Overall Frequency Distribution by Total Score
Grade 3
Mean=26.60; S.D.=8.45
800
700
Frequency
600
500
400
300
200
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
Third Grade – 2003
pg.
1
Level Frequency Distribution Chart and Frequency Distribution
2003 - Numbers of students
tested:
Level % at ('99)
1
2
3
4
25%
29%
34%
12%
% at least
('99)
100%
75%
46%
12%
Level
% at ('02)
1
2
3
4
8%
20%
31%
41%
Grade 3:
14609
Grade 3 1999 - 2001
% at ('00)
% at least
('00)
19%
100%
28%
81%
35%
53%
18%
18%
Grade 3 2002 - 2003
% at least ('02) % at ('03)
100%
92%
72%
41%
9%
14%
33%
44%
% at ('01)
% at least
('01)
100%
86%
66%
25%
14%
20%
41%
25%
% at least
('03)
100%
91%
77%
44%
7000
6000
Frequency
5000
4000
3000
2000
1000
0
Frequency
0-13
1 Minimal Success
14-20
2 Below Standard
21-29
3 At Standard
30-40
4 High Standard
1256
2117
4815
6421
Third Grade 2003
pg.
2
3rd grade
Task 1
Even and Odd Numbers
Student
Task
Solve to complete given number sentences using even and odd
addends. Explain effects of adding odd numbers.
Core Idea
1
Number
Properties
Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
• Develop a sense of whole numbers and represent and use them
in flexible ways, including relating, composing, and
decomposing numbers
Core Idea 2
Understand the meanings of operations and how they relate to
each other, make reasonable estimates and compute fluently.
• Understand the effects of adding and subtracting whole
numbers
• Develop fluency in adding and subtracting whole numbers
Number
Operations
Third Grade – 2003
pg.
3
Third Grade 2003
pg.
4
Third Grade – 2003
pg.
5
Looking at Student Work – Even and Odd
The most common justification for why three odd numbers are needed to make 15 is
because two odd numbers make an even number. The nice thing about the work of
Student A is that she goes further to explain what will happen when an even and an
odd number are added.
Student A
Student B can’t make a generalization about odds and evens. Instead the student
states,” You can’t use an odd number”. Then the student gives an example of how to
make 15 with an odd and even number and how to replace the even number with 2
more odd numbers. It is important for students to use examples to help clarify their
thinking, but not confuse an example with a complete justification.
Third Grade 2003
pg.
6
Student B
Many students think that it’s just the size of 15 that means they need to use so many
odd numbers. Maybe they forget that two digit numbers can also be odd. Student C
thinks that its just because 15 is big. Student D has the idea that an even and an odd
are needed to make an odd, but then uses two even numbers to get the even.
Student C
Student D
Student E knows how to put three odd numbers together to make 15 in the beginning
of part 2. However in his explanation he seems to only think about 7 + 8 making 15
and implies that somehow an 8 is needed to make 15. His examples,however, do not
reflect this strategy.
Student E
Third Grade – 2003
pg.
7
Student F knows that it takes an odd and an even to get an odd answer. The student
uses an example to show that this is true. But the student can’t complete the argument
by explaining how to use odd numbers to get the even addend.
Student F
Student G seems to have made an exhaustive list of all the ways to combine the
numbers 1,3,5, and7 and found that you can’t make 15. Student G may also have
thought you needed to double the number, so 7+7 would be too small and 9+9 would
be too large. The student doesn’t seem to be paying attention to the fact that those
answers are all even.
Student G
Teacher Notes:
Third Grade 2003
pg.
8
Frequency Distribution for each Task – Grade 3
Grade 3 – Even and Odd
Even and Odd
Mean: 3.11, S.D.: 1.44
5000
Frequency
4000
3000
2000
1000
0
Frequency
0
1
2
3
4
5
877
1528
2132
3127
4474
2471
Score
Score:
0
1
2
3
4
5
%<=
6.0%
16.5%
31.1%
52.5%
83.1%
100.0%
%>=
100.0%
94.0%
83.5%
68.9%
47.5%
16.9%
The maximum score available for this task is 5 points.
The cut score for a level 3 response is 3 points.
Most students (about 94%) could add even numbers to get a sum of 8. Many students
(about 70%) could add even numbers to get 8 and find at least one combination of odd
numbers to sum to 15. More than half the students could find multiple combinations
for making 8 and 15, while paying attention to the constraints in the task. More than
15% could also give a justification about adding odd numbers. About 6% of the
students scored no points on this task.
Third Grade – 2003
pg.
