbers to divide by 100. Student B Eighth Grade 2003 pg. 39
Student C does not understand these relationships clearly. In part 1 the student multiplies by 100 instead of dividing. In part 2 the student doesn t multiply the decimal number by 100 to convert to a percent. Student C Eighth Grade 2003 pg. 40
Many students with a score of zero demonstrate some conceptual understanding of the problem, but don t have the formal skills to calculate the exact answer. They rely on estimation to tackle the task. Student D uses pictures and a knowledge of degrees and percents to get very good estimates in part 1. Student D uses pictures in part 2 and also adjusts estimates to make sure the total adds to 100%. Student D Eighth Grade 2003 pg. 41
Student D, part 2 Eighth Grade 2003 pg. 42
While other students do not show their estimation skills as clearly as Student D, a close look at their work reveals a fairly good sense of the relationships involved. Student E has a fairly good estimate for on the landing, and then further answers in part 1 get smaller to match the decrease in degrees. Student E s total does equal 360 degrees. In Part 2 Student E again makes fairly reasonable estimates, which total to 100%. Student E Eighth Grade 2003 pg. 43
Not all students with a score of zero show an understanding of the situation. Student F chooses to find an average number of degrees to use for all the percentages in part 1. In part 2 Student F uses the degrees from table 1, failing to recognize that the tables represent different sports. Student F Eighth Grade 2003 pg. 44
Grade 8 Sports Injuries Sports Injuries Mean: 2.45; S.D.: 2.30 4000 3500 3000 Frequency 2500 2000 1500 1000 500 0 0 1 2 3 4 5 Frequency 3446 320 270 599 342 3201 Score Score: 0 1 2 3 4 5 % < = 42.1% 46.1% 49.4% 56.7% 60.9% 100.0% % > = 100.0% 57.9% 53.9% 50.6% 43.3% 39.1% The maximum score available for this task is 5 points. The cut score for a level 3 response is 2 points. About half the students (51%) could correctly convert three or more of the percentages to degrees in a circle in part one of the task. Almost 40% of the students could convert percentages to degrees and degrees to percentages in the context of a circle graph. More than 40% of the students scored no points on this task. More than 80% of those students attempted the task. Eighth Grade 2003 pg. 45
Sports Injuries Points Understandings Misunderstandings 0 More than 40% of the students scored no points on this task. Some students showed an understanding of the situation by giving fairly good estimates. The degrees may have added to 360 or the percents may have added to 100. 2 Students could convert three of the percentages to degrees. 3 Students could convert all of the percentages to degrees in part one of the task. 4 Students made some errors in part 2, converting degrees to percentages. 5 Students showed a variety of strategies for converting percentages to degrees and degrees to percentages. Students were comfortable with the relationship between percentages and decimals. Teacher Notes: Other students simply exchanged the numbers from the two tables, putting the degrees from table 2 in table 1 and vice versa. Other students did not know how to change from percents to decimals. These students might multiply when then should divide or leave numbers in decimal form. They did not understand the inverse relations to go from degrees to percents. Based on teacher observations, this is what eighth grade students seemed to know and be able to do: Estimate the percentage of a circle given the number of degrees or estimate the degrees when given a percentage. Convert percentages to degrees. Areas of difficulty for eighth graders, eighth grade students struggled with: Calculating percentages when given the number of degrees. Converting percentages to decimals and vice versa. Eighth Grade 2003 pg. 46
Questions for Reflection on Sports Injuries: What types of experiences have students in your class had this year with circle graphs? What strategies did students who were successful on this problem use? How many of your students did not understand that the two tables represented different situations? (Look to see if they just interchanged the numbers between the graphs.) Many students showed some conceptual understanding of the situation and had some skills for making sense of the situation. Look carefully at student work to check for estimation skills, and knowledge about degrees and percents. How many of your students: Made a fairly accurate estimate for some parts of question 1 (check on landing Degrees in part 1 totaled 360/ were very close to 360 / Made fairly accurate estimate for some parts of question 2 ( check sharp twists or falling ) Percentages added to 100. What kinds of activities or experiences do students need who made good estimates to develop the ability to calculate correctly? How is this different from the types of activities or experiences needed by students who couldn t calculate or estimate correctly? Teacher Notes: Implications: Students need to know that there are 360 degrees in a circle. At this grade level they should be proficient at calculating with percents. Students, who lack a conceptual understanding of percents, have difficulty remembering and applying procedural rules when solving problems or determining if their answers are reasonable. Some students have conceptual understanding and can do estimation, but students at this grade level should be fluent with these conversions. A few students still do not understand the relationship between decimals and percents. Eighth Grade 2003 pg. 47
