School Transformation | Evidence-based Whole … · Web viewMathematically proficient students can...

World of PolygonsAligned to the Common Core Standards

Written byJonathan Katz, Ed. D.

Joseph WalterISA Mathematics Coaches

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Dear Math Teacher,

What is mathematics and why do we teach it? This question drives the work of the math coaches at ISA. We love mathematics and want students to have the opportunity to begin to have a similar emotion. We hope this unit will bring some new excitement to students.

This unit is the first part of the second unit for a full-year geometry course that is aligned to the common core standards. It is a unit about triangles that revisits some concepts and procedures students experienced in middle school and also an introduction of new concepts but with an expectation that students will leave with deeper understanding. Essential to this work is an inquiry approach to teaching mathematics where students are given multiple opportunities to reason, discover and create. Problem solving is the catalyst to the inquiry process so as you look closely at this unit you will see that students are constantly placed into problem solving situations where they are asked to think for themselves and with their classmates.

The first four Common Core Standards of Practice are central to this unit. Through the constant use of problematic situations students are being asked to develop perseverance and independent thought, to reason abstractly and quantitatively, and to critique the reasoning of others.

Mathematical modeling is present throughout the unit as students are asked to describe and analyze different bare number problems and real world situations leading to geometric ideas. Students are also asked to create models including the final project, which is to create a city plan based on the ideas of lines and angles.

The other four Standards of Practice are also present in this unit. Two of them are central to the inquiry approach. You will see these two statements in the last two standards.

Mathematically proficient students look closely to discern a pattern or structure. Mathematically proficient students notice if calculations are repeated, and look both for

general methods and for shortcuts.

We believe, as do many mathematicians, that mathematics is the science of patterns. This underlying principle is present in all the work we do with teachers and studentsIn this unit you will see that students are often asked to discern a pattern within a particular situation. This leads students to making conjectures and possibly generalizations that are both conceptual and procedural.

Thank you for looking at this unit and we welcome feedback and comments.

Sincerely,

Dr. Jonathan Katz(For the ISA math coaches)

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Unit 2 – World of Polygons: Triangles

Essential Questions: What makes a triangle unique? Interim Assessments/Performance TasksTriangle Dilemma - Lesson 4Possible Triangle Lengths - Lesson 5Triangular Inheritance – Lesson 12

Final Assessment: Logo Design

What will students understand and be able to do at the end of the unit?

Students will develop a working definition of a polygon

Students will develop an understanding of the big idea of “reasoning with relationships” as means of understanding the similarities and differences between different triangles.

Students will develop a complete understanding of triangles including proving theorems such as the measure of the interior angles of a triangle sum to 180 degrees, the inequality theorem, relationship of exterior angles to the two non-adjacent angles, joining midpoints of two sides of a triangle is parallel to the third side and half the length.

Students will develop an understanding of concurrence through investigations and constructions with angle bisectors, medians and altitudes.

What enduring understanding will students have?

One can develop understanding of theorems of triangles through investigations, conjecturing and proof.

Triangles can be thought about in multiple ways making for a deeper understanding of its many possibilities.

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Common Core Content Standards in the Unit

G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

GCO.12 Make formal geometric constructions with a variety of tools andmethods (compass and straightedge, string, reflective devices,paper folding, dynamic geometric software, etc.). bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment;

Common Core Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems,

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and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

4. Model with mathematics.

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Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

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7. Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

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World of Polygons: Triangles

Teacher GuideLesson 1

What is a Polygon?

Opening Activity

In middle school, you might have worked with polygons. Write down all the things you know about polygons.

(To the teacher: List all the comments students have and save for a conversation we will have after the next activity. Let any misconceptions remain on the list without comment.)

Second Activity

We are going to revisit an activity we did in the last unit with some additions. But this time, we are going to ask you to place things into two different groups, those which you think are polygons and those which are not. Be able to explain why you made the choices you did.

(To the teacher: Walk around the room and observe what the students are doing, because you want to decide in which order students will present. You might want to have a group which has some misconceptions go first so that it can add to the discussion. If there is a group that has a strong understanding, you might want to save them for last. Within this discussion, you might want to bring up the concepts of curve, a closed curve, and a simple closed curve for things that are not polygons. One of the definitions that might result is that a polygon is a simple closed curve made up of line segments. If a group was called simple closed curves, it would include 2, 3, 5, 7, 9, 11, and 12. But by adding the short statement, “made up of line segments,” the group would only include the polygons.)

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(11) (12)

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Activity Three

In your group, create three new figures that are polygons based on our agreed upon definition.

1. How many different polygons do there exist in this world?

2. Did you think of placing the circle into this group of polygons? Why might someone argue that it should be included?

(To the teacher: These questions are connected. If students see that by adding more sides to a polygon, there must exist an infinite number of different polygons. This might result in suggesting, that as the number of sides increases, the figure approaches a circle.)

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World of Polygons: TrianglesStudent Activity Sheet

Lesson 1What is a Polygon?

Name_______________________Date________________________

Opening Activity

In middle school, you might have worked with polygons. Write down all the things you know about polygons.

Second Activity

We are going to revisit an activity we did in the last unit with some additions. But this time, we are going to ask you to place things into two different groups, those which you think are polygons and those which are not. Be able to explain why you made the choices you did.

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(11) (12)

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Activity Three

In your group, create three new figures that are polygons based on our agreed upon definition.

