Overall Frequency Distribution by Total Score
Grade 8
Mean=18.07; S.D.=11.01
300
Frequency
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
Eighth Grade – 2003
pg.
1
Level Frequency Distribution Chart and Frequency Distribution
2003 - Numbers of students Grade 8: 8178
tested:
Grade 8 2000 - 2001
Level % at ('00) % at least
% at ('01) % at least
('00)
('01)
1
19%
100%
19%
100%
2
47%
81%
42%
81%
3
26%
34%
31%
39%
4
8%
8%
8%
8%
Grade 8 2002 - 2003
Level % at ('02) % at least
('02)
1
27%
100%
2
37%
73%
3
23%
36%
4
13%
13%
% at ('03)
31%
26%
19%
24%
% at least
('03)
100%
69%
43%
24%
3000
2500
Frequency
2000
1500
1000
500
0
Frequency
0-10
1 Minimal Success
11-19
2 Below Standard
20-27
3 At Standard
28-40
4 High Standard
2509
2158
1556
1955
Eighth Grade – 2003
pg.
2
8th grade
Task 1
Pete’s Numbers
Student
Task
Use mathematical reasoning to solve number problems.
Core Idea 2
Mathematical
Reasoning
Employ forms of mathematical reasoning and justification
appropriately to the solution of a problem.
• Extract pertinent information from situations and determine
what additional information is needed
• Verify and interpret results of a problem.
• Invoke problem-solving strategies
• Use mathematical language and representations including
numerical tales and equations, simple algebraic equations,
formulas, and graphs to make complex situations easier to
understand
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantity
and change.
• Recognize and generate equivalent forms of simple
algebraic expressions and solve linear equations
Core Idea
3
Algebra and
Functions
Eighth Grade – 2003
pg.
3
Eighth Grade – 2003
pg.
4
Eighth Grade – 2003
pg.
5
Looking at Student Work – Pete’s Numbers
Students did very well on Pete’s Numbers, with more than 1/3 getting perfect scores.
Looking carefully at the methods or strategies these students were using can give us
insight into how students are progressing in their use and understanding of symbolic
algebra, algebra as problem-solving tool, and understanding of variable.
Student A successfully translates the problem statements into equations and uses
symbol manipulation to solve systems of equations.
Student A
Eighth Grade – 2003
pg.
6
Student B has a clear understanding of constraints and how to systematically check
that the demands of each constraint have been met. In part 3 Student B uses number
sentences that mirror the language from the prompt. The student uses guess and
check and substitution to find a correct solution to fit both constraints.
Student B
Eighth Grade – 2003
pg.
7
Student C appears to understand how to use symbolic algebra to write down the
constraints of the problem in the form of equations. Examining the student’s
thinking, there is no evidence of using the algebra to solve the equations. Instead
Student C uses a systematic guess and check approach for checking the equations.
What questions might a teacher ask to help the student connect his strategy to the
concept of a variable?
Student C
Eighth Grade – 2003
pg.
8
Student D shows an understanding of how to write equations appropriate for the
constraints of the problem. Student D can successfully solve equations if there is the
possibility for substitution (see part 2), but struggles or doesn’t have the skills for
correctly manipulating the variables when simple substitution does not work.
Student D
Eighth Grade – 2003
pg.
9
Student E is still struggling with the idea of constraints. The student is manipulating
numbers, but unclear as to what the answers represent. In part 2 Student E solves
correctly for both constraints, but confuses the numbers (variables) for the partial
answers in the second equation. In part 3 the student only solves for one of the two
constraints and therefore does not reach the unique solution defined in the problem.
Student E
Eighth Grade – 2003
pg.
10
Sometimes students with low scores demonstrate a strong foundation that will assist
them in solving problems as they gain new skills. Student F shows the ability to use
algebraic notation to write the appropriate equations for part 2 and 3. The student is
still in the acquiring skills stage of working with symbol manipulation and does not
solve the equations accurately.
Student F
Eighth Grade – 2003
pg.
11
Other students do not understand to how look at constraints and figure out all the
demands presented. In the work of Student G only one of the two constraints is
addressed in each part of the task. The student does not connect the two sentences in
the problem statement.
Student G
Teacher Notes:
Eighth Grade – 2003
pg.
12
Frequency Distribution for Each Task – Grade 8
Grade 8 – Pete’s Numbers
Pete's Numbers
Mean:4.49, S.D.: 2.99
3000
2500
Frequency
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
1310
283
1343
320
340
1410
293
387
2492
5
61.2%
56.0%
6
64.8%
38.8%
7
69.5%
35.2%
8
100.0%
30.5%
Score
Score:
%<=
%>=
0
16.0%
100.0%
1
19.5%
84.0%
2
35.9%
80.5%
3
39.8%
64.1%
4
44.0%
60.2%
The maximum score available for this task is 8 points.
