# Circles in Triangles. This problem gives you the chance to: use algebra to explore a geometric situation

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1 Circles in Triangles This problem gives you the chance to: use algebra to explore a geometric situation A This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units, 4 units, and 5 units long r Y r O r X r r B r Z C 4 1. Explain why triangles AOX and AOY are congruent. 2. What can you say about the measures of the line segments CX and CZ? Copyright 2007 by Mathematics Assessment Page 48 Circles in Triangles Test 10

2 3. Find r, the radius of the circle. Explain your work clearly and show all your calculations. 4. This diagram shows a circle that just touches the sides of a right triangle whose sides are 5 units, 12 units, and 13 units long Draw construction lines as in the previous task, and find the radius of the circle in this 5, 12, 13 right triangle. Explain your work and show your calculations. _ Copyright 2007 by Mathematics Assessment Page 49 Circles in Triangles Test 10 9

3 Circles in Triangles Rubric The core elements of performance required by this task are: use algebra to explore a geometric situation Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Triangle AOY is congruent to triangle AOX (SAS) or (SSS) or (Hypotenuse, leg) Partial credit For a partially correct explanation 2. States that: CZ = CX 1 3. Gives correct answer r = 1 Shows correct work such as: CX = 4 r = 3 = CZ CX = 5 - AX 5 = 3 r + 4 r or r 2 + r(3- r) + r(4 r) = 6 Partial credit Shows partially correct work or uses guess and check. Alternatively, may use area reasoning 4. Draws in construction lines and uses a similar method to Question #3, Gives correct answer r = 2 Shows correct work such as: 13 = 5 r + 12 r or r 2 + r(5- r) + r(12 r) = 30 Alternatively, may use area arguments. 2 (1) (1) 3 Total Points Copyright 2007 by Mathematics Assessment Page 50 Circles in Triangles Test 10

5 Half the students in the sample did not attempt this part of the task? How many of your students fell into this category? What might be some reasons for this? Do you think they don t understand the mathematical ideas needed for the solution? Do you think they don t have enough experience applying the geometrical ideas in non-routine, problem-solving situations? Almost 2/3 of the students did not attempt part 4 of the task including drawing in the construction lines. How did your students do on this section of the task? What opportunities do students have that require them to draw in lines on a diagram to help them think about the problem and identify a solution path? What are some favorite problems that you might give students to help them develop this skill? Geometry

6 Looking at Student Work on Circles in Triangles The significant story of this task is the quality of explanations and strategies students at this level are capable of using. Notice that the successful students mark up their diagrams to help them think (use the diagram as a tool). Successful students write complete explanations instead of just making shorthand statements about the situation. Students A and B use area strategies to solve for the radius in part 3 and 4. Student C uses a substitutions and a series of equations to solve for r. Student A Geometry

7 Student A, part 2 Geometry

8 Student B Geometry

9 Student B, part 2 Geometry

10 Student C Geometry

12 The maximum score available for this task is 9 points. The minimum score for a level 3 response, meeting standards, is 4 points. Many students, about 85%, could identify that line segments CZ and CX were equal. More than half the students, 63%, could explain with justification why two triangles were congruent and explain that segments CZ and CX were equal. Almost half, 45%, were also able to draw construction lines on a diagram. 20% of the students met all the demands of the task including finding the radius of circles inscribed in right triangles by using properties of congruence or area. 15% of the students scored no points on this task. 30% of the students with this score attempted the task. Geometry

13 Circles in Triangles Points Understandings Misunderstandings 0 30% of the students with this score attempted the problem. Students could not make a correct statement about segments CX and CZ. Comments included they re tangents, but not equal, not congruent, both intersect at C, and they re flat. 1 Students knew that CX=CZ Students could not give evidence to support why triangles were congruent in part 1. 12% assumed that the line through A was a bisector, with no supporting evidence. About 10% named a theorem with no supporting evidence. 3 Students could prove that the triangles were congruent and that CX=CZ. 4 Students could solve part 1 and 2 and guess the size of the radius in 3 or draw in the construction lines in 4. More than half the students did not attempt part 3 of the task or did not attempt to draw construction lines in 4. Those who tried to draw construction lines made errors, usually they did not make the lines from the center to the opposite side form right angles or they left out the lines from the vertex to the center. 72% of the students did not try to solve for the radius in part 4. 7 Students could not solve for the radius in part 4, even though could solve a similar problem in part 3. 9 Students could read and interpret a diagram. Use geometric relationships to prove congruency of triangles in a complex diagram, and use substitution or area for the side measure in the diagram. Students at this level understood the importance of evidence to back up their claims and the logic needed to make a complete and convincing argument. Geometry

14 Implications for Instruction For students to be successful on this task, they would need to have knowledge and utility with proofing congruent triangles. This might require more than a basic familiarity with two-column proofs and using fundamental theorems of congruency such as SSS, SAS, ASA, SAA, HL and LL. Students need to understand how valid geometric arguments are made. They also need to learn to apply this knowledge to non-routine situations. The problem may appear complex to the students because it involved more than one set of congruent triangles. In addition the inscribed circle may have distracted students from visualizing the sets of congruent triangles. Students need experiences with applying their knowledge to complex diagrams. Students must understand the importance in terms of flexibility having a right angle in a triangle. If the angle in the diagram was not a right angle then there would not have been enough information to justify that the triangles were congruent due to the fact that there is no SSA theorem. Providing opportunities for students to sort through when there is enough or not enough information is important to developing the reasoning and logic that underlies justification. Another challenge to the task was taking the information in parts one and two and using that information to solve part three of the task. Parts one and two could have provided a scaffold for the students to prove that Triangle AYO was congruent to Triangle AXO and that Triangle CZO was congruent to Triangle CXO. Students also needed to use the knowledge that the corresponding parts of the triangles were congruent. The final step of logic was to use algebraic reasoning to determine a value for the radius. Three steps of reasoning often challenge students. They must have opportunities to reason over multi-step problems. Teachers should resist always breaking the reasoning sequence into parts for students, but rather teach them to think through the steps and develop a capacity to handle complex logic. Part four of the problem involved generalizing from the initial problem and applying that generalization to a similar but different size triangle. Generalization and application are higher level thinking skills. Students need to be given opportunities to generalize and then apply the findings to a new situation. Often problems are solved for the sake of answers. In geometry, finding patterns, processes or formulas should be an initial step that then is utilized in new situations and/or proved to be a general tool for further exploration. Geometry

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Sec 1.6 CC Geometry Triangle Proofs Name: POTENTIAL REASONS: Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point

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The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

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Elements of Plane Geometry by LK These are notes indicating just some bare essentials of plane geometry and some problems to think about. We give a modified version of the axioms for Euclidean Geometry