I. Add the following integers:
(i) + 84 and + 45
(ii) – 63 and – 23
(iii) -44 and + 35
(iv) +12 and -20
(v) + 2245 and -1013
(vi) -260 and 0
(vii) -57 and -476
(viii) 274 and – 342
(ix) + 145 and +264
(x) – 814 and +415
II. Fill in the blanks:
(i) 15 + 27 = ………..
(ii) 27 + 15 = ………..
(iii) 27 + 0 = ………..
(iv) 0 + 15 = ………..
(v) 7 + ……….. = 0
(vi) (-9) + ……….. = 0
(vii) 17 + (-17) = ………..
(viii) (-7) + ……….. = -5
III. Add the following:
(i) 7 + 9
(ii) (-4) + 9
(iii) 2 + (-2)
(iv) (-7) + (-6)
(v) (-8) + (-4) + (-9)
IV. Evaluate the following:
(i) (+423) + (253)
(ii) (-423) + (+253)
(iii) (+423) + (-253)
(iv) (-423) + (-253)
V. Find the value of the following:
(i) 3 + 4 + (-6) + (-6) + 2
(ii) 11 + 53 + (-40) – 29
(iii) (-98) + (-43) + 69 + 77
(iv) 41 + (-73) + 23 – 85
VI. State whether the following statements are true (T) of false (F).
(i) The sum of two integers can be zero.
(ii) The addition of three distinct integers is zero if one of integer is zero.
(iii) The addition of a negative (-ve) integer and a positive (+ve) integer is always a negative (-ve) integer.
(iv) The addition of an integer and its opposite is zero.
(v) The addition of two negative (-ve) integers is always a positive (+ve) integer.
Subtraction:
I. Subtract the following:
(i) + 9 from + 12 (ii) + 15 from – 21 (iii) – 42 from + 74 (iv) – 10 from + 25 (v) 7 from 15 (vi) +7 from -15 (vii) -7 from +15 (viii) -7 from 15 |
(ix) 346 from 293
(x) -80 from 0 (xi) 0 from+39 (xii) -59 from -100 (xiii) -350 from 200 (xiv) 63 from -63 (xv) 0 from -247 (xvi) -55 from +55 |
II. Fill in the blanks:
(i) (-9) – ……….. = 15 (ii) (-17) – ……….. = -4 (iii) (-9) – ……….. = 3 (iv) -15 – 27 = ……….. (v) 15 – 27 = ……….. |
(vi) -15 – 0 = ………..
(vii) 0 – 27 = ……….. (viii) 0 – (-27) = ……….. (ix) (-8) – ……….. = -3 (x) ……….. – (-5) = +13 |
III. Subtract the following integers:
(i) 8 – 9
(ii) (-5) – 9
(iii) 6 – (-8)
(iv) (-4) – (-6)
(v) (-2) – (-4) – (-6)
(vi) (+18) – (-12) – (+6) – (-9)
IV. Fill in the blanks:
(i) To subtract (- 8) from 16, we add ……………… to 16.
(ii) To subtract 8 from – 16, we add ……………… to – 16.
(iii) The opposite (negative) of a negative integer is a ……………… integer.
(iv) If a and b are two integers then a + b is also ……………… .
]]>Prerequisites
It is assumed that the reader is familiar with the linear algebra concepts of dot products, orthogonal vectors and cross products.
The full contents can be found at my main website here.
The featured image is from https://chortle.ccsu.edu/VectorLessons/vch07/acuteORobtuse.gif.
]]>
How do you visualize empirical data in a way that is honest, comprehensive and comprehensible? One popular method that is commonly used involve the use of bar charts. However, these can give the misleading impression of precision, obscure distribution of the data, not be all that useful for certain distributions and be strongly affected by outliers.
One potential solution to these problems are box plots. They are insensitive to extreme outliers and can be combined with a ton of different kinds of aspects of the data to ensure that the readers knows everything that they need to know. So what are box plots and how are they calculated and created in statistical programs?
Visualizing Samples With Box Plots is an article in the “Points of Significance” series written by Krzywinski and Altman and published in the Nature Methods journal in 2014. This installment focuses on the benefits of using box plots to show empirical data.
