Overall Frequency Distribution by Total Score
Grade 5
Mean=24.15; S.D.=8.79
700
600
Frequency
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
Fifth Grade – 2003
pg.
1
Level Frequency Distribution Chart and Frequency Distribution
2003 - Numbers of students
tested:
Level % at ('99)
1
2
3
4
25%
43%
22%
10%
% at least
('99)
100%
75%
32%
10%
Grade 5: 14401
% at ('00)
12%
36%
34%
18%
% at least
('00)
100%
88%
52%
18%
Grade 5 2002 - 2003
Level % at ('02) % at least
% at ('03)
('02)
1
22%
100%
10%
2
31%
78%
25%
3
25%
47%
33%
4
22%
22%
32%
% at ('01)
% at least
('01)
100%
89%
59%
23%
11%
30%
36%
23%
% at least
('03)
100%
90%
65%
32%
5000
4500
4000
3500
Frequency
3000
2500
2000
1500
1000
500
0
0-11
1 Minimal Success
12-21
2 Below Standard
22-29
3 At Standard
30-40
4 High Standard
1479
3562
4715
4645
Frequency
Fifth Grade – 2003
pg.
2
5th grade
Task 1
Number Story Time
Student
Task
Solve a multi-step money story problem. Write a story problem to
illustrate a division problem with a remainder amount.
Core Idea 2
Number
Operations
Understand the meanings of operations and how they relate to
each other, make reasonable estimates and compute fluently.
• Reason about and solve problem situations that involve more
than one operation in multi-step problems
• Develop fluency in dividing whole numbers
• Understand the meaning of remainders by modeling division
problems
Fifth Grade – 2003
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3
Fifth Grade – 2003
pg.
4
Fifth Grade – 2003
pg.
5
Looking at Student Work – Number Story Time
Many students had trouble with monetary notation. Students don’t know that you
can’t use the decimal point with a cent sign. While there are three major types of
division problems, most students wrote sharing (partitive) problem types. Their
stories were all about cookies, marbles, baseball cards. Even the contexts showed
little variation. Student A has a story problem that requires division by 5 and
addresses the remainder.
Fifth Grade – 2003
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6
Fifth Grade – 2003
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7
Student D seems to create a division story situation, but does not include all the
information to solve the problem. The story does not say how many students can fit
on a bus or how many students there were altogether. The story does include a
question about the remainder.
Fifth Grade – 2003
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8
Student E does not know where the decimal is placed in whole numbers and therefore
does not line up amounts correctly when adding. The student also adds all the
numbers in the problem, failing to recognize that finding change is a subtraction
situation. The student needs more work on understanding the meaning of operations.
This is again pointed out by the number story in part 2, where Student E reverses
whole and parts in the division story.
Student F does not understand the idea of a word problem or story problem. The
student instead writes a different number sentence that gives the same answer.
Fifth Grade – 2003
pg.
9
Frequency Distribution for each Task – Grade 5
Grade 5 – Number Story Time
Number Story Time
Mean: 3.63, S.D.: 2.09
4000
3500
Frequency
3000
2500
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
2249
211
1608
2791
859
3077
3606
4
53.6%
52.4%
5
75.0%
46.4%
6
100.0%
25.0%
Score
Score:
%<=
%>=
0
15.6%
100.0%
1
17.1%
84.4%
2
28.2%
82.9%
3
47.6%
71.8%
The maximum score available for this task is 6 points.
The cut score for a level 3 response is 4 points.
Most students (about 85%) could show a correct process for finding the change.
About half the students could solve correctly the number story problem about change
and write their own number story that required dividing by 5. 25% of the students met
all the demands of the task. About 15% of the students scored no points on this task.
Fifth Grade – 2003
pg.
10
Number Story Time
Points
0
2
3
5
6
Understandings
Misunderstandings
Most of the students attempted this Students with this score do not
problem.
understand the meaning of
operations. They might have added
all the numbers in part one instead
of subtracting or written a
multiplication problem for part 2.
Students with this score could
show a correct process for finding
change. They made a calculation
error or error in monetary notation.
