text:
Logic for Computer Scientists/Propositional Logic/Horn clauses

Horn clauses
In this subsection we introduce a special class of formulae which are of particular interest for logic programming. Furthermore it turns out that these formulae admit an efficient test for satisfiability.

Definition 9
A formula is a Horn formula if it is in CNF and every disjunction contains at most one positive literal. Horn clauses are clauses, which contain at most one positive literal.

$$F = (A \lor \lnot B)$$      $$ \land$$   $$  (\lnot C \lor \lnot A \lor D) $$
 * $$\land (\lnot A \lor \lnot B)$$      $$ \land$$   $$  D \land \lnot E $$

$$F \equiv (B \to A)$$            $$ \land $$   $$ (C \land A \to D) $$

 * $$ \land (A \land B \to false)$$   $$ \land $$   $$ (true \to D)\land (E \to false)$$

where $$true$$ is a tautology and $$false$$ is an unsatisfiable formula.

In clause form this can be written as $$ \{\{ A,\lnot B\}, \{ \lnot C,  \lnot A, D\}, \{\lnot A, \lnot B  \} , \{  D  \},  \{  \lnot E\}\} $$ and in the context of logic programming this is written as:

$$\{ A  \gets B $$
 * $$D \gets A,C $$
 * $$\gets A, B$$
 * $$ \gets E $$

$$D \gets $$     $$ \}$$

For Horn formula there is an efficient algorithm to test satisfiability of a formula $$F$$ :

Theorem 6
The above algorithm is correct and stops after $$n$$ steps, where $$n$$ is the number of atoms in the formula.

As an immediate consequence we see, that a Horn formula is satisfiable if there is no subformula of the form $$A_1 \land \cdots \land A_n \to false$$.

Horn formulae admit unique least models, i.e. $$\mathcal{A}$$ is a unique least model for $$F$$ if for every model $$\mathcal{A}' $$ and for every atom B in F holds: if $$\mathcal{A}(B) = true$$ then $$\mathcal{A}'(B) = true$$. Note, that this unique least model property does not hold for non Horn formulae: as an example take $$A \lor B$$ which is obviously non Horn. $$\mathcal{A}_1(A) = true, \mathcal{A}_1(B) = false$$ is a least model and $$\mathcal{A}_2(A) = false, \mathcal{A}_2(B) = true$$ as well, hence we have two least models.

Problem 20 (Propositional)
Let $$F$$ be a propositional logical formulae and $$S$$ a subset atomic formula occurring in $$F$$. Let $$T_S(F)$$ be the formula which results from $$F$$ by replacing all occurrences of an atomic formulae $$A \in S$$ by $$\lnot A$$. Example: $$T_{\{A\}}(A \land B) = \lnot A \land B$$. Prove or disprove: There exists an $$S$$ for
 * 1) $$F = A \stackrel{\cdot} \vee B$$ or
 * 2) $$F = A \stackrel{\cdot} \vee B \stackrel{\cdot} \vee C$$, so that $$T_S(F)$$ is equivalent to a Horn formula $$H$$ (i.e. $$T_S(F) \equiv H$$).

$$\Box$$

Problem 21 (Propositional)
Apply the marking algorithm to the following formula F.

Which is a least model? $$ (A \lor \lnot D \lor \lnot E) \land (B \lor \lnot C) \land (\lnot B \lor \lnot D) \land D \land (\lnot D \lor E) $$

$$\Box$$
 * 1) $$F = (\lnot A \lor \lnot B \lor \lnot C) \land \lnot D \land(\lnot E \lor A) \land E \land B \land (\lnot F \lor C) \land F$$
 * 2) $$F = (A \lor \lnot D \lor \lnot E) \land (B \lor \lnot C) \land (\lnot B \lor \lnot D) \land D \land (\lnot D \lor E) $$

