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

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

Dec 19, 2022 393 likes •1.05k views

Successive Approximations as a tool to Measure Distances Hctor Ochoa Grimaldo Lorenza Illanes Daz Rivera Video 1: Definition of the Problem We see ants walking on the curved side of the Puente Atirantado. Video 1: Definition of the

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Successive Approximations as a tool to Measure Distances

Héctor Ochoa Grimaldo Lorenza Illanes Díaz Rivera

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Video 1: Definition of the Problem

› We see ants walking on the curved side of the Puente Atirantado.

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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

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Video 1: Definition of the Problem

› We notice that the ants that are walking are getting smaller and smaller.

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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?

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Video 2. Sheets of Paper

  • Activity. Successive

Approximations.

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7

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8

1 r/2=0.5 0.8660

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9

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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

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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

  • n a smaller problem.
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Video 3. Rope Activity: Formula for Distance.

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Video 1: Definition of the Problem

› We notice that the ants that are walking are getting smaller and smaller.

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How much does the longest rope measure?

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

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¿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.

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Formula for Distance (Video of its construction)

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Formula for Distance (Video of its construction)

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Formula for Distance (Video of its construction)

X y

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Formula for Distance (Video of its construction)

X y (5,6) (9,2)

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Formula for Distance (Video of its construction)

X y (5,6) (9,2)

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Formula for Distance (Video of its construction)

X y (5,6) (9,2) 9

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Formula for Distance (Video of its construction)

X y (5,6) (9,2) 9 5

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Formula for Distance (Video of its construction)

X y (5,6) (9,2) 9 5 9-5=4

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Formula for Distance (Video of its construction)

X y (5,6) (9,2) 9 5 9-5=4

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Formula for Distance (Video of its construction)

X y (5,6) (9,2) 9 5 9-5=4 6

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Formula for Distance (Video of its construction)

X y (5,6) (9,2) 9 5 9-5=4 6 2

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Formula for Distance (Video of its construction)

X y (5,6) (9,2) 9 5 9-5=4 6 2 6-2=4

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Formula for Distance (Video of its construction) Let’s apply the Pythagorean Theorem

X y (5,6) (9,2) 9 5 9-5=4 6 2 6-2=4

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Formula for Distance (Video of its construction) Let’s apply the Pythagorean Theorem

X y (5,6) (9,2) 9 5 9-5=4 6 2 6-2=4 Ley de Pitgoras

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Formula for Distance (Video of its construction) Let’s apply the Pythagorean Theorem

X y (5,6) (9,2) 9 5 9-5=4 6 2 6-2=4 d=6.41

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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.

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

d1 d2 d2 d3 d3 d3 d3

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Video 4. The Scale

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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

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

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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?

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Video 5. The Solution

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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)

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Puente Atirantado

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

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Puente Atirantado

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Puente Atirantado

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(1195, 1930) (750, 0)

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Puente Atirantado

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

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Puente Atirantado

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(707, 965)

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Puente Atirantado

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

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Puente Atirantado

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Puente Atirantado

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d3 d3 d3 d3

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Puente Atirantado

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Puente Atirantado

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d4 d4 d4 d4 d4 d4 d4 d4

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Puente Atirantado

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Puente Atirantado

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d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5 d5

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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.

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

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Now let’s apply the scale.

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Scale: Héctor is 1.70 meters tall

(725, 10) (725, 30) (750, 10) (740, 30)

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Scale: Héctor is 1.70 meters tall

(740 10) (740, 30) (750, 10) (740, 30)

When we move Professor Héctor horizontally, the points change and we can apply the Pythagorean Theorem.

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Scale: Professor Héctor is 1.70 meters tall

(740 10) (740, 30) (750, 10) (740, 30)

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

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

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Teacher Guide Segment

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Puente Atirantado

LORENZA ILLANES DÍAZ RIVERA 63

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Approximate Length of Arch (Excel)

A B C 1 x f(x) =x^3 distance 1 2 0 0 3 1 1 =SQRT(POWER ((A3-A2), 2)+POWER((B3- B2),2)) L= =SU M (C2: C5) 4

LORENZA ILLANES DÍAZ RIVERA 64

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Succesive Approximations as a Tool to Measure Distances

END

Héctor Ochoa Grimaldo Enrique Miguel Arroyo Chavelas Lorenza Illanes Díaz Rivera