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Applications of functions in real life

Published by Ludmila Cojocari, 2022-07-14 19:31:38

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APPLICATIONS OF FUNCTIONS IN REAL LIFE Worksheets Collaborative e-Book MATH TRAVELLERS eTwinning project 2021-2022

Autors: students from 9 schools coordinated by teachers, as follows: School Teacher 1. Gaudeamus High School, Moldova Ludmila Cojocari 2. Jean Monnet High School, Romania Mihaela Git 3. Osman Nuri Hekimoğlu Anatolian High Yasemin School,Turkey Çetinkaya 4. EB23 Caniço, Madeira, Portugal Maria José Garapa 5. AE Virgínia Moura, Guimarães, Sónia Ribeiro Portugal 6. Technical School Virovitica, Croatia Vlatka Hižman- Tržić 7. Baruthane Ortaokulu, İlkadım, Turkey Tuğba Varol 8. Etiler Anatolian High School, Istanbul, Arzu Çalık Turkey Seydim 9. 1st Vocational high school, Greece Yolanda Christopoulou

Applications of functions in real life Florentina Stancu, Vlad Niculescu 10C Grade - 2022 Teacher Mihaela Git Guitar string A guitar string is pulled at point P a distance of 3 cm above its rest position. It is then released and vibrates in damped harmonic motions with a frequency of 165 cycles per second. After 2 s, it is observed that the amplitude of the vibration at point P is 0.5 cm. a) Find the damping constant c. b) Find an equation that descibes the position of point P above its rest position as a function of time. Take t=0 to be the instant that the string is released. ������������������ ������������������������������������������������������������ ������������������ ������������������ ������������������������������������������������������������������������������ ������������������������������������������������: y=ke–ctcosωt , c > 0 Solution: y=ke–ctcosωt , c > 0 a) In point P, for t=0 the displacement is 3. Thus, y(0)= ke–ctcos(ω·0)=k, so k=3. 3������−2������ = 0.6 ⟹ ������−2������ = 0.2 ⟹ −2c = ln(0.2) ⟹ 2c = ln5 ⟹ c = ln5 ⟹ ������ = ������. ������ 2 is f=165Hz, and since ω=2πf, we get ω=2π·165= 330π b) ������(������) = ������������−������.������������������������������ (������������������������������)

Worksheet Topic: String Art Made by the student: Mădălina P. Gaudeamus High School, Moldova Teacher: Ludmila Cojocari Nr. Activity Icon used Observations / Demonstration crt. We observe the splitting of the 1. Use the sequence segments into n times in equal function for both Parts. segments ● Sequence(i/n*A + We observe the formation of the (n - i)/n*B, i, 0, n) parabola. ● Sequence(i/n*B + (n - i)/n*C, i, 0, n) 2. Use the function: ● Sequence(Segmen t(Element(L_1, i), Element(L_2, i)), i, 1, n + 1) 3. I placed 3 points ������: ������ → ������, ������(������) = ������������������ + ������������ + ������ belonging to the parabola A (0; 0), B (4; 2), ������ = ������ ������ = ������ C (8; 0),and determine the function. {������������������ + ������������ = ������ ⟺ { ������������������ = −������ ⟺ ������������������ + ������������ = ������ ������������������ + ������������ = ������ ������ = ������ ������ = ������ ������ ������ ⟺ {������ = − ������ ⟺ {������ = − ������ https://www.geogebra.org/m/ ������������ = ������ ������ = ������ mvnqdgfa ������: ������ ⟶ ������, ������(������) = − ������ ������������ + ������ ������ 4. Duble the objet with – We observe the doubling of this string art and we obtain the final Rotate around point with result. 45°.

Worksheet Topic: String Art Student: Dolinschi Nicoleta Gaudeamus High School, Moldova Teacher: Ludmila Cojocari Nr Activity Icon used Observations / Demonstration .cr t. -I observe an automatic instrument running to the right 30 times. 1. -Use the option -I observe that the two segments formed a right angle. –Segment -I observe the splitting of the And build the segments segments into n times in equal [AB] ,[BC] Parts. -Use the sequence function for both I observe the formation of the segments parabola. ● Sequence(i/n*A + (n - i)/n*B, i, 0, n) ● Sequence(i/n*B + (n - i)/n*C, i, 0, n) 2. Use the function: Sequence(Segment(Ele ment(L_1, i), Element(L_2, i)), i, 1, n + 1) 3 Determination of the second degree function, the graph of which is represented in the image (-4;0) (4;0) https://www.geogebra.org/m/mc (0;2) 276bxj 4. Duble the objet with I observe the doubling of this string art and we obtain the final result –Rotate around point with 45°

Worksheet Topic: String art & parabola The student: Katarina Šimić Teacher: Vlatka Hižman-Tržić Nr. Activity Icon used Observations / crt. Demonstration 1. I create a triangle ΑΒC Observe that the sides of and a slider n with value the triangle are separate 0 - 50. in the equal pieces. Use the sequence to separate the sides. L1=Sequence(B+i/n(A- B),i,1,n) L2=Sequence(C+i/n(A- C),i,1,n). 2. Insert an instruction Changing the slider n, Sequence(Segment(Ele observe a shape of ment(L1,i), parabola and how it is Element(L2,n-i),i,1,n) to changing. draw segments which connect points on segment AC and AB. 3. I create three sliders a, Observe that the b, c and insert the parabola has the branch upwards, so the slider a quadratic function must have nagative value ������(������) = ������������������ + ������������ + ������

4. Change the values of the The quadratic function slider to find the most that describes parabola fitting parabola the best is ������(������) = −������. ������������(������ − ������)������ + ������. ������

Worksheet Topic: String Art Student: Sudenaz ONHAL Osman Nuri Hekimoğlu Anatolian High School,Turkey Teacher: Yasemin Çetinkaya Nr. Activity Icon used Observations Demonstration cr t. -I observe an automatic instrument running to the 1. Use the option right 30 times. -I observe that the two segments –Segment And build the formed a right angle. -I segments [AB] ,[BC] observe the splitting of the -Use the sequence segments into n times in function for both equal Parts. segments ● Sequence(i/n*A + (n - I observe the formation of the parabola. i)/n*B, i, 0, n) ● Sequence(i/n*B + (n - i)/n*C, i, 0, n) 2. Use the function: Sequence(Segment(Ele ment(L_1, i), Element(L_2, i)), i, 1, n + 1) 3. Determination of the A(-12,0) F(-2,0) second degree function, ������(������) = ������(������ + ������������)(������ + ������) the graph of which is represented in the E(-4,2) image A (-12;0) ������(−������ + ������������)(−������ + ������) = ������ F (-2;0) E (-4;2) −������ ������ = ������ https://www.geogebra.org/classic/jcc −������ ������(������) = ������ (������ + ������������)(������ + ������) −������ ������(������) = ������ (������������ + ������������������ + ������������) vtr6n 4. Duble the objet with I observe the doubling of this string art and we –Rotate around point with 45° obtain the final result

Worksheet Topic: String Art The student: Frunză Andreea Teacher:Cojocari Ludmila Nr. Activity Icon used Observations / Demonstration crt. -I observed the formation of the 2. -Click on the segment parabola made by segment’s motion. that is placed between and choose the option: -show trace 5. -Do the same action as -I observed the moving of 4 parts of the in rule number 3 so as work and we obtain the final result to colorate our object -Repeteat rule number 4 so as to have a final and complete work,and colorate other 2 parts. 6. I placed 3 points belonging to the parabola A (-6; 0), B (-2; 1), C (0; 4),and determine the function.

Worksheet Icon used Topic: String Art & Curves The student: Mădălina Gușanu Gaudeamus High School, Moldova Teacher: Cojocari Ludmila Nr. Activity 1. A circle was built, then 3 diameters were drawn. Making Strin Art, I made the product. The coordinates of 3 points belonging to the graph of the function obtained were determined. A(0;1) B(1,5;2) C(3;1) 2. The system was solved, after which the function that determines the graph of the function was obtained. ������(0) = 1 ������ = 1 ������(3) = 1 ⟺ 9������ + 3������ = 0 ⟺ ������ + 3 = 3 9 2 ������ + 1 2 ������ (2) = 2 {4 { ������ = 1 ⟺ 9������ + 3������ = 0 93 {4 ������ + 2 ������ = 1 ������ = 1 ⟺ 9������ + 3������ = 0 9������ + 6������ = 4 { ������ = 1 ⟺ ������ = 1 ⟺ 4 3������ = 4 ������ = 3 9������ + 3������ = 0 4 { {������ = − 9 4 4 ������: ������ → ������, ������(������) = − 9 ������2 + 3 ������ + 1 3. Conclusions:  The graph of the function represents a parabola with the branches oriented downwards;  The function has 2 zeros;  The function is strictly increasing on (−∝; 3) 2  Functions is strictly decreasing on (3 ; +∝) 2

Worksheet Topic: Graphics in real life The student: Elif ONHAL Teacher: Yasemin Çetinkaya Nr. Activity Icon used Observations crt. / 1. I placed an image with a Demonstratio tourist place placed on the n axis I placed three points: You can Ayasofya Mosque,İstanbul observe the three points 2. A=(-5,6,-4,4) placed on the B=(6,4,-4,4) axis that will C=(0,0) help to calculate and Ayasofya Mosque,İstanbul define the function. 3. Calculations: ������: ������ → ������, ������(������) = ������x2 I obtained the f(0)= 0, f(-5.6)= -4,4 definition of f(6,4)= -4,4 the function => f:R->R ,f (x)= -0,85 x2 and by placing this function we obtained the parabola corresponding to the image. You can see the parabola that was formed after this step.

Worksheet Topic: Art with functions The student: Duzinchevici Mariela Gaudeamus High School, Moldova Teacher: Cojocari Ludmila Nr. Activity Icon used Observations / crt. Demonstration 1. I programmed the first parabola and a vector. I made the second parabola and programmed it to be translated with the movements of the arrow. 2. I let the second parabola trace it’s movements and rotated the arrow to test it. 3. Final result: I learned how to make a heart with basic functions and parabolas.