Problem 22 (Propositional)
Decide which one of the indicated CNFs are Horn formulae and transform then into a conjunction of implications: $$\Box$$
 * 1) $$(A \lor B \lor C) \land (A \lor \lnot C) \land (\lnot A \lor B) \land \lnot B$$
 * 2) $$(S \lor \lnot P \lor Q) \land (S \lor \lnot P \lor \lnot R)$$
 * 3) $$(A \lor \lnot A)$$
 * 4) $$A \land (\lnot A \lor B) \land (\lnot B \lor \lnot C \lor D) \land (\lnot E) \land (\lnot A \lor \lnot C) \land D$$


text:
Logic for Computer Scientists/Propositional Logic/Resolution

Resolution
In this subsection we will develop a calculus for propositional logic. Until now we have a language, i.e. a set of formulae and we have investigated into semantics and some properties of formulae or sets of clauses. Now we will introduce an inference rule, namely the resolution rule, which allows to derive  new clauses from given ones.

Definition 10
A clause $$R$$ is a resolvent of clauses $$C_1$$ and $$C_2$$, if there is a literal $$L \in C_1$$ and $$\overline{L}\in C_2$$ and $$ R= (C_1 - \{L\} ) \cup (C_2 -\{\overline{L}\}) $$ where $$\overline{L} = \begin{cases} \;\;\,\lnot A \; \text{ if }  L = A \\ \;\;\,A \; \text{ if }  L = \lnot A \end{cases}$$

Note that there is a special case, if we construct the resolvent out of two literals, i.e. $$L$$ and $$\overline{L}$$ can be resolved upon and yield the empty set. This empty resolvent is depicted by the special symbol $$\square$$. $$\square$$ denotes an unsatisfiable formula. We define a clause set, which contains this empty clause to be unsatisfiable,

In the following we investigate in properties of the resolution rule and the entire calculus. The following Lemma is stating the correctness of one single application of the resolution rule.

Theorem 7
If $$S$$ is a set of clauses and $$R$$ a resolvent of $$C_1,C_2 \in S$$, then $$S \equiv S \cup \{R\} $$ Proof: Let $$\mathcal{A}$$ be an assignment for $$S$$; hence it is an assignment for $$S \equiv S \cup \{R\}$$ as well. Assume $$\mathcal{A} \models S$$: hence for all clauses $$C$$ from $$S$$, we have that $$\mathcal{A} \models C$$. The resolvent $$R$$ of $$C_1$$ and $$C_2$$ looks like: $$ R= (C_1 - \{L\} ) \cup (C_2 -\{\overline{L}\}) $$ where $$L\in C_1$$ and $$\overline{L} \in C_2$$. Now there are two cases:
 * $$\mathcal{A} \models L$$: From  $$\mathcal{A} \models C_2$$ and  $$\mathcal{A} \not\models  \overline{L}$$, we conclude  $$\mathcal{A} \models (C_2  -\{\overline{L}\})$$ and   hence $$\mathcal{A} \models R$$.
 * $$\mathcal{A} \not \models L$$: From  $$\mathcal{A} \models C_1$$ we conclude  $$\mathcal{A} \models (C_1   -\{L\})$$ and   hence $$\mathcal{A} \models R$$. The opposite direction of the lemma is obvious.

Definition 11
Let $$S$$ be s set of clauses and $$ Res(S) = S \cup \{ R \mid R \text{ is a resolvent of two clauses in } S\}$$ then

$$Res^0(S)   =  S$$ $$Res^{n+1}(S) = Res(Res^n(S)), n\geq  0 $$ $$Res^*(S)   =  \bigcup_{n\geq 0} Res^n(S)$$

If we understand the process of iterating the Res-operator as a procedure for deriving new clauses from a given set, and in particular to derive possibly the empty clause, we have to ask, under which circumstances we get the empty clause, and vice versa, what does it mean if we get it. These properties are investigates in the following two Theorems.

Theorem 8 (Correctness)
Let $$S$$ be a set of clauses. If $$\square \in Res^*(S)$$ then $$S$$ is unsatisfiable.

Proof: From $$\square \in Res^*(S)$$ we conclude, that $$\square$$ is obtained by resolution from two clauses $$C_1 = \{L\}$$ and $$C_2 = \{\overline{L} \}$$. Hence there is a $$\exists n \geq 0$$ such that $$\square \in Res^n(S)$$ and $$C_1, C_2 \in Res^n(S)$$ and therefore $$Res^n(S)$$ is unsatisfiable. From Theorem (resolution-lemma) we conclude that $$Res^n(S) \equiv S$$ and hence $$S$$ is unsatisfiable.

