Overall Frequency Distribution by Total Score
Grade 8
M e an=17.23; S.D.=8.73
500
Frequency
400
300
200
100
0
0
1 2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Frequency
Eighth Grade – 2004
pg. 1
Level Frequency Distribution Chart and Frequency Distribution
2004 - Numbers of students tested in 8th grade:
10758
Grade 8 2000 - 2001
% at
% at least
% at
% at least
Level
('00)
('00)
('01)
('01)
1
19%
100%
19%
100%
2
47%
81%
42%
81%
3
26%
34%
31%
39%
4
8%
8%
8%
8%
Level
1
2
3
4
% at
('02)
27%
37%
23%
13%
% at least
('02)
100%
73%
36%
13%
Grade 8 2002 - 2004
% at
% at least
('03)
('03)
31%
100%
26%
69%
19%
43%
24%
24%
% at
('04)
27%
26%
31%
17%
% at least
('04)
100%
73%
48%
17%
3500
3000
Frequency
2500
2000
1500
1000
500
0
0-10
1 Minimal Success
11-17
2 Below Standard
18-26
3 At Standard
27-40
4 High Standard
2852
2745
3303
1858
Frequency
Frequency
Eighth Grade – 2004
2852
2745
3303
1858
pg. 2
8th grade
Student
Task
Core Idea
3
Algebra and
Functions
Task 1
Merritt Bakery
Find the length of ribbon necessary to decorate cakes of various
sizes. Generate, apply, and test a rule or formula to find the
amount of ribbon needed to decorate any sized cake.
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
4
Geometry and
Measurement
Analyze characteristics and properties of two-dimensional
geometric shapes; develop mathematical arguments about
geometric relationships; and apply appropriate techniques,
tools, and formulas to determine measurements.
Core Idea
2
Mathematical
Reasoning
Solve problems that make significant demands in one or more
of these aspects of the solutions process: problem formulation,
problem implementation, and problem conclusion. Students
communicate their knowledge of basic skill, conceptual
understanding, and problem solving.
• Formulate conjectures, and argue why they must be or
seem to be true.
Eighth Grade – 2004
pg. 3
Eighth Grade – 2004
pg. 4
Eighth Grade – 2004
pg. 5
Looking at Student Work on Merritt Bakery
Student A writes a complete formula for finding the length of the ribbon in part 2 as
well as defining the variables and parts of the formula. Student A uses calculations to
check the size of the cakes with the 2 different diameters to find the exact effect of
doubling the diameter and then reasons about the bow to make a convincing argument
in part 2.
Student A
Eighth Grade – 2004
pg. 6
Student B uses mathematical calculations to show that doubling the diameter is not
the same as doubling the ribbon.
Student B
Eighth Grade – 2004
pg. 7
Student C has a process for finding the length of the ribbon in part 1 and can use this
process to calculate and show that doubling the diameter doesn’t double the length of
the ribbon in the final part of the task. The student makes the common error of
forgetting to add the constant of 20 in the formula for part 2.
Student C
Eighth Grade – 2004
pg. 8
While Student D can calculate and find a formula for total length of the ribbon, the
student cannot use this reasoning to think about the amount of ribbon in the final part
of the task. Student D makes a common error about not seeing how π effects the size.
The π makes the sizes incomparable.
Student D
Eighth Grade – 2004
pg. 9
Student E shows clearly a lack of understanding of basic geometric definitions in part
2, where she confuses diameter with circumference. The student does not attempt to
write a formula for finding the ribbon and has trouble comprehending the statement
about doubling the diameter of the cakes.
Student E
Many students do no understand how formulas are constructed. 29% of the students
put D and L on the same part of the formula. Of those many considered L to be the
length of the bow instead of the length of the ribbon. Almost 25% ignored dealing
with circumference and π, and just added the diameter + 20, which they may have
expressed as L+D, D+20, or L+20. 14% of the students forgot the bow and wrote πD.
Teacher Notes:
Eighth Grade – 2004
pg. 10
Frequency Distribution for each Task – Grade 8
Grade 8– Merritt Bakery
M erritt Bakery
M e an:3.56, S.D.: 3.07
3000
2500
Frequency
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
2519
1689
661
1098
432
688
1204
130
2337
Score
Score:
%<=
%>=
0
23.4%
100.0%
1
39.1%
76.6%
2
45.3%
60.9%
3
55.5%
54.7%
4
59.5%
44.5%
5
65.9%
40.5%
6
77.1%
34.1%
7
78.3%
22.9%
8
100.0%
21.7%
The maximum score available for this task is 8 points.
The cut score for a level 3 response, meeting standards, is 5 points.
Many students (70%) could multiply the diameter by the pi to find the circumference
of the cake in part 1. A little less than half the students (45%) could find the length of
the ribbon including the bow in part 1 and write a partially correct equation involving
the circumference in part 2. Some students (41%) could find the total length of ribbon
for part 1 and explain why doubling the diameter does not double the amount of string
in part 2. More than 20% of the students could reason about how circumference and
bow affected the total length of ribbon including writing a complete equation. More
than 28% of the students scored no points on this task. About 95% of them attempted
the task.
