Illustrative Math Alignment: Grade 6 Unit 9

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

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

The laws of rational exponents describe how to handle exponents that are fractions, including rules for multiplication, division, and raising a power to a power.

Description

The Laws of Rational Exponents are vital in the study of Rational Numbers, Expressions, Equations, and Functions. These laws provide a framework for simplifying expressions involving exponents that are fractions. For example, the law

 $$a^{m/n} = \sqrt[n]{a^m}$$ 

Topic

Rationals and Radicals

Definition

Factoring a radical involves expressing it as a product of simpler expressions.

Description

Factoring radicals is a key technique in the study of Radical Numbers, Expressions, Equations, and Functions. It simplifies complex radical expressions, making them easier to work with. This process is essential for solving equations and for performing algebraic manipulations involving radicals.

Understanding how to factor radicals is also important for simplifying expressions in calculus and higher-level mathematics. It helps in breaking down complex problems into more manageable parts.

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

Partial fraction decomposition is a method used to express a rational expression as a sum of simpler fractions.

Description

Partial Fraction Decomposition is a powerful tool in the study of Rational Numbers, Expressions, Equations, and Functions. It involves breaking down a complex rational expression into a sum of simpler fractions, which are easier to integrate or differentiate. For example, the rational function 

$$\frac{2x+3}{(x+1)(x-2)}$$

can be decomposed into 

Topic

Rationals and Radicals

Definition

An asymptote is a line that a graph approaches but never touches.

Description

Asymptotes are significant in the study of Rational Numbers, Expressions, Equations, and Functions. They help in understanding the behavior of graphs of rational functions, particularly as the values of the variables approach certain limits. Horizontal asymptotes indicate the value that the function approaches as the input grows infinitely large or small. Vertical asymptotes show the values that the function cannot take because they cause division by zero.

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

Rationals and Radicals

Definition

A horizontal asymptote is a horizontal line that a rational function graph approaches as the input values become very large or very small.

Description

Horizontal asymptotes are an important concept in the study of Rational Numbers, Expressions, Equations, and Functions. They indicate the value that a function approaches as the input grows infinitely large or small. Understanding horizontal asymptotes is crucial for graphing rational functions accurately and for analyzing their long-term behavior.

Topic

Rationals and Radicals

Definition

An oblique asymptote is a diagonal line that the graph of a function approaches as the input values become very large or very small.

Description

Oblique Asymptotes are important in the study of Rational Numbers, Expressions, Equations, and Functions. They occur in rational functions where the degree of the numerator is one more than the degree of the denominator. For example, the function 

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

Topic

Rationals and Radicals

Definition

Rational equations are equations that involve rational expressions, which are fractions containing polynomials in the numerator and denominator.

Description

Rational Equations are a fundamental aspect of Rational Numbers, Expressions, Equations, and Functions. These equations involve rational expressions, which are fractions containing polynomials in the numerator and denominator. Solving rational equations typically requires finding a common denominator, clearing the fractions, and then solving the resulting polynomial equation. For example, to solve 

$$\frac{1}{x} + \frac{1}{x+1} = \frac{1}{2}$$

Topic

Rationals and Radicals

Definition

Radical equations are equations in which the variable is inside a radical, such as a square root or cube root.

Description

Radical Equations are a fundamental aspect of Radical Numbers, Expressions, Equations, and Functions. These equations involve variables within radical signs, such as square roots or cube roots. Solving radical equations typically requires isolating the radical on one side of the equation and then squaring both sides to eliminate the radical. For example, to solve 

$$\sqrt{x+3} = 5$$

one would square both sides to obtain 

$$x + 3 = 25$$

Topic

Rationals and Radicals

Definition

Graphs of radical functions are visual representations of equations involving radicals.

Description

Graphs of radical functions are crucial in the study of Radical Numbers, Expressions, Equations, and Functions. They provide a visual understanding of how these functions behave, including their domains, ranges, and key features such as intercepts and asymptotes.

Topic

Polynomials

Topic

Polynomials

Description

The example illustrates the process of finding rational roots for the polynomial. Find the rational roots of the polynomial P(x) = x3 - 3x2 + 7x - 5 using the Rational Root Theorem. 1. Identify factors of the constant term (-5), which are ±1 and ±5, and factors of the leading coefficient (1), which is ±1. 2. Form possible values of p/q: ±1 and ±5. 3. Test these values to find the roots. Roots found are x = 1. 4. The polynomial can be factored as P(x) = (x - 1)(x2 - 2x + 5). The answer is P(x) = (x - 1)(x2 - 2x + 5). There are no additional rational roots.