1. How many different polygons do there exist in this world?

2. Did you think of placing the circle into this group of polygons? Why might someone argue that it should be included?

(To the teacher: These questions are connected. If students see that by adding more sides to a polygon, there must exist an infinite number of different polygons. This might result in suggesting, that as the

number of sides increases, the figure approaches a circle.)

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World of PolygonsLesson 2

Teacher GuideWhat do you know about triangles?

Opening Activity

We are going to spend the next few weeks with triangles. Throughout this unit you should think about our essential question. “What makes triangles unique?”

1. With your partner, write down all the ideas you know about triangles.

(To the teacher: Make a list of all the ideas that students have. If students mention that they know that the sum of the angles of a triangle is 180 degrees, ask them how do they know.)

2. Choose one of the ideas on the class list and prove the statement true or false. You can use a compass, ruler or protractor to aid your proof.

(To the teacher: Give the pairs about 15 minutes to work on this. You might ask two to three of the groups, based on your observations to share their proof to the class. This is an opportunity for you to learn how your students understand the idea of proof. We are not expecting a formal proof, nor should you push for that. This will come later. Informal proof is a very important idea in mathematics, where students give evidence to support their thinking. For example, if a pair states that the angles of a triangle sum to 180 degrees and their evidence is that they drew one triangle and measured its angles, you might respond, “is one triangle sufficient?” Would two triangles be sufficient? “How do you know that this is always true?”)

Second Activity

Journal Writing: Which student presentation was the best example of a proof and why did you think so?

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World of PolygonsStudent Activity Sheet

Lesson 2What do you know about triangles?

Name_______________________Date________________________

Opening Activity

We are going to spend the next few weeks with triangles. Throughout this unit you should think about our essential question. “What makes triangles unique?”

1. With your partner, write down all the ideas you know about triangles.

2. Choose one of the ideas on the class list and prove the statement true or false. You can use a compass, ruler or protractor to aid your proof.

Second Activity

Journal Writing: Which student presentation was the best example of a proof and why did you think so?

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Homework

You will be given three photographs of structures in the world. Your task is to choose one of them, observe the triangles, and write a couple of paragraphs that answers the following questions. We recommend that you use the internet to learn about their uses in these structures.

1. Describe where you see triangles used in your chosen structure?2. Why were triangles used rather than some other polygon?

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One tower of the George Washington Suspension Bridge

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The Great Pyramids of Egypt

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A Geodesic Dome Tent Construction

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Teacher GuideSums of the angles of a triangle

Opening Activity

Discovery: What is the sum of the angles of a triangle?

Step 1 – Measure the three angles of each triangle on the next page with a protractor.

Triangle A – Angle 1 – ________, Angle 2 – ________, Angle 3 – ________. Sum = _________

Triangle B – Angle 1 – ________, Angle 2 – ________, Angle 3 – ________. Sum = _________

Triangle C – Angle 1 – ________, Angle 2 – ________, Angle 3 – ________. Sum = _________

Step 2 – What hypothesis can you draw about the sum of the angles from the 3 triangles?

Step 3 – Cut the three angles off each triangle on the worksheet.

Step 4 – Prove your hypothesis using the 3 cut off pieces from each triangle. (Tape them side by side on the line below without any space between)

Step 5 – What do you notice? Does it support your hypothesis? Why?

(To the teacher: In this activity, you notice, we are asking students to add the angles of a triangle, but that is not sufficient for a proof. But by having them cut off the angles of a triangle, placing them side by side, and noticing that they form a straight angle, we are seeing that we have gone beyond just the numbers to show a visual representation of a proof. We are going to have students compare this proof to the next one.)

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Discovery: Worksheet

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Triangle A

Triangle B

Triangle C

Angle 1

Angle 3

Angle 2

Angle 1

Angle 2

Angle 3

Angle 2

Angle 3

Angle 1

Second Activity

1. Draw a triangle (any kind). Use a straightedge.2. Draw a line touching one of the vertices (corners) and parallel to one of the sides.3. Label the three angles of the triangle (1, 2, and 3).4. How could you use this drawing to prove the sum of the three angles equals 180? Why?

(Hint: Use a few ideas you learned about parallel lines in the previous unit to assist you.) Explain, in a few sentences, your conclusions.

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Student Activity SheetSums of the angles of a triangle

Name_______________________Date________________________

Opening Activity

Discovery: What is the sum of the angles of a triangle?

Step 1 – Measure the three angles of each triangle on the next page with a protractor.

Triangle A – Angle 1 – ________, Angle 2 – ________, Angle 3 – ________. Sum = _________

Triangle B – Angle 1 – ________, Angle 2 – ________, Angle 3 – ________. Sum = _________

Triangle C – Angle 1 – ________, Angle 2 – ________, Angle 3 – ________. Sum = _________

Step 2 – What hypothesis can you draw about the sum of the angles from the 3 triangles?

Step 3 – Cut the three angles off each triangle on the worksheet.

Step 4 – Prove your hypothesis using the 3 cut off pieces from each triangle. (Tape them side by side on the line below without any space between)

Step 5 – What do you notice? Does it support your hypothesis? Why?

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Triangle A

Triangle B

Triangle C

Angle 1

Angle 3

Angle 2

Angle 1

Angle 2

Angle 3

Angle 2

Angle 3

Angle 1

Second Activity

1. Draw a triangle (any kind) using a straightedge.2. Draw a line touching one of the vertices (corners) and parallel to one of the sides.3. Label the three angles of the triangle (1, 2, and 3).4. How could you use this drawing to prove the sum of the three angles equals 180? Why?

