Illustrative Math Alignment: Grade 8 Unit 3

Topic

Linear Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the linear function y = x - 8. The image displays both a graph and a table representing this function. The table includes coordinate pairs (0, -8), (1, -7), (2, -6), (3, -5), and (4, -4), illustrating how the y-value increases by 1 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = x. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 0), (1, 1), (2, 2), (3, 3), and (4, 4), showing how the y-value increases by 1 for each unit increase in x.

Topic

Volume

Description

A rectangular prism with dimensions labeled: length = 30, width = 10, and height = 8. The image shows how to find the volume of the prism using the formula for volume of a rectangular prism. This image illustrates Example 1: The caption explains how to calculate the volume of the rectangular prism using the formula V = l * w * h. The given dimensions are substituted into the formula: V = 30 * 10 * 8 = 2400..

Topic

Volume

Description

A green cylinder with a general radius y and height x. The radius is marked on the top surface, and the height is marked on the side. This image illustrates Example 10: The task is to find the volume of this cylinder. The volume formula V = πr2h is used, and substituting r = y and h = x, the volume is calculated as V = xy2π.

Topic

Volume

Description

A hollow green cylinder with an outer radius of 10 units, an inner radius of 9 units, and a height of 15 units. The radii are marked on the top surface, and the height is marked on the side. This image illustrates Example 11: The task is to find the volume of this hollow cylinder. The volume formula for a hollow cylinder V = πr12h1 - πr22h2 is used. Substituting values, the result is V = 285π.

Topic

Volume

Description

A hollow green cylinder with an outer radius y, an inner radius y - 1, and a height x. The radii are marked on the top surface, and the height is marked on the side. This image illustrates Example 12: The task is to find the volume of this hollow cylinder. Using V = π(r12h1 - r22h2), substituting values gives: V = πx(y2 - (y - 1)2= πx(2y - 1).

Topic

Volume

Description

A rectangular-based pyramid is shown with dimensions: base length 10, base width 8, and height 30. The image demonstrates how to calculate the volume of this pyramid. This image illustrates Example 13: The caption provides a step-by-step solution for calculating the volume of a pyramid with a rectangular base using the formula V = (1/3) * Area of Base * h. Substituting values: V = (1/3) * 8 * 10 * 30 = 800.

Topic

Volume

Description

A general rectangular-based pyramid is shown with variables x, y, and z representing the base dimensions and height. This example shows how to calculate the volume of a pyramid using variables instead of specific numbers. This image illustrates Example 14: The caption explains how to calculate the volume of a pyramid with a rectangular base using the formula V = (1/3) * Area of Base * h, which simplifies to V = (1/3) * x * y * z.

Topic

Volume

Topic

Volume

Description

A truncated rectangular-based pyramid is shown with variables x, y, and z representing dimensions. The smaller virtual pyramid has reduced dimensions by 3 units for both width and length and reduced height by z - 20. The image demonstrates how to calculate the volume in terms of variables. This image illustrates Example 16: The caption explains how to find the volume of a truncated pyramid using variables for both pyramids' dimensions. Formula: V = (1/3) * xy(z + 20) - (1/3) * (y - 3)(x - 3)(z), which simplifies to V = (1/3) * (xyz + 60x + 60y - 180).

Topic

Volume

Description

A green sphere with a radius labeled as 3. The image is part of a math example showing how to calculate the volume of a sphere. This image illustrates Example 17: The text describes finding the volume of a sphere. The formula used is V = (4/3) * π * r3, where r = 3. After substituting, the result is V = 36π.

Volume is a fundamental concept in geometry that helps students understand the space occupied by three-dimensional objects. In this collection, each example uses various geometric shapes to calculate volume, showcasing real-life applications of volume in different shapes.

Topic

Volume

Description

A green sphere with a radius labeled as x. This image is part of a math example showing how to calculate the volume of a sphere using an unknown radius. This image illustrates Example 18: The text explains how to find the volume of a sphere with an unknown radius x. The formula used is V = (4/3) * π * r3, and substituting r = x gives V = (4/3) * x3 * π.

Topic

Volume

Description

A green cube with side length labeled as 7. The image illustrates how to calculate the volume of a cube with known side length. This image illustrates Example 19: The text describes finding the volume of a cube. The formula used is V = s3, where s = 7. After substituting, the result is V = 343.

Volume is a fundamental concept in geometry that helps students understand the space occupied by three-dimensional objects. In this collection, each example uses various geometric shapes to calculate volume, showcasing real-life applications of volume in different shapes.

Topic

Volume

Description

A rectangular prism with dimensions labeled as x, y, and z. The image shows a general example of calculating the volume of a rectangular prism using variables instead of specific numbers. This image illustrates Example 2: The caption describes how to find the volume of a rectangular prism using variables for length (x), width (y), and height (z). The formula is given as V = x * y * z, but no specific values are provided.

