Lernergebnisse:

1. Students should be able to identify and explain the fundamental counting principle.

2. Students are expected to use the counting principle to solve simple counting problems.

3. Students will become familiar with using technology to support counting operations.

Lernergebnisse:

1. Define factorial as the product of all positive integers less than or equal to a given number, with 0! defined as 1.

2. Understand how factorials relate to permutations and how to compute permutations using the formula nPr = n! / (n - r)!.

3. Apply theoretical knowledge of factorials and permutations to solve practical counting problems using calculator operations.

4. Integrate factorial calculations with the fundamental counting principle and other counting methods to solve complex permutation and combination problems.

Lernergebnisse:

1. Students are expected to grasp that a combination is the process of selecting items from a larger set without regard to the order.

2. Learners will be able to apply the fundamental counting principle to solve problems involving combinations.

3. Students will learn to compute combination values using the standard formula and verify their results by using appropriate calculator functions.

4. Learners are expected to extend the concept beyond theoretical examples to practical applications, including solving selection problems and using visual tools to represent outcomes.

Lernergebnisse:

1. Understanding the Variation Function: Students learn that when the independent variable x changes by an amount h, the corresponding change in the function value is given by the variation function: V(h) = f(x + h) – f(x).

2. Computing the Average Rate of Change: The lesson emphasizes calculating the average rate of change over an interval using the formula A(h) = [f(x + h) – f(x)] / h.

3. Deriving the Instantaneous Rate of Change: Students explore how the average rate of change approaches the instantaneous rate of change as h tends to zero, defining the first derivative of the function: f '(x) = limₕ→0 A(h).

Lernergebnisse:

1. Mastery of the Fundamental Definition: Understanding the derivative as the limit of the difference quotient and computing the rate of change of a function.

2. Application of Differentiation Rules: Skillful application of various differentiation rules, including basic functions, sum or difference, product and quotient rules, and the chain rule.

3. Geometrical Interpretation: Ability to interpret the derivative as the slope of the tangent line to a curve and to determine instantaneous rates of change.

4. Conceptual and Practical Applications: Application of differentiation skills to model and solve real-world problems in fields such as physics, economics, and engineering.

Lernergebnisse:

1. Differentiate constant functions and simple power functions (for example, f(x) = xⁿ) using the power rule.

2. Apply the sum and difference rules to find the derivative of functions presented as sums or differences.

3. Use the product rule for differentiating the product of two functions.

4. Employ the quotient rule to differentiate ratios of functions.

5. Utilize the chain rule for composite functions—including expressing derivatives for functions written in forms such as y = (f(x))ⁿ.

6. Explain the geometrical meaning of the derivative as the slope of the tangent to the curve at a point.

7. Use differentiation rules to find the equations of tangents and normals to given curves.

8. Recognize and correctly apply the variety of derivative rules in calculus.

9. Solve problems that require finding the derivative of various types of functions.

10. Interpret the derivative in both algebraic and geometric contexts for practical applications.

Lernergebnisse:

1. Define integration and explain its role as the inverse process of differentiation.

2. Compute integrals by using foundational rules and techniques.

3. Interpret integration outcomes in both a geometrical and practical context.

4. Apply integration to calculate areas under curves and to model cumulative processes.

Lernergebnisse:

1. Recognize the concept of random experiment and distinguish all possible outcomes that may arise before its execution.

2. Identify and describe the sample space (or set of outcomes) of an experiment, recognizing that it is the set of all possible results.

3. Define and differentiate between simple events (composed of a single outcome) and compound events (which contain multiple outcomes), including the cases of certain event and impossible event.

4. Recognize and work with mutually exclusive events, understanding that two events are mutually exclusive when they cannot occur at the same time.

5. Apply basic operations on events such as union, intersection, difference, and complement to describe and analyze combinations of outcomes.

6. Use the axioms of probability to calculate the probability of occurrence of events, including solving applied and real-life problems.