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Balanced Assessment Test –Geometry 2009
Core Idea
Task
Geometry and Measurement
Triangles
This task asks students to show their understanding of geometry by finding the number of
rotations of triangle to make a polygon. Students are asked to explain their thinking
using knowledge about angle relationships.
Geometric Formulas and
Hanging Baskets
Measurement
This task asks students to work with volumes of pyramids and hemispheres.
Mathematical Reasoning
Pentagon
This task asks students to use mathematical reasoning and proof to explain similarity and
congruence in embedded geometric figures. Successful students proved equality using
geometric properties, rather than relying on the visual diagram to assume relationships.
Successful students thought about all the steps needed to complete an argument.
Geometry and Measurement/ Algebra Circle Pattern
This task asks students to explore fraction patterns in the context of area of a circle.
Successful students were able to use algebra to calculate the change in area and note
salient features of the numeric pattern.
Geometry/ Measurement and
Circle and Squares
Mathematical Reasoning
This task asks students to use geometric properties to find the angles of geometric figures
formed by overlapping squares. Successful students could also make mathematical
arguments about parallel sides, using arguments about congruency and similarity.
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Geometry
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Triangles
This problem gives you the chance to:
• show your understanding of geometry
• explain your reasoning
This diagram shows a right triangle with
angles of 60° and 30°.
The second diagram shows five copies of the same triangle fitted together.
If this continues, a regular polygon with a hole in the middle will be formed.
Use the angles of the triangle to calculate the number of sides the regular polygon will have.
Explain all your reasoning carefully.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
5
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Triangles
4
2009 Rubrics Grade 10
Triangles
Rubric
The core elements of performance required by this task are:
• show your understanding of geometry
• explain your reasoning
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answers: 12 sides
2
Gives correct explanation which may involve the external angles of the
polygon.
3
Partial credit
Incomplete explanation
(1)
Total Points
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5
5
Triangles
Work the task and look at the rubric. What are the key mathematical ideas that a student needs to
use to be successful on this task?______________________________________
Look at student work. How many of your students put:
12
10
16
Other
For this task there were 3 solution strategies used by successful students. How many of your
students used these strategies:
• Exterior angles to add to 360° (360°/30°)?________________
• Finding a common multiple of 180° and 150°(a regular polygon has equal interior
angles)?____________
• Using the formula 180(n-2)/n = size of the interior angle for a regular
polygon?_____________
When you looked at student work, what was missing in their explanations that you would like to
have seen?
What are some of the strategies used by unsuccessful students? What misconceptions do these
strategies show?
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Looking at Student Work on Triangles
Student A is able to use a formula for exterior angles to solve the task. The student uses the diagram
as a tool to mark in sides, extend angles, and think about the given information.
Student A
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While Student A seemed to just know a formula, Student B is able to arrive at the correct answer by
using prior knowledge to come up with the same solution.
Student B
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Student C uses a different approach. The student knows that polygons can divided into an even
number of triangles to find the total degrees in the polygon. Using this knowledge and the size of
the interior angles, the student can find the number of sides.
Student C
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Student D uses a similar solution strategy to reason about the number of sides, developing the logic
into a formula.
Student D
Student E is able to pull the formula from memory.
Student E
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Student F is able to find the answer, but is unable to fully describe how he got the answer. The
student may have used the diagram to guess about the number of triangles or may have used a
successful strategy.
Student F
Student G misinterprets the diagram, putting the 60° angle in the wrong location. The student then
tries to work with the wrong interior angle.
Student G
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Geometry
Student Task
Core Idea 4
Geometry and
Measurement
Task 1
Triangles
Show understanding of geometry and explain reasoning in a problem
situation.
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.
Mathematics of the task:
• Reasoning about interior and exterior angles
• Understanding that the sum of the exterior angles for a polygon is always 360°
• Reasoning that the interior angles of a polygon are always a multiple 180°
• Developing a convincing argument or justification
Strategies used by successful students:
• 50% used exterior angles 360/30
• 34% used the formula- 180 (n-2)/n
• 6% used the interior angles 1800/12 = 150
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The maximum score available for this task is 5.
The minimum score for a level 3 response, meeting standards, is 3 points.
Most students, 89%, could find the number of sides for the figure make of rotating triangles. Many
students, 80%, could meet all the demands of the task including explaining how they used the given
measurements to find the number of signs. 8% of the students scored no points on this task. 75% of
the students in the sample with this score attempted the task.
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Triangles
Points
0
2
5
Understandings
Misunderstandings
75% of the students with this
score attempted the task.
A common error was to think the new figure
would have only 10 sides. Students may
have guessed using the diagram or
incorrectly assigned the 60° angle to the
wrong part of the triangle.
Students could find the number Students could not put together a complete
of sides for the new figure.
explanation of how they figured it out.
They may have just made some marks on
their diagrams and used no words. They
may have left out steps in making the
justification.
Students could reason about
fitting together copies of a 3060 degree right triangle to
make a regular polygon.
Students could find the number
of sides for the polygon and
give a convincing justification
for the solution.
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Implications for Instruction
Students need help with diagram literacy. Some students assigned the 60° angle to the smallest
angle of the triangle. They may have then used a correct process for finding the solution, but got an
incorrect answer because of the first incorrect assumption.
Students needed to understand something about the sum of interior or exterior angles to make a
convincing argument. Some students did not know how to complete an argument. Students should
have frequent opportunities to make convincing arguments.
Ideas for Action Research
While most students did very well on this task, it still is interesting enough to have students
investigate some misconceptions or solve the problem for different perspectives.
Re-engagement – Confronting misconceptions, providing feedback on thinking, going deeper
into the mathematics. (See overview at beginning of toolkit).
1. Start with a simple problem to bring all the students along. This allows students to clarify
and articulate the mathematical ideas.
2. Make sense of another person’s strategy. Try on a strategy. Compare strategies.
3. Have students analyze misconceptions and discuss why they don’t make sense. In the
process students can let go of misconceptions and clarify their thinking about the big ideas.
4. Find out how a strategy could be modified to get the right answer. Find the seeds of
mathematical thinking in student work.
I might start with the misconception that the smallest angle is 60° and show the work of Student G.
