task of the week – Page 10

The present task of the week is about geometric figures. In the task “Cylinder on the Rhine”, located in Cologne, the aim is to determine the radius of a cylinder by means of measurements or the relationship between radius and circumference of a circle.

Task: Cylinder on the Rhine (task number: 1183)

Determine the radius of the cylinder. Give the result in m.

The task can be solved in different ways. One possibility is to use the relationship between the circumference of the circle and the diameter or radius. The result is then obtained by measuring the circumference. Alternatively, the radius can be determined by means of the inch post and suitable application (here the right angle plays a role). The task can therefore be classified into the topic circle, in particular the formula for the calculation of the circumference. The task shows that mathematical tasks can often be solved in various ways without calculation. Although the task does not require any profound knowledge of the cylinder (apart from the fact that the base is circular), it can be precisely in this aspect, and a connection of planar and spatial geometry can be made clear.

It can be used starting from class 9.

The present task of the week is about a sculpture in form of a hand. In this form, it can be found in the Dillfeld Trail in Wetzlar. The aim of the task is to determine the body size of the person to whom this hand would fit.

Task: The Hand (task number: 1092)

How big would a man be in meters with a hand of this size?

The pupils should measure a finger that is easily accessible to them. Especially the thumb offers itself for it. How can the thumb size be related to the body size? For conversion, the own body can play a role by correlating the thumb size and the body size. Then the body size of the human with the shown hand can be determined. The students use the idea of measuring their own body and the hand sculpture. In particular, relations and sizes play a role. The task can be used from class 6 on with the development of relations.

The current “Task of the Week” is about the hydrant sign, which might have been noticed frequently in everyday life. By means of them, hydrants can be quickly and precisely located, e.g. for fire-fighting operations. But how exactly is such a sign read? With this question, the students are confronted in the task “Hydrant sought” from the trail “Campus Griebnitzsee” in Potsdam.

Task: Hydrant sought (task number: 1047)

On the house is a reference to the next hydrant attached (red-white sign). How far is the hydrant from the sign in meters? Determine the result to the second decimal place.

In order to solve the problem, the sign must at first be interpreted correctly. If the students do not know it, the hints help them. The indication on the sign is to be read so that one runs a certain length in meters in one direction (left/right) and then turns at right angles and again runs the length of the second number in meters. The situation can thus be described and solved using a right-angled triangle. The two indications on the hydrant sign (in the picture, they are made unrecognizable in order to ensure the presence of the pupils) are the cathets, while the direct distance corresponds with the hypotenuse. This can be determined by means of the Theorem of Pythagoras. Further, the solution can be determined and valued through measuring the distance to the hydrant. The problem is therefore to be assigned to geometry and can be used as a practical application for this from class 9 onwards with the development of the Theorem of Pythagoras. Since hydrant signs can be found in many places, the task can easily be transferred to other sites and allows mathematical operations in the environment in an easy way.

This week we would like to present the task “On a big Foot”. It is located close to the main railway station in Hamburg and is part of the trail “In and around St. Georg”.

Task: On large feet (task number: 647)

These figures are created by the contemporary German sculptor Stephan Balkenhol. I would like to know from you: What shoe size does the man have? For shoe sizes, there are four common systems worldwide. In Germany, European shoe sizes are the usual measure. They are based on the so-called “Parisian Stitch”. The stitch is a length measure with which a shoemaker specifies the length of a stitch and thus also the shoe size of the complete shoe. A French stitch or Parisian stitch is ⅔ centimeters long. The shoe last is a piece of wood, plastic or metal which is modeled on the shape of a foot and used to build a shoe. Since the feet should have some space, the length of the shoe last corresponds approximately to the foot length + 15 mm.

For the task, the pupils first measure the length of the man’s shoe and calculate the length in “stitches” so that the European shoe size can be specified. A major component of the task is the measurement and conversion of quantities. In doing so, the unity of the stitch, which should be unknown to most students, is used. It can be used from class 6 onwards. In addition, the first proportional basic ideas can be formulated for the conversion and could be a suitable transition to the proportionality and the rule of three.

