Download Grade 6 - Noyce Foundation

Overall Frequency Distribution by Total MARS Raw Scores, Grade 6
Mean: 20.92
StdDev: 8.91
Total MARS Raw Scores
Grade 6 – 2005
pg.
1
MARS Test Performance Level Frequency Distribution Chart and Bar Graph
2005 – Numbers of Students tested in 6th grade: 7992
Frequency Distribution of MARS Test Performance Levels, Grade 6
2000
Perf.
Level
1
2
3
4
% at
26%
47%
17%
10%
Year of Testing
2001
% at least
100%
74%
27%
10%
% at
16%
43%
28%
13%
2003
Perf.
Level
1
2
3
4
% at
21%
28%
33%
18%
% at least
100%
79%
50%
18%
% at least
100%
84%
41%
13%
2002
% at
21%
25%
36%
19%
2004
% at
17%
27%
31%
25%
% at least
100%
83%
56%
25%
% at least
100%
79%
55%
19%
2005
% at
20%
22%
31%
26%
% at least
100%
80%
58%
26%
Bar Graph of 2005 MARS Test Performance Levels, Grade 6
Total Student Count: 7,992
Grade Six – 2005
pg.
2
6th grade
Task 1
Crystal Earrings
Student
Task
Describe and extend a pattern of crystal earrings that is increasing in size.
Make generalizations around this pattern.
Core Idea
3
Algebra and
Functions
Understand relations and functions, analyze mathematical situations,
and use models to solve problems involving quantity and change.
• Represent, analyze, and generalize a variety of relations and
functions with tables, graphs, and words
• Model and solve contextualized problems using various
representations such as graphs, tables, and equations
Grade 6 – 2005
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3
Grade Six – 2005
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4
Crystal Earrings
Grade 6
Rubric
The core elements of performance required by this task are:
• to describe, extend, and make generalizations about a number pattern
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Draws Pattern #4 correctly:
points
section
points
1
1
2.
Gives correct answer: 15
1
Shows work such as:
1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15
3.
4.
1
2
Completes table to show:
Pattern #4 has 20 crystals.
Pattern #5 has 30 crystals.
1
1
Shows work such as:
Pattern #7 will be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
and 28 x 2 = 56, not 50
or
(accept 7 x 8)
1
1
or
2
Gives correct answer: $56
1
Total Points
Grade 6 – 2005
2
3
8
pg.
5
Looking at Student Work on Crystal Earrings
The purpose of this task is to give students an opportunity to show their algebraic thinking.
Student A is able to meet all the demand of the task, but relies heavily on drawing and
counting. For doubling the student uses repeated addition instead of multiplication. While
these strategies work, the will not help the student develop generalizations and these
strategies can be cumbersome and lead to errors. Student A is making use of labels to
clarify the thinking and the equal size groups.
Student A
Grade Six – 2005
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6
A different solution path is for students to notice that each row increases in size by one.
This again is a pattern that will not lead to a formula and allows for more possibility of
computation error. Student B has noticed the numerical structure of the pattern and uses
drawing to verify his solution.
Student B
Grade 6 – 2005
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7
Student C notices a different numerical pattern. In the table the numbers go up by an
increasing even number each time. The student is able to make the argument for why fifty
dollars is incorrect, by showing the number of crystals for 1 earring if $50 were correct and
then showing the correct solution by continuing the table. In this solution path, the
doubling has been included in the design of the table.
Student C
Grade Six – 2005
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Students at this grade level should start to look for relationships between the independent
variable (pattern number) and the dependent variable (number of crystals). The earring
design is made of triangular numbers whose algebraic expression would be [n(n+1)]/2.
Notice that Student D has found that relationship, although it is not expresses in symbolic
notation to answer for part 2. Because the earrings are doubled for part 3 and 4 of the task,
the expression could be simplified to n(n+1). Student D sees this relationship and uses it to
find the value in part 4.
Student D
Grade 6 – 2005
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Student E makes the common mistake of justifying the answer for Pattern 4 in part 2
instead of showing the way to find the crystals for Pattern 5. Student E sees that the pattern
adds the pattern number for each new row to the previous total. However, the student still
seems to be more comfortable with drawing and counting than using the pattern.
Student E
Grade Six – 2005
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Student F is starting to look for generalizable patterns. The wording in part 2 gives a
recursive pattern for finding the number of crystals. The wording in part 4 gives a
generalizable pattern, which could be written in the form n(n+1).
Student F
Grade 6 – 2005
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Student G sees the numerical pattern in the table [ n (n+1) ], but fails to realize that the
table has already doubled the crystals to make the pair of earrings. Student G then
incorrectly doubles the values a second time.
Student G
Grade Six – 2005
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Student H shows the problem of relying on drawing strategies. The student omits one of
the rows when attempting to draw pattern #5. For many students it is difficult to
understand how tables made by other people relate to the meaning of the problem.
Students often look for any common numerical pattern to fill in the table. Student H
notices that 3 x 4 =12, then assumes all values will be 4 times the pattern number. The
student could have found the mistake by looking at the total for 2 or by checking other
values in the table, e.g. 2 x 4 ≠ 6.
Student H
Grade 6 – 2005
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Student I makes the assumption that all earrings will have 3 rows. The student’s answer for
part 2 would be correct if that were the pattern. However, like Student H, this student
looks for a numerical pattern in the table rather than continuing her found pattern. The
student sees the number of crystals going from 6 to 12, or increasing by 6. The student
then continues the table by going up in groups of 6. The student did not check either the
number that was found in part 2 or look at the another part of the table to test the
conjecture. 2+6 ≠ 6. The student does not understand the table is for the pair of earrings,
rather than the individual earring. So the student doubles the number in the table.
Student I
Teacher Notes:
Grade Six – 2005
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Frequency Distribution for Task 1 – Grade 6 – Crystal Earrings
Crystal Earrings
Mean: 4.93
StdDev: 2.77
MARS Task 1 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
845
10.6%
100.0%
499
16.8%
89.4%
441
22.3%
83.2%
815
32.5%
77.7%
658
40.8%
67.5%
701
49.5%
59.2%
1016
62.2%
50.5%
774
71.9%
37.8%
2243
100.0%
28.1%
The maximum score available for this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, 91%, could extend the visual pattern to #4. Many students, 76%, could
extend the pattern in pictures, find the correct number of crystals for pattern #5, and show
how they figured it out. More than half the students, 59%, could also extend the pattern in
the table for a pair of earrings, for pattern # 4 and #5. 30% of the students could meet all
the demands of the task including proving why $50 would not be the cost of a pair of
earrings if the pattern were extended to 7. 9% of the students scored no points on this task.
All the students in the sample attempted the problem.
Grade 6 – 2005
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15
Crystal Earrings
Points
Understandings
All
students
in the sample
0
attempted the problem.
1
Students could extend the pattern
by drawing.
3
Students could extend the pattern
in a drawing and could find the
number of crystals for #5 and
explain how they figured it out in
words, pictures, or calculations.
Students could extend the pattern
of crystals to pattern #4 and #5
and they could extend the pattern
in the table for a pair of earrings
for pattern # 4 and #5.