What is the student thinking? Where do the numbers come from? If the angle really were 60
degrees would the solution be correct? Why or why not? As with any re-engagement, I would want
to give students first some individual think time, then pair/share before going to a whole group
discussion. I would put think/discuss break after the first two questions and the second set of
questions.
Next I might give a partial strategy to have students rethink the task. I might use some of the work
from Student A. I want would to show only a snippet of the work to really make students think hard
about what was going on. I might start with the calculations and have students try to guess where
the numbers came from and what the student was thinking by showing:
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or I might start with the diagram and ask students how this might lead to a solution:
You might try each version with a different class and compare how the discussions are different.
Which start provided the most interesting discussions? Why? Do you think the prompt was more
interesting or was it the students in the class?
Now I might use the idea of the formula. I might introduce the idea from the work of student D, by
saying that I overheard someone say that the sum of the interior of any polygon is equal to (n-2)
180. What does this mean? Can you give me some examples to convince me that this is true?
After some discussion I might follow up with part of the work of Student E.
I am confused when I look at this formula. Where does the 150 come from? Why is the student
dividing by n? I want students to think about the formula from a different perspective and relate it to
the problem at hand. I am hoping to provide a small amount of disequilibrium.
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Next I might use a statement by Student D:
What is the student thinking? How would this help the student? What might the student do next?
Next I might show the work of Student B:
If the student didn’t know the formula, how might this information help the student find the solution?
What do you think the student did next?
This re-engagement lesson is about letting students look at a problem from different perspectives or
points of view. It helps students follow different reasoning paths. Why is this important for
students? How does it help further their skills?
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Hanging Baskets
This problem gives you the chance to:
• work with the volumes of pyramids and hemispheres
Hugo sells hanging baskets. They have different shapes and sizes.
Hugo needs to know their volume so that he can sell the correct amount
of potting compost to fill them.
1. There are pyramid shaped hanging baskets with a
square opening at the top.
The square has sides of 25 cm and the basket is 30 cm deep.
25 cm
30 cm
Volume of a pyramid = 1/3 area of base x height
Calculate the volume of this basket.
Show your work.
__________________cm3
2. Hugo makes a tetrahedron shaped basket with the same volume
as the square based pyramid.
He decides to make it with an opening that is an equilateral
triangle with sides that measure 30 cm.
Find the area of the equilateral triangle.
_____________ cm2
d cm
30 cm
How deep will this basket have to be?
Show how you figured it out.
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__________________cm
Hanging Baskets
18
3. Hugo also sells hemispherical baskets with a diameter of 30 centimeters.
Volume of a sphere = 4/3πr3
30 cm
Calculate the volume of this basket.
Show your work.
__________________cm3
9
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Hanging Baskets
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Hanging Baskets
Rubric
• The core elements of performance required by this task are:
• work with volumes of pyramids and hemispheres
Based on these, credit for specific aspects of performance should be assigned as follows
1.
points
Gives correct answer: 6250
1
Shows correct work such as: 252 x 30 / 3
1
section
points
2
2.
Gives correct answer: 390 ± 1
2
Gives correct answer: approx 48
1 ft
Shows correct work such as: 390/3 = 130
6250 / 130
3.
1 ft
Gives correct answer: 7069 or 2250π
4
2
Partial credit:
14137 or 4500π, 56549 or 18000π
(1)
Shows work such as: 2/3 π 153 or 4/3 π 153 or 2/3 π 303
1
3
9
Total Points
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Hanging Baskets
Work the task and look at the rubric. What are the key mathematical ideas that students
need to solve this task? ____________________________________________________
How often do students in your class have opportunities to decompose complex figures?
What do you think they understand about 3-dimensional shapes?
When working with formulas, are expectations for students to understand why they make
sense? What kinds of activities or questions help promote this understanding?
Look at student work for part 1, finding the volume of the pyramid. How many of your
students:
• Correctly calculated the area (6250 cm3)?_________________
• Gave an answer of 250 cm3? ____________________
What are students not understanding about the formula to arrive at this answer?
Now look at student work on part 2, finding the area of an equilateral triangle. How
many of your students put:
390
780
450
375
318
Other
How might they get an answer of 780? 450? 375? How are these misconceptions
different?
Now look at student work on the second part of 2, finding the height of the basket. How
many of your students put:
48
26
25
24
No answer
Other
What were some of the misconceptions leading to these errors?
Now look at student work in part 3. How many of your students put:
• A correct volume of 7069 or 2250π cm3 ? _____________________
• Forgot to divide the answer by 2 ( half a sphere)?________________
• Some other answer?_________________________
Look at some of these errors to see what caused students difficulty.
• Did they use the wrong radius?
• Did they square the radius instead of cubing the radius?
• Did they make arithmetic errors?
• What other things did you notice?
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Looking at Student Work on Hanging Baskets
Student A lays out the calculations in a clear, orderly fashion. The student marks the
diagrams and makes additional diagrams to clarify the thinking, showing the
decomposition of the shapes.
Student A
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Student A, continued
Student B confuses the height of the basket with the height of the triangle on the base of
the pyramid. The student does not use the information that the second basket should have
the same volume as the first basket.
Student B
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Student C does not understand what the height is for finding the area of the triangle and
uses the side instead. Do you think the student knows the Pythagorean theorem? In the
second part of 2, finding the depth of basket, the student knows that the volume needs to
be the same as the volume above, but assumes the height is 30 and ignores the rest of the
formula. What strategies might help this student think through the process? What
questions would you ask to get the student to rethink their process?
Student C
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Student D confuses the side of the base for the area of the base in calculating the volume
in part 1. In the second part of 2, the student uses the area of the base for basket one,
instead of the volume. What suggestions might you make to help this student organize his
work? Where would you go next with this student?
Student D
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Student E has trouble sorting out what the base is for each figure. What questions could
you pose for this student to push his thinking?
Student E
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Geometry
Student Task
Core Idea 4
Geometry and
Measurement
Core Idea 3
Algebraic
Properties and
Representations
Task 2
Hanging Baskets
Work with the volumes of pyramids and hemispheres.
Apply appropriate techniques, tools, and formulas to determine
measurements.
• Understand and use formulas for the area, surface area, and
volume of geometric figures, including spheres and cylinders.
• Visualize three-dimensional objects from different
perspectives and analyze their cross sections.
Represent and analyze mathematical situations and structures
using algebraic symbols.