The cut score for a level 3 response is 5 points.
Most students (about 84%) could find the numbers using the clues in part 1 of the task
and about 80% could show the proof for picking those numbers. Many students (over
60%) could find the numbers and prove how they met the conditions for part 1 and
find the numbers to fit the clues in part 2. About 30% of the students could use the
clues to find the missing numbers in all 3 parts of the task and show how the numbers
fit the clues. About 16% of the students scored no points on this task. 91% of the
students with a score of zero attempted the task.
Eighth Grade – 2003
pg.
13
Pete’s Numbers
Points
0
Understandings
90% of the students with this
score attempted the problem.
2
Students with this score could
find the missing numbers in part
one of the task and show how
they checked the numbers with
the clues.
4
Students could find the missing
numbers for 2 of the 3 parts of
the task and show the work to
prove that one of those sets was
correct.
5
Students with this score could
find the numbers using the clues
in parts one and two and show
how the numbers fit the clues.
7
8
Misunderstandings
Students usually found numbers that
worked for only one of the two clues
in each part of the problem. They
did not make a connection between
the two clues.
While many students can use guess
and test and do mental computations
for finding the numbers, they have
not formed a habit of checking
numbers with the clues to prove that
the numbers meet all the demands of
the task. They do not see the need to
justify their answers.
Students do not consistently check
their answers against constraints. In
some cases they would check one,
but not both constraints or they could
write algebraic equations, but
students lacked the skills to solve
them. About 1/3 of the students made
some attempt to use algebra.
Students with this score could not
find the missing numbers for part 3
or show the work to prove why they
were correct.
Students did not justify either part
1,2, or 3.
Students could use number clues
to find missing numbers.
Students may have used algebra,
substitution in number sentences,
or guess and check to show why
the numbers were correct.
Teacher Notes:
Eighth Grade – 2003
pg.
14
Based on teacher observations, this is what eighth grade students seemed to know and
be able to do:
• Find numbers to fit number clues
• Use guess and check strategies to find missing numbers
• Write symbolic expressions or number sentences to fit problem statements
Areas of difficulty for eighth grade students, eighth grade students struggled with:
• Solving two equations for two unknowns
• Justifying solutions
Questions for Reflection:
Look carefully at the work of students with scores above 5. How many of these
students:
Used guess
and check
•
•
•
•
Forgot part of
a justification
Wrote number
sentences and/or
used substitution
Wrote
appropriate
equations
Used rules of
algebra to
correctly solve
equations
How often do students in your classrooms have the opportunity to make
justifications or get credit for showing their work?
What types of experiences do students have identifying constraints within a
problem? Are they asked how they know they have met all the constraints?
What activities or experiences have students had with the concept of variable?
In what ways to do you help students make the transition or connections from
guess and test to algebra? What might you like to do or remember for next
year?
Do students in your algebra classes see the connections between the
procedures they are learning to their usefulness in solving problems or are they
still more comfortable with more familiar strategies? What might be holding
them back?
Implications for Instruction:
Students at this grade level should be comfortable with the idea of constraints and
finding numbers that meet multiple conditions. They should also have experience
with the need to justify how a solution fits the conditions in the problem. Students
should ask themselves have I checked for each condition in the problem? Just as
students at lower grades need help transitioning from the concrete to numeric and
symbolic representations, students in eighth grade need help transitioning from guess
and check to the use of algebra as a problem-solving tool. What questions or
experiences can help them make that transition? How can we help them connect the
multiple guesses to the concept of a variable? How can we get them to see the
connection between the guesses and a unique solution in equations with multiple
variables? Could graphing help them to see some of these connections or
relationships? Perhaps their experiences in Pre-Algebra need to be more focused on
some of these big ideas around variables and justification.
Eighth Grade – 2003
pg.
15
8th grade
Student
Task
Core Idea
4
Geometry
and
Measurement
Task 2
Squares and Rectangles
Use the properties of shapes to find similar shapes.
Analyze characteristics and properties of two- and threedimensional geometric shapes; develop mathematical arguments
about geometric relationships; apply transformations and use
symmetry to analyze mathematical situations; and apply
appropriate techniques tools, and formulas to determine
measurements.
• Understand relationships among the angles, side lengths,
perimeter, and area of similar objects
• Describe sizes, positions, and orientations of shapes under
informal transformations such as flips, turns, slides, and
scaling.
Core Idea
Employ forms of mathematical reasoning and justification
2
appropriately to the solution of a problem.
