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Lesson 1: Why Move Things Around? Date: 9/17/13
S.1
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 1
Lesson 1: Why Move Things Around?
Classwork
Exploratory Challenge 1
1. Describe, intuitively, what kind of transformation will be required to move the figure on the left to each of the figures (1â3) on the right. To help with this exercise, use a transparency to copy the figure on the left. Note that you are supposed to begin by moving the left figure to each of the locations in (1), (2), and (3).
Lesson 1: Why Move Things Around? Date: 9/17/13
S.2
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 1
2. Given two segments đŽđ” and đ¶đ·, which could be very far apart, how can we find out if they have the same length without measuring them individually? Do you think they have the same length? How do you check? In other words, why do you think we need to move things around on the plane?
Lesson 1: Why Move Things Around? Date: 9/17/13
S.3
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 1
Problem Set 1. Using as much of the new vocabulary as you can, try to describe what you see in the diagram below.
2. Describe, intuitively, what kind of transformation will be required to move Figure A on the left to its image on the right.
Lesson Summary
A transformation of the plane, to be denoted by đč, is a rule that assigns to each point đ of the plane, one and only one (unique) point which will be denoted by đč(đ).
So, by definition, the symbol đč(đ) denotes a specific single point. The symbol đč(đ) shows clearly that đč moves đ to đč(đ)
The point đč(đ) will be called the image of đ by đč
We also say đč maps đ to đč(đ)
If given any two points đ and đ, the distance between the images đč(đ) and đč(đ) is the same as the distance between the original points đ and đ, then the transformation đč preserves distance, or is distance-preserving.
A distance-preserving transformation is called a rigid motion (or an isometry), and the name suggests that it âmovesâ the points of the plane around in a ârigidâ fashion.
A
B
F(A)
F(B)
Figure A
Image of A
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 2
Lesson 2: Definition of Translation and Three Basic Properties
Classwork
Exercise 1
Draw at least three different vectors and show what a translation of the plane along each vector will look like. Describe what happens to the following figures under each translation, using appropriate vocabulary and notation as needed.
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
S.5
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 2
Exercise 2
The diagram below shows figures and their images under a translation along đ»đŒïżœïżœïżœïżœâ . Use the original figures and the translated images to fill in missing labels for points and measures.
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 2
Problem Set 1. Translate the plane containing Figure A along đŽđ”ïżœïżœïżœïżœïżœâ . Use your
transparency to sketch the image of Figure A by this translation. Mark points on Figure A and label the image of Figure A accordingly.
2. Translate the plane containing Figure B along đ”đŽïżœïżœïżœïżœïżœâ Use your transparency to sketch the image of Figure B by this translation. Mark points on Figure B and label the image of Figure B accordingly.
Lesson Summary
Translation occurs along a given vector:
A vector is a segment in the plane. One of its two endpoints is known as a starting point; while the other is known simply as the endpoint.
The length of a vector is, by definition, the length of its underlying segment.
Pictorially note the starting and endpoints:
A translation of a plane along a given vector is a basic rigid motion of a plane.
The three basic properties of translation are:
(T1) A translation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(T2) A translation preserves lengths of segments.
(T3) A translation preserves degrees of angles.
Lesson 2: Definition of Translation and Three Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 2
3. Draw an acute angle (your choice of degree), a segment with length 3 cm, a point, a circle with radius 1 in, and a vector (your choice of length, i.e., starting point and ending point). Label points and measures (measurements do not need to be precise, but your figure must be labeled correctly). Use your transparency to translate all of the figures youâve drawn along the vector. Sketch the images of the translated figures and label them.
4. What is the length of the translated segment? How does this length compare to the length of the original segment?
Explain.
5. What is the length of the radius in the translated circle? How does this radius length compare to the radius of the original circle? Explain.
6. What is the degree of the translated angle? How does this degree compare to the degree of the original angle? Explain.
7. Translate point đ· along vector đŽđ”ïżœïżœïżœïżœïżœâ and label the image đ·âČ. What do you notice about the line containing vector đŽđ”ïżœïżœïżœïżœïżœâ , and the line containing points đ· and đ·âČ? (Hint: Will the lines ever intersect?)
8. Translate point đž along vector đŽđ”ïżœïżœïżœïżœïżœâ and label the image đžâČ. What do you notice about the line containing vector đŽđ”ïżœïżœïżœïżœïżœâ , and the line containing points đž and đžâČ?
Lesson 3: Translating Lines Date: 9/17/13
S.8
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 3
Lesson 3: Translating Lines
Classwork
Exercises
1. Draw a line passing through point P that is parallel to line đż. Draw a second line passing through point đ that is parallel to line đż, that is distinct (i.e., different) from the first one. What do you notice?
2. Translate line đż along the vector đŽđ”ïżœïżœïżœïżœïżœâ . What do you notice about đż and its image đżâČ?
Lesson 3: Translating Lines Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 3
3. Line đż is parallel to vector đŽđ”ïżœïżœïżœïżœïżœâ . Translate line đż along vector đŽđ”ïżœïżœïżœïżœïżœâ . What do you notice about đż and its image, đżâČ?
4. Translate line đż along the vector đŽđ”ïżœïżœïżœïżœïżœâ . What do you notice about đż and its image, đżâČ?
Lesson 3: Translating Lines Date: 9/17/13
S.10
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 3
5. Line đż has been translated along vector đŽđ”ïżœïżœïżœïżœïżœâ resulting in đżâ. What do you know about lines đż and đżâ?
6. Translate đż1 and đż2 along vector đ·đžïżœïżœïżœïżœïżœâ . Label the images of the lines. If lines đż1 and đż2 are parallel, what do you know about their translated images?