• Solve equations involving radicals and exponents in
contextualized problems such as use of Pythagorean Theorem.
Mathematics of this task:
• Recognizing the base in a pyramid
• Being able to decompose a formula into smaller parts to derive the needed
numbers such as area of the base
• Understanding that the height of a triangle is not always equal to the side length
• Being able to use Pythagorean theorem in a practical application
• Understanding equality to set up an equation to make the volume of one figure
equal to the volume of another
• Understanding circles and spheres, recognizing the difference between radius and
diameter, a half sphere and a whole sphere
Based on teacher observation, this is what geometry students know and are able to do:
• Use the formula the find the volume of a sphere
• Find the volume of a pyramid with a square base
Areas of difficulty for geometry students:
• Confusing the base of the square for the area of the base in the pyramid
• Understanding that the side length is not the height of an equilateral triangle
• Understanding that the area of a 3-dimensional figure requires an area of the base
times the height
• Using the height from the base triangle as the height for the pyramid
• Using the area from part 1 instead of the volume from part 1 to solve in part 2
• Forgetting that the formula was for the area of a sphere and that the figure was a
hemisphere
• Squaring the radius when finding the volume of a sphere instead of cubing the
radius
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The maximum score available for this task is 9 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, 97%, could use the formula to find the volume of a sphere. 91%
recognized that the figure was a hemisphere and divided the volume by 2. Many students,
70%, could either find the volume of a pyramid with a square base or use Pythagorean
theorem to find the area of an equilateral triangle and find the volume of a hemisphere.
20% of the students could meet all the demands of the task, including finding the volume
of a pyramid with a square base, finding the height of a triangular prism with an
equilateral triangle for the base and volume equal to the square pyramid, and find the
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volume of a hemisphere. 1.6% of the students scored no points on this task. All the
students in the sample with this score attempted the task.
Hanging Baskets
Points
Understandings
All the students in the sample
0
with this score attempted the
task.
2
3
Students could find the volume
of a sphere.
Students could calculate the
volume of a hemisphere.
5
Students could calculate the
volume of the hemisphere and
either find the volume of the
square pyramid or find the area
of an equilateral triangle using
Pythagorean theorem.
8
Students could compose and
decompose geometric shapes
and formulas to find the volume
of a square pyramid, find the
area of an equilateral triangle,
find the height needed to make
the volume of a triangular
pyramid equal to the volume of
the square pyramid, and
calculate the volume of a
hemisphere.
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Misunderstandings
Students had trouble using the formula for
volume of a sphere. Some students
squared the radius instead of cubing the
radius. Some students used the diameter
instead of the radius. Students had trouble
calculating with the 4/3.
26% of the students didn’t recognize it that
it was a hemisphere.
Students couldn’t calculate the volume of
the pyramid. 39% multiplied the base of
the square times the height of the pyramid
instead of the using the area of the square.
Many students, 14%, thought height of the
triangle was the side length. 10% forgot
that the area of a triangle is length time
width divided by 2. Students also struggled
with finding the height of the triangular
pyramid. They used the area of the base in
part 1 instead of the volume (22%). They
thought the height for the pyramid was the
same as the height of the triangular base
(18%).
29
Implications for Instruction
Students need more opportunities to compose and decompose figures. Students need to
be able to identify what they know and what they need to find out in complex problems.
Students need rich tasks, using a variety of past knowledge and skills. Will they know to
pull out Pythagorean theorem or can they use it only during the chapter it is introduced?
Students seem to struggle with ways to organize their information, when dealing with a
multi-step problem. Students need opportunities to work tasks with longer reasoning
chains. Using labels or diagrams might be helpful tools to track their thinking process.
Students should learn material, such as formulas, for sense making, so that they can see
how the parts of the formula relate to the parts of the shape.
Some students need work or review on basic computational skills, such as understanding
exponents and working with fractions in multiplication.
Ideas for Action Research
Students in this task had difficulty understanding the formulas. They didn’t seem to be
able to identify the base or understand why that is critical to making sense of the formula.
One of the SVMI Resources are some video case studies developed by Cathy Humphreys
and Jo Boaler, Middle School Mathematics Teaching Cases. In the lesson students first
make sense of the formula for volume of a rectangular prism. Where do the letters come
from? How do they relate to other geometric ideas? Then students are asked if they
could use some of those ideas to come up with ideas about how to find the volume of
cylinder. This would be a good case to view and discuss with colleagues.
Then working together as a team, how could you further develop the lesson ideas to help
students make sense of the formulas in Hanging Baskets? What questions would you
pose? What are the essential understandings that you want geometry students to have
about the formulas?
One of the critical ideas in the Japanese approach to understanding geometry is the idea
of composing and decomposing shapes. In this task, students are not understanding
where the base of the figures is located and what are the measurements needed to think
about finding the area. So, after developing a lesson on making sense of volume
formulas for 3-dimensional shapes students may also need to do a re-engagement lesson
with the Hanging Basket task and try to decompose the figures.
For example, a teacher might show the work of Student E and pose the question:
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I saw this work on a student paper. What do you think the student was thinking? What
was the student confused about? Give students individual think time, then allow them to
pair/share. The idea is to maximize student thinking and talking. Give all students a
chance to verbalize their ideas before starting a class discussion. In this case, the
question is designed to help them recognize a misconception and why and doesn’t work.
The teacher might follow up with questions, such as:
What is the base of this shape? What are the dimensions of the base? How can we find
the area?
Next the teacher might look at work on part 2.
Again, ask the students what is going on with this work. Where are the numbers coming
from? What is the base of this shape? What are the dimensions of the base? How can we
find the area?
How does this type of probing help students develop their ideas about formulas and
decomposing shapes?
Now work with colleagues to look at other pieces of student work. How could you use
snippets of the work to help further the discussion? What big mathematical ideas do you
want students to develop in this follow up session? What generalizations do you want
students to have at the end of the lesson?
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Pentagon
This problem gives you the chance to:
• show your understanding of geometry
• write mathematical proofs
The diagram shows a regular pentagon.
0
108°
1. Explain why each inside angle of the regular pentagon is 108°.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
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Pentagon
32
P
This diagram shows a pentagram,
a shape of mystical significance.
It is centered on a regular pentagon.
