ALEX Classroom Resource

  

Geometry Module 2, Topic C: Similarity and Dilations

  Classroom Resource Information  

Title:

Geometry Module 2, Topic C: Similarity and Dilations

URL:

https://www.engageny.org/resource/geometry-module-2-topic-c-overview

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

Students learn what it means for two figures to be similar in general, and then focus on triangles and what criteria predict that two triangles will be similar. Length relationships within and between figures are studied closely and foreshadows work in Module 2, Topic D. The topic closes with a look at how similarity has been used in real-world application.

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
16 ) Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
17 ) Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [G-SRT4]

Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
27. Given two figures, determine whether they are similar by identifying a similarity transformation (sequence of rigid motions and dilations) that maps one figure to the other.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given two figures, determine if they are similar by demonstrating whether one figure can be obtained from the other through a dilation and a combination of translations, reflections, and rotations.
Teacher Vocabulary:
  • Similarity transformation
  • Similarity
  • Proportionality
  • Corresponding pairs of angles
  • Corresponding pairs of sides
  • Rigid Motion
Knowledge:
Students know:
  • Properties of rigid motions and dilations.
  • Definition of similarity in terms of similarity transformations.
  • Techniques for producing images under a dilation and rigid motions.
Skills:
Students are able to:
  • Apply rigid motion and dilation to a figure.
  • Explain and justify whether or not one figure can be obtained from another through a combination of rigid motion and dilation.
Understanding:
Students understand that:
  • A figure that may be obtained from another through a dilation and a combination of translations, reflections, and rotations is similar to the original.
  • When a figure is similar to another the measures of all corresponding angles are equal, and all of the corresponding sides are in the same proportion.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.27.1: Establish a sequence of similarity transformations between two similar polygons.
GEO.27.2: Determine if two triangles are similar based on their corresponding parts.
GEO.27.3: Develop a similarity statement for two similar polygons.
GEO.27.4: Identify corresponding angles and sides based on similarity statements.

Prior Knowledge Skills:
  • Recognize dilations.
  • Recognize translations.
  • Recognize rotations.
  • Recognize reflections.
  • Define rotation, reflection, and translation.
  • Recognize translations (slides), rotations (turns), and reflections (flips).
  • Distinguish between lines and line segments.
  • Identify parallel lines.
  • Demonstrate how to locate points on a vertical or horizontal number line.
  • Define ordered pairs.
  • Show how to plot points on a Cartesian plane.
  • Locate the origin on the coordinate plane.
  • Identify the length between vertices on a coordinate plane.
  • Recall how to read a graph or table.
  • Draw and label a coordinate plane.
  • Plot independent (input) and dependent (output) values on a coordinate plane.
  • Plot pairs of integers and/or rational numbers on a coordinate plane.
  • Arrange integers and/or rational numbers on a horizontal or vertical number line.
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Calculate the distances between points having the same first or second coordinate using absolute value.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
34. Use congruence and similarity criteria for triangles to solve problems in real-world contexts.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a contextual situation involving triangles,
  • Determine solutions to problems by applying congruence and similarity criteria for triangles to assist in solving the problem.
  • Justify solutions and critique the solutions of others.

  • Given a geometric figure, establish and justify relationships in the figure through the use of congruence and similarity criteria for triangles
Teacher Vocabulary:
  • Congruence and similarity criteria for triangles
Knowledge:
Students know:
  • Criteria for congruent (SAS, ASA, AAS, SSS) and similar (AA) triangles and transformation criteria.
  • Techniques to apply criteria of congruent and similar triangles for solving a contextual problem.
  • Techniques for applying rigid motions and dilations to solve congruence and similarity problems in real-world contexts.
Skills:
Students are able to:
  • Accurately solve a contextual problem by applying the criteria of congruent and similar triangles.
  • Provide justification for the solution process.
  • Analyze the solutions of others and explain why their solutions are valid or invalid.
  • Justify relationships in geometric figures through the use of congruent and similar triangles.
Understanding:
Students understand that:
  • Congruence and similarity criteria for triangles may be used to find solutions of contextual problems.
  • Relationships in geometric figures may be proven through the use of congruent and similar triangles.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.34.1: Develop an equation from given information to prove congruence or similarity.
GEO.34.2: Illustrate congruence and similarity in geometric figures.
GEO.34.3: Apply proportional reasoning to real-world scenarios.
GEO.34.4: Solve proportions.

Prior Knowledge Skills:
  • Analyze an image and its dilation to determine if the two figures are similar.
  • Identify similar figures.
  • Define similar.
  • Identify congruent figures.
  • Identify attributes of two-dimensional figures.
  • Compare rotations to translations.
  • Compare reflections to rotations.
  • Compare translations to reflections.
  • Define congruent and sequence.
  • Apply the rule of proportional relationship to real-world context.
  • Recognize similar triangles.
  • Define similar triangles, intercept, slope, vertical, horizontal, and origin.
  • Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
  • Analyze the graph to determine the rate of change.
  • Generate the slope of a line using given ordered pairs.
  • Graph a function given the slope-intercept form of an equation.
  • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
  • Graph a linear equation given the slope-intercept form of an equation.
  • Recognize that two sets of points with the same slope may have different y-intercepts.
  • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
  • Recall that for a relationship to be proportional, the graph must pass through the origin.
  • Demonstrate how to graph on a Cartesian plane.
  • Recall that for a relationship to be proportional, both variables must start at zero.
  • Identify the unit rate of two quantities.
  • Recall how to write a ratio of two quantities as a fraction.
  • Recall equivalent ratios and origin on a coordinate (Cartesian) plane.
  • Define proportional, independent variable, dependent variable, and unit rate.
  • Identify proportional relationships.
  • Locate/use scale on a map.
  • Define scale, scale drawings, length, area, and geometric figures.
  • Use a table or graph to determine whether two quantities are proportional.
  • Define equivalent ratios and origin.
  • Define unit rate, proportions, area, length, and ratio.
  • Recognize polygons. M. 6.3.4: Restate real-world problems or mathematical problems. M. 6.3.3: Calculate unit rate or rate by using ratios or proportions. M. 6.3.2: Create a ratio or proportion from a given word problem, diagram, table, or equation. M. 6.3.1: Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table. M. 6.3.16: Form a ratio. M. 6.3.15: Solve a proportion using part over whole equals percent over 100. M. 6.3.14: Identify a proportion from given information. M. 6.3.13: Calculate a proportion for missing information. M. 6.3.10: Create a proportion or ratio from a given word problem.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.


Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
36. Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a real-world object,
  • Select an appropriate geometric shape to model the object.
  • Provide a description of the object through the measures and properties of the geometric shape which is modeling the object.
  • Explain and justify the model which was selected.
Teacher Vocabulary:
  • Model
Knowledge:
Students know:
  • Techniques to find measures of geometric shapes.
  • Properties of geometric shapes.
Skills:
Students are able to:
  • Model a real-world object through the use of a geometric shape.
  • Justify the model by connecting its measures and properties to the object.
Understanding:
Students understand that:
  • Geometric shapes may be used to model real-world objects.
  • Attributes of geometric figures help us identify the figures and find their measures. therefore, matching these figures to real-world objects allows the application of geometric techniques to real-world problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.36.1: Estimate the dimensions of a given object.
GEO.36.2: Discuss the properties of a given object.
GEO.36.3: Identify the relationship of geometric representations to real-life objects.

Prior Knowledge Skills:
  • Recognize attributes of geometric shapes.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.


Tags: congruence, geometry, similarity, theorems, transformations, triangles
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There are nine lessons in this topic.

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Author: Hannah Bradley