Box plots do not require the assumption of any particular data distribution and thus give the reader a lot more information about the sample and observed values than alternatives such as bar charts. Box plots uses medians, quartiles, interquartile range and are thus not sensitive to extreme outliers. Crucially, when using box plots, it is important to mention what sample size was used and if the whiskers are based on the Tukey or Spear style. The former puts the ends of the whiskers at the data point that is maximum 1.5 times the interquartile range, whereas the latter puts the ends at the most extreme data point.
Box plots should not be used when the sample size is under 5, and as always, the bigger the sample size the better in terms of accurately estimating the population parameter. Box plots can also have information about mean and error bars that are commonly used in traditional bar charts. The paper also recommends a box plot software that can accomplish all of the aspects discussed in the paper.
This paper about box plots can be read online or downloaded in PDF format. Cached versions of the online paper can be found here and here. A cached version of the PDF article can be found here.
Also check out other resources in this series about reasoning under uncertainty, error bars, statistical tests and statistical power. Also consider checking out a useful article on using effect sizes and confidence intervals in science. If you prefer, you can check out a full introductory statistics textbook if you need more material or are interested in learning statistics form the ground and up.
Support more articles like this at patreon.com/emilskeptic.
Follow Defending Science on Facebook, Twitter or Instagram for new updates.
Follow Emil Karlsson on Facebook, Twitter or Instagram for broader perspectives.
Math is all around us. While we all have complained at sometime “when are we going to use this?” (or at least heard it from students) the fact is that math how we are able to understand and define the universe around us. Today’s infographic looks at ten mathematical equations from history that have fundamentally changed how we see the world. [VIA]
]]>
As we have seen earlier in our post, 360-degree view at Maths Anxiety some major causes of Maths Anxiety. In this post, we would like to check our or our children’s level of Maths Anxiety. So here is a simple questionnaire:
Put a number from 1 to 5 next to each of these statements according to whether it is…
Never true (Disagree) | 1 |
Sometimes true | 2 |
Usually true | 3 |
Almost always true | 4 |
Always true (Strongly Agree) | 5 |
1. I am afraid to ask questions in maths class. | |
2. When maths starts I get a physical reaction in my body, like a headache. | |
3. I’m not sure I can trust my answers, even on simple problems. | |
4. I’m afraid I won’t be able to keep up with the rest of the class. | |
5. It’s clear to me in maths class, but when I go home it’s like I was never there. | |
6. Being asked to “go to the board” to explain a maths idea in a class – even for maths I am able to do at desk – scares me. | |
7. When I meet students who love maths or do it well, I either think they are little weird or I envy them. | |
8. Maths never seems to stick, and after I learn it or even get a good grade on it, I still don’t think I know it. | |
9. I don’t know how to study for a maths test. | |
10. I understand maths now, but I worry that it’s going to get really difficult soon. |
Now add up all of the numbers : _________
Here are your math anxiety estimates:
Less than 20: Wow! possibly you can get a major in mathematics.
20 – 25: You are not too far to be called a mathematician.
26 – 30 Some maths discomfort.
31 – 40 Quite a bit of fear, anxiety, and discomfort with maths.
41 – 50 Very anxious about maths. Talking about and working on this with your teacher, and may be with another adult you trust will help you a lot.
Above 50: Death by numbers. The Einstein’s quote mentioned above doesn’t apply to you. You are just about paralyzed by maths, yet still alive!
Stay tuned, solutions ‘How to overcome Maths Anxiety’ coming soon!
Image courtesy – Brainy Quote
]]>We all know about our friend – ‘The sinusoid’.
y becomes 0 whenever sin(x) = 0 i.e
Now the form of the travelling sine wave is as follows:
When does the value for y become 0 ? Well, it is when
As you can see this value of x is dependent on the value of time ‘t’, which means as time ticks, the value of x is pushed forward/backward by a .
When the value of , the wave moves forward and when , the wave moves backward.
Here is a slowly moving forward sine wave.
]]>Is it me? Is it the weather? Is it the crystal? Today I am in a black&white state of mind.
