Unit 3

Three Dimensions

 

Geometry

Suggested Unit Length and Description:

 

15 days

 

Studentsí experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line.(Mathematics Appendix A, p.32)

 

        Explain volume formulas and use them to solve problems.

        Visualize the relation between two-dimensional and three-dimensional objects.

        Apply geometric concepts in modeling situations.

        Explain volume formulas and use them to solve problems.

 

Standards:

 

G-GMD.A.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. Use dissection arguments, Cavalieriís principle, and informal limit arguments.

 

G-GMD.A.2 (+) Give an informal argument using Cavalieriís principle for the formulas for the volume of a sphere and other solid figures.

 

G-GMD.A.3 Use volume formulas for cylinders, pyramids, cones and spheres to solve problems.

 

G-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

 

G-MG.A.1 †† Use geometric shapes, their measures, and their properties to describe objects (e.g. modeling a tree trunk or a human torso as a cylinder).

 

G-MG.A.2 †† Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

 

G-MG.A.3 †† Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

G-GMD.A.1

  • I can recognize cross-sections of solids as two-dimensional shapes
  • I can recognize formulas for area and circumference of a circle and volume of a cylinder, pyramid, and cone
  • I can use the techniques of dissection and limit arguments
  • I can recognize Cavalieriís principle
  • I can decompose volume formulas into area formulas using cross-sections
  • I can apply dissection and limit arguments (e.g. Archimedesí inscription and circumscription of polygons about a circle) and as a component of the informal argument for the formulas for the circumference and area of a circle
  • I can apply Cavalieriís Principle as a component of the informal argument for the formulas for the volume of a cylinder, pyramid, and cone

G-GMD.A.2(+)

  • I can calculate the volume of a sphere and other solids using Cavalieriís principle.

G-GMD.A.3

  • I can utilize the appropriate formula for volume depending on the figure
  • I can use volume formulas for cylinders, pyramids, cones, and spheres to solve contextual problems

G-GMD.B.4

  • I can use strategies to help visualize relationships between two-dimensional and three-dimensional objects
  • I can relate the shapes of two-dimensional cross-sections to their three-dimensional objects
  • I can discover three-dimensional objects generated by rotations of two-dimensional objects

G-MG.A.1

  • I can use measures and properties of geometric shapes to describe real-world objects to solve geometric problems
  • I can, given a real-world object, classify the object as a known geometric shape; use this to solve problems in context

G-MG.A.2

  • I can define density
  • I can apply concepts of density based on area and volume to model real-life situations (e.g., persons per square mile, BTUs per cubic foot)

G-MG.A.3

  • I can describe a typographical grid system
  • I can apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid system based on ratios)

 

Enduring Understandings:

 

        Three-dimensional figures have relationships to specific tow-dimensional figures.

        Geometric objects may be used to model various physical phenomena.

        Representation of geometric ideas and relationships allows multiple approaches to geometric problems and connects geometric interpretations to other contexts.

        Reasonable estimates and sensible judgment about the precision and accuracy of measurement values is important.

 

Essential Questions:

 

        How are two-dimensional and three-dimensional space related?

        How can geometric properties and relationships be applied to solve problems that are modeled by geometric objects?

        How can you find the volume of solids for which no formula is available?

        How can you make sound decisions about how quantities should be measured and represented?