Lesson 3: Translating Lines Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 3
Problem Set 1. Translate â đđđ, point đŽ, point đ”, and rectangle đ»đŒđœđŸ along vector đžđčïżœïżœïżœïżœïżœâ Sketch the images and label all points using
prime notation.
2. What is the measure of the translated image of â đđđ. How do you know?
3. Connect đ” to đ”âČ. What do you know about the line formed by đ”đ”âČ and the line containing the vector đžđčïżœïżœïżœïżœïżœâ ?
4. Connect đŽ to đŽâČ. What do you know about the line formed by đŽđŽâČ and the line containing the vector đžđčïżœïżœïżœïżœïżœâ ?
5. Given that figure đ»đŒđœđŸ is a rectangle, what do you know about lines đ»đŒ and đœđŸ and their translated images? Explain.
Lesson Summary
Two lines are said to be parallel if they do not intersect.
Translations map parallel lines to parallel lines.
Given a line L and a point đ not lying on đż, there is at most one line passing through đ and parallel to đż.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 4
Lesson 4: Definition of Reflection and Basic Properties
Classwork
Exercises
1. Refect âđŽđ”đ¶ and Figure đ· across line đż. Label the reflected images.
2. Which figure(s) were not moved to a new location on the plane under this transformation?
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 4
3. Reflect the images across line đż. Label the reflected images.
4. Answer the questions about the image above.
a. Use a protractor to measure the reflected â đŽđ”đ¶.
b. Use a ruler to measure the length of image of đŒđœ after the reflection.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 4
5. Reflect Figure R and âđžđčđș across line đż. Label the reflected images.
Basic Properties of Reflections:
(Reflection 1) A reflection maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(Reflection2) A reflection preserves lengths of segments.
(Reflection 3) A reflection preserves degrees of angles.
If the reflection is across a line L and P is a point not on L, then L bisects the segment PPâ, joining P to its reflected image Pâ. That is, the lengths of OP and OPâ are equal.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 4
Use the picture below for Exercises 6â9.
6. Use the picture to label the unnamed points.
7. What is the measure of â đœđŸđŒ? â đŸđŒđœ? â đŽđ”đ¶? How do you know?
8. What is the length of segment đ đđđđđđĄđđđ(đčđ»)? đŒđœ? How do you know?
9. What is the location of đ đđđđđđĄđđđ(đ·)? Explain.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 4
Problem Set 1. In the picture below, â đ·đžđč = 56°, â đŽđ¶đ” = 114°, đŽđ” = 12.6 đąđđđĄđ , đœđŸ = 5.32 đąđđđĄđ , point đž is on line đż and
point đŒ is off of line đż. Let there be a reflection across line đż. Reflect and label each of the figures, and answer the questions that follow.
Lesson Summary
A reflection is another type of basic rigid motion.
Reflections occur across lines. The line that you reflect across is called the line of reflection.
When a point, đ, is joined to its reflection, đâČ, the line of reflection bisects the segment, đđâČ.
Lesson 4: Definition of Reflection and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 4
2. What is the size of đ đđđđđđĄđđđ(â đ·đžđč)? Explain.
3. What is the length of đ đđđđđđĄđđđ(đœđŸ)? Explain.
4. What is the size of đ đđđđđđĄđđđ(â đŽđ¶đ”)?
5. What is the length of đ đđđđđđĄđđđ(đŽđ”)?
6. Two figures in the picture were not moved under the reflection. Name the two figures and explain why they were not moved.
7. Connect points đŒ and đŒâ Name the point of intersection of the segment with the line of reflection point đ. What do you know about the lengths of segments đŒđ and đđŒâ?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 5
Lesson 5: Definition of Rotation and Basic Properties
Classwork
Example 1
Let there be a rotation around center đ,đ degrees.
If đ = 30, then the plane moves as shown:
If đ = â30, then the plane moves as shown:
Exercises
1. Let there be a rotation of đ degrees around center đ. Let đ be a point other than đ. Select a đ so that đ â„ 0. Find đâ (i.e., the rotation of point đ) using a transparency.
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 5
2. Let there be a rotation of d degrees around center đ. Let đ be a point other than đ. Select a đ so that đ < 0. Find đâ (i.e., the rotation of point đ) using a transparency.
3. Which direction did the point đ rotate when đ â„ 0?
4. Which direction did the point đ rotate when đ < 0?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 5
5. Let L be a line, đŽđ”ïżœïżœïżœïżœïżœâ be a ray, đ¶đ· be a segment, and â đžđčđș be an angle, as shown. Let there be a rotation of đ degrees around point đ. Find the images of all figures when đ â„ 0.
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 5
6. Let đŽđ” be a segment of length 4 units and â đ¶đ·đž be an angle of size 45Ë. Let there be a rotation by đ degrees, where đ < 0, about đ. Find the images of the given figures. Answer the questions that follow.
a. What is the length of the rotated segment đ đđĄđđĄđđđ(đŽđ”)?
b. What is the degree of the rotated angle đ đđĄđđĄđđđ (â đ¶đ·đž)?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 5
7. Let đż1, đż2 be parallel lines. Let there be a rotation by d degrees, where â360 < đ < 360, about đ. Is (đż1)âČ â„ (đż2)âČ?
8. Let đż be a line and đ be the center of rotation. Let there be a rotation by đ degrees, where đ â 180 about đ. Are the lines đż and đżâ parallel?
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 5
Problem Set 1. Let there be a rotation by â 90Ë around the center đ.
