Unit 1 Rigid Transformations And Congruence — Unit Plan

Title	Assessment
Lesson 1 Moving in the Plane	Frame to Frame Here are positions of a shape: Describe how the shape moves from: Frame 1 to Frame 2. Frame 2 to Frame 3. Frame 3 to Frame 4. Show Solution Slide down Turn counterclockwise 90 degrees (or one quarter of a full turn) Slide up
Lesson 2 Naming the Moves	Is It a Reflection? What type of move takes Figure A to Figure B? Explain your reasoning. Show Solution Sample responses: The move is 1 rotation. If Figure A is turned around the point shared by Figures A and B, it can land on Figure B. The move is 2 reflections. If Figure A is flipped over line $\ell$ and then flipped over again so that the shared points and angle line up, then it can land on Figure B.
Lesson 3 Grid Moves	Triangle Images Translate triangle $ABC$ so that $B$ goes to $B’$ . Reflect triangle $ABC$ over line $\ell$ . Show Solution
Lesson 4 Making the Moves	What Does It Take? For each description of a transformation, identify what information is missing. Translate triangle $ABC$ to the right. Rotate triangle $ABC$ $90^\circ$ around point $C$ . Reflect triangle $ABC$ over a line. Show Solution Sample responses: Distance—how many units to the right Direction—clockwise or counterclockwise A drawing or description of where the line is
Lesson 5 Coordinate Moves	Rotation or Reflection One of the triangles pictured is a rotation of triangle $ABC$ and one of them is a reflection. Triangle A B C reflected on a coordinate plane, origin O. Horizontal axis scale negative 6 to 6 by 1’s. Vertical axis scale negative 5 to 5 by 1’s. Triangle A B C is blue and has coordinates: A(1 comma 1), B(3 comma 2) and C(2 comma 5). The green triangle has coordinates: (negative 1 comma 1), (negative 2 comma 3) and (negative 5 comma 2). The red triangle has coordinates: (1 comma negative 1), (3 comma negative 2) and (2 comma negative 5). Label the rotated image $PQR$ . Label the reflected image $XYZ$ . Show Solution
Lesson 6 Describing Transformations	Describing a Sequence of Transformations Triangle T' is the image of Triangle T. Han gave this information to Jada to describe the sequence of transformations. Triangle T is reflected over line $\ell$ . Triangle T is translated 2 units to the left. The order of the sequence of transformations is translation, then reflection. Which of these figures shows the correct Triangle T'? Figure 1 Figure 2 Show Solution Figure 2
Section A Check Section A Checkpoint	Problem 1 Here is line segment $AB$ and a point $C$ . Reflect line segment $AB$ across the $x$ -axis. What are the coordinates of the new endpoints? Point $C$ is translated 3 units to the left and 2 units up. Plot this point on the grid and label it $C’$ . Show Solution The image of $A$ is at $(\text-4,\text-5)$ and the image of $B$ is at $(3, \text-2)$ . Problem 2 Here are 2 figures. Describe a sequence of transformations that takes triangle $ABC$ to triangle $DEF$ . Show Solution Sample response: Translate triangle $ABC$ so that $A$ moves to $D$ . Rotate 90 degrees counterclockwise around point $D$ .

Lesson 7 No Bending or Stretching	Translated Trapezoid Trapezoid $A’B’C’D’$ is the image of trapezoid $ABCD$ under a rigid transformation. Trapezoid A B C D and its image, trapezoid A prime B prime C prime and D prime. Angle A is 130 degrees, angle B is 50 degrees and angles D and C are right angles. Side A prime D prime is 6 units and side D prime C prime is 4 units. Label all vertices on trapezoid $A’B’C’D’$ . On both figures, label all known side lengths and angle measures. Show Solution
Lesson 8 Rotation Patterns	Is It a Rotation? Triangle $ABC$ is rotated $180^\circ$ around point $C$ . Will the image line up with triangle $CDE$ ? Explain how you know. Show Solution No. Sample response: If triangle $CDE$ was a $180^\circ$ rotation of triangle $ABC$ , then line segment $AB$ would be parallel to line segment $DE$ .
