Auntie Em’s Cookies
Auntie Em is making interesting cookies for Holly and David. She
wants to cut the cookies so each child gets half a cookie.
1. Use a dotted line to show Auntie Em how to cut each cookie
below.
2. Auntie Em cut a cookie in half. Here is David’s half. Draw the
whole cookie.
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3. There are 12 cookies on a tray. David gets home first and eats
half of them. How many cookies are left?
_______cookies
4. When Holly gets home, she eats half of the cookies that were
left. How many cookies does Holly eat?
_______cookies
5. Auntie Em eats half of the cookies that are still on the tray.
How many cookies does Auntie Em eat?
________cookies
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Auntie Em’s Cookies
Mathematics Assessment Collaborative
Performance Assessment Rubric Grade 2: 2009
Auntie Em’s Cookies: Grade 2: 2009
Points
Section
Points
The core elements of the performance required by this task are:
• Represent commonly used fractions such as 1/2 in a
variety of ways
• Understand situations that entail equal groups of objects
and equal sharings
• Communicate reasoning using words, numbers or
pictures
1
Based on these credit for specific aspects of performance should
be assigned as follow:
All 5 shapes correctly divided into two fair shares
2
3 or 4 shapes correctly divided into two fair shares
Draws correct shape for whole cookie, such as:
2
(1)
2
2
2
Draws 2 equal parts but not the whole cookie
3
6 cookies
(1)
1
1
4
3 cookies
1ft
1ft
5
1 1/2 cookies
or
2
Special Case: Correctly finds half of an odd number using their
answer in part 4.
2
2
8
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5
Test 2
2nd Grade – Task 5: Auntie Em’s Cookies
Work the task and examine the rubric.
What do you think are the key mathematics the task is trying to assess? _______________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
For Part 1, sort papers into students who were able to draw a correct line of symmetry for all five
shapes, and those who did not. For those students who did not draw a correct line of symmetry for
each shape, identify which shapes were incorrectly halved. How many incorrect shapes for:
Shape 1
Shape 2
Shape 3
Shape 4
Shape 5
• Look at the orientation of the ‘cuts’ students have drawn. For students who got one or more
wrong in this section, did they seem to favor vertical over horizontal orientations?
• Look at successful students. Were they able to move between a vertical and horizontal
interpretation of the shapes?
• What experiences and explorations do students need to make sense of the attributes of a
shape, and to see a shape on more than one axis?
For Part 2, how many students recreated the “whole cookie” as two physically separated pieces?
What does this indicate about their understanding of unit (see Implications for Instruction)? How
many students did not estimate an appropriate approximation of equal size to the half given? What
does this indicate about their understanding of halving?
Look at Parts 3, 4, and 5 together. Sort students by their answers, respectively, for the three parts:
6, 3, 0
6, 3, 3
6, 6, 0
4, 4, 4
11, 10, 9
Other
• What does each of these answer sets indicate about the student’s understanding around
halving? Around equal parts? Around the relationship between subtraction and division?
• Although the students were not required to show their work, is there any evidence of student
thinking on the page? For students who chose to work on the page, did they use the picture?
Did they draw their own models? What number sentences did students use?
o When analyzing the number sentences, did any students use double facts to help solve
halving in context?
o Did students who drew their own models have better success than those who used the
picture provided?
Look at the answers for Part 5. For students who could successfully halve an odd number, what
models or number sentences did they use? Were there students who could draw an accurate model
for halving three cookies, but then were unable to use correct notation to write 1 ½? Which notations
do you value? Why? What do different notations (1 ½, one and a half, 1 + ½) demonstrate about the
student’s understanding of the value represented by the number?
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Student A represents the almost 12% of second graders who met all the
demands of this task. These students were able to make sense of half using line
symmetry for regular and irregular shapes, work backwards from a half shape to
a whole shape, and halve sets of even and odd numbered sets of objects.
