MathCityMap was used as part of a cooperation project of the University of Paderborn under the management of Max Hoffmann with the Pelizaeus Gymnasium in Paderborn. 9th grade students created a mathematical city tour with MathCityMap. Here, you can find further information.
MathCityMap was used as part of a cooperation project of the University of Paderborn under the management of Max Hoffmann with the Pelizaeus Gymnasium in Paderborn. 9th grade students created a mathematical city tour with MathCityMap. Here, you can find further information.
In addition to a variety of geometric issues, combinatorial and stochastic problems also play an important role in MathCityMap. Today, we would like to introduce you to the most common generic tasks concerning combinatorics and probability. Two combinatorial questions that can be created quickly and easily with the Task Wizard are tasks asking for combination […]
In addition to a variety of geometric issues, combinatorial and stochastic problems also play an important role in MathCityMap. Today, we would like to introduce you to the most common generic tasks concerning combinatorics and probability. Two combinatorial questions that can be created quickly and easily with the Task Wizard are tasks asking for combination options of stairs and bike stands.
In how many ways can you climb the stairs by taking one or two steps?
There are various possibilities for solving the problem. On the one hand, it is possible to systematically record different combinations of 1 and 2 steps. In doing so, the students can use the stairs directly and conclude which combinations are possible. In another consideration, the fact that the last step comprises either one step or two steps is used. Leaving this last step, the number of possibilities for a staircase with n steps can be determined using the possibilities for n-1 and n-2 steps. This reasoning leads to the Fibonacci numbers, a recursive sequence in which a number results from the addition of its two predecessors.
How many possibilities do you have to lock k bikes?
In this task, it is necessary to determine the number of possibilities to lock k bicycles at n spaces. For the first bike, there exist n possibilities to lock it. As this space is full afterwards, the number of possibilities to lock the second bike is n-1. Analogous, the possibilities for bike k is n-(k+1). This combinatorial problem is a situation where repetition is not allowed and order matters. With help of the possibilities’ product, one can calculate the total number of possibilities.
It is important to formulate the tasks precisely and to make clear which object or which part of the object is concerned (for example, in the case of a very long staircase, the lowest part). The activity of the task solver initially refers to the counting of the stairs or parking spaces of the bicycles. Therefore, when photographing, it should be noted that this number can not already be taken from the photo.
Further, probabilities can be realized by MCM, for example, the question for the probability of arriving at a traffic light during a green phase or to wait at a bus stop less than 5 minutes for the next bus. Both types of tasks focus on the Laplace probability (favorable events divided by all possible events).
We have compiled both emphases for you in the following document with detailed mathematical background and hints.
Although the focus of many MCM tasks is on lower secondary maths, some upper secondary level tasks can also be realised. So our current Task of the Week, which was created in the context of a teacher training at the commercial schools Hanau. Task: Parabolic Slide (task number: 2241) The shape of the slide is […]
Although the focus of many MCM tasks is on lower secondary maths, some upper secondary level tasks can also be realised. So our current Task of the Week, which was created in the context of a teacher training at the commercial schools Hanau.
The shape of the slide is the part of a parabola. Determine the compression factor. 1m equals 1 unit of length. You can assume that the slide is almost horizontal at the end.
The slide is approximated according to the task with the help of the equation f (x) = ax² of a parabola. To solve the task, the students first have to transfer the situation to a suitable coordinate system. Since only the compression factor is asked, it is not necessary to specify this in the task. It makes sense to set the coordinate system so that the origin lies at the lower end of the parabola, but leaves out the horizontal end. Through such a choice, it is sufficient to determine another point on the slide, so the change in the x- and y-coordinate. By inserting this into the equation, the compression factor a results.
At the beginning of the year, MCM was successfully presented in Mumbai. Of course, in this context the first Indian Math Trail was created, from which our current task of the week originates. Task: Grass Field (Task number: 2459) Calculate the area of the grass field. Give the result in m²! First, a mathematical model […]
At the beginning of the year, MCM was successfully presented in Mumbai. Of course, in this context the first Indian Math Trail was created, from which our current task of the week originates.
Calculate the area of the grass field. Give the result in m²!
First, a mathematical model has to be found that represents the area most accurately. This is best done by dividing the total area into several individual areas. The obvious choice is the division into two halfcircles and a rectangle. For this, the rectangular side lengths and the circle’s radius must be measured, the areas calculated and all partial areas added.
The task belongs to the topic of compound surfaces, whereby calculations on the circle must already be known in order to solve the task as exactly as possible. In the German school system, it would therefore be solvable from class 8.
Our first focus on generic tasks, meaning tasks that can be created in any location with similar objects, is on the subject of slopes. This topic has relevance for math lessons in different grades up to the upper secondary level. In particular, the slope of a straight line or linear function makes it possible to […]
Our first focus on generic tasks, meaning tasks that can be created in any location with similar objects, is on the subject of slopes. This topic has relevance for math lessons in different grades up to the upper secondary level.
In particular, the slope of a straight line or linear function makes it possible to determine the slope of various objects, such as ramps or handrails, with mathematics from lower secondary school. The result can be expressed either in percent or in degrees, including trigonometric relationships.
