Illustrative Math Alignment: Grade 6 Unit 9

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of terms and definitions related to trigonometry concepts.

This is part of a collection of terms and definitions related to trigonometry concepts.

This is part of a collection of terms and definitions related to trigonometry concepts.

This is part of a collection of terms and definitions related to trigonometry concepts.

This is part of a collection of terms and definitions related to trigonometry concepts.

This is part of a collection of terms and definitions related to trigonometry concepts.

This is part of a collection of terms and definitions related to trigonometry concepts.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

This is part of a collection of math examples that focus on ratios and rates.

Topic

Angles

Description

The image shows a unit circle with a highlighted arc and a point labeled (0.707, 0.707) on the circumference. The solution steps are shown on the right, explaining how to find the angle.

Example 1: The tangent ratio is calculated as tan(Θ) = 0.707 / 0.707 = 1. The inverse tangent is found as tan-1(1) = 0.785 radians. This is converted to degrees: Θ = 0.785 * (180 /π) = 45°.

Topic

Angles

Description

The image shows a unit circle with a highlighted arc and a point labeled (0.866, 0.5) on the circumference. The solution steps are displayed on the right, showing how to find the angle measure.

Example 2: The tangent ratio is calculated as tan(Θ) = 0.5 / 0.866 = 0.577. The inverse tangent is found as tan-1(0.577) = 0.524 radians. This is converted to degrees: Θ = 0.524 * (180 /π) = 30°.

Topic

Angles

Description

The image shows a unit circle with a highlighted arc and a point labeled (0.5, 0.866) on the circumference, connected by a tangent line from the origin to this point. The solution process for finding the angle measure is shown on the right side.

Example 3: The tangent ratio is calculated as tan(Θ) = 0.866 / 0.5 = 1.732. The inverse tangent is found as tan-1(1.732) = 1.047 radians. This is converted to degrees: Θ = 1.047 * (180 /π) = 60°.

Topic

Angles

Description

The image shows a unit circle with a highlighted arc and a point labeled (0, 1) at the top of the circle, representing an undefined tangent ratio as it lies on the y-axis directly above the origin. Solution steps are listed on the right side.

Example 4: The tangent ratio is undefined because tan(Θ) = 1 / 0, but tan-1(undefined) gives an angle of Θ = 1.57 radians, or 90°.

Topic

Angles

Description

This image shows a unit circle with a highlighted arc in the second quadrant. The point (-0.5, 0.866) is marked on the circle. The caption explains how to find the radian measure of the highlighted arc using trigonometric ratios. The arc is in red, and the radius is drawn.

Example 5: The tangent ratio is calculated using tan(Θ) = 0.866 / -0.5 = -1.732. The inverse tangent is found as tan-1(-1.732) = -1.047 radians, and converting this to degrees gives Θ = -60° or 120°.

Topic

Angles

Description

This image shows a unit circle with a highlighted arc in the second quadrant. The point (-0.707, 0.707) is marked on the circle. The caption explains how to find the radian measure of the highlighted arc using trigonometric ratios. The arc is in red, and the radius is drawn.

Example 6: The tangent ratio is calculated using tan(Θ) = 0.707 / -0.707 = -1. The inverse tangent is found as tan-1(-1) = -0.785 radians, and converting this to degrees gives Θ = -45 degrees or 135 degrees.

Topic

Angles

Description

This image shows a unit circle with a highlighted arc in the second quadrant. The point (-0.866, 0.5) is marked on the circle. The caption explains how to find the radian measure of the highlighted arc using trigonometric ratios. The arc is in red, and the radius is drawn.

Example 7: The tangent ratio is calculated using tan(Θ) = 0.5 / -0.866 = -0.577. The inverse tangent is found as tan-1(-0.577) = -0.524 radians, and converting this to degrees gives Θ = -30° or 150°.

Topic

Angles

Description

This image shows a unit circle with a highlighted arc along the negative x-axis (180 degrees). The point (-1, 0) is marked on the circle at this position on the unit circle's circumference, representing an angle of either 0 or 180 degrees depending on direction.

Example 8: The tangent ratio is calculated using tan(Θ) = 0 / 1 = 0. The inverse tangent is found as tan-1(0) = 0 radians, and converting this to degrees gives Θ = 0° or 180° depending on direction along the unit circle's circumference from origin to point (-1, 0).

Topic

Angles

Description

The image shows a unit circle with a highlighted arc in the fourth quadrant. The coordinates of the point on the circle are (0.866, -0.5). The tangent ratio is calculated, followed by the inverse tangent to find the angle in radians and degrees.

Example 9: The x-y coordinates are (0.866, -0.5). The tangent ratio is calculated as tan(Θ) = -0.5 / 0.866 = -0.577. The inverse tangent gives Θ = -0.524 radians, which is converted to degrees as Θ = -30°.

Topic

Angles

Description

This image displays a unit circle with a highlighted arc in the fourth quadrant. The coordinates of the point on the circle are (0.707, -0.707). The tangent ratio is calculated, followed by the inverse tangent to find the angle in radians and degrees.

Example 10: The x-y coordinates are (0.707, -0.707). The tangent ratio is calculated as tan(Θ) = -0.707 / 0.707 = -1. The inverse tangent gives Θ = -0.785 radians, which is converted to degrees as Θ = -45°.

Topic

Angles

Description

This image shows a unit circle with a highlighted arc in the fourth quadrant. The coordinates of the point on the circle are (0.5, -0.866). The tangent ratio is calculated, followed by the inverse tangent to find the angle in radians and degrees.