Database resource by J Burt, N. Ross, R. Angel & M. Koch.
]]>Instead, it appears in this form:
It seems daunting but the above is the same as the LDE. We can arrive at it by taking and proceeding as follows:
Now, applying chain rule, we obtain that
Now simplifying the above expression, we obtain that:
Plugging in the values of and into the Legendre Differential Equation,
Now if we do some algebra and simplify the trigonometric identities, we will arrive at the following expression for the Legendre Differential Equation:
If we take the solution for the LDE as , then the solution to the LDE in the above form is merely .
]]>In this post we will give some more important kinds of sheaves.
We will start with the twisting sheaf. For this we need the notion of projective space, which we introduced in Projective Geometry. We know that projective space provides us with many advantages, in particular “points at infinity”, but they come at the cost of some new language – for example, we require that our polynomials be homogeneous, which means that every term of such a polynomial must be of the same degree. The zero set of such a polynomial then defines a projective variety.
The definition of the sheaf of regular functions on a projective variety also has some differences compared to that on an affine space. To protect our definition of projective space, we need the numerator and the denominator to always have the same degree. This has the effect that the only regular functions defined everywhere on a projective variety are the constant functions.
The twisting sheaves, written for an integer , are made up of expressions where and are homogeneous polynomials and the degree of is equal to , where is the degree of . We also require for each open set that never be zero on any point of , as in the definition of the regular functions on . The sheaf of regular functions on is then just the twisting sheaf when . Twisting sheaves are isomorphic to the sheaf of regular functions “locally“, i.e. on open sets of the space, but not “globally“.
Twisting sheaves can be thought of as sheaves of modules, with the sheaf of regular functions serving as their “scalars”. More generally, sheaves of modules play an important part in algebraic geometry. In the same way that a ring determines the sheaf of regular functions on the affine scheme , -modules can always give rise to sheaves of -modules on . However, not all sheaves of -modules come from -modules. In the special case that they do, they are referred to as quasi-coherent sheaves. Quasi-coherent sheaves are interesting because we have ways of constructing new modules from old ones, for instance using the tensor product or direct product, hence, we can also construct new sheaves of modules from old ones.
A quasi-coherent sheaf such that is isomorphic to the quasi-coherent sheaf ,i.e. a direct sum of copies of the sheaf of regular functions, is called a locally free sheaf of rank . Locally free sheaves correspond to vector bundles, which we have already discussed in the context of differential geometry and algebraic topology (see Vector Fields, Vector Bundles, and Fiber Bundles). A locally free module of rank is also known as a line bundle. As we have mentioned earlier, a twisting sheaf is locally isomorphic to the sheaf of regular functions, therefore, it is an example of a line bundle.
We now discuss the concept of differentials. As may be inferred from the name, this concept is somewhat related to concepts in calculus, such as tangents. However, in algebraic geometry we want to be able to define things algebraically, as this contributes to the strength of algebraic geometry in relating algebra and geometry. In addition, in algebraic geometry we may consider not only real and complex numbers but also rational numbers or even finite fields, and some of the methods we have developed in calculus may not always be immediately applicable to the case at hand. Therefore we must “redefine” these objects algebraically, even if they are going to be conceptually inspired by the objects we are already familiar with from calculus.
We now give the definition of differentials (which in the context of algebraic geometry are also called Kahler differentials). Given a homomorphism of rings , the module of relative differentials, denoted , to be the free -module generated by the formal symbols , modulo the following relations:
for
for
for
If we have schemes and whose open subsets are given by the spectrum of some ring , and a morphism , we have for each of these open subsets a module of relative differentials which we can “glue together” to form a quasi-coherent sheaf called the sheaf of relative differentials, written using the symbol . If is a point, i.e. it is the spectrum of some field , we simply write .
The sheaf of relative differentials is also known as the cotangent bundle, since it is dual to the tangent bundle. From the cotangent bundle we can form the canonical bundle by taking exterior products. The exterior product is the tensor product of and modulo the relation . The canonical bundle is then the top exterior power of the cotangent bundle, i.e. . It is yet another example of a line bundle.
