In this article, we propose five activities in the field of mathematics to be carried out with your smartphone. They can be offered to middle and high school students during math lessons or for homework.

**1. Why do math with a smartphone?**

Some students may find math too abstract and difficult to understand, especially if the curriculum focuses primarily on theory and solving abstract problems without showing their real-world relevance. This can make math boring and frustrating for some students.

The use of the smartphone or the tablet, a tool that all the pupils have in their pocket or their satchel, makes it possible to show on concrete cases of everyday life what the purpose of mathematics is. These digital devices have many sensors such as the accelerometer, the GPS, the camera or the microphone which can be used to make precision measurements thanks to free applications like FizziQ.

By using their smartphone to apply the theory of the course or to approach problems by the inquiry based science education method, students become more engaged, understand the interest of science and learn scientific reasoning by practice. Here we offer five activities that can easily be done in class or at home and which allow the student to put into practice the concepts he has learned about angles, trigonometric relations, or statistics. Of course this list is not exhaustive and we encourage everyone to share new activities and protocols with us!

**2. The law of sines**

The law of sines remains a very theoretical formula for students. This mathematical law that describes the relationship between the angles and the lengths of the sides of any triangle:

In a triangle ABC, the ratios between the lengths of the sides and the sines of the opposite angles are equal:

a / sin(A) = b / sin(B) = c / sin(C)

where a, b and c are the lengths of the sides of the triangle, and A, B and C are the opposite angles respectively.

In other words, the law of sines indicates that the ratio between each side of the triangle and the sine of the angle opposite to this side is constant for all the sides and all the angles of the same triangle. This means that if you know the measure of two angles and one side of the triangle, you can use the law of sines to calculate the measures of the other sides and angles of the triangle.

An interesting activity for students is to have them measure a distance that they cannot measure with a regular surveying tool. For example because there is an obstacle between the two points such as an impassable stream or river.

In the activity that we propose, the students use the theodolite of the FizziQ application and the law of sines to measure the lengths of a triangle in the playground, we can symbolize a stream in the middle of the course to make clear the usefulness of the method. This practice allows a rapid and experimental acquisition of the concept, shows students a practical application of a very abstract formula, and can be carried out either with a tablet or a smartphone.

Download the Law of Sinus activity: __https://www.fizziq.org/team/loi-des-sinus__

Our video on measuring a distance with the theodolite:

**3. Calculation of the height of a building**

Who does not know the apocryphal story of Niels Bohr and the calculation of the height of a building using a barometer. It describes the inventiveness of the young Niels who, in response to a statement in physics, finds a number of solutions that are technically correct, but intentionally irrelevant.

In this activity, students will use several ways to use, not the barometer, but FizziQ's theodolite to calculate the height of a school building. By comparing their results, they will be able to determine the best solution. Be careful, dropping your laptop is not an option!

Read the hands-on educational sheet: __https://fondation-lamap.org/sites/default/files/sequence_pdf/Defi-mesure-batiment.pdf__

La vidéo Billes de Sciences par "La physique autrement" :

## 4. Study of a cycloid

When mathematics meets physics! A cycloid is the curve representing the trajectory of a point fixed to a circle which rolls without slip and at constant speed on a road.

The cycloid has many interesting properties. For example, the length of the cycloid is 4 times the radius of the circle, and the area under the cycloid is 3 times the area of the circle. Also, the cycloid is a brachistochronous curve, i.e. any object that follows the curve under the effect of gravity will reach the end point faster than if it followed any other trajectory.

In this activity, the student will film a dot on the wheel of a bicycle, or they can also use the Cycloid video downloadable from the Cinematic Video Library, to perform a kinematic analysis of the motion of the dot. This analysis allows him to visualize the trajectory followed by a point on the wheel, or other points on the spoke. It can also export data to an Excel spreadsheet. This simple and quick experiment to implement allows the student to familiarize himself with kinematic analysis, the equation of curves, and he can make his own film of the wheel of a moving bicycle.

Download the Cycloid Study activity: https://www.fizziq.org/team/cycloide

## 5. Triangulation

The activity on the law of sines can be continued by a calculation of distance by triangulation.

Leon Battista Alberti is credited as one of the first to devise a method for calculating far distances, but it was with the mapping of the earth and research into its shape that the triangulation method became an extremely powerful tool for measure. We can first make a historical reminder on the calculation of the length of the meridian by the astronomers, Pierre Méchain and Jean-Baptiste Delambre and the definition of the meter.

In the form of an investigative session, we will also address the various problems posed by length measurement on uneven terrain and the usefulness of the triangulation method. As a session of practical work, in the playground, or better on a large field, we will then use the theodolite to calculate a significant length, and we will check the results on a satellite mapping site. For example, students can be asked to calculate the greatest possible length around their home using triangulation.

The protocol is based on the notations described in this video:

Download the activity in FizziQ : __https://www.fizziq.org/team/triangulation__

## 6. Measurement variability

This activity proposes to make the students work on the variability of the measurement of a physical quantity. It can also be used in the experimental and digital project. FizziQ can record a large number of data produced by the sensors (acceleration, magnetism, frequency, sound volume). This data can be analyzed by the students directly in the workbook or exported to a spreadsheet. They make it possible to answer different types of questions: how accurate is a measurement? how to improve accuracy? which smartphone is the most accurate? how to account for the dispersion of a measurement?

Download the Uncertainty activity: __https://www.fizziq.org/team/certainty__

Consult the La main à la pâte educational sheet on "Measurements and uncertainty": __https://fondation-lamap.org/sites/default/files/sequence_pdf/mesures-et-incertitudes-defi-fizziq.pdf__

## 7. In conclusion

Although its name does not indicate it, FizziQ can be used in many fields other than physics. Mathematics, SVT, chemistry, music, geography, sports, FizziQ instruments allow students to make measurements and analyze work in many fields and thus better understand the world around them.

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P.S. Many thanks to Guillaume Lefranc, Julien Bobroff, Frédéric Bouquet, Aline Chaillou and Pauline Bacle.

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