9
Even and Odd
Points
0
1
3
4
5
Understandings
Misunderstandings
Most students attempted the
problem. They were able to
write number sentences that
equaled 8 or 15.
They did not pay attention to the
constraints of using only even or
only odd numbers. They may have
repeated a combination more than
one time.
Students could make a few
Students may have used one
correct combinations to make combination twice or provided one
8 in part 1.
example with an odd number in part
one. In part 2 students may have
used zero as an odd number or used
a combination of odd and even.
Students could correctly
They could not find enough
complete part one of the task. different combinations using only
They also had some correct
odd numbers to complete part 2.
combinations in part 2.
They had difficulty explaining why
3 odd numbers were needed to
make 15. They tried things like
because it’s a big number, odd
numbers are bigger than evens, you
can’t use an even number, or it’s
confusing.
Students could complete all of Students could not write an
part 1 and part 2 of the task.
explanation of why 3 odd numbers
are needed to make 15.
Students could find multiple
Students often did not finish the
solutions to problems
argument by explaining that a third
following constraints like
number needed to be used to get the
using only odd or only even
odd total.
numbers. They could also
explain that two odd numbers
make an even.
Based on teacher observations, this is what third grade students seemed to know and
be able to do:
• Add accurately.
• Find multiple combinations to get a specific sum.
Areas of difficulty for third graders, third grade students struggled with:
• Using constraints, such as “only even numbers” or “only odd numbers”
• Making a generalization about adding odd and even numbers
• Writing a mathematical justification
• Noting which combinations had already been used
Third Grade 2003
pg. 10
Questions for Reflection on Even and Odd:
•
When working problems in class, do you often tell students to highlight or to
underline certain words to be sure they remember all the rules for the
problem? How often do students get the opportunity to find these rules for
themselves or make sense of directions for themselves?
Looking at student answers to make 15, how many of them made correct totals:
Using only
odd
numbers
•
•
•
•
Using 0
Repeating a Using even
combination and odd
numbers
Using an
operation
besides
addition
Incorrect
total
Do you think students are confused about whether zero is an odd or even
number?
What opportunities do students in your classroom get to make mathematical
arguments or justifications? Do they know that one example does not make a
proof?
Do students get opportunities to explain their strategies and find out if they
will work for other problems?
What are some examples of interesting problems where students in your class
had to explain their thinking?
Teacher Notes:
Implications for Instruction:
While many students were very successful with this problem, some students struggled
with the directions. It is important for students to pick out for themselves the
constraints of the problems, to develop the skill of finding key ideas. Students need to
have a variety of rich problems that require them to pay attention to several demands
such as equals 15, use only addition, and use only odd numbers. Students need
practice with solving problems, where the numbers used are limited in some way
(greater than 5, odd, multiples of 3, etc.) to develop their problem solving skills as
well as computational practice. Some students are still unclear about whether “0” is
odd or even. This is an interesting question to ask students and have them develop
convincing arguments about. Students also need opportunities to explore how
numbers work and explain the patterns made by using certain kinds of numbers. The
task gives students the opportunity not just to repeat information like an odd plus an
odd equals an even, but to construct a convincing argument connecting that
information with why 3 numbers are needed to get an odd answer. However students
Third Grade – 2003
pg. 11
need many different kinds of opportunities to stretch their thinking and ability to
make justifications and should not be limited to just problems about odd and even. To
help them develop the ability to make convincing arguments, teachers might pose
questions like “Will that always work? Why? How can we show why?...”
Teacher Notes:
Third Grade 2003
pg. 12
3rd grade
Student
Task
Core Idea
2
Number
Operations
Core Idea 3
Patterns
Functions
And
Algebra
Task 2
Addition Trains
Given a sum and specific numbers, find the addends to correctly
complete a number sentence.
Understand the meanings of operations and how they relate to
each other, make reasonable estimates and compute fluently.
• Understand the effects of adding and subtracting whole
numbers
• Develop fluency in adding and subtracting whole numbers
• Develop a sense of whole numbers and represent and use them
in flexible ways, including relating, composing, and
decomposing numbers.
Understand patterns and use mathematical models to represent
and to understand qualitative and quantitative relationships.
• Illustrate general principles and properties of operations using
specific numbers
Third Grade – 2003
pg. 13
Third Grade 2003
pg. 14
Third Grade – 2003
pg. 15
Third Grade 2003
pg. 16
Looking at Student Work – Addition Trains
The majority of students did quite well on this task. They could correctly compute the
sums in their head and put down the correct answers. Very few students showed any
work or thinking on this task. Student A is one of the few students who shows a
counting strategy using tallies to solve the problem. Many students, like Student B,
used a guess and check strategy with several erasures on their paper. Student B is
also an example of a typical error for part 4. Many students did not fill in all the
boxes if they could find a shorter solution.