Worksheet Topic: Graphics in real life The student: Mădălina P. Gaudeamus High School, Moldova Teacher: Ludmila Cojocari Nr. Activity Icon used Observations / crt. Demonstration 1. I place the appropriate image. Observe the image with the bridge placed on the axis. 2. I place three points: C (0.0), D (- I notice the three points 6, -6), E (6, -6). placed on the axis that will help to calculate 3. Calculating: and define the function. We obtained the definition of the function and by placing this function we obtained the parabola corresponding to the image. 4. I enter the function and the And we get the final name of the bridge. result.

Worksheet Topic: Graphics in real life The student: Petcu Sofia Teacher: Cojocari Ludmila Nr. Activity Icon used Observations crt. These 4 dots 1. I selected a cute image of a rainbow are the main to represent the parabola source for I picked 4 points to place on the calculation image C(0;0,8); We can D(-1,6;0); observe that the E(1,6;0) function we have 2. Calculating the function: calculated, is defenating the ������: ������ → ������, ������(������) = ������������2 + ������������ + ������ image. ������(0) = 4 4 The graph of 5 ������ = 5 the function is a 8 parabola with ������ (− 5) = 0 ⟺ 64 8 4 the branches 25 ������ − 5 ������ + 5 = 0 facing 8 64 8 4 downwards. { ������ (5) = 0 {25 ������ + 5 ������ + 5 = 0 The peak of 4 ⟺ ������ 4 parabola C (0; ������ = 5 { ������ = 5 0.8) is the 64 128 8 0 maximum point. {25 25 ������ = − 5 5 84 = − 16 ������ + 5 ������ + 5 = ������ = 0 ������: ������ ⟶ ������, ������(������) = − ������ ������������ + ������ ������������ ������

Worksheet Topic: Graphics in real life The student: Dolinschi Nicoleta Gaudeamus High School, Moldova Teacher: Ludmila Cojocari Nr. Activity Icon used Observations / crt. Demonstration 1. I place the appropriate image. Observe the image with the bridge placed on the axis. 2. I place three points: C (0,0), I notice the three D (-6, -14), E (6, -14). points placed on the axis that will 3. Calculating: help to calculate and define the ������: ������ → ������, ������(������) = ������������������ + ������������ + ������ function. ������ = ������ We obtained the {������������������ − ������������ = −������������ definition of the ������������������ + ������������ = −������������ function and by placing this ������ = ������ function we ⇔ { ������������������ = ������������ − ������������ obtained the parabola ������������ − ������������ + ������������ = −������������ corresponding to ������ = ������ the image. ⇔ { ������ = ������ ������������������ = ������������ − ������������ ������ = ������ ������ = ������ ������ = ������ ������ = ������ ⟺{ ⟺{ ������ = ������ ⟺ { ������ = ������ ������������ ������ ������������������ = −������������ ������ = − ������������ ������ = − ������������ ������: ������ → ������, ������(������) = − ������ ������������ ������������

Worksheet Topic: Graphics in real life The student: Duzinchevici Mariela Gaudeamus High School, Moldova Teacher: Cojocari Ludmila Nr. Activity Icon used Observations / crt. Demonstration 1. I placed an image with a You can monument placed on observe the the axis three points I placed two points: placed on the C(0;0), D(3;0,5). axis that will help to calculate 2. Calculations: and define the ������: ������ → ������, ������(������) = ������������ + ������ function. ������(������) = ������ ������ = ������ I obtained the definition of the {������(������) = ������ ⟺ {������������ = ������ function and by ������ ������ placing this function we ������ = ������ obtained the line ������ corresponding to ⟺ {������ = ������ the image. ������: ������ → ������, ������(������) = ������ ������ You can see the ������ line that was formed after this 3. I put the function and a step. bit of introduction of the monument. ������: ������������������������ ������(������) This is the final result.

Worksheet Topic: Graphics in real life The student: Parthena Lazaridou Teacher: Yolanda Christopoulou Nr. Activity Icon used Observations / crt. Demonstration 1. I create a triangle ΑΒC, a Observe that the slider n with value 0 - 30. sides of the triangle Use the sequence to are separate in the separate the sides equal pieces Sequence(i/n*A + (n - i)/n*B, i, 0, n) When change the Sequence(i/n*B + (n - i)/n*C, i, 0, n) slider I observe the Sequence(i/n*C + (n - i)/n*B, i, 0, n) formation of the 2. Insert the instruction parabola. Sequence(Segment(Ele ment(L_1, i), Element(L_2, i)), i, 1, n + 1) For different sides 3. I change the top of the The parabola is triangle up to the axis create 4. I create three sliders a, b, c Observe that the and insert the function y = parabola has the ax^2 + bx + c branches upwards, so the slider a have 5. Change the value of the to take nagative slider a to the negative value The branches of the parabola is down 6. Change the values of the The parabola is slider to find the right y = -0.1x^2 + 3 parabola

Worksheet Topic: Graphics in real life The student: Efe ONHAL Teacher: Yasemin Çetinkaya Nr. Activity Icon used Observations / crt. Demonstration 1. I placed an image with a You can observe tourist place placed on the three points the axis I placed three placed on the points: axis that will help A=(0,0) to calculate and B=(1,- 3) define the function. 2 C=(-1,- 3) 2 Galata Tower, İstanbul I obtained the 2. definition of the function and by Galata Tower, İstanbul placing this function we 3. Calculations: ������: ������ → ������, obtained the parabola ������(������) = ������x2 corresponding to the image. f(0)= 0, f(1)= - 3 , f(-1)= - 3 2 2 You can see the => f:R->R ,f (x)= - 3 x2 parabola that was formed after this step. 2

