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Illustrative Math-Media4Math Alignment

 

 

Illustrative Math Alignment: Grade 7 Unit 9

Putting it All Together

Lesson 12: Using a Trundle Wheel to Measure Distances

Use the following Media4Math resources with this Illustrative Math lesson.

Thumbnail Image Title Body Curriculum Topic
Definition--Circle Concepts--Circular Functions Definition--Circle Concepts--Circular Functions Circular Functions

Topic

Circles

Definition

Circular functions are trigonometric functions that relate angles to ratios of sides in a right triangle.

Definition of a Circle
Definition--Circle Concepts--Circular Models Definition--Circle Concepts--Circular Models Circular Models

Topic

Circles

Definition

Circular models are representations of circular phenomena using mathematical equations and diagrams.

Definition of a Circle
Definition--Circle Concepts--Circular Numerical Models Definition--Circle Concepts--Circular Numerical Models Circular Numerical Models

Topic

Circles

Definition

Circular numerical models use numerical methods to analyze and simulate circular phenomena.

Definition of a Circle
Definition--Circle Concepts--Circumcenter of Triangle Definition--Circle Concepts--Circumcenter of Triangle Circumcenter of Triangle

Topic

Circles

Definition

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect, equidistant from the vertices.

Definition of a Circle
Definition--Circle Concepts--Circumference Definition--Circle Concepts--Circumference Circumference

Topic

Circles

Definition

The circumference of a circle is the distance around the circle, calculated as C = 2πr.

Description

The circumference is a fundamental concept in geometry, representing the perimeter of a circle. It is widely used in fields such as engineering, design, and manufacturing, where precise measurements of circular objects are required. The formula 

C = 2πr 

Definition of a Circle
Definition--Circle Concepts--Circumscribed Angle Definition--Circle Concepts--Circumscribed Angle Circumscribed Angle

Topic

Circles

Definition

A circumscribed angle is an angle formed outside a circle by two intersecting tangents.

Definition of a Circle
Definition--Circle Concepts--Concentric Circles Definition--Circle Concepts--Concentric Circles Concentric Circles

Topic

Circles

Definition

Concentric circles are circles that share the same center but have different radii.

Definition of a Circle
Definition--Circle Concepts--Diameter Definition--Circle Concepts--Diameter Diameter

Topic

Circles

Definition

The diameter of a circle is a line segment that passes through the center and has its endpoints on the circle, calculated as D = 2πr.

Description

The diameter is a fundamental concept in geometry, representing the longest distance across a circle. It is widely used in fields such as engineering, design, and manufacturing, where precise measurements of circular objects are required. The formula 

D = 2πr

Definition of a Circle
Definition--Circle Concepts--Equation of a Circles Definition--Circle Concepts--Equation of a Circles Equation of a Circle

Topic

Circles

Definition

The equation of a circle in a plane is (x − h)2 + (y − k)2 = r2, where (h , k) is the center and r is the radius.

Definition of a Circle
Definition--Circle Concepts--Hexagon Inscribed in a Circle Definition--Circle Concepts--Hexagon Inscribed in a Circle Hexagon Inscribed in a Circle

Topic

Circles

Definition

A hexagon inscribed in a circle is a six-sided polygon where each vertex touches the circle.

Definition of a Circle
Definition--Circle Concepts--Incenter of a Triangle Definition--Circle Concepts--Incenter of a Triangle Incenter of a Triangle

Topic

Circles

Definition

The incenter of a triangle is the point where the angle bisectors intersect, equidistant from the sides.

Definition of a Circle
Definition--Circle Concepts--Inscribed Angle Definition--Circle Concepts--Inscribed Angle Inscribed Angle

Topic

Circles

Definition

An inscribed angle is an angle formed by two chords in a circle that share an endpoint.

Definition of a Circle
Definition--Circle Concepts--Inscribed Angle Theorem Definition--Circle Concepts--Inscribed Angle Theorem Inscribed Angle Theorem

Topic

Circles

Definition

The inscribed angle theorem states that an inscribed angle is half the measure of the central angle that subtends the same arc.