Lesson Summary
Rotations require information about the center of rotation and the degree in which to rotate. Positive degrees of rotation move the figure in a counterclockwise direction. Negative degrees of rotation move the figure in a clockwise direction.
Basic Properties of Rotations:
(R1) A rotation maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(R2) A rotation preserves lengths of segments.
(R3) A rotation preserves degrees of angles.
When parallel lines are rotated, their images are also parallel. A line is only parallel to itself when rotated exactly 180Ë.
Lesson 5: Definition of Rotation and Basic Properties Date: 9/17/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 5
2. Explain why a rotation of 90 degrees never maps a line to a line parallel to itself.
3. A segment of length 94 cm has been rotated đ degrees around a center đ. What is the length of the rotated segment? How do you know?
4. An angle of size 124Ë has been rotated đ degrees around a center đ. What is the size of the rotated angle? How do
you know?
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
S.25
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
Lesson 6: Rotations of 180 Degrees
Classwork
Example 1
The picture below shows what happens when there is a rotation of 180Ë around center đ.
Example 2
The picture below shows what happens when there is a rotation of 180Ë around center đ, the origin of the coordinate plane.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
Exercises
1. Using your transparency, rotate the plane 180 degrees, about the origin. Let this rotation be đ đđĄđđĄđđđ0. What are the coordinates of đ đđĄđđĄđđđ0(2,â4)?
2. Let đ đđĄđđĄđđđ0 be the rotation of the plane by 180 degrees, about the origin. Without using your transparency, find đ đđĄđđĄđđđ0(â3, 5).
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
3. Let đ đđĄđđĄđđđ0 be the rotation of 180 degrees around the origin. Let đż be the line passing through (â6, 6) parallel to the đ„-axis. Find đ đđĄđđĄđđđ0(đż). Use your transparency if needed.
4. Let đ đđĄđđĄđđđ0 be the rotation of 180 degrees around the origin. Let đż be the line passing through (7,0) parallel to the đŠ-axis. Find đ đđĄđđĄđđđ0(đż). Use your transparency if needed.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
S.28
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
5. Let đ đđĄđđĄđđđ0 be the rotation of 180 degrees around the origin. Let đż be the line passing through (0,2) parallel to the đ„-axis. Is đż parallel to đ đđĄđđĄđđđ0(đż)?
6. Let đ đđĄđđĄđđđ0 be the rotation of 180 degrees around the origin. Let đż be the line passing through (4,0) parallel to the đŠ-axis. Is đż parallel to đ đđĄđđĄđđđ0(đż)?
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
S.29
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
7. Let đ đđĄđđĄđđđ0 be the rotation of 180 degrees around the origin. Let đż be the line passing through (0,â1) parallel to the đ„-axis. Is đż parallel to đ đđĄđđĄđđđ0(đż)?
8. Let đ đđĄđđĄđđđ0 be the rotation of 180 degrees around the origin. Is đż parallel to đ đđĄđđĄđđđ0(đż)? Use your transparency if needed.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
S.30
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
9. Let đ đđĄđđĄđđđ0 be the rotation of 180 degrees around the origin. Is đż parallel to đ đđĄđđĄđđđ0(đż)? Use your transparency if needed.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
S.31
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
Problem Set Use the following diagram for problems 1â5. Use your transparency, as needed.
1. Looking only at segment đ”đ¶, is it possible that a 180Ë rotation would map đ”đ¶ onto đ”âČđ¶âČ? Why or why not?
2. Looking only at segment đŽđ”, is it possible that a 180Ë rotation would map đŽđ” onto đŽâČđ”âČ? Why or why not? 3. Looking only at segment đŽđ¶, is it possible that a 180Ë rotation would map đŽđ¶ onto đŽâČđ¶âČ? Why or why not?
4. Connect point đ” to point đ”âČ, point đ¶ to point đ¶âČ, and point đŽ to point đŽâČ. What do you notice? What do you think that point is?
5. Would a rotation map triangle đŽđ”đ¶ onto triangle đŽâČđ”âČđ¶âČ? If so, define the rotation (i.e., degree and center). If not, explain why not.
Lesson Summary
A rotation of 180 degrees around đ is the rigid motion so that if đ is any point in the plane, đ,đ and đ đđĄđđĄđđđ(đ) are collinear (i.e., lie on the same line).
Given a 180-degree rotation, đ 0 around the origin đ of a coordinate system, and a point đ with coordinates (đ, đ), it is generally said that đ 0(đ) is the point with coordinates (âđ,âđ).
Theorem. Let đ be a point not lying on a given line đż. Then the 180-degree rotation around đ maps đż to a line parallel to đż.
Lesson 6: Rotations of 180 Degrees Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 6
6. The picture below shows right triangles đŽđ”đ¶ and đŽâČđ”âČđ¶âČ, where the right angles are at đ” and đ”âČ. Given that đŽđ” = đŽâČđ”âČ = 1, and đ”đ¶ = đ”âČđ¶âČ = 2, đŽđ” is not parallel to đŽâČđ”âČ, is there a 180Ë rotation that would map âđŽđ”đ¶ onto âđŽâČđ”âČđ¶âČ? Explain.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 7
Lesson 7: Sequencing Translations
Classwork
Exploratory Challenge
1.
a. Translate â đŽđ”đ¶ and segment đžđ· along vector đčđșïżœïżœïżœïżœïżœâ . Label the translated images appropriately, i.e., đŽâČđ”âČđ¶âČ and đžâČđ·âČ.
b. Translate â đŽâČđ”âČđ¶âČand segment đžâČđ·âČ along vector đ»đŒïżœïżœïżœïżœâ . Label the translated images appropriately, i.e., đŽâČâČđ”âČâČđ¶âČâČ and đžâČâČđ·âČâČ.
c. How does the size of â đŽđ”đ¶ compare to the size of â đŽâČâČđ”âČâČđ¶âČâČ?
d. How does the length of segment đžđ· compare to the length of the segment đžâČâČđ·âČâČ?
e. Why do you think what you observed in parts (d) and (e) were true?
