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Art History/Assignments/28 August 2007

PROJECT 2  Point of View: Looking at Art as an Art Historian

After viewing the power point about Looking at Art and discussing what art reviews tell us about an artists work, you will write a review of an exhibition by Richard Serra at the Museum of Modern Art.

Do not read any published reviews of Serra’s work. Remember to capture your thinking in the Reflections portal.Post your review with your name below.

Post by September 14 by class time.

Thoughtful Questions

1. Is it art because of its quality, or the fame of the artist? 2. What is the relationship between art and design? 3. Is it art because he can make up a complicated explanation to explain what he's created? 4. Why is Serra an artist? Why is he considered "great"? 5. If I made a piece of metal art and sent it to MOMA, would be show it? 6. How does an artist become famous? 7. How much thought went into the artwork? 8. Why does it take so long to make the work? 9. Is it original if someone else executes it?

Emma Wilson Richard Serra Review Sept. 9th

Richard Serra’s retrospective at the Museum of Modern Art is a magnificent display of minimalist sculpture. A pioneer for the minimalist movement, Serra has forged (no pun intended) new ways to work with space, time, and steel. With monstrous twisted steel giants standing in both the sculpture garden and a large exhibit space, MoMA has captured the essence of Serra’s vision. A rising artist in the early sixties, Richard Serra was just finding his artistic voice when the minimalist movement was just beginning. Shifting his focus from painting to sculpture, Serra became one of the original minimalist sculptors. Serra began by experimenting with neon, lead, and rubber, and then moved on to working smaller steel pieces, all of which are represented in MoMA’s exhibition. As Serra became more comfortable with the smaller-scaled pieces, he started to test the limits of the palpability of steel. His pieces grew from smaller, welded steel pieces to colossal, forged steel plates, standing over 13 feet tall and at times hundreds of feet long. By manipulating the steel’s curvature and direction, Serra began to manipulate viewers’ perception of size and space, creating a unique perspective for the viewer. Serra uses bare, unpainted steel in keeping with his minimalist style, but this also causes the viewer to focus on the special illusions Serra creates. By displaying Serra’s pieces in a simple white room, the Museum of Modern Art has created a stark contrast between the twisted steel and the sharp right angles of both the museum and New York itself. This contrast is exactly what Serra wished to capture as he worked and lived in New York for most of the 40 years this retrospective is based upon. This exhibition is a must see for any fan of the visually simple but the expressively complex style of minimalist sculpture. “Richard Serra Sculpture: Forty Years” is a fantastic and perfect retrospective of an artistic pioneer.

Anna Konieczny

Richard Serra Sculpture: Forty Years

According to Dictionary.com, the definition of art is “the quality, production, expression, or realm, according to aesthetic principles, of what is beautiful, appealing, or of more than ordinary significance”. Many art lovers, on first glance, would assume that the works of Richard Serra are not art. I, myself, thought that this was the case when I first laid eyes on Serra’s pieces. His works lack the beauty, grace, and most importantly, realism of many of his colleagues. Yet, after listening to Serra explain his pieces, the beauty and grace of his works shown through. Serra, born in 1939, began his life as a painter. Yet, something was missing to him. He says that, I wanted to reduce things to pure process and activity, so I wrote down a list of verbs – to lift, to curl, to roll, to bend, to tie, to curve, to inlay, to splash…And I decided to work those verbs in relationship to material and place and times. This is exactly what Serra does, as he incorporates these specific verbs into his works, some even bearing the same name, such as To Lift (1967) and Bent Pipe Roll (1968). Serra prefers to work with raw materials, such as lead and rubber, and takes every opportunity to search for a new material. While living in New York, during the destruction of one of the piers, Serra took the opportunity to rummage through the trash in order to find materials of use for any of his works. Each piece in this exhibit does not seem as if it is created by the same sculptor. The sixth floor contains smaller pieces involving pipes and small lead plates, whereas the second floor contains thirteen to fourteen feet tall curved pieces of massive steel. Yet, Serra had almost two different aspirations while creating both of these two “categories”. Serra focuses on an art called Process Art (part of Minimalism), defined by Grove Art Online as, “the process of a works creation is presented as its subject”. With his smaller pieces, Serra played with light and space, attempting at making his viewers feel differently as they walk into the room filled up with his pieces versus walking into that same room empty. However, with his larger pieces, Serra creates situations in which the viewer can walk into his work and enter a whole new world. These steel plates may look massive from the outside, but once a viewer walks in, he or she feels enclosed in some parts, bringing back memories in which they have felt captured, yet open in other parts, giving the viewer a sense of freedom and relief. These huge pieces were built exactly for the room in which they are held, to achieve Serra’s goal of changing the space inside the room. Not only does Serra play with the feeling of space, he also attempts to play with shapes and the eye’s perceptions of shapes. For example, in his work Torqued Elipse IV (1998) the eye gives different perceptions of the shape depending on where the viewer is standing inside. One of Serra’s goals in this piece was to defy what was acceptable in the everyday world by not including one right angle. He went to talk to an Aerospace engineer in order to find out if what he dreamed could become a reality. After three years of building models and working with computers, Serra finally built the real thing and defied the ordinary. Serra’s works definitely push boundaries in terms of what is accepted in everyday life. After researching the exhibit, I do believe that what Serra has created is art. However, to understand and truly feel the exhibit, the viewer must interact with his pieces in real life. Then, the viewer will understand the power of Serra.

Bibliography

“Process Art.” Grove Art Online 2006 Oxford University Press. September 13, 2007 < http://www.groveart.com/shared/views/article.html?section=art.069795>.

Richard Serra Sculpture: Forty Years. 2007. The Museum of Modern Art. September 13, 2007. .

Semin, Didier. “Serra, Richard.” Grove Art Online 2007 Oxford University Press. September 13, 2007 .

Want, Christoper. “Minimalism.” Grove Art Online 2007 Oxford University Press. September 13, 2007 .

“Richard Serra Sculpture: Forty Years” A Review

-By Kristen Robinson

Richard Serra, the 68 year old sculptor from San Francisco, has arrived at MOMA to present the full brunt of his work. I was skeptical at first at his ability to really call what he does during this process ‘art’. Bending metal in my mind wasn’t really a big deal; anyone with a few tools in a steel mill can do that. Also, I felt as though Mr. Serra might be using his reputation to build up attention for his artwork so that more of his fame covered for his pieces. I must admit though, somehow the real emotion that he put into his work came through. After doing a bit of research myself on his pieces and looking through his collection, I was stunned not only at the large scale masterpieces that were on display, but also the smaller ideas. One of the pieces that initially made me feel skeptical was Equal (Corner Prop Piece), worked on by Mr. Serra from 1969 to 1970. The more I began to go over the piece, the more I realized that this work was not melded. There were two parts of heavy steel somehow propped up together perfectly, prohibiting them from falling down. That was when the wheels started turning and the thought process in my brain went from, “This cannot be art!” to “How on earth did he do that?” He has quite a few other prop pieces held up simply by a pole or the corners of a room. These pieces seemed to convey a sense acceptance and general interest, but not like the landscape work Richard Serra also has on display. Very few pieces of art in this world give you the ability to interact with the art so you can acutely feel a difference. The words ‘look but don’t touch’ ring true for many contributions to art, but that is not the case here. That would only be one of the many properties that make his collection more appealing. Each piece of artwork has the ability to evoke an emotion with every step you take throughout the process. How Mr. Serra can take four pieces of steel from Intersection II (1992-93) and make you feel a range of emotions from closed in and uncomfortable to carefree and relatively safe in a matter of moments is unreal. Quite a few of the pieces like Band (2006) and Torqued Torus Inversion (2006) are enough to make simply looking at them seem like they go on forever and never end. I could actually see myself tripping over my own feet if someone simply stuck me in the crazy world that must lie in the middle of his piece Sequence (2006). It is incredible and also eye opening to realize that Mr. Serra must have taken a lot of time going through each planning phase of his work. If one of his pieces were simply tilted at an angle that didn’t fit, the illusion would not only be ruined, but so would the safety of the museum’s patrons. So how should we as mere viewers of Richard Serra’s mind interpret his most interesting visions? Should we look at what the art really brings out of us and go from there? These pieces bring to surface not only the magnitude of what he’s created but just how small we are next to such monumental effort. The ability to demonstrate his minimalist movement over a 40 year span and still be able to capture people’s attention is truly a very important gift. Few people are given the magic to inspire others, but seeing as it is Mr. Serra’s goal to continue working, we will continue to have younger artists building off the illusions he has created for many years to come.

"Richard Serra Sculpture: Forty Years". 2007. The Museum of Modern Art. September 13th, 2007. .

Semin, Didier. “Serra, Richard.” Grove Art Online 2007 Oxford University Press. September 13th, 2007

Want, Christoper. “Minimalism.” Grove Art Online 2007 Oxford University Press. September 13th, 2007 .

Richard Serra Sculpture- 40 Years

By Connor McCleskey The Richard Serra Sculpture- 40 Years Exhibit is a retrospective of Serra’s work, currently at the Museum of Modern Art in New York City. In addition to a few sculptures, created just for this exhibit, it contains pieces from all points of Serra’s long career, from little known pieces involving only a pole leaning against a wall, to giant twenty ton sculptures made out of steel. The story of how Serra came to make these pieces is a very interesting one.

Richard Serra was born in San Francisco, California in 1939. He then went to University of California – Berkeley, where he majored in English. After attending Yale’s University of Art, where he was trained as a painter, he had to support himself by working in a steel mill. This had a tremendous effect on his art, as is obvious. His first work was an unusual one. Serra took molten lead and threw it at the walls of his studio, where it cooled. He then went on to experiment with many other industrial objects, such as rubber, lead, steel, concrete, and fiberglass. Over time, Serra came to be at the forefront of a movement called “Process Art” which is a form of Minimalism. Process Art is where you can see every step of the artist’s creation, and the creation itself is as much as a main subject as what the piece is trying to convey. However, unlike many other minimalists, Serra preferred to play around with the sense of space, and made sculptures that would only have their effect when placed in a certain site. Understandably, Richard’s work has been controversial, and he has had several works removed. Tilted Arc, for example is a piece he made for a park in New York. After a long battle, the city said it blocked people walking through the area, and it was taken down by workers, melted and used for scrap.