(Hint: Use a few ideas you learned about parallel lines in the previous unit to assist you.) Explain, in a few sentences, your conclusions.

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Teacher GuideSide/Angle Relationships

Opening Activity

Observe the three triangles on the following page. Think about the measurements of the angles in comparison to the sides (e.g. the shortest, the longest, etc.). What do you predict would be true about their relationship? Write down your prediction before you continue.

Experiment: What is the relationship between the sides and angles of any triangle?

Step 1 – Measure the sides of each of the triangles on the following worksheet, in centimeters using a ruler and using a protractor for each angle.

Triangle 1 – Side AB = __________, Side BC = __________, Side AC = __________

Angle 1 = _________, Angle 2 = _________, Angle 3 = _________





Step 2 – Observe each of the triangles by looking at the measures of the sides and the angles. Do you see any patterns? Does the data you’ve collected support your hypothesis?

Step 3 - Do you think this will be true for all triangles? Why? Create a visual demonstration of this idea.

(To the teacher: We treated this as a science experiment to learn what students observe about triangles, collect data, and make conclusions. While this idea is somewhat obvious, students don’t immediately see this. But it is so important for them to understand that as you change the angle measurement it affects the side opposite and vice versa. Be prepared to give them straws, spaghetti, or something similar as a means to do the demonstration in step 3. You want students to see that as you change the dimensions of one angle, the other two must be affected because of the invariability of the sum of the angles.)

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C

B

C

B

3

B

A1

C

2

22

1

1

3

3

Performance Task: Triangle Dilemma

Based on our previous experiment about side/angle relationships, your task is to find the MINIMUM amount of dimensions that would have to change if you change ONE of the dimensions (either a side or an angle.)

Possible Approach

1. With pieces of spaghetti, create a triangle. Measure the three sides and the three angles.

2. Change one of the six dimensions which will form a new triangle. Measure all of the dimensions of the new triangle. What do notice about the other five dimensions of the triangle?

3. Do my results represent the minimum number of changes or could I have done something differently?

Show all your work and explain your thinking.

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Student Activity SheetSide/Angle Relationships

Name_______________________Date________________________

Opening Activity

Observe the three triangles on the following page. Think about the measurements of the angles in comparison to the sides (e.g. the shortest, the longest, etc.). What do you predict would be true about their relationship? Write down your prediction before you continue.

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Experiment: What is the relationship between the sides and angles of any triangle?

Step 1 – Measure the sides of each of the triangles on the following worksheet, in centimeters using a ruler and each angle using a protractor.







Step 2 – Observe each of the triangles by looking at the measures of the sides and the angles. Do you see any patterns? Does the data you’ve collected support your hypothesis?

Step 3 - Do you think this will be true for all triangles? Why? Create a visual demonstration of this idea.

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31

C

B

C

B

3

B

A1

C

2

22

1

1

3

3

Performance Task: Triangle Dilemma

Based on our previous experiment about side/angle relationships, your task is to find the MINIMUM amount of dimensions that would have to change if you change ONE of the dimensions (either a side or an angle.)

Possible Approach

1. With pieces of spaghetti, create a triangle. Measure the three sides and the three angles.

2. Change one of the six dimensions which will form a new triangle. Measure all of the dimensions of the new triangle. What do notice about the other five dimensions of the triangle?

3. Do my results represent the minimum number of changes or could I have done something differently? Show all your work and explain your thinking.

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Teacher GuideInequality Theorem

What is true about the sides of a triangle?

Opening Activity: Am I telling the truth?

Observe the following diagram, read the information and decide if I’m telling the truth.

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My Grandmother’s House

SchoolMy House

I live in a house ten blocks from school.I live four blocks from my grandmother’s house and her house is five blocks from School. All the blocks are the same length.Am I telling the truth? Explain your thinking.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

10 Blocks

4 Blocks5 Blocks

(To the teacher: Discuss the students’ ideas. Your goal is to help students develop a conceptual understanding of the inequality theorem. Further work in this lesson will help clarify the theorem. )

Second Activity

Make a change to the length of one of the sides that you think would now make this a true story. Draw the figure below.

(To the teacher: Have five students display their figures. Have a discussion about the different triangles, if they are true or not true and why. This should lead to a discussion of the generalization that leads to the inequality theorem.)

Which of the following would not be possible representations of the story? Why?

4 Blocks 15 Blocks

10 Blocks

5 blocks 9 blocks

10 blocks

16 blocks 5 blocks

10 blocks

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Performance Task: Possible Triangle Lengths

If the lengths of two sides of a triangle are 3 inches and 7 inches, what is the possible range for the length of the missing side to the nearest tenth?

(To the Teacher: The students might struggle. You want to understand how they’ve understood the inequality theorem. Here are some thoughts about solving this problem.

**Students might make 3 the smallest side and 7 the middle side and say that it can’t be bigger than 9.9 because 3 + 7 = 10 and the sum of the two shortest sides has to be greater than the third.