Students could calculate accurately
the amount of change. This
included the ability to put a
decimal point in the proper place
for a whole number.
Students could calculate change
and write a division number story
to fit the parameters in the number
sentence. Almost all the number
stories were partitive (sharing).
Successful students used correct
monetary notation. Students could
calculate accurately with decimals
to find change. They could write a
number story with division and
include a question about the
remainder.
14% of the students made addition
or subtraction errors in part 1. 15%
made errors in monetary notation in
part 1.
Students wrote multiplication
problems, wrote “87 divided by
five” in words instead of number
story, used situations that did not fit,
or made up problems with numbers
not in the given number sentence.
The most common error was to
forget to address the issue of the
remainder. The next common error
was incorrect monetary notation.
Based on teacher observations, this is what fifth grade students seemed to know and
be able to do:
• Determine a correct process for adding expenditures and finding change using
decimals
• Write a sharing or partitive number story
• Understand the purpose of number operations and choose them appropriately
when creating and solving problems
Areas of difficulty for fifth graders, fifth grade students struggled with:
• Monetary notation
• Writing questions to include dealing with the remainder
• Knowing or being comfortable with a variety of division situations and
contexts
Fifth Grade – 2003
pg.
11
Questions for Reflection on Number Story Time
•
Do your students solve most of their problems on worksheets that have preprinted
the dollar sign for them?
• Do you address the difference between using dollars and decimal point versus
using the cent sign? Do students have opportunities to compare and contrast the
value or relative size of quantities like $.75 and 75⊄?
• How often do you provide opportunities for students to make up their own story
problems? How does that type of experience contribute to their mathematical
understanding? How is it different from working a problem where the numbers
are already set up for the student?
When looking at student number stories, how many stories were:
Division
Subtraction Multi.
Used words
instead of
numbers
Number
sentence
= to 87
Used
other
numbers
Other
When looking at the work that used division problems, did the number story require
them to:
Divide by 5
Divide by 17
Divide by 87
Address the
remainder
Does not
address the
remainder
Were there any examples of problem situations that were not sharing or partitive
situations?
Does your textbook address the different variety of types of division situations? Do
you ask a variety of types of division problems? (A good reference is Children’s
Mathematics, Cognitively Guided Instruction published by Heinemann.
Fifth Grade – 2003
pg.
12
Teacher Notes:
Instructional Implications:
Students need to learn to use proper monetary notation. Students also need to develop
an understanding of different situations that require division of whole numbers with
remainders. Their classroom experiences should include a variety of all the major
types of division problems. Besides sharing or partitive, another important type of
division is the measurement model, “How many groups fit into . . .?” A third type of
division is products and factors (e.g. if a rectangle has an area of 60 sq. m. and one
side is 20 m. How long is the other side?) Students need more experiences with
writing word problems to match number sentences. Problem posing is an important
skill to develop a deeper understanding of operations.
Teacher Notes:
Fifth Grade – 2003
pg.
13
5th grade
Task 2
Raspberry Cake
Student
Task
Core Idea
1
Number
Properties
Halve the amounts of ingredients in a recipe. Solve a simple faction
problem in a practical context.
Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.
• Use models, benchmarks, and equivalent forms to judge the
size of fractions
• Understand the place-value structure of the base-ten number
system including being able to represent and compare rational
numbers
Core Idea
2
Number
Operations
Understand the meanings of operations and how they relate to
each other, make reasonable estimates and compute fluently.
• Develop and use strategies to solve problems involving
number operations with fractions and decimals relevant to
students’ experience
Fifth Grade – 2003
pg.
14
Fifth Grade – 2003
pg.
15
Fifth Grade – 2003
pg.
16
Fifth Grade – 2003
pg.
17
Looking at Student Work – Raspberry Cake
Many students had difficulty finding half of a fraction. Even more had difficulty
taking half of a mixed number. They also had difficulty thinking of strategies to
convert a recipe for 8 people to a recipe for 20 people. Very few students could solve
this task using operations with fractions. Student A is able to use multiplication of
fractions to check his division in part 1. In part 3 Student A also multiplies mixed
numbers to take 2 and 1/2 times the first recipe.