Applications of functions in real life Robert Firescu 10C Grade - 2022 Teacher Mihaela Git VOLCANIC ERUPTIONS Due to a volcanic eruption, 18 ������������3 of ash reached the air. Knowing that every day the amount of ash in the atmosphere decreases by 4%: a) establish the function that shows the decrease in the amount of ash and draw a graph; b) determine the amount of ash two weeks after the eruption; c) calculate how many days after the eruption the amount of ash has reached half. The modelation use the exponential function: N (x) = ������0������������������, where ������0 = 18 and ������ = 0,04 ⟹ ������ (������): [������; ������������] → ������, ������ (������) = ������������ ∙ ������0,04������ a) The amount of ash follows an exponential decrease and N(x) = ������0������������������, where ������0 = 18 and ������ = – 0,04. ������(������): [������; ������������] → ������, ������(������) = ������������ ∙ ������− 0,04������ b) To find out the amount of ash 14 days after the eruption, calculate N (14): ������ (14) = 18������−0.04∙ 14 = 18 ∙ ������−0.56 = 10.58 ������������3 c) Solve the equation ������(������) = 9 ⟹ 18 ∙ ������−0,04 ∙������ = 9 ⟹ ������−0,04 ∙������ = 0,5 ⟹ −0,04 ∙ ������ ≈ − 0,69314 ������ ≈ ������������, ������������

Applications of functions in real life Florentina Stancu -10C Grade-2022 Teacher Mihaela Git Biology A mammal’s surface area S (in cm2) can be approximated by the model S=k·m2/3 where m is the mass –in grams- of the mammal and k is a constant. The value of k for some mammals are shown below. Mammal Sheep Rabbit Horse Human Monkey Bat k 8.4 9.75 10 11 11.8 57. 5 a) Approximete the surface area of a rabbit that has a mass of 3.4 kilograms. b) Define the function that expresses the dependency of the surface area by mass for a horse. Solution: a) The surface area for a rabbit is: ������ = ������. ������������ ∙ ���√��� ������������, where m=3.4 kg ⟹ ������ =22.6 cm2 b) ������: (0; ∞) → (0; ∞), ������(������) = ������������ ∙ ���√��� ������������ , for a horse ������: (0; ∞) → (0; ∞), ������ = ������������ ∙ ���√��� ������������ , for a human ������: (0; ∞) → (0; ∞), ������ = ������������. ������ ∙ ���√��� ������������ , for a monkey ������: (0; ∞) → (0; ∞), ������ = ������������. ������ ∙ ���√��� ������������ , for a bat Remark. The maximal domain of definition of the function is R.

Applications of functions in real life Daria Teodorescu, Maria Feleaga 10C Grade - 2022 Teacher Mihaela Git Richter's scale In 2021, in Vrancea, a 5.2 magnitude earthquake struck. A year later (in 2022), also there was a 3.5 magnitude earthquake. How many times was the earthquake in 2021 more intense than the one next year? The ratio is needed to find out what is 2021: 5,2 = lg ������1 required ������1, where: ������0 ������2 2022: 3,5 = lg ������2 ������1 =is the intensity of the first earthquake ������0 ������2 =the intensity of the second. lg ������1 – lg ������2 =5,2 – 3,5 ⇒ ������0 ������0 lg(������1 : ������2)= 1,7 ⟹ ������������ ������1 = 1,7 ⟹ ������0 ������0 ������2 101,7 = ������1 ⇒ ������������ ≈ ������������ ⇒ ������������ ≈ ������������������������ ������2 ������������ → The earthquake in 2021 was 50 times more intense than in 2022. f:(0;+∞) → R, f(x) = lg ������ ������0

Population growth Diana Ilie, Raisa Bedau -10C Grade One city has a population of 20000 inhabitants at 1.01.2000. Population growth is expected at a continuous rate of 8% per year. What will be the population of the city in 5/10/15/20/25 years? The exponential function that shapes population growth is: N(x) = ������0������������������, where ������0= 20000 and k=0,08 N(x): [0;20] → ������, N(x) = 20000∙ ������0.08������ N(5) = 20000∙ ������0.4 ; N(10) = 20000∙ ������0.8; N(15) = 20000∙ ������1.2; N(20) = 20000∙ ������1.6; N(25) = 20000∙ ������2;

Exponential Function -The amount of money Iris Luican -10C Grade We have an initial deposit, in a bank of 20000 lei, the interest rate of this deposit fixed by the bank is 4.7% per year. Determine the value of the deposit after 5 years. Find the function that describe the amount of money. First year: 200.000 ∙ 4.7% = 20000 ∙ (1 + 4,7 ) = 20000 ∙ 1,047 = 20940 100 2 Second year: 200.000 ∙ 4.7% = 20000 ∙ (1 + 4,7 = 20000 ∙ 1,0472 = 21924.18 ) 100 3 Third year: 200.000 ∙ 4.7% = 20000 ∙ (1 + 4,7 = 20000 ∙ 1,0473 = 22954.61 ) 100 Fourth year: 200.000 ∙ 4.7% = 20000 ∙ 1,0474 = 24033.48 Fifth year: 200.000 ∙ 4.7% = 20000 ∙ 1,0475 = 25163 ������(������) = ������������������������������ ∙ ������. ������������������������

Radioactive decay Raisa Bedau and Diana Ilie -10C Grade The function ������: [0; 300] → ������, ������(������) = 425������−0,05������ give the amount of a drug, in milligrams, from the blood circuit after x minutes of administration. a) Determine the amount of the drug in a person's body 30/60/90/120 minutes after administration. ������(30) = 425������−1,5 ≈ 95 (mg) ������(60) = 425������−3 ≈ 21 (mg) ������(90) = 425������−4,5 ≈ 4,7 (mg) ������(120) = 425������−6 ≈ 1 (mg)

Applications of functions in real life Daria Teodorescu, Maria Feleaga 10C Grade - 2022 Teacher Mihaela Git Bacteria&Antibiotics In medicine, the exponential function is used when testing an antibiotic. If you put antibiotic on a plate with a certain number of bacteria, one can find out how quickly it destroys those bacteria and in how long. Determine the function that shows how many bacteria kill an antibiotic x minutes after administration, knowing that there are 7640 bacteria on a plaque and that 10 minutes after administration their number has dropped to 6400. a) Find out how many minutes after administration the number of bacteria halves. Make the corresponding graph. b) Find out the number of bacteria 40 minutes after administration. The function has the form: N(x) = ������������ ������������������ , where ������0 = 7640 si N(30) = 6400. Din N(10) = 6400 ⟹ 6400 = 7640 ∙ ������10������ ⟹ ������10������ = 6400 ⟹ k = 1 ∙ ln67644000 ⟹ k=≈ 7640 10 −0,017 deci: N(x): [0;200] → ������, N(x) = ������������������������������−������,������������������������ a) To find out how many minutes the number of bacteria halves is equal 76400������−0,017������ cu 3820, from where x ≈ 40 minutes. b) To find out the number of bacteria 60 minutes after administration, calculateN(60). N(60) = 7420������−0,017 ∙60 = 7420������−1,02 = 2754 bacteria

Applications of functions in real life Andrei Ciocoiu, David Prunescu 10C Grade - 2022 Teacher Mihaela Git Shot Put The shot used in men’s put has a volume of about 905 cubic centimeters. a) Find the radius of the shot (use the formula for the volume of a sphere). b) Find a function that models the radius r of a shot in terms of its volume V. ������ = 4 ������������3 ⟹ ������������������ ������������������������������������������������������������ ������������������ ������������������ ������������������������������������������������������������ ������������������������������������������������: ������ = ���√��� ������������ 3 ������������ Solution: a) ������ = 4 ������������3 ⟹ ������ = 3√3������ ⟹ ������(������) = ������ ���√��� ������������ ⟹ ������(������������������) ≅ ������ ���√��� ������ ∙ ������������������ = ������. 3 4������ ������ ������ ������ ������.������������ b) The function that models the radius r of a shot in terms of its volume V is: ������: [������; +∞) → [������; +∞), ������(������) = ������ ���√��� ������������ ������ ������ Remark. The maximal domain of definition of the function is R. The graph of the function is:

Worksheet Topic: Graphics in real life The student: Elif ONHAL Teacher: Yasemin Çetinkaya Nr. Activity Icon used Observations crt. / 1. I placed an image with a Demonstratio tourist place placed on the n axis I placed three points: You can observe the three points placed on the axis that will help to calculate and define the function. 15 Temmuz Şehitler Bridge,İstanbul 2. A=(-16,8,-6,17) I obtained the B=(7,16,-6,05) definition of C=(0,0) the function and by 15 Temmuz Şehitler Bridge,İstanbul placing this function we 3. Calculations: ������: ������ → ������, obtained the ������(������) = ������x2 parabola f(0)= 0, f(-16,8)= -6,17 corresponding f(7,16)= -6,05 to the image. => f:R->R ,f (x)= -0,05 x2 You can see the parabola that was formed after this step.

FATHER AND CHILD (The power of exponential function) Isabela Gaciu -10C Grade The father complains to the child that the amount of pocket money each week - $ 5 - is too high. The child comes up with the idea of receiving 1 cent on the first day of the month, 2 cents on the second, 4 cents on the third, 8 cents on the next day, and so on until the end of the month. The father agreed. Who was smarter - the father or the child? See. On day 31 alone, the child should receive $ 10,737,418.24. If you look closely, you notice that the power of exponentiality shows its fangs only on the 26th. Graph in Excel Day Power of 2 Sum in $ Sum in $ (Sum in cents) 1 0,010 5500000,000 2 1 0,020 3 2 0,040 4500000,000 4 4 0,080 5 8 0,160 3500000,000 6 16 0,320 7 32 0,640 Sum in $ 2500000,000 8 64 1,280 9 128 2,560 1500000,000 10 256 5,120 11 512 10,240 500000,000 12 1024 20,480 13 2048 40,960 -500000,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 14 4096 81,920 15 8192 163,840 Days in a month 16 16384 327,680 17 32768 655,360 18 65536 1310,720 19 131072 2621,440 20 262144 5242,880 21 524288 10485,760 22 1048576 20971,520 23 2097152 41943,040 24 4194304 83886,080 25 8388608 167772,160 26 16777216 335544,320 27 33554432 671088,640 28 67108864 1342177,280 29 134217728 2684354,560 30 268435456 5368709,120 31 536870912 10737418,240 1073741824