Lesson 9 Moves in Parallel	Finding Unknown Measurements Points $A’$ and $B’$ are the images of $A$ and $B$ after a $180^\circ$ rotation around point $O$ . Answer each question and explain your reasoning without measuring segments or angles. Name a segment whose length is the same as segment $AO$ . What is the measure of angle $A'OB'$ ? Show Solution Segment $A’O$ , because $A’$ is the image of $A$ after a $180^\circ$ rotation with center at $O$ . This rotation preserves distances and takes segment $AO$ to segment $A’O$ . $79^\circ$ , the same measure as $\angle AOB$ , because the $180^\circ$ rotation with center at $O$ takes $\angle AOB$ to $\angle A’OB’$ . The rotation preserves angle measures.
Section B Check Section B Checkpoint	Problem 1 Here is a line segment $CD$ with midpoint $M$ . Rotate segment $CD$ $90^\circ$ clockwise around point $M$ , and label the image as $FG$ . Rotate segment $CD$ $180^\circ$ around point $E$ and label the new segment $HJ$ . Which segment is parallel to segment $CD$ ? Show Solution See image See image $HJ$ is parallel to $CD$ Problem 2 Triangle $EDC$ is the image of triangle $ABC$ after a rigid transformation. Describe a rigid transformation that takes $ABC$ to $EDC$ . Name 2 angles that have the same measure and explain how you know. Name 2 side lengths that must be the same and explain how you know. Show Solution Sample responses: Rotate triangle $ABC$ $180^\circ$ around point $C$ . Angle $A$ is the same as angle $E$ , or angle $B$ is the same as angle $D$ , or angle $ACB$ is the same as angle $ECD$ ; since triangle $EDC$ is the image of triangle $ABC$ after a rigid transformation, the corresponding angles are the same measure. $AB$ is the same length as $ED$ , or $BC$ is the same length as $DC$ , or $AC$ is the same length as $EC$ ; since triangle $EDC$ is the image of triangle $ABC$ after a rigid transformation, the corresponding side lengths are the same.

Lesson 11 What Is the Same?	Mirror Images Figure B is the image of Figure A when reflected across line $\ell$ . Are Figure A and Figure B congruent? Explain your reasoning. Show Solution Yes, they are congruent. There is a rigid transformation that takes one figure to the other, so they are congruent.
Lesson 13 Congruence	Explaining Congruence Are Figures A and B congruent? Explain your reasoning. Show Solution These figures are not congruent. Sample reasoning: If they were congruent, the longest horizontal distances between two points would be the same. However, for A it is less than 4 units, and for B it is about 4 units.
Section C Check Section C Checkpoint	Problem 1 Which shape is congruent to Shape A? Describe a rigid transformation that takes A to that figure. Which shape is not congruent to Shape A? Explain how you know. Show Solution Shape B is congruent to Shape A. Sample response: Translate Shape A 4 units right, then rotate $90^\circ$ clockwise around $(2, 1)$ . Shape C is not congruent to Shape A. Sample response: Shape C has 2 side lengths of 3, but Shape A has no side lengths of 3, so they cannot be congruent.

Lesson 14 Alternate Interior Angles	All the Rest The diagram shows two parallel lines cut by a transversal. One angle measure is shown. Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angles are marked in clockwise order as a degrees, b, degrees, c degrees, and 54 degrees. At the second intersection, angles are marked in clockwise order as e degrees, f degrees, g degrees, and d degrees. Find the values of $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , and $g$ . Show Solution $a$ : $126$ , $b$ : $54$ , $c$ : $126$ , $d$ : $54$ , $e$ : $126$ , $f$ : $54$ , $g$ : $126$
Lesson 15 Adding the Angles in a Triangle	Three Angles Tyler has 3 right angles. Can he use them to make a triangle? Explain your reasoning. Show Solution No. Sample reasoning: 3 right angles sums to more than 180 degrees, since $3\boldcdot90=270$ .
Lesson 16 Parallel Lines and the Angles in a Triangle	Angle Sum What is the sum of the angle measures of triangle $ABC$ ? How do you know? Show Solution $180^\circ$ . Sample response: Since the base and vertex lie on grid lines, we can see the line parallel to $BC$ through $A$ is a straight line. The three angles around point $A$ add up to a straight angle. Using alternate interior angles, two angles are congruent to angle $B$ and angle $C$ , and the third angle is the same as angle $A$ . So angles $A$ , $B$ , and $C$ add up to $180^\circ$ .
