Illustrative Math Alignment: Grade 6 Unit 3

Unit Rates and Percentages

Lesson 3: Measuring with Different-Sized Units

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Nodes
Closed Captioned Video: Percents: Applications of Percent -- Grade	Closed Captioned Video: Percents: Applications of Percent -- Grade Topic Percents Description This video applies percents to measure slopes or grades, using ratios for rise over run. Examples include road inclines, ski slopes, and cycling paths. Applications showcase percents in engineering, sports, and transportation, emphasizing their practicality in measuring inclines accurately.	Percents
Closed Captioned Video: Percents: Calculating Commissions and Tips	Closed Captioned Video: Percents: Calculating Commissions and Tips Topic Percents Description The video demonstrates calculating tips and commissions using percentages. Examples include restaurant tips, real estate commissions, and catering costs. Applications focus on service industry calculations and earnings management.	Percents
Closed Captioned Video: Percents: Calculating Tax	Closed Captioned Video: Percents: Calculating Tax Topic Percents Description This video covers tax calculations for purchases, property, and VAT. Examples include grocery bills, iPad purchases, and real estate property taxes. Applications highlight the importance of understanding tax rates for budgeting and financial management.	Percents
Closed Captioned Video: Percents: Multiple Percents	Closed Captioned Video: Percents: Multiple Percents Topic Percents Description This video demonstrates how to handle multiple percentage changes applied sequentially, including discounts and price fluctuations. Examples include double-discounted items, stock price variations, and gasoline price changes. Applications emphasize consolidating multiple percents into single calculations for efficiency in financial and retail scenarios.	Percents
Closed Captioned Video: Percents: Percent Decrease	Closed Captioned Video: Percents: Percent Decrease Topic Percents Description This video covers calculating percent decreases in scenarios such as price reductions, vehicle depreciation, and cube dimension changes. Examples demonstrate the relationship between linear decreases and their exponential effects on surface area and volume. Applications include consumer goods, real estate, and geometry.	Percents
Closed Captioned Video: Percents: Percent Increase	Closed Captioned Video: Percents: Percent Increase Topic Percents Description The video explains how to calculate percent increases, focusing on changes in school enrollments, asset values, and cube dimensions. Examples highlight the non-linear effects of changes on surface area and volume. Applications include finance, education, and geometry, emphasizing understanding growth trends.	Percents
Closed Captioned Video: Percents: Percents and Multiples of Numbers	Closed Captioned Video: Percents: Percents and Multiples of Numbers Topic Percents Description The video explains the distinction between percent growth and multiple growth. Examples include doubling heights, stock price increases, and oil price changes. Applications emphasize clear understanding of percent versus multiplier concepts in finance and everyday math.	Percents
Closed Captioned Video: Percents: Probabilities and Percents	Closed Captioned Video: Percents: Probabilities and Percents Topic Percents Description The video explores percent-based probabilities in scenarios like dice rolls. Examples include calculating probabilities of specific outcomes and estimating occurrences over multiple trials. Applications include games of chance and statistical analysis.	Percents
Closed Captioned Video: Percents: Simple Interest	Closed Captioned Video: Percents: Simple Interest Topic Percents Description The video explains how to calculate simple interest using a formula involving principal, rate, and time. Examples include savings account growth, stock investments, and rare coin valuation. Applications include financial planning and historical item appraisal.	Percents
Closed Captioned Video: Ratios: Application of Ratios: Roofs and Ramps	Closed Captioned Video: Ratios: Application of Ratios: Roofs and Ramps Topic Ratios Description The video covers practical applications of ratios for measuring slopes of roofs and ramps. Examples include comparing roof pitches, calculating base lengths of roofs, and determining ramp heights. Ratios provide clarity for gradual slopes.	Ratios and Rates and Applications of Ratios, Proportions, and Percents
Definition--Ratios, Proportions, and Percents Concepts--Calculating Tax	Calculating Tax 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	Applications of Ratios, Proportions, and Percents
Definition--Ratios, Proportions, and Percents Concepts--Calculating Tips	Calculating Tips 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	Applications of Ratios, Proportions, and Percents
Definition--Ratios, Proportions, and Percents Concepts--Converting Units	Converting Units 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	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Dimensional Analysis	Dimensional Analysis 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:	Applications of Ratios, Proportions, and Percents