Graph in GeoGebra Coord.Theacher Mihaela Git

Worksheet Topic: Graphics in real life The student: Valentina Galović Teacher: Vlatka Hižman-Tržić Nr. Activity Icon used Observations / crt. Demonstration 1. Choose the picture of I observe many parabola, place it in parabolas in this coordinate system and set it picture, so I choose as a backgroung picture in one. GeoGebra. Create three sliders a, b, c 2. Insert the quadratic function Observe that the ������(������) = ������������������ + ������������ + ������ parabola has the branch upwards, so the slider a must have negative value 3. Change the values of the The quadratic function slider to find the most fitting that describes parabola parabola the best is ������(������) = −������. ������������������������ + ������. ������������ + ������. ������

Worksheet Topic: Graphics in real life The student: Manolis Nazemllaris Teacher: Yolanda Christopoulou Nr. Activity Icon used Observations / crt. Demonstration 1. I place the image in the Observe that the orthogonal coordinate tower that the tower system at the bottom is in the form of a parabole 2. I create three sliders a, b, c Observe that when change the sliders and insert the function we have different y = ax2 + bx +c functions. We will try to find the best options of the funtions 3. I change the value of the Observe that the slider until I take the best option of the funtions funtion is y = -0,3x2 + 0,2

Worksheet Topic: Graphics in real life The student: Duzinchevici Mariela Gaudeamus High School, Moldova Teacher: Cojocari Ludmila Nr. Activity Icon used Observations / crt. Demonstration 1. I placed an image with a I notice the monument placed on the coordinates of the axis. image on the axis. I placed two points: C(0,5), I observe the D(4,13). three points placed on the axis 2. Calculations: that will help to ������: ������ → ������, ������(������) = ������������ + ������ calculate and define the {������������((40))==153 ⟺ {4������ ������ = 5 13 function. + 5 = I obtained the ⟺ {4������������==58 ⟺ {������������ = 5 definition of the = 2 function and by placing this ������: ������ ⟶ ������, ������(������) = ������������ + ������ function we obtained the line 3. I put the function and a bit of corresponding to introduction of the the image. monument. You can see the line that was formed after this step. This is the final result.

Worksheet Topic: Graphics in real life The student: Letícia Martins Teacher: Sónia Ribeiro Nr. Activity Icon used Observations / crt. Demonstration 1. I placed a photo and drew the Observe that the semicircle (Palácio da Pena, tower that the tower Sintra, Portugal). at the bottom is in the form of a semicircle. 2. I set the diameter length of Calculate the the circle. distance between points A and B. A B 3. Circumference of the circle; Observe the d = 4,5 m. perimeter is 14,14 m and the area 15,91 m2.

Worksheet Topic: Graphics in real life The student: Ana Cojocari Gaudeamus High School, Moldova Teacher: Cojocari Ludmila Nr. Activity Icon used crt. 1. I placed the image that shows the trajectory of the ball thrown by the 2 children. I placed 3 points that belong to the graph that defines the trajectory of the ball. C(3;2) D(2;2) E(1;1) 2. Calculations: ������: ������ → ������, ������(������) = ������������������ + ������������ + ������ ������(������) = ������ ������������ + ������������ + ������ = ������ {������(������) = ������ ⟺ {������������ + ������������ + ������ = ������ ⟺ ������(������) = ������ ������ + ������ + ������ = ������ ������ = ������ − ������ − ������ ⟺ {������ − ������������ − ������������ + ������������ + ������ = ������ ⟺ ������ − ������������ − ������������ + ������������ + ������ = ������ ������ = ������ − ������ − ������ ������ = ������ − ������ − ������ ⟺ {−������������ − ������������ = −������ ⟺ {−������������ − ������������ = −������ −������������ − ������������ = −������ −������������ − ������������ = −������ ������ = −������ ������ = ������ − ������ − ������ ������ ⟺ ������ = −������ ⟺ ������ = ������ −������������ + ������ = −������ ������ { {������ = − ������ ������: ������ ⟶ ������, ������(������) = − ������ ������������ + ������ ������ − ������ ������ ������ Remarks: The graph of the trajectory of the ball represents the parabola with the branches oriented downwards. The tip of the parabola is the maximum value of the function. Top of the parable: ������ ������ ������ ������ ������������ = − ������������ = − ������ : (������ ∙ (− ������)) = ������ ∆ ������������ ������ ������������ ������������ = − ������������ = − ������ : (������ ∙ (− ������)) = ������ ������ ������������ ������ (������ ; ������ )

