Illustrative Math Alignment: Grade 7 Unit 5

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

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.

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

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

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

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

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 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

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

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

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

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.

Use this activity to explore slope as a rate of change. In this Desmos activity, the slope of the line is the rate (cost per pound) for purchasing fruit. Students manipulate the slider for m to see the impact on the cost.

Topic

Geometric Models

Topic

Geometric Models

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

Library of Instructional Resources

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

In this 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.

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

Library of Instructional Resources

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

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

Library of Instructional Resources

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

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

Library of Instructional Resources

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

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

Library of Instructional Resources

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

Topic

Functions

Description

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

In these clip art images, show students the difference between ratios and rates. These images are useful when talking about slope as both a ratio and a rate. In particular, this is useful when talking about slope as a rate of change.

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Create a Slide Show

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Create a Slide Show

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Create a Slide Show

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Create a Slide Show

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Related Resources

Create a Slide Show

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Related Resources

Create a Slide Show

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Related Resources

Create a Slide Show

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Related Resources

Create a Slide Show

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Topic

Rational Functions

Description

This example demonstrates the graph of the rational function y = 1 / x. The graph is a hyperbola with vertical and horizontal asymptotes, with the vertical asymptote occurring at x = 0. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = (x - 5) / (x + 4). The graph features a vertical asymptote at x = -4 and displays two branches, one in the first quadrant and one in the third quadrant. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = (x - 1) / (x - 3). The graph features a vertical asymptote at x = 3 and displays two distinct branches, one in the first quadrant and another in the third quadrant. Students are tasked with graphing the function and identifying its vertical asymptote.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x + 1) / ((x + 2)(x + 4)). The graph features two vertical asymptotes at x = -2 and x = -4, and displays three distinct branches, with one in each of the first, second, and third quadrants. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = (x - 3) / ((x + 3)(x + 5)). The graph features vertical asymptotes at x = -3 and x = -5, illustrating how the function behaves near these lines. Students are tasked with graphing the function and determining its vertical asymptotes.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x + 3) / ((x - 1)(x + 4)). The graph features vertical asymptotes at x = -4 and x = 1, displaying how the function behaves near these lines. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = (x + 4) / ((x + 2)(x - 5)). The graph features vertical asymptotes at x = -2 and x = 5, demonstrating the behavior of the function near these asymptotes. Students are tasked with graphing the function and determining its vertical asymptotes.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x - 9) / ((x - 1)(x + 3)). The graph features vertical asymptotes at x = -3 and x = 1, showing the function's behavior around these asymptotes. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = (x - 1) / ((x - 2)(x - 3)). The graph features vertical asymptotes at x = 2 and x = 3, marked with dashed red lines. Students are tasked with graphing the function and determining its vertical asymptotes.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x - 8) / ((x + 3)(x - 3)). The graph features vertical asymptotes at x = -3 and x = 3, highlighted by dashed red lines. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = -(x + 1) / (x + 2). The graph features a single vertical asymptote at x = -2, indicated by a dashed red line. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -1 / x. The graph is a hyperbola with vertical and horizontal asymptotes, with the vertical asymptote at x = 0. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = -(x - 3) / (x + 4). The graph features a vertical asymptote at x = -4, but it is incorrectly labeled as x = -2. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -(x - 5) / (x - 4). The graph features a vertical asymptote at x = 4, which is correctly labeled. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = -(x + 2) / ((x + 4)(x + 1)). The graph features vertical asymptotes at x = -4 and x = -1. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -(x - 4) / ((x + 5)(x + 2)). The graph features vertical asymptotes at x = -5 and x = -2. Students are tasked with graphing the function and determining its vertical asymptotes.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = -(x + 2) / ((x - 3)(x + 7)). The graph features vertical asymptotes at x = -7 and x = 3. Students are asked to graph the function and identify its vertical asymptotes.