Topic

Polynomials

Description

The example illustrates the process of finding rational roots for the polynomial. Find the rational roots of the polynomial P(x) = 3x3 - 7x2 + 4 using the Rational Root Theorem. 1. Identify factors of the constant term (4), which are ±1, ±2, and ±4, and factors of the leading coefficient (3), which are ±1 and ±3. 2. Form possible values of p/q: ±1/3, ±2/3, ±1, ±4/3, ±2, and ±4. 3. Test these values to find the roots. Roots found are x = -2/3, 1, and 2. 4. The polynomial can be factored as P(x) = 3x3 - 7x2 + 4. The answer is P(x)=(x + 2/3)(x - 1)(x - 2).

Topic

Polynomials

Topic

Polynomials

Description

The example illustrates the process of finding rational roots for the polynomial. Find the rational roots of the polynomial P(x) = x3 - 7x2 + 14x - 8 using the Rational Root Theorem. 1. Identify possible factors of the constant term (-8), which are ±1, ±2, ±4, and ±8. 2. Test each factor by substituting them into the polynomial until a root is found. Roots found are x = 1, 2, and 4. 3. The polynomial can be factored as P(x)=(x - 1)(x - 2)(x - 4). The answer is P(x) = x3 - 7x2 + 14x - 8.

Topic

Polynomials

Description

The example illustrates the process of finding rational roots for the polynomial. Find the rational roots of the polynomial P(x) = x3 + 7x2 + 14x + 8 using the Rational Root Theorem. 1. Identify possible factors of the constant term (8), which are ±1, ±2, ±4, and ±8. 2. Test each factor by substituting them into the polynomial to find the roots. Roots found are x = -1, -2, and -4. 3. The polynomial can be factored as P(x) = x3 + 7x2 + 14x + 8. The answer is P(x) = (x + 1)(x + 2)(x + 4).

Topic

Polynomials

Description

The example illustrates the process of finding rational roots for the polynomial. Find the rational roots of the polynomial P(x) = x3 + 4x2 + x - 6 using the Rational Root Theorem. 1. Identify possible factors of the constant term (-6), which are ±1, ±2, ±3, and ±6. 2. Test each factor as possible roots by substituting them into the polynomial until a root is found. Roots found are x = 1, -2, and -3. 3. The polynomial can be factored as P(x) = x3 + 4x2 + x - 6. The answer is P(x)=(x - 1)(x + 2)(x + 3).

Topic

Polynomials

Definition

The Rational Root Theorem states that any rational root of a polynomial equation is a factor of the constant term divided by a factor of the leading coefficient.

Description

The Rational Root Theorem is a powerful tool in algebra that provides a systematic way to identify possible rational roots of a polynomial equation. According to the theorem, any rational root of a polynomial equation is a factor of the constant term divided by a factor of the leading coefficient. This theorem is particularly useful for solving higher-degree polynomial equations and for analyzing polynomial functions.

This is part of a collection of video tutorials on the topic of Ratios and Proportions. This series includes a complete overview of ratios, equivalent ratios, rates, unit rates, and proportions. The following section will provide additional background information for the complete series of videos.

What Are Ratios?

A ratio is the relationship between two or more quantities among a group of items. The purpose of a ratio is find the relationship between two or more items in the collection.

Let's look at an example.

This is part of a collection of definitions on the topic of closure. This collection includes closure under the four basic operations, as well as for even numbers, odd numbers, whole numbers, integers, fractions, rational numbers, real number, and complex numbers.

Topic

Math Properties

Definition

The closure property for subtraction of rational numbers states that the difference between any two rational numbers is always another rational number.

Description

The closure property for subtraction of rational numbers is a key concept in mathematics that demonstrates the consistency and completeness of the rational number system. This property ensures that when we subtract one rational number from another, the result is always another rational number, keeping us within the same number system.

Topic

Math Properties

Definition

The closure property for multiplication of rational numbers states that the product of any two rational numbers is always another rational number.

Description

The closure property for multiplication of rational numbers is a fundamental concept in mathematics that illustrates the consistency and completeness of the rational number system. This property ensures that when we multiply any two rational numbers, the result is always another rational number, keeping us within the same number system.

Topic

Math Properties

Definition

The closure property for division of rational numbers states that the quotient of any two rational numbers (where the divisor is not zero) is always another rational number.

Description

The closure property for division of rational numbers is a crucial concept in mathematics that demonstrates the robustness of the rational number system. This property ensures that when we divide one rational number by another (non-zero) rational number, the result is always another rational number, keeping us within the same number system.

Use the properties of operations to multiply numbers.

Related Resources

Worksheet Library

Topic

Ratios, Proportions, and Percents

Definition

Equivalent ratios are ratios that express the same relationship between quantities.

Description

Equivalent ratios are fundamental in understanding proportions and scaling in mathematics. They represent the same relationship between quantities, even though the numbers themselves may differ. This concept is crucial in various applications, such as cooking, map reading, and creating models.