Student A
Fifth Grade – 2003
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18
Fifth Grade – 2003
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19
Student B shows the work of dividing by 2 to find half of the first recipe. To get
enough cake for 20 people, the student uses 2 of the first recipe and one of the second
recipe. Student B makes a clear explanation for this strategy.
Student B
Fifth Grade – 2003
pg.
20
Student B, part 2
Fifth Grade – 2003
pg.
21
Student C uses a similar strategy; she multiplies the first recipe by 2 and adds in the
second recipe to make enough cake for 20 people
Fifth Grade – 2003
pg.
22
Student E used repeated addition as a strategy to find half of the quantity in the first
recipe.
Student E
A few students used drawings to help them make sense of dividing the quantities in
half. See the work of student F.
Student F
Fifth Grade – 2003
pg.
23
Some students interpreted part 2 of the task differently. Student G thinks the three
people are sharing 2 pieces of cake, instead of the more common idea that 3 people
had 2 pieces of cake each. Student G’s diagram is correct for this interpretation. A
common error was to make extra cake and simply multiply the first recipe by 3.
Fifth Grade – 2003
pg.
24
Some students forgot that the first recipe was for 8 people. To find the amount for 20
people they just multiplied the first recipe by 20. See the work of Student H.
Student H
Teacher Notes:
Fifth Grade – 2003
pg.
25
Grade 5 – Raspberry Cake
Raspberry Cake
Mean: 5.22, S.D.: 2.29
4000
3500
Frequency
3000
2500
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
9
460
669
904
1471
1413
1557
3774
2193
730
1230
Score
Score:
%<=
%>=
0
1
2
3
4
5
6
7
8
9
3.2%
100.0%
7.8%
96.8%
14.1%
92.2%
24.3%
85.9%
34.1%
75.7%
45.0%
65.9%
71.2%
55.0%
86.4%
28.8%
91.5%
28.8%
100.0%
8.5%
The maximum score available for this task is 9 points.
The cut score for a level 3 response is 5 points.
Most students (about 86 %) could cut the recipe in half. They may have made an
error in finding the amount of cream or raspberries, which required working with
fractional parts. More than half the students (about 66%) could cut the recipe in half
with error in cream and/or raspberries and draw a model to represent fraction parts of
a cake in a story problem. About 15% of the students could correctly find all the
values for halving the recipe, draw and diagram and find the amount of raspberries
needed to make a cake for 20 people. About 8% of the students met all the demands
of the task. About 3% of the students scored no points on the task.
Fifth Grade – 2003
pg.
26
Raspberry Cake
Points
Understandings
In
the
sample,
no one got a score
0
1
of zero. Everyone made an
attempt at the task.
Students with this score could find
half the recipe for all the
ingredients except raspberries and
cream.
3/4
Students could find half the recipe
for the raspberries, but not the
cream. For 4 points they could
also draw in the diagram and write
that 7/8 of the cake had been
eaten.
6/7
These students still missed the
quantities for raspberries and/or
cream in part one. They could
then use that information to find a
solution to part 3.Successful
students made drawings to help
divide the fractional parts or
converted fractions to decimals.
To find the amount of raspberries
and cream in part 3, the best
strategies were to multiply the
recipe in part 1 by 5 or add 2 of
the given recipe and 1 of the recipe
for 4 people.
Students did not have good strategies
to work with fractions or mixed
numbers. 75% of the students did
not show work in either part 1 or part
3.
Students had more difficulty working
with the mixed number 3 1/2, while
many just knew that 1/2 of 1/2 is 1/4.
Few students could use multiplying
fractions in any part of the task.
About 10% of the students multiplied
by 3 in part 3, making cake for 24
people instead of 20 people. Almost
the same number of students took 20
times the first recipe.
Students, who had a good strategy
for finding the amount of raspberries,
had difficulty working with mixed
numbers to find the amount of cream
in part 3.
