Objective
Describe multiple rigid transformations using coordinate points.
Common Core Standards
Core Standards
The core standards covered in this lesson
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.
Geometry
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.
Geometry
8.G.A.3— Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Criteria for Success
The essential concepts students need to demonstrate or understand to achieve the lesson objective
- 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,$${(x, y) \rightarrow (x+a, y+b)}$$.
- 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 $${(x,y) \rightarrow (-x,y)}$$.
- Determine new coordinates of points that undergo transformations.
Tips for Teachers
Suggestions for teachers to help them teach this lesson
- 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.
Lesson Materials
- Optional: Patty paper (transparency paper)
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Anchor Problems
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Problem 1
Point $$A$$is located at $${ (2, 4)}$$. Perform the following transformations on point $$A$$, and label each new point with its letter and coordinates.
What impact does each transformation have on the coordinates of point $$A$$?
a.Translate point$$A$$2units to the right and 4units down. Label it point $$W$$.
b.Rotate point $$A$$ $${90^{\circ}}$$counter-clockwise about the origin. Label it point $$X$$.
c.Reflect point $$A$$over the $$x$$-axis. Label it point $$Y$$.
d.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 |
| $$D(-2,-2)$$ | $$D'(-2,-4)$$ | $$D''(3,-2)$$ |
| $$E(-2,-3)$$ | $$E'(-2,-5)$$ | $$E''(3,-3)$$ |
| $$F(-3,-3)$$ | $$F'(-3,-5)$$ | $$F'' ($$___ , ___$$)$$ |
a.Describe the translation from$${{DEF}}$$to$${D'E'F'}$$.
b.What is thecoordinate 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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Target Task
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
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.
a.Explain how you can determine the coordinates for point$${E'}$$after the two transformations.
b.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$${{(-5,5)}}$$. IsVictoria correct? Explain why or why not.
Student Response
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Additional Practice
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or 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 coordinates$$(x, y)$$. If line segment$$AB$$is rotated$${90^{\circ}}$$counterclockwise, then what are the coordinates of point$$A'$$?
- Open Up Resources Grade 8 Unit 1 Practice Problems—Lesson 5
- Grade 8 Mathematics Sample ER Item Form Claim 3—Page 2
Lesson 8
Lesson 10
