methodological guide

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cross curricular guide (maths applied on different fields of science)

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    Cross-curricular Approach of Mathematics

    Methodological Guide

    QED Quality in Europe's Diversity

    Comenius Multilateral Partnership 2011 - 2013

    11-PM-75-BT-RO

  • 2

    Chapters:

    I. Applications of Mathematics in Science ..page 3

    QED team of Borgarholtsskli from Reykjavik, Iceland

    II, Maths in cultural and social life ..page 12

    QED team of Geniko Likeio from Kissamos, Greece

    III. Mathematics &Economy ..page 24

    QED team of Ahmet Eren Anadolu Lisesi from Kayseri Turkey

    IV. ICT applications in math teaching ..page 28

    QED teams of Goetheshule Wetzlar, Germany, and CETCP Botosani,

    Romania

    External:

    http://www.geogebratube.org/collection/show/id/3309

    http://www.slideshare.net/effiefil/maths-in-culturar-life-new

    http://new-twinspace.etwinning.net/c/portal/layout

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    I. Applications of Mathematics in Science

    The QED team from Borgarholtsskli, Iceland

    Natural sciences are important for daily life. It is important that students choose to study

    natural sciences, health sciences and engineering. In Iceland today there is a lack of students

    who study these subjects. Students seem to choose something else.

    Mathematics is closely related to science. Natural sciences can not be without

    mathematics. For students, natural sciences and mathematics, tend to be difficult and they are

    intimidated by them. For those who like to study these subjects it is often difficult to transfer the

    knowledge of them to other subjects.

    The exercises in this pamphlet are based on to link together mathematics and physics

    through assignments which are done manually! The goal by combining mathematics and

    physics is to integrate natural sciences with mathematics. By using practical and manual

    exercises the students should increase their understanding and also they might be more positive

    towards natural sciences. In addition these exercises are important for physics.

    The exercises have been used in the first and second level in physics in

    Borgarholtsskli, Iceland. The uniqueness of the school is that the background of the students is

    very different and that means that the teaching is very individually based.

    The exercises can easily be changed, that is, their focus can be changed, they can be

    made more difficult, they can be less time consuming or more time consuming.

    Exercise 1: Errors in measurements

    Purpose:

    When doing measurements in science there is never a 100% certainty in the numbers we

    measure. There is always some minimum resolution we can get, determined by the

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    instruments we use or by the users doing the measurements. In this exercise student will

    practice measuring objects using simple tools, determining the uncertainty or error in the

    measurements and then use errors in simple calculations. As a prerequisite, students need

    to know how to determine errors in measurements and to do calculations using errors.

    Materials

    Stopwatch

    Ruler or caliper

    A few pencils or sticks

    Procedure

    Part 1: Sticks, length, addition

    Groups get one stick/pencil for each member of the group

    a) Find an empty page in your workbook. Write down the name of the exercise, the date

    and the names of the students in your group

    b) Give each stick a name or make sure to be able to distinguish between them.

    c) Each student measures the length of one stick and determines the uncertainty/error in

    the measurements. Write all the results down, including the name of the person doing

    each measurement

    d) Calculate the total length of the sticks and determine the total error in that

    determination

    e) Calculate the average length of all the sticks and the error in that value using the

    MIN/MAX method of determining total errors. Make sure to document everything in

    your workbook

    Part 2: Box, volume, multiplication, % error

    Each group of students gets one small box

    a) Write down a name for the box, draw a picture and mark each edge so its certain

    what side is what

    b) Measure the length of the edges and determine the error in the measurements.