Part of what makes this exhibit so good is that it comprehensively looks at all of his works, from his widely popular ones to the ones that take a specific person to see. Due to the wide range of pieces that are in the exhibit, the quality of the work varies. Some of the pieces, such as Band or Sequence (both new pieces made for MoMA) are spectacular, and the space altering illusion they cause, are fantastic, while many of the smaller pieces, I find rather pointless. One of his pieces, titled Belts, is made up of piles of belts, lined up on hooks, and one has neon tubing inside, causing it to glow. Compared to some of his other enormous works, a row of belts don’t deserve to be in the same museum as Band. In addition, you can see the meticulous detail Serra puts into his sculptures, some of which take him three years to create. On some of his works, if the sculpture was any other length than what it was, it would ruin the effect. Also, the way the sculptures interact with the room they are in is fantastic. The way Torqued Torus Inversion dominates the room, and completely changes an ordinary white room in to something completely different, is something close to magical.

This exhibit is not for everyone. Nearly everyone can appreciate the awe of seeing a feat of engineering such as Torqued Ellipse II, which is an oval that rotates at it goes higher. The sense that the floor is moving is amazing. However, his smaller pieces, specifically those on the 6th floor, may cause a few blank stares from the more casual visitor, and I admit that I am frequently lost when looking at some of them. But, you owe it to yourself to at least walk through his bigger sculptures, and sense what I am talking about.

Bibliography

"Richard Serra." Answers. 13 Sep 2007 .

"Richard Serra Sculputure-40 Years." MoMA. Museum of Modern Art. 13 Sep 2007 .

(Sorry, the formatting got ridiculously messed up when I put this into the wiki)

Alison Bennett-

The Museum of Modern Art has a new exhibition of Richard Serra’s greatest sculptures from over a span of forty years. Richard Serra is a famous sculptor whose work is on permanent display at one of the greatest modern art museums in the country, The Museum of Modern Art. Though he is a famous sculptor now, he didn’t begin as one. He was born in San Francisco in 1939. He went to the University of California at Santa Barbara to get his B.A. in English Literature after going to the University of California at Berkeley. Then he went on to study painting at Yale University’s School of Art and Architecture. Only upon returning to New York in 1966 did he first start sculpting. This exhibition will contain some of his greatest works using metal slabs and pipes. At first glance, Richard Serra’s sculptures just look like nothing. But as you take deeper looks and think a little harder you will realize that they are beautiful sculptures that have hidden purposes. For example take one of his two huge walk-through pieces in the Sculpture Garden, Intersection II. Seeing this for the first time may send some critical viewers over the edge. Intersection II is made up of four identical slanted steel plates placed parallel to each other. This critical audience might argue that anyone could do that or this isn’t special. Well Serra isn’t just anyone, and he isn’t just doing anything. His intended purpose of this sculpture was to give the viewer a sense of openness and captivity that is truly unbelievable. If you go online to moma.org, The Museum of Modern Art’s website, viewers can see the pieces and listen to audios of Richard Serra talk about his own sculptures. This is where I first came to understand how art critics call these 13 feet high steel plates art. On the second floor of the museum there are three more intricate walk through metal plate sculptures. The sixth floor contains some of his smaller works. The pieces on the sixth floor are smaller pieces of bended metal and pipes. Though you can’t walk through and experience these pieces the same way, they still have their deeper meaning. Richard Serra’s exhibition is a must see for any modern art lover. I encourage the skeptics to give his works a try and to try to understand how truly magnificent his pieces are.

Bibliography:

The Museum of Modern Art. Richard Serra Sculpture: Forty Years. September 13, 2007. http://www.moma.org.

Bibliography:

The Museum of Modern Art. Richard Serra Sculpture: Forty Years. September 13, 2007. http://www.moma.org.

Richard Serra- 40 years Rawson Haverty

The Museum of Modern Art is presenting an exhibition on the sculptures of Richard Serra. Serra, born in San Francisco in 1939, is a minimalist sculptor whose work is appreciated by many. Minimalism is defined as when a work is stripped down to its original features. Minimalism incorporates some aspects of modernism and is usually interpreted as a reaction against Abstract Expressionism. Also, it is often viewed as a connector to post modern practices. Minimalist style includes features such as: cubic and rectangular forms purged of all metaphor, equality of parts, repetition, neutral surfaces, and industrial materials, which lead to an immediate visual impact. Serra also took part in the Process Art Movement which focused more on the process of building the piece and lesser emphasis on the final work. The gathering, collecting, and creativity of the journey of making the piece should be more important than the final product. Serra started off as a painter and received training at Yale University, but after to years in Europe, following his influence of Robert Smitherson, became a sculptor. Serra, a leader in the minimalist movement, uses his specific style to create simple, but immense sculptures. His exterior steel goes through an initial oxidation process, but after 8-10 years, the steel settles to one color that will remain relatively stable over the rest of the pieces life. He uses his sculptures to manipulate and displace space in such an incredible way that it is impossible not to value his innovation. At first, I had a hard time connecting to Serra’s works because I am so used to the idea of art as traditional realism, and I had a negative reaction to Serra’s sculptures. However, after hearing Serra talk about his works, and the thought process behind them, I stated to appreciate the pieces. Serra hopes that his inventiveness will inspire younger artists in the minimalist movement to experiment for themselves and create different types of art. Serra has expanded the boundaries of art, and his works rightfully deserve to be displayed in the Museum of Modern Art.

Bibliography:

The Museum of Modern Art. Richard Serra Sculpture: Forty Years. September 13, 2007. http://www.moma.org.

Wikipedia. Richard Serra. 2007. World Wide Web. 14 Sep. 2007. http://en.wikipedia.org/wiki/Richard_Serra

Elizabeth - The Museum of Modern Art is known for its presentation of works that further define what can be classified as art. The museum's newest exhibition, "Richard Serra Sculptures: Forty Years" does just that. Located on the second floor Contemporary Galleries, this vast array of sculptures (some supported by merely corners) blend the processes of sculpting, mathematics, and defiance of gravity perfectly. It seems unlikely that works thirteen feet high made of cortez steel 4 inches thick would be able to stand by themselves. This intriguing phenomena creates a "vacuum effect" where viewers, as well as art critics like myself, can not withstand to look away. Serra, a man of great engenuity, was surrounded with the geometric architecture of row houses in San Francisco as a child. It is no wonder that a man who grew up in a city whose bridge building ideas were so modern, that they were put down by the society (but were later labeled as a way to internationally recognize San Francisco) would have the desire of modernizing the world, in his own way. (Serra was born two years after the Golden Gate Bridge was constructed.) Artists like Serra are vital to the growth of the art society because they strive for the development of aspiring artists. In one of his interviews, Richard Serra stated that his work is directed to affect young artists and that he challenges them to make art out of objects such as tin cans or even water. With a mind that is always thinking, Richard Serra is a man that has successfully worked his way onto the "hit list" of any art-lover across the globe. Inspiration that reverberates off these sculpted masterpieces can form a new generation of artists who have the power to take art to the next level.

Forty Years of Richard Serra- Yeshwanth Kandimalla

Process art, a major concept that emerged from the evolutionary 1960s and 1970s, places a great deal of importance on the process of the artist’s creation of the work. The idea blends minimalism and its threadbare standards and use of unconventional materials. Richard Serra’s work does fit such a description. The metalworking involved in the creation of his sculptures particularly deserves notice. Serra, in an assessment of his own work, acknowledges that the public see such industrial materials everyday, but, that in a confined space within a museum, these materials take on an entirely new meaning. That’s certainly true of Richard Serra Sculpture: Forty Years, which just finished its run at the Museum of Modern Art. The sheer size of many of Serra’s sculpture greatly diminishes the visual experience when one views them in photographs. Experiencing such pieces by moving around them, seeing all their curvature and metallic color, fosters a nonverbal connection between Serra’s and the viewer’s mind. Serra’s extraordinary combination of mathematical principles and optical illusion open the public’s eyes and force individuals, including me, to realize that Serra’s work belongs in a museum, not a steel mill. In particular, the works in the outdoor sculpture garden have the most evocative presence, given their great contrast with the angular buildings surrounding them. The work on the Second Floor features similar works and offers the perspective of Serra’s work within an interior space. The view within this area provides a spatial and geometric Serra’s work, which is his intention. This area also features his most recent works, including Band, Sequence, and Torqued Torus Inversion. Serra does include the shape of the room in the design of his exhibition. Unfortunately, the portion of the exhibition on the sixth floor seems lacking in such aspects. Serra himself does not discuss these earlier works, which are much smaller dimensionally compared to their younger counterparts. These works also lack the spatial quality that radiates from the large sculptures. Serra’s roots in Minimalism certainly defined those early works, as he had been influenced by the noted artists of the Minimalist heyday of the 50s and 60s. Serra began in painting and moved towards sculpture. Despite some deficiencies, Serra deserves commendation for the originality and the accompanying power and beauty of his work. The level of effort that such sculptures require is not lost on me; regardless of the fact the Serra doesn’t make the sculpture by hand. In fact, Serra should receive further acclaim for his ability to effectively create his vision without total control of the physical production of his work. For forty years, Serra has pushed the boundaries of art and MoMA’s decision to exhibit his work is nothing short of phenomenal. Bibliography

Didier Semin: “Serra, Richard.” Grove Art Online. Oxford University Press, 9/13/2007, http://www.groveart.com/ Serra, Richard. Richard Serra Sculpture: Forty Years. 2007. 9/13/2007. http://www.moma.org

“Process Art.” Grove Art Online. Oxford University Press, 9/13/2007, http://www.groveart.com

Tusa, John. “Interview with Richard Serra.” The John Tusa Interviews. BBC Radio. BBC London. Transcript https://archive.is/20130707021240/www.bbc.co.uk/radio3/johntusainterview/serra_transcript.shtml

title: Art History/Assignments/28 August 2007


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User talk:Hkt 6654

Hello Hkt 6654, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz (evaluate me!, discussion ) 17:26, 27 August 2007 (UTC)

P.S. have fun with the Construction Engineering

How to edit Contents
I am new to Wiki. Pl. help me in editing the Contents Table. I have ample material on constrution equipements, and am interested in using wiki as a platform to give others what I have. I am a lecturer in Civil Engineering teaching this subject since long.