8 th grade Task 4 Dots and Squares Student Task Core Idea 3 Algebra and Functions Core Idea 2 Mathematical Reasoning Find and table number patterns in a geometric content. Find and use rules or formulas to answer questions. Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Use tables to analyze the nature of changes on quantities in linear relationships Recognize and generate equivalent forms of simple algebraic expressions and solve linear equations. Represent, analyze, and generalize a linear relationship (7 th grade) Use symbolic algebra to represent situations to solve problems (7 th grade) Employ forms of mathematical reasoning and justification appropriately to the solution of a problem. Use mathematical language and representations to make situations easier to understand Eighth Grade 2003 pg. 48
Eighth Grade 2003 pg. 49
Eighth Grade 2003 pg. 50
Eighth Grade 2003 pg. 51
Looking at Student Work Dots and Squares Dots and Squares requires students to identify patterns and generalize the patterns in the form of a rule or formula. Student A looks at the relationship between the geometric pattern and how it affects the relationship between the shapes and the number patterns. Student A is able to find a rule for both part 2 and part 5 of the task. The rule for part 5 will work for any size rectangle and is not restricted to rectangles where the length and width vary by one unit. Eighth Grade 2003 pg. 52
Student B does a nice job of working with the inverse relationships in part 3 and explaining how to use the formula to find the side if given the number of inside dots. Student B is able to generate a formula for part 5 which will for all rectangles, but can t apply to finding the dimensions of the rectangle when given the interior dots.. The student can t use the formula to find the dimensions of a rectangle with 63 inside dots. The student has noticed another pattern. In the examples in the table, all interior dots are even numbers. Sometimes students try to generalize about things that are not true for all cases. Eighth Grade 2003 pg. 53
Student C has a similar problem to Student B. The student is not able to use the formula to find the dimensions of a rectangle with 63 inside dots. The student focuses on the numbers in the table rather than thinking about the properties in the geometry of the pattern to find out what will hold true for all cases. Student C Eighth Grade 2003 pg. 54
Student D has found a formula that only works for one case of rectangles. The formula will only work for those rectangles whose length is one unit longer than the width. Therefore Student D cannot use the formula to help find the dimensions of a rectangle with 63 interior dots. Student D Eighth Grade 2003 pg. 55
Student E has also found a formula that works only for rectangles where the length is one more than the width. While the generalization is incorrect (the task was looking for a formula for all cases of rectangles), the student was able to use the formula to find numbers to fit the pattern in part 6. However the numbers will not map back to the geometric situation. Student E Eighth Grade 2003 pg. 56
Student F makes a common mistake of finding a recursive relationship of adding the next higher odd or even number each time. This is a cumbersome relationship to use because it requires generating the entire list to solve for a particular solution. Student F Eighth Grade 2003 pg. 57
Grade 8 Dots and Squares Dots and Squares Mean: 3.45, S.D.: 2.77 2000 1500 Frequency 1000 500 0 0 1 2 3 4 5 6 7 8 9 10 Frequency 810 1722 1273 869 1238 557 423 222 341 438 285 Score Score: 0 1 2 3 4 5 6 7 8 9 10 % < = 9.9% 31.0% 46.5% 57.2% 72.3% 79.1% 84.3% 87.0% 91.2% 96.5% 100.0% % > = 100.0% 90.1% 69.0% 53.5% 42.8% 27.7% 20.9% 42.8% 13.0% 8.8% 3.5% The maximum score available for this task is 10 points. The cut score for a level 3 response is 6 points. Most students (about 90%) could fill in the table with the correct perimeter for each square. Many students (about 70%) could also find the number of inside dots for a square. A little less than half the students (43%) could find the perimeter and the inside dots for the squares and the rectangles. About 21% could meet standards by filling in the tables for perimeter and dots and find the dimensions of square with 49 inside dots. Less than 5% of the students met all the demands of the task. Almost 10% of the students scored no points on this task. Of those more than 60% attempted some part of the task. Eighth Grade 2003 pg. 58