Student A
Third Grade – 2003
pg. 17
Student B
Third Grade 2003
pg. 18
Student C has noticed an important number pattern, that all the sums are one more
than a multiple of 5. Even though the constraint of using only 3 and 5 is bolded, the
student ignores that direction. This was the most common error.
Student C
Student D used the constraints, limiting number choices to 3 and 5, where convenient;
but picked other choices when he could not find the answer easily.
Student D
Third Grade – 2003
pg. 19
A small number of students picked a wide assortment of convenient numbers to get
the correct sum, ignoring both the 3 and the 5 as part of problem. Student E is an
example of this type of problem solving.
Student E
Teacher Notes:
Third Grade 2003
pg. 20
Grade 3 – Addition Trains
Addition Trains
Mean: 4.11, S.D.: 1.55
11000
10000
9000
Frequency
8000
7000
6000
5000
4000
3000
2000
1000
0
Frequency
0
1
2
3
4
5
808
1054
641
560
1721
9825
Score
Score:
0
1
2
3
4
5
%<=
5.5%
12.7%
17.1%
21.0%
32.7%
100.0%
%>=
100.0%
94.5%
87.3%
82.9%
79.0%
67.3%
The maximum score available on this task is 5 points.
The cut score for a level 3 response is 3 points.
Most students (about 95%) could use both constraints to find a sum equal to eleven
and a sum equal to 18.Students meeting standards (83%) could find addition trains for
three of the given target numbers. More than 65% of the students could meet all the
demands of the task. A little more than 5% of the students scored no points on this
task.
Third Grade – 2003
pg. 21
Addition Trains
Points
0
Understandings
All students surveyed attempted
this problem. Most showed
correct calculations to get the
specified totals.
1
Students with this score could
find the combination of all 3’s to
make 18.
2
Students could use 3’s and 5’s to
make 11 and one combination of
18.
Students could use 3’s and 5’s to
make 11 and 21 and either make
16 or find the combination of 18
using only 3’s.
Students could do most of the
task with only one error.
3
4
5
Misunderstandings
Students ignored the constraints of
using only 3’s and 0’s. About 7% of
the students used many other
numbers. More than 10% of all
students used a combination of 5’s
and 1’s.
In part 4 many students left empty
boxes. They found the combination
of 5’s and 3’s and used it in both
trains.
The most difficult part seemed to be
finding two combinations to make
18. Students left empty cars or used
other numbers for one of the
combinations. About 25% of the
students missed question 3. These
errors were generally a combination
of 3’s and 5’s with an incorrect
total.
Students could identify and use
the constraints to find a variety of
sums using only 3’s and 5’s.
Based on teacher observations, this is what third grade students seemed to be able to
do:
• Students were most successful on using a combination of 3’s and 5’s to make
18.
• Students were good at using addition to make the correct sums, even if they
did not use the constraint.
• Many students recognized that problems 1 through 3 were multiples of 5 plus
one.
Areas of difficulty for third graders, third grade students struggled with:
• Many students had difficulty using the constraints of “either 3 or 5” as the
addends.
Third Grade 2003
pg. 22
Questions for Reflection on Addition Trains:
•
•
•
•
What evidence do you see of students using counting strategies or guess and
check to solve this problem? How can you help students make the transition
from counting to using addition?
How many of the student errors were caused by addition errors? How many
of the errors were caused by ignoring the constraints? What are the
implications for instruction?
Do students in your class get enough practice interpreting directions and using
multiple constraints to solve problems? Do you have a good resource for this
type of problem?
What types of problems have your students had to develop their logical
reasoning?
Teacher Notes:
Implications for Instruction:
Students need practice with interpreting problems and their constraints. Teachers
might ask questions like, “How many different sums less than 20 can you make using
two addends?” Students need to have fluency in addition of more than two whole
numbers. Students would often use numbers of convenience, rather than trying to get
the answer using only the choices given. Students need practice with solving
problems, where the numbers used are limited in some way (greater than 5, odd,
multiples of 3, etc.) to develop their flexibility with numbers. It is important for them
not only to be able to add, but also to be able to work backwards from the total to
finding a variety of ways to get that total. Students need to be able to reason about the
number system and explore how different numbers can combine to make a given total.
Part of problem solving is the ability to find solutions with limited choices.
Teacher Notes:
Third Grade – 2003
pg. 23
3rd grade
Student
Task
Core Idea
5
Data
Analysis
Core Idea
3
Patterns,
Functions,
and Algebra
Task 3
Vending Machines
Interpret a pictograph and add additional information. Complete a
chart with information provided.
