M2 Worksheet - EngageNY

Grade 8 Module 2 Lesson Excerpts

Lesson 3, Exercises 1-‐4 Draw a line passing through point P that is parallel to line 𝐿. Draw a second line passing through point 𝑃 that is parallel to line 𝐿, that is distinct (i.e., different) from the first one. What do you notice?

Translate line 𝐿 along the vector 𝐴𝐵. What do you notice about 𝐿 and its image 𝐿′?

Line 𝐿 is parallel to vector 𝐴𝐵. Translate line 𝐿 along vector 𝐴𝐵. What do you notice about 𝐿 and its image, 𝐿′?

Translate line 𝐿 along the vector 𝐴𝐵. What do you notice about 𝐿 and its image, 𝐿′?

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 “A above” and “B above” should be replaced by “Lesson 2” 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.)

𝐿


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.

§ Let there be a reflection across line 𝐿, let 𝑃 be a point not on line 𝐿, and let 𝑃’ represent 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑃 . Let the line 𝑃𝑃’ intersect 𝐿 at 𝑂, and let 𝐴 be a point on 𝐿 distinct from 𝑂, as

shown.

§ Because 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑃𝑂 = 𝑃′𝑂, (Reflection 2) guarantees that segments 𝑃𝑂 and 𝑃’𝑂 have the same length.

§ In other words, 𝑂 is the midpoint (i.e., the point equidistant from both endpoints) of 𝑃𝑃’. § In general, the line passing through the midpoint of a segment is said to “bisect” the segment.

Lesson 5, Problem Set 1

Let there be a rotation by – 90˚ around the center 𝑂.

During the lesson, be sure to show students how to use the transparency to rotate in multiples of 90˚.


Lesson 6, Exit Ticket 1

Let there be a rotation of 180 degrees about the origin. Point 𝐴 has coordinates −2,−4 , and point 𝐵 has coordinates (−3, 1), as shown below.

What are the coordinates of 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝐴)? Mark that point on the graph so that 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝐴) = 𝐴′.

What are the coordinates of 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝐵)? Mark that point on the graph so that 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝐵) = 𝐵′.

Lesson 7, Discussion

§ What need is there for sequencing transformations?

§ 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.

§ Specifically, if a figure undergoes two transformations 𝐹 and 𝐺, and ends up in the same place as it was originally, then the figure has been mapped onto itself.

§ Suppose we translate figure 𝐷 along vector 𝐴𝐵.


§ How do we undo this move? That is, what translation of figure 𝐷 along vector 𝐴𝐵 that would bring 𝐷′ back to its original position?

Lesson 8, Discussion

§ Does the order in which we sequence rigid motions really matter?

§ 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?

§ Let there be a reflection across line 𝐿 and let 𝑇 be the translation along vector 𝐴𝐵. Let 𝐸 represent the ellipse. The following picture shows the reflection of E followed by the translation of 𝐸.

§ Before showing the picture, ask students which transformation happens first: the reflection or the translation?

ú Reflection

§ 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 𝐸 followed by the reflection of E.

§ 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 𝐸 is not the same as the translation followed by the reflection of 𝐸; therefore a translation followed by a reflection and a reflection followed by a translation are not equal.

𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐸)

𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛, 𝑡ℎ𝑒𝑛 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓(𝐸)

𝑇𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛, 𝑡ℎ𝑒𝑛 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 (𝐸)

𝑇𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛(𝐸)


Lesson 9, Exploratory Challenge 2

a. Rotate △ 𝐴𝐵𝐶 𝑑 degrees around center 𝐷 and then rotate again 𝑑 degrees around center 𝐸. Label the image as △ 𝐴′𝐵′𝐶′ after you have completed both rotations.

b. Can a single rotation around center 𝐷 map △ 𝐴′𝐵′𝐶′ onto △ 𝐴𝐵𝐶? c. Can a single rotation around center 𝐸 map △ 𝐴!𝐵!𝐶! onto △ 𝐴𝐵𝐶? d. Can you find a center that would map △ 𝐴′𝐵′𝐶′ onto △ 𝐴𝐵𝐶 in one rotation? If so, label the

center 𝐹.


Lesson 10, Exercise 4

In the following picture, we have two pairs of triangles. In each pair, triangle 𝐴𝐵𝐶 can be traced onto a transparency and mapped onto triangle 𝐴!𝐵!𝐶!. Which basic rigid motion, or sequence of, would map one triangle onto the other?

Scenario 1:

Scenario 2:


Describe the sequence of basic rigid motions that shows 𝑆! ≅ 𝑆!. Describe the sequence of basic rigid motions that shows 𝑆! ≅ 𝑆!. Describe the sequence of basic rigid motions that shows 𝑆! ≅ 𝑆!.


Congruence is transitive!


In the figure below, 𝐿! ∥ 𝐿!, and 𝑚 is a transversal. Use a protractor to measure angles 1–8. List the angles that are equal in measure.

What did you notice about the measures of ∠1 and ∠5? Why do you think this is so? (Use your transparency, if needed).

What did you notice about the measures of ∠3 and ∠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 ∠4 and ∠6? Why do you think this is so? (Use your transparency, if needed). Is there another pair of angles with this same relationship?



The figure below shows parallel lines 𝐿! and 𝐿!. Let 𝑚 and 𝑛 be transversals that intersect 𝐿! at points 𝐵 and 𝐶, respectively, and 𝐿! at point 𝐹, as shown. Let 𝐴 be a point on 𝐿! to the left of 𝐵, 𝐷 be a point on 𝐿! to the right of 𝐶, 𝐺 be a point on 𝐿! to the left of 𝐹, and 𝐸 be a point on 𝐿! to the right of 𝐹.

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 180˚.

Write your proof below.



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


Find the length of the segment 𝐴𝐵.


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M2 Worksheet - EngageNY