Rigorous Curriculum Design
Rigorous Curriculum Design
Unit Planning Organizer
|Subject(s) |Middle Grades Mathematics |
|Grade/Course |8th |
|Unit of Study |Unit 2: Congruency & Similarity, Lines & Angles |
|Unit Type(s) |❑Topical X Skills-based ❑ Thematic |
|Pacing |22 days |
|Unit Planning Organizer Content |
| Unit Abstract | EQ’s/ Corresponding Big Ideas |
|NCSCOS State Standards |Vocabulary |
|Standards for Mathematical Practice |Language Objectives |
|“Unpacked Standards” |Information and Technology Standards |
|Concepts/Skills/DOK’s |Instructional Resources and Materials |
|Unit Abstract |
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|Students will discover rigid transformations and explore properties that are preserved by translations, rotations, and reflections. They will |
|use coordinates to explore transformations on the coordinate plane. Students will continue their study of the basic rigid motion |
|transformations: rotations, reflections, and translations. They will discover how these basic transformations can be combined. Students will |
|confirm that key properties of rotations, reflections, and translations are preserved when transformations are combined. They will identify a |
|sequence of rigid motion transformations that will map one figure onto another when given the pre-image and image, including figures in the |
|coordinate plane. Students will extend their work with scale figures and similarity. They will explore the images of figures under a dilation |
|as they work to determine which geometric properties are preserved under dilations. Students will discover how scale factors greater than 1 |
|and less than 1 impact the size of a pre-image. They will connect their understanding of congruence and similarity by seeing that similar |
|figures with a scale factor of 1 are also congruent. |
|Students will study and describe the interior and exterior angles of a triangle, as well as develop the relationships among angles described |
|by the angle sum theorem and the exterior angle theorem. They will use transformations to create parallel lines and to identify and apply the |
|relationship between the lines and the angles formed when parallel lines are cut by a transversal. |
|Vertical Articulation: In seventh grade students used facts about supplementary, complementary, vertical, and adjacent angles to solve |
|multi-step equations for an unknown angle in a figure. In Math I students will work on horizontal and vertical translations with linear and |
|exponential functions. |
|NCSCOS |
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|Domains: Geometry (8.G) |
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|Clusters: Understand congruence and similarity using physical models, transparencies, |
|or geometry software. |
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|Standards: |
|8.G.1 VERIFY experimentally the properties of rotations, reflections, and translations: |
|a. Lines are taken to lines, and line segments to line segments of the same length |
|b. Angles are taken to angles of the same measure. |
|c. Parallel lines are taken to parallel lines. |
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|8.G.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. |
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|8.G.3 DESCRIBE the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. |
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|8.G.4 UNDERSTAND that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, |
|reflections, translations, and dilations; given two similar two- dimensional figures, DESCRIBE a sequence that exhibits the similarity between|
|them. |
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|8.G.5 USE informal arguments to ESTABLISH facts about the angle sum and exterior angle of triangles, about the angles created when parallel |
|lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same |
|triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. |
|Standards for Mathematical Practice |
|1. Make sense of problems and persevere in solving them. | |
|2. Reason abstractly and quantitatively. |5. Use appropriate tools strategically. |
|3. Construct viable arguments and critique the reasoning of |6. Attend to precision. |
|others. |7. Look for and make use of structure. |
|4. Model with mathematics. |8. Look for and express regularity in repeated reasoning. |
|Unpacked Standards |
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|8.G.1 Students use compasses, protractors and rulers or technology to explore figures created from translations, reflections and rotations. |
|Characteristics of figures, such as lengths of line segments, angle measures and parallel lines, are explored before the transformation |
|(pre-image) and after the transformation (image). Students understand that these transformations produce images of exactly the same size and |
|shape as the pre-image and are known as rigid transformations. |
|8.G.2 This standard is the students’ introduction to congruency. Congruent figures have the same shape and size. Translations, reflections |
|and rotations are examples of rigid transformations. A rigid transformation is one in which the pre-image and the image both have exactly the|
|same size and shape since the measures of the corresponding angles and corresponding line segments remain equal (are congruent). |
|Students examine two figures to determine congruency by identifying the rigid transformation(s) that produced the figures. Students recognize|