Student A
Student B is typical of students scoring at the cut score, or meeting standards. These students
can generally find half of 12, either using the drawing or a double fact, but their notions of
“halfness” are confused or fragile. In this example, the student gets confused when there’s more
than one apparent line of symmetry in Part 1. The picture for Parts 3, 4, and 5 is evidence of the
“twoness” s/he associates with halving, and they seem to have also associated subtraction with
halving. S/He has not formalized “equal shares”.
Student B
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Student C successfully drew an
attached, congruent shape, to work
backwards from a half to a whole.
Why did this student add the
drawing across the top, to fill in the
missing space? What does this tell
us about their understanding of
what “a whole” needs to look like?
Student C
Students working at the lower levels of the task offer evidence about how they are
thinking of “halfness” and “wholeness”. In Part 1, Student D uses line symmetry to
bisect a shape into halves. For many students, this is their first understanding of what
half is. If they interpret line symmetry as “cut down the middle”, they can struggle with
shapes that are irregular or that have a horizontal line of symmetry.
In Part 2, there is further evidence of how this student makes sense of region (or area)
fractions. Although Student D makes congruent shapes, there is no indication that s/he
understands that those congruent shapes need to be attached into one whole shape.
Student D
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Students E and F are associating “halving” with “getting smaller”, but only Student E
provides evidence that s/he is thinking about “halving” as “fair shares”. How can Student
E’s answer for Part 3 be used to begin an investigation or reengagement lesson around
linking the subtractive quality of taking away the eaten cookies to the idea that the cookies
being taken away, or eaten, are in two equal groups?
Student E
Student F
Student G kindly provides a drawing so we can understand why s/he might answer “4” for
Parts 3, 4, and 5. What does this student understand about fair shares? What does this
student understand about taking part of a part, and making the remaining part the new
‘whole’? What does this student need in order to make sense of “half” as two equal parts,
specifically?
Student G
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Students H and I demonstrate a common understanding of halving as related to doubling.
Young children often have access to double facts with facility, and while Student H connects
to the subtractive nature of division, Student I moves between addition and subtraction to
show the two doubles make up the whole set. Both of these students understood that the
remaining part becomes the new whole for the next problem. In what ways does Student H’s
work make this especially clear? How can this piece of student work be used to help children
confused about this aspect of the task?
Student H
Look at the work for Student H for Part 5. In
what ways does the notation s/he chose to use
help or hinder the understanding of how to halve
an odd number? Look at the work for Student J
for the same problem. This student’s drawing
provides evidence that “three halves” would
have been an accurate and sufficient answer for
how they were halving three cookies.
Student I
Student J
How might Student H benefit from modeling or
drawing their thinking? How might Student J
benefit from learning different forms of notation,
rather than strictly relying on numerical?
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Students K and L provide examples of different notations for 1 ½. What does each
reflect about the meaning of a mixed number? How could each of these be used to
introduce options to students who are struggling with numerical representation of
mixed numbers? In what ways does each form deepen an understanding of what a
mixed number is and what it means?
Student K
Student L
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Both Student M and Student N began their modeling of these problems by circling 6
“cookies” in the drawing provided. Both students then circled the remaining six cookies
when answering Part 4. However, Student N erased the second markings in the problem
and chose instead to redraw the remaining six cookies into a new drawing for Part 4, a
drawing which models the double fact of 3 + 3 inside of those six remaining cookies.
How did Student N’s decision to redraw the model help show the taking away of the
eaten cookies and making the remaining cookies the new whole for the next part in the
problem? How could Student M benefit from seeing this particular strategy?
Student M
Student N
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Just as Student N used their own model, so did Students O and P. In this case, they didn’t use
the drawing at all, but rather created their own models from the start. In what ways can the
labeling strategy used by Student P be used to help students understand what is happening in
the problem?
How do Students O and P handle the modeling of half of three in Part 5? In what ways do
their models match their answers? While both students handily model using doubles to halve
the even number sets, they seem unsure how to proceed when the set is an odd number of
objects. Although modeling can be a very powerful strategy, in what ways might these students
benefit from a conceptual experience where they act out sharing 10 brownies among 6 people?
