task of the week – Page 5

As part of the ICM conference in Rio de Janeiro, Iwan Gurjanow created the first tasks in South America. In the resulting trail, our current Task of the Week is included.

Task: Cobblestones (Task number: 4505)

How many cobblestones are approximately in the highlighted area?

While we have often used the task on circular or rectangular surfaces, the parallelogram is focused here. It makes sense to determine the surface area and to count the number of stones in a certain range, e.g. in a square of size 60x60cm marked with the folding ruler. This number is then multiplied up to the total area, the area which quickly results from side length and height of the parallelogram.

A time-saving alternative to tedious counting, because the solution is highly above 1000.

With our GPS tasks, such as the walking of a north-south line, we have already presented a first way how to connect the directions with MathCityMap. But many statues also offer the opportunity to implement the theme, for example, the monument of Maximilian of Bavaria in Munich.

Task: Maximilian’s Pointer (Task number: 4483)

In which direction does the right finger of Maximilian point? Give the result in degrees. 0 °corresponds to the exact north direction and 90° to the exact east direction.

With the use of the smartphone and compass app, the task can be solved quickly and accurately. Without compass, creativity is required. The direction could be determined by means of the position of the sun or the north directed mathtrail map. If it is clear that students should work without a compass, it makes sense to limit the question to the direction of the compass with multiple choice

The city center of Münster provides several different tasks as one can notice within the portal map. Also in the trail around the Aasee, Münster presents itself from a various mathematical side.

Aufgabe: Vertebra (Task Number: 4096).

The work of art by Henry Moore created in 1974 represents several idealized vertebrae. These vertebrae are deliberately close, yet unconnected to each other. Imagine, they were being put together and seen as part of an adult’s human spine. In reality, an average vertebra of a 1.80 m adult is about 2 cm long. Guess how big a giant would be whose spine would consist of vertebrae of this size (in m).

In this task, especially estimation and modelling are forced. Through the detail on the relation of body size and size of a human vertebra in the task, the size of the vertebra can be determined as an object of reference. An adequate measurement and calculation by means of the realtion results the questioned size.

At the beginning of the month, Iwan Gurjanow presented the MathCityMap project in Sweden at the PME conference. Of course, around the location, different tasks were created as our current Task of the Week.

Task: Height of the Statue (Task Number: 4303)

How high is this statue? Give the result in meters.

The task can be solved via different approaches. On the one hand, it is possible to estimate how often a person with a known height fits into the heigth of the statue.

A more elaborated approach is the application of the intercept theorem, which is shown in the hint picutre. One can use the folding ruler as object of reference.

This week, Carmen Monzo, teacher in Spain gives us an inside into her task “Colums in the Parc”. It is created in a parc in Albacete, which ” is full of mathematical elements, though people are not aware of them until they are in math-vision mode.”

Task: Columns in the Parc (Task Number: 3981)

Calculate the lateral surface (in m²) of one of the columns of this structure.

“I especially love this structure. Parallel and perpendicular lines can be easily identified, as well as a set of columns (cylinders) whose lateral surface can be easily calculated by using a folding ruler or a measuring tape, and a calculator to introduce the data and the formula. The height of the cylinder is easy to get, but to calculate the radius of the base as accurate as posible, students first have to measure the circumference and then divide by 2*pi.

As this structure has a dozen columns, the activity can be done by around 20 students, comparing their results and thinking about the importance of the accuracy when measuring. To solve this task, students should have previously studied 2D and 3D shapes, the concept of the lateral surface and some formula to calculate it.

As a secondary mathematics teacher, I think that our students need to handle things, measure, count, touch, feel, use their senses… MathCityMap provides the motivation students and teachers need to do those things with the help of the mobilephone technology.”

This week, we focus on a task that can be used to realize linear functions in the environment. It was created by Kim Biedebach in Kassel.

I became aware of MathCityMap during a didactics lecture as part of my teaching studies that I attended. The idea for the task actually came to me by chance. I am from Kassel and had in mind that I have to design a modelling task for the lecutre. When I passed the figure, I spontaneously decided that this might be a suitable task.

Task: Man walking to the Sky (Task Number 3832)

How many meters is the man on the pole above the ground?