5
6
8
Most students with this score
could extend the geometric
pattern with a drawing and
numbers, continue the table for
pairs of earrings, and find that
pattern #7 would have 28
crystals.
Students could extend a pattern,
use a table to think about a pair
of earrings, and find the cost for
a pair of earrings for pattern #7.
26% of all students relied on a
drawing and counting strategy.
15% of the students could
continue the table to solve the
problem in part 4. 9% of the
students added all the rows
(1+2+3+4+5+6+7=28) and
doubled that answer. 6% saw a
pattern of n(n+1). 3% noticed
that the table was going up by the
next larger even number each
time.
Grade Six – 2005
Misunderstandings
Students did not understand the visual
pattern. Half the students, who missed the
drawing for pattern #4, put only 3 rows of
beads for each earring.
Many students thought pattern #5 should
have 12 crystals (13% of the total sample).
8% of all students counted the number of
crystals for pattern #4 instead of for pattern
#5, putting an answer of 10 crystals.
Some students did not relate the table to the
context of the pattern. They made
conjectures about the number patterns in the
table and so filled out patterns like 12, 18,
and 24 (8%); 12,24,48 (5.5%); and 12,16,20
(5%).
Students had difficulty with doubling to find
for a pair of earrings. 7% of the students left
the answer as $28. 4% of the students
doubled twice, $112, not realizing the table
was for a pair of earrings. Answers varied
from $1 to $300.
pg.
16
Based on teacher observation, this is what sixth graders knew and were able to do:
• Describing and extending a geometric pattern
• Doubling
• Using drawing and counting strategies to extend pattern
Areas of difficulty for sixth graders:
• Relating table to problem context
• Writing a justification for why something is not true
Strategies used by successful students:
• Drawing and counting
• Continuing a table
• Finding a pattern for part 4: goes up by larger even numbers each time or n(n+1)
• Finding a pattern for part 2: 1 + 2+3+4…n, increases by the pattern # each time, or
[n(n++1)]/2
Questions for Reflection on Crystal Earrings:
What kind of experiences do your students have with different types of patterns?
Did students in your class have good visualization skills? Were they able to draw
the pattern for pattern #4? What types of questions might help them to focus on the
important attributes of the pattern? Do students get opportunities to discuss patterns
and make convincing arguments for different interpretations of a pattern? What
evidence might they give for why the pattern would have more than 3 rows?
• What kinds of questions do you ask which might push students to look for
numerical patterns instead of relying solely on drawing and counting? Why is this
important for students?
• When working on problem-solving activities do students in your class have the
opportunity to make their own tables to record information? What evidence do you
see that students did not relate the table to the context of the problem?
Look at student work on the table. How many of your students put:
•
•
20,30
18,24
24,48
16,20
9,12
Other
What patterns were students thinking about to make these errors? What
evidence were they ignoring? What does this show you that they don’t
understand about either tables or testing conjectures?
• What are classroom norms for testing conjectures? Do students know that it
takes more than one case or example to make a pattern?
Look at the solutions paths that students used to solve for part 4. Did they:
•
Draw and count
Continue the table
Add (1+2+…+7) and double
Add a larger even number each time (+2,+4,+6,+8…)
Use a numerical pattern: n(n+1)
Forget to double (answer of 28)
Double twice (answer of 112)
Use strategy not appropriate to mathematics of problem
Grade 6 – 2005
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•
Do you see evidence of students looking for numerical patterns? Are some students
ready to start thinking with algebraic symbols?
Teacher Notes:
Implications for Instruction:
Students at younger grades can solve problems using drawing and counting strategies.
Students in middle school should start to be looking for numerical patterns. This task gives
them a chance to work with triangular numbers. Some students can look at the earring
pattern to notice that the pattern is increasing by the row number each time, which gives
them a recursive rule. Some students may notice that the table grows by an increasing even
number each time, which is another recursive rule. Other students may notice that the
crystals in each earring is the sum of the counting numbers from 1 to the pattern number,
which is a more generalizable rule, which will work for any earring in the sequence. A few
students may notice that the pattern for earrings is equal to [n(n+1)]/2. This simplifies the
process for finding the number of crystals and reduces the tendency for an arithmetic slip.
Some students notice that to find any value in the table, just multiply the pattern number by
the pattern number plus one. This again is a generalizable rule.
Students at this grade need many opportunities to work with patterns and should be striving
for generalizable rules, rather than drawing and counting or using recursive rules. When
thinking about generalizable rules, students need to develop the habit of mind of testing
their rule against all the evidence. So just because 6 x 2= 12 could explain the jump in the
table from pattern 3 to pattern 4, other evidence proves that this is an incorrect conjecture.
Students should be presented with a variety of conjectures and asked which one best fits the
evidence and why. Discussions will engage students in finding the most convincing
argument and developing their logic skills. This will also help them start to look at the
tables in relationship to the context of the problem, rather than a mindless exercise in
continuing an arithmetic pattern. Students need to experience that cognitive dissonance,
that there are a variety of possibilities and therefore they need to check which possibility
fits the context and evidence of the problem.
Teacher Notes:
Grade Six – 2005
pg.
18
6th grade
Student
Task
Core Idea
5
Statistics
Grade 6 – 2005
Task 2
Money
Analyze and interpret bar chart information to determine how much
money was spent by four children. Write a description to fit a fifth child’s
bar chart.
Select and use appropriate statistical methods to display, analyze,
compare and interpret different data sets.
• Interpret data to answer questions about a situation
• Compare data sets using measures of center and spread to
understand what each indicates about the data sets
• Communicate mathematical thinking clearly and coherently
• Use representations to interpret physical, social, and mathematical
phenomena
pg.
19
Grade Six – 2005
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20
Grade 6 – 2005
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21
Money
Grade 6
Rubric
The core elements of performance required by this task are:
• interpret bar charts
points
Based on these, credit for specific aspects of performance should be assigned as follows
section
points
1. Gives correct answers:
Danny
Chris
Ben
Ali
All four answers correct
Partial credit
Three correct answers
Two correct answers
3
(2)
3
(1)
2. a. Danny
1
b. Ali
1
Compares values on at least two graphs to show that the sum of Ali’s bars is
larger than the other bars
2
Partial credit
Makes a sensible verbal comment about the total amounts spent
(1)
4
3. Gives a reasonable description such as:
“I spent more and more each week” or similar
1
1
8
Total Points
Grade Six – 2005
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22
Looking at Student on Money:
Student A assigns a scale to the graphs, finds the total money spent for each person, and
then concludes that Ali spent the most. Student A also writes a good description for Ernest,
showing the trend of the graph in the context of the problem.
Student A
Grade 6 – 2005
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Student A, continued
Student B realizes is able to think about the bars on the graph as a measurement. The scale
is not important as long as it’s the same for all the graphs. The student just needs to
compare the lengths of the bars for the four graphs. Student B notices the shape of the
graph for Ernest, but is unable to relate it to a mathematical trend, increasing, or to the
context spending more money.
Grade Six – 2005
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Student B
Grade 6 – 2005
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25
Student B, continued
Student C makes an observation about Ernest’s graph similar to that of Student B.