Definition--Ratios, Proportions, and Percents Concepts--Equivalent Ratios	Equivalent Ratios 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Part-to-Part Ratios	Part-to-Part Ratios 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Part-to-Whole Ratios	Part-to-Whole Ratios 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Percent	Percent Topic Ratios, Proportions, and Percents Definition A percent is a ratio that compares a number to 100. Description Percentages are a fundamental concept in mathematics, representing a ratio out of 100. They are used in various applications, including finance, statistics, and everyday calculations such as discounts and interest rates. For example, if you score 45 out of 50 on a test, your percentage score is (45/50) × 100 = 90%	Percents
Definition--Ratios, Proportions, and Percents Concepts--Percent Decrease	Percent Decrease 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%.	Percents
Definition--Ratios, Proportions, and Percents Concepts--Percent Increase	Percent Increase 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%	Percents
Definition--Ratios, Proportions, and Percents Concepts--Percent of a Number	Percent of a Number 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.	Percents
Definition--Ratios, Proportions, and Percents Concepts--Percent of an Unknown	Percent of an Unknown 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	Percents
Definition--Ratios, Proportions, and Percents Concepts--Percents as Decimals	Percents as Decimals 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.	Percents
Definition--Ratios, Proportions, and Percents Concepts--Proportion	Proportion 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.	Proportions
Definition--Ratios, Proportions, and Percents Concepts--Proportional	Proportional Topic Ratios, Proportions, and Percents Definition Proportional refers to the relationship between two quantities where their ratio is constant. Description Proportional relationships are fundamental in mathematics and science, describing how one quantity changes in relation to another. This concept is used in various fields, including physics, economics, and engineering. For example, if the speed of a car is proportional to the time it travels, doubling the time will double the distance covered. Understanding proportionality helps students solve complex problems and apply mathematical reasoning in real-world situations.	Proportions
Definition--Ratios, Proportions, and Percents Concepts--Rate	Rate Topic Ratios, Proportions, and Percents Definition A rate is a ratio that compares two quantities with different units. Description Rates are used to compare different quantities, such as speed (miles per hour) or price (cost per item). Understanding rates is essential for interpreting data and making informed decisions in various contexts, such as travel and budgeting. For instance, if a car travels 60 miles in 2 hours, the rate is 30 miles per hour. Learning about rates helps students analyze real-world situations and apply mathematical reasoning to everyday problems.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Ratio	Ratio Topic Ratios, Proportions, and Percents Definition A ratio is a comparison of two quantities by division. Description Ratios are used to express the relationship between two quantities, providing a way to compare different amounts. They are fundamental in various fields, including mathematics, science, and finance. For example, the ratio of 4 to 5 can be written as 4:5 or 4/5. Understanding ratios helps students analyze data, solve problems, and make informed decisions in real-world situations.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Ratios and Fractions	Ratios and Fractions 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Ratios and Slope	Ratios and Slope 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Ratios in Simplest Form	Ratios in Simplest Form 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.	Applications of Ratios, Proportions, and Percents
Definition--Ratios, Proportions, and Percents Concepts--Ratios with Decimals	Ratios with Decimals 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Ratios with Fractions	Ratios with Fractions 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Ratios with Percents	Ratios with Percents 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Scale Drawing	Scale Drawing Topic Ratios, Proportions, and Percents Definition A scale drawing is a representation of an object or structure with dimensions proportional to the actual object or structure. Description Scale drawings are essential in fields like architecture, engineering, and cartography, where accurate representations of large objects or areas are needed. For example, an architect might create a scale drawing of a building where 1 inch on the drawing represents 10 feet in reality. This allows for detailed planning and visualization without needing a full-sized model.	Proportions
Definition--Ratios, Proportions, and Percents Concepts--Scale Factor	Scale Factor 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.	Proportions
Definition--Ratios, Proportions, and Percents Concepts--Similar Figures	Similar Figures 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.	Proportions