B
A
T
Q
E
C
D
S
R
2. Show that triangle PCE is similar to triangle ACE.
3. Prove that triangle PAB is congruent to triangle DAB.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
8
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Pentagon
33
Pentagon
Rubric
The core elements of performance required by this task are:
• understanding of geometrical situations
• construction of mathematical proofs and explanations
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct explanation such as:
Recognition that sum of exterior angles is 3600
Calculation of each exterior angle (720)
3
Calculation of each interior angle (1800 - 720)
Accept alternative methods such as: (5 – 2)180 = 108
5
Partial credit
Incomplete explanation.
2.
(2)
Determines the measures of the angles in each triangle. (720, 720, 360)
Recognition that triangles are therefore similar.
3
Partial credit
Incomplete explanation
3.
3
(2)
Recognition and explanation of similarity of triangles
Explanation that as triangles have a corresponding common side, they must
be congruent
2
2
8
Total Points
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Pentagon
Work the task and look at the rubric. What are the key mathematical ideas being
assessed in this task?___________________________________________________
How do you help students build up their ability to make logical arguments or
justifications?
How do you help students distinguish between what they know and what appears to be
true from looking at the diagram?
Look at student work on part 2, showing similarity. How many of your students:
• Made a complete justification?____________________________
• Assumed sides were proportional or assumed parallel sides, without
justification?________
• Assumed the triangles were isosceles without first making convincing arguments
about the sides or angles?____________
• Assumed the stars trisected the angle without justification?______________
• Made statements about the diagonals without justification?_____________
Now look at student work on part 3, proving congruency. How many of your students:
• Made a complete justification?_________________
• Assumed the triangles were isosceles without first making a convincing argument
about angle measure?_________________
• Used some type of looks like argument?________________
• Assumes angle measurements without justification?_________________
How can you plan a class discussion to help bring up some of these flaws in justification
to help students see the faulty logic in their arguments? How can you use student work to
help build the level of proof within the classroom?
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Looking at Student Work on Pentagon
Student A uses a formula to find the measure of the interior angle of a regular pentagon.
The student is able to present a concise argument, supported by calculations and the
diagram to prove similarity and congruence.
Student A
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Student B uses two solution paths to make a convincing argument in part 1. The student
uses exterior angles and supplementary angles to show that it is 108 degrees. The student
also uses a formula to check the solution. In part 2 and 3 the student uses justifications
by how things look, “if I flip this piece” they will be the same. While the statements may
be true, this is not a sufficient justification mathematically. How would you help this
student? What types of experience does this student need?
Student B
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Student B, continued
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Student C is another example of a convincing argument for part 2 and 3 of the task.
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Student D starts a convincing argument in part 2, but doesn’t explain why the information
is helpful or complete the argument. How could you use the information developed to
help prove the similarity? In part 3 the student is assuming sides are equal by
congruency, rather than proving the sides are equal to show congruency. It’s a circular
argument.
Student D
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Student E uses the diagram to show the angles, but there is no explanation of how the
numbers were derived. Did the student assume the angles were trisected? Did the student
assume that triangle PAB is isosceles to get the angles or did the student use
supplementary angles? There is too much unexplained to make a complete justification.
What suggestions might you make to help this student?
Student E
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Student F makes assumptions about parallel lines. While this appears to be true in the
diagram, there is no supporting justification. How do we help students suspend what they
see in order to develop convincing arguments?
Student F
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Student G does not use clearly defined angles, making the arguments difficult to follow.
There is no justification for why angle A is equal and congruent to angle C. What might
be your next steps to help this student? What types of experience does the student need?
The student knows the appropriate theorems for similarity and congruency but can’t build
the logical progression of ideas to support the theorem.
Student G
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Student H makes comments about diagonals, assuming they are equal and assuming they
are parallel to the sides. While these may be true, they need to be proved. These are not
givens. What are the points valued in this explanation? What points do you think need
further clarification?
Student H
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Student I has no points on parts 2 and 3. The student puts in a lot of effort, including
making a new diagram to clarify the ideas. Where does the student’s thinking break
down? What further experiences does the student need?
Student I
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Geometry
Student Task
Core Idea 2
Mathematical
Reasoning and
Proof
Task 3
Pentagon
Show understanding of geometry and write mathematical proofs.
Employ forms of mathematical reasoning and proof appropriate
to the solution of the problem at hand, including deductive and
inductive reasoning, making and testing conjectures and using
counter examples and indirect proof.
• Show mathematical reasoning in solutions in a variety of
ways, including words, numbers, symbols, pictures, charts,
graphs, tables, diagrams and models.
• Explain the logic inherent in a solution process.
• Identify, formulate and confirm conjectures.
• Establish the validity of geometric conjectures using
deduction; prove theorems, and critique arguments made by
others.
Mathematics of this task:
• Show why the interior angle of a regular pentagon is 108°
• Decompose a complex figure
• Make a convincing justification about similarity and congruency using given
information
• Understand the difference between what “looks to be true” and what needs to be
proven
Based on teacher observation, this is what geometry students know and are able to do:
• Find the interior angle of a regular figure
• Know the theorems for similarity and congruency
• Identify corresponding parts in similar and congruent figures
• Use properties of supplementary angles
• Use reflexive property
Areas of difficulty for geometry students:
• Assuming sides of a triangle are isosceles
• Assuming parallelism
• Assuming angles within a figure are divided equally
• Assuming properties of diagonals with making a justification
• Using properties of similarity or congruency to make the proof of similarity or
congruency, e.g. if the triangles are congruent the sides are equal, the sides are
equal because its congruent, therefore its congruent
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The maximum score available for this task is 8 points.
The minimum score for a level three response, meeting standards, is 5 points.
Most students, 96%, could explain why the interior angles of a regular pentagon are 108°.
A little less than half the students, 42%, could explain why two triangles are similar with
convincing justification. 23% could meet all the demands of the task including proving
congruency of two triangles. About 2% of the students scored no points on the task.
None of the students in the sample had this score.
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Pentagon
Points
Understandings
No
students
in the sample had
0
3
5
6
8
this score.
Students could make a complete
explanation for size of the
interior angle of a regular
pentagon.
Students could give the size of
the interior angles of a regular
pentagon and make a convincing
argument for congruency or a
partially correct argument for
similarity.
