Geometry Online Resources
THE STANDARD (NCTM - PSSM)
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
Specify locations and describe spatial relationships using coordinate geometry and other representational systems:
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
- 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.
Specify locations and describe spatial relationships using coordinate geometry and other representational systems:
- use coordinate geometry to represent and examine the properties of geometric shapes;
- use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
- 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.
- draw geometric objects with specified properties, such as side lengths or angle measures;
- use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume;
- use visual tools such as networks to represent and solve problems;
- use geometric models to represent and explain numerical and algebraic relationships;
- recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.
ISOMETRIC DRAWING TOOL
Which standard?
This applet provides a way for students to construct 3-D models of block (cube) figures through visualization via three different views: top, side (right), and front. The applet provides the number of cubes that are needed to construct the 3-D model, including color-coding of the cubes within the three different views so that students can visualize where each cube belongs in the 3-D model construction. There is an extension opportunity for students to compute surface area and volume of the constructed model. By constructing the model and analyzing the views, students can “see” how much of each surface of the cube is visible, thereby, given the ability to solve the computation of the surface area. Knowing the number of cubes enables the students to use their knowledge of volume formulas.
What mathematical content is being learned (or intended to be learned)?
Applying knowledge of surface area and volume formulas with irregular 3-D models.
Is the focus on instrumental or relational understanding?
The focus is mostly instrumental since students are using their knowledge of area and volume for a polygon (cube), but applying the formulas to irregular 3-D figures where students have to dissect views from an irregular figure into subsets of cubes (squares) in order to calculate the surface area. For example, 6 blocks that are connected into the shape of an “L” can be deconstructed into two rectangles and it is only necessary to compute the area of these two rectangles rather than each individual block.
What role does technology play?
The technology plays a visual and “hands-on” role, similar to what students could achieve by actually building irregular 3-D figures with manipulatives such as “snap-cubes,” and provides an opportunity for students to “draw” the figures as they would on actual isometric dot paper. The technology allows the 3-D figure to be rotated, just as students would do with their own hands if they constructed the model using snap-cubes. The “view” feature allows the students to see the actual view of the model they have created so that they can assess whether or not what they are constructing actually matches up to the three views that they are initially given.
What instructional function(s) does the resource serve?
This resource primarily serves a “practicing skills” function, there is no direct instruction. Students have to approach this educational resource once they have already learned how to utilize the surface area and volume formulas. First, students have to learn how to apply area formulas to 2-D objects before moving on to 3-D objects, specifically in the case of finding the surface area of irregular 3-D figures where figures have to be deconstructed in order to calculate surface area.
What kinds of representations of the mathematics are used?
This resource provides an excellent combination of visual/spatial, concrete/real-world objects, and dynamic representations. Use of cubes, similar to “snap-cubes” and the 2-D isometric dot paper provide a visual means of creating 3-D figures. The dynamic feature of seeing the “views” of the 3-D object replicates the actual object. The feature allows the object to be turned and flipped (rotated), just as a student would manipulate it in their own hands.
- Use visualization, spatial reasoning, and geometric modeling to solve problems.
This applet provides a way for students to construct 3-D models of block (cube) figures through visualization via three different views: top, side (right), and front. The applet provides the number of cubes that are needed to construct the 3-D model, including color-coding of the cubes within the three different views so that students can visualize where each cube belongs in the 3-D model construction. There is an extension opportunity for students to compute surface area and volume of the constructed model. By constructing the model and analyzing the views, students can “see” how much of each surface of the cube is visible, thereby, given the ability to solve the computation of the surface area. Knowing the number of cubes enables the students to use their knowledge of volume formulas.
What mathematical content is being learned (or intended to be learned)?
- Surface Area
- Volume
Applying knowledge of surface area and volume formulas with irregular 3-D models.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
The focus is mostly instrumental since students are using their knowledge of area and volume for a polygon (cube), but applying the formulas to irregular 3-D figures where students have to dissect views from an irregular figure into subsets of cubes (squares) in order to calculate the surface area. For example, 6 blocks that are connected into the shape of an “L” can be deconstructed into two rectangles and it is only necessary to compute the area of these two rectangles rather than each individual block.
What role does technology play?