Also, if the two given sides are the smaller of the three, then the least the largest side could be is 7.1. (So 7.1 to 9.9 are possible lengths)

**Students might make 3 the shortest of the three and 7 the largest of the three. They might say that the least the third side could be is 4.1 because 3 + 4.1 is 7.1 which is larger than 7. Since 7 is the largest side, the longest the middle side could be is 6.9. (So 4.1 to 6.9 are possible lengths)

**The last possibility the students might suggest is that 3 is the shortest side and instead of there being a longest side, the two missing sides are both 7. ( 7 )

**ANSWER – THEREFORE THE RANGE IF YOU PUT ALL THREE POSSIBILITIES TOGETHER WOULD BE 4.1 TO 9.9.)

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Student Activity SheetInequality Theorem

Name_______________________

Date________________________

What is true about the sides of a triangle?

Opening Activity: Am I telling the truth?

Observe the following diagram, read the information and decide if I’m telling the truth.

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My Grandmother’s House

SchoolMy House

I live in a house ten blocks from school.I live four blocks from my grandmother’s house and her house is five blocks from School. All the blocks are the same length.Am I telling the truth? Explain your thinking.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

10 Blocks

4 Blocks5 Blocks

Second Activity

Make a change to the length of one of the sides that you think would now make this a true story. Draw the figure below.

Which of the following would not be possible representations of the story?

4 Blocks 15 Blocks

10 Blocks

5 blocks 9 blocks

10 blocks

16 blocks 5 blocks

10 blocks

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I live in a house ten blocks from school.I live four blocks from my grandmother’s house and her house is five blocks from School. All the blocks are the same length.Am I telling the truth? Explain your thinking.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Performance Task: Possible Triangle Lengths

If the lengths of two sides of a triangle are 3 inches and 7 inches, what is the possible range for the length of the missing side to the nearest tenth?

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Teacher GuideExterior Angles

Opening ActivityExterior Angles of Triangles: Today we will be looking at the angles formed by a side and an extension of a side. These angles are called exterior angles.

DISCOVERY 1: In each of the following triangles, measure each of the labeled angles.

1) Measure of 1=1 Measure of 2=

Measure of 3= Measure of 4=

2

4 3

2) 2 Measure of 1=Measure of 2=Measure of 3=

3 Measure of 4= 4

1

3.) 4 3

Measure of 1= Measure of 2= Measure of 3=

Measure of 4=

1 2

(To the teacher: This lesson is about conjecture and proof. Bring students together once they have made conjectures. Then have them engage in the two proofs. How do these proofs justify the claim about the relation of exterior angles and the two non-adjacent interior angles?)

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4. Observe the angle measurements in the three triangles. Look at the relationship between the exterior angles that you measured and the interior angles. What conjectures would you like to make based on your observations?

Second Activity

Now you are going to try to prove your hypothesis. Follow the directions and then be ready to

explain how this proof supports your conjecture.

PICTORIAL PROOF OF YOUR CONJECTURE

Prove your hypothesis for the relationship between the exterior angle of a triangle and the remote

interior angles of a triangle.

• First, on a separate sheet of paper draw a triangle and one exterior angle. • Second, using a scissor cut the picture out of the separate sheet of paper. • Third, before gluing the cut out picture onto the paper below, cut off each of the two remote interior angles of the triangle. • Fourth, glue the cut out picture onto the paper below. • Fifth, try to arrange the cut off remote interior angles inside the exterior angle to prove your hypothesis. • Finally, glue the remote interior angles inside the exterior angle.• Does this prove your conjecture?

Third Activity: Now you are going to do an algebraic proof. Be ready to explain how this proof does or does not justify your hypothesis.

Algebraic Proof

Using the diagram below, how can you use the following questions to create an algebraic proof of the agreed upon hypothesis of the class?

1. What do we know about m¿ 1 + m¿ 2 + m¿ 3 ?2. What do we know about m¿ 3 + m¿ 4 ?3. How can you use this information to prove the hypothesis?

1

2 3 4

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Student Activity SheetExterior Angles

Name_______________________Date________________________

Exterior Angles of Triangles: Today we will be looking at the angles formed by a side and an extension of a side. These angles are called exterior angles.

DISCOVERY 1: In each of the following triangles, measure each of the labeled angles.

1) Measure of 1=1 Measure of 2=

Measure of 3= Measure of 4=

2

4 3

2) 2 Measure of 1=Measure of 2=Measure of 3=

3 Measure of 4= 4

1

3.) 4 3

Measure of 1= Measure of 2= Measure of 3=

Measure of 4=

1 2

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4. Observe the angle measurements in the three triangles. Look at the relationship between the exterior angles that you measured and the interior angles. What conjectures would you like to make based on your observations?

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Second Activity

Now you are going to try to prove your hypothesis. Follow the directions and then be ready to

explain how this proof supports your conjecture?

PICTORIAL PROOF OF YOUR CONJECTURE

Prove your hypothesis for the relationship between the exterior angle of a triangle

and the remote interior angles of a triangle.

• First, on a separate sheet of paper draw a triangle and one exterior angle. • Second, using a scissor cut the picture out of the separate sheet of paper. • Third, before gluing the cut out picture onto the paper below, cut off each of the two remote interior angles of the triangle. • Fourth, glue the cut out picture onto the paper below. • Fifth, try to arrange the cut off remote interior angles inside the exterior angle to prove your hypothesis. • Finally, glue the remote interior angles inside the exterior angle.

• Does this prove your conjecture?

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Third Activity: Now you are going to do an algebraic proof. Be ready to explain how this proof does or does not justify your conjecture.

Algebraic Proof

Using the diagram below, how can you use the following questions to create an algebraic proof of the agreed upon hypothesis of the class?