However, Student C is able to tie the observation back to the mathematics and context of
the task.
Student C
Grade Six – 2005
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Student D chooses a different scale from most students. Because the relative values of the
bars remains unchanged, this gives the student a successful way of making the comparison.
Notice that the student starts the scale at 10 instead of 0. Again, because this is done for all
the graphs, the relative values are not effected. In part 3, Student D confuses description
for total. The task was asking for the mathematical trend of the graph.
Student D
Grade 6 – 2005
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Student E gives factual information about the graph for Ernest rather than summarizing the
information or giving a trend of the data.
Student E
Student F writes a good description for the shape of Ali’s graph, but in order to justify who
spent the most, the student needs to make a comparison to the values of the other graphs.
In part 3, Student F describes the shape of the graph, without reference to mathematical
trend or context.
Student F
Grade Six – 2005
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28
Student G redraws the graph for Ali in an attempt to show who spent the most. The student
does not give a reason why this particular shape is significant. In part 3, the student seems
to confuse decreasing with small instead of the trend “getting smaller”.
Student G
Teacher Notes:
Grade 6 – 2005
pg.
29
Frequency Distribution for Task 2 – Grade 6 - Money
Money
Mean: 6.41
StdDev: 1.62
MARS Task 2 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
60
0.8%
100.0%
67
1.6%
99.2%
103
2.9%
98.4%
155
4.8%
97.1%
375
9.5%
95.2%
1257
25.2%
90.5%
2097
51.5%
74.8%
915
62.9%
48.5%
2963
100.0%
37.1%
The maximum score available for this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, about 94%, could match all the descriptions to the appropriate graphs and
read and compare information from the graph to identify the student who spent the most
money in the first week. Many students, 89%, could also identify who spent the most
money overall. A majority of the students, 73%, could match the descriptions, decide who
spent the most the first week and who spent the most overall, and write a description of the
mathematical trend for Ernest’s graph. 37% of the students could meet all the demands of
the task including justifying why Ali spent the most by comparing significant graphical
features for all students. Less than 1% of the students scored 0 points on this task.
Grade Six – 2005
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30
Money
Points
Understandings
All
students
in the sample
0
attempted the task.
4
Students could match the verbal
descriptions to the appropriate
graphs. Students could compare
the graphs to identify who spent
the most the first week.
5
Students could match the
descriptions to the graphs,
compare who spent the most the
first week, identify who spent
the most overall, and write a
description for Ernest’s graph.
Students could match
descriptions to graphs, write
description of the mathematical
trend of a graph, and compare
graphs by identifying who spent
the most the first week and who
spent the most overall.
6
8
Misunderstandings
Students often confused the
descriptions for Danny and Ali. Some
students chose Ernest to fit one of the
descriptions.
Some students thought that Chris spent
the most, because of the 3 high bars.
Others though Ben spent the most,
because he bought an expensive
present. Less than 5% made addition
errors.
Students, who had difficulty with the
description for Ernest, tended to give
information about the shape of the data,
it looks like a staircase, or give factual
information like the total or the amount
spent each week.
Students had difficulty writing a
justification for who spent the most.
12% of the students did not write a
reason at all. 16% gave reasons for
students other than Ali. Students who
chose Ali, often did not compare her
scores to the other students. (e.g. She
spent a lot, she spent the same every
week, or she has the most.
Students could match
descriptions to graphs, write
description of the mathematical
trend of a graph, and compare
graphs by identifying who spent
the most the first week and who
spent the most overall. Students
could also justify their
comparisons by assigning a scale
to the graph and totaling the
values for each student.
Grade 6 – 2005
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31
Based on teacher observations, this is what sixth graders know and are able to do:
• Read and compare graphs to match verbal descriptions of the trends to the
appropriate graph
• Read and compare graphs to identify who spend the most the first week
• Write a verbal description to of the trend of a graph and put that trend in context
with the problem
• Identify who spent the most overall
Areas of difficulty for sixth graders:
• Assigning a scale to the graphs
• Making a mathematical comparison, knowing that to justify why one graph
represents the most there must be reference to why the other graphs are less
Strategies of successful students:
• Labeling the graph with appropriate, consistent scales
• Showing the addition of the bars
• Looking for trends in the graph, trying to summarize the data
Questions for Reflection on Money:
What opportunities have students in your class had to make comparisons of data
from different graphs? What types of comparisons have they been asked to make?
Why might this be important?
• Were students in your class able to assign a scale to the graphs? Why is this
important?
Look at student justifications for part 2, who spent the most overall. How many of your
students:
•
Assigned a
scale and
totaled all
the graphs
•
•
•
Ali has
the most
or spent a
lot every
week
Ali
spent
about
the same
every
week
Chris has
the highest
bars or
Chris has
the 3
highest
Ben
bought an
expensive
present
Too general, Other
e.g. I looked
at graph or
added but
no values
When working with data and graphs, are your students regularly asked to give a
trend of the data? Are students regularly asked to give a summary of the
information shown on the graph?
Were your students able to write a mathematical description of the trend for the data
for Ernest or did they give factual information, like the total or the size of individual
bars?
What other questions could you ask using these graphs to probe the understanding
of your students?
Grade Six – 2005
pg.
32
Implications for Instruction:
Students need more experience identifying the trends in data. Is the data increasing,
decreasing, or staying the same? Is there a relationship or correlation between the two
variables being graphed? What is the purpose of the graph? What is the message the
maker of the graph trying to convey? Students should also be in the habit of trying to
quantify information from the graphs, by making statements like how much high or
how much lower when comparing features of the graph or making comparisons
between graphs. Students need experiences working with graphs with and without
scale, so that they start to be able to assign relative values to compare relationships.
This also helps them to identify ways that graphs are misleading, e.g. graphs don’t start
at zero or the scale distorts minor differences in size.
Comparison is a big mathematical idea for middle school. Students should think of a
variety of ways to make comparisons. The first big idea is that to make a comparison,
there needs to be a reference to all the items being compared – how are they different,
how are they alike. At this grade level, students should be starting to make a variety of
comparisons: subtraction comparison, multiplicative comparisons ( A is 5 times larger
than B), and using percents to compare ratios of different size quantities ( 80% score on
a test of 20 questions, 80% on a test with 5 questions instead of missing 4 questions
versus 1 question).
Teacher Notes:
Grade 6 – 2005
pg.
33
6th grade
Task 3
Winning Spinners
Student
Task
Work with two spinners to find the probability of winning a prize.
Design two new spinners that will increase the likelihood of winning.
Core Idea
2
Probability
Demonstrate understanding and the use of probability in problem
situations.
• Determine theoretical probabilities and use these to make
predictions about events
• Understand that the measure of the likelihood of an event can be
represented by a number from 0 to 1
• Represent probabilities as ratios, proportions, decimals or
percents
• Represent the sample space for a given event in an organized way
(e.g. table, diagram, organized list)
• Use representation to model and interpret mathematical
phenomena
Grade Six – 2005
pg.
34
Grade 6 – 2005
pg.