Students could explain the
interior angles of a regular
pentagon and make a convincing
argument for similarity.
Misunderstandings
Students struggled with making a complete
argument for similarity. 17% made
assumptions about isosceles triangles
without justification. Students also made
assumptions about parallelism, trisecting of
angles, properties of diagonals without
justification. In some cases students used
vertical angles properties for non-vertical
angles
Students struggled with arguments for
congruency. Students made assumptions
about isosceles triangles, without giving a
justification. Students assumed angles
were trisected without justification. Many
knew that AB =AB, but didn’t know the
next steps in completing the argument.
Students could make convincing
arguments with justification for
the interior angles of a regular
pentagon and prove similarity
and congruency for triangles
embedded in a complex figure.
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Implications for Instruction
Students at this grade level need to develop their ability to make a convincing chain of
reasoning to complete a logical argument. Students need to be able to distinguish what
they know and what needs to be proved. Students should understand that diagrams can
not be relied upon as facts. Just because a shape appears to be an isosceles triangle, there
needs to be a justification for the assertion. Students can’t assume that lines are parallel
or that angles are bisected or trisected. The heart of geometry is that these things are not
given and need to be proved.
Some students are making circular arguments. They take the properties of similarity or
congruency to prove things about the sides or angles of given shapes. Then because
those angles or sides are equal, use it to prove similarity or congruency.
The cognitive demands in developing these types of arguments are high. Students need
to be able to examine the arguments of others and decide where the proof is missing. It is
often easier to look critically at the work of someone else to see the flaws in thinking,
than it is to see the errors in your own work. Class discussions about different chains of
reasoning help students develop stronger internal guidelines for making a justification.
Ideas for Action Research
Re-engagement – Confronting misconceptions, providing feedback on thinking,
going deeper into the mathematics. (See overview at beginning of toolkit).
1. Start with a simple problem to bring all the students along. This allows students to
clarify and articulate the mathematical ideas.
2. Make sense of another person’s strategy. Try on a strategy. Compare strategies.
3. Have students analyze misconceptions and discuss why they don’t make sense. In
the process students can let go of misconceptions and clarify their thinking about
the big ideas.
4. Find out how a strategy could be modified to get the right answer. Find the
seeds of mathematical thinking in student work.
In this task students had difficulty making justifications and jumping to conclusions
without backup information. This is a good lesson for students to look at work and see
where these flaws occur and discuss how to improve the justifications. Secondly students
had difficulty using the diagrams effectively. This examination of thinking allows
students to better develop their own internal values for what constitutes a good argument.
In an article, “Keeping Learning on Track: Formative assessment and the regulation of
Learning” by Dylan Wiliam feedback is the most important factor of keeping students
interested in learning and helping them improve substantially the quality of their
performance. A re-engagement lesson can help give that pointed helpful feedback.
In looking at the task, the re-engagement work might start by examining explanations in
part 2, proving similarity. The teacher might start by have several copies of the diagram
available for students as think sheets, places to put down ideas.
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Now the teacher might pose a question. “I noticed that some students in my other class
had difficulty using the diagrams to show their information or keep track of what they
had proved. For example, Paul wrote, “Pentagon ABECD is a regular pentagon and by
the definition of a regular polygon all angles and sides are equal.” Could you show Paul
how to mark that on his diagram?” Give students an opportunity to work individually
and then discuss their ideas in pairs. The idea is to maximize the amount of student
conversation.
“Next Paul wrote that triangle ΔDCE and ΔABE and ΔDAC are isosceles triangles. How
would Paul mark this information on the diagram? What new information does that give
Paul to help him continue his justification?” Here students need to think about the
logical conclusions that can be made from the statement. The teacher might press to see
if they can quantify any of the information. After a good discussion on this piece, ask
students to see if they can figure out what Paul should do next to prove that the triangles
on similar. While the original work by Student D stopped here, the work can be pushed
to a solution. This allows students to see the seeds of mathematical thinking, without
focusing that the original work was incomplete.
Another line of questioning might start with the work of Student E. The teacher might
say, “I spilled some coffee on Sara’s paper. She said that ΔPCE has angles of 36°, 72°
and 72°, but I can’t read how she figured it out. Can you help me decide what was under
the stain? How do you think Sara might have come to this conclusion?” The point is that
many students gave this information with no justification, but during the discussion we
hope they start to see why this is important. Follow up with saying that Sara’s next
statement is that ΔCEA has angles of 36°, 72°, 72° and ask them to again help you figure
out how Sara might have reached this conclusion. After time for discussion, ask them
how this information might have helped Sara make the argument for similarity. What
might her ending statement have been?
Now that students have had a couple of opportunities to make complete arguments or
justifications, give students some incomplete or unsatisfactory arguments and see if they
can identify why they are incomplete or faulty. For example say, “I say this explanation
for part 2 on one student’s paper. “ΔPAB≅ΔCEA if you flip the triangle onto it, so by
SSS they are equal and congruent.” Do you think this is a convincing argument? Why or
why not?” Give students a chance to work by themselves and then in pairs before
opening up the discussion to the class.
Now look through some of the papers from your students or examples from the toolkit.
How might you develop a follow up discussion for part 3? What are some of the key
mathematical ideas that you want to highlight in the discussion?
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Circle Pattern
This problem gives you the chance to:
• explore fractions in context
Here is a developing circle pattern.
Here is one black circle.
Two white circles of half the radius have been
added to the diagram.
1. Show that the fraction of the diagram that is now
black
is one half.
_________________________________
_________________________________
_________________________________
_________________________________
Four black circles have now been added.
2. What fraction of the diagram is now black?
_________________________________
_________________________________
_________________________________
_________________________________
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3. Fill in the table to show what happens as the
pattern continues.
Pattern
Black fraction
White fraction
One black circle
1
0
Two white circles
1
2
1
2
Four black circles
Eight white circles
Sixteen black circles
4. Write a description of what is happening to the black and white fractions as the pattern
continues.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
9
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52
Circle Pattern
Rubric
• The core elements of performance required by this task are:
• explore fractions in context
explore fractions in context
section
Based on these, credit for specific aspects of performance should be assigned as follows
points points
1.
2.
Gives correct explanation such as:
Let radius white circle be r, then area = πr2
Radius black circle is 2r, then area = 4 πr2
Area of two white circles is 2 πr2
2
Gives correct answer: 3/4
2
2
2
3.
