Illustrative Math Alignment: Grade 7 Unit 5

Topic

Ratios and Rates

Description

This example focuses on unit conversion using the height of the Statue of Liberty. The image shows the Statue of Liberty, and students are asked to convert its height from feet to inches.

Understanding unit conversions is crucial in many fields, including science, engineering, and everyday life. This example demonstrates how to use a conversion factor to change units, illustrating the practical application of multiplication in real-world measurement scenarios.

Topic

Ratios and Rates

Description

This example focuses on unit conversion, specifically converting the height of the Statue of Liberty from feet to yards. The image shows the iconic Statue of Liberty, providing a real-world context for the mathematical problem.

Understanding unit conversions is crucial in many fields, including engineering, science, and everyday life. This example demonstrates how to use a conversion factor to change units, illustrating the practical application of division and multiplication in real-world measurement scenarios.

Topic

Ratios and Rates

Description

This example explores unit conversion, specifically converting the height of the Statue of Liberty from feet to miles. The image depicts the Statue of Liberty, providing a tangible reference for the mathematical problem.

Understanding unit conversions, especially between widely different scales like feet and miles, is important in various fields such as geography, engineering, and urban planning. This example showcases how to use a conversion factor to change units, demonstrating the practical application of division and multiplication in real-world measurement scenarios.

Topic

Ratios and Rates

Description

This example focuses on unit conversion, specifically converting the height of columns from inches to feet. The image shows columns, providing a visual context for the mathematical problem.

Understanding unit conversions between inches and feet is crucial in many practical applications, including construction, interior design, and everyday measurements. This example demonstrates how to use a conversion factor to change units, illustrating the practical application of division and multiplication in real-world measurement scenarios.

Topic

Ratios and Rates

Description

This example explores unit conversion in a sports context, specifically converting a quarterback's pass distance from yards to feet. The image shows a silhouette of a quarterback throwing a football, providing a real-world scenario for the mathematical problem.

Understanding unit conversions between yards and feet is crucial in many sports, especially American football. This example demonstrates how to use a conversion factor to change units, illustrating the practical application of multiplication in sports-related measurement scenarios.

Topic

Ratios and Rates

Description

This example focuses on unit conversion in a geographic context, specifically converting the distance between San Antonio and Austin from miles to feet. The image shows a map of the route between these two Texas cities, providing a real-world scenario for the mathematical problem.

Understanding unit conversions between miles and feet is important in various fields, including geography, transportation, and urban planning. This example demonstrates how to use a conversion factor to change units, illustrating the practical application of multiplication in distance-related measurement scenarios.

Topic

Ratios and Rates

Description

This example focuses on time unit conversion, specifically converting hours to minutes in the context of a standardized test duration. The image shows a clock, providing a visual representation of time for the mathematical problem.

Understanding time unit conversions is crucial in many aspects of daily life, including scheduling, time management, and test-taking. This example demonstrates how to use a conversion factor to change units, illustrating the practical application of multiplication in time-related scenarios.

Topic

Ratios and Rates

Description

This example explores time unit conversion, specifically converting hours and minutes to seconds in the context of baking a pie. The image shows a steaming pie, providing a real-world scenario for the mathematical problem.

Understanding time unit conversions, especially when dealing with mixed units, is important in various fields such as cooking, manufacturing, and project management. This example demonstrates how to use conversion factors to change units, illustrating the practical application of multiplication in time-related scenarios.

Topic

Ratios and Rates

Description

This example focuses on time unit conversion in a sports context, specifically converting minutes and seconds to seconds for a horse race lap time. The image shows a horse, providing a visual context for the mathematical problem.

Understanding time unit conversions is crucial in many sports, especially those involving racing and timed events. This example demonstrates how to use conversion factors to change units and add different time units, illustrating the practical application of multiplication and addition in sports-related time scenarios.

Topic

Ratios and Rates

Description

This example explores time unit conversion in an IT context, specifically converting seconds to minutes for user session duration. The image shows a computer tower, providing a real-world scenario for the mathematical problem.

Understanding time unit conversions is crucial in many technological fields, especially in IT and user experience analysis. This example demonstrates how to use a conversion factor to change units, illustrating the practical application of division in time-related measurement scenarios. It also introduces the concept of mixed numbers in the result.

Topic

Ratios and Rates

Description

This example focuses on ratios using black and green socks. The image displays a collection of socks, and students are tasked with determining the ratio of black socks to green socks. The solution shows that there are 2 pairs of black socks and 6 pairs of green socks, resulting in a ratio of 2 : 6, which simplifies to 1 : 3.

Topic

Ratios and Rates

Description

This example focuses on time unit conversion in an IT context, specifically converting seconds to hours for user session duration. The image shows a computer tower, providing a real-world scenario for the mathematical problem.