4. What do we know about m¿ 1 + m¿ 2 + m¿ 3?5. What do we know about m¿ 3 + m¿ 4 ?6. How can you use this information to prove the hypothesis?

1

2 3 4

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Teacher GuideDifferent Types of Triangles

Opening Activity

In your groups observe the following triangles, noting their side and angle measurements.

45o

A B

60o 60o 45o

6 6

C D

121 o 32o 6

12 3

5 E F 8

12 5 10

12 G 13

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Step 2 – Based on your observations and the measurements, which triangles can be grouped together? Look at angle measurements. Look at side measurements. Give evidence for your chosen groups. Some triangles will be in more than one group.

(To the teacher: Have students present their findings on chart paper to the whole class. Some of these classifications will be fairly simple but some will require a greater level of sophistication. Some may see that “B” is right but not that it is isosceles. Will they know that “C” and “F” are both scalene since in one case angles are given and in the other sides are given. This is an opportunity for you to learn about their understanding of the different classifications as well as definition.)

(To the teacher: If your students don’t know the base angle theorem, we recommend that you have them do the following investigation. Place the students in groups of four with each student given one of the following angle measurements: 30, 45, 70 and 80.)Second Activity: Investigation

You will be given an angle measurement. Your task is to create two angles with that measurement on the ends of a line segment facing each other. (See diagram)

Extend the sides so that a triangle is formed. Measure all three sides.

What observations can you make?

(To the teacher: Have students share out their results and their observations. Students should see that all created isosceles triangles leading to a statement of the base angles theorem. It’s important also that it be stated that right isosceles, acute isosceles and scalene isosceles triangles were formed. You can give (students the name of the theorem. Now we will follow up with an investigation with equilateral triangles. You might use one inch, two inch, three inch and four inch line segments)

Third Activity: Investigation

1. In your groups, you will be given a length of a line segment. Draw your line segment, and place the vertex of a 60 degree angle on the ends of each of the line segments facing each other similar to what you did before.

2. Extend the sides so that a triangle is formed.3. Measure all three sides.


(To the teacher: Have students share out their results and their observations. Students should see that all created equilateral triangles and that all equilateral triangles are equiangular. Now you may want to return to the original activity with this new information to extend the groupings. Groupings may now extend to include both sides and angle, e.g. right isosceles, right scalene, or an acute scalene versus an obtuse scalene.)

(To the teacher: A conclusion that should come out of this is: if two sides are congruent the opposite angles are congruent, if three sides are congruent then three angles are congruent, if no sides are

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congruent, then no angles are congruent and vice versa. This idea about scalene triangles can just be a discussion led by you through questioning and based on the patterns within the two previous investigations.)

Fourth Activity

In your groups discuss the following questions. Be ready to defend your thinking with evidence.

1. What are the most obtuse angles you can create in a triangle?_________________

2. What are the most right angles you can create in a triangle? _________________

3. What are the most acute angles you can create in a triangle? _________________

4. Can a triangle be both isosceles and obtuse? Give an example. _______________

5. Why can’t an equilateral triangle be obtuse?______________________________

6. Is there such a thing as a right isosceles triangle? Give an example.___________

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Student Activity SheetDifferent Types of Triangles

Name_______________________Date________________________

Opening Activity

In your groups observe the following triangles, noting their side and angle measurements.

45o

A B

60o 60o 45o

6 6

C D

121 o 32o 6

12 3

5 E F 8

12 5 10

12 G 13

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Step 2 – Based on your observations and the measurements, which triangles can be grouped together? Look at angle measurements. Look at side measurements. Give evidence for your chosen groups. Some triangles will be in more than one group.

Second Activity: Investigation In your groups, you will be given an angle measurement. Your task is to create two

angles with that measurement on the ends of a line segment facing each other. (See diagram)

Extend the sides so that a triangle is formed. Measure all three sides.


Third Activity: Investigation

1. In your groups, you will be given a length of a line segment. Draw your line segment, and place the vertex of a 60 degree angle on the ends of each of the line segments facing each other similar to what you did before.

2. Extend the sides so that a triangle is formed.3. Measure all three sides.


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Fourth Activity

In your groups discuss the following questions. Defend your thinking with evidence.

1. What are the most obtuse angles you can create in a triangle?

2. What are the most right angles you can create in a triangle?

3. What are the most acute angles you can create in a triangle?

4. Can a triangle be both isosceles and obtuse? Give an example.

5. Why can’t an equilateral triangle be obtuse?

6. Is there such a thing as a right isosceles triangle? Give an example.

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Teacher GuidePlaying with the different ideas about triangles

In today’s lesson you are going to get the chance to learn how well you understand the ideas we have been talking about in class. There are going to be 10 problems. You should work with a partner but I recommend that for each problem first work by yourself and then join up with your partner when you both feel you have ideas about what to do or questions you would like to ask.

1. You have an isosceles triangle. If the unequal angle is 38 degrees, what are the measures of the other angles?

2. How many different equiangular triangles can you create? Justify your answer

3. In triangle ABC, angle B is 40 less than angle A and angle C is two times bigger than angle B. What is the measure of each angle?

4. If the measure of the unequal angle in an isosceles triangle is xo what is the measure of each base angle?

5. If the measure of each base angle in an isosceles triangle is xo what is the measure of the unequal angle?

6. The direct distance between city A and city B is 200 miles. The direct distance between city B and city C is 300 miles. Which could be the direct distance between city C and city A? Justify your answer(1) 50 miles (3) 550 miles(2) 350 miles (4) 650 miles

7. In the accompanying diagram, ABCD is a straight line, and angle E in triangle BEC is a right angle. What does a° + d° equal?