8
9
Misunderstandings
Students could work with fractions
and mixed numbers to increase or
decrease a recipe. Many moved
easily between fractions and
decimals. Students could use a
proportional reasoning strategy to
increase the recipe.
Fifth Grade – 2003
pg.
27
Questions for Reflection on Raspberry Cake
Look at the strategies your students used to make sense of taking half a recipe. How
many of them:
Made
pictures
Divided by
2
Added 2
equal
quantities
Used
decimals
Multiplied
by 1/2
Showed no
work
In part 3, what strategy did students use to increase the recipe? Did they:
Add 2 of
the recipe
for 8 and 1
of the recipe
for 4
•
•
•
•
Times the
first recipe
by 2 and
add 1 of the
second
recipe
Multiple the Multiple the Multiply
Other
recipe for 4 recipe for 8 either recipe
by 5
by 3
by 20
How frequently do your students work word problems with fractions?
The numbers in this problem are designed to be friendly enough for many
students to do the operations on fractions as mental math. How often do your
students get the opportunity to make sense of fractions and solve problems
with fractions as a mental math or number talk activity?
Are your students comfortable with proportional reasoning? Could they find a
variety of proportional ways to get from 8 or 4 to 20?
How many of your students showed a facility with decimal equivalents and
could use them to solve this problem?
Teacher Notes:
Instructional Implications:
Students need more practice working with mixed numbers and understanding the
quantity represented by mixed numbers. Students at this grade level need to
understand proportional reasoning and apply this in a practical context (e.g. increasing
or decreasing quantities such as doubling, halving, and two-and-a-half times whole
numbers, fractions, and mixed numbers). Students should have a variety of strategies
to help them make sense of fractions, such as drawing pictures, decomposing them
Fifth Grade – 2003
pg.
28
into simpler parts, and using algorithms. Students at this grade level should develop a
facility with doing operations with common fractions and mixed numbers as mental
math activities.
5th grade
Student
Task
Core Idea
3
Patterns
and
Functions
Task 3
Buttons
Use a button arrangement pattern to describe, extend, and make
generalization about its numeric pattern.
Understand patterns and use mathematical models such as
algebraic symbols and graphs to represent and understand
quantitative relationships.
• Describe and extend numeric patterns (3rd grade)
• Represent and analyze patterns and functions using words (4th
grade)
• Investigate how a change in one variable relates to a change in
a second variable
Fifth Grade – 2003
pg.
29
Fifth Grade – 2003
pg.
30
Fifth Grade – 2003
pg.
31
Fifth Grade – 2003
pg.
32
Looking at Student Work – Buttons
Many students at this grade level are able to see the pattern and form a generalization
in words or in number algorithms. These generalizations could easily be converted to
algebraic symbols at later grade levels. Student A has a nice description of how the
pattern grows and an algorithm for find the total number of buttons in part 4.
Fifth Grade – 2003
pg.
33
Student A, part 2
For further examples of making generalizations and algorithms, look at the work of
Student B and C.
Student B
Fifth Grade – 2003
pg.
34
Student C
While Student D does a thorough explanation of the growing pattern and finding her
answers in part 2 and 3, the explanation about multiplying is unclear in part 4.
However the work at the top of page shows a understanding of the process for finding
the number of buttons in pattern 24.
Student D
Fifth Grade – 2003
pg.
35
Student D, part 2
Student E has a very good generalization of the process for finding the total buttons in
pattern 24. Unfortunately the student does not seem to use it to get the final answer.
Student E
Fifth Grade – 2003
pg.
36
Some students can find the answer or come close to a solution, but use strategies that
are inefficient, tedious, and may lead to errors. Student F uses a drawing and
counting strategy. Student G does not see the complete pattern and solves for the
white buttons every time. Student G uses repeated addition, which is a correct
process, but makes a calculation error. Student H uses graphing, but because of the
inaccuracies of the graph, the student’s answer is incorrect.
Student F
Fifth Grade – 2003
pg.
37
Student G
Fifth Grade – 2003
pg.
38
Student H
Teacher Notes:
Fifth Grade – 2003
pg.