Definition--Ratios, Proportions, and Percents Concepts--Solving Proportions	Solving Proportions Topic Ratios, Proportions, and Percents Definition Solving proportions involves finding the value of a variable that makes two ratios equal. Description Solving proportions is a key skill in algebra and is used in various applications, such as scaling recipes, converting units, and solving real-world problems. For example, if you know that 2/3 = x/6 you can solve for x by cross-multiplying to get 2 * 6 = 3 * x leading to x = 4	Proportions
Definition--Ratios, Proportions, and Percents Concepts--The Constant of Proportionality	The Constant of Proportionality 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).	Proportions
Definition--Ratios, Proportions, and Percents Concepts--The Golden Ratio	The Golden Ratio 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.	Applications of Ratios, Proportions, and Percents and Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--The Percent One Number Is of Another	The Percent One Number Is of Another 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%.	Percents
Definition--Ratios, Proportions, and Percents Concepts--Unit Rate	Unit Rate 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Visual Models for Percents	Visual Models for Percents 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.	Percents
Definition--Ratios, Proportions, and Percents Concepts--Visualizing Equivalent Ratios	Visualizing Equivalent Ratios 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.	Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Visualizing Ratios	Visualizing Ratios 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.	Ratios and Rates
INSTRUCTIONAL RESOURCE: Algebra Application: Linear Functions: Circumference vs. Diameter	INSTRUCTIONAL RESOURCE: Algebra Application: Linear Functions: Circumference vs. Diameter In this Algebra Application, students study the direction between diameter and circumference of a circle. Through measurement and data gathering students analyze the line of best fit and explore ways of calculating pi. The math topics covered include: Mathematical modeling, Linear functions, Data gathering and analysis, Ratios, Direct variation. This is a great back-to-school activity for middle school or high school students. This is also a great crossover activity that ties algebra and geometry.	Applications of Linear Functions, Applications of Ratios, Proportions, and Percents and Applications of Circles
INSTRUCTIONAL RESOURCE: Algebra Application: Why Are Receipts So Long?	INSTRUCTIONAL RESOURCE: Algebra Application: Why Are Receipts So Long? In this Algebra Application, students gather data to study the lengths of paper receipts. They also learn to estimate the total number of receipts produced worldwide using industry data. From this they can study the environmental impact of having to produce so many paper receipts. The math topics covered include: Measures of central tendency (specifically the mean), Scientific notation, Rates, Converting different units. The culminating activity is for students to make a persuasive case to retailers about finding alternatives to paper receipts, using the data they gathered and analyzed. This is a great back-to-school activity for middle school or high school students. A relevant real-world application allows them to review math concepts.	Laws of Exponents and Data Analysis
INSTRUCTIONAL RESOURCE: Algebra Application: Why Are Wildfires So Dangerous?	INSTRUCTIONAL RESOURCE: Algebra Application: Why Are Wildfires So Dangerous? In this Algebra Application, students learn about wildfires and the measurement of air quality. The math topics covered include: Scientific notation, Rates, Density, Data Analysis. The specific focus of this investigation is the health hazards from wildfire smoke. This includes a discussion of air density, measurement in microns, and measurement of air quality. Links to various web sites, including the EPA's site, provide relevant background information and data. The culminating activity is a case study of the wildfires in the Lake Tahoe area. Students analyze historical data and make a recommendation on the air quality. This is a great back-to-school activity for middle school or high school students. A relevant real-world application allows them to review math concepts.	Laws of Exponents and Applications of Ratios, Proportions, and Percents
INSTRUCTIONAL RESOURCE: Pumpkin PI	In this Slide Show, do a hands-on geometry activity to measure π. Note: The download is a PPT file.	Ratios and Rates
Interactive Math Game--DragNDrop--The Point-Slope Form	Interactive Math Game--DragNDrop Math--The Point-Slope Form In this drag-and-drop game, match the linear equation in point-slope form with its slope-intercept counterpart. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Point-Slope Form
Interactive Math Game--DragNDrop--The Slope Formula	Engaging Math Activities & Games For Learning Slope In this drag-and-drop game, match the coordinates to the slope of the line that passes through the two points. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. Improve math skills and have fun at the same time with our selection of interactive games and activities that teach the concept of slope. These entertaining games make learning enjoyable and engaging, helping young learners understand this mathematical process better.	Slope