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8. In the accompanying diagram of ▲ABC, segment AB is extended through D, m<CBD = 30 and AB is congruent to BC. What is the measure of m<A?

9. Hakim says if a triangle is an obtuse triangle, then it cannot also be an isosceles triangle. Using a diagram, show that Hakim is incorrect, and indicate the measures of all the angles and sides to justify your answer.

10. If m<C = 3y – 10, m<B = y + 40, and m<A = 90 what type of right triangle is triangle ABC? Justify your answer.

(To the teacher: The goal of this lesson is to see how the students can think about the different ideas you have discussed in this unit. Please explain the procedure of how you want students to work on these problems There are multiple ways you can approach this lesson. You can have students in pairs work on a set of 2 or 3 problems and then have students present their ideas. You should choose the pairs that present that would make for an interesting discussion. It could be how they thought about the problem, the error that arose or clarity of thinking. When students present you should expect the students to be ready to ask the presenters questions. You can have them pairs work on all of them and ask each pair to be ready to present on chart paper their work. You can follow that up with a gallery walk where students comment on what they see.)

Journal Writing: Assess your understanding of the work with triangles. What do you understand and what is confusing you? What questions would you like to ask?

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Student ActivityPlaying with the different ideas about triangles

In today’s lesson you are going to get the chance to learn how well you understand the ideas we have been talking about in class. There are going to be 10 problems. You should work with a partner but I recommend that for each problem first work by yourself and then join up with your partner when you both feel you have ideas about what to do or questions you would like to ask.

1. You have an isosceles triangle. If the unequal angle is 38 degrees, what are the measures of the other angles?

2. How many different equiangular triangles can you create? Justify your answer.

3. In triangle ABC, angle B is 40 less than angle A and angle C is two times bigger than angle B. What is the measure of each angle?

4. If the measure of the unequal angle in an isosceles triangle is xo what is the measure of each base angle?

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5. If the measure of each base angle in an isosceles triangle is xo what is the measure of the unequal angle?

6. The direct distance between city A and city B is 200 miles. The direct distance between city B and city C is 300 miles. Which could be the direct distance between city C and city A? Justify your answer(1) 50 miles (3) 550 miles(2) 350 miles (4) 650 miles

7. In the accompanying diagram, ABCD is a straight line, and angle E in triangle BEC is a right angle. What does a° + d° equal?

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8. In the accompanying diagram of ▲ABC, segment AB is extended through D, m<CBD = 30 and AB is congruent to BC. What is the measure of m<A?

9. Hakim says if a triangle is an obtuse triangle, then it cannot also be an isosceles triangle. Using a diagram, show that Hakim is incorrect, and indicate the measures of all the angles and sides to justify your answer.

10. If m<C = 3y – 10, m<B= y + 40, and m<A = 90 what type of right triangle is triangle ABC? Justify your answer.

Journal Writing: Assess your understanding of the work with triangles. What do you understand and what is confusing you? What questions would you like to ask?

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Teacher GuideMedians

Open Activity

You are going to be given a line segment j. Using a compass and straight edge your job is to come up with a method of drawing a line perpendicular to line segment j that passes through the middle of the line segment (called a perpendicular bisector.) This is similar to the activity we did in Unit 1, Lesson 14. You can look back in your notes to guide you.

(To the teacher: In unit 1 students were given a line and now they are given a segment. This is simpler since we have the endpoints. Take time to discuss what a person needs to do to draw a perpendicular line that intersects the segment at its midpoint. This activity is presented here to support students to understand and construct medians and midsections (midsegments)of triangles.)

Second Activity

You are going to be given triangle ABC. Using your understanding from the previous activity, construct a line from one of the vertices to the midpoint of the opposite side. You may choose any of the three vertices to make your construction.

B

A C

Explain how you did your construction. Use a ruler measure to see if you’ve actually found the midpoint of the side.

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(To the teacher: Please have a discussion about the methods used by students. These activities are leading to tomorrow’s lesson which has students discover that the weighted center of a triangle is the same point as the point of concurrency (centroid) of the medians.)

Third Activity

Questions to answer

Explain how to construct a median. Why does this construction work? Why did the perpendicular bisector construction help you to do this construction?

Will the perpendicular bisector and median ever be the same? Explain your thinking?

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Student Activity Medians

Open Activity

You are going to be given a line segment j. Using a compass and straight edge your job is to come up with a method of drawing a line perpendicular to line segment j that passes through the middle of the line segment (called a perpendicular bisector.) This is similar to the activity we did in the previous unit.

Second Activity

You are going to be given triangle ABC. Using your understanding from the previous activity, construct a line from one of the vertices to the midpoint of the opposite side. You may choose any of the three vertices to make your construction.

B

A C

Explain how you did your construction. Using a ruler measure to see if you’ve actually found the midpoint of the side.

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Third Activity

Questions to answer

Explain how to construct a median. Why does this construction work? Why did the perpendicular bisector construction help you to do this construction?

Will the perpendicular bisector and median ever be the same?

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Teacher GuideBalancing a triangle

Opening Activity

Your group will be given a triangle and your initial task is to locate the point that would make it possible to balance the triangle on a pencil point. Be ready to explain why you picked this point.