Worksheet Icon used Topic: Graphics in real life The student: Cojocari Ana Gaudeamus High School Teacher: Cojocari Ludmila Nr Activity 1. The drawing was placed in GeoGebra, then the coordinates of 3 points belonging to the function graph were determined. C(3; 2,5) D(2;3,5) E(4; 2) 2. The system was formed from 3 equations, and the coefficients a, b, c were found. ������(������) = ������, ������ ������(������) = ������, ������ ������(������) = ������ { ������������ + ������������ + ������ = ������, ������ ⟺ ������������ + ������������ + ������ = ������, ������ ������������������ + ������������ + ������ = ������ { ������ = ������, ������ − ������������ − ������������ ⟺ ������������ + ������������ + ������, ������ − ������������ − ������������ = ������, ������ ������������������ + ������������ + ������, ������ − ������������ − ������������ = ������ { ������ = ������, ������ − ������������ − ������������ ⟺ −������������ − ������ = ������ ������������ + ������ = −������, ������ { ������ = ������ ������ = ������, ������ − ������������ − ������������ ⟺ ������ ������ ������: ������ ⟶ ������, ������(������) = ������ ������������ − ������ ������ + ������ { ������ = ������ ⟺ ������ = ������ ������ ������ ������ ������ ������ = − ������ {������ = − ������ 3.  The graph of the function represents a parabola with the branches oriented upwards;  The function has no zeros;  The function takes positive values;  The peak is the minimum point;  The function is strictly descending on (−∝; 29);  The function is strcit ascending (29 ; +∝).

Worksheet Topic: Graphics in real life The student: Mădălina G. Gaudeamus High School, Moldova Teacher: Ludmila Cojocari Nr. Activity Icon used Observatio crt. ns 1. I place the appropriate image. I notice the I place three points: three points C (-5,5;0), D (- 0,5;0), E (-3;4). placed on the axis that 2. Calculating: will help to calculate 3. I enter the function and the name of and define the bridge. the function. The graph of the function represents a parabola, with the branches oriented downwards. The peak is the maximum point. And we get the final result.

Worksheet Topic: Graphics in real life The student: Yağmur ONHAL Teacher: Yasemin Çetinkaya Nr. Activity Icon used Observations / crt. Demonstration 1. I placed an image with a You can tourist place placed on the observe the axis I placed three points: three points placed on the 2. A=(-2,8,-2,6) axis that will B=(2,6,-2,5) help to calculate C=(0,0) and define the function. I obtained the definition of the function and by placing this function we obtained the parabola corresponding to the image. 3. Calculations: ������: ������ → ������, You can see the ������(������) = ������x2 parabola that f(0)= 0, f(-2.8)= -2,6 was formed after this step. f(2,6)= -2,5 => f:R->R ,f (x)= - 4,25 x2

Worksheet Topic: Graphics in real life The student: Buştiuc Arina Teacher: Cojocari Ludmila Nr.c Activity Icon used Observations / rt. Demonstration 1. You can observe I placed an image with a tourist the three points place placed on the axis placed on the axis that will help I placed two points: C(-4,-2), to calculate and D(4,-2). define the function. 2. Calculations: I obtained the ������: ������ → ������, ������(������) = ������������ + ������ definition of the function and by placing this function we obtained the line corresponding to the image. You can see the line that was formed after this step.

Worksheet Topic: Graphics in real life The student: Sevginur D-Etiler Etiler Etiler Anatolian High School Teacher: Arzu Çalık Seydim Nr. Activity Icon Used Observations / Demonstration Crt. You can observe 1. I placed an image with the three points a tourist place placed placed on the axis on the axis I placed that will help to three points: calculate and 2. L'oceanografic, Spain define the L'oceanografic, Spain function. A=( -57.96039,-107.505) B=(120.91497,-107.505) I obtained the C=(0,0) definition of the function and by 3. Calculations: ������������: ������������ → ������������, placing this ������������(������������) = ������������x² function we f(0)= 0, f(-57.96039)= - obtained the 107.505 f(120.91497)= -107.505 parabola => f:R->R ,f (x)= - 0.031x² corresponding to the image. You can see the parabola that was formed after this step.

Worksheet Topic: Graphics İn Real Life The student:: Yiğitalp A-Etiler Etiler Anatolioin High School Teacher:Arzu Çalık Seydim Nr. Activity Icon used Observations / crt. Demonstration 1. I placed an image with a You can do tourist place placed on the placed on three axis I placed three points: dots to observe will be the axis and 2. B=(6.53055,-5.29439) calculate to help F(x)=-0.2 x² and C=(0,0) It will be very D=(-6,-6) functional. E=(6,-6) İn me 3. Calculations: ������: ������ → ������, definition functionally f(x)=-0.20 x² and by me f(0)= 0, f(-5.6)= -4,4 f(6,4)= -4,4 => to place f:R->R ,f (x)= -0,85 x² can be used functionally made parabola corresponding image You can see the parabola that was formed after this step.