39
Grade 5 – Buttons
Buttons
Mean: 5.40, S.D.: 2.36
4000
Frequency
3000
2000
1000
0
Frequency
0
1
2
3
4
5
6
7
8
651
879
587
828
1313
1769
3020
1808
3546
Score
Score:
%<=
%>=
0
1
2
3
4
5
6
7
8
4.5%
100.0%
10.6%
95.5%
14.7%
89.4%
20.4%
85.3%
29.6%
79.6%
41.9%
70.4%
62.8%
58.1%
75.4%
37.2%
100.0%
37.2%
The maximum score available for this task is 8 points.
The cut score for a level 3 response is 4 points.
Most students (about 89%) could draw pattern 4 and give the correct number of
buttons for pattern 5 and pattern 6. Many students (about 80%) could draw and
extend the pattern and explain in words how the patterned worked. More than half the
students (about 60%) could draw the pattern, extend the pattern for 5 and 6 and
explain how it grew, find and explain the number of white buttons for pattern 11, and
could find the total buttons for pattern number 24. Almost 26% of the students met
all the demands of the task. About 5% of the students scored no points on this task.
Fifth Grade – 2003
pg.
40
Buttons
Points
0
3
6
Understandings
Most students attempted this task.
Students could make a drawing of
the next pattern in a sequence.
They could also find the number
of white buttons for pattern 5 and
6.
Students could find and extend a
pattern. To find a solution in part
4,
7
31% of all students used the
expression of 24 times 3 plus 1 to
solve for part 4. Another 7% used
24+24+24+1.
8
Students could continue a pattern
using a drawing. They could also
find rules for extending it without
drawing.
Misunderstandings
Students had difficulty explaining
how they figured out the white
buttons in the pattern. They said
things like, “I added”, without
specifying what or why.
They did not convert from white
buttons to finding the pattern for all
buttons in part 3 and 4. 37% of all
the students multiplied 24 by 3 in
part 4, but did not add on the black
button.
Some students used inefficient
strategies, which may have led to
errors. 13% drew a picture. 8%
made a table and extended it, 3% just
kept adding 3, and some students
drew a graph.
Based on teacher observations, this is what fifth grade students know and are able to
do:
• Continue a pattern using pictures and numbers
• Explain how a pattern grows and use that algorithm to solve for larger
numbers in the pattern
Areas of difficulty for fifth graders, fifth grade students struggled with:
• Distinguishing between part of a pattern and the whole pattern
• Explaining a pattern in words
Teacher notes:
Fifth Grade – 2003
pg.
41
Questions for Reflection on Buttons:
•
•
•
What are some rich problems that your students have done this year? What
are some good resources for pattern problems?
Do you ask questions like: “What stays the same?” and “What changes?” to
help students develop the ability to form generalizations?
Do students have opportunities to connect their number sentences to geometric
patterns and share how they visualize the growth pattern?
Look carefully at your student work. What strategies did they use in Part 4?
Draw a
24 x 3 or
24 x 3 + 1 or Extend a
Repeated
Used a
picture
24+24+24
24+24+24+1 table
addition of doubling
3
strategy
from a
previous
part of the
problem
Teacher Notes:
Instructional Implications:
Fifth grade students need more experiences that require them to move beyond
drawing the next figure in the pattern so that they analyze the pattern and represent
the growth numerically. Fifth graders need to move beyond thinking about “what
comes next?” to thinking about the problem as a whole: this involves generalizing
what is happening with the growth, but need not necessarily involve variables or
algebraic equations. Being able to see what remains the same and what changes in a
pattern helps students develop algebraic thinking and the ability to make a
generalization. Asking questions about how the pattern changes helps students to
move beyond counting and drawing strategies to rules that will solve for any number
in the pattern.
Fifth Grade – 2003
pg.
42
5th grade
Student
Task
Core Idea
5
Data
Analysis
Core Idea
3
Patterns,
Functions,
And
Algebra
Core Idea 2
Number
Operations
Task 4
Winter Sports
Use two tables of winter sports information to interpret data and to
make calculations.