Mathematical
• Formulate conjectures and test them for validity
Reasoning
Eighth Grade – 2003
pg.
16
Eighth Grade – 2003
pg.
17
Eighth Grade – 2003
pg.
18
Eighth Grade – 2003
pg.
19
Looking at Student Work – Squares and Rectangles
Student A gives a completely acceptable response. In part one the student gives a
good explanation for rectangle. The student can find the square on the diagonal in
part 3 and correctly name the coordinates. The student gives an answer to part 7
based on the assumption that the vertices of the rectangle need to be on an intersection
of the grid lines, and even includes the squares as member of the set of rectangles.
Student A
Eighth Grade – 2003
pg.
20
Student A
Eighth Grade – 2003
pg.
21
Part 7 was a more open question allowing students to make more than one
interpretation. Student B makes that assumption that the vertices do not need to be on
the intersections of grid, but can appear anywhere along the appropriate parallel lines.
Student B
Eighth Grade – 2003
pg.
22
Student C assumes the rectangle must stay in the ratio of 2/3.Student C does not
consider the possibility that the horizontal dimension could be the height.
Student D also assumes the ratio must stay 2:3, but considers the possibility of
fractional sides with a size of 2 2/3.
Eighth Grade – 2003
pg.
23
Many students make some false assumptions. Students E and F struggle with the idea
that squares are not rectangles.
Student E
Student F
Eighth Grade – 2003
pg.
24
Some students with low scores show quite a bit of understanding. Student G gives a
good definition for rectangle. Student G can draw and correctly name the coordinates
for two of the squares. Student G also demonstrates misconceptions or difficulties
with measurements. When trying to draw the rectangle in part 5, the student counts
the width as 3 instead of 4, giving a common mistake of 6 for the area. The student’s
explanation in part 7 is unclear.
Student G
Eighth Grade – 2003
pg.
25
Student G, part 2
Eighth Grade – 2003
pg.
26
A few students did not understand that the lengths of diagonal lines are not the same
as perpendicular lines. Some students made a diagonal rectangle for part 5. See the
work of Student H. Student H also demonstrates a common misconception that
rectangles have 2 short sides and 2 long sides. Student H reverses the coordinates in
part 4.
Student H
Eighth Grade – 2003
pg.
27
Student H, part 2
Teacher Notes:
Eighth Grade – 2003
pg.
28
Grade 8 – Squares and Rectangles
Squares and Rectangles
Mean: 5.24, S.D.: 3.56
1000
Frequency
800
600
400
200
0
Frequency
0
1
2
3
4
5
6
7
8
9
10
11
12
875
611
692
826
659
911
572
740
527
525
510
232
498
7
8
9
10
11
Score
Score:
%<=
%>=
0
1
2
3
4
5
6
12
10.7% 18.2% 26.6% 36.7% 44.8% 55.9% 62.9% 72.0% 78.4% 84.8% 91.1% 93.9% 100.0%
100.0% 89.3% 81.8% 73.4% 63.3% 55.2% 44.1% 37.1% 28.0% 28.0% 44.1% 8.9% 6.1%
The maximum score available for this task is 12 points.
The cut score for a level 3 response is 6 points.
Most students (more than 80%) could define a square and find 1 or 2 squares in part 3
of the task. More than have the students could define a square, draw 2 squares on the
grid, and correctly give the coordinates for the vertices. 20% of the students could
define a square, find and give the coordinates for 2 of the squares, draw a rectangle
with sides in a given ratio and find the area of the rectangle. 6% of the students met
all the demands of the task. 11% of the students scored no points on this task. Of
those students, almost 94% attempted the task.
Eighth Grade – 2003
pg.
29
Squares and Rectangles
Points
Understandings
Almost
94%
of the students with
0
this score attempted the problem.
1
Students could give a definition
for square.
3
Students could give a definition
for square and find 2 of the
squares in the grid.
Students could give the definition
for a square, find 2 squares on the
grid and name the appropriate
coordinates.
5
6
8
Students could give the definition
for a square, find squares on the
gird, give coordinates for some of
the squares, find the rectangle on
the grid given a ratio of sides.
Students could define a square,
find the non-diagonal squares, give
their coordinates, draw the
rectangle with the proper ratio,
give the area of the rectangle, and
find the total number of rectangles
that could be made in the grid.
10
12
Students could define rectangle
and square, locate squares on a
grid, give coordinates of vertices,
find a rectangle with sides in a 2:3
ration find the area, and find the
total number of rectangles.
Eighth Grade – 2003
Misunderstandings
Students with this score often counted
all the little squares in the grid and
gave definitions, like rectangles have 3
sides.
18% of the students thought that
rectangles have 2 long and 2 short
sides (with about half stating the
opposite sides should be parallel).