|the symbol for congruency (≅) and write statements of congruency. |
|Example 1: |
|Is Figure A congruent to Figure A’? Explain how you know. |
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|Solution: |
|These figures are congruent since A’ was produced by translating each vertex of Figure A, 3 to the right and 1 down. |
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|Example 2: |
|Describe the sequence of transformations that results in the transformation of Figure A to Figure A’. |
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|[pic] |
|Solution: |
|Figure A’ was produced by a 90º clockwise rotation around the origin. |
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|8.G.3 Students identify resulting coordinates from translations, reflections, and rotations (90º, 180º and 270º both clockwise and |
|counterclockwise), recognizing the relationship between the coordinates and the transformation. |
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|Translations |
|Translations move the object so that every point of the object moves in the same direction as well as the same distance. In a translation, the|
|translated object is congruent to its pre-image. Triangle ABC has been translated 7 units to the right and 3 units up. To get from A (1, 5) to|
|A’ (8, 8), move A 7 units to the right (from x = 1 to x = 8) and 3 units up (from y = 5 to y = 8). Points B and C also move in the same |
|direction (7 units to the right and 3 units up), resulting in the same changes to each coordinate. |
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|Reflections |
|A reflection is the “flipping” of an object over a line, known as the “line of reflection”. In the 8thgrade, the line of reflection will be |
|the x-axis and the y-axis. Students recognize that when an object is reflected across the y-axis, the reflected x-coordinate is the opposite |
|of the pre-image x-coordinate (see figure below). |
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|Likewise, a reflection across the x-axis would change a pre-image coordinate (3, -8) to the image coordinate of (3, 8) -- note that the |
|reflected y-coordinate is opposite of the pre-image y-coordinate. |
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|Rotations |
|A rotation is a transformation performed by “spinning” the figure around a fixed point known as the center of rotation. The figure may be |
|rotated clockwise or counterclockwise up to 360º (at 8th grade, rotations will be around the origin and a multiple of 90º). In a rotation, |
|the rotated object is congruent to its pre-image. |
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|Consider when triangle DEF is 180˚ clockwise about the origin. The coordinate of |
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|triangle DEF are D(2,5), E(2,1), and F(8,1). When rotated 180˚ about the origin, the new coordinates are D’(-2,-5), E’(-2,-1) and F’(-8,-1). |
|In this case, each coordinate is the opposite of its pre-image (see figure below). |
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|Dilations |
|A dilation is a non-rigid transformation that moves each point along a ray which starts from a fixed center, and multiplies distances from |
|this center by a common scale factor. Dilations enlarge (scale factors greater than one) or reduce (scale factors less than one) the size of |
|a figure by the scale factor. In 8th grade, dilations will be from the origin. The dilated figure is similar to its pre-image. |
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|The coordinates of A are (2, 6); A’ (1, 3). The coordinates of B are (6, 4) and B’ are |
|(3, 2). The coordinates of C are (4, 0) and C’ are (2, 0). Each of the image coordinates is [pic] the value of the pre-image coordinates |
|indicating a scale factor of[pic]. |
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|The scale factor would also be evident in the length of the line segments using the |
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|Ratio = image length |
|pre-image length |
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|Students recognize the relationship between the coordinates of the pre-image, the image and the scale factor for dilation from the origin. |
|Using the coordinates, students are able to identify the scale factor (image/pre-image). |
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|Students identify the transformation based on given coordinates. For example, the pre-image coordinates of a triangle are A (4, 5), B (3, 7),|
|and C (5, 7). The image coordinates are A (-4, 5), B (-3, 7), and C (-5, 7). What transformation occurred? |
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|8.G.4 Similar figures and similarity are first introduced in the 8th grade. Students understand similar figures have congruent angles and |
|sides that are proportional. Similar figures are produced from dilations. Students describe the sequence that would produce similar figures,|
|including the scale factors. Students understand that a scale factor greater than one will produce an enlargement in the figure, while a |
|scale factor less than one will produce a reduction in size. |
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|Example1: |
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|Is Figure A similar to Figure A’? Explain how you know. |
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|Solution: |
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|Dilated with a scale factor of ½ then reflected across the x-axis, making figures A and A’ similar. |