Collect, organize, represent and interpret numerical and categorical
data, and clearly communicate their findings. Display, analyze,
compare, and interpret different data sets.
• Interpret data to answer questions about a situation (4th grade)
• Compare different sets of data.
Understand quantitative relationships.
• Identify and describe situations with rates and compare them.
Understand the meanings of operations and how they relate to
each other, make reasonable estimates and compute fluently.
• Reason about and solve problem situations that involve more
than one operation in multi-step problems
Fifth Grade – 2003
pg.
43
Fifth Grade – 2003
pg.
44
Fifth Grade – 2003
pg.
45
Fifth Grade – 2003
pg.
46
Looking at Student Work – Winter Sports
Making a comparison is a very difficult skill for students. Most students want to
name the best choice or in this case choice with highest score, but they fail to point
out how they know it’s the best or highest. Student A does a good job of calculating
the scores for the top three countries and additionally comparing them to the
remaining countries in the table.
Student A
Fifth Grade – 2003
pg.
47
Student B finds the scores for two countries to show why Norway is the highest, but
does not make a reference to remaining countries or why only these two were chosen.
Student C makes a fairly good case for why Norway is the highest without actually
calculating the totals. A small reference to relative point values for the various
medals would have completed this argument.
Student B
Student C
Fifth Grade – 2003
pg.
48
Student D does a nice job of showing how the missing values in the table were found.
The circle under total medals and round 7 bronze medals is probably a mental effort
to compare scores for different countries, but that information is not mentioned in part
5 for explaining why Norway has the highest point value.
Student D
Fifth Grade – 2003
pg.
49
Many students did not understand how to use a weighted total in part 4. The most
common error, like the work of Student E, was to just add the points in medal chart
together. Some students, like Student F, are still looking for key words. When they
see the word total in the question, the students add all the totals from the table on page
6 completely ignoring the relationship between medals and points or even that the
question is about the country Finland.
Student E
Student F
Teacher Notes:
Fifth Grade – 2003
pg.
50
Grade 5 – Winter Sports
Winter Sports
Mean: 5.32, S.D.: 1.99
5500
5000
4500
Frequency
4000
3500
3000
2500
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
9
391
265
739
955
2921
1359
2005
4917
558
291
Score
Score:
%<=
%>=
0
1
2.7%
4.6%
100.0% 97.3%
2
3
4
5
6
7
8
9
9.7%
95.4%
16.3%
90.3%
36.6%
83.7%
46.0%
63.4%
60.0%
54.0%
94.1%
40.0%
98.0%
5.9%
100.0%
2.0%
The maximum score available for this task is 9 points.
The cut score for a level 3 response is 5 points.
Most students (about 81%) were able to use information on the table to find missing
values in the table and answer simple questions about the table. A little more than
half the students (about 54%) could fill in the table, answer questions about the table,
use a weighted point system to find the points for Finland and explain how they
solved for it. A little less than half of the students could also find the point total for
Norway. Few students (less than 3%) could make a mathematical comparison to
show how they knew Norway won. Less than 3% of the students scored no points on
this task.
Fifth Grade – 2003
pg.
51
Winter Sports
Points
O
2/3
Understandings
Most students tried the task.
Students could answer simple
questions about the table in part 2
and 3. Students could fill in
missing values in the table.
4
Students could fill in the table and
answer simple questions about the
table.
6
Students could fill in the table,
answer questions relating to the
table, and find the weighted point
value for Finland.
7/8
Students could find the point value
for Norway. They may also have
found the value for Germany of
the US, but not both.
9
Misunderstandings
Some students did not fill in the
missing values in the table or they
could not find the second number in
the table. Students were more likely
to find the country with the most
bronze medals than the country with
the highest total medals.
Many students just added the medal
points together to get 10. They were
not familiar with a weighted point
value. This error prevented them
from comparing point values in part
5.
These students made calculation
errors in finding the total points for
Norway or made an argument about
total medals. They generally did
not mention any of the other
countries.
They failed to compare how
Norway compared to other
countries or how the top two
countries compared to the rest.
They usually only did calculations
and did not attempt to explain the
comparison.