(To the teacher: You need to create triangles for the groups out of matted paper. The triangles need to have thickness and weight. In the discussion, it will be interesting if anyone (no one) mentions the use of medians to accomplish this. You may want to mention the task in the next activity to generate conversation.)

Second Activity

Each member of the group should draw a triangle. They should be different. On your triangle, construct a median from each of the vertices. What do you observe? Can you draw any conclusion about your observation? Is there any connection you can make to the first activity?

(To the teacher: In their observations, we would expect that the groups would see that there is a common point of intersection for the medians. No one may make a useful connection to the first activity. But you can ask them to return to their matted triangle and find the centroid. After they have done this, ask the question is the centroid the center of gravity? Can they balance the triangle on their pencil? You can share vocabulary with the students once they have made sense out of the idea. This should include concurrence and centroid. The students should be able to state that the medians of a triangle are concurrent and the point of intersection is a centroid.)

Third Activity

Each of the three medians of your triangle can be seen as two separate line segments formed by intersection with the other medians. Measure each median and its pieces. Write down your observations.

(To the teacher: Make a table on the board which has three columns labeled: median, part 1, part 2. Enter students’ data so they have a lot of data to make their observations and conjectures. A discussion should follow leading to the 2:1 ratio of the parts of the median.)

Fourth Activity

Journal Writing: Today we’ve done a few activities. Write about what you’ve understood through the different investigations. Make as many connections as you can.

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Student Activity SheetBalancing a Triangle

Name_______________________Date________________________

Opening Activity

Your group will be given a triangle and your initial task is to locate the point that would make it possible to balance the triangle on a pencil point. Be ready to explain why you picked this point.

Second Activity

Each member of the group should draw a triangle. They should be different. On your triangle, construct a median from each of the vertices. What do you observe? Can you draw any conclusion about your observation? Is there any connection you can make to the first activity?

Third Activity

Each of the three medians of your triangle can be seen as two separate line segments formed by intersection with the other medians. Measure each median and its pieces. Write down your observations.

Fourth Activity

Journal Writing: Today we’ve done a few activities. Write about what you’ve understood through the different investigations. Make as many connections as you can.

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Teacher GuideWhat can you discover about the segment joining midpoints of two sides of a triangle?

Opening Activity

You will be given three triangles. Do the following:

Measure the length of each side of the three triangles. Find the midpoint of each side of the three triangles. Use a ruler or a compass. Mark the

midpoints. In each triangle connect the midpoints of two sides. You should have created three

segments inside each triangle. Measure the lengths of each segment inside the triangles

Now you are going to observe the lines and the data you collected. Think about the following two questions. Be prepared to discuss your findings with the rest of the class.

What do you observe about the relationship between the segments formed by joining the midpoints and the sides of the triangles?

What do you observe about the lengths of the segments formed by joining the midpoints and the lengths of the sides of the triangles?

(To the teacher: Students are asked to find the midpoint. They can use a ruler and just measure or they can use a compass. The compass would be more exact. How can they use constructing a perpendicular bisector to get the midpoint? There should be a discussion of the findings where you also share language with the students. This should include the term “midsegment” and the midpoint theorem. This activity will get the students ready to think about upcoming lessons about medians, angle bisectors and altitudes.

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Second Activity

Given: D, E and F are midpoints

A

D E

B F C

1) Given the triangle and information above describe all the things you know and the reason that supports it. This should be information about sides and angles.

2) If m<DBF = 47o and m<FCE = 68o then what is m<DAE? Explain your thinking.

3) If the perimeter of triangle ABC is 42 units, segment DE = 8 units and segment AB = 14 units find the length of segment DF.

4) Revisit your data from the earlier investigation. Observe the perimeters for both the given triangle and the triangle formed by joining the midpoints. What observations can you make? Will this always be true?

(Have a discussion about these questions. Please discuss question 1 before students move to question 2 Encourage students to use their understanding of parallel lines and transversals to talk about question 1.)

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Student ActivityWhat can you discover about the segment joining midpoints of two sides of a triangle?

Opening Activity

You will be given three triangles. Do the following:

Measure the length of each side of the three triangles. Find the midpoint of each side of the three triangles. Use a ruler or a compass. Mark the

midpoints. In each triangle connect the midpoints of two sides. You should have created three

segments inside each triangle. Measure the lengths of each segment inside the triangles

Now you are going to observe the lines and the data you collected. Think about the following two questions. Be prepared to discuss your findings with the rest of the class.

What do you observe about the relationship between the segments formed by joining the midpoints and the sides of the triangles?

What do you observe about the lengths of the segments formed by joining the midpoints and the lengths of the sides of the triangles?

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Second Activity

Given: D, E and F are midpoints

A

D E

B F C

1) Given the triangle and information above describe all the things you know and the reason that supports it. This should be information about sides and angles.

2) If m<DBF = 47o and m<FCE = 68o then what is m<DAE? Explain your thinking.

3) If the perimeter of triangle ABC is 42 units, segment DE = 8 units and segment AB = 14 units find the length of segment DF.

4) Revisit your data from the earlier investigation. Observe the perimeters for both the given triangle and the triangle formed by joining the midpoints. What observations can you make? Will this always be true?

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Teacher GuideAngle Bisector

Opening Activity

What do you think the term angle bisector means?

(To the teacher: We want to have some beginning understanding so students can go on with the following investigation. Remind the students the definition of an angle that we created/discussed in unit 1.)