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 7
2. Translate âđŽđ”đ¶ along vector đčđșïżœïżœïżœïżœïżœâ and then translate its image along vector đœđŸïżœïżœïżœïżœâ . Be sure to label the images appropriately.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 7
3. Translate figure đŽđ”đ¶đ·đžđč along vector đșđ»ïżœïżœïżœïżœïżœïżœâ . Then translate its image along vector đœđŒïżœïżœïżœâ . Label each image appropriately.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 7
4.
a. Translate Circle đŽ and Ellipse đž along vector đŽđ”ïżœïżœïżœïżœïżœâ . Label the images appropriately.
b. Translate Circle đŽâČ and Ellipse đžâČ along vector đ¶đ·ïżœïżœïżœïżœïżœâ . Label each image appropriately.
c. Did the size or shape of either figure change after performing the sequence of translations? Explain.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 7
5. The picture below shows the translation of Circle A along vector đ¶đ·ïżœïżœïżœïżœïżœâ . Name the vector that will map the image of Circle A back onto itself.
6. If a figure is translated along vector đđ ïżœïżœïżœïżœïżœâ , what translation takes the figure back to its original location?
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 7
Problem Set 1. Sequence translations of Parallelogram đŽđ”đ¶đ· (a quadrilateral in which both pairs of opposite sides are parallel)
along vectors đ»đșïżœïżœïżœïżœïżœïżœâ and đčđžïżœïżœïżœïżœïżœâ . Label the translated images.
2. What do you know about đŽđ· and đ”đ¶ compared with đŽâČđ·âČ and đ”âČđ¶âČ? Explain.
3. Are đŽâČđ”âČ and đŽâČâČđ”âČâČ equal in length? How do you know?
4. Translate the curved shape đŽđ”đ¶ along the given vector. Label the image.
Lesson Summary
Translating a figure along one vector then translating its image along another vector is an example of a sequence of transformations.
A sequence of translations enjoys the same properties as a single translation. Specifically, the figuresâ lengths and degrees of angles are preserved.
If a figure undergoes two transformations, đč and đș, and is in the same place it was originally, then the figure has been mapped onto itself.
Lesson 7: Sequencing Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 7
5. What vector would map the shape đŽâČđ”âČđ¶âČ back onto đŽđ”đ¶?
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 8
Lesson 8: Sequencing Reflections and Translations
Classwork
Exercises
Use the figure below to answer Exercises 1â3.
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 8
1. Figure đŽ was translated along vector đ”đŽïżœïżœïżœïżœïżœâ resulting in đđđđđ đđđĄđđđ(đčđđđąđđ đŽ). Describe a sequence of translations that would map Figure đŽ back onto its original position.
2. Figure đŽ was reflected across line đż resulting in đ đđđđđđĄđđđ(đčđđđąđđ đŽ). Describe a sequence of reflections that would map Figure đŽ back onto its original position.
3. Can đđđđđ đđđĄđđđđ”đŽïżœïżœïżœïżœïżœâ undo the transformation of đđđđđ đđđĄđđđđ·đ¶ïżœïżœïżœïżœïżœâ ? Why or why not?
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 8
Use the diagram below for Exercises 4- 5.
4. Let there be the translation along vectorđŽđ”ïżœïżœïżœïżœïżœâ and a reflection across line đż. Let the black figure above be figure đ. Use a transparency to perform the following sequence: Translate figure đ, then reflect figure S. Label the image đâČ.
5. Let there be the translation along vectorđŽđ”ïżœïżœïżœïżœïżœâ and a reflection across line đż. Let the black figure above be figure đ. Use a transparency to perform the following sequence: Reflect figure đ, then translate figure S. Label the image đâČâČ
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 8
6. Using your transparency, show that under a sequence of any two translations, đđđđđ đđđĄđđđ and đđđđđ đđđĄđđđ0 (along different vectors), that the sequence of the đđđđđ đđđĄđđđ followed by the đđđđđ đđđĄđđđ0 is equal to the sequence of the đđđđđ đđđĄđđđ0 followed by the đđđđđ đđđĄđđđ. That is, draw a figure, đŽ, and two vectors. Show that the translation along the first vector, followed by a translation along the second vector, places the figure in the same location as when you perform the translations in the reverse order. (This fact will be proven in high school). Label the transformed image đŽâČ. Now draw two new vectors and translate along them just as before. This time label the transformed image đŽâČâČ. Compare your work with a partner. Was the statement that âthe sequence of the đđđđđ đđđĄđđđ followed by the đđđđđ đđđĄđđđ0 is equal to the sequence of the đđđđđ đđđĄđđđ0 followed by the đđđđđ đđđĄđđđâ true in all cases? Do you think it will always be true?
7. Does the same relationship you noticed in Exercise 6 hold true when you replace one of the translations with a reflection. That is, is the following statement true: A translation followed by a reflection is equal to a reflection followed by a translation?
Lesson 8: Sequencing Reflections and Translations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 8
Problem Set 1. Let there be a reflection across line đż and let there be a translation along vector đŽđ”ïżœïżœïżœïżœïżœâ as shown. If đ denotes the
black figure, compare the translation of đ followed by the reflection of đ with the reflection of đ followed by the translation of đ.