13% stated that 2 sides were equal and
the other 2 sides were equal. 9% just
stated rectangles have 4 sides.
Many students had difficulty giving
the coordinates correctly, often
reversing the x and y coordinates.
Students had difficulty measuring the
sides for the rectangle in part 5. They
may have counted the two given
coordinates as 3 apart instead of 4.
They may have attempted a diagonal
rectangle, not realizing that diagonal
lines are longer than horizontal or
vertical lines between grid points.
They could not find the square on the
diagonal, had difficulty defining
rectangle, and finding the total number
of rectangles on the grid.
Student assumptions about question 7
varied. Almost 10% gave an answer of
13, 14% gave an answer of 12, 9%
gave an answer of 3, and only 3% gave
an answer of infinity. Another 14%
gave a response of 1, followed by
incorrect responses like 2,8, and 6.
Students had trouble with the
definition for rectangle or giving the
number of rectangles in part 7 and
some other small error.
Many students with this score did not
think that the vertices could go in
between the intersections of the grids.
pg.
30
Based on teacher observations, this is what eighth grade students seemed to know and
be able to do:
• Define a square.
• Locate squares on a grid, whose sides were located on grid lines.
• Give coordinates of points on a grid.
• Find area of a rectangle.
Areas of difficulty for eighth graders, eighth grade students struggled with:
• Defining a rectangle.
• Finding squares whose sides were not parallel to the sides of the grid.
• Finding the dimensions of a shape when given the ratio of the sides.
• Finding the total number of rectangles that would fit on a grid.
Questions for Reflection on Squares and Rectangles:
•
•
What types of logic problems have students worked on? How might these
help students to be more precise in making their geometric definitions?
What experiences have students had with classifying and sorting? Why do
you think students have such difficulty understanding that squares are
rectangles? What other shapes might give them similar confusion?
Look at your student answers for the definition of rectangle. How many of them
gave answers such as:
4 right
Parallel 2 sets of 4 sides 2 equal 3 sides 2 short/ All
angles
sides, 2 parallel
sides, 2
2 long
sides
long –2 lines
other
sides
equal
short
equal
sides
•
•
•
When presenting shapes to the class, how often do they look at irregular
shapes? Shapes with sides not parallel to the sides of the paper?
Have your 8th graders worked with Pythagorean theorem? Did you have
students who did not know how to find the length of diagonal lines? While
not needing to measure precisely, do you think most of you students have an
intuitive notion that diagonals are longer than the sides of rectangle?
What other types of measurement errors did you see in your students’ work?
Students also had trouble defining their assumptions in part 7 of the task. How many
of your students gave answers, such as:
Infinite
13
12
3
1
4
2
What did students need to think about to get those answers? What would you like to
clear up about their thinking around rectangles?
Eighth Grade – 2003
pg.
31
•
Many students who worked the parts of the problem (almost 15%) were
uncomfortable with giving definitions or explaining their thinking in part one
and 7 of the task. How frequently are students in you class asked to give
justifications for their answers? Are students in your class comfortable with
the demands of making a mathematical argument?
Teacher Notes:
Implications for Instruction:
Students at this level need to know mathematical definitions of common geometric
shapes like squares and rectangles and be able to analyze the properties of twodimensional figures. More experiences with classifying and sorting, working with
attributes, or logic problems might help them become more precise in the use of
definitions. They should be able to find and to draw those shapes on a coordinate
graph, including those shapes with sides not parallel to the sides of the grid. They
also need to be able to use coordinates to locate points a graph. Students should be
comfortable with the idea that not all points in a graph are located on the grid
intersections. Points can lie anywhere on the continuum and represent fractional
distances. Students at this grade level should also be comfortable working with
simple ratios to find different sides. Their thinking should be flexible enough to see
that the horizontal axis could be the width or height of a shape. Students should work
with shapes in a variety of orientations, including those with sides not parallel to the
side of the paper.
Eighth Grade – 2003
pg.
32
8th grade
Task 3
Sport Injuries
Use real data to interpret a circle graph regarding sports injuries.
Student
Task
Core Idea
Formulate questions that can be addressed with data and collect,
5
organize, analyze, and display relevant data to answer them.
Data Analysis
• Use graphical representations of data
Core Idea
Employ forms of mathematical reasoning and justification
2
appropriately to the solution of a problem.
Mathematical
• Extract pertinent information from situations and determine
Reasoning
what additional information is needed
• Invoke problem-solving strategies
Core Idea
1
Number and
Operation
•
Eighth Grade – 2003
Work flexibly with fractions, decimals, and percents to solve
problems.
pg.
33
Eighth Grade – 2003
pg.