Theorem 9 (Completeness)
Let $$S$$ be a finite set of clauses. If $$S$$ is unsatisfiable then $$\square \in Res^*(S)$$.

Proof: Induction over the number $$n$$ of atomic formulae in $$S$$. With $$n=0$$ we have $$S = \{ \square\} $$and hence $$\square \in S \subseteq Res^*(S)$$.

Assume $$n$$ fixed and for every unsatisfiable set of clauses $$S$$ with $$n$$ atomic formulae $$A_1, \cdots, A_n$$ it holds that $$\square \in Res^*(S)$$.

Assume a set of clauses $$S$$ with atomic formulae $$A_1, \cdots ,A_n,A_{n+1}$$. In the following we construct two clause sets $$S_f$$ and $$S_t$$:
 * $$S_f$$ is received from $$S$$ by deleting every occurrence of  $$A_{n+1}$$ in a clause and by deleting every clause which contains an occurrence of $$\lnot A_{n+1}$$. This transformation obviously corresponds to interpreting the atom   $$A_{n+1}$$ with $$false$$,
 * $$S_t$$ results from a similar transformation, where occurrences  of  $$\lnot A_{n+1}$$ and clauses containing   $$A_{n+1}$$ are deleted, hence  $$A_{n+1}$$ is interpreted with $$true$$.

Let us show, that both $$S_f$$ and $$S_t$$ are unsatisfiable: Assume an assignment $$\mathcal{A}$$ for the atomic formale $$\{ A_1, \cdots, A_n\}$$  which is a model for $$S_f$$. Hence the assignment $$ \mathcal{A}'(B) = \begin{cases} \;\;\, \mathcal{A}(B) \; \text { if } B \in \{ A_1, \cdots, A_n\} \\ \;\;\, false \text{ if } B = A_{n+1} \end{cases} $$ is a model for $$S$$, which leads to a contradiction. A similar construction shows that $$S_t$$ is unsatisfiable. Hence then we can use the induction assumption to conclude that $$\square \in Res^*(S_t)$$ and $$\square \in Res^*(S_f)$$. Hence there is a sequence of clauses $$ C_1, \cdots, C_m = \square $$ such that $$\forall 1 \leq i \leq m $$ it holds $$C_i \in S_f$$ or $$C_i$$ is a resolvent of two clauses $$C_a$$ and $$C_b$$ with $$a,b < i$$.There is an analog sequence for $$S_t$$: $$ C_1', \cdots, C_t' = \square $$

Now we are going to reintroduce the previously deleted literals $$A_{n+1}$$ and $$\lnot A_{n+1}$$ in the two sequences: where $$\overline{ C_m}$$ is either $$\square$$ or $$A_{n+1}$$. In any of the above cases we get  $$\square \in Res^*(S)$$ latest after one resolution step with  $$\overline{ C_m}$$ and  $$\overline{ C_t'}$$.
 * Clause $$C_i$$ which has been the result of deleting  $$A_{n+1}$$ from  the original clause in $$S$$ are again  modified to $$C_i \cup \{A_n+1\} $$. This results in a sequence  $$ \overline{C_1}, \cdots, \overline{ C_m}$$
 * Analogous we introduce $$ \lnot A_{n+1}$$ in the second sequence, such that $$\overline{ C_t'}$$ is either $$\square$$ or $$\lnot A_{n+1}$$

Based on the  theorems for correctness and completeness,  we give a procedure for deciding the  satisfiability of propositional formulae.

Theorem 10
If $$S$$ is a finite set of clauses, then there exists a $$k \geq 0$$ such that $$Res^k(S) = Res^{k+1}(S)$$

Until now, we have been dealing with sets of clauses. In the following it will turn out, that it is helpful to talk about sequences of applications of the resolution rule.

Definition 12
A deduction of a clause $$C$$ from a set of clauses $$S$$ is a sequence $$ C_1, \cdots, C_n$$, such that
 * $$C_n = C$$ and
 * $$\forall 1 \leq i \leq n : ( C_i \in S \text{ or } \exists l,r < i : C_i   \text{ is a resolvent of  }  C_l = and = C_r)$$

A deduction of the empty clause $$\square$$ from $$S$$ is called a refutation of $$S$$.

