Unit 3: Transformations and Angle Relationships
Students investigate congruence and similarity by studying transformations of figures in the coordinate plane, and apply these transformations to discover new angle relationships.
In Unit 3, eighth-grade students bridge their understanding of geometry from middle school concepts to high school concepts. Until now, standards in the geometry domain have been either supporting or additional to the major standards. In eighth grade, geometry standards, especially the standards highlighted in this unit, are part of the major work of the grade and play a critical role in setting students up well for success in high school.
In this unit, students begin their work with transformations using patty paper (transparency paper) to experiment with, manipulate, and verify hypotheses around how shapes move under different transformations (MP.5). They use precision in their descriptions of transformations and in their justifications for why two figures may be similar or congruent to each other (MP.6). Students then apply their understanding of transformations to discover new angle relationships in parallel line diagrams and triangles.
Prior to eighth grade, students developed their understanding of geometric figures and learned how to draw them, calculate measurements, and model real-world situations. In seventh grade, students were introduced to the concept of scaling through scale drawings, and they solved for various measurements using proportional reasoning. Students will draw on these prior skills when they investigate dilations and similar triangles.
In high school geometry, students spend significant time studying congruence and similarity in-depth. They build off of the informal proofs and reasoning developed in eighth grade to hone their definitions of transformations, prove geometric theorems, and derive trigonometric ratios.
Pacing: 26 instructional days (22 lessons, 3 flex days, 1 assessment day)
For guidance on adjusting the pacing for the 2021-2022 school year, see our 8th Grade Scope and Sequence Recommended Adjustments.
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This assessment accompanies Unit 3 and should be
given on the suggested assessment day or after completing the
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Suggestions for how to prepare to teach this unit
The central mathematical concepts that students will come to understand in this unit
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 8th Grade Course Material Overview.
Terms and notation that students learn or use in the unit
alternate interior and exterior angles
To see all the vocabulary for this course, view our 8th Grade Vocabulary Glossary.
Topic A: Congruence and Rigid Transformations
Understand the rigid transformations that move figures in the plane (translation, reflection, rotation).
Describe and perform translations between congruent figures. Use translations to determine if figures are congruent.
Describe and apply properties of translations. Use coordinate points to represent relationships between translated figures.
Describe and perform reflections between congruent figures. Use reflections to determine if figures are congruent.
Describe sequences of transformations between figures using reflections and translations. Use coordinate points to represent relationships between reflected figures.
Describe and perform rotations between congruent figures.
Describe sequences of transformations between figures using rotations and other transformations.
Describe a sequence of rigid transformations that will map one figure onto another.
Describe multiple rigid transformations using coordinate points.
Review rigid transformations and congruence between two figures.
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Topic B: Similarity and Dilations
Define a dilation as a non-rigid transformation, and understand the impact of scale factor.
Describe and perform dilations.
Describe a sequence of dilations and rigid motions between two figures. Use coordinate points to represent relationships between similar figures.
Determine and informally prove or disprove if two figures are similar or congruent using transformations.
Find missing side lengths in similar figures. Find scale factor between similar figures.
Use properties of similar triangles to model and solve real-world problems.
Topic C: Angle Relationships
Define and identify corresponding angles in parallel line diagrams. Review vertical, supplementary, and complementary angle relationships.
Define and identify alternate interior and alternate exterior angles in parallel line diagrams. Find missing angles in parallel line diagrams.
Solve for missing angle measures in parallel line diagrams using equations.
Define and use the interior angle sum theorem for triangles.
Define and use the exterior angle theorem for triangles.
Define and use the angle-angle criterion for similar triangles.
The content standards covered in this unit
— Verify experimentally the properties of rotations, reflections, and translations:
— Lines are taken to lines, and line segments to line segments of the same length.
— Angles are taken to angles of the same measure.
— Parallel lines are taken to parallel lines.
— 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.
— Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
— 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.
— 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 covered in previous units or grades that are important background for the current unit
— Solve linear equations in one variable.
— Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
— Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
— Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
— Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
— Recognize and represent proportional relationships between quantities.
— Use proportional relationships to solve multistep ratio and percent problems.
Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Standards in future grades or units that connect to the content in this unit
— Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
— Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
— Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
— Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
— Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
— Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
— Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
— Prove theorems about triangles.
Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
— Prove theorems about lines and angles.
Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
— Verify experimentally the properties of dilations given by a center and a scale factor:
— Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
— Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
— Prove theorems about triangles.
Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
— Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
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