Lernergebnisse:

1. Students are expected to learn what the counting principle is and its two main forms: the Fundamental Counting Principle and the Conditional Counting Principle.

2. Students will apply what they learn by using tree diagrams or similar visual tools to illustrate and solve counting problems.

3. Students will utilize scientific calculators and computers to perform counting operations, including calculating permutations and combinations.

4. Learners are encouraged to see the relationship between counting principles and other topics such as permutations, combinations, and factorial calculations.

Lernergebnisse:

1. Students learn that a factorial (n!) is the product of an integer and all positive integers less than it.

2. They develop the skill to compute factorial values and understand its notation and significance in counting operations.

3. Learners apply the concept of factorials to calculate the number of different arrangements (permutations) of a set of objects.

4. They become familiar with the permutation formula, which is generally expressed as: nPr = n! / (n – r)!

5. Exercises show how to determine the number of ways to arrange or select items—illustrated by problems such as forming three-digit numbers or seating arrangements.

6. The unit guides students through solving specific problems by using factorial calculations to determine permutations.

7. They practice with worked examples and 'try to solve' exercises that reinforce the connection between factorial computation and permutation principles.

8. These exercises build the students’ abilities to tackle real-world counting and arrangement problems using a scientific calculator and other educational tools.

Lernergebnisse:

1. Understand the definition and concept of combinations and recognize that selecting items when order is unimportant is a distinct counting process.

2. Learn how to calculate the number of combinations, typically using the combinations formula (nCr), which is closely related to permutations and factorial notation.

3. Explore practical applications of combinations, such as solving problems that involve selecting committees or groups from a larger set.

4. Gain insights into related concepts such as Pascal’s triangle, which provides a visual and numerical method for understanding combinations.

Lernergebnisse:

1. Students are expected to learn and clearly identify key concepts such as the function of variation, average rate of change, and the rate of change.

2. Students will deduce the first derivative of a function and identify its geometric interpretation as the slope of a tangent line.

3. Learners should investigate differentiability, check right-hand and left-hand derivatives, and understand the relation between differentiation and continuity.

4. Students will determine and apply various differentiation rules including the derivative of a constant, power functions, linear functions, and the sum, product, and quotient of functions.

Lernergebnisse:

1. Mastering the Fundamental Concept of Differentiation: Understanding the process of finding the derivative of a function and its representation as the instantaneous rate of change or slope of the tangent line.

2. Applying Differentiation Rules and Techniques: Ability to apply various rules of differentiation to compute derivatives for diverse types of functions and determine slopes of tangents to curves.

3. Analyzing Differentiability and Continuity: Learning the relationship between differentiability and continuity, including understanding conditions that cause a function to be non-differentiable.

4. Exploring One-Sided Derivatives: Investigating right-hand and left-hand derivatives and their equality to reinforce the definition of differentiability.

5. Visualizing the Calculus Unit Structure: Understanding how Differentiation fits into the broader study of calculus, with reference to concept maps and diagrams.

Lernergebnisse:

1. Students are expected to learn how to apply key derivative rules—such as the product rule, quotient rule, and chain rule—to a variety of functions.

2. Learners should understand that the derivative represents the slope (or instantaneous rate of change) of a function at any given point.

3. Students will work on exercises that involve computing derivatives of algebraic and trigonometric functions using the established rules.

4. The lesson emphasizes the use of tools such as a scientific calculator and graphical programs for verifying computed derivatives.

5. Students are expected to explain why the rules (such as the chain rule) work, linking these procedures back to the definition of the derivative.

Lernergebnisse:

1. Students are expected to be able to explain how to determine the derivative of trigonometric functions using the limit definition.

2. Demonstrate that d/dx (sin x) = cos x and understand how other derivatives (such as for cosine and tangent) can be deduced using trigonometric identities.

3. Apply differentiation rules to composite trigonometric functions using the chain rule.

4. Correctly compute derivatives like d/dx (sin (g(x))) = cos (g(x)) · g′(x).

5. Appreciate the geometric interpretation of derivatives as the slope of the tangent line to the trigonometric curve at a given point.

6. Reinforce familiarity with core trigonometric identities by deriving the formulas for the derivatives of trigonometric functions.

7. Ensure students can explain and apply the underlying principles of differentiation in both theoretical and practical contexts.

Lernergebnisse:

1. Demonstrate a deep understanding of the concept of the derivative, including deriving the definition using limits.

2. Compute derivatives of algebraic and trigonometric functions by applying rules such as the chain rule and product rule.

3. Recognize that the derivative represents the instantaneous rate of change as well as the slope of the tangent line to a curve.

4. Apply derivative rules to find the slopes of tangents at specific points on curves.

5. Interpret the derivative function as a measure of rate of change in various contexts.

6. Practice determining derivatives from first principles as well as by using known differentiation rules.

7. Compare alternative solution methods to verify that the same result is reached.

8. Analyze the relationship between a function and its antiderivative to understand the reverse process of differentiation.

Lernergebnisse:

1. Understanding Antiderivatives: Students learn the concept of an antiderivative—that is, if a function F is an antiderivative of f, then F′(x) = f(x) for every x in the domain. This includes the idea that the set of all antiderivatives of a function is represented by the indefinite integral (∫ f(x) dx = F(x) + C).

2. Integration of Algebraic Functions: The lesson covers how to integrate algebraic functions. Students see examples and are guided through the process of deducing the original function from its derivative using power rules and constant adjustments.

3. Integration of Trigonometric Functions: The integration of certain trigonometric functions is also addressed. Through specific examples, students learn techniques to integrate functions such as sin x and cos x, deepening their understanding of how integration applies to a broader set of functions.

4. Practical Applications of Integration: The exercise examples in the lesson illustrate how integration is not only a theoretical operation but also has practical applications in reconstructing functions from rates of change. This reinforces the link between differentiation and integration.

Lernergebnisse:

1. Clearly define an angle of elevation as the angle formed when looking upward from a horizontal line, and an angle of depression as the angle formed when looking downward.

2. Set up and solve problems involving triangles where one or more angles of elevation or depression are given. For instance, using the measured angles to determine unknown distances, heights, or other dimensions in a triangle.

3. Apply trigonometric relationships and appropriate formulas (as introduced later in the unit) to solve real-life problems such as finding the height of a tower or the distance across a river.

Lernergebnisse:

1. Derive and understand the formulas for sin(A + B), sin(A – B), cos(A + B), cos(A – B), and tan(A ± B) based on fundamental trigonometric definitions.

2. Apply these identities to simplify expressions and solve trigonometric equations.

3. Verify the correctness of these identities using differentiation techniques such as the chain rule.

4. Apply the derived identities in solving practical problems.

5. Use a scientific calculator effectively to support the computational aspects of these identities.

Lernergebnisse:

1. Students are expected to learn how to deduce the double‑angle identities for the sine, cosine, and tangent functions.

2. Learners should be able to simplify and solve trigonometric expressions and equations that involve double–angle expressions.

3. This lesson forms part of a unit that connects several trigonometric topics and extends their understanding to real‑world applications involved in trigonometry.

Lernergebnisse:

1. Learners are expected to deduce and fully understand Heron’s Formula, including its derivation from the triangle’s semiperimeter and side lengths.

2. Students will learn to apply Heron’s Formula to calculate the area of a triangle when the measures of all three sides are given.

3. Learners will use Heron’s Formula to solve real-world problems, such as land measurements, by calculating the area of triangular shapes.

4. The lesson reinforces other aspects of triangle geometry and prepares students to integrate Heron’s Formula with other trigonometric concepts.