2. Let đż1 and đż2 be parallel lines and let đ đđđđđđĄđđđ1 and đ đđđđđđĄđđđ2 be the reflections across đż1 and đż2, respectively (in that order). Show that a đ đđđđđđĄđđđ2 followed by đ đđđđđđĄđđđ1 is not equal to a đ đđđđđđĄđđđ1 followed by đ đđđđđđĄđđđ2. (Hint: Take a point on đż1 and see what each of the sequences does to it.)
3. Let đż1 and đż2 be parallel lines and let đ đđđđđđĄđđđ1 and đ đđđđđđĄđđđ2 be the reflections across đż1 and đż2, respectively (in that order). Can you guess what đ đđđđđđĄđđđ1 followed by đ đđđđđđĄđđđ2 is? Give as persuasive an argument as you can. (Hint: Examine the work you just finished for the last problem.)
Lesson Summary
A reflection across a line followed by a reflection across the same line places all figures in the plane back onto their original position.
A reflection followed by a translation does not place a figure in the same location in the plane as a translation followed by a reflection. The order in which we perform a sequence of rigid motions matters.
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 9
Lesson 9: Sequencing Rotations
Classwork
Exploratory Challenge
1.
a. Rotate âł đŽđ”đ¶ d degrees around center đ·. Label the rotated image as âł đŽâČđ”âČđ¶âČ.
b. Rotate âł đŽâČđ”âČđ¶âČ d degrees around center đž. Label the rotated image as âł đŽâČâČđ”âČâČđ¶âČâČ.
c. Measure and label the angles and side lengths of âł đŽđ”đ¶. How do they compare with the images âł đŽâČđ”âČđ¶âČ and âł đŽâČâČđ”âČâČđ¶âČâČ?
d. How can you explain what you observed in part (c)? What statement can you make about properties of sequences of rotations as they relate to a single rotation?
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 9
2.
a. Rotate âł đŽđ”đ¶ d degrees around center đ· and then rotate again đ degrees around center đž. Label the image as âł đŽâČđ”âČđ¶âČ after you have completed both rotations.
b. Can a single rotation around center đ· map âł đŽâČđ”âČđ¶âČ onto âł đŽđ”đ¶?
c. Can a single rotation around center đž map âł đŽâČđ”âČđ¶âČ onto âł đŽđ”đ¶?
d. Can you find a center that would map âł đŽâČđ”âČđ¶âČ onto âł đŽđ”đ¶ in one rotation? If so, label the center đč.
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 9
3.
a. Rotate âł đŽđ”đ¶ 90Ë (counterclockwise) around center đ· then rotate the image another 90Ë (counterclockwise around center đž. Label the image âł đŽâČđ”âČđ¶âČ.
b. Rotate âł đŽđ”đ¶ 90Ë (counterclockwise) around center đž then rotate the image another 90Ë (counterclockwise around center đ·. Label the image âł đŽâČâČđ”âČâČđ¶âČâČ.
c. What do you notice about the locations of âł đŽâČđ”âČđ¶âČ and âł đŽâČâČđ”âČâČđ¶âČâČ? Does the order in which you rotate a figure around different centers have an impact on the final location of the figures image?
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 9
4.
a. Rotate âł đŽđ”đ¶ 90Ë (counterclockwise) around center D then rotate the image another 45Ë (counterclockwise) around center D. Label theâł đŽâČđ”âČđ¶âČ.
b. Rotate âł đŽđ”đ¶ 45Ë (counterclockwise) around center D then rotate the image another 90Ë (counterclockwise) around center D. Label theâł đŽâČâČđ”âČâČđ¶âČâČ.
c. What do you notice about the locations of âł đŽâČđ”âČđ¶âČ and âł đŽâČâČđ”âČâČđ¶âČâČ? Does the order in which you rotate a figure around the same center have an impact on the final location of the figures image?
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 9
5. In the figure below, âł đŽđ”đ¶ has been rotated around two different centers and its image is âł đŽâČđ”âČđ¶âČ. Describe a sequence of rigid motions that would map âł đŽđ”đ¶ onto âł đŽâČđ”âČđ¶âČ.
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 9
Problem Set 1. Refer to the figure below.
a. Rotate â đŽđ”đ¶ and segment đ·đž đ degrees around center đč, then đ degrees around center đș. Label the final
location of the images as â đŽâČđ”âČđ¶âČ and DâEâ.
b. What is the size of â đŽđ”đ¶ and how does it compare to the size of â đŽâČđ”âČđ¶âČ? Explain.
c. What is the length of segment DE and how does it compare to the length of segment đ·âČđžâČ? Explain.
Lesson Summary
Sequences of rotations have the same properties as a single rotation:
âą A sequence of rotations preserves degrees of measures of angles.
âą A sequence of rotations preserves lengths of segments. The order in which a sequence of rotations around different centers is performed matters with respect to
the final location of the image of the figure that is rotated.
The order in which a sequence of rotations around the same center is performed does not matter. The image of the figure will be in the same location.
Lesson 9: Sequencing Rotations Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 9
2. Refer to the figure given below.
a. Let đ đđĄđđĄđđđ1 be a counterclockwise rotation of 90Ë around the center đ. Let đ đđĄđđĄđđđ2 be a clockwise rotation of (â45)Ë around the center đ. Determine the approximate location of đ đđĄđđĄđđđ1(âł đŽđ”đ¶) followed by đ đđĄđđĄđđđ2. Label the image of triangle đŽđ”đ¶ as đŽâČđ”âČđ¶âČ.
b. Describe the sequence of rigid motions that would map âł đŽđ”đ¶ onto âł đŽâČđ”âČđ¶âČ.