34
Eighth Grade – 2003
pg.
35
Eighth Grade – 2003
pg.
36
Looking at Student work – Sports Injuries
Sports Injuries was bimodal. Students either did very well or scored zero. Student A
uses ratios to solve the problems and understands how relationships change when
going from percent to degrees or degrees to percent.
Student A
Eighth Grade – 2003
pg.
37
Eighth Grade – 2003
pg.
38
Student B uses a different, but accurate approach for solving the problem. Student B
shows a good understanding for converting between percents and decimals, using a
slightly unconventional choice of numbers to divide by 100.
Student B
Eighth Grade – 2003
pg.
39
Student C does not understand these relationships clearly. In part 1 the student
multiplies by 100 instead of dividing. In part 2 the student doesn’t multiply the
decimal number by 100 to convert to a percent.
Student C
Eighth Grade – 2003
pg.
40
Many students with a score of zero demonstrate some conceptual understanding of the
problem, but don’t have the formal skills to calculate the exact answer. They rely on
estimation to tackle the task. Student D uses pictures and a knowledge of degrees and
percents to get very good estimates in part 1. Student D uses pictures in part 2 and
also adjusts estimates to make sure the total adds to 100%.
Student D
Eighth Grade – 2003
pg.
41
Student D, part 2
Eighth Grade – 2003
pg.
42
While other students do not show their estimation skills as clearly as Student D, a
close look at their work reveals a fairly good sense of the relationships involved.
Student E has a fairly good estimate for “on the landing”, and then further answers in
part 1 get smaller to match the decrease in degrees. Student E’s total does equal 360
degrees. In Part 2 Student E again makes fairly reasonable estimates, which total to
100%.
Student E
Eighth Grade – 2003
pg.
43
Not all students with a score of zero show an understanding of the situation. Student
F chooses to find an average number of degrees to use for all the percentages in part
1. In part 2 Student F uses the degrees from table 1, failing to recognize that the
tables represent different sports.
Student F
Eighth Grade – 2003
pg.
44
Grade 8 – Sports Injuries
Sports Injuries
Mean: 2.45; S.D.: 2.30
4000
3500
Frequency
3000
2500
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
3446
320
270
599
342
3201
Score
Score:
%<=
%>=
0
1
2
3
4
5
42.1%
100.0%
46.1%
57.9%
49.4%
53.9%
56.7%
50.6%
60.9%
43.3%
100.0%
39.1%
The maximum score available for this task is 5 points.
The cut score for a level 3 response is 2 points.
About half the students (51%) could correctly convert three or more of the
percentages to degrees in a circle in part one of the task. Almost 40% of the students
could convert percentages to degrees and degrees to percentages in the context of a
circle graph. More than 40% of the students scored no points on this task. More than
80% of those students attempted the task.
Eighth Grade – 2003
pg.
45
Sports Injuries
Points
Understandings
Misunderstandings
More than 40% of the students scored Other students simply exchanged
0
no points on this task. Some students
showed an understanding of the
situation by giving fairly good
estimates. The degrees may have
added to 360 or the percents may
have added to 100.
2
3
4
5
Students could convert three of the
percentages to degrees.
Students could convert all of the
percentages to degrees in part one of
the task.
Students made some errors in part 2,
converting degrees to percentages.
Students showed a variety of
strategies for converting percentages
to degrees and degrees to
percentages. Students were
comfortable with the relationship
between percentages and decimals.
the numbers from the two tables,
putting the degrees from table 2 in
table 1 and vice versa. Other
students did not know how to
change from percents to decimals.
These students might multiply
when then should divide or leave
numbers in decimal form.
They did not understand the
inverse relations to go from
degrees to percents.
Teacher Notes:
Based on teacher observations, this is what eighth grade students seemed to know and
be able to do:
• Estimate the percentage of a circle given the number of degrees or estimate the
degrees when given a percentage.
• Convert percentages to degrees.
Areas of difficulty for eighth graders, eighth grade students struggled with:
• Calculating percentages when given the number of degrees.
• Converting percentages to decimals and vice versa.
Eighth Grade – 2003
pg.
46
Questions for Reflection on Sports Injuries:
•
•
•
•
What types of experiences have students in your class had this year with circle
graphs?
What strategies did students who were successful on this problem use?
How many of your students did not understand that the two tables represented
different situations? (Look to see if they just interchanged the numbers
between the graphs.)
Many students showed some conceptual understanding of the situation and
had some skills for making sense of the situation. Look carefully at student
work to check for estimation skills, and knowledge about degrees and
percents. How many of your students:
Made a fairly
accurate estimate
for some parts of
question 1 (check
“on landing”
Degrees in part 1
totaled 360/ were
very close to 360
Made fairly
accurate estimate
for some parts of
question 2 ( check
“sharp twists” or
“falling”)
Percentages added
to 100.
