Transformations and Angle Relationships

Lesson 9

Objective

Describe multiple rigid transformations using coordinate points.

Common Core Standards

Core Standards

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  • 8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

  • 8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Criteria for Success

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  1. Understand that a translation in the horizontal direction adds or subtracts to the $$x-$$coordinate, and a translation in the vertical direction adds or subtracts to the $$y-$$coordinate. In general, $${(x, y) \rightarrow (x+a, y+b)}$$.
  2. Understand that a reflection over the $$x-$$axis will keep the $$x-$$coordinate the same but have the opposite value of the $$y-$$coordinate, or $${(x,y) \rightarrow (x,-y)}$$; and a reflection over the $$y-$$axis will keep the $$y-$$coordinate the same but have the opposite value of the $$x-$$coordinate, or $${(x,y) \rightarrow (-x,y)}$$.
  3. Determine new coordinates of points that undergo transformations. 

Tips for Teachers

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  • Students started some work with coordinate points and transformations in Lessons 3 and 5. This lesson extends on that to look at all three rigid transformations separately and in combination. Standard 8.G.3 will come back later in dilations. 
  • Students do not need to memorize the rules or the formal notation. The focus should be on understanding what impact a transformation has on the points on a figure and how that is represented through the values of the coordinates (MP.7). This will enable students to conceptualize these motions without using a coordinate plane.
  • Use the coordinate plane as a support, with graph paper on hand, but encourage students to try problems first without it.

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Anchor Problems

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Problem 1

Point $$A$$ is located at $${ (2, 4)}$$. Perform the following transformations on point $$A$$, and label each new point. 

  • Translate point $$A$$  $$2$$ units to the right and $$4$$ units down. Label it point $$W$$.
  • Rotate point $$A$$  $${90^{\circ}}$$ counter-clockwise about the origin. Label it point $$X$$.
  • Reflect point $$A$$ over the $$x$$-axis. Label it point $$Y$$.
  • Reflect point $$A$$ over the $$y$$-axis. Label it point $$Z$$.

Guiding Questions

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Problem 2

Figure $${{DEF}}$$ underwent two translations, and the coordinate points after each translation are shown below.

Original figure Translation 1 Translation 2
$$D(-2,-2)$$ $$D'(-2,-4)$$ $$D''(3,-2)$$
$$E(-2,-3)$$ $$E'(-2,-5)$$ $$E''(3,-3)$$
$$F(-3,-3)$$ $$F'(-3,-5)$$ $$F'' ($$ ___ , ___ $$)$$
  1. Describe the translation from $${{DEF}}$$ to $${D'E'F'}$$.
  2. What is the coordinate point for point $${F''?}$$

Guiding Questions

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Problem 3

An isosceles trapezoid has coordinate points $${A(-1,2)}$$$${B(3,2)}$$, $${C(4,-1)}$$$${D(-2,-1)}$$

If the trapezoid is reflected over the $$y$$-axis, what will be the coordinates of point $${B'}$$? Of point $${D'}$$?

Guiding Questions

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Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

  • Examples where students determine the new coordinates of points (of polygons, angles, line segments) after transformations; be sure to include single transformations as well as sequences of transformations 
  • Examples where students are given the resulting image after a sequence of transformations and must give coordinate point of pre-image
  • Error analysis of incorrect translations where not all points undergo the same transformation
  • Always, sometimes, never with concepts of the motion rules (i.e., under a reflection, the y-coordinate becomes the opposite value)
  • Challenge: Line segment $$AB$$ has an endpoint at point $$A$$, given by coordiantes $$(x, y)$$. If line segment $$AB$$ is rotated $${90^{\circ}}$$ counterclockwise, then what are the coordinates of point $$A'$$?

Target Task

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The figure shown below undergoes two transformations. First, it is reflected across the $$x$$-axis. Then the reflected image is translated $$3$$ units to the left and $$4$$ units up.

  1. Explain how you can determine the coordinates for point $${E'}$$ after the two transformations.
  2. Victoria determines that the new coordinates for point $$D$$ after the two transformations will be $${{(-5,5)}}$$. She says that after the reflection, point $$D'$$ is located at $${(-2,1)}$$, and then the translation maps it to $${{(-5,5)}}$$. Is Victoria correct? Explain why or why not.

Mastery Response

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