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|Students need to be able to identify that triangles are similar or congruent based on given information. |
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|Example 2: |
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|Describe the sequence of transformations that results in the transformation of Figure A to Figure A’. |
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|Solution: |
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|90° clockwise rotation, translate 4 right and 2 up, dilation of ½. In this case, the scale factor of the dilation can be found by using the |
|horizontal distances on the triangle (image = 2 units; pre-image = 4 units) |
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|8.G.5 Students use exploration and deductive reasoning to determine relationships that exist between the following: a) angle sums and |
|exterior angle sums of triangles, b) angles created when parallel lines are cut by a transversal, and c) the angle-angle criterion |
|for similarity of triangles. |
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|Students construct various triangles and find the measures of the interior and exterior angles. Students make conjectures about the |
|relationship between the measure of an exterior angle and the other two angles of a triangle, (the measure of an exterior angle of a triangle|
|is equal to the sum of the measures of the other two interior angles) and the sum of the exterior angles (360º). Using these relationships, |
|students use deductive reasoning to find the measure of missing angles. |
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|Students construct parallel lines and a transversal to examine the relationships between the created angles. Students recognize vertical |
|angles, adjacent angles and supplementary angles from 7th grade and build on these relationships to identify other pairs of congruent angles. |
|Using these relationships, students use deductive reasoning to find the measure of missing angles. |
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|Example 1: |
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|You are building a bench for a picnic table. The top of the bench will be parallel to the ground. If m[pic]1 = 148˚, find m[pic]2 and |
|m[pic]3. Explain your answer. |
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|[pic] |
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|Solution: |
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|Angle 1 and angle 2 are alternate interior angles, giving angle 2 a measure of 148º. Angle 2 and angle 3 are supplementary. Angle 3 will |
|have a measure of 32º so the m[pic]2 + m[pic]3 = 180º |
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|Example 2: |
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|Show that m[pic]3 + m[pic]4 + m[pic]5 = 180º if line l and m are parallel lines and t1 and t2 are transversals. |
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|Solution: |
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|[pic]1 + [pic]2 + [pic]3 = 180˚ |
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|[pic]5≅[pic]1 corresponding angles are congruent therefore[pic]1can be substituted for [pic]5. |
|[pic]4 ≅[pic]2 alternate interior angles are congruent; therefore [pic]4 can be substituted for [pic]2. Therefore [pic]3+ [pic]4 + [pic]5 = |
|180˚ |
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|Students can informally conclude that the sum of the angles in a triangle is 180º (the angle-sum theorem) by applying their understanding of |
|lines and alternate interior angles. |
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|Example 3: |
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|In the figure below Line X is parallel to Line [pic]. Prove that the sum of the angles of a triangle is 180º. |
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|[pic] |
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|Solution: |
|Angle a is 35º because it alternates with the angle inside the triangle that measures 35º. Angle c is 80º because it alternates with the |
|angle inside the triangle that measures 80º. Because lines have a measure of 180º, and angles a + b + c form a straight line, then angle b |
|must be 65 º(180 – (35 + 80) = 65. Therefore, the sum of the angles of the triangle is 35º + 65 º + 80 º. |
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|Example 4: |
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|What is the measure of angle 5 if the measure of angle 2 is |
|45º and the measure of angle 3 is 60º? |
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|Solution: |
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|Angles 2 and 4 are alternate interior angles; therefore the measure of angle 4 is also 45º. The measure of angles 3, 4 and 5 must add to |
|180º. If angles 3 and 4 add to 105º the angle 5 must be equal to 75º. |
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|Students construct various triangles having line segments of different lengths but with two corresponding congruent angles. Comparing ratios |
|of sides will produce a constant scale factor, meaning the triangles are similar. Students solve problems with similar triangles. |
|“Unpacked” Concepts |“Unwrapped” Skills |Cognition (DOK) |
|(students need to know) |(students need to be able to do) | |
|8.G.1 | | |
|Properties of rotations, reflections, and translations |I can verify (experimentally) the properties of |2 |
