In Module 2, Topic A students revisit what scale drawings are and discover two systematic methods of how to create them using dilations. The comparison of the two methods yields the Triangle Side Splitter Theorem and the Dilation Theorem.

Content Standard(s):

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

26. Verify experimentally the properties of dilations given by a center and a scale factor.

a. Verify that 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. Verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given a center of dilation, a scale factor, and a polygonal image,

Create a new image by extending a line segment from the center of dilation through each vertex of the original figure by the scale factor to find each new vertex.

Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the same line.

Find the ratio of the length of the line segment from the center of dilation to each vertex in the new image and the corresponding segment in the original image and compare that ratio to the scale factor.

Conjecture a generalization of these results for all dilations.

Teacher Vocabulary:

Dilations

Center

Scale factor

Knowledge:

Students know:

Methods for finding the length of line segments (both in a coordinate plane and through measurement).

Dilations may be performed on polygons by drawing lines through the center of dilation and each vertex of the polygon then marking off a line segment changed from the original by the scale factor.

Skills:

Students are able to:

Accurately create a new image from a center of dilation, a scale factor, and an image.

Accurately find the length of line segments and ratios of line segments.

Communicate with logical reasoning a conjecture of generalization from experimental results.

Understanding:

Students understand that:

A dilation uses a center and line segments through vertex points to create an image which is similar to the original image but in a ratio specified by the scale factor.

The ratio of the line segment formed from the center of dilation to a vertex in the new image and the corresponding vertex in the original image is equal to the scale factor.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.26.1: Define dilation and scale factor. GEO.26.2: Apply a scale factor.
GEO.26.2: Illustrate when given an original figure with a line (e.g., m) through it, not through the center, a parallel line to m will be created when the dilation is performed.
Example: Given a line x=, dilate the graph and line by 2. What happened to the line?
GEO.26.3: Illustrate when given an original figure with a line (e.g., m) through its center the line will remain unchanged when the dilation is performed.
GEO.26.4: Illustrate dilation. Example: Find the distance of line AB, given A (0,0) and B (2,3), after dilating AB by a scale factor of 1/2.
GEO.26.5: Determine the change in length of a line segment after dilation.
GEO.26.6: Discuss the properties of parallel lines.
GEO.26.7: Dilate a line segment.
GEO.26.8: Recognize whether a dilation is an enlargement or a reduction.

Prior Knowledge Skills:

Recall how to name points on a Cartesian plane using ordered pairs.

Recognize ordered pairs (x, y).

Define similar.

Recognize dilations.

Recognize translations.

Recognize rotations.

Recognize reflections.

Identify similar figures.

Analyze an image and its dilation to determine if the two figures are similar.

Define dilation.

Recall how to find scale factor.

Give examples of scale drawings.

Identify parts of the Cartesian plane.

Recognize ordered pairs.

Define function, ordered pairs, input, output.

Demonstrate how to plot points on a Cartesian plane using ordered pairs.

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

38. Use the mathematical modeling cycle involving geometric methods to solve design problems.

Examples: Design an object or structure to satisfy physical constraints or minimize cost; work with typographic grid systems based on ratios; apply concepts of density based on area and volume.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given a contextual situation involving design problems,

Create a geometric method to model the situation and solve the problem.

Explain and justify the model which was created to solve the problem.i

Note: Mathematical Modeling Cycle can be found in the Appendix of the COS document

Teacher Vocabulary:

Geometric methods

Design problems

Typographic grid system

Density

Knowledge:

Students know:

Properties of geometric shapes.

Characteristics of a mathematical model.

How to apply the Mathematical Modeling Cycle to solve design problems.

Skills:

Students are able to:

Accurately model and solve a design problem.

Justify how their model is an accurate representation of the given situation.

Understanding:

Students understand that:

Design problems may be modeled with geometric methods.

Geometric models may have physical constraints.

Models represent the mathematical core of a situation without extraneous information, for the benefit in a problem solving situation.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.38.1: Define density, area, and volume.
GEO.38.2: Illustrate a design conflict (e.g., draw a chair and a desk where the chair will not fit under the desk).
GEO.38.3: Discuss the relationship between units in each modeling situation.
GEO.38.4: Calculate density (D), mass (m) or volume (V) using the formula, D = m/V.
GEO.38.5: Recognize appropriate units for various situations.

Prior Knowledge Skills:

Define volume.

Derive the formulas for the volume of a cone, cylinder, and sphere.

Calculate the volume of three-dimensional figures.

Solve real-world problems using the volume formulas for three-dimensional figures.

Alabama Alternate Achievement Standards

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

Tags:

dilation, line, problem solving, scale, theorems, triangles