Or 6 cookies among 4 people? What needs to happen for students to connect the physical
actions of the acting out with what a model for this action might look like on paper?
Student O
Student P
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2nd Grade
Student Task
Core Idea 1
Number
Properties
Task 5
Auntie Em’s Cookies
Make sense of halving shapes and numbers. Identify a line of symmetry for a variety of
regular and irregular shapes. Work backwards to complete a shape from half to whole. Take
half of a set of objects, both odd and even numbered sets, and use the quantity of half as the
new set to be halved.
Understand numbers, ways of representing numbers, relationships among numbers,
and number systems.
• Understand whole numbers and represent and use them in flexible ways, including
relating, composing, and decomposing numbers.
• Represent commonly used fractions such as ½, 1/3, and ¼ in a variety of ways.
Core Idea 2
Number
Operatons
Understand the meanings of operations and how they relate to each other, make
reasonable estimates, and compute fluently
• Demonstrate fluency in adding and/or subtracting whole numbers.
• Communicate reasoning using pictures, numbers, and/or words.
• Understand situations that entail division such as equal groupings of objects and
equal sharing.
Core Idea 4
Geometry
and
Measurement
Recognize and use characteristics, properties, and relationships of two-dimensional
shapes.
• Develop an understanding of line symmetry.
Mathematics of the task:
• Ability to draw a line symmetry for a variety of regular and irregular shapes
• Ability to work backwards to complete a shape from half to whole
• Understand that half of a set of objects is the same as making two equal groups
• Ability to use the quantity of half the set as the new set, and take half again
• Understand how to represent halves in words, models, or expressions
• Ability to represent mixed numbers in words and/or in expressions
Based on teacher observation, this is what second graders knew and were able to do:
• Use a dotted line to denote the line symmetry of regular and irregular shapes
• Create a whole shape when given half by attaching a congruent shape
• Use double facts or drawings to take half of 12 objects
• Understand equal groups
Areas of difficulty for second graders:
• Taking half of a half
• Halving an odd number
• Notation for mixed numbers
Strategies used by successful students:
• Looking at shapes from a variety of angles to understand their attributes
• Using double facts as addition or subtraction to make equal groups
• Drawing pictures, using the drawing provided
• Labeling the sets of cookies to keep track of “what’s left” after halving
• Using words and expressions to represent mixed numbers
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Frequency Distribution for Task 5 – Grade 2 – Auntie Em’s Cookies
There is a maximum of 8 points for this task.
The cut score for a level 3 response, meeting standards, is 4 points.
Around 72% of the students were able to use symmetry to divide a shape in half, regardless of the
orientation or characteristics of the shape. Two-thirds could work backwards from a half-shape to the
whole shape. More than half the students were able to find half of an even-numbered set of objects,
but only one-third could find half of the remaining set.
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Auntie Em’s Cookies
Points
0
Understandings
Students who attempted to answer the
tasks but scored no points understood
that a line of symmetry bisects a shape.
Misunderstandings
These students may have treated the shape in
Part 2 as the whole, and cut it into two or
more pieces.
2–3
Three-quarters of these students could
correctly identify the line of symmetry
for the regular and irregular shapes.
Two-thirds of these students could
accurately create a whole shape based
on the half shape provided.
Twenty-five percent of these students seem to
interpret line of symmetry as “down the
middle”, which did not work with the
orientation of the irregular shapes, 2 and 5.
They struggled with working backward from
a half to a whole.
4
Twenty percent of the students scored at
the cut score of 4. They were able to
correctly draw in a line of symmetry for
all or most of the shapes in Part 1. They
could work backwards from half a shape
to the whole shape. They may have
used the drawing or a doubles fact to
find half of 12 objects.
Students at the cut score continued to make
some mistakes in Part 1 or Part 2. Twenty
percent received partial credit for Part 2 by
creating a congruent shape to the one given,
but not connecting it to the existing half.
Students at this level may have been able to
find 6 as half of a set of 12 objects.