--Hkt 6654 17:54, 27 August 2007 (UTC)

 * Hello, nice to hear this. Could you give more details ? I assume you mean the table here Construction equipment ? And what exactly would you like to change (e.g. I would like to change x to y) ? YOu can also always look here. Erkan Yilmaz (evaluate me!, discussion ) 18:20, 27 August 2007 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Hkt 6654


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User:JWSchmidt/Blog/27 August 2007

Wikiversity has not had a large amount of vandalism.

 * August 2007- 19 blocks
 * July 12 blocks
 * June 39 blocks
 * May 29 blocks
 * April 29 blocks
 * March 9 blocks -Open Proxies completed
 * February 9 blocks
 * January 2007- 19 blocks
 * December 2006- 9 blocks -Open Proxies started
 * November 15 blocks
 * October 5 blocks
 * September 24 blocks
 * August 2006- 17 blocks (half a month only)

Back in April we started discussing the idea of CheckUser policy for Wikiversity. We are asking all Wikiversity custodians to participate at Nominations for CheckUser.

 * After "March -Open Proxies completed" the number of blocked user accounts is higher than before. Is this due to that Wikiversity attracts now the "real" h.ck.rs - seeking a challenge - after introducing Open Proxies ? Or some other reasons (like press news increased, existing users tell their friends about to join WV, ...) ? Erkan Yilmaz (evaluate me!, discussion ) 08:05, 1 September 2007 (UTC)
 * My guess is that one or two persistent vandals discovered Wikiversity at about the same time (April). --JWSchmidt 16:28, 1 September 2007 (UTC)

Support Staff list
This table is from Support staff:

title: User:JWSchmidt/Blog/27 August 2007


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Nonlinear finite elements/Kinematics - spectral decomposition

Spectral decompositions
Many numerical algorithms use spectral decompositions to compute material behavior.

Spectral decompositions of stretch tensors
Infinitesimal line segments in the material and spatial configurations are related by

d\mathbf{x} = \boldsymbol{F}\cdot d\boldsymbol{X} = \boldsymbol{R}\cdot(\boldsymbol{U} \cdot d\boldsymbol{X}) = \boldsymbol{V} \cdot (\boldsymbol{R} \cdot d\boldsymbol{X}) ~. $$ So the sequence of operations may be either considered as a stretch of in the material configuration followed by a rotation or a rotation followed by a stretch.

Also note that

\boldsymbol{V} = \boldsymbol{R}\cdot\boldsymbol{U}\cdot\boldsymbol{R}^T ~. $$

Let the spectral decomposition of $$\boldsymbol{U}$$ be

\boldsymbol{U} = \sum_{i=1}^3 \lambda_i~\boldsymbol{N}_i\otimes\boldsymbol{N}_i $$ and the spectral decomposition of $$\boldsymbol{V}$$ be

\boldsymbol{V} = \sum_{i=1}^3 \hat{\lambda}_i~\mathbf{n}_i\otimes\mathbf{n}_i ~. $$

Then

\boldsymbol{V} = \boldsymbol{R}\cdot\boldsymbol{U}\cdot\boldsymbol{R}^T = \sum_{i=1}^3 \lambda_i~\boldsymbol{R}\cdot(\boldsymbol{N}_i\otimes\boldsymbol{N}_i)\cdot\boldsymbol{R}^T = \sum_{i=1}^3 \lambda_i~(\boldsymbol{R}\cdot\boldsymbol{N}_i)\otimes(\boldsymbol{R}\cdot\boldsymbol{N}_i) $$ Therefore the uniqueness of the spectral decomposition implies that

\lambda_i = \hat{\lambda}_i \quad \text{and} \quad \mathbf{n}_i = \boldsymbol{R}\cdot\boldsymbol{N}_i $$ The left stretch ($$\boldsymbol{V}$$) is also called the spatial stretch tensor while the right stretch ($$\boldsymbol{U}$$) is called the material stretch tensor.

Spectral decompositions of deformation gradient
The deformation gradient is given by

\boldsymbol{F} = \boldsymbol{R}\cdot\boldsymbol{U} $$ In terms of the spectral decomposition of $$\boldsymbol{U}$$ we have

\boldsymbol{F} = \sum_{i=1}^3 \lambda_i~\boldsymbol{R}\cdot(\boldsymbol{N}_i\otimes\boldsymbol{N}_i) = \sum_{i=1}^3 \lambda_i~(\boldsymbol{R}\cdot\boldsymbol{N}_i)\otimes\boldsymbol{N}_i = \sum_{i=1}^3 \lambda_i~\mathbf{n}_i\otimes\boldsymbol{N}_i $$ Therefore the spectral decomposition of $$\boldsymbol{F}$$ can be written as

\boldsymbol{F} = \sum_{i=1}^3 \lambda_i~\mathbf{n}_i\otimes\boldsymbol{N}_i $$

Let us now see what effect the deformation gradient has when it is applied to the eigenvector $$\boldsymbol{N}_i$$.

We have

\boldsymbol{F}\cdot\boldsymbol{N}_i = \boldsymbol{R}\cdot\boldsymbol{U}\cdot\boldsymbol{N}_i = \boldsymbol{R}\cdot\left(\sum_{j=1}^3 \lambda_j~\boldsymbol{N}_j\otimes\boldsymbol{N}_j\right)\cdot \boldsymbol{N}_i $$ From the definition of the dyadic product

(\mathbf{u}\otimes\mathbf{v})\cdot\mathbf{w} = (\mathbf{w}\cdot\mathbf{v})~\mathbf{u} $$ Since the eigenvectors are orthonormal, we have

(\boldsymbol{N}_j\otimes\boldsymbol{N}_j)\cdot\boldsymbol{N}_i = \begin{cases} 0 & \mbox{if}~ i \ne j \\ \boldsymbol{N}_i & \mbox{if}~ i = j   \end{cases} $$ Therefore,

\left(\sum_{j=1}^3 \lambda_j~\boldsymbol{N}_j\otimes\boldsymbol{N}_j\right)\cdot \boldsymbol{N}_i = \lambda_i~\boldsymbol{N}_i \text{no sum on}~i $$ That leads to

\boldsymbol{F}\cdot\boldsymbol{N}_i = \lambda_i~(\boldsymbol{R}\cdot\boldsymbol{N}_i) = \lambda_i~\mathbf{n}_i $$ So the effect of $$\boldsymbol{F}$$ on $$\boldsymbol{N}_i$$ is to stretch the vector by $$\lambda_i$$ and to rotate it to the new orientation $$\mathbf{n}_i$$.

We can also show that

\boldsymbol{F}^{-T}\cdot\boldsymbol{N}_i = \cfrac{1}{\lambda_i}~\mathbf{n}_i ~; \boldsymbol{F}^T\cdot\mathbf{n}_i = \lambda_i~\boldsymbol{N}_i ~; \boldsymbol{F}^{-1}\cdot\mathbf{n}_i = \cfrac{1}{\lambda_i}~\boldsymbol{N}_i $$

Spectral decompositions of strains
Recall that the Lagrangian Green strain and its Eulerian counterpart are defined as

\boldsymbol{E} = \frac{1}{2}~(\boldsymbol{F}^T\cdot\boldsymbol{F} - \boldsymbol{\mathit{1}}) ~; \boldsymbol{e} = \frac{1}{2}~(\boldsymbol{\mathit{1}} - \left(\boldsymbol{F}\cdot\boldsymbol{F}^T\right)^{-1}) $$ Now,

\boldsymbol{F}^T\cdot\boldsymbol{F} = \boldsymbol{U}\cdot\boldsymbol{R}^T\cdot\boldsymbol{R}\cdot\boldsymbol{U} = \boldsymbol{U}^2 ~; \boldsymbol{F}\cdot\boldsymbol{F}^T = \boldsymbol{V}\cdot\boldsymbol{R}\cdot\boldsymbol{R}^T\cdot\boldsymbol{V} = \boldsymbol{V}^2 $$ Therefore we can write

\boldsymbol{E} = \frac{1}{2}~(\boldsymbol{U}^2 - \boldsymbol{\mathit{1}}) ~; \boldsymbol{e} = \frac{1}{2}~(\boldsymbol{\mathit{1}} - \boldsymbol{V}^{-2}) $$ Hence the spectral decompositions of these strain tensors are

\boldsymbol{E} = \sum_{i=1}^3 \frac{1}{2}(\lambda_i^2 - 1)~\boldsymbol{N}_i\otimes\boldsymbol{N}_i ~; \mathbf{e} = \sum_{i=1}^3 \frac{1}{2}\left(1 - \cfrac{1}{\lambda_i^2}\right)~ \mathbf{n}_i\otimes\mathbf{n}_i $$

Generalized strain measures
We can generalize these strain measures by defining strains as

\boldsymbol{E}^{(n)} = \cfrac{1}{n}~(\boldsymbol{U}^n - \boldsymbol{\mathit{1}}) ~; \boldsymbol{e}^{(n)} = \cfrac{1}{n}~(\boldsymbol{\mathit{1}} - \boldsymbol{V}^{-n}) $$ The spectral decomposition is

\boldsymbol{E}^{(n)} = \sum_{i=1}^3 \cfrac{1}{n}(\lambda_i^n - 1)~\boldsymbol{N}_i\otimes\boldsymbol{N}_i ~; \mathbf{e}^{(n)} = \sum_{i=1}^3 \cfrac{1}{n}\left(1 - \cfrac{1}{\lambda_i^n}\right)~ \mathbf{n}_i\otimes\mathbf{n}_i $$ Clearly, the usual Green strains are obtained when $$n=2$$.