35
Winning Spinners
Grade 6
Rubric
The core elements of performance required by this task are:
• work with probability
Based on these, credit for specific aspects of performance should be assigned as follows
1. Fills in the table correctly:
+
8
4
2
1
7
15
11
9
8
9
17
5
3
section
points
2
Spinner A
Spinner B
points
11 10
13
9 7 6
11 7 5 4
13
Partial credit
No more than 2 mistakes
or no values, only “odd” or “even” given correctly.
(1)
2
2. Gives a correct answer: 4/16 = 1/4 or equivalent
Accept correct decimals, percents and ’out of’.
2ft
Partial credit
denominator 16
(1)
2
3. Fills in spinners correctly:
The even numbers split so there is one even number on one spinner
and two on the other (position of the numbers does not matter)
2
Gives a correct answer: 1/2 or equivalent
2
Gives an explanation or table to show a correct method.
2
Special case
6
Correctly uses their own numbers: maximum 2 points
Total Points
Grade Six – 2005
10
pg.
36
Looking at Student Work on Spinners:
Student A shows a strong understanding of probability. The student defines the number of
possible outcomes and the number of favorable outcomes. In part 3, the student finds two
ways to show the sample space.
Student A
Grade 6 – 2005
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37
Student B is able to figure out the design for the new spinner by reasoning about number
properties (an odd plus an odd makes an even).
Student B
Student C is able to show the sample space for part 3 by making an organized list, which
includes all the possible outcomes.
Student C
Grade Six – 2005
pg.
38
Student D attempts to show the sample space for part 3, but her list is incomplete. Notice
that the student still thinks about probability as whole numbers, how many favorable
outcomes. Students at this grade should know that probability is expressed as a ratio of
favorable outcomes/ all possible outcomes.
Student D
Grade 6 – 2005
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39
Student E is able to fill out the addition chart and correctly rearrange the numbers on the
spinner to get a more favorable probability. The student thinks about number combinations
to reason about the arrangement of the numbers on the spinners. However, like student D
the list of combinations is not exhaustive. While at younger grades students discuss
probabilities in terms of likeliness, at sixth grade students need to be able to quantify the
probability.
Student E
Grade Six – 2005
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40
Student F is able to fill out the addition chart in part 1 and make a new addition chart for
the spinner designed in part 3. However, the student does not realize how to use the chart
to calculate probability. Student F sums each row of the chart to see if its an even number,
which has nothing to do with the mathematics of this task.
Student F
Grade 6 – 2005
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41
While Student G appears to have good understanding of the task, getting 8 out of 10 points,
notice that the student is thinking about individual numbers on the spinner in part 3 rather
than the outcome of combining results of two spinners used together.
Student G
Grade Six – 2005
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42
Student H is able to fill out the addition chart, but does not make a connection between the
results in the chart and finding probability. The student thinks about the spinner as an
individual event, rather than the effect of combining the results of both spinners to
determine an outcome.
Student H
Grade 6 – 2005
pg.
43
Student I thinks of probability as a “how to win”, rather than a condition of likelihood or a
quantity for expressing likelihood.
Student I
Student J does not understand that to improve the probability for winning the numbers must
be switched across spinners. Student J makes the common error of just rearranging the
numbers on the same spinner.
Student J
Grade Six – 2005
pg.
44
Student K changes the numbers for the spinners to improve the probability for winning.
While the new spinners do improve the chances, the mathematics is considerably easier
than rearranging the numbers on the original spinners.
Student K
Student L misinterprets the task in many ways. The student does not understand
probability and writes lists of numbers rather than making a numerical statement about the
chances of winning. The student uses numbers from the addition chart to make the new
spinner, rather than rearranging the numbers on Spinners A and B.
Student L
Grade 6 – 2005
pg.
45
Student M confuses probability with odds. Probably is a ratio of favorable outcomes/ total
outcomes. “Odds” is a ratio of favorable outcomes/unfavorable outcomes. These are not
the same thing mathematically; although they are talking about the same sample space.
Student M
Teacher Notes:
Grade Six – 2005
pg.
46
Frequency Distribution for Task 3 – Grade 6 – Winning Spinners
Winning Spinners
Mean: 5.43
StdDev: 3.37
MARS Task 3 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
9
10
656
8.2%
100.0%
241
11.2%
91.8%
1346
28.1%
88.8%
269
31.4%
71.9%
1325
48.0%
68.6%
179
50.3%
52.0%
1045
63.3%
49.7%
127
64.9%
36.7%
801
74.9%
35.1%
161
77.0%
25.1%
1842
100.0%
23.0%
The maximum score available for this task is 10 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, 89%, were able to fill in the addition chart for combining the values of the
two spinners. More than half the students, 69%, were able to fill in the addition chart and
design a new set of spinners with a higher probability of winning. Almost half, 49%, could
fill in the addition chart for the original spinners, design spinners with a higher probability
of winning, and make an addition chart for the new spinners. 24% of the students could
meet all the demands of the task, including giving numerical probabilities for each set of
spinners. 7% of the students received no points on this task. 92% of those students
attempted the task.
Grade 6 – 2005
pg.
47
Winning Spinners
Points
Understandings
92%
of
the students attempted
0
2
4
6
8
10
Misunderstandings
Students did not understand the purpose
the task.
of the addition chart. Most seemed to
be looking for a pattern to the numbers
in the chart, like 13, 14,15. Students
did not use addition to find values.
Students could use addition to
Students did not know how to express a
accurately fill out the chart.
probability mathematically. 19% of all
students gave the probability as a verbal
description, like not likely, or described
how to win, e.g. “get an even number”.
Students could fill out addition
15% of all students rearranged the
chart and design spinners with
numbers on the same spinner. 15%
better odds.
substituted their own numbers (82% of
these numbers were all even). 6% used
the given numbers, but used some more
than once. Many of these designs did
not improve the odds.
Students could fill out the
13% gave whole numbers for the
addition chart, design spinners
probability. 6% were thinking in
with better odds, and either make eighths because there were eight
a new addition chart for their
numbers on the spinner. 13% gave
spinner or give a probability for
wrong values of sixteenths.
the first set of spinners.
Students could fill out the chart, Some students could make the addition
design spinners, give the
chart, but did not relate the chart to
probability for the first spinner,
finding probability. The chart was not
and either make the chart for the connected to solving the task. Students
new spinners or give the
with the correct probability gave
probability for the new spinners. incomplete explanations, usually not
defining the whole sample space, or
gave explanations not based on the
mathematics of the task.
Students could fill out an
addition chart to show the
sample space for spinning two
spinners, using the same
numbers design a spinner with a
higher probability of winning,
show the sample space for the
new spinner, and give quantify
the probabilities for each set of
spinners. Of the students who
gave correct probabilities: 63%
used fractions,18% used
percents, and 8% “out of”.
Grade Six – 2005
pg.
48
Based on teacher observation, this is what sixth graders know and are able to do:
• Complete an addition chart
• Design a set of spinners, using the given numbers, with a higher probability of
winning
• Give a probability with 16ths as the denominator
Areas of difficulty for sixth graders:
• Quantifying probability instead of using language of likelihood
• Connecting the addition chart to the probability
• Understanding a compound event (Students thought of getting an even number on
one of the spinners, versus the sum of numbers on both spinners.)