Eighth Grade – 2004
pg. 11
Merritt Bakery
Points
0
1
4
Understandings
Misunderstandings
95% of the students with this score
attempted the task.
The most common error was for
students added the diameter to the
bow to find the length of ribbon.
Students did not remember to add in
the bow.
Students could substitute diameter
into the formula to find the
circumference of the cake.
Students could calculate the total
length of ribbon and write a
formula for length of πd.
5
Students could calculate the total
length of ribbon for part 1 and
explain why doubling the diameter
does not double the amount of
string in part 2.
6
Students could calculate the length
of the ribbon in part 1 and write a
formula for finding total length.
Students could find the length of
ribbon including finding the bow
and write a correct formula to
express this process and reason
about why doubling the diameter of
the cake does not double the
amount of ribbon needed. Some
students were able to make a
logical reason about the context, the
size of the bow does not change.
Other students used numbers to
prove that doubling diameter
produced a ribbon that was not
twice as long.
8
Eighth Grade – 2004
Students did not remember to add in
the bow to their formula, even
though they used the 20 inches in
computing their answer in part 1.
Students often wrote formulas with
both variables on the same side of
the equation. They may have
interpreted the L as the length of the
bow instead of the length of the
ribbon.
Students had difficulty reasoning
about how doubling the diameters
affected the length of the ribbon.
pg. 12
Based on teacher observations, this is what eighth graders know and are able to do:
• Find the circumference of a circle and know the value of π
• Multiply
• Use the relationship between the circumference and the bow to find the total
length of the ribbon
Areas of difficulty for eighth graders, eighth graders struggled with:
• Turning a solution strategy into a generalization or rule
• Understanding that the “bow” represented a constant
• Using variables in rules or equations
• Making a mathematical justification in the context of a comparison
Questions for Reflection on Merritt Bakery
How many of your students added 9 + 20 to find the length of the ribbon? Why do
you think they didn’t use the formula for circumference? What are the
mathematical ideas they are missing?
Much of 8th grade is devoted to learning algebra or algebraic concepts about
variables and notation. Look at the types of formulas your students wrote. How
many of them could write a partially correct formula:
ƒ L = πD + 20
ƒ πD + 20
ƒ πD
How many of your students show a lack of understanding about independent
and dependent variables giving formulas like:
L-D=N
LxD
L x 3.14
L+D
L+20 = D
L+ πD
How are students introduced to the idea of variables? When working with
formulas, how are students taught to think about independent and dependent
variables? What are the classroom norms for defining variables when developing
a formula or equation?
What types of things prevented students from reasoning about how doubling the
diameter affects the length of the ribbon?
o Were they confused by the role of pi in the equation?
o Did they make statements about how diameter is not the same as
circumference?
o Did they try to verify the size but used a wrong equation?
o Other?
Eighth Grade – 2004
pg. 13
Teacher Notes:
Implications for Instruction:
Students need more work understanding how algebraic expressions work, being able
to take simple problem situations and write them in terms of variables. Students
should have frequent opportunities to go from specific solutions to generalizations.
This practice of working with algebraic expressions in context helps them to
understand important ideas about variables, e.g. how the independent and dependent
variables operate, and the function of constants in expressions. Students need to have
frequent opportunities to test their expressions using substitution to see if they are
realistic. Many students do not know how to calculate circumference or understand
how circumference relates to an object, such as a cake.
Teacher Notes:
Eighth Grade – 2004
pg. 14
8th grade
Task 2
Odd Numbers
Student
Task
Core Idea
1
Number and
Operations
Sort numbers into sets and explain the reasoning. Identify examples
and counterexamples for statements about odd and even numbers.
Understand number systems, the meanings of operations, and
ways of representing numbers, relationships, and number
systems.
• Compare and contrast the properties of numbers and number
systems including the rational and irrational numbers.
Core Idea
2
Mathematical
Reasoning
Solve problems that make significant demands in one or more of
these aspects of the solutions process: problem formulation,
problem implementation, and problem conclusion. Students
communicate their knowledge of basic skill, conceptual
understanding, and problem solving.
• Extract pertinent information from situations and figure out
and identify what additional information is needed.
• Formulate conjectures, and argue why they must be or seem
to be true.
Eighth Grade – 2004
pg. 15
Eighth Grade – 2004
pg. 16
Eighth Grade – 2004
pg. 17
Eighth Grade – 2004
pg. 18
Looking at Student Work on Odd Numbers
Student A is able to describe the meaning of the overlap region in the Venn diagram,
give examples to prove and disprove statements including commentary about why the
prove or disprove the statement, and can explain using words and pictures why 2 odd
numbers make an even.
Student A
Eighth Grade – 2004
pg. 19
Student A, part 2
Eighth Grade – 2004
pg. 20
Student B does a great job of explaining why the added numbers were put in various
places in the Venn diagram. However the student does not know how to make a
justification for why two odds make an even. It’s just a rule.
Student B
Eighth Grade – 2004
pg. 21
Student B, part 2
Eighth Grade – 2004
pg. 22
Student C makes a common error of trying to use multiplication to prove statements
about addition. The student also makes the most common error in part 4b of restating
the hypothesis instead of providing supporting evidence to make a justification.