6-7
Students were able to flexibly move
between a horizontal and vertical
orientation of the shapes in Part 1. They
could work backwards from a half to a
whole shape. Two-thirds of the students
scoring at this level were able to
successfully halve sets with an even
number of objects. Students used
doubles facts with addition and/or
subtraction to make sense of halving.
Students working at this level struggled with
taking half of an odd number. Only 12% of
these students were able to correctly answer 1
½ for Part 5. Even when students could
conceptualize half of an odd number of
objects, they struggled with the notation for
writing mixed numbers. They may still be
struggling to make sense of “equal shares” in
context.
8
Students working at this level were
successful at all the demands of the task.
Though they were not required to show
their work or explain their thinking in
Parts 3, 4, and 5, students who scored
the maximum points most likely did use
models, the drawing provided, and/or
number sentences to solve these parts.
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Implications for Instruction
John Van de Walle writes about the “big ideas” of fractions for children in the primary grades.
(Teaching Student-Centered Mathematics, K-3, 2006) Among these conceptual ideas is that
fractional parts are equal shares, or equal-sized portions, of a whole or a unit. Perhaps one of the
most confusing ideas for young children first learning how to make sense of fractions, is that “whole”
can be a single object (like a pizza) or a collection of things (like 12 cookies).
Furthermore, the numerator and denominator both have meaning in fractional notation. The
denominator is a divisor; it tells us by what number the “whole” (as a unit or a set) is divided. This
division produces the type of part under consideration (fourth, half, etc.). The numerator is a
multiplier; “it indicates a multiple of the given fractional part”. (Van de Walle, p. 251)
If the denominator is a divisor, it makes sense for us to consider different types of division, partitive
and measurement, both of which may be informing how students are making sense of fractional parts.
In a multiplication number sentence, there are two factors; the number of groups, and the number in
each group. The two parts can be multiplied together to find a total, or product.
Multiplication in context:
There are 4 bags. There are 2 apples in each bag. How many apples are there?
4
x
2
=
8
(number of groups) x (number in each group) = (total number)
In division, we know the total and one of the factors (either the number of groups or the number in
each group) and we divide to find the missing factor. When we know the number of groups, and we
divide to find out the number of groups, we call this “partitive division”.
There are 8 apples. I want to share the apples fairly between 4 bags. How many
apples can I put in each bag?
÷
8
(total number) ÷
4
=
2
(number of groups) = (number in each group)
When we know the number in each group, and we divide to find out the number of groups, we call
this “measurement division”.
There are 8 apples. I want to put 2 apples in each bag. How many bags will I need?
÷
8
(total number) ÷
2
=
4
(number in each group) = (number of groups)
Now we have three ways of looking at fractions as division. In the first way, the “whole” is one
object, and the denominator indicates how many equal pieces to make. The second way is partitive
division, in which the “whole” is a set of objects, and the denominator indicates the number of groups
we should make. The third way is measurement division, where the “whole” is a set of objects, and
the denominator indicates the number of objects to put in each group.
Using the problem posed in Auntie Em’s Cookies, we can model the three ways of looking at
fractions as division, and explore the role of the numerator in each model. In the task, the students
are presented with the problem of thinking about fair shares when making “halves”.
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In Part 3, David eats half of the 12 cookies. If the student makes sense of ‘the whole’ as a single
object, or cookie, it can be modeled as David eats half of each cookie.
X
X
X
X
X
The number of cookies David ate is 6, or
X
X
X
X
X
X
X
12
, which we can see by counting the halves.
2
Then Holly comes and eats half of what is left. In this model, Holly eats half of each half-cookie.
X
X
X
X
X
X
X
X
X
X
X
12
which we can see by combining the halves of the halves, or
4
fourths. When the Aunt comes in, she eats half of the remaining cookies. Now, imagine cutting each
remaining fourth in half. We can determine the number of cookies eaten by the Aunt by adding up the
12
eighths (half of the fourths) as 1 ½ or
.
