Second Grade
Core Idea
Mars 2009
Overview of Exam
Task
Descriptions
Task
Algebra, Patterns, and
Same Number, Same Shape
Functions
The task asks students to think about variables and unknowns in number sentences
that use addition and subtraction. Successful students understand that the same shape
represents the same quantity, and can follow this constraint when composing and
decomposing numbers to 20. They can also successfully add and subtract in a variety
of circumstances, including mixed operations in a single number sentence and
performing missing addend calculations.
Number Operations
Apple Farm Field Trip
The task asks students to interpret story situations in order to set up and perform twodigit addition and subtraction calculations. Students who are successful can
accurately determine if a situation in context is combining, separating, or comparing
quantities, and can choose an appropriate operation and number sentence to solve the
problem. They can use a variety of strategies for performing addition or subtraction
with accuracy and flexibility. They can prove their answer using words, numbers or
pictures.
Data
Top Lunch Choices
The task asks students to read and interpret a pictograph. Students who are successful
can use the relevant features of the graph and the data represented on the graph to
make mathematical statements about quantity and comparison.
Measurement
High Horse
The task asks students to find the number of hands a mom and child will need to
measure the height of a horse. Successful students find the total number of hands by
iterating each hand unit. They can compare units and predict whether the measures
will be greater or smaller when a different unit is used. Successful students can
identify the person to use the fewest and greatest number of hands, and explain their
thinking.
Number Operations
Auntie Em’s Cookies
The task asks students to make sense of halving shapes and numbers. Successful
students can identify a line of symmetry for a variety of regular and irregular shapes.
They can also work backwards to complete a shape from half to whole. They are also
asked to think about halving different quantities, even when the results are not a
whole number.
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Overall Results for Second Grade
Total MARS Raw Scores is the summation of Tasks 1 through 5 on the MARS test.
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2
MARS Test Performance Level Frequency Distribution Table and Descriptive
Statistics
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3
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=
+
+
=8
+
-
=
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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 =
+
+
+
+
-
= 10
= 6
3. Tell how you figured out the number for the
________________________________________________________________
_________________________________________________________________
_________________________________________________________________
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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
1
Based on these credit for specific aspects of
performance should be assigned as follow:
3+3=6
8=4+4
3+4+1=8
4+4–3=5
1 point per correct number sentence
1x4
4
2
2+5=7
16 = 8 + 8
2 + 8 + 0 = 10
8+2–4=6
1x4
4
3
Gives an explanation such as:
The two sides must be equal because 6 = 10 – 4.
Well, you had 8 and you add 2 and you have to take
away 4 to make 6.
1 f.t.
1
9
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2nd Grade – Task 1: Same Shape, Same Number
Work the task and examine the rubric.
What do you think are the key mathematics the task is trying to assess? _______________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Look at the student work for Part 1 (equations 1 and 2) and Part 2 (equation 2) that require the use of
doubles. Sort the student work into two piles; those who were able to use a double fact, and those
that weren’t.
• Of the students who were not able to use a double fact, were there any who used 4 + 2 to
make 6? Or 1 + 7 to make 8? Or 10 + 6 to make 16? Are there other combinations that they used,
to make the 6, 8, or 16 required in the equations? What does this tell you about their understanding
of equality? What does it tell you about their understanding of the constraints of the shape?
• Of the students who were not able to use a double fact, were there any who filled the shapes,
but with numbers that did not total the 6, 8, or 16 required in the equations? What does this tell
you about their understanding of equality? What does it tell you about their understanding of the
constraints of the shape?
Look at the student work for Part 1 (equation 4). Sort the papers into students who could correctly
create the expression “4 + 4 – 3”, and those who could not. Looking at the students who correctly
created the expression “4 + 4 – 3”, how many of them totaled to:
5
6
8
11
Other
What does each of these answers tell you about how the student is making sense of equality?
What does each of these answers tell you about how the student is making sense of the
operations?
Look at the student work in Part 2 (equation 4) and Part 3. Sort the student work into the following
categories:
Correct equation,
adequate
explanation
Correct equation,
incomplete or
incorrect
explanation
Incorrect
equation, adequate
explanation
Incorrect
equation,
incomplete or
incorrect
explanation
Other
• Which of these explanations did you value? Why?
• Of the explanations that concerned you, what do you think confused the students?
• What evidence do you have of any understanding? In what ways can those understandings be
used to re-engage with the task? What experience(s) might be next for these students?
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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
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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
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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
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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
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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. 4 + 3 = 7 can be expanded into 4 +
3 = 7 + 0 (to help them see the format of a + b = c + d) or 4 + 3 + 0 = 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?
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Student I In the student work, 20% of the students could successfully follow
the shape constraints to set up the fourth expression (4 + 4 – 3) 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
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All three students below (Student K, Student L, and Student M) provided an explanation
in Part 3 that includes both 8 + 2 = 10 and 6 + 4 = 10. Student K and Student L both
include an additional statement that indicates that are thinking about the expression 6 + 4 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 6 + 4 = 10 to
explain why they chose “4” to subtract.
