Illustrative Math Alignment: Grade 8 Unit 3

Topic

3D Geometry

Definition

A net for an icosahedron is a two-dimensional flat pattern that, when folded along its edges, forms the surface of a three-dimensional icosahedron.

Description

The net for an icosahedron is a crucial concept in three-dimensional geometry, particularly in the study of polyhedra. An icosahedron is a regular polyhedron with 20 faces, each of which is an equilateral triangle. The net provides a visual representation of how these 20 triangular faces are connected and arranged in a flat pattern that can be folded to create the three-dimensional shape.

Topic

3D Geometry

Definition

A net for an octahedron is a two-dimensional figure that can be folded to form a three-dimensional octahedron.

Description

In the realm of three-dimensional geometry, a net is a crucial concept that helps in visualizing and constructing 3D shapes from 2D representations. Specifically, a net for an octahedron consists of eight equilateral triangles arranged in a specific pattern. When these triangles are folded along the edges, they form an octahedron, which is a polyhedron with eight faces, twelve edges, and six vertices.

Topic

3D Geometry

Definition

A Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space, where each face is a congruent regular polygon, and the same number of faces meet at each vertex.

Description

Platonic solids are fundamental constructs in the study of three-dimensional geometry. These solids are unique because they are the only five regular polyhedra that exist. Each Platonic solid has faces that are congruent regular polygons, and the same number of faces meet at each vertex, making them highly symmetrical and aesthetically pleasing. The five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, each named for its number of faces.

Topic

3D Geometry

Definition

A prism is a three-dimensional solid object with two identical polygonal bases connected by parallelogram faces.

Description

In the realm of three-dimensional geometry, a prism is a polyhedron with two parallel, congruent bases connected by rectangular or parallelogram faces. The defining characteristic of a prism is that its cross-sections parallel to the bases are identical to the bases themselves. This property makes prisms a significant topic of study in geometry. 

Topic

3D Geometry

Definition

A pyramid is a three-dimensional geometric figure with a polygonal base and triangular faces that converge to a single point called the apex.

Description

In the realm of three-dimensional geometry, a pyramid is a significant shape due to its unique properties and applications. A pyramid consists of a base that can be any polygon, such as a triangle, square, or pentagon, and triangular faces that connect each edge of the base to a single apex point. This structure results in a solid figure that is both aesthetically pleasing and structurally efficient.

Topic

3D Geometry

Definition

A rectangular prism is a three-dimensional figure with six rectangular faces, where opposite faces are congruent and parallel.

Description

The rectangular prism is a fundamental shape in three-dimensional geometry, serving as a building block for understanding more complex 3D structures. It is characterized by its three dimensions: length, width, and height, which are clearly labeled in the image. This shape is ubiquitous in both natural and man-made environments, making it a crucial concept for students to grasp.

Topic

3D Geometry

Definition

Slant height is the distance measured along a lateral face from the base to the apex of a three-dimensional figure, such as a pyramid or a cone.

Description

In the context of three-dimensional geometry, the slant height is a crucial measurement for various solid figures, particularly right pyramids and right circular cones. It represents the shortest path along the surface of the figure from the apex (top point) to the base, distinguishing it from the vertical height which measures the perpendicular distance from the apex to the center of the base.

Topic

3D Geometry

Definition

A sphere is a perfectly round three-dimensional geometric object in which every point on the surface is equidistant from the center.

Description

In the realm of three-dimensional geometry, a sphere is a fundamental shape characterized by its symmetry and uniformity. It is defined mathematically as the set of all points in space that are at a constant distance, known as the radius, from a fixed point called the center. This distance is the same in all directions, making the sphere a unique object with no edges or vertices.

Topic

3D Geometry

Definition

A square pyramid is a three-dimensional geometric figure with a square base and four triangular faces that converge at a single point called the apex.

Topic

3D Geometry

Definition

Surface area is the total area that the surface of a three-dimensional object occupies.

Description

In the realm of three-dimensional geometry, surface area is a fundamental concept that quantifies the extent of a 3D shape's exterior surface. This measure is crucial for various applications, including engineering, architecture, and everyday tasks. For example, when painting a room, the surface area of the walls, ceiling, and floor must be calculated to determine the amount of paint required.

Topic

3D Geometry

Definition

A triangular prism is a three-dimensional geometric solid with two congruent triangular bases and three rectangular faces.

Description

The triangular prism is a fundamental shape in three-dimensional geometry, playing a crucial role in understanding the properties of polyhedra and their applications in various fields. This prism is characterized by its unique structure, consisting of two parallel triangular bases connected by three rectangular faces. The shape of the triangular bases can vary, allowing for right, equilateral, isosceles, or scalene triangular prisms.

Topic

3D Geometry

Definition

A triangular pyramid, also known as a tetrahedron, is a three-dimensional geometric figure with four triangular faces, six edges, and four vertices.

Description

In the realm of three-dimensional geometry, the triangular pyramid holds significant relevance due to its unique properties and structural simplicity. Each triangular face of the pyramid converges at a single point known as the apex, forming a solid figure that is both symmetrical and aesthetically pleasing. This geometric shape is the simplest form of a pyramid and is often used in various fields such as architecture, molecular chemistry, and computer graphics.

Topic

3D Geometry

Definition

A vertex is a point where three or more edges meet in a three-dimensional figure.