Example We want to show, that the formula $$ K = ((B \land \lnot A) \lor C)$$ is a logical consequence of $$F = ( (A \lor (B \lor C)) \land (C \lor \lnot A))$$. For this negate $$K$$ and prove the unsatisfiability of $$ F \land \lnot K$$

For this you can use the interaction in this book in various forms:
 * Use the interaction Truth Tables for proving the unsatisfiability, or
 * use the interaction CNF Transformation for transforming   the formula into CNF, and then
 * use the following interaction Resolution.

Problem 23 (Propositional)
Compute $$Res^n(M)$$ for all $$n \geq 0$$ and $$Res^*(M)$$ for the following set of clauses: Which formula is satisfiable or which is unsatisfiable? $$\Box$$
 * 1) $$M = \{\{A\}, \{B\}, \{\lnot A,C\},\{B,\lnot C,\lnot D\}, \{\lnot C,D\},\{\lnot D\}$$
 * 2) $$M = \{\{A,\lnot B\},\{A,B\},\{\lnot A\}\}$$
 * 3) $$M = \{\{A,B,C\},\{\lnot B,\lnot C\},\{\lnot A,C\}\}$$
 * 4) $$M = \{\{\lnot A,\lnot B\},\{B,C\},\{\lnot C,A\}\}$$

Problem 24 (Propositional)
Indicate all resolvent of the clauses in S, where $$S =\{ \{A, \lnot B, C\}, \{A, B, E\}, \{\lnot A, C, \lnot D\}, \{A, \lnot E\}\}$$ $$\Box$$

Problem 25 (Propositional)
Prove: A resolvent $$R$$ of two clauses $$C_1$$ and $$C_2$$ is a logical consequence from $$C_1$$ and $$C_2$$. Note: Use the definition of "consequence". $$\Box$$

Problem 26 (Propositional)
Let $$M$$ be a set of formulae and $$F$$ a formula. Prove: $$M \models F$$ iff $$M \cup \{\lnot F\}$$ is unsatisfiable. $$\Box$$

Problem 27 (Propositional)
Compute $$Res^n(S)$$ with $$n = 0, 1, 2 $$ and $$S = \{ \{A, \lnot B, C\}, \{B, C\}, A \{\lnot, C\}, \{B, \lnot C\}, \{\lnot C\} \}$$ $$\Box$$

Problem 28 (Propositional)
Show that the following set $$S$$ of formulae is unsatisfiable, by giving a refutation. $$S = (B \lor C \lor D) \land (\lnot C) \land (\lnot B \lor C) \land (B \lor \lnot D)$$ $$\Box$$

Problem 29 (Propositional)
Show by using the resolution rule, that $$\lnot A \land \lnot B \land C$$ is an inference from the set of clauses $$F = \{ \{A, C\}, \{\lnot B, \lnot C\}, \{\lnot A\} \}$$. $$\Box$$

Problem 30 (Propositional)
Show by using the resolution rule, that $$((P \to Q \text{ is }) \land P ) \to Q$$ is a tautology. $$\Box$$

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


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User:Whiteknight/Page Nav Header

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Talk:Horticulture/An Invitation to Contribute

is this page necessary? Hoogli 16:13, 29 May 2007 (UTC)


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Talk:Horticulture/Book Development Strategies

I think this should be moved to Talk:A_Wikimanual_of_Gardening. Hoogli 16:15, 29 May 2007 (UTC)


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Horticulture/Shrubs

 * Privet Honeysuckle (Lonicera pileata)

/Finder/Types/Shrubs, Large /Finder/Types/Shrubs, Small

title: Horticulture/Shrubs


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Talk:Horticulture/Modifying Wikipedia Articles for Use in Horticulture

Shouldn't this page go into Talk:AWOG? Hoogli 16:24, 29 May 2007 (UTC)