Topic

Volume

Description

A green cube with side length labeled as x. This image is part of a math example showing how to calculate the volume of a cube using an unknown side length. This image illustrates Example 20: The text explains how to find the volume of a cube with an unknown side length x. The formula used is V = s3, and substituting s = x gives V = x3.

Topic

Volume

Description

A hollow cube with an outer edge of 9 and an inner hollow region with an edge of 7. The image shows how to calculate the volume by subtracting the volume of the inner cube from the outer cube. This image illustrates Example 21: Find the volume of a hollow cube. The formula used is V = s13 - s23, where s1 is the outer edge (9) and s2 is the inner edge (7). The solution calculates 9^3 - 7^3 = 386..

Topic

Volume

Description

A hollow cube with an outer edge of x and an inner hollow region with an edge of x - 2. The image shows how to calculate the volume by subtracting the volume of the inner cube from the outer cube. This image illustrates Example 22: Find the volume of a hollow cube. The formula used is V = s13 - s23, where s1 = x and s2 = x - 2. Expanding and simplifying gives V = 6x2 - 12x + 8.

Topic

Volume

Description

A cone with a height of 12 and a radius of 4. The image shows how to calculate its volume using the cone volume formula (V = 1/3 * π * r2 * h). This image illustrates Example 23: Find the volume of a cone. The formula used is V = (1/3) * π * r2 * h, where r = 4 and h = 12. Substituting these values gives V = (1/3) * π * (42) * 12 = 64π.

Topic

Volume

Description

A cone with a height labeled as y and a radius labeled as x. The image shows how to calculate its volume using the cone volume formula (V = 1/3 * π * r2 * h). This image illustrates Example 24: Find the volume of a cone. The formula used is V = (1/3) * π * r2 * h, where r = x and h = y. Substituting these variables gives V = (x^2 * y)/3 * π.

Topic

Volume

Description

A hollow rectangular prism with outer dimensions: length = 60, width = 20, and height = 16. The inner hollow part has dimensions: length = 60, width = 18, and height = 14. The image shows how to subtract volumes to find the hollow volume. This image illustrates Example 3: The caption explains how to calculate the volume of a hollow rectangular prism by subtracting the inner volume from the outer volume. V = (60 * 20 * 16) - (60 * 18 * 14) = 4080.

Topic

Volume

Description

A hollow rectangular prism with outer dimensions labeled as x, y, and z, and inner hollow dimensions labeled as x - 2 and y - 2. The image shows a symbolic calculation for finding the hollow volume using variables. This image illustrates Example 4: The caption describes how to calculate the volume of a hollow rectangular prism by subtracting the inner volume from the outer volume using variables: V = xyz - z(y - 2)(x - 2) = 2z(y + x - 2).

Topic

Volume

Description

The image shows a triangular prism with dimensions labeled as base (7), height (10), and length (25). It is part of an example on how to calculate the volume of a solid triangular prism. This image illustrates Example 5: "Find the volume of this triangular prism." The solution involves substituting the given measurements into the volume formula for a triangular prism: V = 1/2 * b * h * l = 1/2 * 7 * 10 * 25 = 875.

Topic

Volume

Description

The image depicts a triangular prism with dimensions labeled as x, y, and z. The example demonstrates how to calculate the volume using a general formula for a triangular prism. This image illustrates Example 6: "Find the volume of this triangular prism." The solution uses the formula V = 1/2 * b * h * l, which is simplified to V = 1/2 * x * y * z..

Topic

Volume

Description

The image shows a hollow triangular prism with outer dimensions labeled as base (10), height (7), and length (35), and inner dimensions labeled as base (8) and height (5). The example calculates the volume by subtracting the hollow region from the full prism. This image illustrates Example 7: "Find the volume of this hollow triangular prism." The solution calculates the full volume using V = 1/2 * b1 * h1 * l1 - 1/2 * b2 * h2 * l2, which simplifies to V = 1/2 * 10 * 7 * 35 - 1/2 * 8 * 5 * 35 = 525..

Topic

Volume

Description

This image shows a hollow triangular prism with outer dimensions labeled as x, y, and z, and inner dimensions reduced by 2 units each. It demonstrates how to calculate the volume by subtracting the hollow region from the full prism. This image illustrates Example 8: "Find the volume of this hollow triangular prism." The solution uses V = 1/2 * b1 * h1 * l1 - 1/2 * b2 * h2 * l2, which simplifies to V = z(xy - (x - 2)(y - 2)) = z(x + y - 2)..

Topic

Volume

Description

A green cylinder with a radius of 10 units and a height of 8 units. The radius is marked on the top surface, and the height is marked on the side. This image illustrates Example 9: The task is to find the volume of the cylinder. The volume formula V = πr2h is used. Substituting the values r = 10 and h = 8, the volume is calculated as V= 800π.

