Illustrative Math Alignment: Grade 7 Unit 7

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "300 is what percent of 75.5?" The solution involves setting up the equation 75.5 * (x / 100) = 300, then solving for x to get x = 300 * (100 / 75.5), which equals 397.35%. This example introduces a scenario where the resulting percentage is close to 400%, with the first number being significantly larger than the decimal base number.

Topic

Solving Equations

Description

This math example focuses on solving percent equations by asking "400 is what percent of 220?" The solution involves setting up the equation 220 * (x / 100) = 400, then solving for x to get x = 400 * (100 / 220), which equals 181.81%. This example demonstrates how to calculate a percentage when the first number is nearly double the second, resulting in a percentage between 150% and 200%.

Topic

Solving Equations

Description

This math example focuses on solving percent equations, specifically asking "What is 8% of 58?" The solution involves converting 8% to its decimal form, 0.08, and then multiplying it by 58 to arrive at the answer of 4.64. This example introduces a larger whole number as the base value, demonstrating the scalability of the percent-to-decimal conversion method.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "333.5 is what percent of 500.25?" The solution involves setting up the equation 500.25 * (x / 100) = 333.5, then solving for x to get x = 333.5 * (100 / 500.25), which is approximately 66.67%. This example introduces a scenario where both numbers are decimals and the resulting percentage is less than 100%, showing how to handle more complex decimal calculations.

Topic

Solving Equations

Description

This math example focuses on solving percent equations by asking "4 is 0.1% of what number?" The solution involves setting up the equation 4 = 0.001 * x, then solving for x to get x = 4 / 0.001, which equals 4000. This example demonstrates how to calculate the whole when given a very small percentage of it, resulting in a much larger number.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "7 is 1% of what number?" The solution involves setting up the equation 7 = 0.01 * x, then solving for x to get x = 7 / 0.01, which equals 700. This example introduces a scenario where we need to find the whole when given a small percentage of it, resulting in a number 100 times larger than the given value.

Topic

Solving Equations

Description

This math example focuses on solving percent equations by asking "9 is 30% of what number?" The solution involves setting up the equation 9 = 0.3 * x, then solving for x to get x = 9 / 0.3, which equals 30. This example demonstrates how to calculate the whole when given a larger percentage of it, resulting in a number that is only slightly larger than the given value.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "1 is 400% of what number?" The solution involves setting up the equation 1 = 4.0 * x, then solving for x to get x = 1 / 4, which equals 0.25. This example introduces a scenario where we need to find a number that, when increased by 400%, results in 1, leading to a fraction or decimal less than 1.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "15 is 0.25% of what number?" The solution involves setting up the equation 15 = 0.0025 * x, then solving for x to get x = 15 / 0.0025, which equals 6000. This example introduces a scenario where we need to find the whole when given a very small percentage of it, resulting in a number that is significantly larger than the given value.

Topic

Solving Equations

Description

This math example focuses on solving percent equations by asking "30 is 5% of what number?" The solution involves setting up the equation 30 = 0.05 * x, then solving for x to get x = 30 / 0.05, which equals 600. This example demonstrates how to calculate the whole when given a small percentage of it, resulting in a much larger number.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "50 is 80% of what number?" The solution involves setting up the equation 50 = 0.8 * x, then solving for x to get x = 50 / 0.8, which equals 62.5. This example introduces a scenario where we need to find the whole when given a large percentage of it, resulting in a number that is only slightly larger than the given value.

Topic

Solving Equations

Description

This math example focuses on solving percent equations by asking "78 is 150% of what number?" The solution involves setting up the equation 78 = 1.5 * x, then solving for x to get x = 78 / 1.5, which equals 52. This example demonstrates how to calculate the original value when given a percentage greater than 100%, resulting in a number that is smaller than the given value.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "150 is 0.2% of what number?" The solution involves setting up the equation 150 = 0.002 * x, then solving for x to get x = 150 / 0.002, which equals 75,000. This example introduces a scenario where we need to find the whole when given a very small percentage of it, resulting in a number that is significantly larger than the given value.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "What is 6.5% of 45.5?" The solution involves converting 6.5% to its decimal equivalent, 0.065, and then multiplying it by 45.5 to obtain the result of 2.9575. This example introduces both a decimal percentage and a decimal base number, adding complexity to the calculation.