8
The number of cookies Holly ate is 3, or
Do you have students using this model of halving? In order to use this method powerfully,
students need to understand halves, fourths, and eighths, and the relationship between all
three. Without this understanding, the students may just count each part as one, and answer
that David ate 12 (instead of 12 halves) and Holly ate 12 (instead of 12 fourths) and Auntie
ate 12 (instead of 12 eighths).
If the student is using partitive division to make sense of the equal shares in the problem, Part 3 can
be modeled as David makes two [denominator] equal groups of cookies and then eats the cookies in
one [numerator] of the groups.
Holly then arrives and, with the remaining cookies, makes two equal groups of cookies and then eats
the cookies in one of the groups.
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X
Auntie arrives and, with the remaining cookies, makes two equal groups of cookies and then eats the
cookies in one of the groups.
Do you have students who are using this model of halving? In order to use
this model powerfully, students need to understand that each time a group
of cookies is eaten, the remaining half becomes the “new whole” for the
next part of the problem.
If the student is using measurement division to make sense of the equal shares in the problem, Part 3
can be modeled as David makes equal groups of two [denominator] cookies and then eats one
[numerator] cookie from each group. This might be read as a “one out of every two” idea of
fractions.
X
X
X
X
X
X
When Holly comes, she makes equal groups of two cookies and then eats one from each group.
X
X
X
When Auntie comes in, there are three remaining cookies. What does it look like to make groups of
two cookies from 3 cookies?
A second grader may make the first group of two cookies, and then leave the third cookie.
Do you have students who are modeling half of three in this way? What does it say about
their understanding of fractions as division, or halving? Students who use this measurement
model of division also need to understand the remaining half becomes the “new whole” for
the next part of the problem. What does the student need to make sense of that last “groups
3
of two” as ? Or does it make more sense to move flexibly to a different interpretation?
2
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Which model did you value? Why? It may be tempting to explain each of these models for using
fractions to make fair shares, or, on the other hand, it may be tempting to try to limit children’s
exposures to the more “confusing” ones, and only present them one way to solve such a problem.
However, it is more important for us to understand the various models so that we can recognize the
strategies and models the students are using, and provide opportunities for them to make sense of
these in context, and to make connections to the other models. These connections will deepen their
understanding of fractional models of fair shares.
For example, if a student explains that “half of 12 is 6 because 6 + 6 equals 12”, then he or she may
already be using a partitive division model to make sense of the quantity. If a student explains that
“half of 12 is 6 because 1 and 1 and 1 and 1 and 1 and 1 is 6”, then he or she may already be using a
measurement division model to make sense of the quantity. Finally, if a student explains that “half
of 12 is 6 because ½ and ½ is 1, ½ and ½ more is 2, and ½ and ½ more is 3”, then he or she may
already be using a regional model to make sense of the quantity. At this age, students are using bits
and pieces of ideas and information around fractions, division, doubling, halving, and even symmetry
and learning how to model their thinking. Expect lots of fragile and underformed thinking! Certainly
they aren’t naming these strategies as ‘partitive’ or ‘measurement’ or ‘region fractions’. Yet once
we, as educators, recognize the mathematics in their current thinking, we can then reflect on the
questioning strategies and task selection that will move their thinking and push their understanding
when they are stuck in one way of trying to make sense.
Van de Walle provides several thought-provoking part-and-whole activities that present students with
a variety of contexts for working backwards and forwards between parts and wholes, where “whole”
may be either a set of objects, or a single object. These can be found on pages 260 – 263 in
Teaching Student-Centered Mathematics K-3, and samples are given here.
Given the whole
and the fraction,
find the part.
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Given the part
and the fraction,
find the whole.
Given the whole
and the part, find
the fraction.
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Re-Engagement Lesson for Auntie Em’s Cookies
This reengagement was prepared by a group of 1st/2nd grade teachers and coaches. The lessons were
designed by teacher Debbie Stoddard, based on an analysis of the student work for Auntie Em’s
Cookies for 20 second grade students. The results of the student work on the task, both before and
after the reengagement lessons, are included.
Objective for Mathematical Understanding:
• “Halving” is dividing into two equal groups.