Student C
Eighth Grade – 2004
pg. 23
Like many students, Student D tries to prove the generalization in 4b by giving
examples instead of making an argument that will prove it for all cases.
Student D
Eighth Grade – 2004
pg. 24
Student E makes the mistake of many students of thinking that 5 goes in the overlap
region because it’s in the middle. The student is also confused by the labels and uses
the left circle for even numbers. The final justification is also typical, because this is
what I have been taught.
Student E
Eighth Grade – 2004
pg. 25
Student E, part 2
Student F repeats rules learned in school for the justification in 4b.
Student F
Eighth Grade – 2004
pg. 26
Student G uses a rule about pairs with the same label as a justification.
Student G
Student I has an idea about composition of odd numbers that could have led to a good
justification. Student was unable to complete the argument by showing how the one
ups and one downs combined to make a pair.
Student I
Teacher Notes:
Eighth Grade – 2004
pg. 27
Frequency Distribution for each Task – Grade 8
Grade 8– Odd Numbers
Odd Numbers
M e an: 4.66, S.D.: 2.09
3000
2500
Frequency
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
355
645
840
1220
1530
1676
2997
300
1195
Score
Score:
%<=
%>=
0
3.3%
100.0%
1
9.3%
96.7%
2
17.1%
90.7%
3
28.4%
82.9%
4
42.7%
71.6%
5
58.2%
57.3%
6
86.1%
41.8%
7
88.9%
13.9%
8
100.0%
11.1%
The maximum score available for this task is 8 points.
The cut score for a level 3 response, meeting standards, is 4 points.
Most students (91%) could place all the numbers but 15 into the Venn diagram and
give an example to prove that 2 odd numbers don’t add to an odd number. Many
students (72%) could explain why 5 goes in the overlap region in the Venn diagram,
place all numbers correctly in the diagram, and give an example to show that 2 odd
numbers don’t add to an odd answer. A little less than half the students (42%) could
use and interpret the Venn diagram and give examples to prove and disprove
statements. 11% of the students could meet all the demands of the task including
making a mathematical justification for why tow odd numbers always add to make an
even number. 3% of the students scored no points on this task. About 50% of the
students with this score attempted the task.
Eighth Grade – 2004
pg. 28
Odd Numbers
Points
0
1
2
4
5
6
8
Understandings
Misunderstandings
About 50% of the students with
this score attempted the task.
Many thought 5 was the number in
the middle between 1 and 10. They
also confused the right circle for even
numbers instead of multiples of 5.
Some students gave examples that
Students could give a numerical
were not odd numbers or used
example to show that two odds
multiplication instead of addition.
don’t add to an even number.
Students could put 4 numbers into Students were very confused about
the Venn diagram and give an
what to do with the 15. Most put the
example showing 2 odd numbers 15 with the circle labeled 5 times
don’t add to an even number.
tables. Some put it with the odd
numbers.
Students could interpret and fill in
a Venn diagram and give an
example about 2 odd numbers
adding to an even number.
Students could fill in most of the Students who did not give correct
examples to support the statements in
Venn diagram, explain the
part three generally used
location of the 5 in the diagram,
multiplication instead of addition.
and give numerical examples to
Some students used numbers that
prove statements.
were not multiples of 5.
Students could not make a
justification for why two odd numbers
always add to make an even number.
22% of the students tried to prove the
statement by giving one or more
examples. Almost 20% just restated
the hypothesis. 15% did not attempt
this part of the task.
Students could interpret and fill in
a Venn diagram, give numerical
examples to prove and disprove
statements, and give a
mathematical justification for
why two odd numbers always add
to and even number.
Based on teacher observations, this is what eighth graders know and are able to do:
• Give examples to prove or disprove statements about odd numbers
• Use a Venn diagram based on number properties
Eighth Grade – 2004
pg. 29
Areas of difficulty for eighth graders, eighth graders struggled with:
• Mathematical justification and generalizations about even and odds
• Understanding the logic of having the overlap area in the Venn diagram,
having numbers fall into more than one category
Questions for Reflection on Odd Numbers
•
•
•
•
•
•
•
Have your students worked with Venn diagrams this year?
What experiences have students had with number properties such as multiples,
factors, odds, evens? What surprised you when you looked at their errors in
part 1 and 2?
Why do you think students had difficulties providing examples in part 3?
What opportunities do students have making mathematical justifications?
Do students have an understanding of why using examples is not enough to
prove for all cases?
What are the classroom norms for making a good justification?
What are some types of problems students have had to make justifications for
in class?
Teacher Notes:
Implications for Instruction:
Students need frequent opportunities to sort numbers and shapes and develop their
own classification systems to understand the logic of classification. Students need to
be asked questions to help them see why objects can fit into multiple categories.
Students do not always check their rules against all the evidence provided. Students
need more practice with finding evidence to support or to disprove a rule. Students at
this age should be developing their skills at mathematical justification to prepare the
way for fully developed proofs at later grades. Rich classroom discourse around
interesting tasks allows students to learn the arguments of others, discuss what about
the argument convinced them, and deal with issues about whether an argument holds
for all cases. This type of mathematical thinking requires frequent practice in order to
develop and to become a natural part of the student’s mathematical repertoire.