For instance, the ratios 2:3 and 4:6 are equivalent because they both simplify to the same ratio when reduced.

Topic

Ratios, Proportions, and Percents

Definition

Visualizing ratios involves using diagrams or models to represent and understand the relationship between two quantities.

Description

Visualizing ratios is essential in various fields, such as mathematics, science, and economics, where understanding the relationship between quantities is crucial. For example, using a bar model or a double number line can help illustrate the ratio of 3:4.

Topic

Ratios, Proportions, and Percents

Definition

Visualizing equivalent ratios involves using diagrams or models to show that two ratios are equivalent.

Description

Visualizing equivalent ratios is important in fields such as mathematics and engineering, where understanding proportional relationships is crucial. For example, using a double number line or a ratio table can help illustrate that the ratios 2:3 and 4:6 are equivalent.

Topic

Ratios, Proportions, and Percents

Definition

A proportion is an equation that states that two ratios are equal.

Description

Understanding proportions is essential in mathematics, as it is used to solve problems involving ratios and fractions. Proportions are commonly seen in real-world applications such as cooking, map measurements, and scale models.

To illustrate, if there are 2 apples for every 3 oranges, the proportion can be expressed as 2:3. Solving proportions involves finding and solving an equivalent ratio.

Topic

Ratios, Proportions, and Percents

Definition

Part-to-whole ratios compare one part of a whole to the entire whole. These ratios are more commonly known as fractions.

Description

Part-to-whole ratios are used to compare a part of a whole to the entire whole, providing insights into the composition of a dataset or population. This type of ratio, more commonly referred to as fractions, is widely used in statistics, finance, and everyday decision-making.

Topic

Ratios, Proportions, and Percents

Definition

Part-to-part ratios compare different parts of a whole to each other.

Description

Part-to-part ratios are used to compare different parts of a whole, providing a way to understand the relationship between different components. This type of ratio is essential in fields such as statistics, biology, and economics.

For example, if a class has 10 boys and 15 girls, the part-to-part ratio of boys to girls is 10:15, which simplifies to 2:3.

Topic

Ratios, Proportions, and Percents

Definition

A scale factor is a number that scales, or multiplies, some quantity.

Description

Scale factors are used in various applications, such as resizing images, models, and maps. For instance, if a model car is built at a scale factor of 1:24, it means the model is 1/24th the size of the actual car. This concept is crucial in fields requiring accurate scaling, such as architecture and engineering.

Topic

Ratios, Proportions, and Percents

Definition

The slope of a line is a ratio that represents the change in y over the change in x.

Description

Understanding the relationship between ratios and slope is essential for interpreting graphs and solving problems in algebra and geometry. The slope is a measure of how steep a line is, calculated as the ratio of the vertical change to the horizontal change between two points.

For example, if a line rises 2 units for every 3 units it runs horizontally, the slope is 2/3. This concept is crucial for understanding linear relationships and analyzing data in various fields.

Topic

Ratios, Proportions, and Percents

Definition

Ratios in simplest form are ratios that have been reduced to their smallest whole number terms.

Description

Reducing ratios to their simplest form is similar to the process of simplifying fractions, making it easier to compare and interpret data. A ratio is in simplest form when the greatest common divisor of the terms is 1.

For example, the ratio 8:12 simplifies to 2:3 by dividing both terms by their greatest common divisor, 4. This skill is essential for solving problems involving proportions and understanding relationships between quantities.

Topic

Ratios, Proportions, and Percents

Definition

Ratios and fractions are both ways of comparing quantities, with fractions representing a part of a whole.

Description

Understanding the connection between ratios and fractions is crucial for solving problems involving proportions and scaling. Ratios can be expressed as fractions, providing a way to understand the relationship between quantities. A fraction is a part-whole ratio.

Topic

Ratios, Proportions, and Percents

Definition

Visual models for percents are diagrams or illustrations that represent percentages to help visualize and understand them.

Description

Visual models for percents are useful tools in various fields, such as education, finance, and statistics, to represent data and make it more comprehensible. For example, pie charts and bar graphs are common visual models that help illustrate percentages and proportions effectively.

Topic

Ratios, Proportions, and Percents

Definition

A unit rate is a comparison of any two separate but related measurements when one of the measurements is reduced to a single unit.

Description

Unit rates are commonly used in everyday life, such as calculating speed (miles per hour), cost per item, or efficiency (miles per gallon). For example, if a car travels 300 miles on 10 gallons of gas, the unit rate is 30 miles per gallon.

Topic

Ratios, Proportions, and Percents

Definition

Converting units involves changing a measurement from one unit to another using a conversion factor.

Description

Converting units is essential in various fields such as science, engineering, and everyday life. It involves using ratios and proportions to switch between different measurement systems, such as converting inches to centimeters or gallons to liters.