Topic

Ratios and Rates

Description

This example focuses on temperature unit conversion, specifically converting Celsius to Fahrenheit for the boiling point of water. The image shows a beaker of boiling water, providing a visual context for the mathematical problem.

Understanding temperature unit conversions is crucial in many scientific fields, including chemistry, physics, and meteorology. This example demonstrates how to use a conversion formula to change units, illustrating the practical application of mathematical equations in real-world temperature scenarios.

Topic

Ratios and Rates

Description

This example explores temperature unit conversion, specifically converting Fahrenheit to Celsius for the freezing point of water. The image shows icicles, providing a visual representation of the freezing temperature for the mathematical problem.

Topic

Ratios and Rates

Description

This example explores ratios using purple and red socks. The image shows a collection of socks, and students are asked to determine the ratio of purple socks to red socks. The solution reveals that there is 1 pair of purple socks and 4 pairs of red socks, resulting in a ratio of 1 : 4, which is already in its simplest form.

Topic

Ratios and Rates

Description

This example focuses on ratios using black and green socks among a variety of colored socks. The image displays pairs of socks in black, green, orange, yellow, blue, and red. Students are asked to determine the ratio of black socks to green socks. The solution shows that there are 2 pairs of black socks and 10 pairs of green socks, resulting in a ratio of 2 : 10, which simplifies to 1 : 5.

Topic

Ratios and Rates

Description

This example explores ratios using geometric shapes of different colors. The image displays various shapes in yellow, blue, red, and green. Students are asked to determine the ratio of circular shapes to yellow objects. The solution reveals that there are 2 circular shapes and 1 yellow object, resulting in a ratio of 2 : 1, which is already in its simplest form.

Topic

Ratios and Rates

Description

This example focuses on ratios using geometric shapes. The image displays various shapes in yellow, blue, red, and green. Students are asked to determine the ratio of triangles to quadrilaterals. The solution shows that there are 4 triangles and 6 quadrilaterals, resulting in a ratio of 4 : 6, which simplifies to 2 : 3.

Topic

Ratios and Rates

Description

This example explores ratios using geometric shapes and colors. The image displays various shapes in yellow, blue, red, and green. Students are asked to determine the ratio of octagons to blue shapes. The solution reveals that there are 2 octagons and 5 blue shapes, resulting in a ratio of 2 : 5, which is already in its simplest form.

Topic

Ratios and Rates

Description

This example introduces the concept of ratios in genetics using a Punnett square. The image shows genetic combinations for a flower's color, including RR, Rw, and ww, with corresponding images of red and white flowers. Students are asked to determine the ratio of different gene combinations.

Topic

Equations

Topic

Equations

Topic

Equations

Topic

Equations

Topic

Equations

Topic

Equations

This Teacher's Guide provides an overview of the 28 worked-out examples that show how to graph Rational Functions.

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This Teacher's Guide provides an overview of the 28 worked-out examples that show how to simplify a variety of rational expressions.

Related Resources

To see resources related to this topic click on the Related Resources tab above.

March 2014. In this issue of Math in the News we look at the Iditarod Race in Alaska. This gives us an opportunity to analyze data on average speed. We look at data in tables and line graphs and analyze the winning speeds over the history of the race.

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To see resources related to this topic click on the Related Resources tab above.

All living things take up a certain amount of space, and therefore have volume. They also have a certain amount of surface area. The ratio of surface area to volume, which is a rational function, reveals important information about the organism. Students look at different graphs of these functions for different organisms. Note: The download for this resources is the Promethean Flipchart.

To view the full video [Algebra Applications: Rational Functions, Segment 2: Biology]: https://www.media4math.com/library/algebra-applications-rational-functions-segment-2-biology

The Hubble Telescope, like all telescopes, relies on the lens formula to focus an image. The lens formula results in a rational equation that can be solved for determining the settings on the lens. Note: The download for this resources is the Promethean Flipchart.

To access the full video [Algebra Applications: Rational Functions, Segment 3: Hubble Telescope]: https://www.media4math.com/library/algebra-applications-rational-functions-segment-3-hubble-telescope

In spite of their massive size, submarines are precision instruments. A submarine must withstand large amounts of water pressure; otherwise, a serious breach can occur. Rational functions are used to study the relationship between water pressure and volume. Students graph rational functions to study the forces at work with a submarine. Note: The download for this resources is the Promethean Flipchart.

To access the full video [Algebra Applications: Rational Functions, Segment 1: Submarines]: https://www.media4math.com/library/algebra-applications-rational-functions-segment-1-submarines

After briefly reviewing the concept of inverse variation, this video explores Boyle’s law, a real world example of an inversely proportional relationship between pressure and volume of a gas. Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, it goes on to examine similarities and differences among rational functions and numbers. Finally, it takes a look at rational functions graphs and ends with a delightful example merging Euclidean and analytic geometry, thanks to the TI-Nspire technology. Concepts explored: functions, rational expressions, rational functions, asymptotes Note: The download for this resources is the Promethean Flipchart.

