Exploring Proportion Properties

Objective:

Students will be able to identify proportion properties, solve proportions using cross-multiplication, and apply the concept to real-world scenarios.

Assessment:

Create a set of 5 proportions for students to solve independently. Include a real-world scenario where students need to set up and solve a proportion to find a solution.

Key Points:

• Understanding what a proportion is: A proportion is two ratios that are set equal to each other. Usually the ratios in proportions are written in fraction form.

• Identifying proportional relationships

• Cross-multiplication method for solving proportions: To solve a proportion, you need to cross-multiply. This involves multiplying the numerator of one ratio by the denominator of the other ratio and vice versa.

• Cross Multiplication Theorem: The Cross Multiplication Theorem states that in a proportion, the product of the extremes (first and last terms) is equal to the product of the means (middle terms). This can be written as ( a/b = c/d ) is equivalent to ( ad = bc ).

• Corollaries of the Cross Multiplication Theorem:

  1. If ( a/b = c/d ), then ( a/c = b/d )

  2. If ( a/b = c/d ), then ( b/a = d/c )

  3. If ( a/b = c/d ), then ( a/(b+c) = c/(d+a) )

  4. If ( a/b = c/d ), then ( (a+c)/b = (c+d)/d )

  5. If ( a/b = c/d ), then ( (a-c)/b = (c-d)/d )

• Applying proportions to solve real-world problems

Opening:

• Start the lesson by showing students a recipe that serves 4 people.

• Ask students to determine how much of each ingredient would be needed to make the same recipe for 8 people.

Introduction to New Material:

• Define the term "proportion" as two ratios set equal to each other.

• Explain the concept using examples of proportional relationships.

• Common Misconception: Students may confuse ratios with proportions. Reinforce that proportions are specifically when two ratios are equal to each other.

Guided Practice:

• Provide examples of proportions for students to solve with guided assistance.

• Scaffold questioning from simple to complex proportions.

• Monitor student performance by circulating the room and providing guidance as needed.

Independent Practice:

• Assign a worksheet with various proportions for students to solve independently.

• Include real-world scenarios where students must set up and solve proportions.

• Monitor student progress and provide individual support as necessary.

Closing:

• Have students share their answers to the independent practice questions.

• Summarize the key points of the lesson and clarify any remaining questions or misconceptions.

Extension Activity:

• For students who finish early, provide a set of challenging word problems that require multiple steps to solve using proportions.

Homework:

• Homework: Ask students to find and bring in examples of proportions from newspapers, magazines, or online sources.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.CO.A.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs.

• CCSS.MATH.CONTENT.HSG.CO.A.2: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.