The technology plays a visual and “hands-on” role, similar to what students could achieve by actually building irregular 3-D figures with manipulatives such as “snap-cubes,” and provides an opportunity for students to “draw” the figures as they would on actual isometric dot paper. The technology allows the 3-D figure to be rotated, just as students would do with their own hands if they constructed the model using snap-cubes. The “view” feature allows the students to see the actual view of the model they have created so that they can assess whether or not what they are constructing actually matches up to the three views that they are initially given.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
- learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings)
This resource primarily serves a “practicing skills” function, there is no direct instruction. Students have to approach this educational resource once they have already learned how to utilize the surface area and volume formulas. First, students have to learn how to apply area formulas to 2-D objects before moving on to 3-D objects, specifically in the case of finding the surface area of irregular 3-D figures where figures have to be deconstructed in order to calculate surface area.
What kinds of representations of the mathematics are used?
- symbolic(i.e., numerals, symbols)
- visual/spatial (e.g., circles or squares with lines to show fractions)
- concrete or real-world objects (e.g., images of base-10 blocks, puppies, or jars)
- dynamic (mathematical ideas represented through motion or sound)
This resource provides an excellent combination of visual/spatial, concrete/real-world objects, and dynamic representations. Use of cubes, similar to “snap-cubes” and the 2-D isometric dot paper provide a visual means of creating 3-D figures. The dynamic feature of seeing the “views” of the 3-D object replicates the actual object. The feature allows the object to be turned and flipped (rotated), just as a student would manipulate it in their own hands.
SURFACE AREA AND VOLUME
Which standard?
This activity allows students to manipulate three-dimensional rectangular prisms, or triangular prisms to experiment with surface area and volume. Students will be able to illustrate notions of surface area and volume through the use of visuals within the applet. Students will be able to practice skills at estimating surface area and volume. There is a slider where students can manipulate the width, height and depth of the prism. From there, they can either practice calculating the surface area and volume, or the program can calculate it for them so students can focus on recognizing the relationship between surface area and volume.
What mathematical content is being learned (or intended to be learned)?
This activity is designed for the student to manipulate triangular and rectangular prisms in order to practice estimating volume and surface area as well as recognize the relationship between surface area and volume.
Is the focus on instrumental or relational understanding?
This activity begins with a “Learner” page where important vocabulary is introduced such as depth, width, height, faces, surface area and volume. Extra links are included at the bottom of the page to give students examples of how to find surface area and volume. The “Activity” page gives students the opportunity to develop a relational understanding within the topic. Students can manipulate the width, height and depth of the prisms in order to calculate the surface area and volume. Once these calculations are done, students can look at trends within the changes in the surface area and volume. Once they have explored the relationship between surface area and volume in rectangular prisms, they can take what they have noticed and try and apply it to the triangular prism and develop an understanding between the surface area and volume of that shape. All that is given to the students is the calculations- it is up to the students to determine the relationship between the data and how it changes within the two shapes.
What role does technology play?
The technology within this applet allows students to see a visual of triangular and rectangular prisms. The focus of this lesson is not how to draw these figures, instead find the relationships between surface area and volume. Therefore, this applet provides the prisms in a way that students can use them as a visual in order to determine relationships within the measurements. Students are able to move a slider in order to easily, accurately and quickly alter the dimensions of the shape without having to draw the shapes. They can also see how increasing or decreasing a certain dimension on a prism alters its overall shape.
What instructional function(s) does the resource serve?
Direct instruction is given on the “Learner” page, however it is not the focus of the applet. Therefore, this resource’s primary function is learning through exploration. Students are to use the dimension of the prism to recognize the relationship between surface area and volume. They can alter the dimensions in order to help confirm or support their reasoning.
What kinds of representations of the mathematics are used?
The prisms are represented on the applet in a three-dimensional model. There is an option to display the prism with grid lines in order to provide a different visual. The grid lines help make the prisms look more concrete, and in some way resemble base-ten blocks or cubes. As the dimensions are changed within the prisms, the prism stretches or shrinks providing movement and motion to the applet.
- Analyze characteristics and properties of geometric shapes and develop mathematical arguments about geometric relationships
This activity allows students to manipulate three-dimensional rectangular prisms, or triangular prisms to experiment with surface area and volume. Students will be able to illustrate notions of surface area and volume through the use of visuals within the applet. Students will be able to practice skills at estimating surface area and volume. There is a slider where students can manipulate the width, height and depth of the prism. From there, they can either practice calculating the surface area and volume, or the program can calculate it for them so students can focus on recognizing the relationship between surface area and volume.
What mathematical content is being learned (or intended to be learned)?
- Relationship between surface area & volume
This activity is designed for the student to manipulate triangular and rectangular prisms in order to practice estimating volume and surface area as well as recognize the relationship between surface area and volume.