Gives correct answers: 3/4, 1/4, 5/8, 3/8, 11/16, 5/16
3
Partial credit
4 or 3 correct two points
2 correct one point
4.
(2)
(1)
Gives correct explanation such as:
Each time a half of the previous fraction is added or subtracted from the
black fraction.
(The limit of the black fraction is 2/3.)
Partial credit
For a partially correct explanation that either addresses change by half or the
oscillating adding or subtracting.
3
2
(1)
2
Total Points
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53
Circle Patterns
Work the task and look at the rubric. What are the big mathematical ideas being assessed?
How do you build in opportunities for students to maintain algebraic and arithmetic skills
within the context of geometry?
As students develop increasing skills at making geometrical arguments and justifications, how
can we help them also work on noticing and describing more sophisticated and complex
mathematical patterns?
Look at student work on part one, comparing the areas of white and black circles. How many
of your students:
• Made a complete justification using area formula?________________
• Did not square the 1/2 when finding the area of the white circle(s)?_________
• Thought the area for white was 1/4?__________
• Tried an argument based on the 2 radius were 1/2 the diameter?___________
Look at student work on part three, completing the table. How many of your students could:
• Fill in the table correctly?___________
• Jumped to an incorrect pattern rather than continuing the mathematical exploration, so
gave answers for 8 white circles, such as:
o 1/2 and 1/2
or for 16 black circles:
• 9/16 and 7/16?_________ or 3/4 and 1/4?_____________
What types of misconceptions may have lead to these errors?
Did you notice any other common mistake? Why do you think students struggled with this
part of the task?
Look at part 4, where students needed to describe the pattern in the table. How many
students:
• Gave a complete explanation?_______
• Talked about the denominator doubling, halving, or being the same as the
circles being added?___________
• Thought the black was always increasing?__________• Talked about the alternating decrease and increase, but without trying to
quantify the pattern?__________
• Gave descriptive information, such as black is always larger?________
Pick out two or three good explanations. What were the qualities that you valued in a good
explanation?
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Looking at Student Work on Circle Pattern
Student A gives a complete explanation of why the area is 1/2 in part1. Notice that for both
part 1 and 2 the student changes the diameters of the black circle to avoid using fractions. The
student then sees and quantifies a pattern in part 4.
Student A
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Student A, continued
Student B uses an incorrect formula for finding area in part 1 forgetting to square the radius,
but is able to see the pattern of halving and use it to correctly complete the rest of the task
with little or no calculation.
Student B
Student B, continued
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Student C is able to do the calculations in part 1,2 and continue it in the table for 8 white
circles. However the student makes a mistake for 16 circles. The student then finds an
incorrect pattern for relationship between the black and white circles based on the
denominator.
Student C
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Student D, like many students, tries to find a pattern to soon without doing the calculations.
The student uses an alternating pattern. What might the student have been considering to get
this conclusion?
Student D
Student E does not understand the effect of squaring a number in the area formula. In part 4
the student notices some general attributes in the pattern, but doesn’t quantify how to continue
the pattern.
Student E
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Student F does not look at the full pattern when making her conclusion. She is thinking about
a constant difference rather than a changing difference. What are some of the mathematical
reasons why this pattern does not have a constant difference?
Student F
Student G does not notice the alternating size of the fractions. Why might a student think only
the black is growing?
Student G
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Geometry
Student Task
Core Idea 3
Algebraic
Properties and
Representations
Core Idea 2
Mathematical
Reasoning
Core Idea 4
Geometry and
Measurement
Task 4
Circle Pattern
Use algebra patterns to explore a geometric situation.
Explore fractions in context
Represent and analyze mathematical situations and structures
using algebraic symbols.
• Solve equations involving radicals and exponents in
contextualized problems.
Employ forms of mathematical reasoning and proof appropriate
to the solution of the problem at hand, including deductive and
inductive reasoning, making and testing conjectures and using
counter examples and indirect proof.
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.
Mathematics of the task:
• Use area formula to make a generalization for any size circle
• Notice a pattern about area using fractional parts
• Be able to look at features of a pattern to make a generalization about how it grows
Based on teacher observations, this is what geometry students know and are able to do:
• Find the black fractional area for a circle with 4 small black circles
• Fill in some of the numbers in the table
• Find some key attributes of the pattern
Areas of difficulty for geometry students:
• Using the area formula to prove that in the first case, the white circles are half the area
of the black circle
• Completing all the lines of the pattern, often because of generalizing the pattern too
quickly
• Generalizing the growth pattern for the areas of the circles
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The maximum score available for this task is 9 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, 97%, could find the fraction of black when there were 4 black circles and do
some of the calculations in the table. Many students, 84%, could explain why the area in part
1 is 1/2, find the area in part 2, and fill in part of the table. More than half the students, 54%,
could fill out the entire table. 17% of the students could meet all the demands of the task
including generalizing about how the pattern grows. 2% of the students scored no points on
this task. None of the students in the sample had this score.
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Circle Pattern
Points
0
Understandings
None of the students in the
sample had this score.
3
Students could find the fraction
of black in the pattern with 4
black circles and fill in part of
the table.
5
Students could explain why the
beginning pattern was 1/2 and
why the second pattern was 3/4
and fill in part of the table.
7
Students could calculate the
pattern for all the stages and fill
in the table.
9
Students could use the area
formula to reason about a
pattern of inscribed circles and
calculate the fractional
relationships between to the two
colors of circles. Students
could also make a
generalization about how the
pattern grows, noting that the
growth number is always 1/2
the previous amount.
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Misunderstandings
About 8% of the students had difficulty
with part 2 of the task, finding the fraction
of black with 4 black circles. There was no
pattern in their errors.
20% of the students thought the pattern
went back to 1/2 for 8 white circles. 12%
thought the pattern was 3/4 – 1/4 for 16
black circles. 10% thought the pattern was
9/16 – 7 /16 for the 16 black circles.
Some students did not square the radius
when using the area formula. Some
students used an argument about the two
white radius equaling the black radius, but
couldn’t complete the argument.
They had difficulty generalizing the
pattern. 16% noticed the alternating
pattern: up, down. 8% noticed that the
denominator is always the same as the
number of the smallest circles. 10% noticed
that the denominator doubled or was half as
large each time. 10% thought the black was
growing each time.