Definition of a Circle
Definition--Circle Concepts--Point of Tangency to a Circle Definition--Circle Concepts--Point of Tangency to a Circle Point of Tangency to a Circle

Topic

Circles

Definition

The point of tangency is the point where a tangent line touches the circle.

Definition of a Circle
Definition--Circle Concepts--Radius of a Circle Definition--Circle Concepts--Radius of a Circle Radius of a Circle

Topic

Circles

Definition

The radius of a circle is a line segment from the center to any point on the circle.

Definition of a Circle
Definition--Circle Concepts--Ratio of Circumference to Diameter Definition--Circle Concepts--Ratio of Circumference to Diameter Ratio of Circumference to Diameter

Topic

Circles

Definition

The ratio of the circumference to the diameter of a circle is the constant π.

Definition of a Circle
Definition--Circle Concepts--Secant to a Circle Definition--Circle Concepts--Secant to a Circle Secant to a Circle

Topic

Circles

Definition

A secant is a line that intersects a circle at two points.

Description

Secants are significant in geometry, representing lines that intersect a circle at two distinct points. These lines are used in various applications, such as in design and architecture, where precise measurements of angles and distances are essential. In mathematics, secants are explored in the context of circle theorems, providing insights into the properties of lines and circles. In education, understanding secants helps students develop geometric reasoning and problem-solving skills, which are essential for advanced studies in geometry and trigonometry.

Definition of a Circle
Definition--Circle Concepts--Sector Definition--Circle Concepts--Sector Sector

Topic

Circles

Definition

A sector is a portion of a circle enclosed by two radii and the arc between them.

Description

Sectors are fundamental in the study of circles, representing a portion of the circle's area. These shapes are used in various applications, such as in design and architecture, where precise measurements of angles and areas are essential. The area of a sector is calculated using the formula 

A = 1/2r2θ

Definition of a Circle
Definition--Circle Concepts--Square Inscribed in a Circle Definition--Circle Concepts--Square Inscribed in a Circle Square Inscribed in a Circle

Topic

Circles

Definition

A square inscribed in a circle has its vertices touching the circle.

Definition of a Circle
Definition--Circle Concepts--Tangent Circles Definition--Circle Concepts--Tangent Circles Tangent Circles

Topic

Circles

Definition

Tangent circles are two or more circles that intersect at exactly one point.

Definition of a Circle
Definition--Circle Concepts--Tangent Line Definition--Circle Concepts--Tangent Line Tangent Line

Topic

Circles

Definition

A tangent line is a straight line that touches a circle at exactly one point.

Definition of a Circle
Definition--Circle Concepts--Triangle Inscribed in a Circle Definition--Circle Concepts--Triangle Inscribed in a Circle Triangle Inscribed in a Circle

Topic

Circles

Definition

A triangle inscribed in a circle has its vertices on the circle.

Definition of a Circle
Definition--Circle Concepts--Unit Circle Definition--Circle Concepts--Unit Circle Unit Circle

Topic

Circles

Definition

The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane.

Definition of a Circle
Definition--ComplexFraction.jpg Definition--Complex Fraction Complex Fraction

Topic

Fractions

Definition

A complex fraction is a fraction where the numerator, the denominator, or both, contain a fraction themselves.

Description

Complex fractions are an important concept in the study of fractions and algebra. They appear in various mathematical contexts, particularly when dealing with rational expressions and advanced algebraic operations. Understanding complex fractions is essential for simplifying expressions and solving equations that involve multiple layers of fractions.

Fractions and Mixed Numbers
Geometric Construction Instructional Resource--Geometric Construction--Constructing an Egg Shape Instructional Resource | Geometric Construction | An Egg Shape

In this instructional resource, we show the steps in constructing an egg shape. In the process, students learn about using common points of tangency to create smooth curves from two different shapes. This activity can be done with pencil, compass, ruler, and grid paper. 

—PRESS PREVIEW TO VIEW THE ACTIVITY— To see the complete collection of Instructional Resource, click on this link.
Geometric Constructions with Circles
INSTRUCTIONAL RESOURCE: Math Examples--Circular Area and Circumference INSTRUCTIONAL RESOURCE: Math Examples 1 INSTRUCTIONAL RESOURCE: Math Examples--Circular Area and Circumference

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

This is part of a collection of math examples for a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file.