Collect, organize, display, and interpret data about themselves
and their surroundings.
• Represent and interpret data using pictograph
• Describe important features of a set of data
Understand patterns and use mathematical models to represent
and to understand qualitative and quantitative relationships.
• Show quantitative change
• Use concrete, pictorial, and verbal representations to develop
an understanding of invented and conventional symbolic
notations
• Model problem situations using representations such as graphs
and tables to draw conclusions
Third Grade 2003
pg. 24
Third Grade – 2003
pg. 25
Third Grade 2003
pg. 26
Third Grade – 2003
pg. 27
Looking at Student Work – Vending Machines
Most students did very well on this task. One of the most common errors was
forgetting to add another apple juice to the graph. Student A does a nice job of
showing the mathematics for finding the answer in part 4 and labeling what each
number represents. Student A is one of the few students who put a scale on the
vertical axis to make it easier to quantify the symbols on the graph. Student A is also
one of the few students to use the least number of symbols when making his own
graph in part 6. Student A seems to think that the equal sign means “the answer”
follows” , rather than showing an understanding of equality. (see work in #4)
Student A
Third Grade 2003
pg. 28
Student A
Student B also shows the addition for finding the total number of drinks and
explaining why she did the addition. Her graph is more typical, using many circles to
simplify the filling in the graph.
Third Grade – 2003
pg. 29
Student B
Third Grade 2003
pg. 30
Student C gives another example of clarifying where the numbers in the equation
came from.
Student C
Many students went to great lengths to make number sentences to verify their
answers. Student D makes the table into an array and subtracts the empties, a strategy
which fits the context. Student E uses a number sentence that gives the correct
answer, but the numbers do not match back to the graph or the situation.
Student D
Student E
Third Grade – 2003
pg. 31
More than 50% of the students showed the addition for finding all the drinks sold.
More than 10% of the students are obviously still relying on counting, as did Student
F and Student G. Additionally Student G has not learned to do modeling and relied
on drawing every cup to show what was counted.
Student F
Student G
A few students have difficulty understanding what is required for a mathematical
explanation. Student H explains about reading the question, rather what mathematical
operation or process helped to get the answer. Student H did not understand the
symbol scales in part 6. The student seems to want to fill all the spaces in the graph
with a symbol of some kind. The symbols for salsa are actually correct. However the
symbols for the other three do not fit the numbers in the problem.
Third Grade 2003
pg. 32
Student H
Teacher notes:
Third Grade – 2003
pg. 33
Grade 3 – Vending Machines
Vending Machines
Mean: 8.03, S.D.: 2.39
6000
5000
Frequency
4000
3000
2000
1000
0
Frequency
0
1
2
3
4
5
6
7
8
9
10
203
184
249
384
608
711
630
941
2033
3793
4873
6
7
8
9
10
Score
Score:
%<=
%>=
0
1
2
3
1.4%
2.6%
4.4%
7.0%
4
5
11.1% 16.0% 20.3% 26.8% 40.7% 66.6% 100.0%
100.0% 98.6% 97.4% 95.6% 93.0% 88.9% 84.0% 79.7% 73.2% 59.3%
33.4%
The maximum score available on this task is 10 points.
The cut score for a level 3 response is 6 points.
Most students (about 89%) could answer simple questions about the drink graph and
find the total number of drinks. More than half the students (about 60%) could
answer questions about the drinks graph, explain how they found the total number of
drinks, and correctly fill out most of the graph for chips. More than 30% could also
fill in an additional apple juice on the drink graph to meet all the demands of the task.
Approximately 1% of the students scored no points on this task.
Third Grade 2003
pg. 34
Vending Machines
Points
0
1-4
5
9
10
Understandings
No students in the sample
received a zero.
Students in this range could
answer something in the first
four questions. Generally 1,2,
and 4 were the easiest. Some
students were able to put in the
correct symbols for salsa or
corn chips.
Students with this score could
also explain how they found
the total number of drinks.
About 54% used addition to
find the drinks and 13% used
counting.
Students with this score could
do most of the task, but forgot
to add the additional apple
juice to the drink graph.
Students could answer
questions about a graph,
including comparison
subtraction and explaining
their thinking for the finding
all the drinks, and use symbols
with scales to make their own
graphs and add data to existing
graphs.
Misunderstandings
Less than 10% of the errors in part 4
were miscalculations. About 13 %
of the students missed the graph
because they didn’t attempt it, tried
to fill in all the spaces, or missed or
ignored the value of the symbols.
13% did not attempt to explain how
they found the total number of
drinks.
