# Second Grade Mars 2009 Overview of Exam. Task Descriptions. Algebra, Patterns, and

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2 Overall Results for Second Grade Total MARS Raw Scores is the summation of Tasks 1 through 5 on the MARS test. pg. 2

3 MARS Test Performance Level Frequency Distribution Table and Descriptive Statistics pg. 3

4 Same Shape, Same Number Fill in the shapes with numbers. Same shapes have same numbers. Different shapes have different numbers. 1. Write the numbers in the shapes to make the number sentences true. + = 6 8 = = = pg. 4

5 Fill in the shapes with numbers. Same shapes have same numbers. Different shapes have different numbers. 2. Write the numbers in the shapes to make the number sentences true. + 5 = 7 16 = = = 6 3. Tell how you figured out the number for the pg. 5

6 Same Shape, Same Number Mathematics Assessment Collaborative Performance Assessment Rubric Grade 2: 2009 Same Shape, Same Number: Grade 2: 2009 Points Section Points The core elements of the performance required by this task are: Use concrete, pictorial and verbal representations to develop an understanding of symbolic notations Demonstrate fluency in adding and subtracting whole numbers. Communicate reasoning using words, numbers or pictures Based on these credit for specific aspects of performance should be assigned as follow: = 6 8 = = = 5 1 point per correct number sentence 1 x = 7 16 = = = 6 3 Gives an explanation such as: 1 x The two sides must be equal because 6 = Well, you had 8 and you add 2 and you have to take away 4 to make 6. 1 f.t. 1 9 pg. 6

8 Student A represents the 23% of the students who were able to successfully complete all parts, including solving for unknowns, attending to the ongoing constraints, and writing a mathematical justification using words, symbols, and/or pictures. Student A pg. 8

9 Student B is typical of students who score at the cut score of 5; these students are meeting standards. They are able to use doubles and fact families to solve for unknowns, but struggle with carrying the constraint of the shapes through to other equations. There is evidence that they haven t completely sorted out how to keep track of the operations, or how to solve for unknowns in more complex equations. In order to meet the cut score, they had to negotiate both the constraints and the operations for at least one number sentence that did not involve doubles or single missing addend for a basic fact. Student B pg. 9

10 Student C is typical of students who were unable to meet the minimum standards for this task. These students over relied on the doubles strategy, even when it didn t make sense for the equation. Student C pg. 10

11 Students D and E are not following the constraints of the numbers in the shapes established, accurately, in previous equations. However, they are making sense of equality to solve the equations. One of these students is using both relational thinking and operations to solve for unknowns, and the other student is focusing on operations to solve for unknowns. What evidence can you use to identify which student is also thinking relationally? Student D Student E Student F How might the explanation from Student F be used in a re-engagement lesson to help Students D and E begin to attend to the constraints of the values of shapes established in previous equations? Student F pg. 11

12 Students G and H represent the students who accurately follow the constraints of the values created in the shapes from previous equations, but then cannot negotiate the demands of the equality statement. Although Student H writes the correct equation in Part 2, both Student H and Student G are only balancing parts of the expression on the left with the expression on the right. Student G ignores the + 2 portion of the expression, and states that 8 2 = 6. Student H sees (8 + 2) separately from (2 4), and incorrectly resolves 2 4 as 6. Student G Student H In the book Thinking Mathematically (Carpenter, 2003), the authors outline the use of True/False Statements and Open Number Sentences as a way of helping students make sense of equations that fall outside of the a + b = c format. Using these two formats, what experiences might you plan for these students? One of the ways Carpenter suggests that teachers can introduce students to new expression formats (outside of a + b) is to begin with a zero addend = 7 can be expanded into = (to help them see the format of a + b = c + d) or = 7 (to help them see the format of a + b + c = d). What evidence is there in the student work presented that either of these students (Student G or Student H) is able to use that zero addend to think about alternate formats of expressions? What numbers and open number sentences could you plan to push each student further along the spectrum of understanding equality? pg. 12

13 Student I In the student work, 20% of the students could successfully follow the shape constraints to set up the fourth expression ( ) but, when solving the equation, added 3 instead of subtracting. How many students were not able to make accurate calculations because the were misreading signs? What previous understandings do they need to make sense of an expression format that includes more than one operation? (See: Action Research Idea for this task in the Tool Kit.) Student J provides an explanation in Part 3 that reflects an understanding of the need for balance in this equation, but clearly shows a lack of understanding on the constraints of the problem, both in terms of the shape values, and the operations represented. Student I Student J pg. 13