Library of Instructional Resources

To see the complete library of Instructional Resources , click on this link.

Area and Circumference
INSTRUCTIONAL RESOURCE: Pumpkin PI INSTRUCTIONAL RESOURCE: Pumpkin PI

In this Slide Show, do a hands-on geometry activity to measure π. Note: The download is a PPT file.

Ratios and Rates
Interactive Crossword Puzzle--Circles Interactive Crossword Puzzle--Circles Interactive Crossword Puzzle--Circles

This interactive crossword puzzle tests knowledge of key terms on the topic of circles.

This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide.

Related Resources

To see additional resources on this topic, click on the Related Resources tab.
Definition of a Circle
Math Example--Area and Perimeter--Circular Area and Circumference: Example 1 Math Example--Area and Perimeter--Circular Area and Circumference: Example 1 Math Example--Area and Perimeter--Circular Area and Circumference: Example 1

Topic

Geometry

Description

This example presents a circle with a radius of 5 units. The task is to calculate the area of the circle using the given radius. The solution involves substituting the radius value into the area formula: A = π * r^2 = π * (5)^2 = 25π.

Understanding circular area and circumference is integral to mastering geometry. Concepts such as calculating areas and circumferences of circles are fundamental, and exercises like these examples not only provide practice but also deepen the understanding of theoretical concepts in a practical way.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 10 Math Example--Area and Perimeter--Circular Area and Circumference: Example 10 Math Example--Area and Perimeter--Circular Area and Circumference: Example 10

Topic

Geometry

Description

This example presents a circle with radius x and a shaded sector with central angle θ (theta). The task is to express the area of the shaded sector in terms of x and θ. The solution involves using the central angle to find the fractional amount of the total area: A = (θ / 360) * π * x2 = πx2θ / 360.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 11 Math Example--Area and Perimeter--Circular Area and Circumference: Example 11 Math Example--Area and Perimeter--Circular Area and Circumference: Example 11

Topic

Geometry

Description

This example presents a circle with radius x and a shaded sector with central angle θ (theta). The task is to express the arc length of the shaded region in terms of x and θ. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (θ / 360) * (2 * π * x) = (θ * x / 180) * π.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 12 Math Example--Area and Perimeter--Circular Area and Circumference: Example 12 Math Example--Area and Perimeter--Circular Area and Circumference: Example 12

Topic

Geometry

Description

This example features a circle with a radius of 5 units and a shaded sector with central angle θ (theta). The task is to express the area of the shaded sector in terms of θ. The solution involves using the central angle to find the fractional amount of the total area: Area = (θ / 360) * (π * 52) = 25π / 360 * θ = 5π / 72 * θ.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 13 Math Example--Area and Perimeter--Circular Area and Circumference: Example 13 Math Example--Area and Perimeter--Circular Area and Circumference: Example 13

Topic

Geometry

Description

This example presents a circle with a radius of 5 units and a shaded sector with central angle θ (theta). The task is to express the arc length of the shaded region in terms of θ. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (θ / 360) * (2 * π * 5) = θ / 18 * π.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 14 Math Example--Area and Perimeter--Circular Area and Circumference: Example 14 Math Example--Area and Perimeter--Circular Area and Circumference: Example 14

Topic

Geometry

Description

This example features a circle with radius x and a shaded sector with a central angle of 30 degrees. The task is to express the area of the shaded sector in terms of x. The solution involves using the central angle to find the fractional amount of the total area: Area = (30 / 360) * (π * x2) = π * x2 / 12.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 15 Math Example--Area and Perimeter--Circular Area and Circumference: Example 15 Math Example--Area and Perimeter--Circular Area and Circumference: Example 15

Topic

Geometry

Description

This example presents a circle with radius x and a shaded sector with a central angle of 30 degrees. The task is to express the arc length of the shaded region in terms of x. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (30 / 360) * (2 * π * x) = x / 6 * π.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 16 Math Example--Area and Perimeter--Circular Area and Circumference: Example 16 Math Example--Area and Perimeter--Circular Area and Circumference: Example 16