Topic

Solving Equations

Description

This math example focuses on solving percent equations by asking "225 is 3% of what number?" The solution involves setting up the equation 225 = 0.03 * x, then solving for x to get x = 225 / 0.03, which equals 7,500. This example demonstrates how to calculate the whole when given a small percentage of it, resulting in a number that is significantly larger than the given value.

Topic

Solving Equations

Description

This math example focuses on solving percent equations by asking "400 is 40% of what number?" The solution involves setting up the equation 400 = 0.4 * x, then solving for x to get x = 400 / 0.4, which equals 1000. This example demonstrates how to calculate the whole when given a significant percentage of it, resulting in a number that is larger than the given value.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "650 is 130% of what number?" The solution involves setting up the equation 650 = 1.3 * x, then solving for x to get x = 650 / 1.3, which equals 500. This example introduces a scenario where we need to find the original value when given a percentage greater than 100%, resulting in a number that is smaller than the given value.

Topic

Solving Equations

Description

This math example focuses on solving percent equations, specifically asking "What is 30% of 9?" The solution involves converting 30% to its decimal form, 0.3, and then multiplying it by 9 to get the result of 2.7. This example introduces a larger percentage, demonstrating how the method applies consistently across various percentage values.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "What is 28% of 7.2?" The solution involves converting 28% to its decimal equivalent, 0.28, and then multiplying it by 7.2 to obtain the result of 2.016. This example combines a whole number percentage with a decimal base number, further illustrating the versatility of the percent-to-decimal conversion method.

Topic

Solving Equations

Description

This math example focuses on solving percent equations, specifically asking "What is 45% of 68?" The solution involves converting 45% to its decimal form, 0.45, and then multiplying it by 68 to arrive at the answer of 30.6. This example introduces a larger percentage and a larger whole number as the base value, demonstrating the scalability of the percent-to-decimal conversion method.

Topic

Solving Equations

Description

This math example demonstrates solving percent equations by asking "What is 52.3% of 36.9?" The solution involves converting 52.3% to its decimal equivalent, 0.523, and then multiplying it by 36.9 to obtain the result of 19.2987. This example introduces both a decimal percentage and a decimal base number, adding complexity to the calculation and showcasing the versatility of the percent-to-decimal conversion method.

Topic

Solving Equations

Description

This math example focuses on solving percent equations, specifically asking "What is 150% of 8?" The solution involves converting 150% to its decimal form, 1.5, and then multiplying it by 8 to get the result of 12. This example introduces a percentage greater than 100%, demonstrating how the method applies consistently even when dealing with percentages that represent values larger than the whole.

Topic

Quadratics

Description

In this example, we're given a parabola with its focus at (0, 0) and directrix y = 5. Using the locus definition, we can determine that the vertex is at (0, 2.5), halfway between the focus and directrix. The distance p from the vertex to the focus is 2.5. This example demonstrates how the focus and directrix directly determine the equation of the parabola.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Topic

Quadratics

Topic

Quadratics

Description

This example presents a parabola with its focus at (0, 0) and directrix x = -4. The vertex is at (0, -2.5), midway between the focus and directrix. The distance p from the vertex to the focus is 2.5. This example illustrates how a vertically-oriented parabola's equation is derived from its focus and directrix.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

In this example, the parabola has its focus at (0, 1) and directrix y = -2. The vertex is at (0, -0.5), and the parabola opens upward. The distance p from the vertex to the focus is 1.5. This example reinforces the relationship between the focus, directrix, and the resulting equation of the parabola.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

This example shows a parabola with its focus at (0, 1) and directrix y = 2. The parabola opens downward with its vertex at (0, 1.5). The distance p from the vertex to the focus is 0.5. This example illustrates how the position of the focus and directrix determines the orientation and equation of the parabola.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

In this example, the parabola has its focus at (1, 2) and directrix y = 0. The vertex is at (1, 1), and the parabola opens upward. The distance p from the vertex to the focus is 1. 

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Topic

Quadratics

Description

This example presents a parabola with its focus at (2, 3) and directrix y = 3. This parabola is not defined because the focus and directrix cannot have the same y-value.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Topic

Quadratics

Topic

Quadratics

Topic

Quadratics

Topic

Quadratics

Topic

Quadratics

Description

This example illustrates a parabola intersecting the x-axis at two distinct points, indicating two real and distinct solutions. The graph provides a visual representation of the roots and helps in understanding the impact of the quadratic equation's coefficients on the shape and position of the parabola. Graphical solutions are advantageous for quickly identifying the number of solutions, but they may not offer precise numerical solutions without additional calculations.

