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Update 9/15/2015
Curriculum updates are still underway! Follow this curriculum to receive updates.
Update 6/18/2015
Math 2 - Unit 1 has been updated as of 6/18/2015.

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Unit 6 - Similarities and Volume
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Unit 6 - Similarities and Volume

Description

Since the concept of dilations was treated in the Grade 8 standards mostly in the context of hands-on activities, and with an emphasis on geometric intuition, one focus of this unit is to develop a more precise definition for this transformation, as was done in Math 1 for the rigid motions. The rigid motions, when combined with dilations, are referred to as similarity transformations. Specific outcomes of the unit are to develop a precise definition of similarity in terms of similarity transformations, to use this definition to develop criteria for determining when two triangles are similar, and finally, to develop triangle similarity theorems which follow from the definition of similarity in terms of similarity transformations. As in Math 1, attributes of geometric objects observed at earlier grades will now be looked at more precisely through proof; and in many instances, these proofs will make use of triangle similarity and its consequences. Key among the relationships that will be proven through the use of triangle similarity is the one found in the Pythagorean Theorem. Properties of congruent and of similar triangles will also be used to solve a variety of problems that either involve or can be modeled with triangles. Along with physical models, dynamic geometry environments will provide students with tools for investigating, experimenting with, conjecturing about, and modeling geometric phenomena. Students will experience informal arguments for determining various volume formulas and will use volume formulas to solve problems and applications. They will also learn how scaling three dimensional shapes affects their area and volume

Anticipated Timeframe: 4-5 Weeks

Aligned Standards

Major Standards

  • 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.
  • The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
  • 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
  • 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
  • 4. Prove theorems about triangles.
  • 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Supporting Standards

Additional Standards

  • 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
  • 3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.?
Unit 6: Similarity and Volume
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Attribution (CC BY 3.0)
This license lets others distribute, remix, tweak, and build upon your work, even commercially, as long as they credit you for the original creation. This is the most accommodating of licenses offered. Recommended for maximum dissemination and use of licensed materials