Line bundles (including the canonical bundle) on curves are closely related to divisors (see Divisors and the Picard Group). In fact, the set of all line bundles on a curve is the same as the Picard group (the group of divisor classes) of . We will not prove this, but we will elaborate a little on the construction that gives the correspondence between line bundles and divisor classes. Since a line bundle is locally isomorphic to the sheaf of regular functions, a section of the line bundle corresponds, at least on some open set to some regular function on that we denote by . Let be a point in . We define the order of vanishing of the section as the order of vanishing of the regular function at .
A rational section of a line bundle is a section of the bundle possibly multiplied by a rational function (which may not necessarily be a function in the set-theoretic sense but merely an expression which is a “fraction” of polynomials). Similar to the case of ordinary sections of the line bundle and regular functions, there is also a correspondence between rational sections and rational functions. We then define the divisor associated to a rational section by
On the other hand, given a divisor , we may obtain a line bundle by associating to the divisor the set of all rational functions with divisor , such that
.
The notation means that when we formally add the divisors and , the resulting sum has coefficients which are all greater than or equal to . We refer to such a divisor as an effective divisor. Thus we have a means of associating divisors to line bundles and vice-versa, and it is a theorem, which we will not prove, that this gives a correspondence between line bundles and divisor classes.
The correspondence between line bundles and divisor classes will allow us to state the Riemann-Roch theorem (once again, without proof, for now) for the case of complex smooth projective curves. Let denote the dimension of the vector space of global sections of the line bundle , with corresponding divisor . We recall that the degree of a divisor is the sum of its coefficients. The genus of a curve roughly gives the “number of holes” of the curve as a space whose points have coordinates that are complex numbers (recall that the complex points of a curve actually form a surface – for example, an elliptic curve is actually a torus, which has genus equal to ). The Riemann-Roch theorem relates all these concepts. Let denote the divisor corresponding to the canonical bundle of the curve , and let be any divisor on . Then the Riemann-Roch theorem is the following statement:
More on the Riemann-Roch theorem, including its proof, examples of its applications, and generalization to varieties other than curves, will be left to the references for now. It is intended and hoped for, however, that these subjects will be tackled at some later time on this blog.
References:
Kahler Differential on Wikipedia
Sheaves of Modules by Charles Siegel on Rigorous Trivialities
Locally Free Sheaves and Vector Bundles by Charles Siegel on Rigorous Trivialities
Line Bundles and the Picard Group by Charles Siegel on Rigorous Trivialities
Differential Forms and the Canonical Bundle by Charles Siegel on Rigorous Trivialities
Riemann-Roch Theorem for Curves by Charles Siegel on Rigorous Trivialities
Algebraic Geometry by Andreas Gathmann
Algebraic Geometry by Robin Hartshorne
]]>I inked hard straight lines and curves over a softly shaded representation of clouds. The study below portrays the realm of mathematics on the same stage as figurative artistic representation.
]]>1. From a rope 11 m long, two pieces of lengths 13/5 m and 33/10 m are cut off. What is the length of the remaining rope?
2. A drum full of rice weighs 241/6 kg. If the empty drum weighs 55/4 kg, find the weight of rice in the drum.
3. A basket contains three types of fruits weighing 58/3 kg in all. If 73/9 kg of these be apples, 19/6 kg be oranges and the rest pears. What is the weight of the pears in the basket?
4. On one day a rickshaw puller earned $80. Out of his earnings he spent $68/5 on tea and snacks, $51/2 on food and $22/5 on repairs of the rickshaw. How much did he save on that day?
5. Find the cost of 17/5 meters of cloth at $147/4 per meter.
6. A car is moving at an average speed of 202/5 km/hr. How much distance will it cover in 15/2 hours?
7. Find the area of a rectangular park which is 183/5 m long and 50/3 m broad.
8. Find the area of a square plot of land whose each side measures 17/2 meters.
9. One liter of petrol costs $187/4. What is the cost of 35 liters of petrol?
10. An airplane covers 1020 km in an hour. How much distance will it cover in 25/6 hours?
11. The cost of 7/2 meters of cloth is $231/4. What is the cost of one meter of cloth?
12. A cord of length 143/2 m has been cut into 26 pieces of equal length. What is the length of each piece?
13. The area of a room is 261/4 m22. If its breadth is 87/16 meters, what is its length?
14. The product of two rational numbers is 48/5. If one of the rational number is 66/7, find the other rational number.
15. Rita had $300. She spent 1/3 of her money on notebooks and 1/4 of the remainder on stationery items. How much money is left with her?