Teacher Notes:
Eighth Grade – 2004
pg. 30
8th grade
Task 3
Party
Student
Task
Core Idea
3
Algebra and
Functions
Solve problems involving the cost of parties. Relate the formulas
and graphs to this contextual problem.
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.
• Recognize and generate equivalent forms of simple algebraic
expressions and solve linear equations.
Core Idea
2
Mathematical
Reasoning
Solve problems that make significant demands in one or more of
these aspects of the solutions process: problem formulation,
problem implementation, and problem conclusion. Students
communicate their knowledge of basic skill, conceptual
understanding, and problem solving.
• Formulate conjectures, and argue why they must be or seem
to be true.
Eighth Grade – 2004
pg. 31
Eighth Grade – 2004
pg. 32
Eighth Grade – 2004
pg. 33
Eighth Grade – 2004
pg. 34
Looking at Student Work on Party
Student A uses a logical, organized algebraic approach to solving each part of the
task. The student maintains the equalities by separating out the steps. The student is
also able to qualify the expression for cost to show that it is only appropriate for 30 or
more people. Student A gives a good reason about choosing graph 2 to match this
context.
Student A
Eighth Grade – 2004
pg. 35
Student A, Part 2
Eighth Grade – 2004
pg. 36
Student B A is able to make sense of the situation to solve part 1 and uses that
information to use an add-on strategy to solve for part 2. The student does not
discover that the 30 people must be subtracted so the expression is not quite correct in
part 3. While having a good strategy, the student also forgets about the first 30 people
in part 4. Student B knows the cost starts at $750 but ignores that this cost needs to
stay the same for the first 30 people.
Student B
Eighth Grade – 2004
pg. 37
Student B, part 2
Eighth Grade – 2004
pg. 38
Student C also has trouble dealing with the thirty people and finds the cost of 130
instead of 30 people in part 2. Student C does not mention the vertical axis when
discussing the graph
Student C
Eighth Grade – 2004
pg. 39
Student C, part 2
Eighth Grade – 2004
pg. 40
Student D forgets to add in the cost for the first 30 people in part 2. In part 3 the
student only gives a formula for finding the cost of the first 30 people, which is again
forgotten in part 4. The student is close to having a good explanation of the graph in
the final part.
Student D
Eighth Grade – 2004
pg. 41
Student D, part 2
Eighth Grade – 2004
pg. 42
Student E makes the common error of using the $750 for every group of 30 people
this makes the answers incorrect for part 2 and 4. In part 3 the student forgets the
$750 for the first 30 people in trying to derive a formula. This error was made by
17% of the students who missed the formula. Student E’s reason for choosing graph 1
is the most common response for part 3.
Student E
Eighth Grade – 2004
pg. 43
Student E, part 2
Teacher Notes:
Eighth Grade – 2004
pg. 44
Frequency Distribution for each Task – Grade 8
Grade 8– Party
Party
M e an: 4.35; S.D.: 3.08
2000
Frequency
1500
1000
500
0
Frequency
0
1
2
3
4
5
6
7
8
9
10
1793
1164
563
906
581
1603
733
1622
622
948
223
Score
Score:
%<=
%>=
0
16.7%
100.0%
1
27.5%
83.3%
2
32.7%
72.5%
3
41.1%
67.3%
4
46.5%
58.9%
5
61.4%
53.5%
6
68.3%
38.6%
7
83.3%
31.7%
8
89.1%
16.7%
9
97.9%
10.9%
The maximum score available for this task is 10 points.
The cut score for a level 3 response, meeting standards, is 5 points.
Most students (80%) could justify why the cost of 60 people was $1350. Many
students could justify the cost for 60 people and calculate accurately the cost for 100
people. About 50% of the students could justify the cost for 60 people, accurately
calculate the cost of 100 people, and work backward from the cost to the number of
people attending the party. About 32% of the students could also correctly identify
which graph best fit the situation and give a good reason for their choice. More than
10% of the students could write an equation for find the cost with correct use of the
given variables. Almost 17% of the students scored no points on this task. 83% of
the students with this score attempted the task.
Eighth Grade – 2004
pg. 45
10
100.0%
2.1%
Party
Points
Understandings
83% of the students with this
0
1
3
score attempted the task.
Students could show why $1350
was the correct price for $60
people.
Students could show how to
calculate the cost for 60 people
and for 100 people.
5
Students could calculate costs for
a given number of people and
work backwards from cost to
number of people.
7
Students could calculate costs for
different numbers of people, work
backwards from cost to number of
people, and identify a graph and
give a reason for why it matched
the context.
9
Students could meet the demands
of the task including writing a
mathematical equation for
calculating cost.
Students could find the cost for
different numbers of people
attending the party, work
backward from cost to find the
number of people, write an
equation for calculating cost
including adding in constraints for
the size of P, and identify a graph
to match the situation and justify
the reason for their choice.
10
Eighth Grade – 2004
Misunderstandings
19% of the students who missed the
cost of 100 people put a total of $2450.