Logarithmic strain measure
A strain measure that is commonly used is the logarithmic strain measure. This strain measure is obtained when we have $$n \rightarrow 0$$. Thus

\boldsymbol{E}^{(0)} = \ln(\boldsymbol{U}) ~; \boldsymbol{e}^{(0)} = \ln(\boldsymbol{V}) $$ The spectral decomposition is

\boldsymbol{E}^{(0)} = \sum_{i=1}^3 \ln\lambda_i~\boldsymbol{N}_i\otimes\boldsymbol{N}_i ~; \mathbf{e}^{(0)} = \sum_{i=1}^3 \ln\lambda_i~\mathbf{n}_i\otimes\mathbf{n}_i $$


text: Category:Construction engineering

title: Category:Construction engineering


text:
User:Katr67

See my Wikipedia page.

Sister projects
My username is the same on all of Wikipedia's sister projects:

It's best to contact me on my Wikipedia talk page.

title: User:Katr67


text:
Perpendicular and parallel lines

line q has a slope of -6 according to the slopes of perpendicular lines postulate which lines are perpendicular to line q

as long as neither is vertical. Then a and c are the slopes of the two lines. The lines L and M are perpendicular if and only if the product of their slopes is -1, or if ac = − 1.

The angle times the height of another angle equals the sum of one angle. The perpendicular force is equivalent to the base and also the height of the vertex/reflex angle(s). Perpendicular lines have negative reciprocal slopes. So perpendicular means a 90 degree angle at at least two sides of the symbol. It has to have a line sticking straight up above a horisontal line.

To construct the perpendicular to the line AB through the point P using compass and straightedge, proceed as follows (see Figure 2).

* Step 1 (red): construct a circle with center at P to create points A' and B' on the line AB, which are equidistant from P.   * Step 2 (green): construct circles centered at A' and B', both having a radius greater than A'P. Let Q be the other point of intersection of these two circles. * Step 3 (blue): connect P and Q to construct the desired perpendicular PQ.

To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for triangles QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal.

As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

In Figure 3, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:

* One of the angles in the diagram is a right angle. * One of the orange-shaded angles is congruent to one of the green-shaded angles.As shown in Figure 3, if two lines (a and b) are both perpendicular to a third line (c), all of the angles formed on the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

If lines a and b are parallel, any of the following conclusions leads to all of the others:

* One of the angles in the diagram is a right angle. * One of the orange-shaded angles is congruent to one of the green-shaded angles. * Line 'c' is perpendicular to line 'a'. * Line 'c' is perpendicular to line 'b'.

* Line 'c' is perpendicular to line 'a'. * Line 'c' is perpendicular to line 'b'.

Algebra

In algebra, for any linear equation y=mx + b, the perpendiculars will all have a slope of (-1/m), the opposite reciprocal of the original slope. It is helpful to memorize the slogan "to find the slope of the perpendicular line, flip the fraction and change the sign." Recall that any whole number a is itself over one, and can be written as (a/1)

To find the perpendicular of a given line which also passes through a particular point (x, y), solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b. Calculus

First find the derivative of the function. This will be the slope (m) of any curve at a particular point (x, y). Then, as above, solve the equation y = (-1/m)x + b, substituting in the known values of m, x, and y to solve for b.

In the Unicode character set, the perpendicular sign has the codepoint U+27C2 and is part of the Miscellaneous Mathematical Symbols-A range: ⟂.

Parallel Lines
Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember: Parallel lines always the same distance apart and never touch.

Parallel lines also point in the same direction.

When parallel lines get crossed by another line (which is called a Transversal), you can see that many angles are the same.

These angles can be made into pairs of angles which have special names.

title: Perpendicular and parallel lines


text:
Nonlinear finite elements/Kinematics - volume change and area change

Volume change
Consider an infinitesimal volume element in the reference configuration. If a right-handed orthonormal basis in the reference configuration is $$(\mathbf{E}_1, \mathbf{E}_2, \mathbf{E}_3)$$ then the vectors representing the edges of the element are

d\mathbf{X}_1 = dX_1~\mathbf{E}_1 ~; d\mathbf{X}_2 = dX_2~\mathbf{E}_2 ~; d\mathbf{X}_3 = dX_3~\mathbf{E}_3 $$ The volume of the element is given by

dV = d\mathbf{X}_1\cdot(d\mathbf{X}_2 \times d\mathbf{X}_3) = dX_1~dX_2~dX_3~\mathbf{E}_1\cdot (\mathbf{E}_2\times\mathbf{E}_3) = dX_1~dX_2~dX_3 $$ Upon deformation, these edges go to $$(d\mathbf{x}_1, d\mathbf{x}_2, d\mathbf{x}_3)$$ where

d\mathbf{x}_1 = \boldsymbol{F}\cdot d\mathbf{X}_1 = \cfrac{\partial \mathbf{x}}{\partial \mathbf{X}} \cdot d\mathbf{X}_1~;~ d\mathbf{x}_2 = \boldsymbol{F}\cdot d\mathbf{X}_2 = \cfrac{\partial \mathbf{x}}{\partial \mathbf{X}} \cdot d\mathbf{X}_2~;~ d\mathbf{x}_3 = \boldsymbol{F}\cdot d\mathbf{X}_3 = \cfrac{\partial \mathbf{x}}{\partial \mathbf{X}} \cdot d\mathbf{X}_3 $$ or,

d\mathbf{x}_1 = dX_1~\cfrac{\partial \mathbf{x}}{\partial \mathbf{X}} \cdot \mathbf{E}_1 = dX_1~\cfrac{\partial \mathbf{x}}{\partial X_1} ~; d\mathbf{x}_2 = dX_2~\cfrac{\partial \mathbf{x}}{\partial \mathbf{X}} \cdot \mathbf{E}_2 = dX_2~\cfrac{\partial \mathbf{x}}{\partial X_2} ~; d\mathbf{x}_3 = dX_3~\cfrac{\partial \mathbf{x}}{\partial \mathbf{X}} \cdot \mathbf{E}_3 = dX_3~\cfrac{\partial \mathbf{x}}{\partial X_3} $$ Therefore, the deformed volume is given by

dv = d\mathbf{x}_1 \cdot (d\mathbf{x}_2 \times d\mathbf{x}_3) = dX_1~dX_2~dX_3~\cfrac{\partial \mathbf{x}}{\partial X_1} \cdot \left(       \cfrac{\partial \mathbf{x}}{\partial X_2} \times       \cfrac{\partial \mathbf{x}}{\partial X_3}       \right) $$ Now,

J = \det(\boldsymbol{F}) = \det\left(\frac{\partial \mathbf{x}}{\partial \mathbf{X}}\right) = \cfrac{\partial \mathbf{x}}{\partial X_1} \cdot \left(       \cfrac{\partial \mathbf{x}}{\partial X_2} \times       \cfrac{\partial \mathbf{x}}{\partial X_3}       \right) $$ Hence,

Recall that from conservation of mass we have

\rho_0~dV = \rho~dv $$ Therefore, an alternative form of the conservation of mass is

Distortional component of the deformation gradient
For many materials it is convenient to decompose the deformation gradient in a volumetric part and a distortional part. This is particularly useful when there is no volume change in the material when it deforms - for example in muscles, rubber tires, metal plasticity, etc.

Let us assume that the deformation gradient can be decomposed into a volumetric part and a distortional part, i.e,

\boldsymbol{F} = \boldsymbol{F}_v\cdot\boldsymbol{F}_d $$ Then

J = \det(\boldsymbol{F}) = \det(\boldsymbol{F}_v)~\det(\boldsymbol{F}_d) $$ Since there is no volume change due to the pure distortion, we have

\det(\boldsymbol{F}_d) = 1 $$ If $$\boldsymbol{F}_d = J^{-1/3}~\boldsymbol{F}$$ we have

\det(\boldsymbol{F}_d) = (J^{-1/3})^3~\det(\boldsymbol{F}) = J^{-1}~J = 1 $$ Therefore the distortional component of the deformation gradient is given by

We can use this result to find the distortional components of various strain and deformation tensors. For example, the right Cauchy-Green deformation tensor is given by

\boldsymbol{C} = \boldsymbol{F}^T\cdot\boldsymbol{F} $$ If we define its distortional component as

\boldsymbol{C}_d = \boldsymbol{F}_d^T\cdot\boldsymbol{F}_d $$ we have

Area change - Nanson's formula
Nanson's formula is an important relation that can be used to go from areas in the current configuration to areas in the reference configuration and vice versa.

This formula states that where $$da$$ is an area of a region in the current configuration, $$dA$$ is the same area in the reference configuration, and $$\mathbf{n}$$ is the outward normal to the area element in the current configuration while $$\mathbf{N}$$ is the outward normal in the reference configuration.

Proof: To see how this formula is derived, we start with the oriented area elements in the reference and current configurations:

d\mathbf{A} = dA~\mathbf{N} ~; d\mathbf{a} = da~\mathbf{n} $$ The reference and current volumes of an element are

dV = d\mathbf{L}\cdot d\mathbf{A} ~; dv = d\mathbf{l} \cdot d\mathbf{a} $$ where $$d\mathbf{l} = \boldsymbol{F}\cdot d\mathbf{L}$$.