For example, to convert 5 miles to kilometers, knowing that 1 mile is approximately 1.60934 kilometers, you multiply 

5 × 1.60934 = 8.0467 kilometers

Topic

Ratios, Proportions, and Percents

Definition

Dimensional analysis is a method used to convert one unit of measurement to another using conversion factors.

Description

Dimensional analysis is a powerful tool in mathematics and science for converting units and solving problems involving measurements. It uses the principle of multiplying by conversion factors to ensure that units cancel out appropriately, leading to the desired unit.

For example, to convert 50 meters per second to kilometers per hour, you use the conversion factors  1 meter = 0.001 kilometers and 1 hour = 3600 seconds: 

Topic

Ratios, Proportions, and Percents

Definition

Ratios with fractions compare two quantities where one or both of the quantities are fractions.

Description

Ratios with fractions are essential in various mathematical and real-world contexts, such as cooking, where ingredients are often measured in fractions. Understanding these ratios allows for accurate scaling of recipes or other measurements. For example, if a recipe calls for 1/2 cup of sugar to 1/4 cup of butter, the ratio is 1/2:1/4, which simplifies to 2:1 by multiplying both terms by 4.

Topic

Ratios, Proportions, and Percents

Definition

Ratios with decimals involve comparing two quantities where one or both of the quantities are represented as decimal numbers.

Description

Ratios with decimals are crucial in various real-world applications, particularly in financial calculations, engineering, and scientific measurements. For instance, when calculating financial ratios such as the price-to-earnings ratio, decimals are often involved. Understanding how to work with these ratios allows for more precise and meaningful comparisons.

Topic

Ratios, Proportions, and Percents

Definition

Ratios with percents involve comparing quantities where one or both of the quantities are expressed as percentages.

Description

Ratios with percents are widely used in various fields, including finance, statistics, and everyday life. For example, when comparing interest rates, growth rates, or discount rates, percentages are often used. Understanding these ratios allows for better financial decision-making and data analysis.

Topic

Ratios, Proportions, and Percents

Definition

Percent increase measures the growth in value expressed as a percentage of the original value.

Description

Percent increase is used to quantify the growth in value over time, expressed as a percentage of the original value. It is commonly used in finance, economics, and everyday scenarios such as salary increases and population growth.

For example, if the price of a stock rises from \$50 to \$75, the percent increase is calculated as 

(75 − 50)/50 × 100 = 50%

Topic

Ratios, Proportions, and Percents

Definition

Percent decrease measures the reduction in value expressed as a percentage of the original value.

Description

Percent decrease is used to quantify the reduction in value over time, expressed as a percentage of the original value. It is commonly used in finance, economics, and everyday scenarios such as price reductions and weight loss.

For example, if the price of a jacket drops from $80 to $60, the percent decrease is calculated as 

(80 − 60)/80 × 100 = 25%.

Topic

Ratios, Proportions, and Percents

Definition

Calculating tax involves determining the percentage amount to be added to the base price of a product or service.

Description

Calculating tax is a fundamental application of percentages in real-world scenarios. When purchasing goods or services, the total cost is often the sum of the base price and the tax applied. Understanding how to calculate tax is essential for budgeting and financial literacy.

For example, if a product costs $50 and the tax rate is 8%, the tax amount is calculated as 

50 × 0.08 = 4

Therefore, the total cost is 

Topic

Ratios, Proportions, and Percents

Definition

Calculating tips involves determining the amount of money to give as a gratuity based on a percentage of the total bill.

Description

Calculating tips is a common use of percentages in everyday life, particularly in service industries such as dining. Tips are usually calculated as a percentage of the total bill, and understanding how to compute this is important for both customers and service providers. 

For instance, if a meal costs $80 and you want to leave a 15% tip, the tip amount is calculated as 

80 × 0.15 = 12

Topic

Ratios, Proportions, and Percents

Definition

Similar figures are figures that have the same shape but may differ in size; their corresponding angles are equal, and their corresponding sides are proportional.

Description

Similar figures are fundamental in geometry and are used in various real-world applications, such as creating scale models and maps. For example, two triangles are similar if their corresponding angles are equal and their sides are in proportion. This concept is essential for understanding geometric relationships and solving problems involving shapes and sizes.

Topic

Ratios, Proportions, and Percents

Definition

Percents as decimals involve converting a percentage into its decimal representation.

Description

Converting percents to decimals is a key skill in mathematics, allowing students to perform calculations involving percentages more easily. To convert, divide the percent by 100.

For example, 75% as a decimal is 0.75, calculated by dividing 75 by 100. This conversion is useful in many contexts, such as finance, where calculations are conducted using decimal values. Mastering this concept enables students to approach real-world problems with greater confidence and accuracy.