Successive Approximations as a tool to Measure Distances Hctor - PowerPoint PPT Presentation

Successive Approximations as a tool to Measure Distances Héctor Ochoa Grimaldo Lorenza Illanes Díaz Rivera

Video 1: Definition of the Problem › We see ants walking on the curved side of the Puente Atirantado.

Video 1: Definition of the Problem › We see ants walking on the curved side of the Puente Atirantado. › observa una Hormigas que van caminando por el lado curvo del Puente Atirantado

Video 1: Definition of the Problem › We notice that the ants that are walking are getting smaller and smaller.

Video 1: Definition of the Problem › We want to know what you would do to find: › How much does the curved side measure? › How many ants would cover the curved side if they become increasingly smaller by 0.001 each time?

Video 2. Sheets of Paper Activity. Successive Approximations.

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r/2=0.5 1 0.8660 8

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The video will be the answer (There needs to be an animation of this) › Select two squares and say they are the sheets › Leave one as is, and add halves made from the other square: › + + + +… = 2 sheets

The video will be the answer (There needs to be an animation of this) › One of us continues to explain in the video:: › that › These fractions are known as successive approximations. › How could you use this to solve the problem? › Have them respond orally. Tell them we are going to work on a smaller problem.

Video 3. Rope Activity: Formula for Distance.

Video 1: Definition of the Problem › We notice that the ants that are walking are getting smaller and smaller.

How much does the longest rope measure?

How much does the rope measure, from cross to cross?

¿How much does the rope measure, from cross to cross? The scale is 1 cm = 0.5m. Use successive approximations with the formula for distance.

Formula for Distance (Video of its construction)

Formula for Distance (Video of its construction)

Formula for Distance (Video of its construction) y X 0

Formula for Distance (Video of its construction) y (5,6) (9,2) X 0

Formula for Distance (Video of its construction) y (5,6) (9,2) X 0

Formula for Distance (Video of its construction) y (5,6) (9,2) X 0 9

Formula for Distance (Video of its construction) y (5,6) (9,2) X 0 5 9

Formula for Distance (Video of its construction) y (5,6) (9,2) 9-5=4 X 0 5 9

Formula for Distance (Video of its construction) y (5,6) (9,2) 9-5=4 X 0 5 9

Formula for Distance (Video of its construction) y (5,6) (9,2) 6 9-5=4 X 0 5 9

Formula for Distance (Video of its construction) y (5,6) (9,2) 6 2 9-5=4 X 0 5 9

Formula for Distance (Video of its construction) y (5,6) 6-2=4 (9,2) 6 2 9-5=4 X 0 5 9

Formula for Distance (Video of its construction) Let’s apply the Pythagorean Theorem y (5,6) 6-2=4 (9,2) 6 2 9-5=4 X 0 5 9

Formula for Distance (Video of its construction) Let’s apply the Pythagorean Theorem y Ley de Pitgoras (5,6) 6-2=4 (9,2) 6 2 9-5=4 X 0 5 9

Formula for Distance (Video of its construction) Let’s apply the Pythagorean Theorem y (5,6) d=6.41 6-2=4 (9,2) 6 2 9-5=4 X 0 5 9

How much does the rope measure, from cross to cross? The scale is 1 cm = 0.5m. Use successive approximations with the formula for distance.

Video - Doing successive approximations of the rope, using the distance. d1 d3 d2 d2 d3 d3 d3

Video 4. The Scale

Summarizing the video › We know what successive approximations are › We know the formula for distance › We know how to use successive approximations to measure a curved surface › Now we only need to find the scale

Héctor in the Puente Atirantado We know that Héctor is ??? tall. (Héctor, what is your height?)

Problem › We want to know what you would do to find: › How much does the curved side measure? › How many ants would cover the curved side if they become increasingly smaller by 0.001 each time?

Video 5. The Solution

The solution includes: › Turn the Puente Atirantado around › Apply Successive Approximations with the distance formula (Excel, Mathematica) › Apply the scale › Calculate the number of ants (Excel)

Puente Atirantado 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 42

Puente Atirantado 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 43 0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Puente Atirantado 2000 1900 (1195, 1930) 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 (750, 0) 400 300 200 100 0 44 0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Puente Atirantado Formula for distance in EXCEL = SQRT((B2-B3)^2+(C2-C3)^2) d1 45

Puente Atirantado 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 (707, 965) 300 200 100 0 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 46

Puente Atirantado d2 Dtotal = d1+d2 The formula in EXCEL is =SUM(D2:d5) d2 47

Puente Atirantado 48

Puente Atirantado 2000 1900 1800 1700 1600 d3 1500 1400 1300 1200 d3 1100 1000 900 800 700 d3 600 500 400 300 d3 200 100 0 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 49

Puente Atirantado 50

Puente Atirantado d4 d4 d4 d4 d4 d4 d4 d4 51

Puente Atirantado 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 52

Puente Atirantado d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 53

Note for Roberto: See steps 0-6 to see the algorithm that generates the succesive approximations in the Excel document. The next chart is the last graph generated.

The Excel calculation for the measurement of the curved side of the Puente Atirantado is 2075 Units.

Now let’s apply the scale.

Scale: Héctor is 1.70 meters tall (725, 30) (725, 10) (750, 10) (740, 30)

Scale: Héctor is 1.70 meters tall (740, 30) (740 10) (750, 10) (740, 30) When we move Professor Héctor horizontally, the points change and we can apply the Pythagorean Theorem.

Scale: Professor Héctor is 1.70 meters tall (740, 30) (740 10) (750, 10) (740, 30)

According to Excel calculations, the curved side of the Puente Atirantado measures 176 meters.

According to Excel calculations, the curved side of the Puente Atirantado measures 176 meters and 17600 ants are needed.

Teacher Guide Segment

Puente Atirantado LORENZA ILLANES DÍAZ RIVERA 63

Approximate Length of Arch (Excel) A B C 1 x f(x) distance 1 =x^3 2 0 0 3 1 1 =SQRT(POWER L= =SU ((A3-A2), M 2)+POWER((B3- (C2: B2),2)) C5) LORENZA ILLANES DÍAZ RIVERA 64 4

Succesive Approximations as a Tool to Measure Distances END Héctor Ochoa Grimaldo Enrique Miguel Arroyo Chavelas Lorenza Illanes Díaz Rivera