Post on 23-Oct-2021
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A Story of Ratios:Grade 8 Module 2 Lesson Excerpts
Lesson 1, Exploratory Challenge 1 Describe, intuitively, what kind of transformation will be required to move the figure on the left to each of the figures (1โ3) 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 2, Exercise 2
The diagram below shows figures and their images under a translation along ๐ป๐ผ. Use the original figures and the translated images to fill in missing labels for points and measures.
Grade 8 Module 2 Lesson Excerpts
Lesson 3, Exercises 1-ยญโ4 Draw a line passing through point P that is parallel to line ๐ฟ. Draw a second line passing through point ๐ that is parallel to line ๐ฟ, that is distinct (i.e., different) from the first one. What do you notice?
Translate line ๐ฟ along the vector ๐ด๐ต. What do you notice about ๐ฟ and its image ๐ฟโฒ?
Line ๐ฟ is parallel to vector ๐ด๐ต. Translate line ๐ฟ along vector ๐ด๐ต. What do you notice about ๐ฟ and its image, ๐ฟโฒ?
Translate line ๐ฟ along the vector ๐ด๐ต. What do you notice about ๐ฟ and its image, ๐ฟโฒ?
Understanding that translations of lines produce an image that is either the line itself or a line parallel to the given line rely on the work completed at the end of Lesson 2 about the translation of a point.
Note that references to โA aboveโ and โB aboveโ should be replaced by โLesson 2โ and that the exercise numbers referenced do not match. (Exercise 4 should be Exercise 2, Exercise 5 should be Exercise 3, and Exercise 6 should be Exercise 4.)
๐ฟ
Grade 8 Module 2 Lesson Excerpts
Lesson 4, Example 4
A simple consequence of (Reflection 2: Reflections preserve lengths of segments) is that it gives a more precise description of the position of the reflected image of a point.
ยง Let there be a reflection across line ๐ฟ, let ๐ be a point not on line ๐ฟ, and let ๐โ represent ๐ ๐๐๐๐๐๐ก๐๐๐ ๐ . Let the line ๐๐โ intersect ๐ฟ at ๐, and let ๐ด be a point on ๐ฟ distinct from ๐, as
shown.
ยง Because ๐ ๐๐๐๐๐๐ก๐๐๐ ๐๐ = ๐โฒ๐, (Reflection 2) guarantees that segments ๐๐ and ๐โ๐ have the same length.
ยง In other words, ๐ is the midpoint (i.e., the point equidistant from both endpoints) of ๐๐โ. ยง In general, the line passing through the midpoint of a segment is said to โbisectโ the segment.
Lesson 5, Problem Set 1
Let there be a rotation by โ 90ห around the center ๐.
During the lesson, be sure to show students how to use the transparency to rotate in multiples of 90ห.
Grade 8 Module 2 Lesson Excerpts
Lesson 6, Exit Ticket 1
Let there be a rotation of 180 degrees about the origin. Point ๐ด has coordinates โ2,โ4 , and point ๐ต has coordinates (โ3, 1), as shown below.
What are the coordinates of ๐ ๐๐ก๐๐ก๐๐๐(๐ด)? Mark that point on the graph so that ๐ ๐๐ก๐๐ก๐๐๐(๐ด) = ๐ดโฒ.
What are the coordinates of ๐ ๐๐ก๐๐ก๐๐๐(๐ต)? Mark that point on the graph so that ๐ ๐๐ก๐๐ก๐๐๐(๐ต) = ๐ตโฒ.
Lesson 7, Discussion
ยง What need is there for sequencing transformations?
ยง Imagine life without an undo button on your computer or smartphone. If we move something in the plane, it would be nice to know we can move it back to its original position.
ยง Specifically, if a figure undergoes two transformations ๐น and ๐บ, and ends up in the same place as it was originally, then the figure has been mapped onto itself.
ยง Suppose we translate figure ๐ท along vector ๐ด๐ต.
Grade 8 Module 2 Lesson Excerpts
ยง How do we undo this move? That is, what translation of figure ๐ท along vector ๐ด๐ต that would bring ๐ทโฒ back to its original position?
Lesson 8, Discussion
ยง Does the order in which we sequence rigid motions really matter?
ยง Consider a reflection followed by a translation. Would a figure be in the same final location if the translation was done first then followed by the reflection?
ยง Let there be a reflection across line ๐ฟ and let ๐ be the translation along vector ๐ด๐ต. Let ๐ธ represent the ellipse. The following picture shows the reflection of E followed by the translation of ๐ธ.