64
Implications for Instruction
Students need to be exposed to a wide variety of pattern types. This type of task allows
students to develop their budding reasoning skills, while maintaining their algebraic skills.
Students need to bring the same level of detail to sequences and progressions that they use for
developing a proof.
Ideas for Action Research
Re-engagement – Confronting misconceptions, providing feedback on thinking, going
deeper into the mathematics. (See overview at beginning of toolkit).
1. Start with a simple problem to bring all the students along. This allows students to
clarify and articulate the mathematical ideas.
2. Make sense of another person’s strategy. Try on a strategy. Compare strategies.
3. Have students analyze misconceptions and discuss why they don’t make sense. In the
process students can let go of misconceptions and clarify their thinking about the big
ideas.
4. Find out how a strategy could be modified to get the right answer. Find the seeds of
mathematical thinking in student work.
This task lends it self to a re-engagement lesson. There are some very common patterns in the
incorrect responses, so it is worth taking time to confront those common errors explicitly.
I might start by having students rework part one of the task proving that the area of the black
part of the diagram in 1/2. Then I might pose the question: “Can you find the area of black in
the second diagram in 2 different ways?” Or I might say, “Randy says that he can find the
area of the black without calculating. Do you think this is possible? What do you think Randy
did?” After some discussion, I might have students try to convince me that Randy was correct
by using formulas.
Next I would have students look at some of the incorrect patterns. For example:
What might the student be thinking? Do you agree or disagree with this student? Why?
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Then I might have students look at this pattern:
What is the student thinking? Why doesn’t this pattern work?
Next I might say, “Antonia says that each time the pattern increases or decreases by half of
the previous increase or decrease. What do you think Antonia’s table looks like? What does
Antonia mean by previous increase or decrease?”
Look at some work from students in your class. What other ideas or pieces of student work
could you use to pose questions to the class? What are the big mathematical ideas that you
want students to have regarding this pattern? What generalizations should they be able to
make?
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Floor Pattern
This problem gives you the chance to:
• show your understanding of geometry
• explain geometrical reasoning
The diagram shows a floor pattern.
In the floor pattern, the shaded part is made by
overlapping two equal squares.
The shaded shape can also be seen as a set of eight equal kites.
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1. Find the measures of all four angles of the kites.
Explain how you obtained your answers.
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
________________________________________
A
2. Two of the kites can fit together to make a hexagon.
B
Prove that the quadrilateral ABCD is a parallelogram.
D
C
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
9
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68
Floor Pattern
Rubric
•
• The core elements of performance required by this task are:
• show your understanding of geometry
• explain geometrical reasoning
•
points
1.
3x1
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
Gives correct answers: 90°, 45°, 112.5°, 112.5°
Gives correct explanations such as:
The 90° angle is the corner of a square.
The 45° angle is 360÷8.
The other two angles are equal and the angle sum is 360°.
2.
2x1
5
AB = DC
1
Gives correct explanation showing that ABCD is a parallelogram..
3
Partial credit
Incomplete explanation.
(1)
Total Points
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9
68
Floor Pattern
Work the task and look at the rubric. What are the big mathematical ideas being assessed?
What do you think your students understand about proof?
What types of assumptions do they make?
How do you help them develop longer chains of reasoning?
Look at student work for part 2, proving that a figure is a parallelogram. How many of your
students:
• Made a complete and convincing justification?
• Made claims about angle size without offering a justification?
• Made claims about AD being parallel to BC with a justification?
• Said the two white triangles were congruent without proving the angles or the sides of the
kites?
• Couldn’t go beyond the step that DC=AB?
What other deficiencies did you notice in student explanations?
What would you like to see in student work?
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Looking at Student Work on Floor Pattern
Student A makes a very complete argument for the angles in part 1. The student shows a great deal
of detail in proving the figure is a parallelogram, using 2 sets of opposite equal sides.
Student A
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Student B gives the same information, but very concisely and referring to the diagram.
Student B
Student C understands the theorem needed to prove the figure is a parallelogram (2 sets of opposite
equal sides) and that AD is equal and congruent to CB because of SAS. However, the student does
not justify why the SAS is true. What are the expectations in your classroom for justification?
Student C
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Student D also fails to prove the angle in SAS. But again, the student knows that the opposite sides
must be equal to make the figure a parallelogram.
Student D
Student E tries to use the congruency of the kites to show that AD is equal to BC. The student does
not understand that the triangles must be congruent to prove that AD is equal to BC. The student
also appears to say that all 4 sides must be equal.
Student E
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Student F has added an E and F to the drawing. It is unclear if these points are the vertices of the
white triangles or the lines connecting the outside of the two kites. The student makes a circular
argument claiming properties of a parallelogram to prove something is a parallelogram.
Student F
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Geometry
Student Task
Core Idea 4
Geometry and
Measurement
Core Idea 2
Mathematical
Reasoning
Task 5
Floor Pattern
Show understanding of geometry and explain geometrical reasoning.
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.
Employ forms of mathematical reasoning and proof appropriate
to the solution of the problem at hand, including deductive and
inductive reasoning, making and testing conjectures and using
counter examples and indirect proof.
• Show mathematical reasoning in solutions in a variety of
ways, including words, numbers, symbols, pictures, charts,
graphs, tables, diagrams and models.
• Identify, formulate and confirm conjectures.
• Use synthetic, coordinate, and/or transformational geometry
in direct or indirect proof of geometric relationships.
• Establish the validity of geometric conjectures using
deduction; prove theorems, and critique argument made by
others.