14 All three students below (Student K, Student L, and Student M) provided an explanation in Part 3 that includes both = 10 and = 10. Student K and Student L both include an additional statement that indicates that are thinking about the expression as some form of inverse operation to subtraction. Student K may be making sense of fact families when he/she writes about it being like a triangle and mentions that 10 6 = 4. Student L describes taking 4 from 10, and using the addition fact of = 10 to explain why they chose 4 to subtract. Student K Student L Student M Look at the work for Student M. Is there any indication that this student is also linking the addition of to an inverse operation to make sense of the subtraction statement in the equation = 6? What if the strategy the student is using is to use the knowns (8 + 2) and then find the unknown by thinking + 6 also equals 10? Look at the third equation in Part 2. Would this strategy work for this problem? The knowns are Can the unknown be found by thinking + 10 also equals 10? The strategy works for this equation. What about using that strategy for an equation such as = 5. Does the strategy work? 1+7=8 and + 5 also equals 8. Can the unknown be found? The strategy works for this equation. Does the strategy work for the equation 4+3+ = 5? 4+3=7 and + 5 = 7. Can I put a 2 in the original equation to solve for the unknown? Does = 5? Why doesn t the strategy work for this equation? What experiences does a student need to help connect their strategy (which works a lot of the time) to the deeper understanding of how their strategy is related to inverse operations and maintaining balance (which works all the time)? Can students begin to organize equations into ones that work with this strategy, and ones that don t? What other strategies do they need access to for the ones that don t? pg. 14

15 Student N The explanation reads: The whole is 6 but I put in that shape. I put the 1 in it because is 10 [but] the whole is 6 not 10. So I subtract the 1 and so that makes 6. Student N In the live student work for this example, there is evidence of erasures throughout the page. For example, after solving the first and second equations, the student was able to fill in the third equation with 2 and 8. Unable to negotiate a value for the third shape, however, they erased those values and made new ones that were friendlier for the student to work with. Now that the trapezoid is redesignated as having a value of 3, the student is able to carry that new value to the next equation. What if the student chose to place 7 in the oval, since he/she is now using both a 2 and a 5 for that shape in different places in the problem? Is there enough evidence to demonstrate that the student understands that previously assigned values are carried into the same shapes in other equations? If so, what critical understanding is lacking for him/her to believe that it s okay to start changing those values later, if it suits your purposes? How does the student learn to maintain the value for that shape throughout, once it s assigned? In what ways might strengthening this student s understanding of using the same shape/same number constraint also help them make deeper sense of the part-part-whole relationship that they are currently using? There is evidence that the student knows the equation must balance for the quantity of 6. When does not balance for 6, he/she then turns to 7 1 to balance for 6. Is there a link between knowing that once the oval is solved as 2, it will be 2 in each of the equations in this set, and knowing that once there s a 10 on one side of the equation, it remains on that side of the equation, and we can t just ignore it or change it to suit our purposes? pg. 15

16 Students sometimes tried to make sense of the value of the shape by focusing on the characteristics of the shape. Student O visually confuses the pentagon in Part 2 with the octagon in Part 1, and then incorrectly uses the value from Part 1 in the equation for Part 2. Student P explains in Part 3 that they used the number of sides in the pentagon shape to determine the value for the shape. We can see in the other equations that they didn t use this strategy consistently. Student O Student P pg. 16

18 Frequency Distribution for Task 1 Grade 2 Same Shape, Same Number There is a maximum of 9 points for this task. The cut score for a level 3 response, meeting standards, is 5 points. Most of the students, around 84%, were able to use the double facts or a single missing addend to solve for unknowns in a given equation. Nearly 72% of the students were at or above the level 3 (meeting standards) response. Almost a quarter of the students who did the task were able to successfully complete all parts, including solving for unknowns, attending to the ongoing constraints, and writing a mathematical justification using words, symbols, and/or pictures. pg. 18

22 a. Acting out: Student adds six cubes to a storyboard for the first action, then adds two cubes to the storyboard for the second action, then removes four cubes to represent the third action. b. Reading and interpreting cards and relating them to specific stories and actions: Students might pick between two cards that read either or and justify their choice by referring back to the story context that they had already acted out. In what ways would this approach be different then asking a student to simply read the number sentence carefully? In what ways do our students even understand that it s conceptually possible to both add and subtract in the same sentence? Can you think of a story situation that might result in the expression ? pg. 22

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