16. Adrian earns $16000 per month. He spends 1/4 of his income on food; 3/10 of the remainder on house rent and 5/21 of the remainder on the education of children. How much money is still left with him?
Answers for the worksheet on word problems on rational numbers are given below to check the exact answers of the above rational problems.
Answers:
1. 51/10 m
2. 317/12 kg
3. 145/18 kg
4. $73/2
5. $2499/20
6. 303 km
7. 610 m22
8. 289/4 m22
9. $6545/4
10. 4250 km
11. $33/2
12. 11/4 m
13. 12 m
14. 56/55
15. $150
16. $6400
]]>1. Add the following rational numbers:
(i) -5/7 and 3/7
(ii) -15/4 and 7/4
(iii) -8/11 and -4/11
(iv) 6/13 and -9/13
10. Add the following rational numbers:
(i) 3/4 and -3/5
(ii) -3 and 3/5
(iii) -7/27 and 11/18
(iv) 31/-4 and -5/8
11. Verify the following:
(i) (3/4 + -2/5) + -7/10 = 3/4 + (-2/5 + -7/10)
(ii) (-7/11 + 2/-5) + -13/22 = -7/11 + (2/-5 + -13/22)
(iii) -1 + (-2/3 + -3/4) = (-1 + -2/3) + -3/4
12. Subtract the following rational numbers:
(i) 3/4 from 1/3
(ii) -5/6 from 1/3
(iii) -8/9 from -3/5
(iv) -9/7 from -1
(v) -18/11 from 1
(vi) -13/9 from 0
(vii) -32/13 from -6/5
(viii) -7 from -4/7
5. Simplify the expressions:
(i) -3/2 + 5/4 – 7/4
(ii) 5/3 – 7/6 + (-2)/3
(iii) 5/4 – 7/6 – (-2)/3
(iv) -2/5 – (-3)/10 – (-4)/7
(v) 5/6 + (-2)/5 – (-2)/10
(vi) 3/8 – (-2)/9 + (-5)/36
6. What should be added to (2/3 + 3/5) to get -2/15?
7. What should be added to (1/2 + 1/3 + 1/5) to get 3?
8. What should be subtracted from (3/4 – 2/3) to get -1/6?
9. Multiply the rationals:
(i) -5/17 by 51/(-60)
(ii) -6/11 by -55/36
(iii) -8/25 by -5/16
(iv) 6/7 by -49/36
2. Verify each of the following:
(i) 4/17 × (-7)/9 = (-7)/9 × 4/17
(ii) (-8)/11 × 1/5 = 1/5 × (-8)/11
(iii) (-12)/5 × 7/(-36) = 7/-36 × (-12)/5
(iv) -8 × (-13)/12 = (-13)/12 × (-8)
3. Verify each of the following:
(i) (3/5 × 12/13) × 7/18 = 3/5 ×(12/13 × 7/8)
(ii) (-13)/24 × {(-12)/5 × 35/36} = {(-13)/24 × (-12)/5} × 35/36
(iii) {(-9)/5 × (-10)/3} × 21/-4 = (-9)/5 × {(-10)/3 × 21/-4}
4. Fill in the blanks:
(i) (-23)/17 × 18/35 = 18/35 × (_____)
(ii) -38 × (-7)/19 = (-7)/19 × (_____)
(iii) {15/7 × -21/10} × (-5)/6 = (_____) × {(-21)/10 × (-5)/6}
(iv) (-12)/5 × {4/15 × 25/-16} = {(-12)/5 × (4/15) × (_____)
5. Find the multiplicative inverse of:
(i) -11/-15
(ii) -5
(iii) 0/7
(iv) 2/-5
(v) (-1)/8
6. Verify the following rational numbers using the multiplication properties:
(i) 3/7 × {5/6 + 12/13} = (3/7 × 5/6) + (3/7 × 12/13)
(ii) -15/4 × (3/7 + (-12)/5) = (-15/4 × 3/7) + (-15/4 × (-12)/5)
(iii) (-8/3 + -13/12) × 5/6 = (-8/3 × 5/6) + (-13/12 × 5/6)
(iv) -16/7 × {-8/9 + (-7)/6} = (-16/7 × (-8)/9) + ((-16)/7 × -7/6)
7. Name the property of multiplication illustrated by the following statements:
(i) -11/13 × -17/5 = -17/5 × -11/13
(ii) {(-2)/3 × 7/9} × (-9)/5 = (-2)/3 × {7/9 × (-9)/5}
(iii) (-3)/4 × {(-5)/6 + 7/8 = {(-3)/4 × (-5)/6} + {(-3)/4 × 7/8}
(iv) (-16)/9 × 1 = 1 × (-16)/9 = (-16)/9
(v) (-11)/15 × 15/-11 = 15/-11 × (-11)/15 = 1
(vi) -7/5 × 0 = 0
]]>Accessibility is not only confined to aiding differently-abled students with reduced mobility; it also refers to the design of products, devices, and services, which help create a conducive environment for the differently-abled.
Some of these regulations are as follow:
Under the Individuals with Disabilities Education Act (IDEA), publicly funded schools are obligated to properly accommodate differently-abled students, so that they can have an equal level of participation in the educational environment. Introduced in 1975, this act intends to ensure that the differently-abled students get free of cost, public access to the same level of education as others.
Nowadays, schools are advised to incorporate the braille versions of textbooks, helping visually-challenged students to read and understand content in a much better fashion. As digital consumption in school is increasing, all software vendors are expected to incorporate features that would help differently abled students. For instance, eBook apps are expected to be compatible with screen readers. Screen readers are software programs that help visually-impaired students for whom it’s difficult to read text displayed on the computer screen. This software is equipped with a speech synthesizer or braille display. Its interface performs multiple functions, such as locating the cursor, reading a word, line or the whole text on a screen, locating text in a certain color, and so on. Software like JAWS and NVDA are some examples of screen readers that convert text and components of an operating system into synthesized speech. The output is produced in forms of speech. IDEA ensures that an electronic, screen-reader friendly version of a textbook is always available, in case a visually-challenged student is unable to read the physical copy of the same.
Similarly, if a video is shown in a class of hearing-impaired students, IDEA dictates that visual aid like closed captions should be provided for the differently-abled audience.
Since the usage of technology has multiplied with time, various schools are now required to make electronic information technology (EIT) accessible to every student. As per the Web Content Accessibility Guideline (WCAG) 2.0, following are the attributes that educational technology companies should know about to implement accessibility in their content and products:
Here is your today’s H.W
]]>MCQ
Section B
A number is increased by 5 and the resulting number is 15. Find the number.
If 11 is subtracted from 4 times a number, the result is 89. Find the number.
Dear learner,
Here is today’s homework.
h-w-on-01-03-17
Complete pg. no. 62,63 in I.T .Tools book , pg.no. 24,25 in Art & craft book and submit it on Monday.
With regards,
Preeti Lashkari
]]>h-w-on-01-03-17
Complete pg. no. 62,63 in I.T .Tools book , pg.no. 24,25 in Art & craft book and submit it on Monday.
With regards,
Charu Soni
]]>If you want to learn how to use a compass and map, try looking at this webpage or watching the video below:
]]>
While studying computer science at Stanford University in the 1980s, Hastings said there was an exercise by computer scientist Andrew Tanenbaum in which he had to work out the bandwidth of a station wagon carrying tapes across the US. “It turns out that’s a very high-speed network,” Hastings said, speaking at a Mobile World Congress session in Barcelona. “From that original exercise, it made me think we can build Netflix first on DVD and then eventually the internet would catch up with the postal system and pass it.”