| |reflection, rotation, and translation on a figure, and | |
| |compare the pre-image to the image. | |
|8.G.2 | | |
|Transformational congruency |I can describe a sequence of reflections, rotations, |2 |
| |and/or translations that will map a given shape onto | |
| |another congruent shape. | |
|8.G.3 |Given the coordinates of the vertices of a figure: | |
|Effect of dilations, translations, rotations, and |I can reflect the figure over the x-axis or y-axis. |2 |
|reflections on two-dimensional figures using coordinates |I can rotate a figure around the origin and a multiple of | |
| |90° clockwise or counterclockwise. |2 |
| |I can translate the figure a given number of units |2 |
| |vertically and/or horizontally. | |
| |I can dilate the figure given any scale factor with the |2 |
| |origin as the center. | |
|8.G.4 |Given two similar figures on a coordinate plane: | |
|Similar figures and similarity |I can describe a sequence that would produce similar | |
| |figures including scale factors. |2 |
| |I can demonstrate that two similar figures have congruent | |
| |angles and proportional sides. |2 |
|G.8.5 | | |
|Angle relationships |I can show an informal visual proof to establish the fact | |
| |that the interior angles of any triangle add up to 180° |3 |
| |and the exterior angles of any triangle add up to 360°. | |
| |I can show relationship of angles created by parallel | |
| |lines and a transversal. |2 |
| |I can find measures of missing angles. | |
| |I can show an informal visual proof to establish the fact |2 |
| |that two triangles are similar if at least two sets of | |
| |corresponding angles are congruent. |2 |
| |I can solve problems with similar triangles. | |
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|Essential Questions |Corresponding Big Ideas |
|8.G.1 | |
|How can I verify (experimentally) the properties of a reflection, |Students will verify (experimentally) the properties of reflection, |
|rotation, and translation on a figure, comparing the pre-image to the |rotation, and translation on a figure, and compare the pre-image to |
|image? |the image. |
|8.G.2 | |
|How can I know that a two-dimensional figure is congruent to another? |Students will describe what it means for two shapes to be congruent |
| |using vocabulary of transformational geometry. |
| |Students will describe a sequence of reflections, rotations, and/or |
|How can I describe a sequence that exhibits the congruence between two|translations that will map a given shape onto another congruent shape. |
|congruent figures? |Students will demonstrate two shapes are congruent by using physical |
| |models, transparencies, and geometry software (e.g. Geometer |
| |Sketchpad). |
|8.G.3 |Given the coordinates of the vertices of a figure: |
|How can I describe the effect of dilations, translations, rotations, |Students will reflect the figure over the x-axis or y-axis. |
|and reflections on two-dimensional figures using coordinates? |Students will rotate the figure a multiple of 90° clockwise or |
| |counterclockwise about the origin. |
| |Students will translate the figure a given number of units vertically |
| |and/or horizontally. |
| |Students will dilate the figure given any scale factor with the origin |
| |as the center. |
|8.G.4 | |
|How can I determine that a two-dimensional figure is similar to |Students will describe what it means for two figures to be similar. |
|another? | |
| |Given two similar figures on a coordinate grid, |
| |Students will describe a sequence of reflections, rotations, and/or |
|How can I describe a sequence that exhibits the similarity between two|translations that will map a given figure onto another similar figure. |
|figures? | |
| |Students will demonstrate two figures are similar by using physical |
| |models, transparencies, and geometry software (e.g. Geometer |
| |Sketchpad). |
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|G.8.5 | |
|How can I use informal arguments to establish facts about the angle |Students will show an informal visual proof to establish the fact that |
|sum and exterior angle of triangles? |the interior angles of any triangle add up to 180° and exterior angles |
| |add up to 360°. |
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| |Given parallel lines cut by a transversal: |
| |Students will show an informal proof to establish corresponding angles |
| |are congruent. |
|How can I use informal arguments to establish facts about the angles |Students will show an informal proof to establish alternate interior |
|created when parallel lines are cut by a transversal? |angles are congruent. |
| |Students will show an informal proof to establish alternate exterior |
| |angles are congruent. |
| |Students will show an informal visual proof to establish the fact two |
| |triangles are similar if at least two sets of corresponding angles are |
| |congruent. |
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|How can I use informal arguments to establish facts about the | |
|angle-angle criterion for similarity of triangles? | |
|Vocabulary |
|Tier 2 |Tier 3 |
| |≅, translations, rotations, reflections, line of reflection, center of|
| |rotation, clockwise, counterclockwise, parallel lines, congruence, |
| |reading A’ as “A prime”, similarity, dilations, pre-image, image, |
| |rigid transformations, exterior angles, interior angles, alternate |
| |interior angles, angle-angle criterion, deductive reasoning, vertical |
| |angles, adjacent, supplementary, complementary, corresponding, scale |