Part 1: Take a piece of paper and draw an angle using a straight edge. Cut it out and fold one ray on top of the other and crease the paper. How would you describe the crease?

(To the teacher: Ask the students, “How does this understanding add to our definition of an angle bisector?”)

Part 2: Draw an angle on a piece of paper. Find a method for constructing the bisector. (Hint: Begin by drawing an arc from the vertex of the angle intersecting the rays.) After you’ve completed the construction of the bisector, fold the paper as you did earlier to see if your crease matches the angle bisector. Describe the method you used to make the construction.

(To the teacher: Discuss students’ methods as a way of coming to a correct construction. In the next activity make sure different types of triangles are being drawn.)

Second Activity:

Draw a triangle using a straightedge. For each of the angles construct an angle bisector from the vertex to the opposite side. What do you observe?

(To the teacher: Have a discussion, telling students that the point of concurrence is called the incenter.)

Measure the shortest distance from the incenter to each of the sides. What do you observe? Compare this observation with what you observed for the intersection of the medians.

(To the teacher: It would be good to give students an investigation assignment for homework. Students should be asked to pick a set of points on an angle bisector and measure the distance from that point to the sides of the angle (which is two sides of the triangle.. What observations can they make? They should bring in their work and findings the next day to be discussed. The goal is to get students to see the following: If a point lies on the bisector of an angle, then it is equidistant from the sides of the angle.)

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Student Activity SheetAngle Bisector

Name_______________________Date________________________

Opening Activity

What do you think the term angle bisector means?

Part 1: Take a piece of paper and draw an angle using a straight edge. Cut it out and fold one ray on top of the other and crease the paper. How would you describe the crease?

Part 2: Draw an angle on a piece of paper. Find a method for constructing the bisector. (Hint: Begin by drawing an arc from the vertex of the angle intersecting the rays.) After you’ve completed the construction of the bisector, fold the paper as you did earlier to see if your crease matches the angle bisector. Describe the method you used to make the construction.

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Second Activity

Draw a triangle using a straightedge. For each of the angles construct an angle bisector from the vertex to the opposite side. What do you observe?

Measure the shortest distance from the incenter to each of the sides. What do you observe? Compare this observation with what you observed for the intersection of the medians.

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Teacher GuideAltitudes

(To the teacher: Pass out an envelope with three triangles: an acute, a right and a scalene to groups of three students.)

Opening Activity

Part 1 - In your group, look at the three triangles, have a discussion about how a perpendicular line drawn from a vertex to the opposite side would appear. Would it look the same in each triangle or would they look differently?

(To the teacher: Have a discussion about this before going on since the altitude of a scalene triangle is often confusing to students and that the altitude already exists in a right triangle.)

Part 2 - Divide the triangles, giving one to each of you in the group. Now each of you will draw a perpendicular on your triangle. You will use the same method you learned in unit 1, lesson 14 to draw a perpendicular from a given point to a given line. Within your group, compare your results for the three different triangles and be ready to share your findings with the class.

Second Activity

Using your triangle, construct the other two altitudes. What do you think will happen? Why?

1. What do you observe for your own?2. How does it compare to the others?3. How does it compare to the conclusions about angle bisectors and medians?4. Is there ever an instance when the orthocenter, the incenter and the centroid are the same

point in the triangle? Explain.

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Performance Task: Triangular InheritanceA brother and a sister have inherited a large triangular plot of land. The will states that the property is to be divided along the altitude from the northern most point of the property. However, the property is covered with quicksand at the northern vertex. The will states that the heir who figures out how to draw the altitude without using the northern vertex, gets to choose his or her parcel first. How can the heirs construct the altitude? Is this a fair way to divide the land?

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Student Activity SheetAltitudes

Name_______________________Date________________________

Opening Activity

Part 1 - In your group, look at the three triangles, have a discussion about how a perpendicular line drawn from a vertex to the opposite side would appear. Would it look the same in each triangle or would they look differently?

Part 2 - Divide the triangles, giving one to each of you in the group. Now each of you will draw a perpendicular on your triangle. You will use the same method you learned in unit 1, lesson 14 to draw a perpendicular from a given point to a given line. Within your group, compare your results for the three different triangles and be ready to share your findings with the class.

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Second Activity

Using your triangle, construct the other two altitudes. What do you think will happen? Why?

1. What do you observe for your own?2. How does it compare to the others?3. How does it compare to the conclusions about angle bisectors and medians?4. Is there ever an instance when the orthocenter, the incenter and the centroid are the same

point in the triangle? Explain.

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Performance Task: Triangular InheritanceA brother and a sister have inherited a large triangular plot of land. The will states that the property is to be divided along the altitude from the northern most point of the property. However, the property is covered with quicksand at the northern vertex. The will states that the heir who figures out how to draw the altitude without using the northern vertex, gets to choose his or her parcel first. How can the heirs construct the altitude? Is this a fair way to divide the land?

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Project: Logo DesignObjective: Design a logo based on a movie, book, sports team, musical singer or group.

Requirements:

It must include at least two different kinds of triangles.

It must include at least one of the following: a median, angle bisector or altitude. Each

triangle must have one of these.

There must be a point of concurrence (centroid, incenter or omnicenter) and evidence of

the construction that led to the point.

Write Up:

A description of the logo and how it is connected to the book, team movie, etc.

A detailed mathematical description of the logo.

A mathematical description of how you made the design. What mathematical processes

did you use to create the design?

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