Therefore,

d\mathbf{l} \cdot d\mathbf{a} = dv = J~dV = J~d\mathbf{L} \cdot d\mathbf{A} $$ or,

\boldsymbol{F}\cdot d\mathbf{L} \cdot d\mathbf{a} = dv = J~dV = J~d\mathbf{L} \cdot d\mathbf{A} $$ or,

d\mathbf{L} \cdot (\boldsymbol{F}^T \cdot d\mathbf{a}) = d\mathbf{L} \cdot (J~ d\mathbf{A})~. $$ So we get

d\mathbf{a} = J~\boldsymbol{F}^{-T} \cdot d\mathbf{A} $$ or,

da~\mathbf{n} = J~dA~\boldsymbol{F}^{-T}\cdot \mathbf{N} $$


text:
User talk:Ken924

{subst:W|Erkan Yilmaz (evaluate me!, discussion ) 23:25, 27 August 2007 (UTC)}}

P.S: I hope someone will answer soon your question about Lagrange's Equation. BTW: please always sign, then people know who aska and can contact you faster, see e.g. here

title: User talk:Ken924


text:
User talk:Tofuta

Hello Tofuta, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz (evaluate me!, discussion ) 23:26, 27 August 2007 (UTC)

P.S: have fun with our topics about mathematics

 * about "Do I login with upper case "T" or lower case "t"?"
 * you can use both as you wish. Erkan Yilmaz (evaluate me!, discussion ) 05:32, 28 August 2007 (UTC)

title: User talk:Tofuta


text:
User:Tofuta

I am new to wikii, and a bit dazzled. However, chaos drives me nuts and its going to take alot of patience and browsing before I begin feeling at home. Thanks for the "welcome" message.

I live in Northern California in the foothills at 2000 ft. I am retired after having worked in the computer business since 1966, almost from the begining. I welcome any question but reserve the right to answer or not.

Regards... tofuta

Oh, almost forgot. I created my account in all lower case and noticed it was stored as "Tofuta" note the upper case first letter. Do I login with upper case "T" or lower case "t"?

Rgds... tofuta.

title: User:Tofuta


text: Template:Mad Max:Narrative Music:Pop Quiz part 2:Answers


text:
Open design

The resources on this page enable learners to access materials related to the subject of ‘Open design’. The materials consist of internal contents; sourced from articles in Wikipedia and external hyperlinks. Open design is a disruptive, multifaceted construct that has been shaped by influential conditions and properties. By categorizing the materials into distinct sections, this resource page explores some of these conditions and properties, in a way that informs the reader and nurtures a better understanding of the subject area. Teachers and those who are familiar in this subject area are actively encouraged to extend this page.

Introduction
This section introduces the subject area by first asking ‘What is Open design?’ According to Wikipedia Open design is the development of physical products, machines and systems through use of publicly shared design information. Open design involves the making of both free and open-source software (FOSS) as well as open-source hardware. The process is generally facilitated by the Internet and often performed without monetary compensation. The goals and philosophy are identical to that of the open-source movement, but are implemented for the development of physical products rather than software. Open design is a form of co-creation, where the final product is designed by the users, rather than an external stakeholder such as a private company.

Movements
This section examines some of the influential movements that has helped shape open design. In late 1998, Dr. Sepehr Kiani (a PhD in mechanical engineering from MIT) realized that designers could benefit from open source policies, and in early 1999 he convinced Dr. Ryan Vallance and Dr. Samir Nayfeh of the potential benefits of open design in machine design applications. Together they established the Open Design Foundation (ODF) as a non-profit corporation, and set out to develop an Open Design Definition. The idea of open design was taken up, either simultaneously or subsequently, by several other groups and individuals. The principles of open design are closely similar to those of open-source hardware design, which emerged in March 1998 when Reinoud Lamberts of the Delft University of Technology proposed on his “Open Design Circuits” website the creation of a hardware design community in the spirit of free software. Ronen Kadushin coined the title "Open Design" in his 2004 Master’s thesis, and the term was later formalized in the 2010 Open Design Manifesto.

Moreover, the open source movement has been an influential factor in shaping open design. Rooted in software, the open source movement is a broad-reaching movement of individuals who support the use of open source licenses. Open source software is made available for anybody to use or modify, as its source code is made available. Some open source software is based on a share-alike principle, whereby users are free to pass on the software subject to the stipulation that any enhancements or changes are just as freely available to the public, while other open source projects may be freely incorporated into any derivative work, open source or proprietary. Open source software promotes learning and understanding through the dissemination of understanding. The main difference between open source and traditional proprietary software is in user and property rights, the conditions of use imposed on the user by the software license, as opposed to differences in the programming code.

Open source software
Open design consists of software and hardware components. This section will review the former and list prominent open source concepts in software relative to open design.

Open source hardware
Open design consists of software and hardware components. This section will review the latter and list prominent open source concepts in hardware relative to open design.

Models of innovation
Open design may use different models to innovate and develop its products. This section lists and reviews some of these prominent open models.

Processes, procedures and guidelines
This section lists and reviews some of the prominent open working processes, procedures and guidelines that are practiced by stakeholders in open design digital platforms, organisations and projects.

Procurement and manufacturing
This section lists and reviews some of the prominent open procurement and manufacturing companies and organisations that are related to open design.

Organizations
This section lists and reviews some of the prominent open organisations that support open source practices relative to open design.

Collectives, communities and projects
This section lists and reviews some of the prominent open collectives, communities and projects that have employed open source practices relative to open design. Some of these collectives and communities have digital repositories (open source files which are freely shared to other community users).

Intellectual property and licensing systems
This section lists and reviews some of the prominent intellectual property and licensing systems employed in open design to allow further dissemination of the open source works.

Discussion questions

 * Could open source designs for vertical farms be created?

title: Open design


text:
Bloom Clock/Keys/Southeastern Pennsylvania/September/White Flowers/DPL

This page is used to help create a graphical list.

 category=White flowered plants category=BCP/SEPA/9 order=descending ordermethod=lastedit


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Bloom Clock/Keys/Southeastern Pennsylvania/September/Blue Flowers/DPL

This page is used to help create a graphical list.

 category=Blue flowered plants category=BCP/SEPA/9 order=descending ordermethod=lastedit


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Bloom Clock/Keys/Southeastern Pennsylvania/September/Yellow Flowers/DPL

This page is used to help create a graphical list.

 category=Yellow flowered plants category=BCP/SEPA/9 order=descending ordermethod=lastedit


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Bloom Clock/Keys/Southeastern Pennsylvania/September/Orange Flowers/DPL

This page is used to help create a graphical list.

 category=Orange flowered plants category=BCP/SEPA/9 order=descending ordermethod=lastedit


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Bloom Clock/Keys/Southeastern Pennsylvania/September/Red Flowers/DPL

This page is used to help create a graphical list.

 category=Red flowered plants category=BCP/SEPA/9 order=descending ordermethod=lastedit


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Bloom Clock/Keys/Southeastern Pennsylvania/September/Pink Flowers/DPL

This page is used to help create a graphical list.

 category=Pink flowered plants category=BCP/SEPA/9 order=descending ordermethod=lastedit


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Bloom Clock/Keys/Southeastern Pennsylvania/September/Purple Flowers/DPL

This page is used to help create a graphical list.

 category=Purple flowered plants category=BCP/SEPA/9 order=descending ordermethod=lastedit


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BCP/Heuchera americana

Heuchera americana

title: BCP/Heuchera americana


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BCP/Heuchera americana/Logs

2008 Logs

 * (no archived logs yet)

Recent Logs

 * SB Johnny-LM 11:17, 7 July 2007 (UTC)
 * SB Johnny-LM 10:46, 25 July 2007 (UTC)
 * SBJ 08:30, 25 August 2007 (UTC)
 * SB Johnny-LM 09:23, 8 September 2007 (UTC)
 * --SB_Johnny | talk 11:04, 29 June 2008 (UTC)
 * SB Johnny-LM 09:52, 20 July 2008 (UTC)

title: BCP/Heuchera americana/Logs


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BCP/Oenothera

Oenothera

title: BCP/Oenothera


text:
BCP/Oenothera/Logs

2007 Logs

 * --SB_Johnny | PA! 11:27, 23 August 2007 (UTC)
 * --Jomegat 13:47, 23 August 2007 (UTC)
 * --Luai lashire 15:41, 26 August 2007 (UTC)
 * --Jomegat 17:17, 31 August 2007 (UTC)
 * --SB_Johnny | talk 10:39, 1 September 2007 (UTC)
 * --Luai lashire 21:52, 1 September 2007 (UTC)
 * --Luai lashire 21:52, 5 September 2007 (UTC)
 * --Luai lashire 22:04, 6 September 2007 (UTC)
 * --Jomegat 16:53, 10 September 2007 (UTC)
 * --Luai lashire 18:00, 15 September 2007 (UTC)
 * --Jomegat 01:59, 17 September 2007 (UTC)
 * --Jomegat 23:05, 21 September 2007 (UTC)
 * --SB_Johnny | talk 08:26, 1 October 2007 (UTC)
 * --Jomegat 04:20, 10 October 2007 (UTC)
 * --Jomegat 16:42, 18 October 2007 (UTC)
 * --SB_Johnny | talk 13:07, 24 October 2007 (UTC)
 * --SB_Johnny | talk 09:41, 2 November 2007 (UTC)
 * Cormaggio talk 15:09, 5 November 2007 (UTC)
 * --Jomegat 00:54, 7 November 2007 (UTC)

2008 Logs

 * Jomegat 14 July, 2008 (from photos)
 * Jomegat 13:08, 18 July 2008 (UTC)
 * Jomegat 17:11, 22 July 2008 (UTC)
 * Jomegat 01:51, 27 July 2008 (UTC)
 * --SB_Johnny | talk 10:54, 29 July 2008 (UTC)
 * Jomegat 22:00, 2 August 2008 (UTC)
 * Jomegat 18:01, 9 August 2008 (UTC)
 * Jomegat 01:13, 17 August 2008 (UTC)

Recent Logs

 * Jomegat 03:13, 21 August 2008 (UTC)
 * Jomegat 12:41, 2 September 2008 (UTC)
 * Jomegat 22:48, 10 September 2008 (UTC)
 * Jomegat 21:18, 30 September 2008 (UTC)

title: BCP/Oenothera/Logs


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BCP/Agrimonia

Agrimonia

title: BCP/Agrimonia


text:
BCP/Agrimonia/Logs

2008 Logs

 * (no archived logs yet)

Recent Logs

 * --SB_Johnny | PA! 10:55, 25 August 2007 (UTC)
 * --SB_Johnny | talk 09:15, 1 September 2007 (UTC)
 * --SB_Johnny | talk 10:30, 29 July 2008 (UTC)
 * --SB_Johnny talk 10:28, 17 August 2008 (UTC)

title: BCP/Agrimonia/Logs


text:
File:Dorothybaez Seduced by the dark side.pdf

Used with the WikiU Film School Lesson on how to format a movie script


text:
User:By MeRSaVi~enwikiversity

www.mersavi.com is my website

title: User:By MeRSaVi~enwikiversity


text:
Introduction to graph theory/Problems 1

Okay... here's my algorithm for determining graph isomorphims.