| |factor, transversal |
|Language Objectives |
|Key Vocabulary |
| |SWBAT Define and give examples of the specific vocabulary for this standard: translations, rotations, |
|8.G.1- 5 |reflections, line of reflection, center of rotation, clockwise, counterclockwise, parallel lines, |
| |congruence, ≅, reading A’ as “A prime”, similarity, dilations, pre-image, image, rigid transformations, |
| |exterior angles, interior angles, alternate interior angles, angle-angle criterion, deductive reasoning, |
| |vertical angles, adjacent, supplementary, complementary, corresponding, scale factor, transversal, parallel |
|Language Function |
|8.G.2 |SWBAT compare two two-dimensional figures and using a graph, explain to a partner why they are congruent. |
|8.G.3 |SWBAT describe to a partner the effect of dilations, translations, rotations, and reflections on |
| |two-dimensional figures using coordinates. |
|Language Skill |
|8.G.1 - 4 |SWBAT explain the sequence of rotations, reflections, and translations applied from one two-dimensional |
| |figure to another. Explain the sequence to a partner. |
|8.G.5 |SWBAT informally write an argument to establish facts about the angle sum and exterior angle of triangles, |
| |about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for |
| |similarity of triangles. |
|Grammar and Language Structures |
|8.G.1 |SWBAT describe the properties of rotations, reflections, and translations to a small group using graphs. |
|8.G.3 |SWBAT describe to a partner the effects of dilations, translations, rotations, and reflections on |
| |two-dimensional figures using coordinates. |
|8.G.4 |SWBAT describe to a partner what it means for two figures to be similar. |
|8.G.5 |SWBAT create an informal argument to establish facts about interior/exterior angles in triangles and about |
| |corresponding/alternating angles in parallel lines cut by a transversal. Explain informal argument to a |
| |partner. |
|Language Tasks |
|8.G.1 |SWBAT perform a reflection, rotation, and a translation on a figure, and compare the pre-image to the image |
| |to verify that… |
| |The corresponding segments are the same length. |
| |The corresponding angles have the same measure. |
| |Parallel lines in the pre-image remain parallel in the image. |
|8.G.2 |SWBAT describe to a partner (in terms of transformational geometry) what it means for two shapes to be |
| |congruent. |
| |SWBAT demonstrate and describe a sequence of reflections, rotations and/or translations that will map a |
| |given shape onto another congruent shape by using physical models, transparencies, and geometry software. |
|8.G.3 |SWBAT (when the coordinates of the vertices of a figure are given/known) give the coordinates of |
| |corresponding coordinates of its image as a result of… |
| |A reflection over the x-axis or y-axis |
| |A reflection over the line y = x or y = -x |
| |A rotation of 90˚ and 180˚ about the origin |
| |A translation of a given number of units to the left/right and/or up/down |
| |A dilation of any scale factor with the origin as the center |
|8.G.4 |SWBAT describe to a partner what it means for two shapes to be similar. |
| |SWBAT (When given two similar figures on a coordinate grid), demonstrate and describe to a partner a |
| |sequence of reflections, rotations, translations, and/or dilations that will map one figure onto the other |
| |to verify that they are similar by using physical models, transparencies , and geometry software |
|8.G.5 |SWBAT show an informal visual proof to a partner to establish the fact that the interior angles of any |
| |triangle add up to 180˚ |
| |SWBAT show an informal visual proof to establish the fact two triangles are similar is at least two sets of |
| |corresponding angles are congruent. |
|Language Learning Strategies |
| |SWBAT identify and interpret language that provides key information to solve real-world and mathematical |
|8.G.1 - 5 |word problems using visual and graphical supports. |
|Information and Technology Standards |
|8.RP.1 |Apply a research process to complete project-based activities. |
|8.TT.1.1 |Use technology and other resources for assigned tasks. |
|Instructional Resources and Materials |
|Physical |Technology-Based |
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|Discovery Education |WSFCS Math Wiki |
|Unit 3: Congruence and Similarity | |
|Unit 6: Intersecting Lines and Angles |NCDPI Wikispaces Eighth Grade |
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|Connected Math 2 Series |MARS |
|Common Core Investigations 3, 4 | |
|Kaleidoscopes, Hubcaps, and Mirrors, Inv. 5 |Georgia Unit |
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| |Illustrative Math |
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|Partners in Math |Illuminations |
|Similar Triangle Activity | |
|Shrinking Circles |Illuminations.NCTM.Archimedes' Puzzle |
|Enlarging a Drawing | |
|Movin' On Task |Illuminations.NCTM.Algebraic Transformations |
|Look Alike Rectangles |Video Ohio State Marching Band |
|Follow the Path | |
|Tangram Activity | |
|Sierpinski Triangle | |
|The Koch Snowflake | |
|Alice in Wonderland Deli Dilations | |
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|Mathematics Assessment Project (MARS) | |
|Representing and Combining Transformations | |
|Identifying Similar Triangles | |
|Sampling & Estimating: How Many Jellybeans | |
|Aaron’s Designs, task | |
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|Video Ohio State Band | |
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