week 06 notes

8/9/2019 Week 06 Notes

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an o a r ver. e r verdepositss so a new an s orme ariverside. How is this new fertilesoil to be dividedup ?

Mevius

F LUMEN'\, alluvio

GaIus Lucius Tc u sFIGURE1.

A possible solution is to constructthe boundariesin the deposit bextendingin a straight ine the boundariesbetween the properties n thold land. This solution however caused trouble and could lead tconflicts; imagine if in figure 1 Lucius insisted on this procedure! Ifact Bartolusbecame involved in just such a conflict while on holida


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cons ruc e .Figure 5 shows the manuscript rom which this trnamely from the words ponas duas lineas at the e

perfindunt n puncto .n. in line 18. Interesting ooline 11: ut probatur per x. primi Euclidis whichProposition 10 of book 1 of Euclid's Elements.Euclid shows how "to bisect a given finite straightby applyingProposition9 ('To bisect a given rectilin

G N

H

FIGURE4.

FIGURE2.

FIGURE 3.

with Bartolus's rule, for in all the cases in figures 1, 2 and 3 hereduce his claims.

IFit UMENal Iu vio

Ga ius Luc iu s

FIGURE2.

FIGURE 3.


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c.k*na).?^KXT?l .pftA L?? ttv^ .., t?~fwt??


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.in the classroom

The goals set for the work in the class (three first-yearclasses ofgrammar school, pupils about 11 years old who follow of 3 hours of Latin and 4 hours of mathematicsa week) weri) to demonstrate heimportanceof mathematicsn society;ii) to show how profitable t can be to work togethersolving opiii) to integratedisciplines(in this case Latin andmathematics)

specificallyrelatedto the mathematicscurriculum;iv) to let pupils discover a numberof constructions with ruler a

(in The Netherlandsthis is no longer a subject in the babut it is a very naturalextension of work on symmetryav) to let pupils solve somejuridicalproblemsusing the constructitheyhaddiscovered earlier.

The project was split into a series of three lessons about ruler aconstructionsand a monthlatera series of five lessons (two


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http://www.math.psu.edu/dlittle/java/calculus/area.htmlhttp://www.math.psu.edu/dlittle/java/calculus/area.htmlhttp://www.math.psu.edu/dlittle/java/calculus/area.htmlhttp://www.math.psu.edu/dlittle/java/calculus/area.htmlhttp://www.math.utah.edu/~alfeld/Archimedes/Archimedes.htmlhttp://www.math.utah.edu/~alfeld/Archimedes/Archimedes.htmlhttp://www.math.utah.edu/~alfeld/Archimedes/Archimedes.htmlhttp://www.math.utah.edu/~alfeld/Archimedes/Archimedes.htmlhttp://www.math.utah.edu/~alfeld/Archimedes/Archimedes.htmlhttp://www.math.utah.edu/~alfeld/Archimedes/Archimedes.htmlhttp://polymer.bu.edu/java/java/montepi/montepiapplet.htmlhttp://polymer.bu.edu/java/java/montepi/montepiapplet.htmlhttp://polymer.bu.edu/java/java/montepi/montepiapplet.htmlhttp://polymer.bu.edu/java/java/montepi/montepiapplet.htmlhttp://polymer.bu.edu/java/java/montepi/montepiapplet.htmlhttp://polymer.bu.edu/java/java/montepi/montepiapplet.html


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Navigating through Measurement in Grades 912 Copyright 2005 by the National Council of Teachers of Mathematics, Inc.www.nctm.org. All rights reserved.

Counting on CommensurabilityName _____________________________________

Quest for the Golden RulerPart 1

We know that a segment of length feet can be measured as 10 inches, and a segment of length

meters can be measured as 75 centimeters. The Pythagoreans, a group of early mathematicians (ca. 550

B.C.), supposed that the counting numbers would always suffice for measurement if the ideal units could be

found, as in the above examples.

1. Suppose that you have a blank ruler and a line segment that is as long as your ruler. Also sup-

pose that to measure the segment, you divide your blank ruler into 360 equal units that you will call

jarboos.

a. How long is your line segment, as measured in jarboos? _______

b. Suppose that you have a second line segment that is as long as the ruler. How long is this sec-ond segment if you measure it in jarboos? _______

2. Suppose that you have two different line segments, one as long as your ruler from step 1, and the

other as long as the ruler.

a. Why would it not be convenient for you to measure these two segments in jarboos?

b. What is the smallest number of equal units into which you would need to divide the ruler if youwanted to measure both of these segments with whole numbers of units? ________

3. Suppose that you have two other line segments, one of which is as long as the ruler, and the

other is as long as the ruler.

a. Could you divide the ruler into yet another set of units that would allow you to measure both ofthese segments with whole numbers of those units? _________

b. How would you determine how many units your ruler would have?

You can measure the pairs of line segments in steps 2 and 3 in whole numbers of units as long as you

choose the right units. This result means that the line segments in each pair are commensurable. The

Pythagoreans believed that any two segments are commensurable.

Is this true? Part 2, Thats Irrational, continues the investigation.

1379

482

5893

3798

4

11

5

7

5

6

271

360

3

4

5

6


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What is the mathematical definition/description of pi?=Ratio of circumference of a circle to its diameter

How have people represented pi throughout history?Fractions or Mixed numbers - between 3+10/71 and 3+10/70, using the symbolic

symbol pi, decimal approximations, 22/7.

Why has there been so attention paid to pi throughout history?Since pi is irrational, well never know all the digits of the decimal. There also is no

patterns found in the digits. Challenge to find more digits. Ratio is used so often (so

popular) because of its relation to a circle. Also, possible discovers await if we can

discover about the nature of irrational numbers within the digits.