Let G1 and G2 be graphs. G1 has m,n vertices and edges; G2 has x,y vertices and edges.

First, does m =x ? If yes = they are potentially isomorphims (as they have the same number of vertices) Next, does n = y ? If yes = they are potentially isomorphims (as they have the same number of edges)

Next, compute eatch vertex degree in G1, and G2. (that is, count how many edges are associated with each vertex).

For G1, sort the vetrex degrees in decending order - we'll call this S1 For G2, sort the vertex degrees in decending order - we'll call this S2

If S1 = S2, they could be isomorphic as they are equal in the sense that the vertex weights are the same.

Loop:

Now from G1 remove all vertices with the weight of (m-1). Now from G2 remove all vertices with the weight of (m-1). Now the number of edges and the sorted vertex weights may be different! Update the edge count of all vertices

For G1, sort the vetrex degrees in decending order - we'll call this S3 For G2, sort the vertex degrees in decending order - we'll call this S4 If S3 = S4 they could be isomorphic as they are equal in the sense that the vertex weights are the same. Count the number of edges in G1 and G2; if equal, we're still portentially isomorphic

Go back to Loop: and repeat with (m-2), (m-3) until it is 0.

Copyright August 2007, Andrew Walduck. Permission granted to Wikiversity to licences under the GFDL.

andrew.walduck@gmail.com

I tried to prove this: for I = 0 - trival case for I = 1 - trival case assume its true for I = N, then prove N+1. Consider the graph X1 with n, Y1 with n nodes also - they're isomorphisms Add node P to graph X1, and Q to Y1 (the N+1 part) ( If P == Q then the sorted node weight graphs will be the same, if p != Q then the sorted node weight graphs will be different. 3825CJ2N4QJF1
 * *

title: Introduction to graph theory/Problems 1


text:
Talk:Introduction to graph theory/Problems 1

Hi, I dumped an algorithm in here... As far as I can tell its the correct and fastest way to determine graph isomorphism...

Its not in the neatest form... I've tried it on a couple of examples and it seems to work...

Cheers Andrew Walduck P.S. Oh yeah its my algorithm.... ;-) andrew.walduck@gmail.com 38859X2N4QJCF1


text:
Talk:Technical writing/Edits

Adam
You use 'planning' as a heading twice. What is the difference between the two?

title: Talk:Technical writing/Edits


text:
Talk:History of Scotland

Possible topics
AlistairReece 14:41, 28 August 2007 (UTC)
 * Pre-Celtic Scotland
 * Dalriada
 * The Picts
 * Origins of Christianity in Scotland
 * Wars of Independence
 * Stuarts
 * Reformation
 * Run up to the Union
 * Improvers
 * Clearances
 * Scotland in the Empire

Other topics:
 * Modern Scotland
 * Devolution

I think I'm going to be bold and try and start something on the main page. Berek 22:12, 22 September 2009 (UTC)
 * Yes, please! The Jade Knight (d'viser) 17:55, 28 September 2009 (UTC)

 * Finally added some links to other wikiresources. Berek 22:04, 15 August 2011 (UTC)

A way forward?
Hi all. I'm going to try and develop this further using an approach similar to Open Learn at the Open University. I'm planning a series of activities referring to the main wikipedia page and subsidiary pages and other resources - it's probably easier to see than to explain. I'm not a qualified teacher so I hope you'll bear with me. I'll be grateful for any help or suggestions. In particular, I'd be grateful for advice on whether to link directly to wikipedia pages (which could change over time), or to copy and paste to new pages here on wikiversity, which would be fairly stable. I may well use links at first then create wikiversity pages later on. Berek 09:22, 16 August 2011 (UTC)

title: Talk:History of Scotland


text:
Talk:Nicene Creed (Roman Catholic Theology)

As this resource is dedicated to the theological aspects of the Creed, should we use agree that the original text be the basis for discussion rather than that which is accepted by the Latin churches? The filioque clause comes 1000 years after the original formulation.AlistairReece 14:48, 28 August 2007 (UTC)

I certainly think we should remove the designation in parentheses in the title: "Roman Catholic Theology." Firstly, it's already under the Catholicism heading, so that's assumed. Secondly, the Creed is owned by all three major branches of the Christian Church and interpreted in a similar way by them. (They differ on other points, but not with regard to Trinitarian doctrine.)--gdm 04:18, 19 January 2010 (UTC)


text:
User:Cuchullainn

Conor Neill Professor in Managerial Communication at IESE, University of Navarra IESE

title: User:Cuchullainn


text:
User talk:Adam.fox

Hello,

thank you for participating in Wikiversity. Perhaps you could also sign your posts (see below for a how to):,. Erkan Yilmaz (evaluate me!, discussion ) 15:35, 28 August 2007 (UTC)

Hello Adam.fox, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz (evaluate me!, discussion ) 15:35, 28 August 2007 (UTC)

title: User talk:Adam.fox


text:
User talk:Cuchullainn

Hello Cuchullainn, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

Greetings from Erkan Yilmaz
If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz (evaluate me!, discussion ) 15:37, 28 August 2007 (UTC)

P.S. Did you find already some interesting topics for you?

Greetings from Dionysios
Greetings Cuchullainn and welcome to the Wikiversity School of Rhetoric and Public Speaking! Please be bold, jump right in, and participate in the development of the School. It needs to have much talent applied to it. Thank you in advance for your assistance. -- Dionysios  (talk), Founder of the Wikiversity School Rhetoric and Public Speaking, Date: 2007-10-02 (October 2, 2007) Time:  1905 UTC

title: User talk:Cuchullainn


text:
User talk:By MeRSaVi~enwikiversity

advertisement
Hello "By MeRSaVi", please understand that Wikiversity is not a place for advertising. This does not match our educational mission. As can be seen similar edits were done in Turkish Wikipedia: w:tr:%C3%96zel:Contributions/By_MeRSaVi. Your edit has been shortened now to: "www.mersavi.com is my website" and moved to your user page.

But still we Assume Good Faith - in any case we would love your feeback and until then we give you a welcome message (see below). We do not want to lose someone interested in learning by just such a miscommunication, because you might not be aware of it (yet). Erkan Yilmaz (evaluate me!, discussion ) 16:14, 28 August 2007 (UTC)
 * Also a message was left at Turkish Wikipedia. Erkan Yilmaz (evaluate me!, discussion ) 16:24, 28 August 2007 (UTC)

Welcome
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title: User talk:By MeRSaVi~enwikiversity


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target of serial vandal

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Greek Language/Two-letter Consonants

Like two-letter vowels, the Greek language is also "equipped" with two-letter consonants (Greek: δίψηφα σύμφωνα), in other words, combinations of two consonants having one sound. The two-letter consonants seem to have been introduced to the language to cover the lack of specific phthongs (sounds), like [b],[d] and [g]. Below is a table presenting the two-letter consonants and some instructions about how to pronounce them.

title: Greek Language/Two-letter Consonants


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Primary mathematics/Numbers

Introducing Numbers
Numbers should be introduced as physically as possible. Primary student should understand that numbers are associated to values. For a teacher and a guardian its better to approach to student for creating these values using the physical daily life object by letting them count it. Psychologically this approach would be helpful for them to realizing the daily application of numbers, and let them be more practical to it, but formal figure of number should not be skipped rather it should be taught simultaneously.

Developing a sound concept of numbers
Children typically learn about numbers at a very young age by learning the sequence of words, "one, two, three, four, five" etc. Usually, in chanting this in conjunction with pointing at a set of toys, or mounting a flight of steps, for example. Typically, 'mistakes' are made. Toys or steps are missed or counted twice, or a mistake is made in the chanted sequence. Very often, from these sorts of activities, and from informal matching activities, a child's concept of numbers and counting emerges as their mistakes are corrected. However, here, at the very foundation of numerical concepts, children are often left to 'put it all together' themselves, and some start off on a shaky foundation. Number concepts can be deliberately developed by suitable activities. The first one of these is object matching.

Matching Activities
As opposed to the typical counting activity children are first exposed to, matching sets of objects gives a firm foundation for the concept of number and numerical relationships. It is very important that matching should be a physical activity that children can relate to and build on.

Typical activities would be a toy's tea-party. With a set of (say) four toy characters, each toy has a place to sit. Each toy has a cup, maybe a saucer, a plate etc. Without even mentioning 'four', we can talk with the child about 'the right number' of cups, of plates etc. We can talk about 'too many' or 'not enough'. Here, we are talking about number and important number relations without even mentioning which number we are talking about! Only after a lot of activities of this type should we talk about specific numbers and the idea of number in the abstract.

Numbers and Numerals
Teachers should print these numbers or show the children these numbers. Ideally, the numbers should be handled by the student. There are a number of ways to achieve this: cut out numerals from heavy card stock, shape them with clay together, purchase wooden numerals or give them sandpaper numerals to trace. Simultaneously, show the definitions of these numbers as containers or discrete quantities (using boxes and small balls, e.g. 1 ball, 2 balls, etc. Note that 0 means "no balls"). This should take some time to learn thoroughly (depending on the student).

0 1 2 3 4 5 6 7 8 9

Children should be alerted to the use of numbers in a variety of contexts, prices, measurements of various types (eg. length, weight, volume), the speedometer in a car, labels on products, road signs, dates and time.