3. Refer to the figure given below. Let đ be a rotation of (â90)Ë around the center đ. Let đ đđĄđđĄđđđ2 be a rotation of (â45)Ë around the same center đ. Determine the approximate location of đ đđĄđđĄđđđ1(âđŽđ”đ¶) followed by đ đđĄđđĄđđđ2(âđŽđ”đ¶). Label the image of triangle đŽđ”đ¶ as đŽâČđ”âČđ¶âČ.
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 10
Lesson 10: Sequences of Rigid Motions
Classwork
Exercises
1. In the following picture, triangle đŽđ”đ¶ can be traced onto a transparency and mapped onto triangle đŽâČđ”âČđ¶âČ. Which basic rigid motion, or sequence of, would map one triangle onto the other?
2. In the following picture, triangle đŽđ”đ¶ can be traced onto a transparency and mapped onto triangle đŽâČđ”âČđ¶âČ. Which basic rigid motion, or sequence of, would map one triangle onto the other?
.
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 10
3. In the following picture, triangle đŽđ”đ¶ can be traced onto a transparency and mapped onto triangle đŽâČđ”âČđ¶âČ. Which basic rigid motion, or sequence of, would map one triangle onto the other?
4. In the following picture, we have two pairs of triangles. In each pair, triangle đŽđ”đ¶ can be traced onto a transparency and mapped onto triangle đŽâČđ”âČđ¶âČ. Which basic rigid motion, or sequence of, would map one triangle onto the other? Scenario 1:
Scenario 2:
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 10
5. Let two figures đŽđ”đ¶ and đŽâČđ”âČđ¶âČ be given so that the length of curved segment đŽđ¶ = the length of curved segment đŽâČđ¶âČ, |â đ”| = |â đ”âČ| = 80°, and |đŽđ”| = |đŽâČđ”âČ| = 5. With clarity and precision, describe a sequence of rigid motions that would map figure đŽđ”đ¶ onto figure đŽâČđ”âČđ¶âČ.
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 10
Problem Set 1. Let there be the translation along vector ïżœâïżœ, let there be the rotation around point đŽ, â90 degrees (clockwise), and
let there be the reflection across line đż. Let đ be the figure as shown below. Show the location of đ after performing the following sequence: a translation followed by a rotation followed by a reflection.
2. Would the location of the image of đ in the previous problem be the same if the translation was performed first
instead of last, i.e., does the sequence: translation followed by a rotation followed by a reflection equal a rotation followed by a reflection followed by a translation? Explain.
Lesson 10: Sequences of Rigid Motions Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 10
3. Use the same coordinate grid, below, to complete parts (a)â(c).
a. Reflect triangle đŽđ”đ¶ across the vertical line, parallel to the đŠ-axis, going through point (1, 0). Label the
transformed points đŽđ”đ¶ as đŽâČ,đ”âČ,đ¶âČ, respectively.
b. Reflect triangle đŽâČđ”âČđ¶âČ across the horizontal line, parallel to the đ„-axis going through point (0,â1). Label the transformed points of đŽâđ”âđ¶â as đŽâČâČđ”âČâČđ¶âČâČ, respectively.
c. Is there a single rigid motion that would map triangle đŽđ”đ¶ to triangle đŽâČâČđ”âČâČđ¶âČâČ ?
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 11
Lesson 11: Definition of Congruence and Some Basic Properties
Classwork
Exercise 1
a. Describe the sequence of basic rigid motions that shows đ1 â đ2.
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 11
b. Describe the sequence of basic rigid motions that shows đ2 â đ3.
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 11
c. Describe the sequence of basic rigid motions that shows đ1 â đ3.
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 11
Exercise 2
Perform the sequence of a translation followed by a rotation of Figure đđđ, where đ is a translation along a vector đŽđ”ïżœïżœïżœïżœïżœâ and đ is a rotation of đ degrees (you choose đ) around a center đ. Label the transformed figure đâČđâČđâČ. Will đđđ â đâČđâČđâČ?
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 11
Problem Set 1. Given two right triangles with lengths shown below, is there one basic rigid motion that maps one to the other?
Explain.
2. Are the two right triangles shown below congruent? If so, describe the congruence that would map one triangle onto the other.
Lesson Summary
Given that sequences enjoy the same basic properties of basic rigid motions, we can state three basic properties of congruences:
(C1) A congruence maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an angle.
(C2) A congruence preserves lengths of segments.
(C3) A congruence preserves degrees of angles.
The notation used for congruence is â .
Lesson 11: Definition of Congruence and Some Basic Properties Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 11
3.
a. Given two rays, đđŽïżœïżœïżœïżœïżœâ and đâČđŽâČïżœïżœïżœïżœïżœïżœïżœïżœâ , describe the congruence that maps đđŽïżœïżœïżœïżœïżœâ to đâČđŽâČïżœïżœïżœïżœïżœïżœïżœïżœâ .
b. Describe the congruence that maps đâČđŽâČïżœïżœïżœïżœïżœïżœïżœïżœâ to đđŽïżœïżœïżœïżœïżœâ .
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 12
Lesson 12: Angles Associated with Parallel Lines
Classwork
Exploratory Challenge 1
In the figure below, đż1 is not parallel to đż2, and đ is a transversal. Use a protractor to measure angles 1â8. Which, if any, are equal? Explain why. (Use your transparency, if needed).
