ALEX Classroom Resource

  

Human Tree: Dilations

  Classroom Resource Information  

Title:

 Human Tree: Dilations

URL:

 https://aptv.pbslearningmedia.org/resource/mgbh.math.rp.humantree8/human-tree-dilations/

Content Source:

 PBS
Type: Audio/Video

Overview:

Watch as the National Museum of Mathematics uses an image of a visitor to create a "Human Tree" using dilations. This video focuses on how similar figures can create dilations and how exponents can be used in an equation to express the proportional relationship in fractals. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
43. Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures. [Grade 8, 23]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Describe the changes occurring to the x- and y-coordinates of a figure after a transformation.
Teacher Vocabulary:
  • Coordinates
  • Congruent
  • Rotation
  • Reflection
  • Translation
  • Dilation
  • Scale factor
Knowledge:
Students know:
  • what it means to translate, reflect, rotate, and dilate a figure.
  • how to perform a translation, reflection, rotation, and dilation of a figure.
  • how to apply (x, y) notation to describe the effects of a transformation.
Skills:
Students are able to:
  • select and apply the proper coordinate notation/rule when given a specific transformation for a figure.
  • Graph a pre-image/image for a figure on a coordinate plane when given a specific transformation or sequence of transformations.
Understanding:
Students understand that:
  • the use of coordinates is also helpful in proving the congruency/proportionality between figures.
  • The relationships between coordinates of a preimage and its image for dilations represent scale factors learned in previous grade levels.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
44. Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence that exhibits the similarity between them. [Grade 8, 24]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Explain how transformations can be used to prove that two figures are similar.
  • Describe a sequence of transformations to prove or disprove that two figures are similar or congruent.
Teacher Vocabulary:
  • Translation
  • Reflection
  • Rotation
  • Dilation
  • Scale factor
Knowledge:
Students know:
  • how to perform rigid transformations and dilations graphically and algebraically (applying coordinate rules).
  • What makes figures similar and congruent.
Skills:
Students are able to:
  • use mathematical language to explain how transformations can be used to prove that two figures are similar or congruent.
  • Demonstrate/perform a series of transformations to prove or disprove that two figures are similar or congruent.
Understanding:
Students understand that:
  • there is a proportional relationship between corresponding characteristics of the figures, such as lengths of line segments, and angle measures as they develop a definition for similarity between figures.
  • The coordinate plane can be used as tool because it gives a visual image of the relationship between the two figures.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 8
23. Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Describe the changes occurring to the x-and y-coordinates of a figure after a transformation.
Teacher Vocabulary:
  • Coordinates
  • Congruent
  • Rotation
  • Reflection
  • Translation
  • Dilation
  • Scale factor
Knowledge:
Students know:
  • What it means to translate, reflect, rotate, and dilate a figure.
  • How to perform a translation, reflection, rotation, and dilation of a figure.
  • How to apply (x, y) notation to describe the effects of a transformation.
Skills:
Students are able to:
  • Select and apply the proper coordinate notation/rule when given a specific transformation for a figure.
  • Graph a pre-image/image for a figure on a coordinate plane when given a specific transformation or sequence of transformations.
Understanding:
Students understand that:
  • the use of coordinates is also helpful in proving the congruence/proportionality between figures.
  • The relationships between coordinates of a preimage and its image for dilations represent scale factors learned in previous grade levels.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.23.1: Define dilation.
M.8.23.2: Recall how to find scale factor.
M.8.23.3: Give examples of scale drawings.
M.8.23.4: Recognize translations.
M.8.23.5: Recognize reflections.
M.8.23.6: Recognize rotations.

Prior Knowledge Skills:
  • Define scale, scale drawings, length, area, and geometric figures.
  • Locate/use scale on a map.
  • Identify proportional relationships.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.23 Recognize the reflection (across the x or y axis) and translation (across quadrants) of a two dimensional figure on a coordinate plane (limited to non-equilateral rectangles and triangles).
Mathematics
MA2019 (2019)
Grade: 8
24. Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence that exhibits the similarity between them.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Explain how transformations can be used to prove that two figures are similar.
  • Describe a sequence of transformations to prove or disprove that two figures are similar or congruent.
Teacher Vocabulary:
  • Translation
  • Reflection
  • Rotation
  • Dilation
  • Scale factor
Knowledge:
Students know:
  • How to perform rigid transformations and dilations graphically and algebraically (applying coordinate rules).
  • What makes figures similar and congruent.
Skills:
Students are able to:
  • Use mathematical language to explain how transformations can be used to prove that two figures are similar or congruent.
  • Demonstrate/perform a series of transformations to prove or disprove that two figures are similar or congruent.
Understanding:
Students understand that:
  • There is a proportional relationship between corresponding characteristics of the figures, such as lengths of line segments, and angle measures as they develop a definition for similarity between figures.
  • The coordinate plane can be used as tool because it gives a visual image of the relationship between the two figures.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.24.1: Define similar.
M.8.24.2: Recognize dilations.
M.8.24.3: Recognize translations.
M.8.24.4: Recognize rotations.
M.8.24.5: Recognize reflections.
M.8.24.6: Identify similar figures.
M.8.24.7: Analyze an image and its dilation to determine if the two figures are similar.

Prior Knowledge Skills:
  • Define ordered pairs.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Identify which signs indicate the location of a point in a coordinate plane.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Define reflections.
  • Calculate the distances between points having the same first or second coordinate using absolute value.
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: dilations, fractal, proportion, similar figures
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
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Author: Kristy Lacks
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