The concept, of sending electronically information physically, is known as a sneakernet. This is how Tanenbaum and co-writer David Wetherall described the problem in their book Computer Analysis (fifth edition, pdf):
One of the most common ways to transport data from one computer to another is to write them onto magnetic tape or removable media (e.g., recordable DVDs), physically transport the tape or disks to the destination machine, and read them back in again. Although this method is not as sophisticated as using a geosynchronous communication satellite, it is often more cost effective, especially for applications in which high bandwidth or cost per bit transported is the key factor.
A simple calculation will make this point clear. An industry-standard Ultrium tape can hold 800 gigabytes. A box 60 × 60 × 60 cm can hold about 1000 of these tapes, for a total capacity of 800 terabytes, or 6400 terabits (6.4 petabits). A box of tapes can be delivered anywhere in the United States in 24 hours by Federal Express and other companies. The effective bandwidth of this transmission is 6400 terabits/86,400 sec, or a bit over 70 Gbps. If the destination is only an hour away by road, the bandwidth is increased to over 1700 Gbps. No computer network can even approach this. Of course, networks are getting faster, but tape densities are increasing, too.
If we now look at cost, we get a similar picture. The cost of an Ultrium tape is around $40 when bought in bulk. A tape can be reused at least 10 times, so the tape cost is maybe $4000 per box per usage. Add to this another $1000 for shipping (probably much less), and we have a cost of roughly $5000 to ship 800 TB. This amounts to shipping a gigabyte for a little over half a cent. No network can beat that. The moral of the story is:
Never underestimate the bandwidth of a station wagon full of tapes hurtling down the highway.
Hastings had an embarrassingly large late fee in the mid-Nineties and became intrigued by the possibility of video rentals by mail, which would have been too costly with VHS tapes—but now the DVD had arrived and was starting to take off. Hastings thought about how his math problem could be applied to video-rental delivery.
“When a friend told me about DVDs and I realized, well that’s 5GB of data, and you know you can mail that very inexpensively, I realized that is a digital-distribution network,” Hastings said.
He and Netflix co-founder Marc Randolph reportedly went down to Logos, a record store in Santa Cruz, California, and bought a used CD to test it out. They picked up a blue greeting-card envelope from a gift shop on Pacific Avenue and went to the nearest post office, where they mailed the CD to Hastings’s house using a single first-class stamp, Randolph told Silicon Valley Business Journal. The next day, Hastings went to see Randolph with the CD in hand—unbroken.
“That was the moment where the two of us looked at each other and said, ‘This idea just might work,'” Randolph said.
Even in those early days, Hastings and Randolph were already thinking about how the internet could change video. But they knew the shift away from physical rentals wouldn’t happen overnight, and didn’t want to tie themselves to specific mode of a delivery.
“If we were to come out and say, ‘This is all about downloading or streaming,’ and we said that in 1997 and ’98, that would have been equally disastrous,” said Randolph. “We had to come up with a positioning which transcends the medium.” Thus, Netflix branded itself as a place to find movies, and now TV shows, you’ll love, while placing itself at the forefront of digital-video delivery.
]]>The focus of Math Fun, Writing Projects, Volume 1.0 is to provide children with opportunities to express creative thoughts and artistic talents as they demonstrate their understanding and knowledge of such mathematical concepts as number, geometry, and measurement. Many of the projects allow students to not only articulate their knowledge but also their feelings towards their learning of mathematical concepts. Throughout the book, the instructional approach and student activities are motivating and engaging for children of all ages. Children are provided with various opportunities to express their creativity and artistic talents as they engage in writing activities to demonstrate their understanding and knowledge of the content and to reveal their feelings towards the mathematical content. The online interactive assessment provide immediate feedback for both the teacher and for the children.
The four chapters are described below.
One example of the writing activities allows students to describe their feelings towards mathematics by creating a poem.
M
A
T
H
Marvelous
Awesome
Tough
Happy
Most of all, I hope through the use of the ideas found in this book teachers, home school providers, and parents find the beauty of mathematics and instill in children a love for learning and doing math. You can purchase the books in the Math Fun series are available for download on your Mac or iPad through the iTunes iBook Store.
For more information about the Math Fun series, visit my website, Ventura Curriculum and Assessment Solutions.
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