The task was created by Dunja Rohenroth. She has already been able to test this task with her students and sees in this task the special advantage that the result cannot be solved by means of an internet search. The aspects of the presence and activity of the pupils are thus particularly emphasized.

Today’s “Task of the Week” leads to Lyon, France, included in the trail “IFE”. It deals with an area calculation of a particular kind and shows in an exciting way which varied mathematical ideas are in everyday objects.

Task: Serpent Surface (task number: 1129)

The metal railing of the fire stairs is in the form of a serpent line. Calculate the surface area in m².

Before the students can begin to solve the problem, preliminary considerations are necessary, e.g. whether the slope of the railing is relevant or which formulas can be used to determine the length of the railing. The pupils should recognize the course of the serpent line as circular. In the case of two rotations of the staircase, the length of the railing corresponds to the double circumference of the circle with the step length as radius. With help of the circumference and height of the railing, the surface area of the serpent surface can be determined.

This is a geometric problem which combines the subcategories “space and form” and “measuring” by recognizing geometric structures in the environment as well as measuring the sizes and using them for calculations. The task is assigned in particular to the theme “circle” and can thus be used with treatment of the formula for the circle circumference from class 8 onwards.

In addition, the task shows that many objects can motivate a wide range of questions. Besides the question of the surface area, it would for example be possible to calculate the slope of the railing.

This week the “Task of the Week” focuses on a typical application of the intercept theorems. In particular, it is about the height determination of objects using the interception theorems. This task type can be transferred to many different objects and can therefore be found in further MathCityMap trails. The here described example is about the height determination of the lanterns in the garden of Erlangen’s castle.

Task: Illumination of the Castle Garden (task number: 709)

Determine the height of the two-armed lamps in the castle garden in the unit cm.

To solve the problem, the second intercept theorem is required. For this purpose, the pupils position themselves a few meters away from the object and fix the object. The intercept theorem can then be applied using the measuring stick. For this, the eye height as well as the distance to the object must be measured. With the arm outstretched, the measuring stick is held so that its tip coincides with the upper end of the lantern. The length of the arm and the scale length, which corresponds to the height of the lantern from the height of the eye, lead to the height of the lantern.

This is a problem-solving situation in which initially missing values have to be determined by a suitable initial situation. The application of the interception theorem can in this case be facilitated through the preparation of a sketch. The task is particularly suited to show students the practical application of the interception theorem and to give a meaningful content to the calculus.

This time, the “Task of the Week” is part of the trail “Rund um den Erlangener Schlosspark”. It is called “Monument Erlangen /Brüx” with task number 704. Thematically, the task can be integrated into the topic parables and is therefore suitable from grade 9.

Task: Monument Erlangen/Brüx

Examine whether the “curve” in the lower quarter of the stone monument is a parable y= -ax². If not, enter a=0 as solution, otherwise enter the calculated value of a.

The task was written by Jürgen Hampp. In the following interview, he gives an insight into the idea behind the task and the aim of the task. At this point, we would like to take the opportunity to thank Mr. Hampp for his answers.

What made you consider including this task into the trail?

My concern was to develop a trail which is on the one hand easy to access in walking distance from our school, the Christian-Ernst-Gymnasium in Erlangen, and on the other hand leads through mostly car-free areas. Of course, the possible objects are limited. Under this perspective, the monument Erlangen/Brüx has an optimal position, the measuring is riskless – one does not have to climb etc. – and only simple resources are needed.

Where do you see the characteristic of the task? Which skills and ideas are especially supported?

I want to train the “mathematical view”, e.g. the recognition of mathematical objects in everyday life, and the activity with these objects with help of the methods which are known from class. This object mainly supports the competence branch K3 “Mathematical modelling”. Here, quadratic functions (topic in grade 9) present themselves. I did not want to use the common tasks with water fountains as they might be out of use, the water pressure may vary and the measuring is difficult. For me, the special attraction of this task is that a plain solution – as for usual schoolbook exercises – does not exist. Inaccurate measuring at the object or discrepancies at the object require skillful forming of averages and approximated values.