• Identifying the constraints of the task ( using the same numbers on the new
spinners, improving the odds)
Strategies used by successful students:
• Making an organized list or new addition chart to find the probability for the second
set of spinners
• Using fractions, percents, and “out of” to quantify probabilities
Questions for Reflection on Winning Spinners:
What experiences have your students had with probability this year?
Have their experiences dealt with strictly the language of probability: likely,
unlikely, equally likely, sometimes, never? Or have students had opportunities to
think about quantifying probability?
• Do your students commonly use ratios, percents, or “out of” to express the value of
a probability?
• Have students’ probability experiences been limited to thinking about a single
event: spinning 1 spinner or rolling one dice? Or have students worked with
compound events?
• Have they used strategies like tree diagrams, organized lists, charts to help them
find all possible outcomes or sample space?
Look at student work on the probability for the first set of spinners. How many of your
students could:
•
•
4/16 or 1/4
Express
probability
in words
(likely,
unlikely)
Give rules
of the game
(get an even
number)
Used a
Used a
Other
denominator denominator
of 16
of 8
What does this show about their understanding of probability and compound events?
Grade 6 – 2005
pg.
49
Look at the design of their spinners. How many of the students could:
Design
spinner with
higher prob.
Used their
own numbers
Repeated
Rearranged
some numbers the numbers
within the
same spinner
Did not
improve their
spinner
What constraints were students missing? What probability concepts were students
missing?
In part 3, did your students:
• Make or attempt to make an organized list of possible outcomes? Did they make an
addition chart for the new spinners?
• Did they think about number properties (an even plus and odd equals an odd)?
• Did they think just about combinations that make even numbers?
• Did they ignore the combination of the two spinners and just look at values on the
spinners (3/8 or 2 out of 4)
• Did they appear to guess?
What experiences will help students start to develop an understanding of sample space for
compound events?
Implications for Instruction:
Students at this grade level should transition from thinking about probability in terms of
likelihood to being able to quantify probabilities as a value between 0 and 1. Students
might start by trying to give a variety of probability words, such as always, never, likely,
equally likely, seldom and trying to assign possible values to the numbers. Students should
then start to think about more exact values such as getting heads on a coin or a certain color
on a single spinner. Students then need to move to compound events, like tossing two
coins or spinning a spinner and flipping a coin. These combined events then need to be
first expressed as possible outcomes. Students might make factor trees, organized lists,
charts or tables, or use a combinatorics formula to find all the possible outcomes. Students
should have discussions to find the relationships between all these representations. Having
students design their own games or rules, gets them involved in understanding the context
as well as an opportunity to organize the possible outcomes and find the probabilities.
Giving students a game and asking them to change the rules to improve the chances of
winning, also makes them think about how the events effect each other and how they
change the sample space.
Students should also have opportunities to investigate the difference between theoretical
and experimental probabilities. Playing a game a number of times gives them a small
flavor of this idea, but using a computer simulation can give them the bigger idea of how
the two values become closer over a large number of repetitions. Students often have
misconceptions about how spinners are made. For example they might think that a spinner
50
Grade Six – 2005
pg.
with two separate 1/8 red sections will have a better chance of winning than a single 1/4 red
section because it is more spread out. Computer simulations give them the opportunity to
make and test these conjectures and confront these misconceptions head on.
Teacher Notes:
Grade 6 – 2005
pg.
51
6th grade
Task 4
In The Playground
Student
Task
Find the area of a playground sandbox and make a new design that
will measure twice the area of the first sandbox.
Core Idea
4
Geometry and
Measurement
Analyze characteristics and properties of two-dimensional
geometric shapes and apply the appropriate techniques, tools, and
formulas to determine measurements.
• Develop, understand and use formulas to determine the area of
quadrilaterals
• Select and apply techniques and tools to accurately find length
and area measures to appropriate levels of precision
Grade Six – 2005
pg.
52
Grade 6 – 2005
pg.
53
In the Playground
Grade 6
Rubric
The core elements of performance required by this task are:
• work with areas
Based on these, credit for specific aspects of performance should be assigned as follows
1.
2.
3.
Gives correct answers: 24 square feet (Accept 80 square feet)
56 square feet
points
1
1
2
Draws a correct diagram:
Rectangular area of sand, 48 square feet (12 squares)
2
Surrounded by a row of squares of “rubber matting”
1
Gives correct answer: their number of squares x 4
(dependent on their diagram and following the one row rule)
1ft
80 or 72 (accept 120)
4.
section
points
3
1
Gives a correct answer dependent on their diagram for question 2:
12 and 4
2x1ft
or 8 and 6
2
or 2 and 24
Total Points
Grade Six – 2005
8
pg.
54
Looking at Student Work on In the Playground
Student A highlights the dimensions by labeling the sides of the sandbox and matting with
two’s. The student seems to use a counting by 4’s strategy to find the area of the rubber
matting in part 1. Student A is able to use the information from part 1 to design a new
sandbox with twice the area of the original. The student can think about the scale for linear
and area measures.
Student A
Grade 6 – 2005
pg.
55
Student B uses an interesting technique to find the area of the matting. The student finds
the area of the total play structure and subtracts the area of the sandbox. It seems most
likely that the student found the area of each by multiplying the dimensions together,
because the student is able to reason correctly about the linear dimensions in part 4.
However, the student struggles with the area of the new matting, multiplying the number of
squares by the linear scale of 2 instead of by the area scale of 4.
Student B
Grade Six – 2005
pg.
56
Student C counts squares and multiplies by the scale factor of 4 to find the area of the
sandbox and rubber matting in part 1. The student shows doubling the area to find the area
of 48 sq. ft. for the new sandbox. However, the student cannot work backward from the
area with the scale factor included to the representation of the area on the graph. Instead of
dividing the 48 sq. ft. by 4 to find the number of squares, Student C draws a sandbox of
with 48 squares, which would represent 192 sq. ft. The student is able to use the linear
scale of 2 to find the dimensions of the sandbox in his design.
Student C
Grade 6 – 2005
pg.
57
Student D is also able to count squares and multiply by the area scale factor of 4 in part 1
and 3. Student D is also able to design a new sandbox with double the area of the sandbox
in part 1. However the student is unable to use the linear scale to translate from the
drawing of the sandbox to the actual dimensions represented by the drawing.
Student D
Grade Six – 2005
pg.
58
Student E is not able to find the area of the sandbox in part 1. The student multiplies the
area by the linear scale factor instead of the area scale factor. The student is still able to
design a new sandbox with double the area, because the number of squares doubles. In the
choice of design the student has made a rectangle, which is equal to original sandboxes side
by side. Student E uses the linear scale factor to find the dimensions of the new sandbox. .
The student misses the constraint of the rubber matting being a strip 2 feet wide. However
the student does correctly find the area for the rubber matting in his drawing.
Student E
Grade 6 – 2005
pg.
59
Student F makes the most common error of counting squares to find the area and ignoring
the scale given in the task. The student is able to double the area, because the scale does
not effect the number of squares in the representation. The student also ignores the scale
for both linear and area measures in part 3 and 4 of the task.
Student F
Grade Six – 2005
pg.