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    c) Calculate the volume of the box and determine the errors in that using the

    MIN/MAX method

    d) Calculate percentage error which is the error divided by volume

    Part 3: Measuring time, % error, instrument error, user error

    Each group of students gets a stopwatch

    a) Each student tries 5 times to start a stopwatch and stop at exactly 10:00 sec. Each

    student writes down his numbers

    b) Each student calculates the average time and determines the error. Also the % error

    which is error divided by average time

    There are 2 main ways to determine the errors in this step

    i) Standard deviation: Use a spreadsheet or a calculator to calculate average time and

    standard deviation. The error is then the standard deviation

    ii) Min/max variation:

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    Exercise 2: Graphs and lines

    Purpose

    The purpose of this exercise is for students to practice matching curves or lines to specific

    equations. Once that is done, its possible to match the curves and values from the curves

    to certain physical laws or rules.

    Materials

    Graphing paper and a ruler

    Procedure

    Part 1: Calculating a trend line

    Draw the points/coordinates in table 2 on a graph. The v

    [m/s ] is on the y-axis and t [s] on the x-axis and make sure

    to mark clearly the axis t or v. Draw a straight line (using a

    ruler) that intersects all the points on the graph and calculate

    the equation for the line. The equation is on the form y = kx

    + c.

    Part 2: Graph matching

    Draw the data in table 2 on a graph and try to determine

    which type of a line/curve fits the data. Then determine the

    equation for the line. I.e. calculate a value for k or a, b and

    c.

    Table 1

    t [s] v [m/s]

    0 1

    1 4

    2 7

    3 10

    4 13

    Table 2

    x-axis y-axis

    0 0

    1 2

    2 8

    3 18

    4 32

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    Suggestions to try out:

    Linear equation

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    x. The force constant (k) of each spring, determines how hard it is to stretch or compress it. A

    stiff spring has a large k but a loose spring has a low k. The purpose of this experiment is to

    determine the force constant in a spring by drawing a graph and calculating the trend line. An

    experiment was done where force was applied to the spring and the stretching of the spring was

    measured.

    Materials

    Graphing paper and a ruler

    Procedure

    Table 1 contains data from a physics experiment. Force was applied to a spring and the length

    that the spring stretched was measured.

    1. Draw a graph and each point on the graph. Mark the axis

    clearly as y-axis Force, F, [N] and x-axis as

    expansion/stretch of the spring, x, [m]. Make sure to keep

    the scale on the graph such that its is detailed. Try to use

    whole page in your book.

    2. Draw the error bars on each point. The errors in F will

    be shown as small lines extending up and down from each

    point. The error in x will be shown from left to right. When

    all is done, the graph should look like it has six small

    crosses.

    3. Use a ruler and try to find out what you feel is the best straight line that intersects the group

    of points the best. Do not connect the dots, the end results should be a graph showing small

    crosses and one straight line intersecting most of them.

    4. Determine the equation for the line you have drawn. Make sure to pick points that are

    actually on the line, to do the calculations. The equation should be on the form y=kx+c, (F=kx)

    where c is 0.

    5. Compare the equation and Hookes law. What is the value of the springs force constant (k)?

    Table 1

    x-axis y-axis

    X [m] +/-

    0,03

    F [N] +/-

    0,5

    0,00 0,0

    0,30 10,0

    0,43 15,0

    0,59 20,0

    0,71 25,0

    0,90 30,0

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    6. Use the equation to forecast how much you would expect the spring to stretch if a force of 23

    N was applied to the spring.

    Exercise 4: Graph matching

    Purpose

    Scientist often do experiments to test hypothesis. A large number of data points are then

    measured and then the scientist tries to make sense of it all and to find the mathematical

    correlation between variables. In this exercise data from physics experiments is collected in

    tables. The data must be entered into graphing software, the relevant graph plotted and matched

    to mathematical functions. When the equations are determined the software shows the relevant

    constants that can then be matched to the variables that are unknown in each physics equation.

    Materials

    Graphic calculator or graphing software

    Method

    Input the data in tables 1-4 in to a graphics calculator or graphing software one at a time. Then

    match each curve to the correct physics equations. I.e. find the equation that matches each table.

    Use the equations for the curves to c

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