 * It should be linked there, but no, it's a "guideline" for the book, so needs to be it's own thing. It's mostly been me doing this so far, so after a lot of trial and error, I made this page to provide advice for new contributors who might be a bit put off by the heavy use of templates in this book.-- SB_Johnny | talk 19:18, 29 May 2007 (UTC)


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User talk:Vbwilson

How Do We Plan Lessons And Why Is It Important?
written by Victoria (Viki) B. Wilson

What Is A Lesson Plan and Why Is It Important?
A lesson plan is a well thought-out plan of what material is to be covered and how it is to be taught to the students by the teacher for the school year. It is a plan that the teacher has developed according to the curriculum and includes the Virginia Standards of Learning Pacing Guidelines. These plans can be designed on an individual basis, but are best when there is a collaborative groups of teachers in the same school who are teaching at the same grade level plotting their goals for the academic year. This is so all classes of this grade will be on the same page throughout the year.

Can you image walking into a class of first graders with no plan? Lesson plans not only remind the teacher where they are in the curriculum, but also help to keep them on track so that all of the vital educational material is covered in the school year. Teachers do not ever want to arrive at school without a prepared plan for their class for the day. Lesson Plans are sometimes part of the teaching job that teachers do not enjoy preparing, but after they have accomplished this task they are thankful for the guidance it provides them with each day in their classrooms. It is a tedious and daunting task, but with today's technology aides and the help of first-year teacher mentors, the job seems less challenging.

Let's Talk About What Are The Ingredients Of A Lesson Plan
In the book Bringing Learning Alive! by The Teachers' Curriculum Institute of Palo Alto, California, it states that there are five rules to planning your course with lesson plans:

(1) Know the standards, essential content, skills, and objectives that must be taught.

(2) Gather a wealth of rich resources -- visual images, music, readings, activity ideas--to make the content more accessible for students.

(3) Collaborate with their fellow teachers to generate or review new ideas about the course.

(4) Plan a realistic master course calendar, making the difficult content choices to ensure that the units they choose are taught thoroughly.

(5) Create a final semester or course assessment that both prepares students for any mandatory standardized testing and also allows students to show what they know about major themes and key standards (Maloney, p.221).

As you can see, the number one item is to know what the curriculum and state requirements are first. Then you can establish your subject matter and divide it into "chunks" or pieces of the pie to determine what you can realistically teach the students well versus how much of the material you can just cover. Begin with your school calendar and count up your exact number of teaching days. This is important when planning your lesson plans, because the calendar will include non-instructional days such as staff development days, testing days, and particularly in high school the days when there are club meetings or pep rallies. These days must be considered, or you will find out that you have short-changed your time in your lesson plans and can not cover the necessary data.

Number two is probably one of the most fun parts of the lesson, and that is collecting the resources to provide your students with a visual such as a video segment, a map, a movie, or transparencies on the over-head projector.

Getting together with the other teachers in number three can be such an eye-opener especially for new teachers. It is very important for elementary teachers of the same grade to get together and discuss their lesson plans and all be working towards the same goals at the same time. An example of this could be teachers introducing poems to the students and the art, music, or even physical education teachers are aware of this and can introduce rhythming songs to sing in music or when jumping rope and even drawing pictures to illustrate a poem could all be integrated during this lesson.

Number four reminds teacher to not over-plan and be realistic with their lesson plans. The teacher must remember to design the lesson around an essential element or question that is the whole meaning of learning the lesson to begin with. The lesson must interest the students and even though not all students are stimulated the same way, the teacher must understand this and find the best methods to make the material interesting to all students. There is a wealth of new technologies such as www.unitedstreaming.com which is a website that brings videos straight into the classroom along with providing data sheets for the students along with lesson plans for the teachers. Make the lesson stimulating and the students will enjoy it and learn.

Last, according to the Teachers' Curriculum Institute prepare the students for the assessment tests of all your lessons for the week or for the year. Make sure that your testing will be a good reflection of the state SOL testing. Another idea here is to allow the student to rate or give feed-back about which lessons they found most interesting during the year and whatever feedback they wish to include about how to change the way something was taught to them during the school year.

