Illustrative Math Alignment: Grade 7 Unit 5

Topic

Rationals and Radicals

Definition

A rational exponent is an exponent that is a fraction, where the numerator indicates the power and the denominator indicates the root.

Description

Rational Exponents are a crucial concept in the study of Rational Numbers, Expressions, Equations, and Functions. These exponents are fractions, where the numerator indicates the power and the denominator indicates the root. For example, the expression 

$$a^{m/n}$$

can be rewritten as 

$$\sqrt[n]{a^m}$$

Topic

Rationals and Radicals

Definition

Rational expressions are fractions in which the numerator and/or the denominator are polynomials.

Description

Rational Expressions are a fundamental aspect of Rational Numbers, Expressions, Equations, and Functions. These expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions often involves factoring the polynomials and canceling common factors. For example, the rational expression 

$$\frac{x^2 - 1}{x - 1}$$

can be simplified to x + 1, provided that 

$$x \neq 1$$

Topic

Rationals and Radicals

Definition

Rational functions are functions that are the ratio of two polynomials.

Description

Rational Functions are a key concept in the study of Rational Numbers, Expressions, Equations, and Functions. These functions are the ratio of two polynomials, such as 

$$f(x) = \frac{P(x)}{Q(x)}$$

where P(x) and Q(x) are polynomials. Understanding rational functions involves analyzing their behavior, including identifying asymptotes, intercepts, and discontinuities. For example, the function 

$$f(x) = \frac{1}{x}$$

Topic

Rationals and Radicals

Definition

Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero.

Topic

Rationals and Radicals

Definition

Rationalizing a radical is the process of eliminating radicals from the denominator of a fraction by multiplying both the numerator and denominator by an appropriate factor.

Description

Rationalizing a Radical is an important technique in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves eliminating radicals from the denominator of a fraction, which simplifies the expression and often makes it easier to work with or compare to other expressions. For example, to rationalize the denominator of 

$$\frac{1}{\sqrt{3}}$$

Topic

Rationals and Radicals

Definition

Rationalizing the denominator is the process of eliminating radicals or complex numbers from the denominator of a fraction by multiplying both the numerator and denominator by an appropriate factor.

Description

Rationalizing the Denominator is a crucial technique in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves removing radicals or complex numbers from the denominator of a fraction, which simplifies the expression and often makes it easier to evaluate or compare with other expressions. For example, to rationalize the denominator of 

Topic

Rationals and Radicals

Definition

Simplifying a radical expression involves reducing the expression to its simplest form by factoring the radicand and removing any perfect square factors (for square roots) or perfect cube factors (for cube roots).

Description

Simplifying a Radical Expression is a fundamental skill in the study of Radical Numbers, Expressions, Equations, and Functions. This process involves reducing a radical expression to its simplest form, which often makes it easier to work with and understand. For example, simplifying 

$$\sqrt{18}$$

results in 

Topic

Rationals and Radicals

Definition

Simplifying a rational expression involves reducing the fraction to its lowest terms by factoring both the numerator and denominator and canceling common factors.

Description

Simplifying a Rational Expression is a crucial skill in the study of Rational Numbers, Expressions, Equations, and Functions. This process involves reducing a rational expression to its simplest form by factoring both the numerator and denominator and canceling common factors. For example, simplifying 

$$\frac{x^2 - 1}{x - 1}$$

results in x + 1 for 

Topic

Rationals and Radicals

Definition

The square root of a number is a value that, when multiplied by itself, gives the number. It is denoted by the radical symbol √.

Topic

Rationals and Radicals

Definition

A vertical asymptote is a vertical line that the graph of a function approaches but never reaches as the input values get closer to a certain point.

Description

Vertical Asymptotes are a crucial concept in the study of Rational Numbers, Expressions, Equations, and Functions. They occur in rational functions when the denominator equals zero for certain input values, causing the function to approach infinity or negative infinity. For example, the function 

$$f(x) = \frac{1}{x-2}$$

Topic

Geometric Models

Topic

Geometric Models

In this Slide Show, learn how to add and subtract rational numbers. Includes links to several Media4Math videos and a math game.

Library of Instructional Resources

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

In this Slide Show, use the Desmos graphing calculator to explore rational functions. To see the complete collection of Desmos Resources click on this link.

Library of Instructional Resources

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

The complete set of 28 examples that make up this set of tutorials.

Library of Instructional Resources

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

The complete set of 28 examples that make up this set of tutorials.

Library of Instructional Resources

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

This set of tutorials provides 28 examples of rational functions in tabular and graph form.

Library of Instructional Resources

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

Topic

Functions

Description

This clip art depicts a rational function graph. Rational functions are quotients of polynomials and can have a wide variety of shapes depending on the degrees of the numerator and denominator. They generally pass the vertical line test and are thus functions, though they may have discontinuities.

This is part of a collection of math clip art images that show money amounts.

This is part of a collection of math clip art images that show money amounts.

This is part of a collection of math clip art images that show money amounts.

This is part of a collection of math clip art images that show money amounts.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math clip art images about different number systems. Included are integers, rational numbers, and real numbers.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

This is part of a collection of math examples that focus on exponents, exponential expressions, and exponential functions.

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