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 12
Exploratory Challenge 2
In the figure below, đż1 â„ đż2, and đ is a transversal. Use a protractor to measure angles 1â8. List the angles that are equal in measure.
a. What did you notice about the measures of â 1 and â 5? Why do you think this is so? (Use your transparency, if needed).
b. What did you notice about the measures of â 3 and â 7? Why do you think this is so? (Use your transparency, if needed.) Are there any other pairs of angles with this same relationship? If so, list them.
c. What did you notice about the measures of â 4 and â 6? Why do you think this is so? (Use your transparency, if needed). Is there another pair of angles with this same relationship?
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 12
Problem Set Use the diagram below to do Problems 1â6.
Lesson Summary
Angles that are on the same side of the transversal in corresponding positions (above each of đż1 and đż2 or below each of đż1 and đż2) are called corresponding angles. For example, â 2 and â 4.
When angles are on opposite sides of the transversal and between (inside) the lines đż1 and đż2, they are called alternate interior angles. For example, â 3 and â 7.
When angles are on opposite sides of the transversal and outside of the parallel lines (above đż1 and below đż2), they are called alternate exterior angles. For example, â 1 and â 5.
When parallel lines are cut by a transversal, the corresponding angles, alternate interior angles, and alternate exterior angles are equal. If the lines are not parallel, then the angles are not equal.
Lesson 12: Angles Associated with Parallel Lines Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 12
1. Identify all pairs of corresponding angles. Are the pairs of corresponding angles equal in measure? How do you know?
2. Identify all pairs of alternate interior angles. Are the pairs of alternate interior angles equal in measure? How do you know?
3. Use an informal argument to describe why â 1 and â 8 are equal in measure if đż1 â„ đż2.
4. Assuming đż1 â„ đż2 if the measure of â 4 is 73Ë, what is the measure of â 8? How do you know?
5. Assuming đż1 â„ đż2, if the measure of â 3 is 107Ë degrees, what is the measure of â 6? How do you know?
6. Assuming đż1 â„ đż2, if the measure of â 2 is 107Ë, what is the measure of â 7? How do you know?
7. Would your answers to Problems 4â6 be the same if you had not been informed that đż1 â„ đż2? Why or why not?
8. Use an informal argument to describe why â 1 and â 5 are equal in measure if đż1 â„ đż2.
9. Use an informal argument to describe why â 4 and â 5 are equal in measure if đż1 â„ đż2.
10. Assume that đż1 is not parallel to đż2. Explain why â 3 â â 7.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 13
Lesson 13: Angle Sum of a Triangle
Classwork Concept Development
â 1 + â 2 + â 3 = â 4 + â 5 + â 6 = â 7 + â 8 + â 9 = 180
Note that the sum of angles 7 and 9 must equal 90Ë because of the known right angle in the right triangle.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 13
Exploratory Challenge 1
Let triangle đŽđ”đ¶ be given. On the ray from đ” to đ¶, take a point đ· so that đ¶ is between đ” and đ·. Through point đ¶, draw a line parallel to đŽđ” as shown. Extend the parallel lines đŽđ” and đ¶đž. Line đŽđ¶ is the transversal that intersects the parallel lines.
a. Name the three interior angles of triangle đŽđ”đ¶.
b. Name the straight angle.
c. What kinds of angles are â đŽđ”đ¶ and â đžđ¶đ·? What does that mean about their measures?
d. What kinds of angles are â đ”đŽđ¶ and â đžđ¶đŽ? What does that mean about their measures?
e. We know that â đ”đ¶đ· = â đ”đ¶đŽ + â đžđ¶đŽ + â đžđ¶đ· = 180°. Use substitution to show that the three interior angles of the triangle have a sum of 180Ë.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 13
Exploratory Challenge 2
The figure below shows parallel lines đż1 and đż2. Let đ and đ be transversals that intersect đż1 at points đ” and đ¶, respectively, and đż2 at point đč, as shown. Let đŽ be a point on đż1 to the left of đ”, đ· be a point on đż1 to the right of đ¶, đș be a point on đż2 to the left of đč, and đž be a point on đż2 to the right of đč.
a. Name the triangle in the figure.
b. Name a straight angle that will be useful in proving that the sum of the interior angles of the triangle is 180Ë.
c. Write your proof below.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 13
Problem Set 1. In the diagram below, line đŽđ” is parallel to line đ¶đ·, i.e., đżđŽđ” â„ đżđ¶đ·. The measure of angle â đŽđ”đ¶ = 28°, and the
measure of angle â đžđ·đ¶ = 42°. Find the measure of angle â đ¶đžđ·. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle.
2. In the diagram below, line đŽđ” is parallel to line đ¶đ·, i.e., đżđŽđ” â„ đżđ¶đ·. The measure of angle â đŽđ”đž = 38° and the measure of angle â đžđ·đ¶ = 16°. Find the measure of angle â đ”đžđ·. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle. (Hint: find the measure of angle â đ¶đžđ· first, then use that measure to find the measure of angle â đ”đžđ·.)
Lesson Summary
All triangles have a sum of interior angles equal to 180Ë.
The proof that a triangle has a sum of interior angles equal to 180Ë is dependent upon the knowledge of straight angles and angles relationships of parallel lines cut by a transversal.
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 13
3. In the diagram below, line đŽđ” is parallel to line đ¶đ·, i.e., đżđŽđ” â„ đżđ¶đ·. The measure of angle â đŽđ”đž = 56°, and the measure of angle â đžđ·đ¶ = 22°. Find the measure of angle â đ”đžđ·. Explain why you are correct by presenting an informal argument that uses the angle sum of a triangle. (Hint: Extend the segment đ”đž so that it intersects line đ¶đ·.)