Free Internet Lesson Plans
In the January/February 2006 issue of "The Social Studies" Journal, there is a wonderful article about designing a lesson plan to shatter stereotypes. This article talks about the problems of attitudes and behavior toward minority groups especially since the terrorist attacks of September 11, 2001. This article includes a free lesson plan for teachers who wish to incorporate this subject into their curriculum. This is the description of the lesson it will teach:

"The lesson plan could be part of a larger unit plan dealing with prejudice, discrimination, and racism in American or world history. It could also be used in government, civics, and sociology courses.  The activities will help students understand the dangerous consequences that stereotypes have had in history, improve their knowledge of minority groups, and develop strategies for combating prejudice and discrimination" (Moore, 2006).

There are many websites with free lesson plans and there are also some websites you can join and get a membership so that you can access the plans. Here is a same list of the ones that are highly recommended:

Google For Educators @ http://www.google.com/educators/index.html LessonPlanZ @ http://www.lessonplanz.com/ WELCOME to A to Z Teacher Stuff @ http://atozteacherstuff.com/ PBS Teachers @ http://www.pbs.org/teachers/

According to an article in an edition of Current Events, a Weekly Reader publication, here are some activites to include in your lesson plans that are sure to get the attention of your students: examine editorial cartoons, analyze photos, debate it, map it, chart it, graph it (Weekly Reader, 2007). All of these will spark your students' creativity and interest.

Field Trip Lesson Plans
Do you think that you should have a lesson plan for a field trip? Of course you do, this is so that you can relate the material that you have covered in class to what the students are seeing while on the field trip.

James Kisiel has written an article for the Science Activities Journal entitled "More than Lions and Tigers and Bears-Creating Meaningful Field Trip Lessons," and it a complete lesson plan with the information for the teacher of how to go about making a field trip connect to the curriculum. His essential question is "What Makes a Mammal?" and he is taking his students to a museum. Mr. Kisiel states: "The approach of a field trip-based unit involves the use of the learning cycle model (Barman and Kotar 1989; Barman 1989).  The observations and experiences of the exploration phase, in which students play an active role in examining the animal exhibits and gathering information, are given additional meaning through the concept introduction phase, in which the teacher plays a more active role in connecting student experiences to particular concepts, such as habitat, mammal, or adaptation.  The concept application phase allows students to test their new ideas by introducing a new situation and asking students to relate their developing ideas and concepts to that new situation" (Kisiel, 2006).

The entire field trip lesson plans can be found in this article by James Kisiel who is the assistant professor of science education at California State University at this website: .

All Teachers Need Lesson Plans
All teachers need lesson plans from preschool to college and beyond. Aristotle said we are all goal oriented and Socrates said we all want happiness. Teachers can achieve their goals and find happiness through the joys of having good working lesson plans.

In their article, Ensuring Lessons Teach the Curriculum with a Lesson Plan Resource, Heddi Craft and Paul D. Bland make this comparison about teachers and how they teach: "Some teachers are like chefs and they serve their student the same basic "recipes" year after year.  These lessons often are tasy and effective, familiar standbys that are relied on and enjoyed.  Other teachers are like experimental chefs, never content to use the same recipe for instruction twice, always looking to try new ingredients and styles" (Craft, 2003).

Think about teaching as serving your students a meal. Wouldn't you get tried of eating the same thing over and over again. Variety of resources and teaching styles will keep your students attentive and learning.

Multiple Choice Questions
''' (1) Why are lesson plans an important part of teaching? '''

(a) Lesson plans guide the teacher in the subject matter.

(b) Lesson plans account for only the instructional days in a calendar.

(c) Lesson plans make sure the curriculum and the state standards are accomplished.

(d) All of the above.

''' (2) How does collaborating with fellow teachers help your lesson plans? '''

(a) It helps you because they will steal your lesson plan.

(b) It helps you teach your students the subject matter, because they can reiterate it in their lessons also.

(c) It makes you proud of your lesson plans to share them with fellow teachers.

(d) Collaborating with other teachers is never helpful.

''' (3) After a year of teaching, your principal wishes to evaluate you. Can you lesson plans show your worth as a teacher? '''

(a) Principals do not care about your lesson plans, so they will not help his evaluation of you.

