Illustrative Math Alignment: Grade 8 Unit 3

Topic

3D Geometry

Description

This animation shows a triangular prism with a horizontal cross-section. It illustrates that slicing the prism parallel to its base results in a rectangular cross-section that varies in size depending on the location of the slice.

Animated math clip art like this helps students understand the internal structure of prisms. Teachers can use this to discuss concepts such as congruence, cross-sectional area, and how this relates to the volume of a prism.

Topic

3D Geometry

Description

This animation depicts a triangular prism with a vertical cross-section. It shows how slicing the prism vertically results in a triangular cross-section, and all such triangles are congruent along the length of the prism.

Using animated math clip art like this provides students with a different perspective on the internal geometry of prisms. Teachers can use this to discuss concepts such as height, lateral area, and how vertical cross-sections relate to the surface area of a prism.

In this presentation we show how to convert a linear equation in Standard Form to a linear function in Slope Intercept Form. We go over the reason for such a conversion and applications that give rise to these equations. Note: The download is a PPT.

Linear Expressions, Equations, and Functions

Linear Expressions, Equations, and Functions

Linear Expressions, Equations, and Functions

Linear Expressions, Equations, and Functions

Linear Expressions, Equations, and Functions

Functions are relationships between quantities that change. Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video explores the definition of a function, its vocabulary and notations, and distinguishes the concept of function from a general relation. Multiple representations of functions are provided using the TI-Nspire, while dynamic visuals and scenarios put them into real-world contexts. Concepts explored: functions, relations, equations, quadratic functions, linear functions, multiple representations.

In this program, internationally acclaimed mathematics educator Dr. Monica Neagoy, explores the nature of linear functions through the use TI graphing calculators. Examples ranging from air travel, construction, engineering, and space travel provide real-world examples for discovering algebraic concepts. All examples are solved algebraically and then reinforced through the use of the TI-Nspire. Algebra teachers looking to integrate hand-held technology and visual media into their instruction will benefit greatly from this series. Concepts explored: Standard form, slope-intercept form, point-slope form, solving linear equations.

In this Investigation we look at linear models for objects moving at a constant speed. This video is Segment 1 of a 4 segment series related to Algebra Nspirations: Linear Functions. Segments 1 and 2 are grouped together.

In this Investigation we look at a linear regression for carbon dioxide emission data. This video is Segment 3 of a 4 segment series related to Algebra Nspirations: Linear Functions. Segments 3 and 4 are grouped together.

In this program we explore the properties of three-dimensional figures. We do this in the context of two real-world applications. In the first, we look at the three-dimensional structure of Mayan pyramids. These stair-step structures provide a unique opportunity to also explore sequences and series. In the second application we look at the Shanghai Tower as an example of cylindrically shaped structures.

Study these animations to learn the basic properties of these 3D figures. In particular, make a note of their sides, edges, and vertices. Look for any symmetries they have. Look for polygon shapes that are familiar. Finally, think of real-world examples that use these  figures.

We visit ancient Greece to learn about the Platonic Solids. This provides an introduction to the more general topic of three-dimensional figures.

Study these animations to learn the basic properties of these 3D figures. In particular, make a note of their sides, edges, and vertices. Look for any symmetries they have. Look for polygon shapes that are familiar. Finally, think of real-world examples that use these  figures.

Below we also include information about Platonic solids and 2D nets of these 3D figures. To get a better understanding of these 3D figures, study these basic forms.

Rectangular Prisms. Mayan pyramids are essentially stacks of rectangular prisms. The volume of each successive level is a percentage decrease of its lower neighbor. This introduces the notion of a geometric sequence and series, including an infinite series.

Study these animations to learn the basic properties of these 3D figures. In particular, make a note of their sides, edges, and vertices. Look for any symmetries they have. Look for polygon shapes that are familiar. Finally, think of real-world examples that use these  figures.

Below we also include information about Platonic solids and 2D nets of these 3D figures. To get a better understanding of these 3D figures, study these basic forms.

The Shanghai Tower in China is a stack of cylindrical shapes, where each successive layer is a percentage decrease of its lower neighbor. As with the previous section, this introduces the notion of a geometric sequence and series.

Study these animations to learn the basic properties of these 3D figures. In particular, make a note of their sides, edges, and vertices. Look for any symmetries they have. Look for polygon shapes that are familiar. Finally, think of real-world examples that use these  figures.

Below we also include information about Platonic solids and 2D nets of these 3D figures. To get a better understanding of these 3D figures, study these basic forms.

