Illustrative Math Alignment: Grade 8 Unit 3

Topic

Geometry

Description

This example features two concentric circles with radii 5 and y, and a shaded sector with a central angle of 30 degrees. The task is to express the area of the shaded region in terms of y. The solution involves calculating the difference between the areas of the larger and smaller circles, accounting for the central angle: Area = π/12 * (25 - y2).

Topic

Geometry

Description

This example features two concentric circles with radii 5 and y, and a shaded sector with a central angle of 30 degrees. The task is to calculate the perimeter of the shaded region. Given a central angle of 30 degrees, radius 5, and unknown radius y, the perimeter is calculated as: Perimeter = π/6 * (5 + y) + 2(5 - y).

Topic

Geometry

Description

This example presents two concentric circles with radii 4 and x, and a shaded sector with a central angle θ (theta). The task is to express the area of the shaded region in terms of x and θ. The solution uses the central angle to find the fractional area difference between the larger and smaller circles: Area = (θ / 360) * (π * x2 - π * 42) = (θ * π / 360) * (x2 - 16).

Topic

Geometry

Description

This example features two concentric circles with radii x and 4, and a shaded sector with a central angle θ (theta). The task is to express the perimeter of the shaded region in terms of x and θ. The solution involves calculating arc lengths based on the central angle and adding straight-line segments between radii: Perimeter = (θ / 360) * (2 * π * x + 2 * π * 4) + 2 * (x - 4) = (π / 180) * θ(x + 4) + 2(x - 4).

Topic

Geometry

Description

This example introduces a circle with an unknown radius represented by x. The task is to express the area of the circle in terms of x. The solution demonstrates how to use the area formula with a variable radius: A = π * r2 = π * (x)2 = π * x2.

Working with variable expressions in geometry helps students transition from concrete to abstract thinking. This example bridges the gap between numerical calculations and algebraic representations, a crucial skill in advanced mathematics.

Topic

Geometry

Description

This example presents a circle with an unknown radius represented by x. The task is to express the circumference of the circle in terms of x. The solution demonstrates how to use the circumference formula with a variable radius: C = 2 * π * r = 2 * π * x = 2πx.

Understanding circular area and circumference is crucial in geometry. This example helps students transition from concrete numerical values to abstract algebraic expressions, fostering a deeper comprehension of the relationship between a circle's radius and its circumference.

Topic

Geometry

Description

This example features two concentric circles with radii of 5 and 4 units. The task is to calculate the area of the shaded region between these circles. The solution involves subtracting the area of the smaller circle from the area of the larger circle: A = π * (5)2 - π * (4)2 = 25π - 16π = 9π.

Concentric circles and shaded regions introduce students to more complex geometric concepts. This example builds upon basic circular area calculations, encouraging students to think about the relationships between different circles and how to find areas of composite shapes.

Topic

Geometry

Description

This example presents two concentric circles with radii of 5 and y units. The objective is to express the area of the shaded region between these circles in terms of y. The solution involves subtracting the area of the smaller circle from the area of the larger circle: A = π * (5)2 - π * y2 = 25π - πy2 = π(25 - y2).

Topic

Geometry

Description

This example features two concentric circles with radii x and y. The task is to express the area of the shaded region between these circles in terms of x and y. The solution involves subtracting the area of the smaller circle from the area of the larger circle: A = π * x2 - π * y2 = π(x2 - y2).

Topic

Geometry

Description

This example presents a circle with a radius of 5 units and a shaded sector with a central angle of 30 degrees. The task is to calculate the area of the shaded sector. The solution involves using the central angle to find the fractional amount of the total area: A = (30 / 360) * π * 52 = 25π / 12.

Topic

Geometry

Description

This example features a circle with a radius of 5 units and a shaded arc corresponding to a central angle of 30 degrees. The task is to calculate the length of the shaded arc. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (30 / 360) * 2 * π * 5 = 5π / 6.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This is part of a collection of math examples that focus on volume.

This set of tutorials provides 23 examples of solving for the area and circumferences of circles and sections of circles.

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

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

Related Resources

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

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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8/1/11. In this issue we look at the physics of air pressure and the forces that give rise to heat waves.

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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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