Is the focus on instrumental or relational understanding?
- instrumental understanding (carrying out procedures)
- relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas)
This activity begins with a “Learner” page where important vocabulary is introduced such as depth, width, height, faces, surface area and volume. Extra links are included at the bottom of the page to give students examples of how to find surface area and volume. The “Activity” page gives students the opportunity to develop a relational understanding within the topic. Students can manipulate the width, height and depth of the prisms in order to calculate the surface area and volume. Once these calculations are done, students can look at trends within the changes in the surface area and volume. Once they have explored the relationship between surface area and volume in rectangular prisms, they can take what they have noticed and try and apply it to the triangular prism and develop an understanding between the surface area and volume of that shape. All that is given to the students is the calculations- it is up to the students to determine the relationship between the data and how it changes within the two shapes.
What role does technology play?
The technology within this applet allows students to see a visual of triangular and rectangular prisms. The focus of this lesson is not how to draw these figures, instead find the relationships between surface area and volume. Therefore, this applet provides the prisms in a way that students can use them as a visual in order to determine relationships within the measurements. Students are able to move a slider in order to easily, accurately and quickly alter the dimensions of the shape without having to draw the shapes. They can also see how increasing or decreasing a certain dimension on a prism alters its overall shape.
What instructional function(s) does the resource serve?
- learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings)
Direct instruction is given on the “Learner” page, however it is not the focus of the applet. Therefore, this resource’s primary function is learning through exploration. Students are to use the dimension of the prism to recognize the relationship between surface area and volume. They can alter the dimensions in order to help confirm or support their reasoning.
What kinds of representations of the mathematics are used?
- Visual/spatial (e.g., circles or squares with lines to show fractions)
- concrete or real-world objects (e.g., images of base-10 blocks, puppies, or jars)
- dynamic (mathematical ideas represented through motion or sound)
The prisms are represented on the applet in a three-dimensional model. There is an option to display the prism with grid lines in order to provide a different visual. The grid lines help make the prisms look more concrete, and in some way resemble base-ten blocks or cubes. As the dimensions are changed within the prisms, the prism stretches or shrinks providing movement and motion to the applet.
BATHROOM TILES
THE STANDARD (NTCM - PSSM):
Which standard?
This applet focuses on relationships between rotational geometry and the coordinate plane. Students are required to identify specific degrees of rotation a portion of a tile must undergo to get from the original point to its image. Students also need to identify equations of lines in order to reflect given objects. Students apply each mathematical transformation in no particular order within the applet. Throughout the investigation, students need to reason through their answers and use geometric models to visualize how each transformation will be produced.
What mathematical content is being learned (or intended to be learned)?
This applet is designed to allow students to use an interactive game to perform transformations using reflections, rotations and translations. Students also need to use this information to apply these transformations in the coordinate plane. Students identify equations of lines and determine slope in order to find lines of reflection or points of rotation.
Is the focus on instrumental or relational understanding?
The focus of this interactive activity is to promote relational understanding between geometric transformations and the coordinate plane. Students use the idea of the coordinate plane to transform geometric figures within the four quadrants of the coordinate plane. If students need a refresher, however, rules and procedures for performing transformations and ideas about the coordinate plane are summarized in the “Key Ideas” section. There are also “Tips” to give the user hints on how to solve a given transformation.
What role does technology play?
Technology provides a visualization tool that allows students to virtually “see” the movement of each transformation. It also gives students practice in defining rotations by degrees, reflections through lines and translations through movement. This technology also transforms the learners experience to a quick and time-saving image that is automated and much more accurate than free hand drawing. Students see the facilitated movement from the original object to its image. This technology also allows students to represent their mathematical thinking through the interactive interface and precise measurements of rotations and reflections. Students cannot simply guess to arrive at the answer.
This technology also supports the role of teacher by providing different levels to enhance differentiated instruction. Each level requires a different challenge to the student all surrounding the basic principles of geometric transformations.
What instructional function(s) does the resource serve?
Students practice performing transformations and use what they know about the coordinate plane to reason through their actions. Direct instruction is provided about each of the geometric transformations as well as some basic information about the coordinate plane (see above). Students learn throughout the activity by exploring different transformations and their applications in geometry. Students begin to realize patterns within the coordinate plane that help them determine degrees of rotation or lines of reflection.
What kinds of representations of the mathematics are used?