15% put $2000. 9% put $2750.
Students had difficulty working
backward from cost to number of
people. Most frequently students
forgot to add in the first 30 people
giving them an answer of 50 instead of
80.
Students had difficulty choosing the
appropriate graph. Many picked graph
one because as the people increases, so
does the cost. Also because graphs are
“supposed to start at zero”. Another
common choice was graph 3 because it
starts at $750. These students did not
factor in the way the number of people
would affect the shape of the graph.
Students had difficulty writing a
mathematical expression. Almost 10%
of the students who missed the
equation, gave an expression for
calculating P (people) instead of C
(cost), like P=21.5C. 18% of the
students thought that cost was equal to
C x P. 10% thought C=P.
Students did not qualify their equation
by mentioning the constraint that P
needed to greater than or equal to 30
for the equation to work.
pg. 46
Based on teacher observations, this is what eighth graders knew and were able to do:
• Find the cost for a given number of people
• Add and multiply
Areas of difficulty for eighth graders, eighth graders struggled with:
• Formulating an equation to represent the situation
• Understanding the domain of the equation
• Using variables in equations
• Connecting parts of graphic representation to the numerical situation
• Understanding slope
Questions for Reflection on Party:
•
•
•
What types of experiences have students had working with problems for
which they need to derive a formula?
What experiences have they had using variables? Do they understand the
difference between independent and dependent variable?
What types of graphing experiences have students had? Do they know how to
plot points? Have they worked with other problems where they match graphs
to mathematical situations?
Look at student work in part 2. How many students gave answers of
$2150
$2000
$2450
$1400
$2750
Other
Can you find how students calculated each of the common wrong answers? How
does each process reveal a different misunderstanding about the problem? What
would be the next steps for each child?
Look at student work in part 4. How many students gave answers of
80
10
50
70
72
Over 100
# with decimal
Other
What misconceptions led to some of the common errors?
Read the reasons students gave for choosing the various graphs. What surprised you?
What might you want to emphasize more or cover differently when you working on
graphing this year?
Eighth Grade – 2004
pg. 47
How did you students do on writing a formula? Were they able to use the variables
correctly? (Some students confused cost with the $20 per person. Some students
reversed the meaning for P and C. Some students wrote expressions for find people
instead of finding cost. ) How many of your students used both P and C on the same
side of the expression? How many students gave a numerical answer for C? What
are the implications of these various types of errors for your instructional practices?
What further experiences do students need to develop their understanding of algebraic
notation, variables, and how they are used?
Teacher Notes:
Implications for Instruction:
Students need more work understanding how algebraic expressions work and being
able to take simple problem situations and write them in terms of variables. Students
did not understand how a fixed cost works. Students also need more practice
graphing problem situations to see how the various parts of an equation affect the
shape of the graph. Students need practice making a table of values to check against
the graph or other appropriate strategies for comparing the graph to the situation.
Some students think that all graphs must start at zero or that graphs always show an
equal change between the two quantities. When making graphs, students need to be
probed about what the different aspects of the graph represent, how do they relate to
the context. Students need to discuss how different events in the context affect the
shape of the graph. Students can then start to make connections about the elements of
the graph and the equation, such as meaning of slope and affect of constant, in a very
real way that dealing just with mathematical exercises does not develop.
Teacher Notes:
Eighth Grade – 2004
pg. 48
8th grade
Student
Task
Core Idea
2
Mathematical
Reasoning
Core Idea
4
Geometry and
Measurement
Task 4
Hexagons
Reason and formulate arguments around the properties of
hexagons and triangles.
Solve problems that make significant demands in one or more
of these aspects of the solutions process: problem formulation,
problem implementation, and problem conclusion. Students
communicate their knowledge of basic skill, conceptual
understanding, and problem solving.
• Extract pertinent information from situations and figure out
and identify what additional information is needed.
• Formulate conjectures, and argue why they must be or
seem to be true.
Analyze characteristics and properties of two-dimensional
geometric shapes; develop mathematical arguments about
geometric relationships; and apply appropriate techniques,
tools, and formulas to determine measurements.
• Understand relationships among the angles, side lengths,
perimeter, and area of shapes.
• Create and critique inductive and deductive arguments
concerting geometric ideas and relationships.
Eighth Grade – 2004
pg. 49
Eighth Grade – 2004
pg. 50
Eighth Grade – 2004
pg. 51
Eighth Grade – 2004
pg. 52
Looking at Student Work on Hexagons
Hexagons was a difficult task for 8th graders. They had difficulty with geometric
terms and making logical arguments. Student A is able to reason about most of the
task, but struggles with the first part of the task. The student sees that the angles join
together to make 360°, but doesn’t complete that line of reasoning to find the degree
of one angle.
Student A
Eighth Grade – 2004
pg. 53
Student A, part 2
Many students repeat geometric definitions instead of offer proofs or justifications.
Student B uses part of the definition of regular hexagon to describe the angles in part
1. In part 2b, Student B is trying to use the elementary description of a right angle
making an “L” shape rather than using reasoning and calculations to prove the angle
measure for the right angle.