This Teacher's Guide provides an overview of the 32 worked-out examples that show how to find the equation of a line parallel or perpendicular to a given line and through a given point.

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This Teacher's Guide provides an overview of the 24 worked-out examples that show how to calculate the surface area of different three-dimensional figures.

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This set of tutorials provides 24 examples of how to find the volume of various 3-dimensional geometric figures.

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This set of tutorials provides 24 examples of how to find the volume of various 3-dimensional geometric figures. NOTE: The download is a PPT file.

6/13/11. In this issue we explore the volcanic eruption in Chile that resulted in a huge plume of smoke and ash that was miles high. We explore the viscosity of lava that makes such eruptions possible.

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6/27/11. In this issue we look at the technology of hydraulic fracking. In particular, we estimate the amount of drilling required for such wells.

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9/19/11. To commemorate the 10-year anniversary of the 911, we look at the geometry and architecture of the Freedom Tower, currently under construction.

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11/14/11. In this issue we commemorate the 125th anniversary of the Statue of Liberty. We also look at the geometry and architecture of this monument.

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11/28/11. In this issue we look at the geometry of parade balloons.

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12/7/11. In this issue we look at the heroic work of salvaging and retrofitting many of the ships sunk during the attack on Pearl Harbor.

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12/12/11. In this issue we look at the MLK Memorial and the special properties of granite that presented some design challenges in the construction of the statue.

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January 2012. In this issue of Math in the News we analyze how the Costa Concordia sank.

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April 2012. In this issue of Math in the News we analyze the sinking of the Titanic. It has been a hundred years, yet the story of the sinkin g of the Titanic still fascinates.

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May 2012. In this issue of Math in the News we look at the architecture of the Freedom Tower, now that it has become the tallest building in NYC.

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August 2013. In this issue we look at the geometry of funnel cakes. You'll like the sweet results.

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August 2013. In this issue we look at Diana Nyad's remarkable swim from Cuba to Florida. It becomes an opportunity to explore Average Speed.

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September 2013. In this issue we look at NASA's newest Moon mission, the LADEE robotic probe. It becomes an opportunity to explore linear and logarithmic equations used in rocketry.

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5/9/11. In this issue we look at tornado season. In particular we study why the swirling winds of a tornado are so damaging.

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5/16/11. In this issue we look at the flooding along the Mississippi. In particular we analyze the flooding along Memphis, Tennessee. We develop a model for predicting the amount of flooding.

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The relationship between slope and grade in cycling is explored. Go on a tour of Italy through the mountains of Tuscany and apply students' understanding of slope. Note: The download for this resources is the Promethean Flipchart.

To access the full video: [Algebra Applications: Linear Functions, Segment 2: Cycling]: https://www.media4math.com/library/video-algebra-applications-linear-functions-segment-2-cycling

This video includes a Video Transcript: [Algebra Applications: Linear Functions, Segment 2: Cycling]: https://www.media4math.com/library/video-transcript-algebra-applications-linear-functions-segment-2-cycling

Exercise needs to become a consistent part of everyone's lifestyle. In particular, aerobic exercises, which vigorously exerts the heart, is an important form of exercise. The maximum heart rate from aerobic exercise is a linear function dependent on age. Students are asked to develop a data table based on the function. Note: The download for this resources is the Promethean Flipchart.

To access the full video: [Algebra Applications: Linear Functions, Segment 4: Exercise]: https://www.media4math.com/library/video-algebra-applications-linear-functions-segment-4-exercise

The potential for oil exploration in the controversial Alaska National Wildlife Refuge (ANWR) sets the scene for this problem. A linear regression of oil consumption data over the past 25 years reveals an interesting pattern. How could new oil fields like ANWR help in breaking our dependence on foreign oil? Note: The download for this resources is the Promethean Flipchart.

To access the full video [Algebra Applications: Linear Functions, Segment 3: Oil Exploration]: https://www.media4math.com/library/video-algebra-applications-linear-functions-segment-3-oil-exploration

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 Note: The download for this resources is the Promethean Flipchart.

This video includes a Teacher's Guide: https://www.media4math.com/library/algebra-nspirations-teachers-guide-linear-functions

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. Note: The download for this resources is the Promethean Flipchart.

For access to the full video [Geometry Applications: 3D Geometry, Segment 2: Pyramids]: https://media4math.com/library/geometry-applications-3d-geometry-segment-2-pyramids

This video includes a Video Transcript [Video Transcript: Geometry Applications: 3D Geometry, Segment 2: Pyramids]: https://www.media4math.com/library/video-transcript-geometry-applications-3d-geometry-segment-2-pyramids