Student K
Student M
2nd Grade – 2009
Student L
Look at the work for Student M. Is there any indication that
this student is also linking the addition of 6 + 4 to an inverse
operation to make sense of the subtraction statement in the equation
8 + 2 – 4 = 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 2 + 8. 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 1+7 ___= 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 4 + 3 + 2 = 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?
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Student N The explanation reads: The whole is 6 but I put 3 + 7 -1 in that shape. I
put the 1 in it because 3 + 7 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 3 + 7 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?
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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
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2nd Grade
Task 1
Same Shape, Same Number
Interpret and solve equations in which there are unknowns. Understand that the same
shape represents the same quantity, and follow this constraint when composing and
decomposing numbers to 20. Add and subtract in a variety of circumstances,
including mixed operations in a single number sentence and performing missing
addend calculations. Justify a strategy for missing part calculations.
Core Idea Understand the meanings of operations and how they relate to each other, make
2
reasonable estimates, and compute fluently
Number
• Demonstrate fluency in subtracting whole numbers
Operations
• Communicate reasoning using pictures, numbers, and/or words
Core Idea Understand patterns and use mathematical models to represent and to
3 Patterns, understand qualitative and quantitative relationships.
Functions,
• Compare principles and properties of operations, such as commutativity, between
and
addition and subtraction.
Algebra
• Use concrete, pictorial, and verbal representations to develop an understanding of
symbolic notations.
Student
Task
Mathematics of the task:
• Ability to compose and decompose numbers to 20
• Ability to follow constraints through the parts of each task
• Ability to solve equations with unknowns
• Ability to solve equations that use two or more operations
Based on teacher observation, this is what second graders knew and were able to do:
• Use their math facts, including turn around facts, fact families, and doubles facts to solve equations
• Fill in with one unknown and simple addition
• Very comfortable using a double fact as the unknown when the two shapes were the same
• Students who successfully switched operations, per the signs in the equation, were able to calculate
accurately
Areas of difficulty for second graders:
• Attending to the operation signs in the equation
• After successfully solving for an unknown shape, using double facts or simple addition with one
missing addend, students did not use that information in subsequent parts of the same task
• Explaining their strategies for solving the final equation, using a justification that includes the
notion of trying to balance between the 10 and the 6 on opposite sides of the equal sign
• Using zero as one of the unknowns
Strategies used by successful students:
• Using double facts and simple addition facts in two addend addition
• Reasoning that the same shape will have the same number throughout that part of the task
• Understanding that the equal sign refers to balancing an equation, rather than announcing that “the
answer” is coming up
• Differentiating between the operation signs
• Using inverse operations, such as counting up, to solve a subtraction problem
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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.
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Same Shape, Same Number
Points
Understandings
0
Students who attempted this task
understood that the shapes needed to be
filled with numbers.
2
Students who scored at this level knew at
least one doubles fact. Some of them may
have known the math fact of 2 +5 = 7.
4
Nearly 84% of the students scored at or
above this level. They successfully used
doubles facts and fact families to solve for
unknowns in two addend equations.
5
Almost three-quarters of the students
scored at the cut score or above. 96% of
these students could use double facts and
basic facts to successfully complete the
equations. 44% could follow the shape
constraints of the previously assigned
values to solve new equations.
7
53% could successfully write and solve an
equation in which one of the addends is
zero. 68% of the students could both
follow the shape constraints established in
the tasks, and accurately negotiate the
mixed operations signs.
9
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.
Misunderstandings
Students who attempted this task, but scored no
points, filled the shapes with random numbers,
numbers in a counting sequence (with no regard to
balancing an equation), or with same number
repeatedly (again, with no regard to balancing an
equation).
Although students at this level were able to do one
or two double facts, they didn’t seem to associate
these facts to the same shape constraint. It was as
though they happened to recall a double fact to 6,
or a double fact to 8, but did not associate the
repeated addends with the repeated shapes.
These students had trouble with one or more
unknowns in a three addend problem. They
struggled with attending to addition and
subtraction in the same number sentence. They
often didn’t carry the constraint of a number
assigned to a shape into the rest of the task. 8% of
the students at this level were able to sum to the
correct amount, but put different numbers into the
same shapes in order to do so.
8% of the students used the correct numbers, but
in the wrong order, for the third equation (3 + 4 +
1 = 8) in Part 1. A third of these students were
able to sum to 10 in the third equation (2 + 8 + 0 =
10) in Part 2, but did not follow the shape
constraints previously established for the 2 and 8.
34% give both an incorrect equation and an
inadequate explanation for Part 3.
Over 20% of the students were able to correctly
follow the constraints of same shape, same
number, but mistakenly added when the equation
called for subtraction. Of the 46% of the students
who successfully wrote and solved the last
equation in Part 2
(8 + 2 – 4 = 6), less than half of them could
provide a mathematical justification for their
work.