Students could use a table, a
weighted point total, and make a
mathematical comparison
referencing not only the lead
scorer but explaining how they
knew it was higher than the other
possibilities.
Fifth Grade – 2003
pg.
52
Based on teacher observations, this is what fifth grade students seemed to know and
be able to do:
• Answer simple questions about a table.
• Fill in missing values in a table.
• Use a weighted point value to find a score.
Areas of difficulty for fifth graders, fifth grade students struggled with:
• Using a weighted point value to find the points for Norway.
• Making a comparison by showing values for countries other than the leader,
not referencing other countries in the comparison set
Questions for Reflection for Winter Sports
•
How many of your students could find the point value for Finland in part 4?
Could they apply this information to find the point value for Norway?
• What experiences have your students had in using a weighted scale?
• Can you think of some interesting problems that might involve this kind of
reasoning? What is the purpose for weighting the points?
• What is a situation with which students might be familiar that uses a weighted
system? I think this is used in some of the reality talent shows, where call in
votes having a higher weight than the judging panel. Can you think of others?
Look at student work in part 5, how many of your students:
Found
Found
points for points for
Norway
Germany
Found
points
for USA
Mentioned
the
remaining
countries
Discussed Other
medals
instead of
points
No
response
What opportunities have your students had to make comparisons? What are some of
your favorite problems for working on this idea?
How can you use student work to make more explicit what is valued in developing a
comparison? What kind of feedback can you give to students to help deepen their
thinking on this topic?
Teacher Notes:
Implications for Instruction:
Students need to be able to interpret and use data from different tables to make
calculations. Students should have experiences working with weighted values and be
exposed to problem-solving situations that highlight the need for them. At this grade
level students need to become familiar with what makes a good comparison. To
convince someone, you need to show more than just the best solution, but also why
other choices are not as favorable.
Fifth Grade – 2003
pg.
53
5th grade
Student
Task
Core Idea
4
Geometry and
Measurement
Fifth Grade – 2003
Task 5
Juan’s Shapes
Use a grid to find the perimeter and area of shapes. Draw shapes
of same area/perimeter on a grid.
Students will analyze characteristics and properties of twoand three-dimensional geometric shapes, understand
attributes, and apply appropriate techniques, tools, and
formulas to determine measurements.
• Understand such attributes as length, area, weight, volume,
and angle size and select the appropriate type of unit for
measuring each attribute
• Develop strategies for estimating or calculating the
perimeters and areas of irregular shapes
• Explore and determine what happens to perimeter and area
of a two-dimensional figure when its shapes changed in
some way.
pg.
54
Fifth Grade – 2003
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55
Fifth Grade – 2003
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56
Fifth Grade – 2003
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57
Looking at Student Work – Juan’s Shapes
Student A uses a counting strategy to show the thinking for finding area and perimeter
of the designed shapes. Student A labels both shapes to clearly mark which one meets
the constraints of part 2 and which one satisfies the conditions in part 3. Student B is
probably using an addition strategy to find the perimeter. The shapes are marked
showing the measurements for area and perimeter. Both shapes meet the demands of
the task, but the Student makes the error of not correctly finding the perimeter for part
2. The student only counts the bottom left square, instead of the two sides of the
shape. Many students have difficulty finding areas for concave angles of a shape.
Student A
Student B
Fifth Grade – 2003
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58
Student C appears to know something about the formula for finding area. However
the student does not realize that it can’t be used for nonrectangular shapes. In
attempting to find something with the same area on the next page, the student uses the
area formula to find a shape that is 2x6. However, when the student realizes the shape
has the same perimeter and erases to add on the bottom square, Student C does not
notice that the area has now changed. An understanding about area as number of
squares, instead of just a formula would have made this a more noticeable error.
Student C
Fifth Grade – 2003
pg.
59
Student C, part 2
Fifth Grade – 2003
pg.
60
Student D may have used a similar misinterpretation of the area formula. The area for
the top figure may be the top times the bottom plus the side. The bottom area may be
3x2x3. Clearly the student understands perimeter, but is confused about area.