Dots and Squares Points Understandings Misunderstandings 0 10% of the students scored no points. Almost 64% of them attempted the problem. 1 Students could fill in the table for the perimeter values for the square. 2 Students could fill in the table for perimeter and inside dots for squares. 4 Students could fill in the tables for squares and rectangles. 6 Students could fill in both tables and find the dimensions of a square if they knew the number of inside dots. 8 Students could fill in both tables, find a rule for inside dots in a square use the rule to find the side length if they knew the number of inside dots. 9 Students could fill in the tables, find a rule for inside dots in squares and rectangles, and use their rule to find the dimensions of a square given the number of inside dots. About 28% of the students did not attempt this task or the final task on the test. Time may or may not have been an issue. Students did not count inside dots, but tried to find numerical patterns like going up by 3 or 4 every time, doubling, or having the inside always 4 less than the perimeter. They looked at only a couple of numbers in the table to find their rule, instead of testing the rule for all values in the table. Most students would not attempt any type of rule or generalization. Of the students who missed part 2, 33% did not attempt it. Of the students who missed part 5, 41% did not attempt it. About 15% of the students found a recursive relationship for the inside dots, like adding the next higher odd number. 11% wanted to count up by 3 or 4 every time. 7% wanted to multiply the number of sides by 4 to find the inside dots. Many students used counting or drawing strategies. They could not find a rule or formula. Students may have found a rule that only works for certain cases of rectangles and so their rule would not help them with part 6. They may have paid attention to more than one possible pattern in the rectangles which limited their thinking in part 6. See the work of Students B and C. 10 Students could make generalizations about geometric patterns to predict the number of inside dots and use the pattern to work backwards from inside dots to dimensions. Eighth Grade 2003 pg. 59
Based on teacher observations, this is what eighth grade students seemed to know and be able to do: Find the perimeter of squares and rectangles. Find the number of inside dots for a square or rectangle. Areas of difficulty for eighth graders, eighth grade students struggled with: Writing rules or formulas for geometric patterns. Using rules to work backwards. Understanding how to check a rule to see if it works for all the cases in the given information. (Making generalizations on too little information.) Questions for Reflection on Dots and Squares: What types of experiences or problems have your students had with making rules or formulas to match geometric patterns? When working with patterns, have the problems focused only on linear patterns? What questions or experiences do you ask students to help them see the relationships between variables instead of looking at how patterns grow (finding recursive relationships)? Look at student responses to part 2. How many of your students: Did not Goes up by attempt a 4 (or 3) rule every time Gave a counting or drawing strategy Goes up by an increasing odd number Multiply the sides by 4 Looked at interior instead of exterior so rules like: SxS or LxL or LxW Look at student responses to part 5. How many of your students: Did not attempt a rule Goes up by 4 every time L x W Gave a counting or drawing strategy Goes up by increasing even number Gave rule that only works for certain rectangles In part 5 many students gave rules that would only work for rectangles where the length was one unit longer than the width. What are some of the formulas? Make a list. Eighth Grade 2003 pg. 60
Why won t these formulas work for all cases of rectangles? How do you know? What would students have needed to focus on in the drawings or tables to know these rules wouldn t apply? What experiences have students had looking at different cases to make a proof? What types of problems have students worked on that required justifications? Have students in your class had opportunities to do investigations on their own and try to make generalizations from the data? Teacher Notes: Instructional Implications: When students look at pattern problems, it is helpful to visualize what is changing and what is staying the same. As they progress through the grades this information could be used to help them write a rule or formula. At this grade level, they can no longer rely on drawing pictures or doing repeated addition to find the solutions to complex problems. Students at this grade level should be proficient at answering a variety of questions about patterns. They need to recognize that patterns can grow in more than one direction and be able to investigate those changes. Students should work with patterns with exponential growth as well as linear growth. Students should develop the habit of verifying their rules or formulas to see if they work for more than one example. Eighth Grade 2003 pg. 61
8 th grade Task 5 Number Pairs Student Task Core Idea 3 Algebra and Functions Identify number pairs on a coordinate grid. 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 Relate and compare different forms of representations for relationships including words, tables, graphs in the coordinate plane and symbols (7 th grade) Eighth Grade 2003 pg. 62
