Similarity and Trigonometry
Suggested Unit Length and Description:
Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. (Mathematics Appendix A, p.29)
· Understand similarity in terms of similarity transformations.
· Prove theorems involving similarity.
· Define trigonometric ratios and solve problems involving right triangles
· Apply geometric concepts in modeling situations.
· Apply trigonometry to general triangles.
G-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.A.2 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.
G-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G-SRT.B.4 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.
G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g. modeling a tree trunk or a human torso as a cylinder).★
G-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★
G-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★
G-SRT.D.9 (+)Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
G-SRT.D.10 (+)Prove the Laws of Sines and Cosines and use them to solve problems.
G-SRT.D.11 (+)Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Focus Standards of 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.
Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:
· I can define image, pre-image, scale factor, center, and similar figures as they relate to transformations
· I can identify a dilation stating its scale factor and center
· I can verify experimentally that a dilated image is similar to its pre-image by showing congruent corresponding angles and proportional sides
· I can verify experimentally that a dilation takes a line not passing through the center of the dilation to a parallel line by showing the lines are parallel
· I can verify experimentally that dilation takes a line not passing through the center of the dilation unchanged by showing that it is the same line
· I can, by using similarity transformations, explain that triangles are similar if all pairs of corresponding angles are congruent and all corresponding pairs of sides are proportional
· I can, given two figures, decide if they are similar by using the definition of similarity in terms of similarity transformations
· I can recall the properties of similarity transformations
· I can establish the AA criterion for similarity of triangles by extending the properties of similarity transformations to the general case of any two similar triangles
· I can recall postulates, theorems, and definitions to prove theorems about triangles
· I can prove theorems involving similarity 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)
· I can recall congruence and similarity criteria for triangles
· I can use congruency and similarity theorems for triangles to solve problems
· I can use congruency and similarity theorems for triangles to prove relationships in geometric figures
· I can name the sides of right triangles as related to an acute angle
· I can recognize that if two right triangles have a pair of acute, congruent angles that the triangles are similar
· I can compare common ratios for similar right triangles and develop a relationship between the ratio and the acute angle leading to the trigonometry ratios
· I can use the relationship between the sine and cosine of complementary angles
· I can identify the sine and cosine of acute angles in right triangles
· I can identify the tangent of acute angles on right triangles
· I can explain how the sine and cosine of complementary angles are related to each other
· I can recognize which methods could be used to solve right triangles in applied problems
· I can solve for an unknown angle or side of a right triangle using sine, cosine, and tangent
· I can apply right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
· I can use measures and properties of geometric shapes to describe real world objects
· I can, given a real-world object, classify the object as a known geometric shape – use this to solve problems in context
· I can define density
· I can apply concepts of density based on area and volume to model real-life situations (e.g., persons per square mile, BTUs per cubic foot)
· I can describe a typographical grid system
· I can apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid system based on ratios)
· I can recall right triangle trigonometry to solve mathematical problems
· I can apply the area of a triangle formula by using the formula A=1/2ab sin(c) to solve real world problems I can derive the formula A=1/2ab sin(c)for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side
· I can use the Laws of Sines and Cosines to find missing angles or side length measurements
· I can Prove the Law of Sines
· I can prove the Law of Cosines
· I can recognize when the Law of Sines or Law of Cosines can be applied to a problem and solve problems in context using them
· I can determine from given measurements in right and non-right triangles whether it is appropriate to use the Law of Sines or Cosines
· I can apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces)
· Transformations, symmetry, and spatial reasoning can be used to analyze and model mathematical situations.
· Proportional relationships express how quantities change in relationship to each other.
· Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial reasoning.
· Similarity can be demonstrated using logical reasoning.
· Similarity in polygons has real-life application in a variety of areas including art, architecture, and sciences.
· Different observed relationships between geometric objects are provable using basic geometric building blocks and previously proven relationships between these building blocks and between other geometric objects.
· Triangles are fundamental aesthetic, structural elements that are useful in many disciplines such as art, architecture, and engineering.
· Developing techniques for measuring indirectly is a useful in many aspects of daily life.
· Proportional relationships express how quantities change in relationship to each other.
· Trigonometry can help us solve real world problems that involve triangles.
· How does geometry explain or describe the structure of our world?
· How does my understanding of algebraic principles help me solve geometric problems?
· How are the concepts of similarity and congruence related to each other?
· What special relationships occur between congruent triangles and similar triangles?
· How can drawings and figures be used to justify arguments and conjectures about congruence and similarity?
· How can proportions be used to solve problem involving similarity?
· What special properties exist in a right triangle that makes it unique?
· How can special segments of a triangle be used to solve real-world problems?
· What connections can be made between algebraic concepts and geometric concepts?
· How are the trigonometric ratios useful in modeling real life situations?