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Introduction to Radians

In this activity, youll use an eye-catching Sketchpad animation to learn aboutradians and discover an interesting connection to estimating the value of

SKETCH AND INVESTIGATE

1. Download from ANGEL and open page 1 of Radians.gsp.2. Press the Animate Pointsbutton. Point A will travel along the circles radius,

and point Bwill move along the circles circumference.

When point A reaches the circles circumference, press the button again tostop the animation. If your timing is off, you can press the Resetbutton and tryagain.

3. PointsA

andB

move at the same speed. Select the arc traced by pointB

,and measure its length. It should be equal, or nearly so, to the radius ofthe circle.

To construct a radian, you sweep out an angle whose corresponding arc length isequal to the radius of the circle. The angle is defined to be one radian.

4. Open page 2 ofthe sketch. Again, press the Animate Pointsbutton. Thistime, let point Btravel around the entire circumference of the circle. Stopthe animation when point Breturns to Start.

Notice that point A leaves a trace of its path. Each trip that point A makes fromthe center of the circle and back produces a petal.

Q1 How many petals are formed during point Bs journey around thecircumference?

3 and a bit extra

Q2 Let rbe the radius of the circle. For each petal formed, how far does point Btravel?

2 radians

Q3 Based on your answers to Q1 and Q2, how many lengths of radius r(approximately) are traced by point Bas it moves once around thecircumference?

6 radians and a little bit extra r


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Q4 Explain why your answer to Q3 makes sense based on the circumferenceformula, C=2**r.

If there are 6 radians and a bit more in a circle, then take that number and divideit by 2 getting 3 radians and a bit more which is roughly pi

3.14159265358979323846 radians.

5. Press the Resetbutton to return points A and Bto theiroriginal locations.

6. Start the animation again. This time let the animationrun for a while, and watch as point A traces acollection of petals. Stop the animation when point Ahas filled the circle with evenly spaced petals. Keeptrack of how many times point Btravels around thecircle.

Q5 How many petals did point A trace? How many times did point Btravel aroundthe circles circumference?

22 petals and 7 times around the circle

Q6 Based on your answer to Q5, fill in the blanks in the following statement withintegers:

______44____ radii = _____7_____ circumferences

Q7 Put your statement from Q6 in equation form, letting r = radius and writing

circumference as 2**r. Isolate on one side of the equation.

44 radians = 7C44r = 7C; C = 2 pi r44r = 7 (2 pi r)44/14 = pi

Q8 What fraction do you obtain for ? Is this an exact value of ? If not, wheremight the inexactness have occurred?

44/14 = 3.142857, pi = 3.14159265358979323, so no because 7 does not divide

evenly into 22. We have a good approxiamtion but not exact. When looking at the

sketch pad, after 22 petals are created, the points begin to almost trace each other

but it is off a tiny bit.


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Introduction to Radians

In this activity, youll use an eye-catching Sketchpad animation to learn aboutradians and discover an interesting connection to estimating the value of

SKETCH AND INVESTIGATE

1. Download from ANGEL and open page 1 of Radians.gsp.2. Press the Animate Pointsbutton. Point A will travel along the circles radius,

and point Bwill move along the circles circumference.

When point A reaches the circles circumference, press the button again tostop the animation. If your timing is off, you can press the Resetbutton and tryagain.

3. PointsA

andB

move at the same speed. Select the arc traced by pointB

,and measure its length. It should be equal, or nearly so, to the radius ofthe circle.

To construct a radian, you sweep out an angle whose corresponding arc length isequal to the radius of the circle. The angle is defined to be one radian.

4. Open page 2 ofthe sketch. Again, press the Animate Pointsbutton. Thistime, let point Btravel around the entire circumference of the circle. Stopthe animation when point Breturns to Start.

Notice that point A leaves a trace of its path. Each trip that point A makes fromthe center of the circle and back produces a petal.

Q1 How many petals are formed during point Bs journey around thecircumference?

3 and a bit extra

Q2 Let rbe the radius of the circle. For each petal formed, how far does point Btravel?

2 radians

Q3 Based on your answers to Q1 and Q2, how many lengths of radius r(approximately) are traced by point Bas it moves once around thecircumference?

6 radians and a little bit extra r


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Q4 Explain why your answer to Q3 makes sense based on the circumferenceformula, C=2**r.

If there are 6 radians and a bit more in a circle, then take that number and divideit by 2 getting 3 radians and a bit more which is roughly pi

3.14159265358979323846 radians.

5. Press the Resetbutton to return points A and Bto theiroriginal locations.

6. Start the animation again. This time let the animationrun for a while, and watch as point A traces acollection of petals. Stop the animation when point Ahas filled the circle with evenly spaced petals. Keeptrack of how many times point Btravels around thecircle.

Q5 How many petals did point A trace? How many times did point Btravel aroundthe circles circumference?

22 petals and 7 times around the circle

Q6 Based on your answer to Q5, fill in the blanks in the following statement withintegers:

______44____ radii = _____7_____ circumferences

Q7 Put your statement from Q6 in equation form, letting r = radius and writing

circumference as 2**r. Isolate on one side of the equation.

44 radians = 7C44r = 7C; C = 2 pi r44r = 7 (2 pi r)44/14 = pi

Q8 What fraction do you obtain for ? Is this an exact value of ? If not, wheremight the inexactness have occurred?

44/14 = 3.142857, pi = 3.14159265358979323, so no because 7 does not divide

evenly into 22. We have a good approxiamtion but not exact. When looking at the

sketch pad, after 22 petals are created, the points begin to almost trace each other

but it is off a tiny bit.

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