Instructional Focus: Use dilations to show figures similar.
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1 Instructional Focus: Use dilations to show figures. Properties of Dilations (G.SRT.1) Suppose we apply a dilation by a factor of 2, centered at the point P to the figure. a. In the picture, locate the images A, B, and C of the points A, B, C under this dilation. b. Based on you picture in part a., what do you think happens to the line l when we perform the dilation? c. Based on your picture in part a., what appears to be the relationship between the distance of A B and the distance of AB? d. Prove your observations in part c. Explain (G.SRT.2) If the two triangles must be, prove this result by describing a sequence of transformations that maps one variable to the other. a. The lengths of AX and XD satisfy the equation 2AAXX = 3XXDD. b. The lengths AX, BX, CX, and DX satisfy the equation c. Lines AB and CD are parallel. d. XAB is congruent to angle XCD. 4 Mastery 3 Proficient 2 - Basic 1 Below Basic 0 No Evidence Can extend thinking Verify that when a side Given an image and the Perform dilation with a Little evidence of beyond the passes through the center pre-image, determine the given center and scale reasoning or standard, including of dilation, the side and its center of dilation factor on a figure in the application to tasks that may image lie on the same line. coordinate plane. solve the involve one of the problem following: Designing Connecting Synthesizing Applying Justifying Critiquing Analyzing Creating Proving Verify that corresponding sides of the pre-image and images are parallel and proportional after dilation. Explain using transformations if two figures are by verifying angles are congruent sides are proportional Verify that corresponding sides of the pre-image and images are proportional by finding the scale factor. Explain if two figures are by verifying angles are congruent sides are proportional Show mathematically if two figures are by verifying angles are congruent sides are proportional Does not meet the criteria in a level 1
2 G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: 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.2 Given two figures, use the definition of in terms of transformations to decide if they are ; explain using transformations the meaning of for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
3 Instructional Focus: Explain and prove triangles. Prove triangles (G.SRT.3) Given that ΔΔMMNNPP is a dilation of ΔΔAABBCC with scale factor k, use properties of dilations to show that the AA criterion is sufficient to prove. Solve and prove relationships (G.SRT.5) In the diagram, quadrilateral PQRS is a parallelogram, SQ is a diagonal, and SQ XY. a. Prove that ΔXYR~ΔSQR. b. Prove that ΔXYR~ΔQSP. 4 Mastery 3 Proficient 2 - Basic 1 Below Basic 0 No Evidence Can extend thinking Identify if triangles are Little evidence of beyond the by: reasoning or standard, including tasks that may AA~ application to solve involve one of the SAS~ the problem following: SSS~ Designing Connecting Synthesizing Applying Justifying Critiquing Analyzing Creating Proving Prove all using transformations of the following theorems: Angle-Angle (AA) criterion for two SAS for two SSS for two Solve and prove geometric problems using congruence and Prove two using transformations of the following theorem: AA criterion for two SAS for two SSS for two Solve geometric problems using congruence and Solve geometric problems using congruence or Does not meet the criteria in a level 1
4 G.SRT.3 Use the properties of transformations to establish the AA criterion for two. G.SRT.5 Use congruence and criteria for triangles to solve problems and to prove relationships in geometric figures.
5 Instructional Focus: Prove and apply theorems using. Prove triangle theorems (G.SRT.4, G.CO.10) Prove that if two triangles are, then the ratio of corresponding altitudes is equal to the ratio of corresponding sides. Solve and prove relationships (G.SRT.5) Calculate the distance across the river, AB. 4 Mastery 3 Proficient 2 - Basic 1 Below Basic 0 No Evidence Can extend thinking Prove all of the Prove 2 of the following Prove1 of the following Little evidence of beyond the following theorems: theorems: theorems: reasoning or standard, including application to solve tasks that may the problem involve one of the following: Designing Connecting Synthesizing Applying Justifying Critiquing Analyzing Creating Proving a line parallel to one side of a triangle divides the other two proportionally if a line divides two sides of a triangle proportionally; then it is parallel to the third side. Pythagorean Theorem proved using triangle a line parallel to one side of a triangle divides the other two proportionally if a line divides two sides of a triangle proportionally; then it is parallel to the third side. Pythagorean Theorem proved using triangle a line parallel to one side of a triangle divides the other two proportionally if a line divides two sides of a triangle proportionally; then it is parallel to the third side. Pythagorean Theorem proved using triangle Does not meet the criteria in a level 1 Solve and prove geometric problems using congruence and Solve geometric problems using congruence and Solve geometric problems using congruence and
6 G.SRT.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. G.CO.10 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. G.SRT.5 Use congruence and criteria for triangles to solve problems and to prove relationships in geometric figures.
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