Illustrative Math Alignment: Grade 6 Unit 9

Use the properties of operations to multiply numbers.

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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.

Topic

Ratios, Proportions, and Percents

Definition

Percent of a number involves calculating the amount represented by a certain percentage of that number.

Description

Understanding percentages is crucial for working with finances, statistics, and data analysis. For instance, to find 20% of 50, multiply 50 by 0.20, resulting in 10. Likewise, it's important for everyday scenarios, such as calculating discounts during shopping.

Topic

Ratios, Proportions, and Percents

Definition

The percent one number is of another is a way to express one number as a percentage of another number.

Description

Understanding how to express one number as a percentage of another is crucial in various real-world applications, such as calculating discounts, tax, and interest rates. For example, if you want to find out what percentage 25 is of 200, you divide 25 by 200 and multiply by 100, resulting in 12.5%.

Topic

Ratios, Proportions, and Percents

Definition

Percent of an unknown refers to solving for an unknown quantity when given a percentage of that quantity.

Description

Knowing how to find a percentage of an unknown variable is essential for solving equations in algebra. This concept appears in various situations, such as when determining discounts or portions of a total amount.

For instance, if 20% of an unknown number equals 15, you can set up the equation: 

0.20x = 15

Topic

Ratios, Proportions, and Percents

Definition

The Golden Ratio is a special number approximately equal to 1.618, often denoted by the Greek letter φ (phi), which appears in various aspects of art, architecture, and nature.

Topic

Ratios, Proportions, and Percents

Definition

The constant of proportionality is the constant value that relates two proportional quantities.

Description

The constant of proportionality is a fundamental concept in mathematics, particularly in linear relationships and direct variation. For example, in the equation y = kx, k is the constant of proportionality that relates y and x. This concept is crucial in fields like physics, where it is used to describe relationships such as speed (distance/time).

Topic

Circles

Definition

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

Topic

Sequences and Series

Definition

The common ratio is the ratio between consecutive terms in a geometric sequence.

Description

The common ratio is a fundamental concept in geometric sequences, representing the consistent factor by which each term is multiplied to obtain the next term. This concept is crucial in understanding exponential growth and decay, which are prevalent in various mathematical and scientific applications.

In this TI-Nspire Activity, use the Geometry and Graphing Tools to explore the ratio of Circumference to Radius as a linear function. 

Note: The Preview is a Google Slide Show and the download is a PPT. Subscribers to Media4Math can download resources.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 1+1. It's a powerful tool for teaching basic addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual link between "1+1" and its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image showcases the numerical expression and corresponding words for 1+2. It's an essential tool for teaching addition and reinforcing the link between mathematical notation and verbal expression. By presenting "1+2" visually alongside its written form, students can more easily understand how addition is expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 1+3. It serves as a valuable tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual representation of "1+3" alongside its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image presents the numerical expression and corresponding words for 1+4. It's an invaluable resource for teaching addition and strengthening the link between mathematical notation and verbal expression. By showcasing "1+4" visually alongside its written form, students can more easily understand how addition is expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 1+5. It serves as a crucial tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual link between "1+5" and its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image showcases the numerical expression and corresponding words for 6+2. It's an essential tool for teaching addition and reinforcing the link between mathematical notation and verbal expression. By presenting "6+2" visually alongside its written form, students can more easily understand how addition is expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 7+1. It serves as a valuable tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual representation of "7+1" alongside its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 8+3. It serves as a valuable tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual representation of "8+3" alongside its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image presents the numerical expression and corresponding words for 9+5. It's an invaluable resource for teaching addition and strengthening the link between mathematical notation and verbal expression. By showcasing "9+5" visually alongside its written form, students can more easily understand how addition is expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 10+1. It serves as a crucial tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual link between "10+1" and its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image showcases the numerical expression and corresponding words for 11+2. It's an essential tool for teaching addition and reinforcing the link between mathematical notation and verbal expression. By presenting "11+2" visually alongside its written form, students can more easily understand how addition is expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 12+3. It serves as a valuable tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual representation of "12+3" alongside its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image presents the numerical expression and corresponding words for 13+4. It's an invaluable resource for teaching addition and strengthening the link between mathematical notation and verbal expression. By showcasing "13+4" visually alongside its written form, students can more easily understand how addition is expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 14+5. It serves as a crucial tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual link between "14+5" and its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 15+6. It serves as a valuable tool in teaching addition and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual representation of "15+6" alongside its written form, students can more easily grasp the concept of addition and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image presents the numerical expression and corresponding words for 1-1. It's an invaluable resource for teaching subtraction and strengthening the link between mathematical notation and verbal expression. By showcasing "1-1" visually alongside its written form, students can more easily understand how subtraction is expressed both numerically and verbally.

Topic

The Language of Math

Description

This image illustrates the numerical expression and corresponding words for 2-1. It serves as a crucial tool in teaching subtraction and reinforcing the connection between mathematical symbols and their verbal representations. By providing a visual link between "2-1" and its written form, students can more easily grasp the concept of subtraction and how it's expressed both numerically and verbally.

Topic

The Language of Math

Description

This image showcases the numerical expression and corresponding words for 3-1. It's an essential tool for teaching subtraction and reinforcing the link between mathematical notation and verbal expression. By presenting "3-1" visually alongside its written form, students can more easily understand how subtraction is expressed both numerically and verbally.