Student B
Eighth Grade – 2004
pg. 54
Student B, part 2
Eighth Grade – 2004
pg. 55
Student C still uses definitions rather than reasoning about the geometrical attributes
in parts a and b. However the student can use the property of symmetry to make an
argument for congruence in part c.
Student C
Eighth Grade – 2004
pg. 56
Student D uses definitions in all the arguments presented.
Student D
Eighth Grade – 2004
pg. 57
Student E is able to make a cohesive argument about why A is an isosceles triangle.
However the student does not have the same level of understanding about right
triangles and congruence.
Student E
Eighth Grade – 2004
pg. 58
Some students do not have an understanding of even the basic geometric definitions.
Student F has a misconception about right and left angles. Other students mentioned
things like isosceles triangles have no equal sides, angle C is a right triangle because it
has straight sides, or B and C are congruent because they are next to each other or
because they have no equal sides.
Student F
Teacher Notes:
Eighth Grade – 2004
pg. 59
Frequency Distribution for each Task – Grade 8
Grade 8– Hexagons
Hexagons
M e an: 2.53, S.D.: 1.47
5000
Frequency
4000
3000
2000
1000
0
Frequency
0
1
2
3
4
5
6
7
8
9
1021
1443
2119
4568
1006
285
91
94
47
84
Score
Score:
%<=
%>=
0
9.5%
100.0%
1
22.9%
90.5%
2
42.6%
77.1%
3
85.1%
57.4%
4
94.4%
14.9%
5
97.1%
5.6%
6
97.9%
2.9%
7
98.8%
14.9%
8
99.2%
1.2%
9
100.0%
0.8%
The maximum score available for this task is 9 points.
The cut score for a level 3 response, meeting standards, is 4 points.
Most students (90%) could identify either a right triangle or name the triangle
congruent to C. More than half the students (57%) could name an isosceles triangle,
identify a right triangle, and name the triangle congruent to triangle C. 15% of the
students could also give either an argument about symmetry or about the sides of the
triangle to explain why triangle C was congruent to triangle B. Few students were
able to provide justification for their choices. Only 5 % or less could meet the
demands of providing justification. Almost 10% of the students scored no points on
the task. 64% of the students with this score attempted the task.
Eighth Grade – 2004
pg. 60
Hexagons
Points
Understandings
Misunderstandings
64% of the students with this score Students did not have good
0
attempted the task.
1
Students could either identify a
right triangle or name the triangle
congruent to triangle C.
2
Students could identify a right
triangle and name the triangle
congruent to C.
3
Students could identify an
isosceles triangle, a right triangle,
and a triangle congruent to C.
4
Students could identify the correct
triangles for 2a, 2b, and 2c and
make an argument for why triangle
C and triangle B are congruent.
5
Students could identify the correct
shapes in part 2, give reasons
about why triangle was isosceles
or congruent.
9
Students could use the tessellation
to find the measure of the angle in
a regular hexagon, identify and
justify why a figure was an
isosceles triangle, a right triangle,
or why two triangles were
congruent. Their justifications
were more than restatements of the
definitions.
Eighth Grade – 2004
understanding of geometric
definitions.
Most students (75%) used the
definition for right triangle to explain
their choice, rather than using
geometrical reasoning and
calculations to prove the size of the
angles.
18% thought B and C were congruent
because they were both right
triangles. 13% thought they were
congruent because they are the same
shape and size. 7% they’re congruent
because they’re congruent. 7%
they’re congruent because the look
alike. 7% think they’re congruent
because they’re next to each other.
75% of the students just repeated the
definition for isosceles triangle to
make their justification without
offering reasons to prove the sides are
the same length.
Because students did not find angle
size in part 1, they were unlikely to do
calculations to prove the size of
angles in part 2b why a shape was a
right triangle.
pg. 61
Based on teacher observations, this is what eighth graders know and are able to do:
• Identify by shape and define a right triangle
• Define geometrical terms, like congruent and isosceles
Areas of difficulty for eighth graders, eighth graders struggled with:
• Trying to quantify relationships
• Making justifications beyond repeating definitions
• Understanding the level of detail to make a convincing argument
• Understanding that “looks like” is not a convincing argument
Questions for Reflection on Hexagons
•
What types of experiences have students had with geometry and geometry
concepts this year?
• Do students seem to know basic geometry definitions for
- Isosceles triangle?
- Right triangle?
- Congruence?
What opportunities do students in your class have to make mathematical
justifications? What are some examples of the types of justifications they have been
asked to make? What criteria do they use to judge or evaluate a justification?
How many of your students could make a reasoned argument for why A or D were
isosceles triangles or why B and C were congruent?
Did any of your students calculate the angle of a regular hexagon? Why do you think
students had difficulty with this part? Do you think this information would have
helped them to prove that the angles in triangle B or triangle C are right angles?
What types of experiences might help them to develop their ability to calculate and
reason about angles?
Teacher Notes:
Eighth Grade – 2004
pg. 62
Implications for Instruction:
Students seem to have a good knowledge of geometrical definitions. Students do not
see how to use these definitions to solve problems and make justifications. Students
need more opportunities to make conjectures and use definitions to make calculations
and develop simple mathematical arguments to support their definitions. Students do
not seem to understand that repeating a definition does not prove why something is
true. Students need more practice quantifying geometric relationships, like finding the
size of the angles in a regular hexagon and then using that quantity to verify other
angles in the diagrams.
