- Create three-dimensional figures.
- Construct a three-dimensional model of a city using similar and congruent figures and geometric transformations.
- Create a two-dimensional representation of their city.
- Find the surface area of three-dimensional figures.
Discovering Math: Exploring Geometryvideo
- Building Construction Chart (see below)
- Surface Area Chart (see below)
- Models of three-dimensional figures
- Modeling clay
- Measuring tools
- Cardboard to use as base for city (size to be determined by teacher)
- Tell students that they will be constructing three-dimensional models of a city. Discuss the directions and parameters of the project.
- Construct a three-dimensional model of a city that has at least 10 buildings. The buildings must include two similar figures, two congruent figures, and two geometric transformations. Each building must be labeled and the figures used identified in the Building Construction Chart.
- Use paper or modeling clay to create the three-dimensional figures that will be used to construct the buildings.
- Add streets to the city using straight edges.
- Students should be able to describe the streets using the terms parallel, perpendicular, and intersecting.
- Ask students to describe surface area Display a three-dimensional figure and model how to calculate its surface area. Remind students to use the Pythagorean theorem when find the length of sides of right triangles.
- Ask students trade cities with a classmate.
- Have students calculate the surface area of at least five buildings in their classmate's city. They should use an appropriate measuring tool and record the surface area of each building on the Surface Area Chart.
- Have students identify parallel, perpendicular, and intersecting streets in the city.
- Ask students to identify the three types of two-dimensional representations presented in the video. Discuss the properties of planar cross sections, scale drawings, and blueprints. Have students create a two-dimensional representation of their city.
- Remind students that two-dimensional representations should be drawn to scale. If needed, drawing to scale should be modeled and explained.
- Have students share their representations with the class. They should compare their two- and three-dimensional representations of the city and discuss the advantages and disadvantages of each type of representations.
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Use the following three-point rubric to evaluate students' work during this lesson.
3 points: Students produced a complete city following all of the parameters; clearly demonstrated the ability to create identify three-dimensional figures; clearly demonstrated the ability to perform geometric transformations; clearly demonstrated the ability to create and identify similar and congruent figures; clearly demonstrated the ability to calculate surface area of three-dimensional figures; and clearly demonstrated the ability to create a two-dimensional representation of a three-dimensional object.
2 points: Students produced an adequate city including at least eight buildings; demonstrated the ability to create and identify three-dimensional figures 80% of the time; demonstrated the ability to perform at least one geometric transformation; demonstrated the ability to create and identify at least one similar or congruent figure; demonstrated the ability to calculate surface area of three to four three-dimensional figures; and satisfactorily demonstrated the ability to create a two-dimensional representation of a three-dimensional object.
1 point: Students created an incomplete city with less than eight buildings; demonstrated the ability to create and identify three-dimensional figures less than 80% of the time; did not demonstrate the ability to perform geometric transformations; did not demonstrate the ability to create and identify similar or congruent figures; demonstrated the ability to calculate surface area of less than three three-dimensional figures; and did not demonstrate the ability to create a two-dimensional representation of a three-dimensional object.
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Definition:Two-dimensional representation of a three-dimensional object that shows the relationships between all the parts of an object from one viewpoint
Context:The architect drew a blueprint to represent the plans for the new house.
Definition:A figure that has the exact shape and size of another figure with a ratio of 1:1 for the corresponding parts
Context:Sue and Mary both drew squares with three-inch-long sides on the board, so they drew congruent figures.
Definition:A geometric transformation that changes a figure's size, but its shape, orientation, and location stay the same
Context:The animator performed a geometric dilation in order to create the illusion that the figure was getting larger without changing its shape, orientation, or location.
planar cross section
Definition:A two-dimensional representation that shows the inside of a three-dimensional object to illustrate the spatial relationships within the object
Context:Seismologists use planar cross sections to shows the layers of the Earth.
Definition:A geometric transformation that changes a figure's orientation and location, but not its size or shape, by rotating it around an axis
Context:The animator performed a geometric rotation to make the figure spin.scale drawing
Definition:A two-dimensional representation of an object using a scale that that is the ratio of the size of the drawing to the actual size of the object
Context:A map is a scale drawing.similar figures
Definition:Figures that have the same shape but not necessarily the same size, congruent corresponding angles, and equal ratios for the lengths of corresponding sides
Context:Triangles ABC and DEF are similar figures.surface area
Definition:The sum of the areas of the outside surfaces of a three-dimensional figure
Context:The surface area of a cube with sides that measure 5 feet is 150 square feet.translation
Definition:A geometric transformation that changes a figure's location, but its size and shape remain the same
Context:The animator performed a geometric translation on the figure to create the illusion of movement.
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Mid-continent Research for Education and Learning (McREL)
McREL's Content Knowledge: A Compendium of Standards and Benchmarks for K–12 Education addresses 14 content areas. To view the standards and benchmarks, visitwww.mcrel.org/compendium/browse.asp.
This lesson plan addresses the following benchmarks:
National Council of Teachers of Mathematics (NCTM)
- Uses geometric methods (i.e., an unmarked straightedge and a compass using an algorithm) to complete basic geometric constructions (e.g., perpendicular bisector of a line segment, angle bisector).
- Understands the defining properties of three-dimensional figures (e.g., a cube has edges with equal lengths, faces with equal areas and congruent shapes, right angle corners).
- Understands the defining properties of triangles (e.g., the sum of the measures of two sides of a triangle must be greater than the measure of the third side).
- Understands geometric transformations of figures (e.g., rotations, translations, dilations).
- Understands the relationships between two- and three-dimensional representations of a figure (e.g., scale drawings, blueprints, planar cross sections).
- Understands the mathematical concepts of similarity (e.g., scale, proportion, growth rates) and congruency.
- Understands the basic concept of the Pythagorean theorem.
The National Council of Teachers of Mathematics (NCTM) has developed national standards to provide guidelines for teaching mathematics. To view the standards online, go tostandards.nctm.org.
This lesson plan addresses the following national standards:
- Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
- Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
- Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
- Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
- Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
- Use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume.
- Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.
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