For this, the pole on which the man steps up is interpreted as a linear function. The point at which the pole starts on the ground is chosen as point (0, 0) for the sake of simplicity. Now, the slope must be determined as the quotient of the change in vertical and the change in the horizontal. If one starts from the chosen origin, and walks e.g.one meter to the side and measures the height there, the slope can be determined.

Afterwards, the slope can be used to determine the equation of function. Then the distance from the origin to the human on the ground has to be determined (corresponds to the x-coordinate). This is best done by positioning oneself under the man and measuring the distance to the origin. By inserting into the function equation the height can be calculated.

The task makes the linear relationship of x and y coordinates particularly clear. Also the slope concept is discussed. Of course, alternative approaches can be chosen, such as using the intercept theorems.

During May, the MathCityMap Team created a trail in Zaryadye Parc in Moscow – in good time for the start of FIFA World Cup in June!

One of the included tasks is in the focus of the “Task of the Week”, not at the latest through the object’s impressive architecture.

Task: Distance (Aufgabennummer 3761)

Calculate the distance between the crosses at the top of the towers! Give the result in meters.

Already in the picture, it becomes obvious that the distance cannot be measured directly. Without the use of special measuring equipment, the task solvers have to develop a creative idea: The distance in the height can be projected on the ground.

This happens best through marked points at the building, or as shown in the picture from a certain distance. With this idea, the beginning problem of the height of the building can be avoided and the task can be solved easily.

In case you search in our MathCityMap portal, you might notice that flowerpots enable various geometric tasks. Solely through the high frequency and the different shapes (cylinder, prismn with hexagonal area, etc.), the question how many liters of soil fit into the flowerpot, can be realised. In today’s Task of the Week, the flowerpot has the shape of a truncated cone.

Task: Flowerpot (Task number: 1219)

How many liters soil fit into the flowerpot, when it is filled until the top?

The formula for the volume of a truncated cone might not be known by all students. Therefore, they need strategies in order to solve the task, e.g. by means of the difference of a big and a small come. Further challenges are the determination of the small radius with help of the circumference and the consideration of the edge/bottom, which is obviously not filled with soil.

On Monday, May 14th, 45 primary students, who are interested in math and take part in an enrichment program, were able to discover the Campus Westend of Goethe University. In the beginning, the weather was thundery, nevertheless the motivated students were able to go on a math trail with their parents and university students. The trail was a combination of combinatory puzzles and measuring tasks which were solved by the participants in cooperation.

Task: Connected Trees (Task Number: 3485)

How many ropes are needed to connect every tree with all the other trees?

The students had a lot of fun while solving the problems. Especially while solving the tree task, they were able to build on prior knowledge from the enrichment program. Here, a similar task with shaking hands was questioned. The solution that the first of the 15 trees is connected with 14, the second with 13 and so on led quickly to the right solution.

Also the questions on the number of possibilities for 6 persons to take a seat, and the number of possiblities to go upstairs a stair with single, double and triple steps could be solved by the children through calculation and testing.

A special highlight was a task whose correct answer opened the lock to a treasure chest with small surprises in it.

Created during a teacher training in Erfurt, today’s Task of the Week fokuses on the topic Archway. The topic allows different questions and problems. Some weeks ago, we already presented an archway task on the weight of an archway’s stones. Today, the maximal height of an archway is focused.

Task: Archway (Task Number: 3090)

Determine the maximal height of the archway at the Krämerbrücke. Give the result in meters with two decimals.

The most elegant solution of this task is to divide the archway into a rectangle and a semi-circle.

With this hint, it is necessary to determine the point at which the semi-circle begins and the rectangle ends. With the height of the rectangle and the radius of the semi-circle (best determined through the diameter), the height of the archway results. In case the topic circle should be highlighted even more, it is possible to ask for the circumference of the archway. Here, the relation of diameter and circumference is further focused.

Tag: task of the week

Task of the Week: Cobblestones

Task of the Week: Direction

Task of the Week: Vertebra

Task of the Week: Height of the Statue

Task of the Week: Columns in the Parc

Task of the Week: Man walking to the Sky

Task of the Week: Distance

Task of the Week: Flowerpot

Task of the Week: Combinatorics

Task of the Week: Archway