Teacher Notes:
Eighth Grade – 2004
pg. 63
8th grade
Student
Task
Core Idea
4
Data Analysis
Task 5
Animals
Analyze and interpret information regarding a species of animals
found in a nature preserve.
Formulate questions that can be addressed with data and collect,
organize, analyze, and display relevant data to answer them.
• Collect data about a characteristic shared by two populations
or different characteristics within one population.
• Find, use, and interpret measures of center and spread.
• Discuss and understand the correspondence between data sets
and their graphical representations.
Eighth Grade – 2004
pg.
1
Eighth Grade – 2004
pg.
2
Eighth Grade – 2004
pg.
3
Eighth Grade – 2004
pg.
4
Looking at Student Work on Animals:
Student A lists the weight categories in order from lowest to highest to find the mean.
It appears that the student started by using the whole numbers like 20 or 30 and then
shortened it to the digit in the 10’s place for convenience. In part 2a and 2b, Student
A attempts to describe the shape of the data in the histograms, but does not quite
achieve a comparison. To push student’s thinking about comparisons and interpreting
graphs, it might be good to ask why it might be significant to note the differences in
the 20-29 gram category.
Student A
Eighth Grade – 2004
pg.
5
Student A, part 2
Student B has memorized a procedure for finding median, by looking at the middle
number. The student knows that if there is an even number of data points, then the
middle two should be averaged. Why doesn’t this algorithm for finding median work
for this graph? Student B makes 3 fairly good comparisons in part 2.
Student B, part 1
Eighth Grade – 2004
pg.
6
Student B, part 2
Student C uses the number of animals as a data point instead of a frequency to find
the median. The student also uses the number of animals to calculate mean.
Student C
Eighth Grade – 2004
pg.
7
Student C, part 2
Student D confuses mode and median. The student identifies the largest number of
animals in any one category. Student D is also using the number of animals as a data
point instead of a frequency. While Student D makes three comparisons in part 2, the
student needs to be pushed to think about is whether there is any significance to
numbers of animals being the same in different categories? Does this help us
understand the data or information in a significant way? How or how not?
Student D
Eighth Grade – 2004
pg.
8
Student D, part 2
Student E uses the frequency as a way to describe the location of the median rather
than using weight range for the group. In part 2a and 2b, Student E is attempting to
identify the what weight range do the most males and most females fall in, but does
not quite highlight why the numbers 10 and 8 are significant. The student is also
really stating two facts, rather than linking the ideas in a comparison. Student E, like
many students, tries to put her own ideas into the comparison, by saying that the
males weigh more.
Student E
Eighth Grade – 2004
pg.
9
Student E, part 2
Student F uses the 10’s digit to simplify the amount of writing for finding the median,
but forgets to decode that shorthand when writing the answer.
Student F
Eighth Grade – 2004
pg.
10
Student F, part 2
Eighth Grade – 2004
pg.
11
The issue of what is important to note about data when making a comparison is a big
one. Work was often scored by whether the statement made any kind of comparison,
rather than did the comparison tell us something significant to note about the data. In
many cases students didn’t make comparisons at all, but just made descriptive
statements about something in the data or descriptive statements about their
assumptions based on their own first hand experience with animals.
Look at the statements below, based on the statements above how
would you score these statements:
1.
2.
3.
4.
5.
Female animals were heavier.
More males were caught.
The graphs had the same weights at the x-axis.
There are more male animals than female animals.
There are as many male animals that weigh 80-89g. as female animals
weighing 50-59 g.
6. The most male animals in any category is 10, while the most female animals
in any category is 8.
7. There’s a lot more males that weigh 50-59 grams than females that weigh 5059 grams.
8. There are twice as many females who way 30-39 grams as males who way 3039 grams in the woodland.
9. Males seem to, on average, weigh more.
10. Females seem to, on average, weigh less.
11. Female weights are much more varied.
12. The similarity is the weight for both range from 20-29 to 80-89.
13. The males have 10 animals in the 50-59 range, where the females only have
two.
14. None of the numbers of animals are the exact same for either males or
females.
15. Most of the females weigh less than the males, who are mostly in the middle.
16. The bar graph for the males starts low, then get high, then low again like a hill.
The bar graph for the females is random.
17. More males weigh 50-59 grams than females.
18. The number of males and females aren’t ever the same as the weight.
19. Overall, there are different numbers of animals.
Eighth Grade – 2004
pg.
12
Frequency Distribution for each Task – Grade 8
Grade 8– Animals
Animals
Mean: 2.13, S.D.: 1.63
2500
Frequency
2000
1500
1000
500
0
Frequency
0
1
2
3
4
5
2362
1861
2162
1916
1301
1156
Score
Score:
%<=
%>=
0
22.0%
100.0%
1
39.3%
78.0%
2
59.4%
60.7%
3
77.2%
40.6%
4
89.3%
22.8%
5
100.0%
10.7%
The maximum score available for this task is 5.