Topic

Geometry

Description

This example features two concentric circles with radii of 5 and 4 units, and a shaded sector with a central angle of 30 degrees. The task is to calculate the area of the shaded region. The solution involves using the central angle to determine the fractional area difference between the larger and smaller circles: Area = (θ / 360) * (π * r12 - π * r22) = (30 / 360) * (π * 52 - π * 42) = π / 12 * (25 - 16) = 3π / 4.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 17 Math Example--Area and Perimeter--Circular Area and Circumference: Example 17 Math Example--Area and Perimeter--Circular Area and Circumference: Example 17

Topic

Geometry

Description

This example presents two concentric circles with radii of 5 and 4 units, and a shaded sector with a central angle of 30 degrees. The task is to calculate the perimeter of the shaded region. The solution involves calculating arc lengths based on the central angle and adding straight-line segments between radii: Perimeter = (θ / 360) * (2 * π * r1 + 2 * π * r2) + 2 * (r1 - r2) = (30 / 360) * (2π * 5 + 2π * 4) + 2 * (5 - 4) = 3π / 2 + 2.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 18 Math Example--Area and Perimeter--Circular Area and Circumference: Example 18 Math Example--Area and Perimeter--Circular Area and Circumference: Example 18

Topic

Geometry

Description

This example features two concentric circles with radii x and y, 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, y, 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 * y2) = (θπ / 360) * (x2 - y2).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 19 Math Example--Area and Perimeter--Circular Area and Circumference: Example 19 Math Example--Area and Perimeter--Circular Area and Circumference: Example 19

Topic

Geometry

Description

This example presents two concentric circles with radii x and y, 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, y, and θ. The solution involves calculating arc lengths based on the central angle and adding straight-line segments between radii: Perimeter = (θ / 360) * (2πx + 2πy) + 2(x - y) = π/180 * θ(x + y) + 2(x - y).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 2 Math Example--Area and Perimeter--Circular Area and Circumference: Example 2 Math Example--Area and Perimeter--Circular Area and Circumference: Example 2

Topic

Geometry

Description

This example features a circle with a radius of 5 units. The objective is to calculate the circumference of the circle using the given radius. The solution involves applying the circumference formula: C = 2 * π * r = 2 * π * (5) = 10π.

Circular area and circumference calculations are fundamental in geometry. These examples provide students with practical applications of theoretical concepts, helping them understand the relationship between a circle's radius and its circumference.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 20 Math Example--Area and Perimeter--Circular Area and Circumference: Example 20 Math Example--Area and Perimeter--Circular Area and Circumference: Example 20

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).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 21 Math Example--Area and Perimeter--Circular Area and Circumference: Example 21 Math Example--Area and Perimeter--Circular Area and Circumference: Example 21

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).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 22 Math Example--Area and Perimeter--Circular Area and Circumference: Example 22 Math Example--Area and Perimeter--Circular Area and Circumference: Example 22

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).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 23 Math Example--Area and Perimeter--Circular Area and Circumference: Example 23 Math Example--Area and Perimeter--Circular Area and Circumference: Example 23

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).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 3 Math Example--Area and Perimeter--Circular Area and Circumference: Example 3 Math Example--Area and Perimeter--Circular Area and Circumference: Example 3

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.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 4 Math Example--Area and Perimeter--Circular Area and Circumference: Example 4 Math Example--Area and Perimeter--Circular Area and Circumference: Example 4

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.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 5 Math Example--Area and Perimeter--Circular Area and Circumference: Example 5 Math Example--Area and Perimeter--Circular Area and Circumference: Example 5

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.

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 6 Math Example--Area and Perimeter--Circular Area and Circumference: Example 6 Math Example--Area and Perimeter--Circular Area and Circumference: Example 6

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).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 7 Math Example--Area and Perimeter--Circular Area and Circumference: Example 7 Math Example--Area and Perimeter--Circular Area and Circumference: Example 7

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).

Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 8 Math Example--Area and Perimeter--Circular Area and Circumference: Example 8 Math Example--Area and Perimeter--Circular Area and Circumference: Example 8

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.

Area and Circumference