Based on teacher observations, this is what third grade students seemed to know and
be able to do:
• Students could correctly answer questions from a graph (e.g. “How many cola
drinks?”, “What kind of drink sold the most?”, and “How many drinks were
sold in all?”
• Most students could do comparison subtraction.
• Most students could successfully either add data to the drink graph or do some
correct portion of making their own graph.
Areas of difficulty for third graders, third grade students struggled with:
• Explaining their thinking for finding the total number of drinks.
• Matching symbols with quantity to represent data on a pictograph.
• Adding data to an existing graph.
Third Grade – 2003
pg. 35
Questions for Reflection on Vending Machines:
Looking at how your students explained their answer in part 4, how many of them:
Showed
Addition
Counting
Gave
words to
describe
the
process
Gave
explanation
not related
to the
solution
Made
addition
errors
Gave no
explanation
Used
number
sentence
that
didn’t fit
the
context
Now look carefully at the types of graphs students made in part 6, how many of them:
Filled in the
appropriate
values with no
more than 1
circle per
column
•
•
•
•
Had correct
values for the
data, but used
more than 1
circle per
column
Tried to fill in
all or almost
all the blanks,
ignoring the
quantities in
the data
For any one
Did not
column, used
attempt the
all circles or all graph
squares
What are the instructional implications based on data from your classroom?
What types of activities or experiences do students seem to need?
How often do students in your classroom get the opportunity to explain their
thinking? Do they realize that explaining the choice of operation is important?
Do they know that part of the explanation should include why they picked
certain numbers?
Which students have not made the transition from counting to addition? What
kinds of activities could help them bridge from the concrete to the more
abstract and why do you think that is valuable?
Teacher notes:
Third Grade 2003
pg. 36
Implications for Instruction:
Students need experience interpreting and completing pictographs and bar graphs.
They need more practice using symbols that represent different quantities of the same
item to complete a graph. If students are using bar graphs, then it is important that the
scale on the vertical axis not always be one. Students need to have a variety of
experiences with data, learning to make their own graphs as well as completing and
interpreting the graphs of others. Asking students a variety of questions about a
graph, such as, “How many more. . . ?” or “How many total . . . are represented on the
graph?” are important to help them understand and analyze what information the
graph is representing. Students also need to have frequent opportunities to explain
how they got their answers, learning to describe which numbers they chose and also
why they picked a particular operation. Hearing, evaluating, and comparing the
explanations of others allows them to learn what is valued in a good response and to
self-correct and to improve the quality of answers they make for themselves.
Teacher notes:
Third Grade – 2003
pg. 37
3rd grade
Task 4
Cherie’s Shapes
Core Idea
3
Patterns,
Functions, and
Algebra
Name some simple geometric shapes. Use multiplication and/or
division to complete given patterns and design their own repeating
patterns.
Understand patterns and use mathematical models to
represent and to understand qualitative and quantitative
relationships.
• Describe and extend geometric patterns
• Represent and analyze patterns using words
Core Idea
2
Number
Operations
Core Idea
4
Geometry and
Measurement
Understand the meanings of operations and how they relate to
each other, make reasonable estimates, and compute fluently.
• Understand multiplication as counting by multiples and
division as forming equal groups
Recognize and use characteristics, properties and relationships
of two-dimensional geometric shapes
• Identify and compare attributes of two-dimensional shapes
and develop vocabulary to describe the attributes
Student
Task
Third Grade 2003
pg. 38
Third Grade – 2003
pg. 39
Third Grade 2003
pg. 40
Third Grade – 2003
pg. 41
Looking at Student Work – Cherie’s Shapes
Almost one third of the students could successfully complete all the parts of the task.
Student A shows an understanding of how fitting equal groups into a larger number
relates to division. Notice in part 3 how the student articulates the number of shapes
in the pattern. Student A used a new pattern of six shapes, which is the number
chosen by about 13% of the students. Approximately 25% of the students picked a
pattern of 5 and 25% picked a pattern of 3. About 10% picked a pattern of 10.
Student A
Third Grade 2003
pg. 42
Student A, part 2
Third Grade – 2003
pg. 43
While many students could use multiplication or division to find the number of
patterns that would fit on the wall, other students are still relying on repeated addition
to get their solution. Student B does a good job of explaining her process for repeated
addition. Student B seems to also know something about how patterns are named,
like AB or AABB. The student has invented her own naming system for the final
pattern.
Student B
Third Grade 2003
pg. 44
Another process for finding the number of patterns needed is to use multiplication.
Student C seems to have some understanding of this relationship, but confuses the
total number with number of patterns needed. The student needs more work on part/
whole relationships. In part four the student draws a pattern of 9, but continues to use
the same arithmetic problem from part 2 and 3.