The mathematical basis is the definition of the slope as a quotient of vertical and horizontal difference, or in practical terms: the use of a gradient triangle. This can e.g. be implemented on ramps, especially if the horizontal length is easy to measure:
Example of a ramp where both, horizontal and vertical changes are easy to detect.
More difficult is the calculation of the slope of handrails, where one should use a water level for the difference in horizontal and vertical change:
Example of a handrail, where the result without a level can be inaccurate.
The spiral staircase takes the topic of slopes to a more complex level and requires imagination and transfer knowledge.For non-linearly rising objects, one may ask for the maximum slope or the slope at a particular point, e.g. as a preparation for the concept of tangent.
Attached you will find our extensive collection of frequently occurring generic tasks on the subject of slopes, the mathematical background as well as hints, compiled by Matthias Ludwig:
By the way: With our Task Wizard you can create the tasks for ramps and handrails with just a few clicks and transfer them to suitable objects in your area!
MathCityMap stands for outdoor mathematics, authentic tasks, physical activity, new technologies and teamwork. And we assert: The combination of these aspects works (almost) everywhere and just like magic … … because in every city there are stairs, buildings, parking lots, ramps, signs and many other recurring objects where you can do math actively. These objects […]
MathCityMap stands for outdoor mathematics, authentic tasks, physical activity, new technologies and teamwork. And we assert: The combination of these aspects works (almost) everywhere and just like magic …
… because in every city there are stairs, buildings, parking lots, ramps, signs and many other recurring objects where you can do math actively. These objects offer the chance to easily and quickly transfer existing tasks to other locations. We call these tasks generic tasks. The idea behind is thus a finished question in which only the object is exchanged and the measuring values are collected.
Very frequent generic tasks can already be created using the Task Wizard. This makes it possible to create complete tasks with just a few clicks, because sample solution and hints are generated automatically and inserted by the wizard.
In the following months, we will present a variety of topics from our catalog, which can be realized through generic tasks, e.g. slope, determine and estimate numbers, combinatorics and probability, speed, areas, volume and weight. It is an open and always growing fund. We are therefore pleased about your ideas for generic tasks!
Today’s object – found in Hamburg – requires the recognition of different quadrilateral shapes as well as a fitting division into several subareas. Task: Glass roofing (task number: 2148) How many square meters of glass were used for the entire roof? The glass roofing consists of a rectangular roof surface (can be divided into three […]
How many square meters of glass were used for the entire roof?
The glass roofing consists of a rectangular roof surface (can be divided into three small rectangles), a rectangular area next to the entrance and three trapezoids on each side. To solve the task, all measurements for the rectangles and trapezoids must be made. Afterwards, the students calculate the individual areas and by adding the entire content of the glass. Due to the individual bars, the decomposition of the surfaces is almost predetermined. Nevertheless, the task requires the recognition of geometric shapes as well as a suitable mathematization of the tasks by formula knowledge of rectangle and trapezoid. This geometric question can be assigned to compound surfaces and can be solved from class 8 onwards.
In the new year, we would like to continue introducing interesting tasks and topics from the MathCityMap task portal. It starts with a task from Qatar, which was created as part of a presentation of MathCityMap. Task: Slope of the Helix (Task number: 2243) Calcaulate the slope of the hailrail of this circular ramp. Give […]
In the new year, we would like to continue introducing interesting tasks and topics from the MathCityMap task portal. It starts with a task from Qatar, which was created as part of a presentation of MathCityMap.
Calcaulate the slope of the hailrail of this circular ramp. Give the result in percentage!
Despite the architectural peculiarity of the building, the task can be solved in a familiar way. One can use the definition of the slope as a quotient of vertical and horizontal change. In particular, with the help of the balusters, distance and horizontal change can be easily detected.
Thus, the task fits thematically in the area of ”slope” – a topic that occurs again and again in almost every location at MathCityMap, whether at railings, ramps or stairs. The task can be solved from grade 7 and serves as a basis for the recognition of functional relationships.
At the beginning of the year, MathCityMap and research results were presented at an international STEM conference in Mumbai, India. Matthias Ludwig spoke at the epiSTEME 7 on 7th January 2018 about the motivational effects of using MathCityMap. Further, he created a small trail together with Xenia Reit, which was enthusiastically tested by the participants […]
At the beginning of the year, MathCityMap and research results were presented at an international STEM conference in Mumbai, India. Matthias Ludwig spoke at the epiSTEME 7 on 7th January 2018 about the motivational effects of using MathCityMap. Further, he created a small trail together with Xenia Reit, which was enthusiastically tested by the participants of the conference. No wonder in this smartphone addicted country.
MathCityMap is very popular in Münster. In an article, written by Simone Mäteling, it is described how MCM is integrated into the project “Expedition Münsterland”. You can read more in the Artikel MS Land Magazin or at the Webseite .
MathCityMap is very popular in Münster. In an article, written by Simone Mäteling, it is described how MCM is integrated into the project “Expedition Münsterland”. You can read more in the Artikel MS Land Magazin or at the Webseite .