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Implications for Instruction:
In the book, Thinking Mathematically – Integrating Arithmetic and Algebra in Elementary School
(Carpenter, et al), the authors provide classroom examples and research-based findings on how
children make sense of equality, and the symbol for equality, the equal sign (=). Fully understanding
the equal sign, including the meaning, syntax, and implications, takes a variety of experiences.
Children often begin creating meaning for equality and the equal sign by associating it with
operations. They may believe that “=” means “here comes the answer” or “do it now!”, and seeing
only number sentences in the format of a + b = c can reinforce that misconception.
Knowing that the operation can be performed is an important part of beginning to work with
equations. In order to be flexible and efficient, however, children need to also develop the
understanding that it’s not just an action, it also represents a relationship. This understanding is what
allows them to make sense of number sentences that fall outside of the a + b = c format. For a
student who is developing their conception of equality, they will need to experience number
sentences such as a + b = d + e (8 + 4 = 7 + 5); a = a (6 = 6); a + b = (a + b) + 0 (4 + 5 = 9 + 0); or
even a + b = b + a (2 + 6 = 6 + 2).
There are times when performing the operation will quickly and simply lead to a correct relationship
in a particular equation. We can see this in the task Same Shape, Same Number, where students can
use their understanding of doubles facts in addition to make quick sense of an equation with
unknowns. Performing the operation is quite helpful and efficient in that case. If we look at the last
example in the previous paragraph, we can see a situation where performing the operation becomes
unnecessary and less efficient. If we understand the nature of equality, and the properties of addition,
we can quickly ascertain that this number sentence is true without doing any addition at all. If we
were to solve for the unknown in the equation 2 + ___ = 6 + 2, we can use our understandings of the
properties of addition and equality to know that the unknown is 6, without adding 6 + 2 and then
subtracting 2 from 8.
This is where we want our students to be working; at a place where they understand both the
relationship expressed with an equal sign in an equation, and the operations being expressed in an
equation. Being mathematical means they have the ability to determine whether performing the
operations, or balancing based on the relationship, or a combination of the two, is the best strategy for
solving any given equation. The authors provide an outline for using True/False statements and Open
Number Sentences to provide thought-provoking experiences that will push students to refine their
understanding of equality.
When we set two quantities as equal to each other, using the equal sign, we create an equation. A
quantity can be represented by a number or an expression, such as (4 x 8), or (2x + y). So, 3 + 5 is a
quantity, represented by an expression. We can set it equal to another quantity, such as 8 (3 + 5 = 8)
or 4 + 4 (3 + 5 = 4 + 4), or a quantity that includes an unknown or a variable (2x = 3 + 5). The
authors caution us against using the equal sign as a type of shorthand, when we are not representing
the relationship between two quantities. Examples of these types of shorthand situations can be
found on p. 20 in the book.
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Action Research Idea
In the task Same Shape, Same Number, at least 20% of the students who took the task were able to
successfully use the constraints of the shapes to create the expression 4 + 4 – 3 in the fourth equation
of Part 1, yet subsequently missed the point for their work because they added three instead of
subtracting three, as indicated.
In Developing Number Concepts (Book 2 for Addition and Subtraction), Kathy Richardson outlines
the goals for each child as they develop the conceptual understanding they will need to add and
subtract as:
1. Interpreting (acting out) addition and subtraction stories
2. Reading and interpreting addition and subtraction cards and relating them to specific stories or
actions
3. Writing addition and subtraction equations to describe problems
4. Writing story problems to go with the equations
~ p. 3
She describes using these four goals when introducing simple situations, such as those in which
students are asked to either combine (first with two addends and then with two or more addends) or
separate groups of objects. At later stages in the construction of their understanding, teachers begin
to introduce more complex situations, such equalizing, missing part, and comparative subtraction.
When each of these new, more complex situations is introduced, the students will need to go right
back to Goal 1 in order to make sense of the math. So even though a student may be able to perform
the more cognitively demanding task of working backwards to write a story problem to go with an
equation for simple addition and subtraction situations (Goal 4), they will need to go back to the
earliest stages of acting out the stories for the more complex story problems.
Where does an equation that includes both the actions of combining and separating fall on this
spectrum of simple to complex? Do we, as teachers, assume that because a student can add, or
subtract, that they should be able to make sense of doing both actions in a single number sentence?
Or should we, in fact, go back to Goal 1 of “acting out” this more complex situation?
What might it look like, to use story boards to act out number sentences that combine adding and
subtracting? Here are some situations to consider:
1. I have three balloons. My dad gives me four more balloons. Then, 2 of my balloons pop.
How many balloons do I have now?
a. Acting out: Student adds three cubes to a storyboard for the first action, then adds
four cubes to the storyboard for the second action, then removes two 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 “3 + 4 – 2” or “3 + 4 + 2” and
justify their choice by referring back to the story context that they had already acted
out.
2. There are six apples growing on my tree. Two more apples grow on my tree. Four apples fall
to the ground. How many apples are in my tree?
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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 “6 + 2 – 4” or “6 + 2 + 4” 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 “5 – 3 + 4”?
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