(b) Teaching is a year to year job and no matter how great a teacher you are, lesson plans will not make or break you.

(c) Carefully planned and executed lesson plans along with the curriculum guide and the SOL pacing guide are definitely great representations of a good teacher performance.

(d) The only thing a principal looks at to evaluate your performance are your students SOL scores.

''' (4) How can a field trip help you in teaching your students the subject matter? '''

(a) Field trips are fun and the kids don't have to learn anything while on the trip.

(b) The field trip that is planned to include the lesson plans will give the students more visual, auditory and kinetic (hands-on) learning.

(c) Students are wide-open on field trips and there is no way to teach them anything.

(d) Field trips can help the teacher by keeping the students occupied.

''' (5) What types of technological resources can a teacher introduce in the lesson to motivate the students? '''

(a) Movies, video streaming, over-head projections, mapping.

(b) Giving them more paperwork to accomplish.

(c) Assigning homework.

(d) Explaining how important the subject matter will be to them someday!

Multiple Choice Question Answers:
(1) (d)

(2) (b)

(3) (c)

(4) (b)

(5) (a)

Essay Response
Sample Essay Response by Victoria (Viki) B. Wilson

The best lesson plan is the plan that is constantly evolving each time the lesson is taught or even during the teaching of the lesson the plan can change by shrinking or expanding. Teachers need to be very flexible with the content and the resources while retaining the curriculum element that is the driving force. An example of this could be that while teaching the subject of electricity, the class and the school is out due to bad weather and the lesson must be cut short to accommodate this shortage of time. The teacher would extract from the lesson a couple of experiments or presentations that would have been great extras if the time had allowed. The material that is subtracted must not take away from the message and necessities of the entire lesson, only the extra reinforcing the data would have provided if the time had been as planned. Lesson plans are just plans, they are not written in stone! All plans, no matter if they are for vacations or for teaching should have lots of flexibility and spontaneity for whatever situations occur. These plans even if never changed from the first time they are taught, should be reevaluated on a regular basis to make sure the content and the structure or time frame should change or not change according to the changes of the day. World events can easily change lessons when comparing history with current events. Science changes daily and some science projects that have previously been used in lesson plans may need tweaking from time to time. Teachers should always be seeking to improve their lessons, and this is why along with current events of the world lesson plans should never become stagnant. To teach is a gift, the students receive a gift in learning and the teacher receives one in teaching. Lesson plans make a teacher's day better!

title: User talk:Vbwilson


text:
Cookbook talk:Mock Hollandaise Sauce I

would be possible to use grams as unit of mensurament or even imperials)? -in Europe I´ve never seen "butter sticks" nor I know how big the cups in your Country are... Thanks --84.190.64.20 18:35, 29 May 2007 (UTC)

 * Done. For reference, a "stick" is a North American term for ¼ pound of butter. Webaware talk 23:46, 29 May 2007 (UTC)

Shouldn't this be a base recipe?
I would assume the simple Hollandaise title would start with the base recipe, and become more complex from there. Here is a very simple Hollandaise sauce recipe.

3 egg yolks 1 tablespoon / 18.25 g / 15 ml lemon juice 1/2 cup / 115 g / 120 ml firm salted butter (cold)

With double boiler: Stir yolks and lemon juice vigorously in top of double boiler. Add half of the butter, and place on stove over medium heat, bringing the water to a steady boil. Stirring constantly, melt the butter completely. Add second half of butter and melt as before, stirring constantly. Continue cooking over boiling water, stirring vigorously, until the yolks cook and the sauce thickens. In 1.5 l/qt sauce pan: Stir yolks and lemon juice vigorously in sauce pan. Add half of the cold/firm butter, and place on stove over very low heat. Stir constantly and melt the butter completely. Add remaining butter and melt as before, stirring vigorously, until the yolks cook and the sauce thickens.

(Be sure the butter melts slowly so the yolks have time to cook and thicken the sauce without curdling.) Serve either hot or room temperature, possibly garnish with a dash of Hungarian paprika.

