ALEX Classroom Resource

  

Rotation and Dilation

  Classroom Resource Information  

Title:

 Rotation and Dilation

URL:

 https://aptv.pbslearningmedia.org/resource/mkcpt-math-g-rotationdilation/rotation-and-dilation/

Content Source:

 PBS
Type: Audio/Video

Overview:

In this video from KCPT, watch an animated demonstration of rotating and dilating a triangle on the coordinate plane. In the accompanying classroom activity, students watch the video; draw rotations and dilations of a triangle; and identify center of rotation, angle of rotation, and scale factors in classmates drawings. To get the most from the lesson, students should be comfortable graphing points on the coordinate plane and reproducing a drawing of a geometric shape at a different scale. Prior exposure to rotation and dilation is helpful.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Unpacked Content
Evidence Of Student Attainment:
Students:

Given a circle,
  • Use repeated reasoning from multiple examples of the ratio of circle circumference to the diameter, to informally conjecture that the circumference of any circle is a little more than three times the diameter.
  • Divide the circle into an equal number of sectors, and rearrange the sectors to form a shape that is approaching a parallelogram.
  • Make conjectures about the area and perimeter of the new shape as the number of sectors becomes larger, and relate those conjectures to the original circle.

  • Given a cylinder, explain how a cylinder could be divided into an infinite number of circles, and the area of those circles multiplied by the height is the volume of the cylinder, and use Cavalieri's Principle to demonstrate that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
  • Given a pyramid or cone, explain that the shapes could be divided into cross-sections, and the area of the cross-sections is decreasing as the cross-sections become further away from the base, and the area of an infinite number of cross-sections is the volume of a pyramid or cone.
Teacher Vocabulary:
  • Dissection arguments
  • Cavalieri's Principle
  • Cylinder
  • Pyramid
  • Cone
  • Ratio
  • Circumference
  • Parallelogram
  • Limits
  • Conjecture
  • Cross-section
Knowledge:
Students know:
  • Techniques to find the area and perimeter of parallelograms.
  • Techniques to find the area of circles or polygons.
Skills:
Students are able to:
  • Accurately decompose circles, cylinders, pyramids, and cones into other geometric shapes.
  • Explain and justify how the formulas for circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone may be created from the use of other geometric shapes.
Understanding:
Students understand that:
  • Geometric shapes may be decomposed into other shapes which may be useful in creating formulas.
  • Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.16.1: Define two-dimensional objects and three-dimensional objects.
GEO.16.2: Identify the two-dimensional figures that result from slicing three-dimensional figures as in plane section of right rectangular prisms and right rectangular pyramids.
GEO.16.3: Identify three-dimensional objects generated by rotations of two-dimensional objects (as in rotating a circle to create a sphere).
GEO.16.4: Distinguish between two-dimensional and three-dimensional objects.

Prior Knowledge Skills:
  • Define three-dimensional figures and nets.
  • Identify three-dimensional figures.
  • Select and create a three-dimensional figure using manipulatives.
  • Define two-dimensional figure, three-dimensional figure, and plane section.
  • List attributes of three-dimensional figures.
  • List attributes of two-dimensional figures.
  • Describe the relationship between two- and three-dimensional figures.
  • Recognize symmetry.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.16 Given a cross section of a three-dimensional object, identify the shapes of two-dimensional cross sections (limited to sphere, rectangular prism, or triangular prism).

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.)
Tags: angle of rotation, dilation center of rotation, rotation, scale factor
License Type: Public Domain
For full descriptions of license types and a guide to usage, visit :
https://creativecommons.org/licenses
AccessibilityVideo resources: includes closed captioning or subtitles
Comments

This resource contains a rotations and dilations activity sheet and problem cards.

  This resource provided by:  
Author: Kristy Lacks
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