/
•
What kinds of activities or experiences do students need who made good
estimates to develop the ability to calculate correctly? How is this
different from the types of activities or experiences needed by students
who couldn’t calculate or estimate correctly?
Teacher Notes:
Implications:
Students need to know that there are 360 degrees in a circle. At this grade level they
should be proficient at calculating with percents. Students, who lack a conceptual
understanding of percents, have difficulty remembering and applying procedural rules
when solving problems or determining if their answers are reasonable. Some students
have conceptual understanding and can do estimation, but students at this grade level
should be fluent with these conversions. A few students still do not understand the
relationship between decimals and percents.
Eighth Grade – 2003
pg.
47
8th grade
Task 4
Dots and Squares
Student
Task
Core Idea
3
Algebra and
Functions
Find and table number patterns in a geometric content. Find and
use rules or formulas to answer questions.
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantity
and change.
• Use tables to analyze the nature of changes on quantities in
linear relationships
• Recognize and generate equivalent forms of simple
algebraic expressions and solve linear equations.
• Represent, analyze, and generalize a linear relationship (7th
grade)
• Use symbolic algebra to represent situations to solve
problems (7th grade)
Core Idea
2
Mathematical
Reasoning
Employ forms of mathematical reasoning and justification
appropriately to the solution of a problem.
• Use mathematical language and representations to make
situations easier to understand
Eighth Grade – 2003
pg.
48
Eighth Grade – 2003
pg.
49
Eighth Grade – 2003
pg.
50
Eighth Grade – 2003
pg.
51
Looking at Student Work – Dots and Squares
Dots and Squares requires students to identify patterns and generalize the patterns in
the form of a rule or formula. Student A looks at the relationship between the
geometric pattern and how it affects the relationship between the shapes and the
number patterns. Student A is able to find a rule for both part 2 and part 5 of the task.
The rule for part 5 will work for any size rectangle and is not restricted to rectangles
where the length and width vary by one unit.
Eighth Grade – 2003
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52
Student B does a nice job of working with the inverse relationships in part 3 and
explaining how to use the formula to find the side if given the number of inside dots.
Student B is able to generate a formula for part 5 which will for all rectangles, but
can’t apply to finding the dimensions of the rectangle when given the interior dots..
The student can’t use the formula to find the dimensions of a rectangle with 63 inside
dots. The student has noticed another pattern. In the examples in the table, all interior
dots are even numbers. Sometimes students try to generalize about things that are not
true for all cases.
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53
Student C has a similar problem to Student B. The student is not able to use the
formula to find the dimensions of a rectangle with 63 inside dots. The student focuses
on the numbers in the table rather than thinking about the properties in the geometry
of the pattern to find out what will hold true for all cases.
Student C
Eighth Grade – 2003
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54
Student D has found a formula that only works for one case of rectangles. The
formula will only work for those rectangles whose length is one unit longer than the
width. Therefore Student D cannot use the formula to help find the dimensions of a
rectangle with 63 interior dots.
Student D
Eighth Grade – 2003
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55
Student E has also found a formula that works only for rectangles where the length is
one more than the width. While the generalization is incorrect (the task was looking
for a formula for all cases of rectangles), the student was able to use the formula to
find numbers to fit the pattern in part 6. However the numbers will not map back to
the geometric situation.
Student E
Eighth Grade – 2003
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56
Student F makes a common mistake of finding a recursive relationship of adding the
next higher odd or even number each time. This is a cumbersome relationship to use
because it requires generating the entire list to solve for a particular solution.
Student F
Eighth Grade – 2003
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57
Grade 8 – Dots and Squares
Dots and Squares
Mean: 3.45, S.D.: 2.77
2000
Frequency
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
9
10
810
1722
1273
869
1238
557
423
222
341
438
285
7
87.0%
42.8%
8
91.2%
13.0%
Score
Score:
%<=
%>=
0
1
9.9% 31.0%
100.0% 90.1%
2
46.5%
69.0%
3
57.2%
53.5%
4
5
6
72.3% 79.1% 84.3%
42.8% 27.7% 20.9%
9
10
96.5% 100.0%
8.8%
3.5%
The maximum score available for this task is 10 points.
The cut score for a level 3 response is 6 points.
Most students (about 90%) could fill in the table with the correct perimeter for each
square. Many students (about 70%) could also find the number of inside dots for a
square. A little less than half the students (43%) could find the perimeter and the
inside dots for the squares and the rectangles. About 21% could meet standards by
filling in the tables for perimeter and dots and find the dimensions of square with 49
inside dots. Less than 5% of the students met all the demands of the task. Almost
10% of the students scored no points on this task. Of those more than 60% attempted
some part of the task.
