 Unit 3

Similarity

Unit Description:

In this unit, students learn about dilation and similarity and use rigid motions learned in Unit 2 to perform complex transformations.  Students will replace the common idea of “same shape, different sizes” with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.  Students will work with two-dimensional figures in general and in the coordinate plane.

Students use similar triangles to solve unknown angle, side length and area problems. They revisit the Pythagorean Theorem from the perspective of similar triangles. In this unit, students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems.

Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

Louisiana Student Standards for Mathematics (LSSM)

 G – Geometry A. Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (Dilations only use the origin as the center of dilation, rotations are only about the origin and reflections are only over the y-axis and x-axis in Grade 8) Adding to the concepts from Unit 2 on transformations: *I can define dilations as a reduction or enlargement of a figure. *I can identify scale factor of the dilation. *I can describe the effects of dilations, translations, rotations, and reflections on 2-D figures using words and coordinates. 8.G.A.4 Explain 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. (Dilations only use the origin as the center of dilation, rotations are only about the origin and reflections are only over the y-axis and x-axis in Grade 8) *I can define similar figures as corresponding angles are congruent and corresponding side lengths are proportional. *I can recognize the symbol for similar. *I can apply the concept of similarity to write similarity statements. *I can reason that a 2-D figure is similar to another if the second can be obtained by a sequence of rotations, reflections, translation or dilation. *I can describe the sequence of rotations, reflections, translations, or dilations that exhibits the similarity between 2-D figures using words and/or symbols. *I can justify similarity of figures using a series of transformations. 8.G.A.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. Building upon parallel lines, transversals, and angle sums: *I can define similar triangles. *I can use Angle-Angle Criterion to prove similarity among triangles. (Give an argument in terms of transversals why this is so). B. Understand and apply the Pythagorean Theorem. 8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse using the area of squares. Pythagorean Theorem: If you have a right triangle, then the square of the longest side = the sum of the squares of the other two sides. Converse of Pythagorean Theorem: Given a triangle, if the square of the longest side = the sum of the squares of the other two sides, then the triangle is a right triangle. *I can define key vocabulary: square root, Pythagorean Theorem, right triangle, legs a and b, hypotenuse, sides, right angle, converse, base, height, proof. *I can identify the legs and hypotenuse of a right triangle. *I can explain a proof of the Pythagorean Theorem. *I can explain a proof of the converse of the Pythagorean Theorem. 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (Some parts of tasks require students to use the converse of the Pythagorean Theorem.) *I can recall the Pythagorean Theorem and its converse in order to apply it to real-world and mathematical problems using 2 and 3 dimensional figures. *I can solve basic mathematical Pythagorean Theorem problems and its converse to find missing length of sides of triangles in two and three dimensions. *I can apply the Pythagorean Theorem in solving real-world problems dealing with two and three dimensional shapes. 8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. *I can recall the Pythagorean Theorem and its converse and relate it to any two distinct points on a graph. *I can determine how to create a right triangle from two points on a coordinate graph. *I can use the Pythagorean Theorem to solve for the distance between the two points.

Enduring Understandings:

*A dilation (stretching/shrinking) is a transformation that changes the size of a figure but not the shape, thus producing similar figures not congruent figures (unless the scale factor is equal to 1).

*If the scale factor of a dilation is greater than 1, the image resulting from the dilation is an enlargement, and if the scale factor is less than 1, the image is a reduction.

*A two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of transformations.

*Two shapes are similar if the lengths of all the corresponding sides are proportional and all the corresponding angles are congruent.

*Two similar figures are related by a scale factor, which is the ratio of the lengths of corresponding sides.

Congruent figures have the same size and shape. If the scale factor of a dilation is equal to 1, the image resulting from the dilation is congruent to the original figure.

Essential Questions:

*What is the relationship of the points, lines, line segments, angles, etc. between the pre-image (the original figure) and the item’s image (the translated figure) generated by translations?

*Where is the origin on a coordinate grid?

*What does the scale factor of a dilation convey?

*What are the similarities and differences between the pre-image (the original figure) and the image (the translated figure) generated by dilations?

*How can transformations be used to determine similarity?

*Does a two-dimensional figure change dimensions when dilated?

*How can you use coordinates to describe the result of a dilation?

*What properties of a two-dimensional figure are preserved under a dilation?

*What is the relationship of the points, lines, line segments, angles, etc. between the pre-image (the original figure) and the item’s image (the transformed figure) generated by transformations?

*Can two figures be both congruent and similar?

*How can dilations be applied to real-world situations?