Section D Check Section D Checkpoint	Problem 1 Line $FG$ is parallel to line $HJ$ and cut by transversal $m$ . Find each angle measure: $a$ $b$ $c$ $d$ Show Solution $45^\circ$ $135^\circ$ $45^\circ$ $135^\circ$ Problem 2 Line $AB$ is parallel to line $CD$ . Explain how you know that the sum of the angles of triangle $ABC$ is $180^\circ$ . Show Solution Sample response: Angles $ECA$ , $ACB$ , and $BCD$ form a straight angle, which is $180^\circ$ . Angle $ECA$ is congruent to angle $CAB$ because they are alternate interior angles. Angle $BCD$ is congruent to angle $CBA$ because they are alternate interior angles. So the angles in triangle $ABC$ are congruent to the angles that make a straight angle and must also sum to $180^\circ$ .

Lesson 17 Rotate and Tessellate	No cool-down
Unit 1 Assessment End-of-Unit Assessment	Problem 1 Select all the true statements. A. Two squares with the same side lengths are always congruent. B. Two rectangles with the same side lengths are always congruent. C. Two rhombuses with the same side lengths are always congruent. D. Two parallelograms with the same side lengths are always congruent. E. Two quadrilaterals with the same side lengths are always congruent. Show Solution A, B Problem 2 Lines $CE$ and $AD$ intersect at $B$ . Select all the true statements. A. The measure of angle $CBA$ is equal to the measure of angle $DBE$ . B. The sum of the measures of angles $CBA$ and $DBE$ is $180^\circ$ . C. The measure of angle $CBD$ is equal to the measure of angle $ABE$ . D. The sum of the measures of angles $CBD$ and $CBA$ is $180^\circ$ . E. The sum of the measures of angles $CBA$ and $DBE$ is $90^\circ$ . Show Solution A, C, D Problem 3 Diego made the shape on the left, and Elena made the shape on the right. Each shape uses 5 circles. <p>Two figures in a grid. One figure has five circles which share a common center. The diameter of the largest circle is 10 units. The diameter of the next circle is 8 units. The diameter of the next circle is 6 units. The diameter of the next circle is 4 units. The diameter of the smallest circle is 2 units. Another figure has five circles which all share a common point. The diameter of the largest circle is 10 units. The diameter of the next circle is 8 units. The diameter of the next circle is 6 units. The diameter of the next circle is 4 units. The diameter of the smallest circle is 2 units.</p> Select all the true statements. A. The smallest circle in Diego's design is congruent to the smallest circle in Elena’s design. B. Diego’s design is congruent to Elena’s design. C. Elena’s design is a translation of Diego's design. D. The largest circle in Elena's design is congruent to the largest circle in Diego's design. E. Each circle in Elena's design has a congruent circle within Diego's design. Show Solution A, D, E Problem 4 Describe a sequence of transformations that shows that Polygon A is congruent to Polygon B. <p>Two pentagons on a grid. Figure A has a vertex at 1 unit right and 3 units up. Another vertex at 1 unit right and 5 units up. Another vertex at 2 units right and 6 units up. Another vertex at 4 units right and 4 units up. Another vertex at 2 units right and 4 units up. Figure B has a vertex at 6 units right and 2 units up. Another vertex at 7 units right and 1 unit up. Another vertex at 9 units right and 1 unit up. Another vertex at 8 units right and 2 units up. Another vertex at 8 units right 4 units up.</p> Show Solution Answers vary. Sample response: Polygon A can be rotated 90 degrees counterclockwise around its furthermost point on the right shown in the picture and then translated 4 units to the right. Minimal Tier 1 response: Work is complete and correct. Sample: Rotate 90 degrees counterclockwise around the rightmost point of Polygon A, translate 4 units right. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: A drawing showing the intermediate transformation (the green polygon in the sample response), but no verbal descriptions; incomplete verbal descriptions (such as reference to a rotation without specifying a center point); the sequence of transformations contains a small, easily identifiable error (such as saying to rotate clockwise when the counterclockwise direction is the one that works); sequence of transformations is correct but does not use proper vocabulary (“turn” instead of rotate; “move” instead of translate). Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Explanation without reference to rigid transformations, or a sequence of transformations that does not take Polygon A to Polygon B (with no obvious small mistakes responsible for this error); descriptions of transformations are very unclear and the intended meaning is not evident. Problem 5 For each pair of shapes, decide whether or not Shape A is congruent to Shape B. Explain your reasoning. First pair: Second pair: Show Solution Congruent. If Shape A is reflected over its right side, then rotated 90 degrees counterclockwise around the lower vertex, it can be placed on top of Shape B with a translation down and to the right. Not congruent. The shapes look congruent, but when Shape A is moved on top of Shape B with a 90-degree counterclockwise rotation and a translation, they do not match up. Minimal Tier 1 response: Work is complete and correct. Acceptable errors: Omitting reference to lines of reflection, centers of rotation, angles of rotation, and distance of translation, provided the visual makes these things clear. Sample: (with accompanying accurate drawing) Congruent, because I can reflect Shape A, rotate it, and then translate it onto Shape B. (with accompanying accurate drawing) Not congruent, because when I rotate Shape A and then translate it, it still doesn’t match up with Shape B. Alternate response: I measured the angles, and they are not the same in the two shapes. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Transformations are shown, but with no written descriptions; in part b, transformations are done mostly correctly but enough accuracy was lost that the shapes appear to coincide. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Work states that shapes are or are not congruent with no justification; descriptions of transformations are unclear and the intended meaning is not evident; vague explanations, such as “the shapes look the same.” Problem 6 Lines $AB$ and $CD$ are parallel. Find the measures of the three angles in triangle $ABF$ . Show Solution $B: 42^\circ$ , $A: 23^\circ$ , $F:115^\circ$ Problem 7 Triangle $CDA$ is the image of triangle $ABC$ after a $180^\circ$ rotation around the midpoint of segment $AC$ . Triangle $ECB$ is the image of triangle $ABC$ after a $180^\circ$ rotation around the midpoint of segment $BC$ . The measure of angle $D$ is $52^\circ$ and the measure of angle $E$ is $94^\circ$ . Identify at least two pairs of congruent angles in the figure and explain how you know they are congruent. What is the measure of angle $CBE$ ? Explain how you know. Name a triangle congruent to triangle $CBE$ . Describe a rigid transformation from triangle $CBE$ to that triangle. Show Solution Any two pairs: Angle $D$ , angle $ECB$ , and angle $CBA$ are congruent; angle $DCA$ , angle $E$ , and angle $BAC$ are congruent; angle $DAC$ , angle $BCA$ , and angle $CBE$ are congruent. Sample reasoning: Triangle $CDA$ and triangle $ECB$ are the images of triangle $ABC$ after a rotation, so there is a rigid transformation from any one of the triangles to the other, and they must be congruent. Since the triangles are congruent, their corresponding angles are also congruent. $34^\circ$ . Angle $ECB$ is $52^\circ$ since it is congruent to angle $D$ . Angle $DCA$ is $94^\circ$ since it is congruent to angle $E$ . Since the angles around point $C$ form a straight angle, they must be $180^\circ$ . Angle $ACB$ is $34^\circ$ since $180-94-52=34$ . Since angle $ACB$ is congruent to angle $CBE$ , angle $CBE$ is also $34^\circ$ . Sample responses: Triangle $DAC$ . Translate left by the directed line segment from point C to point D. Triangle $BCA$ . Rotate $180^\circ$ around the midpoint of $BC$ . Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample: Angle $D$ and angle $ECB$ . Angle $E$ , and angle $DCA$ . Triangle $ACD$ is the image of triangle $BEC$ after a translation, so their corresponding angles are congruent. $34^\circ$ . Angle $ECB$ is $52^\circ$ since it is congruent to angle $D$ . Angle $DCA$ is $94^\circ$ since it is congruent to angle $E$ . Since the angles around point $C$ form a straight angle, they must be $180^\circ$ . Angle $ACB$ is $34^\circ$ since $180-94-52=34$ . Since angle $ACB$ is congruent to angle $CBE$ , angle $CBE$ is also $34^\circ$ . Triangle $DAC$ . Translate left by the directed line segment from point C to point D. Acceptable errors: An argument in part c that relies on congruent pairs of angles named in part a without restating the congruence in part c. Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: Explanations in parts b and c appeal to the diagram and are logically sequenced but do not appeal specifically to transformations; good, complete explanations for parts b and c with incorrect angles identified in part b; no mention that the rotations are $180^\circ$ in part b; one incorrect angle pair in part a; work for part c mentions that the sum of three angle measures is $180^\circ$ but does not justify this by saying they are in a triangle or a straight angle. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: Work for part a does not appeal to rotations; response to part c does not mention congruent angle pairs (see note under Tier 1 response); response to part c does not mention angles adding to $180^\circ$ ; two incorrect angle pairs in part a without excellent parts b and c; three or more error types under Tier 2 response. Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery. Sample errors: Little progress on any of the problem parts; justification with many errors or no justification for parts b and c.