Eighth Grade – 2003
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58
Dots and Squares
Points
Understandings
10%
of
the students scored no
0
points. Almost 64% of them
attempted the problem.
1
Students could fill in the table for
the perimeter values for the
square.
2
Students could fill in the table for
perimeter and inside dots for
squares.
4
Students could fill in the tables for
squares and rectangles.
6
Students could fill in both tables
and find the dimensions of a
square if they knew the number of
inside dots.
Students could fill in both tables,
find a rule for inside dots in a
square use the rule to find the side
length if they knew the number of
inside dots.
Students could fill in the tables,
find a rule for inside dots in
squares and rectangles, and use
their rule to find the dimensions of
a square given the number of
inside dots.
8
Misunderstandings
About 28% of the students did not
attempt this task or the final task on
the test. Time may or may not have
been an issue.
Students did not count inside dots, but
tried to find numerical patterns like
going up by 3 or 4 every time,
doubling, or having the inside always
4 less than the perimeter. They looked
at only a couple of numbers in the
table to find their rule, instead of
testing the rule for all values in the
table.
Most students would not attempt any
type of rule or generalization. Of the
students who missed part 2, 33% did
not attempt it. Of the students who
missed part 5, 41% did not attempt it.
About 15% of the students found a
recursive relationship for the inside
dots, like adding the next higher odd
number. 11% wanted to count up by 3
or 4 every time. 7% wanted to
multiply the number of sides by 4 to
find the inside dots.
Many students used counting or
drawing strategies. They could not
find a rule or formula.
Students may have found a rule that
only works for certain cases of
rectangles and so their rule would not
help them with part 6. They may
have paid attention to more than one
possible pattern in the rectangles
which limited their thinking in part 6.
See the work of Students B and C.
Students
could
make
generalizations
about
geometric patterns to predict the
10
number of inside dots and use the pattern to work backwards from inside
dots to dimensions.
Eighth Grade – 2003
pg.
59
9
Based on teacher observations, this is what eighth grade students seemed to know and
be able to do:
• Find the perimeter of squares and rectangles.
• Find the number of inside dots for a square or rectangle.
Areas of difficulty for eighth graders, eighth grade students struggled with:
• Writing rules or formulas for geometric patterns.
• Using rules to work backwards.
• Understanding how to check a rule to see if it works for all the cases in the
given information. (Making generalizations on too little information.)
Questions for Reflection on Dots and Squares:
•
•
•
What types of experiences or problems have your students had with making
rules or formulas to match geometric patterns?
When working with patterns, have the problems focused only on linear
patterns?
What questions or experiences do you ask students to help them see the
relationships between variables instead of looking at how patterns grow
(finding recursive relationships)?
Look at student responses to part 2. How many of your students:
Did not
Gave a
Goes up by Goes up by Multiply
attempt a
counting or an
4 (or 3)
the sides by
rule
drawing
increasing
every time 4
strategy
odd number
Look at student responses to part 5. How many of your students:
Did not
Gave a
Goes up by Goes up by L x W
attempt a
counting or increasing
4 every
rule
drawing
even
time
strategy
number
Looked at
interior
instead of
exterior so
rules like:
SxS or LxL
or LxW
Gave rule
that only
works for
certain
rectangles
In part 5 many students gave rules that would only work for rectangles where the
length was one unit longer than the width. What are some of the formulas? Make
a list.
Eighth Grade – 2003
pg.
60
Why won’t these formulas work for all cases of rectangles? How do you know?
What would students have needed to focus on in the drawings or tables to know
these rules wouldn’t apply? What experiences have students had looking at
different cases to make a proof? What types of problems have students worked on
that required justifications?
•
Have students in your class had opportunities to do investigations on their
own and try to make generalizations from the data?
Teacher Notes:
Instructional Implications:
When students look at pattern problems, it is helpful to visualize what is changing and
what is staying the same. As they progress through the grades this information could
be used to help them write a rule or formula. At this grade level, they can no longer
rely on drawing pictures or doing repeated addition to find the solutions to complex
problems. Students at this grade level should be proficient at answering a variety of
questions about patterns. They need to recognize that patterns can grow in more than
one direction and be able to investigate those changes. Students should work with
patterns with exponential growth as well as linear growth. Students should develop
the habit of verifying their rules or formulas to see if they work for more than one
example.
Eighth Grade – 2003
pg.
61
8th grade
Student
Task
Core Idea
3
Algebra and
Functions
Task 5
Number Pairs
Identify number pairs on a coordinate grid.