The cut score for a level 3 response, meeting standards, is 3 points.
Many students could make at least one comparison of the two data sets or find the
median for the male animals. More than half the students (61%) could either make
two comparisons of the data or find both of the medians. About 11% of the students
could meet all the demands of the task including finding the median for two different
data sets and making 3 comparison statements about the data sets. The challenge for
finding the median was because students needed to work with scaled intervals (e.g.,
20-30 grams) and with frequencies. 22% of the students scored no points on this task.
About 73% of the students with this score attempted the task.
Eighth Grade – 2004
pg.
13
Animals
Points
0
1
2
3
5
Understandings
Misunderstandings
About 73% of the students with
Many students were confused by
this score attempted the problem. having intervals of weights and
numbers for frequency. Instead of
finding the median interval, they found
the median frequency. Students
thinking this way might have put the
median as males 10, females 2 or
males 5, females 5.
Students could either make one
Some students confused mode and
comparison or find the median
median, so it was more difficult for
for males.
them to find the median for females.
The most frequent errors for median
were males 50-59, females 30-39 or
males 70-79, females 40-49.
Students could either make two
Many students struggled with the
comparison statements or find
meaning of the graphs. Several of the
both medians. The most
students associated the change in
common correct comparisons
weight as occurring over time. The
were more males at the median
males start out weighing not very
weight than females at the
much, then they gain weight, and later
median weight, more males than they lose weight. Other students tried
females, and more males at the
to impose their experience on the data,
50-59 and more females at the
males weigh more. Other students did
30-39 interval.
not make comparison statements.
Some animals weigh medium. Every
weight has to be bigger or less than the
others.
Students could find both
Students made many comparisons that
medians and make one
did not help readers make sense of the
comparison statement.
data or contribute to an understanding
of the data. Example, largest group of
males is 10 and largest group of
females is 8 or males and females have
about the same in 70-70 category.
Students could find the medians Students need to deepen their thinking
using intervals and frequencies
about what makes a good comparison,
and make three comparison
how does it give the reader information
statements about the data.
about the shape of the data and
highlight the important features of the
data such as mode.
Eighth Grade – 2004
pg.
14
Based on teacher observations, this is what eighth graders know and are able to do:
• Procedure for finding median
• Make simple comparisons about the height of the bars
Areas of difficulty for eighth graders, eighth graders struggled with:
• Knowing whether to use the intervals or the frequency numbers to find the
median
• Making comparisons that gave the reader a clearer idea of the findings or
meanings of the graphs
• Using mode, median, mean, and range in context
Questions for Reflection on Animals
Think about how your students learn about measures of central tendency. Do
they get frequent opportunities to discuss which statistic is a better
summarizing the information in the data?
• Do they discuss the shape of the data to see what trends or information the
graph is trying to convey?
• Do they deal with data in context or are most of their experiences dealing with
sets of numbers out of context?
• When students work with data, do they have opportunities to deal with scaled
intervals as well as work with frequency? How do you try to step up the
reasoning demands for eighth grades to reach a deeper level than that expected
at lower grade levels?
Examine your student papers to see how they responded to finding the median.
How many of your students:
•
50-59 male
50-59 fem.
70-79 male
40-49 fem.
60-69 male
40-49 fem.
10 male
2 female
5 male
5 fem.
Other
Can you figure what mathematical procedures they used to arrive at these
answers? What are the big mathematical ideas that they are missing about
intervals and frequencies? About using procedures for median?
•
Do you think students show an understanding of how median helps to
summarize or characterize a set of data or do students seems to be just making
a computation?
Look at the comparisons your students made. Do students seem to have an idea that
mathematical comparisons should help to explain, summarize, or illuminate important
ideas in the data or are they just performing an exercise?
Eighth Grade – 2004
pg.
15
Make a list of your students’ comparison statements. Can you sort them using the
flow chart below?
Is it a comparison?
Yes, its in the form of a comparison
It contributes information
to help make sense of the
of the data.
No, not a comparison.
Does not help clarify, summarize
or illuminate important ideas
about the data.
How does this help you think about the kinds of discussions and work students need
to have with data in order to get a better grasp of using data for sense-making or
decision making? For seeing graphs as a communication tool? For making
mathematical statements that help to communicate significant information about the
data or data sets?
Teacher Notes:
Instructional Implications:
Students need more work with data and graphs. Students at this grade level should
know how to find mean, median, mode and range for a set of data. Students should be
able to compare data sets by discussing these measures, as well as discussing the
shapes of the data. Research shows that students need to work with data in context
and should start to now be able to decide which statistical measure is best for
describing particular sets of data and why. Many students do not understand how to
talk mathematically about the data, what is significant or important to discuss.
Students need to see graphs as tools for making sense of and describing the world
around them, for conveying information, and for making predictions and decisions.
They should be starting to ask questions about whom the audience is for a particular
graph and what types of decisions it could be used for. Students should start to use
proportional reasoning to describe comparisons of data and for analyzing the effect of
scale on visual impact of the data. Students should have frequent opportunities to look
at graphs and describe the important features and summarize the main ideas conveyed
by the graph.
Eighth Grade – 2004
pg.
16