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# Transformations and Angle Relationships

## 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.

#### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problem 1 (benefits from worked example). Find more guidance on adapting our math curriculum for remote learning here.

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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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With Fishtank Plus, you can download a complete problem set and answer key for this lesson. Download Sample

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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