Typical of many students, Student D only draws one of the two figures on the second
page. The student knows perimeter and deals with part three which addresses
perimeter.
Student D
Fifth Grade – 2003
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61
Another common error is to confuse the conditions of the each part of the task.
Student E keys in on area and draws a shape with the same area of 12 then a shape
with an area of 16 instead of a perimeter of 16.
Student E
Teacher Notes:
Fifth Grade – 2003
pg.
62
Grade 5 – Juan’s Shapes
Juan's Shapes
Mean: 4.58, S.D.: 2.78
4000
3500
Frequency
3000
2500
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
1561
1450
860
775
2295
2168
295
1294
3703
Score
Score:
%<=
%>=
0
1
2
3
4
5
6
7
8
10.8%
100.0%
20.9%
89.2%
26.9%
79.1%
32.3%
73.1%
48.2%
67.7%
63.3%
51.8%
65.3%
36.7%
74.3%
34.7%
100.0%
25.7%
The maximum score available for this task is 8 points.
The cut score for a level 3 response is 4 points.
Most students (about 90%) could find the area of shapes on a grid. Many students
(about 70%) could find area of shapes on a grid and draw a different shape with the
same area. A little less than half of the students (about 40%) could find area and
perimeter for shapes on a grid and draw a shape either maintaining the same area or
maintaining the same perimeter. About 26% of the students met all the demands of
the task. About 11% of the students scored no points on this task.
Fifth Grade – 2003
pg.
63
Juan’s Shapes
Points
0
1
4
Understandings
Most students with this score
attempted the problem.
Students could get one of the
areas or perimeters correct in part
one.
Students could find the correct
areas in part one and draw a shape
with the same area but a different
perimeter in part 2.
5
About 10% of all the students
could correctly find all the areas
and perimeters in part one and
draw a different shape with the
same area in part 2. 6% of all the
students could find areas in
perimeters in part one and draw a
shape with the same perimeter
and a different area in part 3.
7
Students with this score usually
missed one of the perimeters in
part one of the task.
Students could find the area and
perimeter of simple shapes.
Students could also design shapes
to meet multiple constraints.
They could follow a logic of
keeping one thing the same and
changing something else.
8
Fifth Grade – 2003
Misunderstandings
A little less than 10% of all students
confused area and perimeter.
Students had trouble counting
perimeters of concave angles.(?)
Some students did not attempt to draw
a shape with the same perimeter and a
different area.
In general, students do not have the
expectation that geometric shapes
should be labeled. So while there
might be two drawings on the grid,
the scorer needed to make a
determination about which shape was
which. In general students had more
difficulty designing a shape with a
given perimeter than with a given
area.
pg.
64
Based on teacher observations, this is what fifth grade students seemed to know and
be able to do:
• Count to find the area and perimeter of simple shapes
• Design a shape to satisfy a constraint about area
Areas of difficulty for fifth graders, fifth grade students struggled with:
• Finding perimeters around concave angles
• Designing a shape with a constraint about perimeter
•
•
•
•
•
•
Questions for Reflection on Juan’s Shapes
What experiences have your students had this year with perimeter?
Do they get opportunities to work with nonstandard shapes to find area?
Why do you think perimeter may have been more difficult for them than area?
What kinds of concrete activities might help them with this concept?
Do students get opportunities in your class to design their own shapes with
given constraints? Do they get opportunities to investigate how shapes with
the same area can have different perimeters or shapes with the same perimeter
can have different areas? What other investigations have they had with area
and perimeter?
What experiences have students had with area this year?
Have they had the opportunity to work with area of nonrectangular shapes,
like triangles, trapezoids and parallelograms?
Teacher Notes:
Instructional Implications:
Students need to understand the difference between area and perimeter. They need to
be able to calculate each for the shapes drawn on the grid. Students also need to be
able to construct simple shapes with certain area and perimeter restrictions. Most
students with this score had difficulty with finding the perimeter of shapes and
designing their own shapes. Practice measuring real objects in different units would
be beneficial.
Fifth Grade – 2003
pg.
65