Student C
Third Grade – 2003
pg. 45
Student D is still struggling with the idea of multiplication. In part two the Student D
makes the same mistake as Student C. In part 4 the student shows a weakness in
multiplication facts. Finally in part 4 the student reverts to a counting strategy to
correctly solve for the number of patterns needed.
Third Grade 2003
pg. 46
Student D
Third Grade – 2003
pg. 47
Student E does not understand the concept of equal groups. The student just adds the
numbers in the problem While the student uses the work “times”, the student
demonstrates no understanding of the meaning of multiplication or making equal
groups. In part 4 the student draws a pattern of 8 or 2 patterns of 4, yet even here
there is no understanding of what constitutes a group. Instead of using 8 or 4 the
student uses 3 because there are three different shapes imbedded within the pattern of
8.
Student E
Third Grade 2003
pg. 48
Grade 3 – Cherie’s Shapes
Cherie's Shapes
Mean: 5.78, 3.42
4000
Frequency
3000
2000
1000
0
Frequency
0
1
2
3
4
5
6
7
8
9
10
405
1187
2203
1538
849
840
858
900
935
1278
3616
6
7
8
9
10
Score
Score:
%<=
%>=
0
2.8%
1
2
3
4
5
10.9% 26.0% 36.5% 42.3% 48.1% 53.9% 60.1% 66.5% 75.2% 100.0%
100.0% 97.2% 89.1% 74.0% 63.5% 57.7% 51.9% 46.1% 57.7% 33.5%
24.8%
The maximum score available on this task is 10 points.
The cut score for a level 3 response is 4 points.
Most students (approximately 90%) could give all the correct shape names or name
all the shapes but hexagon and draw their own pattern. More than half the students
could find how many times all the patterns were needed, including their own design.
Many of the students with scores of 7,8, or 9 struggled with explaining their thinking
in part 2 and 3. Almost 25% of the students could meet all the demands of the task.
Less than 3% of the students scored no points on this task.
Third Grade – 2003
pg. 49
Cherie’s Shapes
Points
0
1
Understandings
All the students with this score
in sample tried the problem.
Students with this score could
name at least three shapes
correctly. These were usually
the triangle, circle and square.
2
Students with this score could
name all the shapes or could
name everything except the
hexagon and make their own
pattern.
3
These students could name all
the shapes and design a pattern
that would go evenly into 30.
4
Students could name the shapes
and draw their own pattern and
find the number of times it
would fit on the wall.
Students could name shapes and
find a solution for the number
of times each pattern would fit
on the wall.
6
9
10
Students showed a good
understanding of multiplication,
using equal groups, and
working with inverse
relationships.
Students could name shapes,
find the number of times
different patterns could fit on a
wall, explain their process, and
design a pattern that would fit
evenly into 30.
Third Grade 2003
Misunderstandings
The most difficult shape for
students was the hexagon, then the
rectangle. The orientation of the
rectangle was confusing for a few
students.
These students do not understand
equal groups and the relationship to
multiplication. Their answers for
the number of times the first pattern
would fit on the wall ranged from 1
and 2 up to 45 or 50 times.
Students could not use counting,
repeated addition, multiplication, or
division to find the number of
groups. They do not have a basic
understanding of the operation of
multiplication.
Students could not articulate how
they got their answers. Their
explanations might say things like,
“I imagined the shapes that were on
the wall.” They may have put
inappropriate number sentences,
like 10+20=30 for the 5 pattern or 3
x 10= 30 for the 6 pattern.
Students with this score made minor
errors with designing their own
patterns or naming hexagons.
A few of these students are still
drawing or counting to find the
answers. Work may still need to be
done to help them bridge to
multiplication.
pg. 50
Based on teacher observations, this is what third grade students seemed to know and
be able to do:
• Identify and name geometric shapes.
• Use counting, repeated addition, or multiplication to find the number of times
equal groups will fit into a larger number.
• Design a pattern that will fit evenly into 30.
Areas of difficulty for third graders, third grade students struggled with:
• Naming hexagons
• Matching a number sentence to the situation of how many groups will fit into
a larger shape, understanding number operation of multiplication and its
inverse.
• Many students have not made the transition from thinking additively to
thinking multiplicatively.
• Students are still having trouble with part/whole relationships. If they get the
correct number sentence, like 5x6=30, they often think the answer is it will fit
30 times.
• Recognition of shapes with nonstandard orientations.
Questions for Reflection on Cherie’s Shapes:
What strategies did your students use to solve parts 2 and 3? Did they –
Draw all
the shapes?