60
Student G is able to count squares and multiply by the scale factor of 4 to find the area for
the sandbox and matting in part 1. The student may have subtracted the sandbox area from
the total area to find the matting. In part 2 the student designs a sandbox that is not the
obvious double of anything from part 1. The student also ignores the idea that the matting
is always 2 feet wide. When calculating the area of the matting, the student leaves out one
set of 4 squares along the side of the sandbox, making the total off by 16 square units. The
student multiplies the dimensions by the area scale factor instead of the linear scale factor.
Student G
Grade 6 – 2005
pg.
61
Student H miscounts the squares in the sandbox or multiplies the linear scale by the area
scale to find the area of the sandbox in part 1. The student gives the width of the matting
instead of the area. The student designs the sandbox by looking at the pattern of filling in
the whole grid except for a surrounding frame of 1 square unit. The student gives the
dimensions of his sandbox without regard to the linear scale factor.
Student H
Teacher Notes:
Grade Six – 2005
pg.
62
Frequency Distribution for Task 4 – Grade 6 – In the Playground
In the Playground
Mean: 2.66
StdDev: 2.24
MARS Task 4 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
1535
19.2%
100.0%
1559
38.7%
80.8%
1246
54.3%
61.3%
1046
67.4%
45.7%
966
79.5%
32.6%
534
86.2%
20.5%
597
93.6%
13.8%
148
95.5%
6.4%
361
100.0%
4.5%
The maximum score available on this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, 79%, can draw a new sandbox that has one row of matting surrounding it.
More than half the students, 62%, can either find the area of the original sandbox and
matting or draw a sandbox with one row of matting and give one of the dimensions for the
new sandbox correctly using the linear scale factor. Some students, about 33%, could find
the area of the original sandbox and matting and either design a new sandbox with double
the area or draw a sandbox with one row of matting and give one dimension for the new
sandbox. Only 4% of the students could meet all the demands of the task including
correctly applying both a linear and area scale factor to a drawing and designing a new
sandbox with double the area of the original while still maintaining only one row of
matting. 21% of the students scored no points on this task. 86% of those students
attempted the task.
Grade 6 – 2005
pg.
63
In the Playground
Points
0
Understandings
Misunderstandings
86% of the students attempted
the task.
Students did not understand how to
apply scale factors to a drawing. 16% of
all students just counted squares in the
drawing to find area for the original
sandbox and matting. 8% found an area
of 48 for the original sandbox, another
8% found an area of 12.
About 14% drew no matting around the
new sandbox. 32% drew matting with
more than one row on at least one
dimension. (Some of these overlap.)
Students frequently confused linear and
area scale factors. They might multiply
the length by 4 instead of two or
multiply area by 2. In some cases
students multiplied the two scale factors
together before applying them to
drawing.
9% of all students drew a rectangle equal
to the entire grid minus one row for
matting, thinking of a pattern rather than
an area. 8% doubled the dimensions
instead of the area, drawing a rectangle
that was 4 x 6. 8% doubled the area of
the entire play area, 80 sq. ft., making a
new sandbox 4 x 10.
1
Students could draw a new
sandbox with one row of
matting along the outside.
2
Students could either find the
area for the original sandbox
and matting or draw a sandbox
with one row of matting and
use the linear scale to find one
of the dimensions of the new
sandbox.
Students could find the area of
the parts of the original
drawing. They could either
design a sandbox with double
the area or draw a sandbox
with the proper matting and
give one dimension of the new
sandbox. Those students who
could draw the sandbox with
double the area were able to do
so because they were thinking
about number of squares.
Students could find areas of
the original sandbox, design a
new sandbox with double the
area, and find the area of the
matting for the new sandbox.
Students could work with
linear and area scale factors to
calculate areas and dimensions
of a sandbox and design a
scale drawing of a sandbox
with double the area.
4
6
8
Grade Six – 2005
Students could not apply the linear scale
factor to find the dimension for their
sandbox. Most of these students counted
squares (3 x4 or 2 x 6).
pg.
64
Based on teacher observation this is what sixth grade students knew and were able to do:
• Students could find the area of the drawing and use the appropriate scale factor to
find the area of the real sandbox and matting
• Many students could draw a sandbox with one row of matting, understanding the
constraint that the size of the matting does not change
Areas of difficulty for sixth graders:
• Doubling the area of the sandbox
• Using a linear scale factor to find the dimensions of the sandbox they designed
• Confusing linear and area scale factors
Questions for Reflection:
•
•
•
What experiences have your students had with finding area? Do they still rely on
counting squares or do they use multiplication?
Have students had multiple experiences with doubling or tripling shapes, so that
they start to see that the dimensions don’t double or is this a new idea for them?
How many of your students, do you think, could make a good justification for why
the dimensions don’t double? What kind of explanation or justification would you
like them to be able to make?
What experiences have your students had with using scale factors? Have students
worked with enlarging or shrinking shapes? Do you think they understand the
difference between a linear and an area scale factor?
Look at student work in part one of the task. How many of your students thought the area
was:
24 sq. ft.
6 sq. ft.
12 sq. ft. 48 sq. ft.
8 sq. ft.
20 sq. ft. 40 sq. ft.
Other
What kind of thinking might have led to those errors? Which errors are related to
understanding the sandbox? To understanding which scale factor to use? To
understanding that a scale factor is needed?
Look at student work on designing a sandbox with double the area. How many of your
students understood the constraint of one row of matting? Did they not draw any matting?
Make the matting more than one row? Now look at the area of the new sandbox. How
many drew a shape with dimensions of:
3x4
or 2 x 6
4 x 12
4x6
4 x 10
Other
What type of logic led to these specific errors? What were students not understanding
about area, the sandbox, and using scale factors? How are these errors different?
Grade 6 – 2005
pg.
65
Implications for Instruction:
Students at this grade level should be comfortable with finding areas of rectangular shapes
by multiplying the dimensions or counting squares. They should have a good
understanding of the difference between linear measures and area measures. A big idea for
middle grade students is to develop proportional reasoning, which would include
understanding scale factors. Students should work with recording dimensions of objects,
rooms, playgrounds, maps, etc. by making their own accurate scale drawings. They should
have frequent opportunities to translate between scale drawings and giving the dimensions
and areas for the real object.
While students are working with growing patterns, they should be challenged to think
about and justify why doubling the dimensions does not double the area. A lesson might be
designed to predict the area of shape when dimensions are doubled or tripled and then
make a model to test this out. Shapes should not be limited to rectangular objects, but
should include triangles, trapezoids, parallelograms, and even compound shapes like an Lshaped room. Students should also work with understanding or visualizing the distortion of
changing only one dimension or changing both dimensions using different scale factors.
Students’ experiences should not be limited to enlarging shapes, but should also include
working with shrinking shapes.
These types of activities help students understand the two-dimensionality of area and build
an understanding of scale. They also lay the foundation for understanding similarity at later
grades.
Teacher Notes:
Grade Six – 2005
pg.
66
6th grade
Task 5
How Much Money?
Student
Task
Work with simple fractions to solve two money problems and use
representations to organize and record the thinking.