Topic

3D Geometry

Definition

An antiprism is a polyhedron composed of two parallel copies of an n-sided polygon, connected by an alternating band of triangles.

Description

Antiprisms are fascinating three-dimensional geometric figures that play a significant role in the study of polyhedra and spatial geometry. Unlike prisms, which have rectangular lateral faces, antiprisms feature triangular faces connecting their two parallel polygonal bases. This unique structure gives antiprisms distinct properties and applications in various fields of mathematics and engineering.

Topic

3D Geometry

Definition

The apex is the highest point or vertex of a three-dimensional figure, typically opposite to and furthest from its base.

Description

In the realm of three-dimensional geometry, the concept of an apex plays a crucial role in defining and understanding various geometric solids. The apex is particularly significant in figures such as cones and pyramids, where it represents the culminating point of the structure. For a cone, the apex is the single point at the top where all the lateral lines or edges converge. In a pyramid, it's the vertex where all the triangular faces meet, forming the pinnacle of the shape.

Topic

3D Geometry

Definition

A geometric base is the face of a three-dimensional figure from which the height is measured.

Description

In the context of three-dimensional geometry, the term "geometric base" is crucial for understanding the structure and properties of various 3D shapes. The base of a 3D figure is typically a flat surface, often a polygon, from which the height of the figure is perpendicular. This concept is fundamental when calculating the volume and surface area of three-dimensional shapes such as prisms, cylinders, pyramids, and cones.

Topic

3D Geometry

Definition

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, circular base to a point called the apex or vertex.

Description

In the realm of three-dimensional geometry, a cone is a significant shape due to its unique properties and applications. A cone is characterized by a circular base and a single apex. The line segments connecting the apex to the base form a curved surface, which is known as the lateral surface of the cone. The height of the cone is the perpendicular distance from the apex to the center of the base.

Topic

3D Geometry

Definition

A cross-section of a cube is the intersection of a plane with the cube, resulting in a two-dimensional shape.

Description

In the study of three-dimensional geometry, understanding the cross-sections of a cube is vital for visualizing how 3D objects interact with planes. When a plane intersects a cube, the resulting shape can vary depending on the angle and position of the plane. Common cross-sections include squares, rectangles, and triangles.

Topic

3D Geometry

Definition

A cross-section of a sphere is the intersection of a plane with the sphere, resulting in a circle.

Description

In three-dimensional geometry, understanding the concept of cross-sections is crucial for visualizing and analyzing the properties of various 3D shapes. A sphere, which is a perfectly round geometrical object in three-dimensional space, can be intersected by a plane in various ways. When a plane cuts through a sphere, the shape of the intersection is always a circle. The size of this circle depends on the position of the plane relative to the center of the sphere.

Topic

3D Geometry

Definition

A cube is a three-dimensional geometric figure with six equal square faces, twelve equal edges, and eight vertices.

Description

In the realm of three-dimensional geometry, the cube stands out as one of the most fundamental and symmetrical shapes. Known for its six identical square faces, a cube is a regular polyhedron, meaning all its faces are congruent squares, and all its angles are right angles. This symmetry makes the cube a crucial shape in various fields, including mathematics, architecture, and computer graphics.

Topic

3D Geometry

Definition

A cylinder is a three-dimensional geometric figure with two parallel circular bases connected by a curved surface at a fixed distance from each other.

Description

In the realm of three-dimensional geometry, a cylinder is a fundamental shape characterized by its two identical, parallel circular bases and a curved surface that connects these bases. The line segment joining the centers of the bases is called the axis of the cylinder, and it is perpendicular to the bases. The distance between the bases is the height of the cylinder, while the radius is the distance from the center to the edge of the base.

Topic

3D Geometry

Definition

An edge in three-dimensional geometry is the line segment where two faces of a 3D shape meet.

Description

In the study of three-dimensional geometry, an edge is a fundamental concept that helps define the structure and properties of 3D shapes. Each edge is formed by the intersection of two faces, and it is a critical component in determining the overall shape and form of a 3D object. For instance, a cube has 12 edges, each formed by the meeting of two square faces.

Topic

3D Geometry

Definition

A face is a flat surface that forms part of the boundary of a three-dimensional figure.

Description

In the realm of three-dimensional geometry, the concept of a face is fundamental to understanding the structure and properties of 3D figures. A face represents one of the flat, two-dimensional surfaces that enclose a three-dimensional object. These surfaces play a crucial role in defining the shape, volume, and surface area of 3D figures.

Topic

3D Geometry

Definition

A horizontal cross-section of a cone is the circular shape obtained when a cone is intersected by a plane that is parallel to its base.