Eighth Grade 2003 pg. 63
Eighth Grade 2003 pg. 64
Eighth Grade 2003 pg. 65
Looking at Student Work Number Pairs More than 36% of the eighth grade students could successfully match descriptions to already made graphs and use descriptions to plot points and make their own graphs. Student A uses the description to graph discrete points on a graph and knows that the graph should pass through the origin. Student A Eighth Grade 2003 pg. 66
Student B can match some description statements to graphs. The student seems to consider the two numbers on axis as making in pair in the upper right graph and when making her own graph, instead of looking at coordinate pairs. Student B Eighth Grade 2003 pg. 67
Student C does not understand that there is only one description for each graph and that the descriptions can t represent the same sets of coordinates. So in making his own graph the first x is on a 12 so the student puts the first number is 12. Then because the last x is also on a 12, the first number is equal to the last number. The student is not showing understanding of functions or coordinates. Student C Eighth Grade 2003 pg. 68
Student D gives statements about the slope of the graphs and does not understand that the descriptors at the beginning of the graph should be matched to the graph. Also Student D plots a second set of points on each graph that demonstrate the same trend. Student D Eighth Grade 2003 pg. 69
Grade 8 Number Pairs Number Pairs Mean: 2.44, S.D.: 2.21 3500 3000 2500 Frequency 2000 1500 1000 500 0 0 1 2 3 4 5 Frequency 2993 689 583 492 460 2961 Score Score: 0 1 2 3 4 5 % < = 36.6% 45.0% 52.2% 58.2% 63.8% 100.0% % > = 100.0% 63.4% 55.0% 47.8% 41.8% 36.2% The maximum score available for this task is 5 points. The cut score for a level 3 response is 3 points. Many students (about 63%) can match one or more descriptions to a graph. A little more than half (55%) can match two or more descriptions to their graphs. 36% of the students can match all descriptions to their graphs and make a coordinate graph from a verbal description. 36% of the students scored no points on this task. About half of these students attempted the task. Eighth Grade 2003 pg. 70
Number Pairs Points Understandings Misunderstandings 0 About 36% of the students scored no points. About half of these students made some attempt at the problem. 1 Most students found it easiest to match The first number is always 12 to its graph. 2 Students could match two of the descriptions to their graphs, with the first number is always 12 being one of the correct choices. About half the students did not try the problem, so time may have been an issue. Those who tried may have given general statements about the graphs, like it goes up or they may have put more than one description for each graph. Students sometimes confused the numbers on the axis with the first number or second number instead of thinking about coordinate pairs. 3 Students could match 3 of the descriptions correctly. The 2 nd easiest match for students was The second number is twice the first. 4 Most students could not make their own graph. They typically made plots going from 12 on the y axis to 12 on the x axis. 5 Students could match descriptions to graphs and make a coordinate graph from a verbal description. The understood the role of the coordinate pairs in the graphing. Teacher Notes: Eighth Grade 2003 pg. 71
Based on teacher observations, this is what eighth grade students seemed to know and be able to do: Match a description about the first number of a coordinate pair to its graph. Understand that graphs of functions go in straight lines. Recognize when a plot of discrete points is appropriate Areas of difficulty for eighth graders, eighth grade students struggled with: Matching descriptions of coordinate pairs to their graphs. Making a graph from a verbal description. Choosing coordinate pairs to match a verbal description. Questions for Reflection on Number Pairs: Did your students have enough time to show all they knew on the test? Do you think students with zeros on questions 4 and 5 in your room ran out of time, were just unmotivated by these types of questions, or lacked the knowledge or skills to be successful? What actions might you or your school take to help improve these scores next year? What opportunities have students in your class had this year with coordinate graphing? Do they seem to understand the relationship between the number pairs that make each point on the graph or are they looking at numbers on the axis when they match descriptions to graphs? What types of errors did students in your class make when making their own graph? What are the instructional implications of those types of errors? What further activities or experiences do they need? Teacher Notes: Instructional Implications: Given a simple rule, students should be able to make an equation and use that equation to develop a table of values. Students should be comfortable using tables to graph simple equations. Students need experiences describing given graphs as well as experiences plotting points. Students should have some system of checking whether different coordinates fit the conditions of the equations. Eighth Grade 2003 pg. 72