•
•
•
•
Show
evidence of
counting?
Use
repeated
addition?
Use
Use
multiplication? division?
Write a
number
sentence
that didn’t
match the
context?
When teaching multiplication, how do you help students make sense of equal
groups? How much work do you do to help them understand the meaning of
the operation? What kinds of activities help build this understanding?
Do you use a variety of question types when working with multiplication?
(e.g. grouping, rate, price, multiplicative comparison)
How do you help students see the relationship between multiplication and
division?
How many of your students are still thinking with repeated addition? What
kinds of activities help them to bridge to thinking multiplicatively? What are
some good resources for this?
Third Grade – 2003
pg. 51
Teacher Notes:
Instructional Implications:
Students need to be able to create repeating patterns being mindful of multiple
constraints. At this grade level students are beginning to use simple division or
multiplication to solve problems. It is important for them to understand the meaning
behind the operation and not just the computation. There is a fundamental idea here
about working with equal groups and understanding part/whole relationships. A
weakness here leads to huge problems when trying to understand fractions,
proportional reasoning and scaling, probability and other mathematical topics.
Students need to have a deep understanding of the operation and multiplicative
relationships to successfully move forward in their mathematical thinking. Students
at this grade level should be able to work flexibly with number fact families. Being
able to see the relationship between multiplication facts and using that relationship to
solve problems is a big idea for students.
Teacher Notes:
Third Grade 2003
pg. 52
3rd grade
Student
Task
Core Idea
4
Geometry and
Measurement
Task 5
Patchwork Quilt
Recognize and describe properties of simple shapes. Mark two
lines of symmetry on a given design. Create a design with two
lines of symmetry.
Recognize and use characteristics, properties, ad relationships
of two-dimensional geometric shapes
• Identify and compare attributes of two-dimensional shapes
and develop vocabulary to describe the attributes
• Recognize geometric ideas and relationships and apply
them to problems
• Use visualization, spatial reasoning, and geometric
modeling to solve problems.
Third Grade – 2003
pg. 53
Third Grade 2003
pg. 54
Third Grade – 2003
pg. 55
Third Grade 2003
pg. 56
Looking at Student Work – Patchwork Quilt
Describing the attributes of a rectangle was very difficult for most students. They did
not understand that the characteristics needed to distinguish a rectangle from other
geometric figures and should be specific. Student A is one of the few students to have
a complete definition. Student A has also demonstrates the knowledge that geometric
shapes should be drawn accurately and with rulers.
Student A
Third Grade – 2003
pg. 57
Student B shows that Zach’s shape is not just a triangle, but a right triangle. The
student’s definition completely defines the rectangle in two clues instead of three.
Student B also uses a ruler to accurately draw a design with 2 lines of symmetry. The
student makes the lines of symmetry larger than the design, to clarify which lines are
part of the design and which are the lines of symmetry.
Student B
Third Grade 2003
pg. 58
Many students want to quantify the size of the sides in the rectangle to differentiate it
from a square. There is a misconception that squares are not rectangles. See the work
of Student C and Student D.
Student C
Student D
Other students are not focusing on geometric attributes of a rectangle, but on how the
rectangle looks or what it resembles. See the work of Student E and Student F.
Student E
Student F
Many students made very complicated shapes. Because the designs were done
without rulers or measurements, it was difficult to tell if in fact there were lines of
symmetry. Student G has probably drawn a symmetrical shape, but because the
corners of the center design are slightly different sizes it is difficult to be sure.
Third Grade – 2003
pg. 59
Student G
A common error for students was to draw a design with a triangle. The work of
Student H has a design with the objects placed symmetrically, but loses points
because the triangles themselves do not have two lines of symmetry.
Student H
Third Grade 2003
pg. 60
Because the directions do not clarify drawing the lines of symmetry in the design or
how to distinguish the lines of symmetry from lines in the design, it is often hard to
tell what the student is thinking. In part 5 Student I has drawn 2 lines that do not
relate to the symmetry of the overall design. They may be showing symmetry for the
rectangles. There is not enough information to tell completely what the student is
thinking about symmetry. Similarly in part 6 the student has drawn 2 lines, which if
interpreted as the design make a picture with 2 lines of symmetry. However if the
lines were drawn to show the symmetry of the square, then the student does not
understand the concept. With student work like this, it might be important for the
teacher to provide some follow up activities or questions to get more in-depth
information.