This applet represents mathematics in a variety of ways. Symbolically, the interactive shows numerals that can be manipulated by the user and geometric shapes that are transformed. Graphically, math is represented through the use of the coordinate plane. Students are able to visualize 2-dimentional objects in motion as they are reflected, rotated, and translated within the coordinate plane. Each shape is a representation of bathroom tile designs, which students are both familiar with and can imagine geometric transformations happening to. All of these transformations are dynamic as students are able to see perpetual movement from the original object to its image. Sound is also apparent within this interactive and a distinct sound is heard differentiating between correct and incorrect answers.
Which standard?
- specify locations and describe spatial relationships using coordinate geometry and other representational systems;
- apply transformations and use symmetry to analyze mathematical situations;
- use visualization, spatial reasoning, and geometric modeling to solve problems.
This applet focuses on relationships between rotational geometry and the coordinate plane. Students are required to identify specific degrees of rotation a portion of a tile must undergo to get from the original point to its image. Students also need to identify equations of lines in order to reflect given objects. Students apply each mathematical transformation in no particular order within the applet. Throughout the investigation, students need to reason through their answers and use geometric models to visualize how each transformation will be produced.
What mathematical content is being learned (or intended to be learned)?
- Geometric Transformations
This applet is designed to allow students to use an interactive game to perform transformations using reflections, rotations and translations. Students also need to use this information to apply these transformations in the coordinate plane. Students identify equations of lines and determine slope in order to find lines of reflection or points of rotation.
Is the focus on instrumental or relational understanding?
- relational understanding (understanding the meaning of mathematical words and symbols; connections among ideas)
The focus of this interactive activity is to promote relational understanding between geometric transformations and the coordinate plane. Students use the idea of the coordinate plane to transform geometric figures within the four quadrants of the coordinate plane. If students need a refresher, however, rules and procedures for performing transformations and ideas about the coordinate plane are summarized in the “Key Ideas” section. There are also “Tips” to give the user hints on how to solve a given transformation.
What role does technology play?
Technology provides a visualization tool that allows students to virtually “see” the movement of each transformation. It also gives students practice in defining rotations by degrees, reflections through lines and translations through movement. This technology also transforms the learners experience to a quick and time-saving image that is automated and much more accurate than free hand drawing. Students see the facilitated movement from the original object to its image. This technology also allows students to represent their mathematical thinking through the interactive interface and precise measurements of rotations and reflections. Students cannot simply guess to arrive at the answer.
This technology also supports the role of teacher by providing different levels to enhance differentiated instruction. Each level requires a different challenge to the student all surrounding the basic principles of geometric transformations.
What instructional function(s) does the resource serve?
- practice (i.e., practicing skills or knowledge already learned)
- direct instruction/explanation (i.e., explaining or presenting content to students)
- learning through exploration (i.e., provides context in which students can see new relationships; come to new understandings)
Students practice performing transformations and use what they know about the coordinate plane to reason through their actions. Direct instruction is provided about each of the geometric transformations as well as some basic information about the coordinate plane (see above). Students learn throughout the activity by exploring different transformations and their applications in geometry. Students begin to realize patterns within the coordinate plane that help them determine degrees of rotation or lines of reflection.
What kinds of representations of the mathematics are used?
- symbolic (i.e., numerals, symbols)
- graphical (i.e., standard graphical notation such as Cartesian (X-Y) coordinate system, bar graph, pie chart)
- visual/spatial(e.g., circles or squares with lines to show fractions)
- concrete or real-worldobjects (e.g., images of base-10 blocks, puppies, or jars)
- dynamic (mathematical ideas represented through motion or sound)
This applet represents mathematics in a variety of ways. Symbolically, the interactive shows numerals that can be manipulated by the user and geometric shapes that are transformed. Graphically, math is represented through the use of the coordinate plane. Students are able to visualize 2-dimentional objects in motion as they are reflected, rotated, and translated within the coordinate plane. Each shape is a representation of bathroom tile designs, which students are both familiar with and can imagine geometric transformations happening to. All of these transformations are dynamic as students are able to see perpetual movement from the original object to its image. Sound is also apparent within this interactive and a distinct sound is heard differentiating between correct and incorrect answers.
POOL GEOMETRY GAMES
Click on image!
A fun, DYNAMIC game of pool that depends on the player’s knowledge of geometry, particularly, angles. The player can experiment with the effect of the cue ball hitting the red balls and observe what direction the impact the ball had on the direction of movement. The observations can help the player determine what angle a ball should be hit in order to get it to move in the desired pocket of the table.