4. What is the measure of â đŽđ¶đ”?
5. What is the measure of â đžđčđ·?
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 13
6. What is the measure of â đ»đŒđș?
7. What is the measure of â đŽđ”đ¶?
8. Triangle đ·đžđč is a right triangle. What is the measure of â đžđčđ·?
Lesson 13: Angle Sum of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 13
9. In the diagram below, lines đż1 and đż2 are parallel. Transversals đ and đ intersect both lines at the points shown below. Determine the measure of â đœđđŸ. Explain how you know you are correct.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
Lesson 14: More on the Angles of a Triangle
Classwork
Exercises 1â4
Use the diagram below to complete Exercises 1â4.
1. Name an exterior angle and the related remote interior angles.
2. Name a second exterior angle and the related remote interior angles.
3. Name a third exterior angle and the related remote interior angles.
4. Show that the measure of an exterior angle is equal to the sum of the related remote interior angles.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
Example 1
Find the measure of angle đ„.
Example 2
Find the measure of angle đ„.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
Example 3
Find the measure of angle đ„.
Example 4
Find the measure of angle đ„.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
Exercises 5â10
5. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
6. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
7. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
S.78
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
8. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
9. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
10. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
S.79
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
Problem Set For each of the problems below, use the diagram to find the missing angle measure. Show your work.
1. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
2. Find the measure of angle đ„.
Lesson Summary
The sum of the remote interior angles of a triangle is equal to the measure of the exterior angle. For example, â đ¶đŽđ” + â đŽđ”đ¶ = â đŽđ¶đž.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
3. Find the measure of angle đ„. Present an informal argument showing that your answer is correct.
4. Find the measure of angle đ„.
5. Find the measure of angle đ„.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
6. Find the measure of angle đ„.
7. Find the measure of angle đ„.
8. Find the measure of angle đ„.
Lesson 14: More on the Angles of a Triangle Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 14
9. Find the measure of angle đ„.
10. Write an equation that would allow you to find the measure of angle đ„. Present an informal argument showing that your answer is correct.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 15
Lesson 15: Informal Proof of the Pythagorean Theorem
Classwork
Example 1
Now that we know what the Pythagorean theorem is, letâs practice using it to find the length of a hypotenuse of a right triangle.
Determine the length of the hypotenuse of the right triangle.
The Pythagorean theorem states that for right triangles đ2 + đ2 = đ2 where đ and đ are the legs and c is the hypotenuse. Then,
đ2 + đ2 = đ2
62 + 82 = đ2
36 + 64 = đ2
100 = đ2
Since we know that 100 = 102, we can say that the hypotenuse đ = 10.
Example 2
Determine the length of the hypotenuse of the right triangle.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 15
Exercises 1â5
For each of the exercises, determine the length of the hypotenuse of the right triangle shown. Note: Figures not drawn to scale. 1.
2.
1.
3.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 15
4.
5.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 15
Problem Set For each of the problems below, determine the length of the hypotenuse of the right triangle shown. Note: Figures not drawn to scale.
1.
2.
Lesson Summary
Given a right triangle đŽđ”đ¶ with đ¶ being the vertex of the right angle, then the sides đŽđ¶ and đ”đ¶ are called the legs of âđŽđ”đ¶ and đŽđ” is called the hypotenuse of âđŽđ”đ¶.
Take note of the fact that side đ is opposite the angle đŽ, side đ is opposite the angle đ”, and side đ is opposite the angle đ¶.
The Pythagorean theorem states that for any right triangle, đ2 + đ2 = đ2.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 15
3.
4.
5.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 15
6.
7.
8.
9.
Lesson 15: Informal Proof of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 15
10.
11.
12.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 16
Lesson 16: Applications of the Pythagorean Theorem
Classwork
Example 1
Given a right triangle with a hypotenuse with length 13 units and a leg with length 5 units, as shown, determine the length of the other leg.
The length of the leg is 12 units.
Exercises
1. Use the Pythagorean theorem to find the missing length of the leg in the right triangle.
52 + đ2 = 132 52 â 52 + đ2 = 132 â 52
đ2 = 132 â 52 đ2 = 169 â 25 đ2 = 144 đ = 12
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 16
2. You have a 15-foot ladder and need to reach exactly 9 feet up the wall. How far away from the wall should you place the ladder so that you can reach your desired location?
3. Find the length of the segment đŽđ”.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 16
4. Given a rectangle with dimensions 5 cm and 10 cm, as shown, find the length of the diagonal.
5. A right triangle has a hypotenuse of length 13 in and a leg with length 4 in. What is the length of the other leg?
6. Find the length of đ in the right triangle below.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 16
Problem Set 1. Find the length of the segment đŽđ” shown below.
2. A 20-foot ladder is placed 12 feet from the wall, as shown. How high up the wall will the ladder reach?
Lesson Summary
The Pythagorean theorem can be used to find the unknown length of a leg of a right triangle.
An application of the Pythagorean theorem allows you to calculate the length of a diagonal of a rectangle, the distance between two points on the coordinate plane and the height that a ladder can reach as it leans against a wall.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 16
3. A rectangle has dimensions 6 in by 12 in. What is the length of the diagonal of the rectangle?
Use the Pythagorean theorem to find the missing side lengths for the triangles shown in Problems 4â8.
4. Determine the length of the missing side.
5. Determine the length of the missing side.
6. Determine the length of the missing side.
Lesson 16: Applications of the Pythagorean Theorem Date: 9/18/13
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NYS COMMON CORE MATHEMATICS CURRICULUM 8âą2 Lesson 16
7. Determine the length of the missing side.
8. Determine the length of the missing side.