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantity
and change.
• Explore relationships between symbolic expressions and
graphs of lines
• Relate and compare different forms of representations for
relationships including words, tables, graphs in the coordinate
plane and symbols (7th grade)
Eighth Grade – 2003
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Eighth Grade – 2003
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63
Eighth Grade – 2003
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64
Eighth Grade – 2003
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65
Looking at Student Work – Number Pairs
More than 36% of the eighth grade students could successfully match descriptions to
already made graphs and use descriptions to plot points and make their own graphs.
Student A uses the description to graph discrete points on a graph and knows that the
graph should pass through the origin.
Student A
Eighth Grade – 2003
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66
Student B can match some description statements to graphs. The student seems to
consider the two numbers on axis as making in pair in the upper right graph and when
making her own graph, instead of looking at coordinate pairs.
Student B
Eighth Grade – 2003
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67
Student C does not understand that there is only one description for each graph and
that the descriptions can’t represent the same sets of coordinates. So in making his
own graph the first x is on a 12 so the student puts the first number is 12. Then
because the last x is also on a 12, the first number is equal to the last number. The
student is not showing understanding of functions or coordinates.
Student C
Eighth Grade – 2003
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68
Student D gives statements about the slope of the graphs and does not understand that
the descriptors at the beginning of the graph should be matched to the graph. Also
Student D plots a second set of points on each graph that demonstrate the same trend.
Student D
Eighth Grade – 2003
pg.
69
Grade 8 – Number Pairs
Number Pairs
Mean: 2.44, S.D.: 2.21
3500
3000
Frequency
2500
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
2993
689
583
492
460
2961
Score
Score:
%<=
%>=
0
1
2
3
4
5
36.6%
100.0%
45.0%
63.4%
52.2%
55.0%
58.2%
47.8%
63.8%
41.8%
100.0%
36.2%
The maximum score available for this task is 5 points.
The cut score for a level 3 response is 3 points.
Many students (about 63%) can match one or more descriptions to a graph. A little
more than half (55%) can match two or more descriptions to their graphs. 36% of the
students can match all descriptions to their graphs and make a coordinate graph from
a verbal description. 36% of the students scored no points on this task. About half of
these students attempted the task.
Eighth Grade – 2003
pg.
70
Number Pairs
Points
0
1
2
3
Understandings
About 36% of the students scored
no points. About half of these
students made some attempt at the
problem.
Most students found it easiest to
match “The first number is always
12” to its graph.
Students could match two of the
descriptions to their graphs, with
the first number is always 12 being
one of the correct choices.
Students could match 3 of the
descriptions correctly. The 2nd
easiest match for students was “The
second number is twice the first.”
About half the students did not try
the problem, so time may have been
an issue. Those who tried may have
given general statements about the
graphs, like it goes up or they may
have put more than one description
for each graph.
Students sometimes confused the
numbers on the axis with the first
number or second number instead of
thinking about coordinate pairs.
Most students could not make their
own graph. They typically made
plots going from 12 on the y axis to
12 on the x axis.
4
5
Misunderstandings
Students could match descriptions
to graphs and make a coordinate
graph from a verbal description.
The understood the role of the
coordinate pairs in the graphing.
Teacher Notes:
Eighth Grade – 2003
pg.
71
Based on teacher observations, this is what eighth grade students seemed to know and
be able to do:
• Match a description about the first number of a coordinate pair to its graph.
• Understand that graphs of functions go in straight lines.
• Recognize when a plot of discrete points is appropriate
Areas of difficulty for eighth graders, eighth grade students struggled with:
• Matching descriptions of coordinate pairs to their graphs.
• Making a graph from a verbal description.
• Choosing coordinate pairs to match a verbal description.
Questions for Reflection on Number Pairs:
•
•
•
•
Did your students have enough time to show all they knew on the test?
Do you think students with zeros on questions 4 and 5 in your room ran out of
time, were just unmotivated by these types of questions, or lacked the
knowledge or skills to be successful? What actions might you or your school
take to help improve these scores next year?
What opportunities have students in your class had this year with coordinate
graphing? Do they seem to understand the relationship between the number
pairs that make each point on the graph or are they looking at numbers on the
axis when they match descriptions to graphs?
What types of errors did students in your class make when making their own
graph? What are the instructional implications of those types of errors? What
further activities or experiences do they need?
Teacher Notes:
Instructional Implications:
Given a simple rule, students should be able to make an equation and use that
equation to develop a table of values. Students should be comfortable using tables to
graph simple equations. Students need experiences describing given graphs as well as
experiences plotting points. Students should have some system of checking whether
different coordinates fit the conditions of the equations.
Eighth Grade – 2003
pg.
72