Core Idea Task - · PDF filesaying that I overheard someone say that the sum of the interior of any polygon is equal to (n-2) 180. What does this mean?

Geometry Copyright © 2009 by Noyce Foundation All rights reserved.

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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 Measurement

Hanging Baskets

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 Mathematical Reasoning

Circle and Squares

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 Triangles Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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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. _____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________ _____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

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Geometry Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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2009 Rubrics Grade 10 Triangles Rubric The core elements of performance required by this task are: • show your understanding of geometry • explain your reasoning Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. Gives correct answers: 12 sides Gives correct explanation which may involve the external angles of the polygon. Partial credit Incomplete explanation

2

3

(1)

5

Total Points 5


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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 Task 1 Triangles Student Task Show understanding of geometry and explain reasoning in a problem

situation. 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.

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 Understandings Misunderstandings

0 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.

2 Students could find the number of sides for the new figure.

Students could not put together a complete 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.

5 Students could reason about fitting together copies of a 30-60 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?

Geometry Hanging Baskets Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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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. 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 How deep will this basket have to be? __________________cm Show how you figured it out.

Volume of a pyramid = 1/3 area of base x height

25 cm

30 cm

30 cm

d cm

Geometry Hanging Baskets Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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3. Hugo also sells hemispherical baskets with a diameter of 30 centimeters.

Calculate the volume of this basket. __________________cm3 Show your work.

30 cm

Volume of a sphere = 4/3πr3

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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

points

section points

1. Gives correct answer: 6250 Shows correct work such as: 252 x 30 / 3

1 1

2

2. Gives correct answer: 390 ± 1

Gives correct answer: approx 48 Shows correct work such as: 390/3 = 130 6250 / 130

2

1 ft

1 ft

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3. Gives correct answer: 7069 or 2250π Partial credit: 14137 or 4500π, 56549 or 18000π Shows work such as: 2/3 π 153 or 4/3 π 153 or 2/3 π 303

2

(1) 1

3

Total Points 9


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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 Task 2 Hanging Baskets Student Task Work with the volumes of pyramids and hemispheres. Core Idea 4 Geometry and Measurement

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.

Core Idea 3 Algebraic Properties and Representations

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 Misunderstandings

0 All the students in the sample with this score attempted the task.

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.

2 Students could find the volume of a sphere.

26% of the students didn’t recognize it that it was a hemisphere.

3 Students could calculate the volume of 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.

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.

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%).

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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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?

Geometry Pentagon Copyright © by 2009 Mathematics Assessment Resource Service. All rights reserved.

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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. 1. Explain why each inside angle of the regular pentagon is 108°.

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

108°

0

Geometry Pentagon Copyright © by 2009 Mathematics Assessment Resource Service. All rights reserved.

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A B

C

D

E

P

Q

RS

T

2. Show that triangle PCE is similar to triangle ACE. 3. Prove that triangle PAB is congruent to triangle DAB.

_____________________________________________________________________________

_____________________________________________________________________________ _____________________________________________________________________________

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This diagram shows a pentagram, a shape of mystical significance. It is centered on a regular pentagon.

Geometry Copyright © by 2009 Mathematics Assessment Resource Service. All rights reserved.

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Pentagon Rubric The core elements of performance required by this task are: • understanding of geometrical situations • construction of mathematical proofs and explanations Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. Gives correct explanation such as: Recognition that sum of exterior angles is 3600 Calculation of each exterior angle (720) Calculation of each interior angle (1800 - 720) Accept alternative methods such as: (5 – 2)180 = 108 5 Partial credit Incomplete explanation.

3

(2)

3

2. Determines the measures of the angles in each triangle. (720, 720, 360) Recognition that triangles are therefore similar.

Partial credit Incomplete explanation

3

(2)

3

3. Recognition and explanation of similarity of triangles Explanation that as triangles have a corresponding common side, they must be congruent

2

2

Total Points 8

Geometry Copyright © by 2009 Noyce Foundation All rights reserved.

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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 Task 3 Pentagon Student Task Show understanding of geometry and write mathematical proofs. Core Idea 2 Mathematical Reasoning and Proof

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


• 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 Misunderstandings

0 No students in the sample had this score.

3 Students could make a complete explanation for size of the interior angle of a regular pentagon.

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

5 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.

6 Students could explain the interior angles of a regular pentagon and make a convincing argument for similarity.

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.

8 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?

Geometry Circle Pattern Copyright © by 2009 Mathematics Assessment Resource Service. All rights reserved. 51

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?

_________________________________

_________________________________

_________________________________

_________________________________

Geometry Circle Pattern Copyright © by 2009 Mathematics Assessment Resource Service. All rights reserved. 52

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

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

1 2

1 2

Geometry Copyright © by 2009 Mathematics Assessment Resource Service. All rights reserved.

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Circle Pattern Rubric • The core elements of performance required by this task are: • explore fractions in context explore fractions in context Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. 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

2

2. Gives correct answer: 3/4 2 2

3. Gives correct answers: 3/4, 1/4, 5/8, 3/8, 11/16, 5/16 Partial credit 4 or 3 correct two points 2 correct one point

3

(2) (1)

3

4. 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.

2

(1)

2

Total Points 9


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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 Task 4 Circle Pattern Student Task Use algebra patterns to explore a geometric situation.

Explore fractions in context Core Idea 3 Algebraic Properties and Representations

Represent and analyze mathematical situations and structures using algebraic symbols.

• Solve equations involving radicals and exponents in contextualized problems.

Core Idea 2 Mathematical Reasoning


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.


• 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


• 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 Understandings Misunderstandings

0 None of the students in the sample had this score.

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.

3 Students could find the fraction of black in the pattern with 4 black circles and fill in part of the table.

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.

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.

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.

7 Students could calculate the pattern for all the stages and fill in the table.

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.

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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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?

Geometry Floor Pattern Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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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.

Geometry Floor Pattern Copyright © 2009 by Mathematics Assessment Resource Service. All rights reserved.

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1. Find the measures of all four angles of the kites. Explain how you obtained your answers.

________________________________________

________________________________________

________________________________________

________________________________________

________________________________________

________________________________________

2. Two of the kites can fit together to make a hexagon. Prove that the quadrilateral ABCD is a parallelogram. _____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

A

B

C

D

9


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Floor Pattern Rubric • • The core elements of performance required by this task are: • show your understanding of geometry • explain geometrical reasoning • Based on these, credit for specific aspects of performance should be assigned as follows

points

section points

1. 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°.

3 x 1

2 x 1

5

2. AB = DC Gives correct explanation showing that ABCD is a parallelogram.. Partial credit Incomplete explanation.

1 3

(1)

4 Total Points 9


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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 Task 5 Floor Pattern Student Task Show understanding of geometry and explain geometrical reasoning. 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


• 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.


• 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 Understandings Misunderstandings

0 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.

5 Students could find all the angles in the kite and explain how they figured it out.

Students did not understand what was needed to prove that the shape was a parallelogram. They 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.

6 Students could find the angles of the kite and knew that AB=DC was important to the proof.

Students made assumptions about congruent triangles or parallel lines without adequate support statements.

7 Students could make some progress on the proof of the parallelogram, but missed some crucial justification or step.

9 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


84

Cameron


85

Cameron, continued


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Dean


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Dean, continued


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Dean, continued

Core Idea Task - · PDF filesaying that I overheard someone say that the sum of the interior of any polygon is equal to (n-2) 180. What does this mean?

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