Mathematics of the task:
• Use geometric properties of circles, squares, and kites to prove the angles of a quadrilateral
• Understand theorems needed to prove congruent triangles
• Understand theorems needed to prove a quadrilateral is a parallelogram
• Develop a justified chain of reasoning to support that a shape is a parallelogram
• Compose and decompose a complex figure
Based on teacher observation, this is what geometry students know and are able to do:
• Find 4 angles of a quadrilateral
• Explain how the angle measure was derived
• Use congruence to show that AB is equal to DC
• Understand that opposites of a quadrilateral need to be equal in order for the shape to be a
parallelogram
Areas of difficulty for geometry students:
• Not making assumptions about congruency of triangles without giving the angle measure
• Taking AD = BC as a given instead of something that needs to be proved
• Assuming that because AB =DC, that the sides must be parallel
• Not understanding the steps needed after AB=DC, stopping at that point
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The maximum score available for this task is 9 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Many students, 82%, could find all the angles in the kite and explain how they figured it out. More
than half the students, 70%, also knew that proving AB=DC was important to proving that the shape
in two was a parallelogram. About half the students, 52%, could make some progress toward an
organized proof for showing the figure made a parallelogram. 32% of the students met all the
demands of the task by finding all the angles of the kite and how they derived the numbers and
showing that two kites formed a parallelogram with complete justification. Less than 2% of the
students scored no points on this task. All the students in the sample with this score did not attempt
the task.
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Floor Pattern
Points
0
5
6
7
9
Understandings
Misunderstandings
All the students in the sample with this score did
not attempt the task. There was only one paper
in the sample between 0 and 5.
Students could find all the angles in
Students did not understand what was needed to
the kite and explain how they figured it prove that the shape was a parallelogram. They
out.
may have made a circular argument or made
assumptions about the embedded figures without
justification. For example 8% of the students
were able to say AD=BC but offered no
justification.
Students could find the angles of the
Students made assumptions about congruent
kite and knew that AB=DC was
triangles or parallel lines without adequate
important to the proof.
support statements.
Students could make some progress on
the proof of the parallelogram, but
missed some crucial justification or
step.
Students could find all the angles in
the kite and explain how they figured it
out. Students could make a complete
proof for showing that figure ABCD
was a parallelogram by showing that
the opposite sides of a quadrilateral
were equal. They took the time to
show that AD=BC by proving that the
triangles were congruent. They could
put together a long chain of reasoning
with justification.
Implications for Instruction
Students need to have exposure to a variety of tasks with rich embedded figures, requiring them to
tease out what they know and what they need to find out. While many students were comfortable
and familiar with the theorems for congruency and parallelograms, they often skipped crucial pieces
of justification. Students need to understand that diagrams and how things look are not acceptable
for justification. They need to find mathematical relationships to show why angles or sides are
equal. Students seemed less sure about how to prove lines are parallel. Many thought that if two
sides were equal then the sides would be parallel. Set up challenges within the class, such as:
construct a shape with 2 opposite sides equal but nonparallel. Students should be given a variety of
challenges, some of which are not possible, so that they learn about relationships and start to see the
necessity of proof.
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Ideas for Action Research
Giving problems that allows students to develop logical reasoning is important. Some interesting
problems can be found in Fostering Geometric Thinking by Mark Driscoll. One that I found
interesting is about the diagonals of trapezoids. I might start by asking groups to draw a variety of
trapezoids with the parallel sides parallel to the bottom of the paper and label the vertices starting at
the bottom left and going clockwise CBED. Then I would have them make diagonals and label the
intersection point A. So they would end up with figures such as:
Then I would make a statement that CBA is always equal to AED. Can you make a convincing
argument to prove me wrong or prove that I am correct?
What are some of your sources of problems for making interesting challenges for students? Discuss
this lesson and your ideas for follow up with colleagues.
Reflecting on the Results for Geometry as a Whole:
Think about student work through the collection of tasks and the implications for instruction. What
are some of the big misconceptions or difficulties that really hit home for you?
If you were to describe one or two big ideas to take away and use for planning for next year, what
would they be?
What were some of the qualities that you saw in good work or strategies used by good stuents that
you would like to help other students develop?
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Three areas stood out for the Collaborative as a whole.
1. Understanding and using formulas: Students had difficulty working with the formulas in
Hanging Baskets. Students did not understand what “the area of the base” meant. Students
were confused by the height of the triangular pyramid. Many students did not use
Pythagorean theorem to find the height of the triangle on the base. Some students had
difficulty decomposing the shapes into the needed parts to use for the formula.
2. Jumping to conclusions: In Circle Pattern, students tried to find a pattern in the numbers too
quickly thinking the numbers would alternate between 1/2 and 3/4 or that the final number in
the pattern would be 9/16 and 7/16. Students did not look at enough detail when trying to
describe the pattern. They might quantify the growth between 2 numbers in the pattern
without noticing that the growth rate was not a constant difference. They might notice that in
the black area was growing, but not notice that it alternated up, down, up, down. Some
noticed features of the pattern, such as the area of black is always larger, but not understand
that an explanation about how the pattern grows is the level of thinking required at this grade
level.
3. Making a complete proof: Students in Pentagon and Floor Pattern had difficulty making a
complete proof. While they might know the appropriate theorems, they made too many
assumptions based on looking at the diagrams. For example, they might assume that
triangles are isosceles or that triangles are congruent without offering pertinent evidence or
justification. Students had difficulty thinking about how to prove all the details to make a
reasoned argument.
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Looking at the Ramp for Geometry
Hanging Baskets
• Part 1 – Identifying the area of the base
o Understanding that the base of the pyramid is a square
o Finding the area of the base before using the formula for volume
• Part 2 – finding the height of the basket
o Understanding that the height for the basket is different from the height of the
triangular base
o Understanding that the height depends on the volume desired
Pentagon
• Part 3 – Proving triangle PAB is congruent to triangle DAB
Circle Pattern
• Part 4 – Generalizing the pattern in the table
Floor Pattern
• Part 2 – Proving that quadrilateral ABCD is a parallelogram
With a group of colleagues look at student work around 30 – 33 points. Use the papers provided or
pick some from your own students.
How are students performing on the ramp?
What things impressed you about their performance?
What are skills or ideas they still need to work on?
Are students relying on previous arithmetic skills rather than moving up to more grade level
strategies?
What was missing that you would hope to see from students working at this level?
How do you help students at this level step up their performance or see a standard to aim for
in explaining their thinking?
Are our expectations high enough to these students?
How do we provide models to help these students see how their work can be improved or
what they are striving for?
Do you think errors were caused by lack of exposure to ideas or misconceptions?
What would a student need to fix or correct their errors?
What is missing to make it a top-notch response?
What concerns you about their work?
What strategies did you see that might be useful to show to the whole class?
Arnie
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Arnie, continued
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Arnie, continued
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Brian
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Brian, continued
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Cameron
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Cameron, continued
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Dean
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Dean, continued
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Dean, continued
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