To access the full video [Algebra Nspirations: Rational Functions and Expressions]: https://media4math.com/library/algebra-nspirations-rational-functions-and-expressions

Topic

Rationals and Radicals

Description

Addition of rational expressions involves combining rational expressions to create a new rational expression, often requiring a common denominator. Examples include 1 / x + 1 / 2x = 3 / 2x and 1 / (x + 1) + 1 / (x - 1) = 2x / (x2 - 1). This term introduces the concept of combining rational expressions, which is fundamental to operations involving fractions with polynomial expressions.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

Graphs of rational functions often feature asymptotes, which are lines the graph approaches but never touches. For example, f(x) = 1 / x has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain and range exclude zero. Explains the unique characteristics of rational function graphs and the significance of asymptotes in defining their behavior.

Topic

Rationals and Radicals

Description

Graphs of rational functions often feature asymptotes, which are lines the graph approaches but never touches. For example, f(x) = 1 / x has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. The domain and range exclude zero. Explains the unique characteristics of rational function graphs and the significance of asymptotes in defining their behavior.

Topic

Rationals and Radicals

Description

Horizontal asymptotes describe the behavior of a graph as x approaches infinity or negative infinity. For example, the equation y = (x + 1) / (x - 2) has a horizontal asymptote at y = 1, demonstrating long-term behavior of the function. Focuses on the role of horizontal asymptotes in describing the long-term trends of rational functions.

Topic

Rationals and Radicals

Description

Horizontal asymptotes describe the behavior of a graph as x approaches infinity or negative infinity. For example, the equation y = (x + 1) / (x - 2) has a horizontal asymptote at y = 1, demonstrating long-term behavior of the function. Focuses on the role of horizontal asymptotes in describing the long-term trends of rational functions.

Topic

Rationals and Radicals

Description

Integers are a subset of rational numbers including positive, negative, and zero values, represented by {..., -3, -2, -1, 0, 1, 2, 3, ...}. This emphasizes their role in forming the basis of discrete mathematics. Introduces integers as a key component of number systems, aiding comprehension of rational numbers.

Topic

Rationals and Radicals

Description

Integers are a subset of rational numbers including positive, negative, and zero values, represented by {..., -3, -2, -1, 0, 1, 2, 3, ...}. This emphasizes their role in forming the basis of discrete mathematics. Introduces integers as a key component of number systems, aiding comprehension of rational numbers.

This video includes Spanish audio and can be used with multilingual learners.

Topic

Rationals and Radicals

Description

Inverse variation describes a non-linear relationship where the product of two variables is a constant k. For instance, xy = k is graphically represented by a hyperbola, illustrating how one variable inversely depends on the other. Describes inverse variation as a key relationship in algebra, foundational for understanding non-linear equations.

Topic

Rationals and Radicals

Description

Inverse variation describes a non-linear relationship where the product of two variables is a constant k. For instance, xy = k is graphically represented by a hyperbola, illustrating how one variable inversely depends on the other. Describes inverse variation as a key relationship in algebra, foundational for understanding non-linear equations.

Topic

Rationals and Radicals

Description

Asymptotes for a rational function are the values a function approaches but does not reach. An example is the function f(x) = 1 / x, where the graph has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. This concept explains the behavior of rational functions at extreme values, helping understand their graphs and limits.

Topic

Rationals and Radicals

Description

Asymptotes for a rational function are the values a function approaches but does not reach. An example is the function f(x) = 1 / x, where the graph has vertical and horizontal asymptotes at x = 0 and y = 0, respectively. This concept explains the behavior of rational functions at extreme values, helping understand their graphs and limits.

Topic

Rationals and Radicals

Description

Multiplication of rational expressions involves multiplying the numerators and denominators of each expression. For example, (x / (x + 1)) * ((x - 1) / 3) simplifies to (x * (x - 1)) / (3 * (x + 1)), illustrating its application in simplifying complex expressions. Covers the multiplication of rational expressions, essential for algebraic simplifications and solving equations.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

An oblique asymptote occurs when, for large values of x, the y-values of the graph approach but do not touch the graph of a linear function. An example is given where y = (x3 - 2x2 + 1) / (x2 - 2x - 3) approaches y = x as x approaches infinity. This term provides insight into the behavior of rational functions, essential for graphing and understanding limits in radical and rational contexts.

Topic

Rationals and Radicals

Topic

Rationals and Radicals

Description

Rational Equations are equations involving fractions where both sides include variables. These often involve steps to solve for variables, like x in (2x + 6) / (x - 3) = 5. Extends the use of fractions by incorporating variables, creating a bridge between fractions and algebra.