Worksheet Topic: The circle and the disc in architecture The student: Betül Ş. Teacher: Tuğba Varol Nr. Activity Icon used Observatio crt. ns / Demonstra tion 1. I placed a photo and drew the circle. 2. I set the radius length of the circle. 3. Circumference of the circle; Π.2,125=6,6725

Worksheet Topic: The circle and the disc in architecture The student: İrem Teacher: Tuğba Varol Nr. Activity Icon used crt. 1. I placed a photo and drew a circle. I have determined the radius of the circle. CD=3,9cm 2. The length of the circle; 2.Π.3,9 = 24,49cm

Worksheet Topic: The circle and the disc in architecture The student: Berre Ç. Teacher: Tuğba Varol Nr. Activity Icon used crt. 1. I placed a photo and drew the circle. I have determined the radius length of the circle. ICDI= 1,00399.. 2. The circumference of the circle; 2.Π.1,00399..= 6,308….

WATER STADIUM (The power of exponential function) Isabela Gaciu -10C Grade The best example of exponential growth - hence the idea of the article - I found recently in the book The Economic Singularity, by Calum Chace. Imagine sitting in the top row in a sealed, sealed football stadium at 12:00 noon. A drop of water falls in the middle of the field, and the drop doubles every minute. How much time do you have to leave? When will you realize it's urgent? The answer is phenomenally surprising. After 6 minutes, it barely fills a cup. 45 minutes into the game, 7% had to leave the pitch due to a knee injury. At 12:49, however, the stadium is 100% full of water. In other words, you have 45 minutes in which everything seems to be under control and only 4 minutes in which you realize that it is too late. Coord.Theacher Mihaela Git

Escola Básica dos 2º e 3º Ciclos do Caniço Ano letivo: Code Week (9 a 24 de out)-Math Tavellers 2021/2022 Disciplinas: Matemática, História, Inglês Nome:__________________________________ Nº___ Turma: ___ A Semana Europeia da Programação é uma iniciativa popular que visa levar a programação e a literacia digital a todos de uma forma divertida e atrativa. O nosso projeto eTwinning está a participar nesta iniciativa, por isso vamos fazer a nossa parte. Para isso, disponibilizamos várias tarefas e poderás optar por fazer pelo menos uma delas. Tarefa 1 Abre a pasta que está na equipa da Oficina de Aprendizagem Interdisciplinar com a designação “Referecial Cartesiano Halloween”, e escolhe uma das imagens para reproduzires em papel ou no geogebra (Se optares pelo Geogebra, no final terás que guardar o teu trabalho à semelhança do trabalho anterior, ou seja, vais guardar na conta [email protected] e a senha é pitagoras5. O nome que irás dar ao teu projeto é Cartesiano_Nome do aluno). Tarefa 2 Abre a pasta que está na equipa da Oficina de Aprendizagem Interdisciplinar com a designação “Referecial Cartesiano Dinossauro”, e reproduz um dos dinossauros em papel ou no geogebra (Se optares pelo Geogebra, no final terás que guardar o teu trabalho à semelhança do trabalho anterior, ou seja, vais guardar na conta [email protected] e a senha é pitagoras5. O nome que irás dar ao teu projeto é Cartesiano_Nome do aluno). 1

Tarefa 3 Como sabes, os seixos tinham um papel muito importante na Vida do Homem. Vais criar o “Desenho” dos seguintes seixos em papel no referencial cartesiano. Deves indicar todas as coordenadas necessárias para representar o teu objeto escolhido. Bom trabalho! ☺ 2

Worksheet Topic: Isometries The student: Ana Margarida Cardoso Teacher: Sónia Ribeiro Nr. Activity Icon used Observations / crt. Demonstration 1. I placed a photo of this Observe that in the photograph that an octagon. tower there are an (Mosteiro da Batalha). octagon. 2. I explain the area of an Calculate the octagon. distance between ������ points A and B. ������ = ������ × ������ This is the final 3. Calculations: result. ������ × ������ ������ = ������ × ������, ������������ ������ = ������������, ������������ ������������������

Escola Básica dos 2º e 3º Ciclos do Caniço Ano letivo: Atividades de Janeiro- Math Tavellers 2021/2022 A Geometria Fractal é uma geometria voltada para o estudo dos objetos que não alteram a sua definição, independente de quanto é ampliado. Isto acontece pelo facto de que cada parte da estrutura da figura é semelhante. Sendo assim, se considerarmos a figura inteira ou uma única parte dela, o que é visto é a mesma imagem. 1. Explora o triângulo Sierpinski: http://pixelartmaker.com/offshoot/b7293f1730b21f0 2. Explora e constroi o teu próprio fractal, tira um print da tua imagem e submete na tarefa do teams. https://www.youtube.com/watch?v=sFEYQMrWNHU https://www.youtube.com/watch?v=eQrj2HbL8Kg, geogebra https://www.youtube.com/watch?v=dabLWk6JOKw, ge 3. O que aprenderam com esta atividade? Bom trabalho! ☺ 1

AGRUPAMENTO DE ESCOLAS VIRGÍNIA MOURA WORKSHEET MATHEMATICS – PROJECT ETWINNING “MATH TRAVELLERS” 8TH GRADE School Year 2021/2022 Name: ___________________________________________ No: ______ Class: ______ Date: _____/_____/_____ Graphic Organizers - Polygons Task: - Choose a quadrilateral; - Choose one of the graphs to organize all the features of that quadrilateral; - Present the graph you built to the class. AEVM 1/4 PEBI 03

AEVM 2/4 DIR07


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