Core Idea
1
Number and
Operation
Understand number systems, the meanings of operations, and ways
of representing numbers, relationships. And number systems.
• Understand fractions as parts of unit wholes
• Select appropriate methods and tools for computing with fractions
from among mental computations, estimation, calculators and
paper and pencil and apply selected methods
• Develop and analyze algorithms for operations on fractions and
develop fluency in their use
• Create and use representations to organize, record, and
communicate mathematical thinking
Grade 6 – 2005
pg.
67
Grade Six – 2005
pg.
68
How Much Money?
Grade 6
Rubric
The core elements of performance required by this task are:
• work with simple fractions
• figure out a money problem
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answer: 4 dollars and 80 cents
1
Finds 1/5 x $6.00 = $1.20
1
Shows work such as:
$6.00 - $1.20
or
4/5 x $6.00
2.
section
points
1
2
1 dollar and 80 cents
1
or
or
one and a half times the incorrect answer that replaced $1.20
1 ft
Shows correct work such as:
Works out that $1.20 represents (1 – 1/3) or 2/3 what Rachelle started with
Works out that Rachelle must have $0.60 = 1/2 of $1.20 left
(adding the $1.20 + .60 = $1.80 is credited in the answer)
3
1
1
3
Total Points
Grade 6 – 2005
pg.
6
69
Looking at Student Work on How Much Money?
Student A is able to make sense of the part/whole relationships in this task. The student
sees that 1/5 plus 4/5 equals a whole. The student also knows that the denominator divides
a quantity into equal-size groups and the numerator describes the number of groups being
represented. The student can think about if two-thirds is equal to $1.20, then dividing by
two will find the amount in 1/3.
Student A
Grade Six – 2005
pg.
70
For some students this task is a straight forward exercise, for others it is a problem-solving
situation. Student B shows a student doing problem-solving and really thinking about the
relationships involved and making sense of the fractional parts. Student B is always
verifying calculations to see if they make sense as evidenced in the work in part one, where
every calculation is rechecked. In part two the student tries several paths to get the
solution. The habit of mind of rechecking each calculation to see if it makes sense in the
context of the problem allows the student to identify strategies that don’t make sense and
finally do four calculations that use all the appropriate relationships to find the answer.
Student B
Grade 6 – 2005
pg.
71
Student C does a nice job of labeling all the appropriate parts. The student uses decimals to
find 1/5 of $6. Notice that the student rewrites the question of what needs to be calculated
before solving the task. In part 2 the student uses symbolic notation to make sense of 1/3,
2/3, and one-whole or 3/3. The student identifies what is known, 2/3, and what is missing
the whole, n.
Student C
Grade Six – 2005
pg.
72
Student D has the habit of mind of clearly documenting work done on a calculator. In part
2 the student makes good use of labels to clarify and organize her thinking. The student
uses the idea of 2/3 to help solve the problem using a bar model, where 1.20 equals two of
three equal parts and showing an unknown third part. This model helps to identify the
operations needed to solve for the missing part and then for finding the total.
Student D
Grade 6 – 2005
pg.
73
Student E is able to work with the idea of 1/5 being one of 5 equal parts to successfully
solve for part 1 of the task. Student E is unable to make sense of the relationship the $1.20
in part 2 must represent 2/3 if after spending that amount only 1/3 of her money remains.
This idea of 2/3 + 1/3 = the total, was very difficult for many students.
Student E
Grade Six – 2005
pg.
74
52% of all students scored no points on this task, so it is important to identify what about
fractions and part/whole relationships they do not understand. Student F is able to meet or
exceed standards on three tasks, but struggles with the idea of fractions. The student
understands how to write mixed and whole numbers as fractions, but doesn’t understand
operations on fractions. The 1/5 being spent is not 1/5 of 1, but 1/5 of the whole $6.00.
The operation should be multiplication instead of subtraction. Just like Student E, Student F
does not see the 1/3 – 2/3 relationship between what is spent and what is left.
Student F
Grade 6 – 2005
pg.
75
Student G is able to make sense of finding 1/5 of $6. However the student does not
understand how to interpret the decimal value of 1.2 and writes $1.02. The student does
not continue the thinking to find the amount left after Chuck spends the money on candy.
The fractions being considered it part 2 appear to have no relationship to the 1/3 – 2/3 of
the task.
Student G
Grade Six – 2005
pg.
76
Even with the low score on task 5, Student H is meeting standards. However the student
shows limited understanding of fractional relationships. Student H is able to find 1/5 of $6
and find the amount needed to add to the 1/5 to make $6. However the student is unable to
recognize which part represents the money spent and which represents the money left over.
In part two the student cannot make sense of the 1/3 left after spending the money. So the
student reverts to a known procedure and finds 1/3 of the original money.
Student H
Grade 6 – 2005
pg.
77
Student J also meets standards. The student can find 1/5 of $6, but cannot convert from
fractional to decimal notation. Student J does not make sense of the relationships described
in the task. While the task is not asking for students to divide with fractions, the work also
allows us to see that the student does not understand how to divide fractions. The student
appears to add the numerators and multiply the denominators. Again the student cannot
convert between fractional and decimal notation.
Student J
Student K is a student working below standards. The student’s model shows that the
student doesn’t have even the basic idea that fractions represent equal-sized parts. Some of
the “fifths” appear to be the same size as 1/4, so the student equates 1/5 with 25¢. It is
unclear what the student was considering in part 2.
Student K
Grade Six – 2005
pg.
78
Student L confuses 1/5 with $1.50 and 1/3 with $1.30. The student has some difficulty
with subtracting decimals.
Student L
Student M makes a similar error. The student adds the denominators and numerators to
find 4/5 = $6.00. So if 1/5 is missing, 3/5 would equal $4.50. The student does not realize
that 5/5 is one whole, a very basic first step in making sense of fractional relationships.
Student M
Grade 6 – 2005
pg.
79
Frequency Distribution for Task 5 – Grade 6 – How Much Money?
How Much Money?
Mean: 1.49
StdDev: 1.97
MARS Task 5 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
4248
53.2%
100.0%
856
63.9%
46.8%
507
70.2%
36.1%
807
80.3%
29.8%
838
90.8%
19.7%
114
92.2%
9.2%
622
100.0%
7.8%
The maximum score available on this task is 6 points.
The minimum score for a level 3 response, meeting standards, is 3 points.
Almost half the students, 48%, were able to find 1/5 of a number. Some students were able
to find 1/5 of a number, recognize that the number should be subtracted from the original
amount in order to correctly calculate the amount of money remaining. 8% of the students
could meet all the demands of the task including understanding that if 1/3 remained, 2/3
was spent. If they knew 2/3, that amount could be divided by 2 to find the size of 1/3. This
information could then be used to find the original amount of money. 52% of the students
scored no points on this task. 87% of the students attempted the task.
Grade Six – 2005
pg.
80
How Much Money?
Points
0
Understandings
87% of the students attempted
the task.
1
Students could find 1/5 of $6.
2
Students could find 1/5 of $6
and attempted to subtract this
amount from $6 to find the
amount of money remaining.
Students could find 1/5 of a
number and subtract accurately
to find the amount of money
remaining.