• Thinking of “doubles” (i.e., 4 + 4 = 8) can be a useful strategy for finding a half. Half can be found by
subtracting out one of the doubles. (Half of 8 is 4, because 8 – 4 = 4).
• Once half is found, the student must think of the remaining half as the new whole in order to find half of
the half. For example, if we start with 8, we can find half by subtracting one of the addends from the
“double fact” (8 – 4 = 4) and then using the remaining half (4) as the new whole. Using double facts (2
+ 2 = 4), the student can then subtract out one of the addends (2) and be left with half of the half (4 – 2 =
2).
Essential questions:
• What is a half?
• How can you find a half of an amount?
• What happens when you take a “half of a half”?
What does the data tell us: Assessed 20 second grade students using Auntie Em’s Cookies (MARS,
2009). Looking specifically at Parts 3 and 4, 9 out of 20 students were able to take half of 12 cookies and
accurately answer “6 cookies” for Part 3. 8 out of 20 were able to also take half of the remaining 6 cookies and
accurately answer Part 4 with “3 cookies”. 3 out of 20 were able to accurately answer the ramp-up question,
Part 5, as 1 ½ cookies.
Student Errors in Parts 3 and 4: Several students answered all Parts ~ 3, 4, and 5 ~ with “4”. There
is evidence that they interpreted “half” as “equal groups”, but not as two equal groups. For these students, “half”
seemed to mean that they should make as many equal groups as there are people (3 equal groups, in this case,
would have four each).
More than 25% of the class used this same idea, but answered Part 3 as 8 (because only one of the
groups of 4 had been eaten by David) and Part 4 as 4 (because now two of the groups of 4 had been eaten by
David and Holly).
There was no call to explain their thinking on Parts 3 and 4, and there were very few students who used
any recordings to explain their thinking.
Activity One
• Read “The Doorbell Rang” with the students. Introduce the story by asking the students to
focus on what is happening to the cookies in the story. The emphasis is on the math action.
o Pause: bottom of Page 3. How many cookies in all? (12) How do you know? (6 + 6 =
12) Teacher records 12 cookies in a 3x4 array.
o Pause: bottom of Page 4 How many friends?(2) How many children? (4) How
would you share the cookies? (Everybody gets 3 cookies. Everybody get’s ¼ of the
cookies.)
o Pause: bottom of Page 9. How many more children came? (2) How do you know?
(Peter and his brother) How many cookies will each child get? (2) How do you know?
(Fair shares. 6 kids each get 2 cookies.)
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o Pause: bottom of Page 13: How many children were at the door? (6. Joy, Simon, 4
cousins) How many children are there now? (12) How many cookies does each child
get? (1 each. Fair shares.)
Activity Two
•
•
•
•
Chocolate Chip Cookies ~ Materials: Each pair of children
needs one cup and 8 mini chocolate chip cookies.
Present: You have baked delicious chocolate chip cookies.
There are 8 on the plate.
o 1. There are 8 cookies. Bill put ½ of them in the
freezer. How many are left? ____ (Students may use
the cup as the freezer.)
o 2. Dad came home and ate half of the cookies that
were left. How many cookies did he eat? ______
o 3. Bill decided to eat ½ of the leftover cookies. How
many did he eat? _______
o 4. Where are all of the cookies?
Students work in pairs to solve each problem presented on the
poster. They record their solutions, and the teacher facilitates
a sharing out of their ideas.
o Guiding questions for the debrief: Where are the
cookies? How did you figure out how many cookies
were left? (on the plate, after he ate them, etc.)? How
did you know it was half? What did you have to do to
figure out what half was?
Students may use numbers, equations, pictures, models, or
words to explain their actions. Look for student thinking
around the following big ideas:
o Using doubles (4 and 4 is 8, they may use this known
fact to measure out 4 at a time.)
o Dividing by 2 (for example, divvying out cookies to the
freezer and the plate until the amounts are equal).
o Subtracting out one of the addends from a double fact.