Topic

3D Geometry

Definition

A horizontal cross-section of a cylinder is the intersection of the cylinder with a plane that is parallel to the base of the cylinder.

Description

In the context of three-dimensional geometry, understanding the concept of cross-sections is crucial. A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. When a horizontal plane intersects a cylinder, the resulting cross-section is a circle. This concept is not only fundamental in geometry but also has practical applications in various fields.

Topic

3D Geometry.

Definition

A horizontal cross section of a square pyramid is a two-dimensional shape obtained by slicing the pyramid with a plane parallel to its base.

Topic

3D Geometry

Definition

A horizontal cross-section of a triangular prism is the shape obtained by cutting the prism with a plane parallel to its base.

Description

In three-dimensional geometry, a cross-section is the intersection of a plane with a solid figure. When this plane is parallel to the base of the solid, the resulting shape is called a horizontal cross-section. For a triangular prism with a rectangular base, this cross-section will always be a rectangle, similar in shape to the base of the prism.

Topic

3D Geometry

Definition

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

Description

In the realm of three-dimensional geometry, a net is an essential concept that helps in visualizing and constructing 3D shapes. A net is essentially a two-dimensional pattern that, when folded along specific lines, forms a three-dimensional object. This concept is particularly useful in understanding the properties and structures of 3D figures such as cubes, pyramids, and prisms.

Topic

3D Geometry

Definition

A net for a cube is a two-dimensional shape that can be folded to form a three-dimensional cube.

Description

In the realm of three-dimensional geometry, a net for a cube is a crucial concept. It represents a flattened out three-dimensional shape that can be folded along the edges to form a cube. This is an essential tool for visualizing and understanding how 3D shapes are constructed from 2D representations.

Topic

3D Geometry.

Definition

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

Description

The dodecahedron is one of the five Platonic solids, characterized by its twelve regular pentagonal faces, thirty edges, and twenty vertices. Understanding the net of a dodecahedron helps in visualizing and constructing this complex polyhedron from a flat sheet.

Topic

3D Geometry

Definition

A net for a pyramid is a two-dimensional representation that, when folded along its edges, forms the three-dimensional shape of a pyramid.

Description

In the study of three-dimensional geometry, understanding the concept of nets is crucial, particularly for complex shapes like pyramids. A net for a pyramid is essentially a flattened version of the 3D shape, showing all its faces in a single plane. This representation is invaluable for visualizing how the pyramid's surfaces connect and for calculating its surface area.

Topic

3D Geometry

Definition

A net for a rectangular prism is a two-dimensional representation that, when folded, forms a three-dimensional rectangular prism.

Description

In the realm of three-dimensional geometry, the concept of a net is crucial for understanding how 3D shapes are constructed from 2D figures. A net for a rectangular prism is essentially a layout of all the faces of the prism in a single plane. This layout includes six rectangles that correspond to the six faces of the prism. By folding along the edges of these rectangles, one can recreate the three-dimensional shape of the rectangular prism.

Topic

3D Geometry

Definition

A net for a tetrahedron is a two-dimensional pattern that can be folded to form a three-dimensional tetrahedron, which is a polyhedron with four triangular faces.

Description

In the realm of three-dimensional geometry, the concept of a net is crucial for understanding the structure and properties of polyhedra. A tetrahedron, one of the simplest forms of polyhedra, consists of four triangular faces, six edges, and four vertices. The net of a tetrahedron is a flat, two-dimensional figure that, when folded along its edges, forms the three-dimensional shape of the tetrahedron.

Topic

3D Geometry

Definition

A net for a triangular prism is a two-dimensional figure that can be folded to form the surface of a three-dimensional triangular prism.

Description

In the context of three-dimensional geometry, a net is a crucial concept for understanding the surface area and structure of 3D shapes. Specifically, the net for a triangular prism consists of two triangular faces and three rectangular faces. When these faces are arranged in a specific two-dimensional pattern, they can be folded along the edges to form a triangular prism.

Topic

3D Geometry.

Definition

A net for an antiprism is a two-dimensional shape that can be folded to form a three-dimensional antiprism, which is a type of polyhedron with two parallel polygonal bases connected by an alternating band of triangles.

Description

In the realm of three-dimensional geometry, an antiprism is a fascinating polyhedron that extends the concept of prisms. Unlike regular prisms, which have two parallel bases connected by rectangular faces, antiprisms have two parallel bases connected by an alternating band of triangles. This unique structure results in a more complex and often more visually interesting shape.

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.