Student I
Third Grade – 2003
pg. 61
Grade 3 – Patchwork Quilt
Patchwork Quilt
Mean: 5.57, S.D.: 2.40
2500
Frequency
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
9
10
675
282
656
1280
1572
2128
2246
2341
1949
1225
255
6
7
8
9
10
Score
Score:
%<=
%>=
0
1
4.6%
6.6%
2
3
4
11.0% 19.8% 30.6%
100.0% 95.4% 93.4% 89.0% 80.2%
5
45.1% 60.5% 76.5% 89.9% 98.3% 100.0%
69.4% 54.9% 39.5% 69.4% 10.1%
1.7%
The maximum score available on this task is 10 points.
The cut score for a level 3 response is 5 points.
Most students (about 90%) could name the three shapes. More than half the students
(about 70%) could name the shapes and draw lines of symmetry. A little less than half
of the students could name two attributes of a rectangle. Only 2% of the students
could meet all the demands of the task. 4% of the students scored no points on this
task.
Third Grade 2003
pg. 62
Patchwork Quilt
Points
Understandings
About half the students with this
0
score attempted the problem.
3
Students with a score of three
could name the three shapes or
name square, triangle and give
one attribute of a rectangle.
4
Students could name the three
shapes and give one characteristic
of a rectangle. The dominant
attribute was having 4 sides.
6
Students with this score generally
gave one correct attribute for
rectangle. They could also do
something with symmetry. They
either drew both lines of
symmetry in part 5 or drew a
design with 2 lines of symmetry,
but not both.
7
Students with this score knew
shape names and understood
symmetry.
Students with this score knew
shape names and understood
symmetry.
Students could name geometric
shapes, give attributes of a
rectangle that distinguish it from
other geometric shapes, draw
lines of symmetry on a design,
and create their own design
having 2 lines of symmetry.
8
10
Third Grade – 2003
Misunderstandings
About half the students with this score
did not attempt the problem. The
beginning of this task is similar
enough to Cherie’s shapes to suggest
than some students may not have had
enough time.
Almost 1/3 of all students talked
about the overall shape all long or
long and thin. Almost 20% compared
it to a square (e.g. its 2 squares, a
stretched square, half a square, 4
squares, etc.)
More than one third of the students
expressed an idea about how the sides
are different (e.g. 2 longs, 2 shorts or
sides are different sizes). About 20%
of all students gave characteristics for
a triangle; like pointy, 3 corners, or 3
sides, all angles less than 90 degrees.
About half the students knew a
rectangle had 4 sides. About half the
students knew it had 4 angles (of
those about half also knew the angles
were right angles). Students had
difficulty with drawing lines of
symmetry or making designs. Their
designs often were too complicated or
included a shape with only one line of
symmetry like a triangle.
65% of the students with this score
got no points in part 4, describing a
rectangle.
Students could only identify one
attribute of a rectangle, which was
usually the number of sides.
pg. 63
Based on teacher observations, this is what third grade students seemed to know and
be able to do:
• Name geometric shapes
• Draw lines of symmetry
• Know that a rectangle has 4 sides or 4 angles
Areas of difficulty for third graders, third grade students struggled with:
• Understanding that squares are rectangles
• Precisely identifying distinguishing attributes of shapes
• Drawing designs with 2 lines of symmetry
• Knowing that the angles in the rectangle were right angles or 90 degrees
Questions for Reflection on Patchwork Quilt:
•
What opportunities have students had this year working with attributes,
classifying and sorting geometric shapes or other objects?
• What kind of language do students in your classroom have for describing
angles? Have they learned or been exposed to terms like angles or right
angles?
• Did many of your students think that squares are not rectangles? What kinds
of activities can help students to develop that understanding?
• What types of geometric drawings have students done this year? Do they
frequently have the opportunity to use rulers or other tools to make their
drawings more exact?
Many student drawings were unclear as to what part was the design and what part
were the lines of symmetry. Do you have a classroom procedure like making the
line go beyond the boundaries of the shape or being dotted which would help to
clarify this issue in future work?
• Did students make designs with triangles? Designs that were too complex?
• What types of work with symmetry have students done this year?
Teacher Notes:
Third Grade 2003
pg. 64
Implications for Instruction:
Students need more practice identifying specific characteristics of simple geometric
shapes and experience with classifying and sorting. Students need to work with the
logic of sorting to find mutually exclusive descriptors, understand the intersections of
two groups, or the concept of subsets. Students also need to demonstrate an
understanding of symmetry by showing lines of symmetry on a given design, and
create a simple design with two lines of symmetry. Students need a variety of
experiences looking at geometric properties, including sorting, paper folding, drawing
designs, and making patterns with blocks. Students at this age need to work with a
variety of symmetrical shapes, including shapes whose lines of symmetry are not
vertical or horizontal.
Third Grade – 2003
pg. 65