(This photo represents one example of how a teacher was recording what her students reported about
their thinking. She helped them to make connections between the various strategies by using
questions, drawings, and equations as part of the debrief.)
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Examples of Teacher Recordings from Activity Two:
From the poster, we can see that the teacher modeled the thinking
shared by students for “8 as double 4” using the picture of the four
circles, with a dividing line to represent that these are two different
groups of four. She then slashes through one group, representing a
“take away” model for subtraction, and also shows this with the
equation, “8 – 4 = 4”. There is also evidence here of how she used
questioning to press on student thinking and clarify their ideas. “Why
4?” asks students to really consider that it has to be four, because that’s
what makes two equal groups from 8.
Here the teacher uses a model of 4 circles, with another
dividing line slash to show that halving the four makes two equal
groups of two. The use of the arrow helps connect where the four
circles originally came from ~ up where the four were half of the
eight!
The teacher also records onto the poster any “big
mathematical understandings” that the students summarize. Here we
see “Half is a fair share.” These are especially important to record,
when it is clear and fresh for the students, because new
understandings can be very fragile and they may need to revisit these
ideas multiple times before they are internalized.
Examples of Student Work from Activity Two:
This pair of students is representing their
halving ideas in a variety of ways. They are
using inverse operations (x2) to think about
division by 2. They are also showing a model of
two groups of four cookies.
They’re showing that they understand
that four (half) of the cookies are going away
(crossing them out in the picture) which leaves
the other group of four as their new ‘whole’.
Although it’s not affecting their
understanding of this current task, it’s interesting
to note that they are also representing the half of
8 as a unit fraction. (Look at the drawing that
looks like a pizza, divided into eights, with four
of the eighths shaded.) Representing fractions of
a whole and parts of a group are two very
different ways of thinking about fractions, yet
they are using these ideas interchangeably here.
2nd Grade – 2009
Copyright ©2009 by Noyce Foundation
All rights reserved.
pg.
111
This pair of students is thinking about
‘fair shares’. They have also
introduced the language of “each”, and
have modeled their two equal groups.
These students are representing the
division with subtraction of equal
groups.
These students use a division sentence
to represent 8 divided by 2, and relates
that to the double fact of 4 + 4 = 8.
In the number sentences, they’ve
demonstrated an understanding of the
inverse relationship between addition
and subtraction, while the pictures
show an understanding of subtracting
out one half to find the new ‘whole’ for
the next part of the problem.
Nicely demonstrates a good labeling
strategy to keep track of where the
cookies are. (See Part 2, where the
students have labeled two as ‘eaten’
and 2 as ‘left’.)
2nd Grade – 2009
Copyright ©2009 by Noyce Foundation
All rights reserved.
pg.
112
This pair of students is really thinking
about the action of the context presented
in the problem. They have clearly
labeled where each group of cookies is
going. They are also using arrows to
represent action.
Although it looks like they’ve shown two
groups of three in Part 4, we can tell that
this picture is really of two cookies that
each have three ‘chocolate chips’. The
label of “half”, and the words and
equation that are part of their
explanation, all combine as evidence of
the depth of these students’ thinking.
After the students experienced the reengagement lessons, Auntie Em’s Cookies was readministered.
Follow-up assessment: Reassessed the same 20 students using Auntie Em’s Cookies (MARS,
2009). Could they answer Parts 3 and 4 specifically, and what evidence is there that they are
using a strategy developed in the re-engagement lesson?
Results: 19 out of 20 students who were reassessed using this task were able to
accurately answer Parts 3 and 4. Students used the following strategies:
o Subtracting an addend from the known double fact.
o Using a drawing and crossing off (taking away) each half.
o Division equations
Although the ramp-up question, Part 5, was not specifically addressed in the reengagement lesson, 12 out of 20 students were able to accurately answer 1 ½ cookies.
Furthermore, there was still no call to explain their thinking on Parts 3 and 4, yet virtually
every student used some sort of recording of words, pictures, or equations.
2nd Grade – 2009
Copyright ©2009 by Noyce Foundation
All rights reserved.
pg.
113