A basic introductory approach to differential calculus is presented here that focuses on building a content-based understanding of differential calculus.

Basic Concepts of the Derivation

The four basic concepts of derivative are derivative as tangent slope, derivative as local rate of change, derivative as amplification factor and derivative as local linear approximation.

Elaborated Teaching Proposal as Moodle Course

Here is a proposal for a series of lessons on the introduction to differential calculus in the state Moodle-RLP, which has been worked out and tested several times in class, and which primarily takes into account the two basic ideas of tangent slope and local rate of change. The course is offered in the Landes-Moodle-RLP. It can be downloaded and uploaded in each school's own Moodle platform.

Link to Moodle Course

Moodle-course: Full speed ahead into differential calculus[Mit Vollgas in die Differentialrechnung]

Note
If you want to view the Moodle course, you can use one of the following login names and associated passwords:

• login name: student1 (continuously numbered until student20)
• password:    student1 (continuously numbered until student20)

Please note that the moodle course is in german.

Idea of the Lesson Series or "Why go Like a Bull at the Gate?"

The idea of the lesson series is that the students find a first access to differential calculus by means of a situation familiar to them from everyday life (here the Porsche task). This serves several purposes:

1. The students can build up an (comprehension) anchor based on a known situation, to which they can come back again and again later in new situations to be solved with the help of differential calculus and solve by analogy to the anchor.
2. The approach enables an introduction to differential calculus without prior treatment of sequences. Limit value processes are intuitively used and grasped directly during the elaboration of the derivation.
3. This is a problem-based introduction. This means, in particular, that it offers the flexibility to respond to the students' ideas and to pick up on the intuitions they use in the problem-solving phase, which are often close to one of the basic (normative) ideas mentioned above. The sequence of topics can thus be tailored to the thinking processes of the course, especially in the "rate of change" block.

Didactic Comments on the Porsche Task as an Anchor

Idea

Everyone understands what the question "How long does it take a Porsche to accelerate from 0 to 100?" means. This way, the students can concentrate on solving the problem right from the start and are not distracted by mathematical terms they do not (yet) understand.

The terms

• absolute change
• average rate of change
• local rate of change

are worked out in terms of content and interpreted in a factual context without using the mathematical terms. These are not conceptualized until the learners have grasped their meaning in terms of content.

The alternative for the basic course: "How fast is the Porsche after 5s?" can also be considered an anchor task. The more narrowly guided question about the instantaneous speed at a certain point in time avoids the unsystematic search for the speed of 100 km/h. The question is also more precise. In addition, the problem of converting units is avoided.

Experience from the Classroom

• Many students initially answered the leading question intuitively by calculating the average speed (mean rate of change) in the given time period [0s; 5s] (which is approximately 100 km/h), which would be far too long an acceleration period for a Porsche.
• Other students try to solve the task with formulas from physics, but apart from the formula for the speed, these do not help either.
• A first hurdle for students is the conversion from km/h to m/s or vice versa (100km/h = 27.8m/s).
• Once students understand the difference between the instantaneous velocity they are looking for and the average velocities they calculate, a key solution idea for high-performing students can also be to approach instantaneous velocity by looking for a distance traveled of about 28m in a time interval of 1s.
• In order to be able to calculate the mean rate of change for smaller time intervals, the given values of the first worksheet are not sufficient. Therefore, the students are provided with both a suitable function equation and the graph of this function, so that a function value can be determined for any value. It is up to the teacher whether to hand out these tools directly to all learners or only after prompting, or asking, by the learning group.

From Anchor to Transfer

In the course of instruction presented here, the development is designed as a group puzzle (group work with expert groups and core groups). It is also conceivable to plan this in the form of station work or similar concepts. In the form of a given table, the students in the core group phase are asked to combine the different interpretations of the terms in the factual contexts. A deepening takes place by the development of own tasks, which can be appreciated then also in the further instruction process as exercise tasks for other groups.

Possible Factual Contexts:

• Flood forecast
• Roller coaster construction
• Elevation profile of a track, installation of suitable traffic signs for the gradient
• Population development (possibly Lotka-Volterra)
• Share price
• Growth rate of body size
• Flow rate of a dam
• Glider barogram
• Tachograph
• Sales figures
• Fever curves
• Filling curves of vessels
• Drug concentration in the body
• Dike inclination
• Terrain slope and vehicles
• download rate
• Awareness (potential votes)
• Supply function (quantity of product offered and price)
• Light absorption (e.g. in the sea)
• ...