At this early stage of numeracy it is also helpful to introduce the idea of the relative size of numbers by asking what comes before or after, or what is a larger or smaller number. For example, a guessing game: the teacher thinks of a number between 1 and 10, the children try to guess the number. After each guess the teacher indicates whether the number is smaller or larger.

Other skills include learning to count to 20 or higher, and counting backwards.

Place Value
The Next step is to learn the place value of numbers.

It is probably true that if you are reading this page you know that after 9 comes 10 (and you usually call it ten) but this would not be true if you would belong to another culture.

Take for example the Maya Culture where there are not the ten symbols above but twenty symbols.

Imagine that instead of using 10 symbols one uses only 2 symbols. For example 0 and 1

Here is how the system will be created:

Or if one uses the symbols A and B one gets:

This may give you enough information to figure the place value idea of any number system.

For example what if you used 3 symbols instead of 2 (say 0,1,2).

If you're into computers, the HEXADECIMAL (Base 16) or Hex for short, number system will be of interest to you. This system uses 4 binary digits at a time to represent numbers from 0 to 15 (decimal). This allows for a more convenient way to express numbers the way computers think - that we can understand. So now we need 16 symbols instead of 2, 3, or 10. So we use 0123456789ABCDEF.

Resources for Early Math

 * 15 Fun Ways and Easy Games for Young Learners Math: Reproducible, Easy-to-Play Learning Games That Help Kids Build Essential Math Skills, by Susan Julio, Scholastic Professional, 2001.
 * Eenie Meenie Miney Math!: Math Play for You and Your Preschooler, by Linda Allison, Little Brown & Co., 1993.
 * Marshmallow Math: Early Math for Toddlers, Preschoolers and Primary School Children, by Trevor Schindeler, Trafford, 2002.
 * Number Wonder: Teaching Basic Math Concepts to Preschoolers, by Deborah Saathoff and Jane Jarrell, Holman Bible, 1999.

title: Primary mathematics/Numbers


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Talk:Primary mathematics/Numbers

Hi,

I am new with this but see that it really has a great potential for teaching and learning and would like to join. Being a Math teacher I am involved with a program that also teaches Basic Math to people who have no idea about Math. I saw the numbers Unit and realized that there could be useful addition to that because the basic numbers 0,1,2...9 are just the basic symbols. The next step is to explain the placement value notation and one can use this opportunity to tie it with different number bases (binary for example).

How do I go by adding a module? Is it reviewed by others before it is being added?

I will appreciate if I could Email to ask specific questions for example:

what does this mean: Place comments that start a new topic at the bottom of the page and give them == A Descriptive Header ==.

Dubimax 15:12, 20 April 2006 (UTC)Dani

 * I'm an ex-maths teacher with experience of 7yr olds to adults. I've been looking for a suitable place to add content. If you are still there, Dubimax, I've taken the liberty of adding to your page. As far as finding our way around, I'll be learning with you. Barry Desborough 18:03, 28 October 2006 (UTC)

Medieval Arabic numerals
This is the new theory about the ideograms created by Abu Ja'far Muhammad ibn Musa al-Khwarizmi (c.778 - c.850).

title: Talk:Primary mathematics/Numbers


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Primary mathematics/Adding numbers

Adding two single digit numbers
When teaching young students how to add you will need to use physical items to help them understand how addition works, use items that are similar to avoid confusing younger children. It is very important that they understand the number system before trying to add or subtract. The following is an example of how to show a child how to add.

I have two blocks, Mary has two blocks, if Mary gives me two blocks how many blocks do I have?

First show that you do in fact have two blocks, then show that Mary (or whatever the name of the person is) also has two blocks. Have the children count the number of blocks you have and the number of blocks Mary has.

Have Mary give you two blocks, now have the children count how many blocks you have. You have four blocks and now Mary has zero blocks. Some children may be able to understand addition better if you teach subtraction at the same time, some may notice that Mary has zero blocks while others may not make this connection, this depends completely on each child.

This block example should be done multiple times until every student understands how to add two single digit numbers together, if some students have trouble understanding the concept have them hold the blocks and do the above block example with each other, help them count the blocks they have.

When the basic concept of adding has been introduced via a physical method children should be encouraged to learn the combinations of single digit numbers - so that addition no longer relies on counting blocks (or fingers). Especially important for mental subtraction is to learn the combinations of numbers that add to give 10.

When these basics have been taught and before moving on to adding 2 digit numbers children must learn place notation (ie that 25=20+5). Once this is learnt children can progress to adding 2 (and more) digit numbers. It needs to be explained that when you have 10 in the units column you must 'carry' that over to be 1 in the 10s column. Note that this means that 25+97 is MUCH harder than 12+13.

Depending on the child it can be useful to do this in binary as there are far fewer number combitations to remember, though this can confuse some children.

Due to place notation addition is usually carried out right to left:

45 +37 === 82 ===  1

With the 1 underneath representing a carried digit.

It is possible to do addition left to right:

45 +37 ==== 70  12 ====  82

(adding the tens first).

Children taught this way will usually work out for themselves the shorter way of writing it out.

title: Primary mathematics/Adding numbers


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Primary mathematics/Subtracting numbers

first 5-2=3 second 40-30=10

45 -32 ===  3  10

then add

45 -32 ===  3  10 ===  13

To get the answer.

When this is grasped move on to harder problems:

52 -35 ====

There are 2 methods that I would like to illustrate, the first uses carrying:

Start by saying that 5 is bigger than 2 so we take 10 from the 50 and add it to the 2 to get 12

5 412 - 3 5 ======

Is the usual way of writing this.

Now subtract 5 from 12 to get 7. This is another 'pair' that needs to be taught - all the numbers up to 10 subtracted from all larger numbers up to 20 are 'required', or it is possible to say that 12-2 is 10 and that 5-2 is 3 and 10-3 is 7, which is how I do it, but is not necessarily the 'easiest' way. Different children will want to do this in different ways and will often develop their own methods for mental subtraction (i.e. when you don't write it down). For instance I do this: 56-38 becomes 50-40+6+2 : note that I have used both the fact that 56=50+6 and that 38=40-2 (this is why I personally feel that every child needs to know the pairs that sum to 10) and that a+b-(c-d) = a-c+b+d although you can 'see' this as the 'big chunk' from 40 to 50 and then the two small end bits added on (6 and 2) without explaining using algebra, again teachers should remember this general truth (that addition can be re-ordered and that minus minus = plus) in order to answer questions and with a view to the child learning it in this form.

To continue:

5 412 - 3 5 ======    7

Then we take 30 from 40 (remember that it is now 40 since we gave the other 10 to the units) to get 10

5 412 - 3 5 ======    7    10

And finally add these two together

5 412 - 3 5 ======    7    10 ======    17

To give the answer.

This is the method I was taught. I was introduced to this second method by being told that a young boy had used it without prompting, proving that young children can (and do) understand negative numbers if told about them:

The same sum:

52 -35 ====

This time we will take 30 from 50 first (working left to right here - much more 'obvious') to get 20

52 -35 ==== 20

Then we will take 5 from 2 to get -3

52 -35 ==== 20   -3

Then we take the 3 from the 20

52 -35 ==== 20   -3 ====  17

To finish the sum. Here students need to know what 20-3 is: This is easiest to look at as 10+10-3 and then know that 10-3 is 7 (again, numbers adding to 10).

title: Primary mathematics/Subtracting numbers


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Primary mathematics/Multiplying numbers

Manipulatives and Models
In the early and middle primary grades, the operation of multiplication is first denoted by the symbol "x", and is normally first taught as repeated addition, where multiplying the numbers "3" and "5" (3 x 5) is the same as adding the number three a total of five times: 3 + 3 + 3 + 3 + 3 = 15, so 3 x 5 = 15. Modern curriculum initially teach this concept with manipulatives and models, which enable students to visualize what multiplication is. For instance, a teacher might model the problem 4 x 6 by putting 4 jelly beans each in 6 bags. Subsequent models become more abstract, but the model that all students eventually learn to understand and use is called an area rectangle:

The above area rectangle models the problem 9 x 4, which equals 36. Note that there are 4 rows and 9 columns and that the height of the rectangle is 4 units and the width (or base) is 9 units. Students find that when they count the 9 unit rows four times, they will count a total of 36 units. Soon, they come to the realization that if they were to count the 4 unit columns nine times, they arrive at the same answer. This becomes one of the ways that they learn the commutative property: a x b = b x a (See:Commutativity)     Eventually, students should memorize all of combinations of multiplying single digit numbers, but it should be noted that just because a student has memorized their multiplication facts does not necessarily conclude that they understand at an abstract level what multiplication represents. Note that the above rectangle has a total of 36 square units, which serves to introduce students to the geometric axiom that area of a rectangle is equal to the base (9 in this case) times the height(4).

By working with area rectangles, students make connections to geometry that serve to strengthen their understanding of multiplication. Eventually students no longer need to see the interior square units and soon come to the understanding that one or both of the lengths of these area rectangles can be expressed as the sum of two lengths. The following area diagram models the problem 16 x 3 :

Because 16 = 10 + 6, students can visualize that the length of the above rectangle is still 16. They then can see that the larger rectangle can be expressed as the sum of the two smaller area rectangles: (3 x 10) + (3 x 6).

In upper elementary grades students are introduced to a more standard mathematical nomenclature where the multiplication symbol "x" is no longer necessary, permitting them to notate an equation for this model as: 3(16) = 3(10) + 3(6). This is the primary model used to teach the distributive property: a(b + c) = a(b) + a(c) (See: Distributivity)

The area model can be used to show the distributive property at an even more complex level. Consider the problem 155x360:

The Use of Different Algorithms to Solve Multiplication Problems
Note that in the above model, the height and width (base) of the larger area rectangle has been broken up into smaller area rectangles based on the place-holder of each digit in each number, making it a representation of the standard algorithm for multiplication:

This standard algorithm is then further modified to make it more efficient.