Topic

Quadratics

Description

This example shows a parabola that is tangent to the x-axis, indicating a single real solution. This scenario occurs when the quadratic equation has a perfect square trinomial, resulting in a discriminant of zero. The graphical approach provides an intuitive understanding of why there is only one solution. However, it may not be sufficient for finding the exact solution without algebraic methods, especially if the vertex is not at a simple coordinate.

Topic

Quadratics

Description

This example shows a parabola that is tangent to the x-axis, indicating a single real solution. This scenario occurs when the quadratic equation has a perfect square trinomial, resulting in a discriminant of zero. The graphical approach provides an intuitive understanding of why there is only one solution. However, it may not be sufficient for finding the exact solution without algebraic methods, especially if the vertex is not at a simple coordinate.

Topic

Quadratics

Description

This example illustrates a quadratic graph with no x-intercepts, indicating no real solutions. The graph provides a straightforward way to see where the solutions lie on the x-axis. This method is particularly advantageous for visual learners and for gaining a conceptual understanding of the roots. However, it may not always provide precise solutions, especially if the intercepts are not integers or easily readable from the graph.

Topic

Quadratics

Description

This example shows a parabola not intersecting the x-axis, indicating no real solutions. The graph helps in visualizing the solutions and understanding how the quadratic equation's coefficients affect the graph's orientation and position. Graphical solutions are beneficial for quickly identifying the nature of the solutions, but they may not provide exact numerical answers without further analytical work.

Topic

Quadratics

Description

This example demonstrates solving a simplified form of a quadratic equation. The equation is in the form ax² = c, where the linear term (bx) is absent. This simplification allows for a straightforward solution by isolating x² and then taking the square root of both sides. The process is more direct than using the quadratic formula, making it easier to solve mentally or with minimal calculations. This method highlights the importance of recognizing special forms of quadratic equations for efficient problem-solving.

Topic

Quadratics

Description

This example showcases another simplified quadratic equation. The equation is in a form where factoring or using the square root method is straightforward. By recognizing this simpler structure, students can solve the equation without resorting to more complex techniques. This approach emphasizes the importance of identifying the most efficient method based on the equation's form, promoting problem-solving skills and mathematical intuition.

Topic

Quadratics

Description

In this example, the quadratic equation is presented in a simplified form that allows for direct solving. The structure of the equation makes it possible to solve without using the quadratic formula or complex factoring techniques. This straightforward approach highlights how recognizing certain forms of quadratic equations can lead to quicker and more intuitive solutions, enhancing problem-solving efficiency.

Topic

Quadratics

Description

This example illustrates solving a quadratic equation in a simplified form. The equation's structure allows for a direct solution method, avoiding the need for more complex techniques like the quadratic formula. By recognizing this simpler form, students can apply basic algebraic skills to solve the equation efficiently, reinforcing the importance of identifying the most appropriate solving method based on the equation's characteristics.

Topic

Quadratics

Description

This example demonstrates solving a quadratic equation in a simplified form that allows for straightforward solution methods. The equation's structure enables direct solving techniques, bypassing the need for more complex approaches. This simplified form emphasizes the importance of recognizing patterns in quadratic equations, allowing for more efficient problem-solving and deepening understanding of quadratic relationships.

Topic

Quadratics

Description

In this example, a simplified form of a quadratic equation is presented, allowing for a more direct solving approach. The equation's structure makes it possible to apply basic algebraic techniques without resorting to complex methods. This straightforward solution process highlights the importance of recognizing special forms of quadratic equations, enabling students to choose the most efficient solving strategy.

Topic

Quadratics

Description

This example showcases solving a quadratic equation in a simplified form that allows for direct solution methods. The equation's structure enables the use of basic algebraic techniques, avoiding the need for more complex approaches like the quadratic formula. This simplified approach emphasizes the importance of identifying the most efficient solving method based on the equation's form, enhancing problem-solving skills and mathematical intuition.

Topic

Quadratics

Description

In this final example, a simplified quadratic equation is presented, demonstrating how certain forms allow for straightforward solving techniques. The equation's structure enables direct solution methods, bypassing the need for complex approaches. This example reinforces the importance of recognizing special forms of quadratic equations, allowing students to apply the most efficient solving strategy and deepen their understanding of quadratic relationships.

Topic

Equations