ยง Before showing the picture, ask students which transformation happens first: the reflection or the translation?
รบ Reflection
ยง Ask students again if they think the image of the ellipse will be in the same place if we translate first and then reflect. The following picture shows a translation of ๐ธ followed by the reflection of E.
ยง It must be clear now that the order in which the rigid motions are performed matters. In the above example, we saw that the reflection followed by the translation of ๐ธ is not the same as the translation followed by the reflection of ๐ธ; therefore a translation followed by a reflection and a reflection followed by a translation are not equal.
๐ ๐๐๐๐๐๐ก๐๐๐(๐ธ)
๐ ๐๐๐๐๐๐ก๐๐๐, ๐กโ๐๐ ๐ก๐๐๐๐ ๐๐๐ก๐๐๐ ๐๐(๐ธ)
๐๐๐๐๐ ๐๐๐ก๐๐๐, ๐กโ๐๐ ๐๐๐๐๐๐๐ก๐๐๐ ๐๐ (๐ธ)
๐๐๐๐๐ ๐๐๐ก๐๐๐(๐ธ)
Grade 8 Module 2 Lesson Excerpts
Lesson 9, Exploratory Challenge 2
a. Rotate โณ ๐ด๐ต๐ถ ๐ degrees around center ๐ท and then rotate again ๐ degrees around center ๐ธ. Label the image as โณ ๐ดโฒ๐ตโฒ๐ถโฒ after you have completed both rotations.
b. Can a single rotation around center ๐ท map โณ ๐ดโฒ๐ตโฒ๐ถโฒ onto โณ ๐ด๐ต๐ถ? c. Can a single rotation around center ๐ธ map โณ ๐ด!๐ต!๐ถ! onto โณ ๐ด๐ต๐ถ? d. Can you find a center that would map โณ ๐ดโฒ๐ตโฒ๐ถโฒ onto โณ ๐ด๐ต๐ถ in one rotation? If so, label the
center ๐น.
Grade 8 Module 2 Lesson Excerpts
Lesson 10, Exercise 4
In the following picture, we have two pairs of triangles. In each pair, triangle ๐ด๐ต๐ถ can be traced onto a transparency and mapped onto triangle ๐ด!๐ต!๐ถ!. Which basic rigid motion, or sequence of, would map one triangle onto the other?
Scenario 1:
Scenario 2:
Lesson 11, Exercise 1
Describe the sequence of basic rigid motions that shows ๐! โ ๐!. Describe the sequence of basic rigid motions that shows ๐! โ ๐!. Describe the sequence of basic rigid motions that shows ๐! โ ๐!.
Grade 8 Module 2 Lesson Excerpts
Congruence is transitive!
Lesson 12, Exploratory Challenge 2
In the figure below, ๐ฟ! โฅ ๐ฟ!, and ๐ is a transversal. Use a protractor to measure angles 1โ8. List the angles that are equal in measure.
What did you notice about the measures of โ 1 and โ 5? Why do you think this is so? (Use your transparency, if needed).
What did you notice about the measures of โ 3 and โ 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.
What did you notice about the measures of โ 4 and โ 6? Why do you think this is so? (Use your transparency, if needed). Is there another pair of angles with this same relationship?
Lesson 13, Exploratory Challenge 2
Grade 8 Module 2 Lesson Excerpts
The figure below shows parallel lines ๐ฟ! and ๐ฟ!. Let ๐ and ๐ be transversals that intersect ๐ฟ! at points ๐ต and ๐ถ, respectively, and ๐ฟ! at point ๐น, as shown. Let ๐ด be a point on ๐ฟ! to the left of ๐ต, ๐ท be a point on ๐ฟ! to the right of ๐ถ, ๐บ be a point on ๐ฟ! to the left of ๐น, and ๐ธ be a point on ๐ฟ! to the right of ๐น.
Name the triangle in the figure.
Name a straight angle that will be useful in proving that the sum of the interior angles of the triangle is 180ห.
Write your proof below.
Lesson 14, Exercise 4
Grade 8 Module 2 Lesson Excerpts
Show that the measure of an exterior angle is equal to the sum of the related remote interior angles.
Lesson 15, Proof of Pythagorean theorem
Lesson 16, Exercise 3
Find the length of the segment ๐ด๐ต.
Grade 8 Module 2 Lesson Excerpts