Matura m Teil 1: Torbogen
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
Short
Video
\(\LaTeX\)
Exercise:
Was ist die theoretische Obergrenze für die Spannweite eines Torbogens welchen man aus identischen Mauersteinen anfertigt?
Solution:
bf Für zwei Steine: Gmein-sam-er Schwerpunkt genau über Tischkante. medskip tikzpicturescale. filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-.cm filldrawcolorblue . circle .; drawdashed colorblue --; tikzpicture medskip bf Für drei Steine: medskip tikzpicture filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-cm filldrawcolorgreen!!black . circle .; filldrawcolorgreen!!black -.. circle .; tikzpicture Schwerpunkte des -Stein-Systems und des neuen Steins grün eingezeichnet. Drehpunkt ist die Tischkante. bf Gleichgewicht: mg leftfracL-xright mg x qquad Rightarrow quad x fracL medskip bf Für n Steine: medskip tikzpicture drawdashed -..---.; drawdashed ..--.; filldrawcolorred!!black -------.---.--cycle; node at .-. x; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; filldrawcolorgreen!!black . circle .; pgftransformyshiftcm pgftransformxshift.cm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift-cm pgftransformxshift-.cm drawcolorred dashed thick .--..; tikzpicture Bei n- Steinen und letztem Stein: mg leftfracL-xright n- mg x qquad Rightarrow quad x fracnfracL Torbogenweite mit n Steinen: d fracL tcbhighmath_k^n frack rightarrow infty harmonische Reihe divergiert lnn+
Was ist die theoretische Obergrenze für die Spannweite eines Torbogens welchen man aus identischen Mauersteinen anfertigt?
Solution:
bf Für zwei Steine: Gmein-sam-er Schwerpunkt genau über Tischkante. medskip tikzpicturescale. filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-.cm filldrawcolorblue . circle .; drawdashed colorblue --; tikzpicture medskip bf Für drei Steine: medskip tikzpicture filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-cm filldrawcolorgreen!!black . circle .; filldrawcolorgreen!!black -.. circle .; tikzpicture Schwerpunkte des -Stein-Systems und des neuen Steins grün eingezeichnet. Drehpunkt ist die Tischkante. bf Gleichgewicht: mg leftfracL-xright mg x qquad Rightarrow quad x fracL medskip bf Für n Steine: medskip tikzpicture drawdashed -..---.; drawdashed ..--.; filldrawcolorred!!black -------.---.--cycle; node at .-. x; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; filldrawcolorgreen!!black . circle .; pgftransformyshiftcm pgftransformxshift.cm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift-cm pgftransformxshift-.cm drawcolorred dashed thick .--..; tikzpicture Bei n- Steinen und letztem Stein: mg leftfracL-xright n- mg x qquad Rightarrow quad x fracnfracL Torbogenweite mit n Steinen: d fracL tcbhighmath_k^n frack rightarrow infty harmonische Reihe divergiert lnn+
Meta Information
Exercise:
Was ist die theoretische Obergrenze für die Spannweite eines Torbogens welchen man aus identischen Mauersteinen anfertigt?
Solution:
bf Für zwei Steine: Gmein-sam-er Schwerpunkt genau über Tischkante. medskip tikzpicturescale. filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-.cm filldrawcolorblue . circle .; drawdashed colorblue --; tikzpicture medskip bf Für drei Steine: medskip tikzpicture filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-cm filldrawcolorgreen!!black . circle .; filldrawcolorgreen!!black -.. circle .; tikzpicture Schwerpunkte des -Stein-Systems und des neuen Steins grün eingezeichnet. Drehpunkt ist die Tischkante. bf Gleichgewicht: mg leftfracL-xright mg x qquad Rightarrow quad x fracL medskip bf Für n Steine: medskip tikzpicture drawdashed -..---.; drawdashed ..--.; filldrawcolorred!!black -------.---.--cycle; node at .-. x; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; filldrawcolorgreen!!black . circle .; pgftransformyshiftcm pgftransformxshift.cm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift-cm pgftransformxshift-.cm drawcolorred dashed thick .--..; tikzpicture Bei n- Steinen und letztem Stein: mg leftfracL-xright n- mg x qquad Rightarrow quad x fracnfracL Torbogenweite mit n Steinen: d fracL tcbhighmath_k^n frack rightarrow infty harmonische Reihe divergiert lnn+
Was ist die theoretische Obergrenze für die Spannweite eines Torbogens welchen man aus identischen Mauersteinen anfertigt?
Solution:
bf Für zwei Steine: Gmein-sam-er Schwerpunkt genau über Tischkante. medskip tikzpicturescale. filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-.cm filldrawcolorblue . circle .; drawdashed colorblue --; tikzpicture medskip bf Für drei Steine: medskip tikzpicture filldrawcolorred!!black -------.---.--cycle; node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformxshift-.cm pgftransformyshift-cm filldrawcolorgreen!!black . circle .; filldrawcolorgreen!!black -.. circle .; tikzpicture Schwerpunkte des -Stein-Systems und des neuen Steins grün eingezeichnet. Drehpunkt ist die Tischkante. bf Gleichgewicht: mg leftfracL-xright mg x qquad Rightarrow quad x fracL medskip bf Für n Steine: medskip tikzpicture drawdashed -..---.; drawdashed ..--.; filldrawcolorred!!black -------.---.--cycle; node at .-. x; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; filldrawcolorgreen!!black . circle .; pgftransformyshiftcm pgftransformxshift.cm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift.cm pgftransformxshiftcm node at .-. fracL; pgftransformxshift-.cm filldrawcolorblack fillblack!!white -----.---.--cycle; pgftransformyshift.cm pgftransformxshiftcm filldrawcolorblack fillblack!!white -----.---.--cycle; node at .-. fracL; pgftransformyshift-cm pgftransformxshift-.cm drawcolorred dashed thick .--..; tikzpicture Bei n- Steinen und letztem Stein: mg leftfracL-xright n- mg x qquad Rightarrow quad x fracnfracL Torbogenweite mit n Steinen: d fracL tcbhighmath_k^n frack rightarrow infty harmonische Reihe divergiert lnn+
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