Example 11: The x-y coordinates are (0.5, -0.866). The tangent ratio is calculated as tan(Θ) = -0.866 / 0.5 = -1.732. The inverse tangent gives Θ = -1.047 radians, which is converted to degrees as Θ = -60°.

Topic

Angles

Description

This image displays a unit circle with a highlighted arc along the negative y-axis (third quadrant). The coordinates of the point on the circle are (0, -1). The tangent ratio is undefined due to division by zero, but the angle is found using radians/degrees.

Example 12: The x-y coordinates are (0, -1). Since tan(Θ) = -1 / 0 is undefined, we use the known value for this position on the unit circle: Θ = -1.57 radians, which is converted to degrees as Θ = -90°.

Topic

Angles

Description

The image shows a unit circle with a point at (-0.5, -0.866), and the highlighted arc is in the third quadrant. The tangent ratio is calculated using these coordinates. The solution involves finding the inverse tangent and converting radians to degrees.

Example 13: The tangent ratio is calculated as tan(Θ) = -0.866 / -0.5 = 1.732. The inverse tangent is tan-1(1.732) = 1.047 radians. Converting this to degrees gives Θ = 60° or -120°.

Topic

Angles

Description

This image shows a unit circle with a point at (-0.707, -0.707) in the third quadrant, and the highlighted arc is shown in red. The tangent ratio is calculated using these coordinates to find the angle in both radians and degrees.

Example 14: The tangent ratio is tan(Θ) = -0.707 / -0.707 = 1, leading to an inverse tangent of tan-1(1) = 0.785 radians. Converting this to degrees gives Θ = 45° or -135°.

Topic

Angles

Description

This image shows a unit circle with a point at (-0.866, -0.5), and the highlighted arc in red appears in the third quadrant of the circle. The tangent ratio is calculated using these coordinates, followed by finding the angle in both radians and degrees.

Example 15: The tangent ratio is tan(Θ) = -0.5 / -0.866 = 0.577, leading to an inverse tangent of tan-1(0.577) = 0.524 radians. Converting this to degrees gives Θ = 30° or -150°.

Topic

Angles

Description

This image shows a unit circle with a point at (-1, 0), located on the negative x-axis, and the highlighted arc covers half of the circle (180¡). The tangent ratio is calculated using these coordinates, followed by finding the angle in both radians and degrees.

Example 16: The tangent ratio is tan(Θ) = 0 / 1 = 0, leading to an inverse tangent of tan-1(0) = 0 radians. Converting this to degrees gives Θ = 0° or -180°.

Topic

Rationals and Radicals

Definition

An irrational number is a number that cannot be expressed as a ratio of two integers. Its decimal form is non-repeating and non-terminating.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

Topic

Exponents

Description

The image shows Example 1 of simplifying a rational exponent expression. The expression is 251/2. The solution involves converting the exponent to a square root. Solution: Convert the rational exponent to an nth root. 251/2 = √(25) = 5. This example involves simplifying expressions with rational exponents, providing a concrete application of exponent properties and the concept of roots. The method breaks down complex expressions into simpler terms, illustrating step-by-step how to handle rational exponents.

Topic

Exponents

Description

The image shows Example 2 of simplifying a rational exponent expression. The expression is 1251/3. The solution involves converting the exponent to a cube root. Example 2. Simplify the expression: 1251/3. Solution: Convert the rational exponent to an nth root. 1251/3 = ∛125 = 5. This example involves simplifying expressions with rational exponents, providing a concrete application of exponent properties and the concept of roots. The method breaks down complex expressions into simpler terms, illustrating step-by-step how to handle rational exponents.

Topic

Exponents

Description

The image shows Example 3 of simplifying a rational exponent expression. The expression is 161/4. The solution involves converting the exponent to a fourth root. Solution: Convert the rational exponent to an nth root. 161/4 = 4√16 = 2. This example involves simplifying expressions with rational exponents, providing a concrete application of exponent properties and the concept of roots. The method breaks down complex expressions into simpler terms, illustrating step-by-step how to handle rational exponents.

Topic

Exponents

Description

The image shows Example 4 of simplifying a rational exponent expression. The expression is 321/5. The solution involves converting the exponent to a fifth root. Solution: Convert the rational exponent to an nth root. 321/5 = 5√32 = 2. This example involves simplifying expressions with rational exponents, providing a concrete application of exponent properties and the concept of roots. The method breaks down complex expressions into simpler terms, illustrating step-by-step how to handle rational exponents.

Topic

Exponents

Topic

Exponents

Topic

Exponents

Topic

Exponents

Topic

Exponents

Description

The image shows Example 9 from a series of math problems on simplifying expressions with rational exponents. It includes a step-by-step solution for simplifying 272/3. The problem asks to simplify 272/3. The solution involves converting the fractional exponent to a root and exponent form: 272/3 = ∛272 = 32 = 9. This example involves simplifying expressions with rational exponents, providing a concrete application of exponent properties and the concept of roots. The method breaks down complex expressions into simpler terms, illustrating step-by-step how to handle rational exponents.

Topic

Exponents

Description

The image shows Example 10 from a series of math problems on simplifying expressions with rational exponents. It includes a step-by-step solution for simplifying 813/4. The solution involves converting the fractional exponent to a root and exponent form: 813/4 = 4√813 = 33 = 27. This example involves simplifying expressions with rational exponents, providing a concrete application of exponent properties and the concept of roots. The method breaks down complex expressions into simpler terms, illustrating step-by-step how to handle rational exponents.

Topic

Fraction Operations