Illustrative Math Alignment: Grade 8 Unit 8

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 

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

In this Algebra Application, students develop a quadratic mathematical model for calculating the speed of a car based on the length of its skid marks. Using this model, students investigate the corresponding parabola. The math topics covered include: Mathematical modeling, Quadratic functions in standard form, Square root formula, Data analysis. The culminating activity is for students to write a program, using either a spreadsheet or a computer language like Python, that can calculate the speed of a car, given the skid mark length and coefficient of friction as the two inputs.

The complete set of 4 examples that make up this set of tutorials.

Library of Instructional Resources

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The complete set of 12 examples that make up this set of tutorials.

Library of Instructional Resources

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

In this Slide Show, learn how to program a function for solving quadratic equations. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.

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

In this Slide Show, learn how to solve a quadratic equation by completing the square.

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Library of Instructional Resources

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

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.

Topic

Exponents

Description

Shows Example 1 with the expression 32. The solution explains how to simplify by multiplying 3 by itself according to the exponent. Example 1: Simplify 32. Multiply 3 by itself two times: 3•3 = 9.

Topic

Exponents

Description

Shows Example 10 with the expression (-3)3•54. The solution uses order of operations to evaluate each term before multiplying. Example 10: Simplify (-3)3•54. Evaluate each exponential term separately, then multiply: -27•625= -16,875.

Topic

Exponents

Description

Example 11 shows how to simplify (1/2)2. The image illustrates multiplying 1/2 by itself due to the exponent of 2. Example 11: Simplify (1/2)2. Multiply one-half two times. Solution: (1/2) * (1/2) = 1/4.

Topic

Exponents

Description

Example 12 shows how to simplify (1/3)4. The image illustrates multiplying 1/3 four times due to the exponent of 4. Example 12: Simplify (1/3)4. Multiply one-third four times. Solution: (1/3) * (1/3) * (1/3) * (1/3) = 1/81.

Topic

Exponents

Description

Example 13 shows how to simplify (-1/3)3. The image illustrates multiplying -1/3 three times due to the exponent of 3. Example 13: Simplify (-1/3)3. Multiply negative one-third three times. Solution: (-1/3) * (-1/3) * (-1/3) = -1/27.

Topic

Exponents

Description

Example 14 shows how to simplify 2-1. The image illustrates rewriting 2-1 as the reciprocal of 2 raised to the first power. Example 14: Simplify 2-1. Negative exponents are written as reciprocals. Solution: 2-1 = 1/2.

Topic

Exponents

Description

Example 15 shows how to simplify 2-2. The image illustrates rewriting 2-2 as the reciprocal of 2 raised to the second power. Example 15: Simplify 2-2. Negative exponents are written as reciprocals. Solution: 2-2 = 1/(22) = 1/4.

Topic

Exponents

Description

Shows Example 2 with the expression 43. The solution explains the simplification by multiplying 4 three times. Example 2: Simplify 43. Multiply 4 by itself three times: 4•4•4 = 64.

In general, the topic of exponents involves understanding how repeated multiplication can be expressed more compactly. The examples provided in this collection allow students to see the step-by-step breakdown of how to simplify various exponential expressions, which can include positive and negative bases, fractional bases, and negative exponents.

Topic

Exponents

Description

Shows Example 3 with the expression 54. The solution details multiplying 5 four times. Example 3: Simplify 54. Multiply 5 by itself four times: 5•5•5•5 = 625.

In general, the topic of exponents involves understanding how repeated multiplication can be expressed more compactly. The examples provided in this collection allow students to see the step-by-step breakdown of how to simplify various exponential expressions, which can include positive and negative bases, fractional bases, and negative exponents.

Topic

Exponents

Description

Shows Example 4 with the expression 106. The solution simplifies by multiplying 10 six times. Example 4: Simplify 106. Multiply 10 by itself six times: 10•10•10•10•10•10 = 1,000,000.

Topic

Exponents

Description

Shows Example 5 with the expression (-1)2. The solution explains the result by multiplying -1 by itself. Example 5: Simplify (-1)2. Multiply -1 by itself two times: (-1)•(-1) = 1.

Topic

Exponents

Description

Shows Example 6 with the expression (-5)3. The solution demonstrates multiplying -5 three times. Example 6: Simplify (_5)3. Multiply -5 by itself three times: (-5)•(-5)•(-5) = -125.

Topic

Exponents

Description

Shows Example 7 with the expression (-6)4. The solution explains multiplying -6 by itself four times. Example 7: Simplify (-6)4. Multiply -6 by itself four times: (-6)•(-6)•(-6)•(-6) = 1296.

Topic

Exponents

Description

Shows Example 8 with the expression 23•32. The solution demonstrates using order of operations to simplify each term separately, then multiply. Example 8: Simplify 23•32. Evaluate each exponential term separately, then multiply: 8•9 = 72.

Topic

Exponents

Description

Shows Example 9 with the expression (-1)2•43. The solution explains simplifying each term individually before multiplying. Example 9: Simplify (-1)2•43. Evaluate each exponential term separately, then multiply: 1•64 = 64.

Topic

Quadratics

Topic

Quadratics

Description

This example illustrates a quadratic with two real roots. The discriminant being positive indicates that there are two real solutions to the equation. This scenario helps students recognize the significance of repeated roots and their graphical representation.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Topic

Quadratics

Description

This example demonstrates a quadratic equation with one distinct real root, highlighting the relationship between the coefficients and the resulting discriminant. By calculating the discriminant, students can identify the nature of the roots, which is crucial for solving quadratics effectively. Skills involved include algebraic manipulation and understanding the graphical implications of roots.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

This example illustrates a quadratic with two real roots. The discriminant being positive indicates that there are two real roots. This scenario helps students recognize the significance of repeated roots and their graphical representation.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

This example demonstrates a quadratic equation with no distinct real roots, highlighting the relationship between the coefficients and the resulting discriminant. By calculating the discriminant, students can identify the nature of the roots, which is crucial for solving quadratics effectively. Skills involved include algebraic manipulation and understanding the graphical implications of roots.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

This example illustrates a quadratic with a double real root. The discriminant being zero indicates that there is exactly one real solution to the equation. This scenario helps students recognize the significance of repeated roots and their graphical representation.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

In this example, the quadratic equation results in two roots, demonstrating the cases where the discriminant is positive. This provides insights into scenarios where quadratic equations intersect the x-axis twice. Skills in algebra are focused on interpreting complex solutions and their geometric implications.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

This example illustrates a quadratic with two real roots. The discriminant being positive indicates that there are two real solutions to the equation. This scenario helps students recognize the significance of two roots and their graphical representation.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

This example demonstrates a quadratic equation with two distinct real roots, highlighting the relationship between the coefficients and the resulting discriminant. By calculating the discriminant, students can identify the nature of the roots, which is crucial for solving quadratics effectively. Skills involved include algebraic manipulation and understanding the graphical implications of roots.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

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.