3
6
Misunderstandings
Students did not understand that 1/5
means to divide a quantity into 5 equal
parts. Many students, 10%, took 1/5 of
$1 instead of 1/5 of $6, thinking Chuck
spent $.20. Others, 6%, thought 1/5
equaled $1.50. Many students tried to
use subtraction instead of multiplication
or division to find 1/5.
10% of the students thought this was
the money left, rather than the money
spent.
Students made calculation errors when
subtracting decimals.
Some students did not subtract
accurately. Students with this score
could not interpret the relationship that
if 1/3 remained, then 2/3 had been
spent. 13% or more just multiplied the
money spent in part 1 times three.10%
gave answers larger than $6.
Students could find 1/5 of a
number, subtract from a starting
amount to find a remaining
amount or change, reason that if
1/3 is left, then 2/3 is spent, and
use inverse operations to go
from the part to find the whole.
Grade 6 – 2005
pg.
81
Based on teacher observations, this is what sixth graders know and are able to do:
• Some students could find 1/5 of a number, usually by dividing by 5 or finding a
number that multiplies by 5 to equal the given number
• Some students understood that spending money required them to subtract the 1/5
from the starting amount
Areas of difficulty for sixth graders:
• Understanding that fractions are equal-size groups
• Converting between fraction and decimal notation
• Recognizing what quantity represented a “whole” (thinking that the whole is always
1)
• Identifying the proper operations when solving problems with fractions
• Seeing that if 1/3 is left, 2/3 was spent (part/part/whole)
Strategies used by successful students:
• Labeling quantities with the fractional amounts being represented (e.g. 1/5= , 4/5 = ,
1/3 = . . .)
• Labeling what is known ( e.g. spent $1.20)
• Changing fractions to percents to simplify calculations
• Drawing bar models to organize information and show relationships
• Checking solutions to see that they match the constraints of the problem
Questions for Reflection on How Much Money?
When designing your unit on fractions, how much thought or time is given to
assessing students’ basic understanding of fractions? How are students introduced
to denominators and numerators? Are students given time to explore a variety of
models for picturing and making sense of fraction; such as, circle graphs, number
lines, area models, bar models, parts of sets?
• When students are introduced to fractions, how much time is spent helping them to
picture all the relationships involved? For example, when looking at a diagram,
many texts will show a bar with 5 equal parts and 3 parts shaded in. The text will
describe the bar as showing 3/5. However, it is important for students to think
about the whole family of relationships. If 3/5 of the bar is shaded, then 2/5 of the
bar is unshaded. 3/5 + 2/5 equals the whole bar.
• Do you students do math talks, discussions about solving problems mentally? Do
they have opportunities to solve problems like 1/5, 1/3, 1/4 of a number mentally
and justify their answers? Do students solve more complicated fraction problems
mentally?
• How comfortable are your students converting between fractions, decimals and
percents? How might this have helped them solve this problem?
Look at student work on part one of the task. How many of your students:
•
$4.80
$5.99 or
$5.95
Grade Six – 2005
$5.80
$5.50
$4.50
Other
pg.
82
Now look at the student thinking in this part of the task. What do your students know
about part/whole relationships?
• What indication do you have that students understand fractions as being divided
into equal parts?
• What indication do you have that students don’t understand fractions as being
divided into equal parts (look at things like diagrams)?
• What evidence do you have that students understood the “whole” as being equal to
$6?
• What evidence do you have that students misunderstood the “whole” (e.g. tried to
find 1/5 of $1)
• What evidence do you have that students understood 1/5 of a number? How many
students used an incorrect operation, like trying to subtract 1/5?
• How many students had difficulty converting from fraction to decimal notation?
(example, trying to make 1/5 into $1.50 or 3/15 = $.20)
After reflecting on student work, what would you change about how you introduce students
to the meaning of fractions or working with unit fractions? What would you do differently
or give more emphasis to?
Look at student work in part 2. What evidence do you have that students could make sense
of the idea that if 1/3 is left, 2/3 is the amount spent?
• Did students label answers to show what they knew or what fractional part was
represented in a given expression?
• Did students make models or diagrams to help them organize or make sense of the
information?
• What obvious evidence do you see that students have no idea what a fraction is (
like trying to add 1/3 + 1/5 to get 2/15, or converting 1/3 to $1.30)
Implications for Instruction:
Students come to middle school without a clear understanding of fractions representing
equal size parts of a whole. They also don’t understand that the “whole” could be one unit,
like 1 dollar, 1 cup, 1 inch, or 1 group, like 1 bag of chips, 1 group of students, or one
value, the starting amount of money, like the $6 in the problem. Students need to have a
clear idea of this definitional representation of a fraction before they can get into problemsolving with fractional amounts. Students should be exposed to a variety of models, such
as pie graphs, bar models, sets of objects, area models, and the number line to help them
reason about the meaning of fractions and their relationships. When introducing a model,
the teacher and student should talk about all the relationships involved. If the shaded part
represents 2/3, then the unshaded part represents 1/3. Together the 2/3 and 1/3 represent or
combine to make one whole. These relationships need be explicit, rather than the text book
representations where the relationships are often implicit.
Having a firm grasp of models helps students to reason about the meaning of operations
with fractions. If 1/5 means that something is divided into 5 equal parts, students should be
able to visualize breaking an object, figure, or number into equal size parts or using
division to find the size of the parts. Students should also be able to reason about spending
1/5 means 4/5 is left or inversely if 1/3 is left 2/3 was spent. Using the definition of
fractions as something divided into equal size parts, students should be able to think about
83
Grade 6 – 2005
pg.
the relationship between 2/3 and 1/3. Students at this grade should be able to do math talks
or mental math around finding fractional parts:
What is 1/5 of 30? 2/5 of 45? How do you know? Why do you think that? How could
you convince your classmates?
Understanding equivalencies is difficult concept for students at this grade level. Many
students think of the equal sign as an indicator that the answer will follow, rather than an
indicator that the two sides of the expression are the same size. While the procedures for
finding equivalent fractions or converting between fractions and decimals is relatively easy,
the understanding of what that means is not. Students need frequent opportunities to
explore equivalency, what it means, and how it can be used to solve problems. It is in
being confronted with discrepancies and trying to make sense of different ideas that
students build and develop their own understanding of the concept. So having students
discuss which is bigger 1/2 or 4/8, is a great activity for students. Having students discuss
is 1/3 equal to $1.30? In what situation might 1/3 be equal to $1.30? In what situation
would 1/3 not be equal to $1.30?
It is important for students to work with fractional concepts in context. It is context that
provides a reason for thinking about the whole as sometimes different from 1. It is the
context that causes the need to convert between fractional notation to decimal notation or
monetary notation. It is the context that helps students see the difference and similarity
between the fraction of a continuous amount like 1/4 of a yard of fabric or 1/8 of a cup of
sugar, versus a fraction of set 1/3 of a dozen donuts, 1/4 of the class, or part of a quantity,
Sam spent 3/8 of his money on bus fare. Students need to have a variety of fraction
contexts and types of fractions to get a good, conceptual understanding of this complex
idea.
Teacher Notes:
Grade Six – 2005
pg.
84