Today’s “Task of the Week” leads to Hamburg, more precisely to the school Am Heidpark. Here, one can find the trail “Am Heidpark” which is a good example to show that already a schoolyard can be made for a MathCityMap trail. The selected “Task of the Week” is called “Climbing Wall” with task number 668.

Task: Climbing Wall

Determine the slope of the climbing wall in percent.

The task enables a suitable embedding of the topic slope of linear functions. The slope of the climbing wall can be determined by recourse of the gradient triangle. In the coordinate system, the slope of a linear function can be calculated with help of two points on it. It is necessary to determine the difference of the y-coordinates (dy) and the difference of the x-coordinates (dx) and divide them afterwards. Corresponding in the real context, it is necessary to measure the height difference (dy) as well as the difference in length (vertical; dx). Afterwards, the slope can be calculated with help of a division and the conversion into percent. The task can be used from grade 8 and supports a basic understanding of the slope of a linear function and its determination with help of a gradient triangle. The task is especially suitable in the beginning of the topic as it already “predefines” a right-angled gradient triangle. Further tasks could for example involve the slope of a stair handrail. The task is a connection of algebra and geometry and can be related to the branches measuring and functional correlation.

In this week, the focus of the “Task of the Week” is on a stochastic problem. The task is called “Permutation at the Bicycle Stand” and is included in the trail “Hubland Nord” located in Würzburg. The task number is 680.

Task: Permutation at the Bicycle Stand

Four bicycles should be locked at the bicycle stands. The bicycles can be locked on the left or on the right side of each stand. How many possibilities exist to lock four bicycles at the stands? You do not have to distinguish whether the bikes are locked “forwards” or “backwards”. You can assume that all stands are free.

In this task, it is necessary to determine the number of possibilities to lock four bicycles at the bicycle stands. Altogether, there are eight stands and therefore 16 possible spaces. On the picture, not all spaces can be seen in order to guarantee the criterion of the students presence (the task can only be solved at this location). For the first bike, there exist 16 possibilities to lock it. As this space is full afterwards, the number of possibilities to lock the second bike is 15. Analogous, the possibilities for bikes three and four amount 14 and 13. 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.

This task enables a suitable embedding of a combinatorial problem into the reality. It belongs to probability calculus and can be used from grade 8 with first combinatorial considerations. Further, it can be especially used in stochastics in grade 12/13 as a repetition of basic combinatorial considerations. Moreover, the task can be transferred easily to similar situations (e.g. parking spaces).

Today’s “Task of the Week” focuses on the “Hammering Man”, a symbol of Frankfurt’s fair. The “Hammering Man” comes to one’s attention through his continuous hammering motion. The task is part of the “Weihnachtstrail” with task number 784.

Task: Hammering Man

The “Hammering Man” hammers continuously. How many hammer blows does the “Hammering Man” carry out in the month December?

To solve this problem, it is necessary to observe the motion of the “Hammering Man” and measure the duration of a blow (in seconds). This can be done through measuring the time for 10 cycles. Afterwards, the number of seconds for one day and for the month December should be determined. With help of a division, the number of hammer blows can be calculated for the month December.

In this task, the main part is to determine the frequence of a periodic motion through measuring. Therefore, the task can be seen as an examplary task which can be adapted to further locations where things move periodically. The focus is especially on the time units second, day and month, as well as their conversion. Further, the arithmetic operations multiplication and division are included. Therefore, the task is in connection with school mathematics and can be used from grade 4.

The task is very suitable, because it requires the presence and activity (measuring of the duration of a blow) of the pupils. Further, it is a realistic problem, which can be solved without special aid. The task offers the possibility to differentiate as the pupils can ask for help if needed. The sample answers can be found with the task in the portal.

Tag: task of the week

Task of the Week: Cylinder on the Rhine

Task of the Week: The Hand

Task of the Week: Hydrant sought

Task of the Week: On large Feet

Task of the Week: Serpent Surface

Task of the Week: Illumination of the Castle Garden

Task of the Week: Monument Erlangen/Brüx

Task of the Week: Climbing Wall

Task of the Week: Permutation at the Bicycle Stand

Task of the Week: Hammering Man