Free 12th Grade Homomorphism And Isomorphism Lesson Plan (Math)

Topic: what is homomorphism and isomorphism

Objectives & Outcomes

Define homomorphism and isomorphism and provide an example of each.

Explain the properties that make a homomorphism and isomorphism a homomorphism or isomorphism.

Materials

Pen and paper for taking notes

Homomorphism and isomorphism worksheet (provided at end of lesson)

Warm-up

Ask students if they have heard of Algebra before. If they have, ask them to describe what it is. If they have not, provide a brief overview of Algebra and its role in mathematics.

Explain that today we will be discussing a type of algebraic structure called a homomorphism, or a isomorphism for short.

Direct Instruction

Start by giving a brief overview of what algebraic structures are and how they can be classified using different types of relationships (e.g. isomorphism, homomorphism, etc.).

Next, explain what a homomorphism is. A homomorphism is a structure-preserving map between two algebraic structures. That is, a homomorphism is a function that takes elements from one algebraic structure and maps them to corresponding elements in a second algebraic structure, while preserving the structure of the second structure.

Give some examples of homomorphisms. For example, if we have two sets, we can consider the homomorphism that maps each element in the first set to the corresponding element in the second set. This is a structure-preserving map because it preserves the set-theoretic structure of the two sets, while mapping each element in the first set to the corresponding element in the second set.

Next, explain what an isomorphism is. An isomorphism is a homomorphism that is also a bijection (that is, a one-to-one and onto function). In other words, an isomorphism is a homomorphism that is also a one-to-one and onto function, such that the function is both injective and surjective.

Give some examples of isomorphisms. For example, if we have two geometric figures (such as triangles or quadrilaterals) we can consider the homomorphism that maps each vertex in the first figure to the corresponding vertex in the second figure. This is a one-to-one and onto function since it preserves the structure of the figures, while mapping each vertex in the first figure to the corresponding vertex in the second figure. Furthermore, since the function is both injective and surjective, it is also an isomorphism.

Guided Practice

Give students some examples of geometric figures, and ask them to identify the isomorphism (if there is one) between the two figures.

Alternatively, give students two algebraic structures (such as two sets or two groups), and ask them to identify the homomorphism (if there is one) between the two structures.

As a class, discuss the different examples and the different homomorphisms and isomorphisms identified.

Independent Practice

Give students a set of geometric figures and a set of algebraic structures, and ask them to find the isomorphism (if there is one) between the two sets.

Alternatively, give students two algebraic structures (such as two sets or two groups), and ask them to find the homomorphism (if there is one) between the two structures.

As a class, discuss the different examples and the different homomorphisms and isomorphisms identified.

Closure

Summarize the key points of the lesson, emphasizing the concept of isomorphism and the importance of being able to recognize it.

Ask students to share their findings from the independent practice, and discuss as a class any additional isomorphisms and homomorphisms that were found.

Assessment

Observe students during the guided practice and independent practice activities to assess understanding of the concepts and ability to apply them in a practical way.

Collect and grade the written assignments for understanding and thoroughness.

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