Description

In the study of three-dimensional geometry, the term vertex is fundamental. A vertex is a critical point in any 3D geometric shape, marking the intersection of edges. For example, in a cube, each corner where the edges converge is a vertex. Vertices are essential in defining the shape and structure of 3D figures, as they help in understanding the spatial relationships between different parts of the figure.

Topic

3D Geometry

Definition

A vertical cross section of a cone is the intersection of the cone with a plane that passes through its vertex and base, resulting in a two-dimensional shape.

Topic

3D Geometry

Definition

A vertical cross-section of a cylinder is the intersection of the cylinder with a plane that is parallel to its axis. This cross-section is typically a rectangle if the plane cuts through the entire height of the cylinder.

Topic

3D Geometry

Definition

A vertical cross section of a square pyramid is the intersection of the pyramid with a vertical plane that passes through its apex and base, resulting in a two-dimensional shape.

Topic

3D Geometry

Definition

A vertical cross section of a triangular prism is a two-dimensional shape obtained by slicing the prism parallel to its height, revealing a triangular face.

Topic

3D Geometry

Definition

Volume is the measure of the amount of space occupied by a three-dimensional object, expressed in cubic units.

Description

Volume is a fundamental concept in the study of three-dimensional geometry. It quantifies the capacity of a 3D object, indicating how much space it occupies. This measurement is crucial in various fields, including mathematics, engineering, architecture, and physical sciences.

Topic

3D Geometry

Definition

Cavalieri's Principle states that if two solids are contained between two parallel planes, and every plane parallel to these planes intersects both solids in cross-sections of equal area, then the two solids have equal volumes.

Description

Cavalieri's Principle is a fundamental concept in three-dimensional geometry that provides a method for determining the volume of solids. Named after the Italian mathematician Bonaventura Cavalieri, this principle is particularly useful for comparing the volumes of solids that might not have straightforward geometric shapes.

Explore Fahrenheit-vs-Celsius graphs using this Desmos graphing calculator activity. In this activity explore linear functions of the form y = mx + b. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources

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Explore Cost-vs-Time graphs using this Desmos graphing calculator activity. In this activity explore linear functions of the form y = mx. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources

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Explore Cost-vs-Time graphs using this Desmos graphing calculator activity. In this activity explore linear functions of the form y = mx + b. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources

To see additional resources on this topic click on the Related Resources tab above.

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Explore Cricket Chirps-vs-Temperature graphs using this Desmos graphing calculator activity. In this activity explore linear functions of the form y = mx + b. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources

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Explore Distance-vs-Time graphs using this Desmos graphing calculator activity. This activity explores linear functions of the form y = mx. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file.

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Desmos Collection

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Explore Distance-vs-Time graphs using this Desmos graphing calculator activity. This activity explores linear functions of the form y = mx + b. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources

To see additional resources on this topic click on the Related Resources tab above.

Desmos Collection

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Explore Hooke's Law using this Desmos graphing calculator activity. In this activity explore linear functions of the form y = mx. There is a companion worksheet for this activity, which subscribers can download. The worksheet is a PDF file. Related Resources To see additional resources on this topic click on the Related Resources tab above. Desmos Collection To see the complete collection of Desmos Resources click on this link.

In this graphing calculator activity, have your students explore linear functions of the form y = mx + b. This Desmos template allows students to explore the effect of changes in the slope (m) and the y-intercept (b) have on the graph of the line. A companion downloadable worksheet uses the graphing calculator template to explore the properties of these linear functions. Note: The download is a PDF worksheet. Related Resources To see additional resources on this topic click on the Related Resources tab above. Desmos Collection To see the complete collection of Desmos Resources click on this link.

This is the Teacher's Guide that accompanies Geometry Applications: 3D Geometry.

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Topic

Geometric Models

Topic

Geometric Models

Topic

Geometric Models

In this Algebra Application, students develop a linear mathematical model based on the maximum heart rate during exercise based on age. Using this model, students investigate heart rate for moderate and vigorous workouts. The math topics covered include: Mathematical modeling, Linear functions, Families of functions, Computational thinking. The culminating activity is for students to create a simple program (using either a spreadsheet or a programming language like Python) to develop an exercise chart. This is a great back-to-school activity for middle school or high school students. A relevant real-world application allows them to review math concepts.

In this Algebra Application, students study the direction between diameter and circumference of a circle. Through measurement and data gathering students analyze the line of best fit and explore ways of calculating pi. The math topics covered include: Mathematical modeling, Linear functions, Data gathering and analysis, Ratios, Direct variation. This is a great back-to-school activity for middle school or high school students. This is also a great crossover activity that ties algebra and geometry.

In this Slide Show, apply concepts of linear functions to the context of circumference vs. diameter.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of cost vs. time data and graphs.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of cricket chirps.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of distance vs. time data and graphs.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of Hooke's law.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of momentum and impulse.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of rectangular perimeter.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of saving money.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of speed and acceleration.

Note: The download is a PPT file.

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In this Slide Show, apply concepts of linear functions to the context of converting Celsius and Fahrenheit temperatures.

Note: The download is a PPT file.

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In this Slide Show, use the Desmos graphing calculator to explore linear equations in slope intercept form. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, use the Desmos graphing calculator to explore linear equations in standard form. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, use the Desmos graphing calculator to explore absolute value functions. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, use the Desmos graphing calculator to explore linear functions. To see the complete collection of Desmos Resources click on this link.

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In this Slide Show, learn about graphs of linear functions.

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In this Slide Show, learn about linear function models.

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The complete set of 32 examples that make up this set of tutorials.

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