The Lattice Method
Some students initially find an algorithm known as the lattice method easier to use because it involves the use of lines that guide the eventual values of the final place-holders. It should be noted that students using this algorithm generally take more time to complete a problem. Additionally, this algorithm tends to become confusing for students when it is used with problems involving decimals. For these reasons, the use of the lattice method can become a liability in later grades. It is recommended that as soon as this model is mastered by the student, it becomes advisable to teach and encourage them to use the standard model.

Test Yourself
 { 9×12={ 108_3 }
 * type="{}"}

{ 4×12={ 48_3 }
 * type="{}"}

{ 10×11={ 110_3 }
 * type="{}"}

{ 11×8={ 88_3 }
 * type="{}"}

{ 4×10={ 40_3 }
 * type="{}"}

{ 3×4={ 12_3 }
 * type="{}"}

{ 9×2={ 18_3 }
 * type="{}"}

{ 11×12={ 132_3 }
 * type="{}"}

{ 2×3={ 6_3 }
 * type="{}"}

{ 10×6={ 60_3 }
 * type="{}"}

{ 8×7={ 56_3 }
 * type="{}"}

{ 10×4={ 40_3 }
 * type="{}"}

{ 7×8={ 56_3 }
 * type="{}"}

{ 10×2={ 20_3 }
 * type="{}"}

{ 9×9={ 81_3 }
 * type="{}"}

{ 8×4={ 32_3 }
 * type="{}"}

{ 3×11={ 33_3 }
 * type="{}"}

{ 8×9={ 72_3 }
 * type="{}"}

{ 12×3={ 36_3 }
 * type="{}"}

{ 10×10={ 100_3 }
 * type="{}"}

Other Issues Related to the Teaching of Multiplication

 * The Use of Calculators

title: Primary mathematics/Multiplying numbers


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Primary mathematics/Dividing numbers

When first teaching young children to divide, you may like to try getting a small group together and sharing out a number of objects evenly between them. This will ensure that the students get a clear understanding of the concept of division before moving on to the written process of short and long division.

The process of division should be taught as the reverse of multiplication, again the times tables are needed. The links between addition and subtraction and multiplication and division become important later when the 'opposite' or inverse operation will be needed to solve algebraic problems. Primary teachers can help with later learning by emphasizing that division is the undoing of multiplication, as well as pointing children in the right direction for actually doing division.

The simplest division is where the multiplication is known:

$$\frac{10}{2} = 5$$

(Note that I will write division as fractions - the formating is easier - the two are the same thing, children need to know this; I have met GCSE students, 15 and 16 years old, who were confused about this. The more normal line with dots above and below is usually used for these simple problems)

Harder are problems where the inverse is not known:

$$\frac{98}{2}$$

As 98 is greater than 10×2, children are unlikely to know the answer. This is where we introduce long division, which we write like this: __ 2/98

The first step is then to divide 9 by 2, however 2 does not go into 9. We find the highest number less than 9 into which 2 does go (8) and divide it by 2. The remainder (9-8=1) we add to the units as a ten (we have taken 10 away from 90 to get 80, we then add the 10 to the 8 to get 18). Having done this we divide 18 by 2 to get 9. This is often set out like this (more often it is set out as a crossed out 9 with a little 1 squished in by the 8, however this setting out is clearer and much better for harder problems):

49 2/98  - 80    18  - 18     0

The 8 under the 9 is 4×2 and is the lowest multiple of 2 less than 9. As the 9 is in the 10s column it is 90 and the 8 is 80. We then subtract the 80 from the original 98 and divide the resulting 18 by 2.

In a longer example we do the same thing more than once:

78487 2/156974  - 140000  (2 does not go into 1 or 15, but 2×7 is 14) 16974    - 16000  (2 does go into 16, 8 times) 974    - 800  (2 does not go into 9 but it does 8) 174    - 160  (not 17 but 16) 14     - 14  (and finally 7×2=14) 0

This method extends directly to division by larger numbers. For numbers larger than 12 writting out the multiples of the number to the side can be useful - only go up to what is required, calculating more as you need them. It also extends to decimals - just continue after the decimal point, allthough if the number you are dividing by is a decimal you need to multiply both it and the number you are dividing it by by 10 or 100 or 1000 etc. in order to make it an integer (whole number).

title: Primary mathematics/Dividing numbers


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Primary mathematics/Negative numbers

 * The intended audience for this lesson is educators and parents, whether in traditional or home-schooling environments. For the same lesson, but aimed directly at students, whether children or adults, please see this Wikibooks page.

Introducing Children to Negative Numbers
The concept of negativity is can be difficult to explain to children. This is in part because, from a child’s perspective, there are no obvious examples of negative numbers naturally found in nature. Temperatures below zero are an artificial construct, as is the idea of below sea-level, or floors in a building below the first floor. The concepts of owing money or borrowing candy bars are abstract ideas as well. Children generally do not understand “overnight” what negativity is. However, introducing children to all of these examples will, over time, help them to make the necessary connections that will lead to a strong basic understanding of the concept.

Children should be encouraged to think of their own examples of negativity. Concepts such as backwards and forwards, up and down, give and take, hot and cold, should be explored.

Models and Manipulatives
There are a few mathematical models used to introduce negative numbers to children but the most commonly used one is most certainly the number line.. Because the number line is the one-dimensional component of the Cartesian coordinate system, it should eventually be modeled both horizontally and vertically, and because the x and y axis' of the Cartesian system run left to right and down to up respectively, number lines should be presented this way as well. Connections can then be made between number lines, negative numbers and thermometers, elevators, rulers (for measuring sea-level).

Another way to teach negative numbers is with a set of manipulatives that look like little plus and minus signs (sometimes known as integer tiles). These tiles can be grouped, combined, and swapped around in ways that model the four basic arithmetic operations. They can then be modeled at a higher level of abstraction by students on paper. These manipulatives and their accompanying visual models are normally introduced in the upper primary grades. Below is an example of addition using these tiles:

Note that in this model's answer set, plus signs and minus signs can be grouped to create null sets, whose value is 0. In this particular problem, 2 null sets have been found and circled in the answer. What remains is the sum of the two initial numbers - in this case a single negative tile. Problems should be set up using this method involving two negative numbers, as well as negative and positive numbers where the sum of these numbers is equal to both positive and negative answers.

Other activities and suggestions:
There is also a "red chip/black chip" manipulative, where red chips are used to show negative numbers and black positve numbers. This manipulative can be confusing for students because both the red chips and the black chips exist.

A lesser known manipulative used to teach negative numbers is The ZeroSum Ruler. The ZeroSum Ruler is a foldable ruler that allows students to add integers of opposite signs wth absolute value. The ZeroSum Ruler also works for subtraction, as "3 - 8", for example, is the sum of 3 and -8.

Give students pictorial representations of using this model and have them identify the associated equation. Have students model three different ways to model the number -5, 3, and 0 using tiles. Give students equations with larger and more complicated numbers until they, in recognizing how cumbersome this model can get, begin to solve the problems by creating their own algorithms or "rules" to speed up the process. In this way, students will understand at a fundamental level why you can just subtract 46.2 from 234.5 and put a negative sign on the answer to solve the problem $$46.2+(-234.5)$$.

Subtraction Involving Negative Numbers
In the earlier primary grades, students use various manipulatives as they explore subtraction. Candy, money, toys, are all things that children relate to. But because negative numbers are not found in nature, the subtraction of negative numbers should be introduced only when students are comfortable with more abstract models such as number lines and plus and minus tiles. The number line will have the advantage in that students will already be familiar with their use for adding and subtracting positive numbers. In the upper primary grades, students should be exposed to the idea that addition and subtraction are inverse operations.

Because students are usually taught to "walk backwards" to model subtraction on a numberline, in order to model the subtraction of a negative integer, we can tell them to "do the opposite" of "walking backwards", thus modeling the idea that subtracting a negative is the same as adding a positive. But this can be very confusing for students and other models should be introduced to give students a broader understanding.

The operation of subtraction can be best understood by most students in terms of the act of "taking away". Teaching a student to say (and eventually think) the words "take away" whenever they see a subtraction sign is a solid teaching strategy that just happens to translate well to working with negative numbers when using plus and minus tiles.

The following illustration models the problem $$-5-(-2)=-3$$

The statement "5 negative tiles take away 2 negative tiles" describes the values and the operation. Note that it is important to stress the words "take away".

Students need to become comfortable with this level of problem before they are exposed to problems where the first number in the equation does not permit the taking away of the tiles specified. Consider the problem $$-1-(-4)$$. It is not possible to "take away" 4 negative tiles when there is only 1 negative tile in the initial set. Students must be taught to model the number -1 in such a way that there are 4 negative tiles available to remove:

Notice that in this illustration, the number -1 has been modeled using three null sets (circled in blue). With the inclusion of these null sets, the value of the number in the rectangle is still -1. Note that the student could put in 10 or 20 null sets and the answer will still be the same. Once students are exposed to this method, they might use more null sets than necessary. This is OK. It is in figuring out for themselves how many null sets are necessary that they truly come to an understanding of the operation.

Students should learn to model problems such as:$$7-(-6)$$, $$-2-5$$, and $$0-(-10)$$. After achieving a certain level of proficiency with this model, students soon come to their own personal understanding of what it means when we say that that subtracting a negative is the same as adding a positive.

Multiplying negative numbers

 * Multiply a positive number by a negative normally. The result is negative.

 * Multiply two negative numbers normally. The result, however, is positive.  Think of it as a double negative making a positive.

Dividing negative numbers

 * Divide a positive number by a negative normally. The result is negative.

 * Divide a negative number by a positive normally. The result is negative.

 * Divide a negative number by a negative normally. The result, however, is positive.  Think of it as a double negative making a positive.

 * This same logic also applies to fractions, where the numerator (top number) and/or denominator (bottom number) is negative. That is, if one of the two numbers is negative, then the fraction is negative.  If both numbers are negative, the minus signs cancel, and the fraction is positive.

title: Primary mathematics/Negative numbers