- Amgine (discuss • contribs) 17:30, 4 May 2011 (UTC)

title: Cookbook talk:Mock Hollandaise Sauce I


text:
The Seven Wonders of the World

The seven wonders of the ancient world are seven man-made structures around the rim of the Mediterranean Sea that have been widely considered to be extraordinary feats of human engineering and design. While the commonly-accepted list of wonders focuses on the Mediterranean area, there are certainly many other wonderful and extraordinary structures from other parts of the world that the writers of this list were likely not familiar with. Some wonders are made for glory or representing their cultures.

This book discusses the classical seven wonders of the ancient world, including their construction, life, and destruction.

Table of contents

 * /The Seven Wonders/
 * /The Great Pyramid/ at Giza
 * /The Hanging Gardens/ at Babylon
 * /The Statue of Zeus/ at Olympia
 * /The Temple of Artemis/ at Ephesus
 * /The Mausoleum/ at Halicarnassus
 * /The Colossus/ at Rhodes
 * /The Pharos/ at Alexandria

Resources

 * /Resources/

title: The Seven Wonders of the World


text:
User:Whiteknight/links.js

function AddPersonalLink(link, text, tag, ibefore) { var li = document.createElement( 'li' ); li.id = tag; var a = document.createElement( 'a' ); a.appendChild( document.createTextNode( text ) ); a.href = link; li.appendChild( a ); if ( ! ibefore ) // append to end (right) of list {    document.getElementById( 'pt-logout' ).parentNode.appendChild( li ); } else {     var before = document.getElementById( ibefore ); before.appendChild( li, before ); } } function WKBFLink { AddPersonalLink('http://en.wikibooks.org/wiki/User:Whiteknight/Book_Foundry',                   'foundry',                   'pt-bookfoundry',                  'pt-mytalk'); AddPersonalLink('http://en.wikibooks.org/wiki/User:Whiteknight/All_Books',                 'books',                  'pt-allbooks',                  'pt-bookfoundry'); AddPersonalLink('http://en.wikibooks.org/wiki/User:Whiteknight/Book_Creator',                 'creator',                  'pt-bookcreator',                  'pt-bookfoundry'); // AddPersonalLink('http://en.wikibooks.org/wiki/User:Whiteknight/Book_Tracker', //                  'Tracker', //                  'pt-tracker', //                  'pt-watchlist'); }

$(WKBFLink);

title: User:Whiteknight/links.js


text:
User:Whiteknight/switcher.js

function AddToolboxLink(text, href, onclick, linkid) { var tb = document.getElementById('p-tb').getElementsByTagName('ul')[0]; var link = document.createElement('a'); link.onclick = onclick; link.href = href; link.appendChild(document.createTextNode(text)); var li = document.createElement('li'); li.id = linkid; li.appendChild(link); tb.insertBefore(li, tb.firstChild); return; }

function WKInsertStyleLink { var url = "http://en.wikibooks.org/w/index.php?title=" + encodeURIComponent(mw.config.get('wgPageName')) + "&useskin=monobook"; AddToolboxLink('Change Stylesheet', url, null, 'tb-ss'); }

$(WKInsertStyleLink);

title: User:Whiteknight/switcher.js


text:
Usenet Explorer User's Manual/Manual/Automation

This chapter explains the automation features of Usenet Explorer.

Associate a Watch Filter with a Newsgroup
By associating a Watch Filter with a Newsgroup, you can automatically download attachments of headers that match a filter.

 * 1) open the Properties window (Edit > Properties or F5), select the Watch/Auto headers tab
 * 2) in the Available watch filters pane, select the watch filter
 * 3) in the Newsgroup(s)/Folder pane, select the newsgroup or folder
 * 4) click the Add button; the filter will show up in the Watch filters pane
 * 5) specify the directory to save the attachments to using the Browse or Default buttons

Now, if headers are downloaded that match the filter, the attachments in the bodies of the messages are saved automatically. Each time new headers are downloaded, additional parts of the attachments are downloaded and saved. There's no need to wait for a posting to complete uploading; it is downloaded automatically when it appears on your news servers.


text:
Usenet Explorer User's Manual/Manual

= Manual =

/Quick Start/
get, unpack, servers, newsgroups, headers, messages, attachments, par2, rar

/Installation/
Download, Unzip, Create Database

/Configuration/
part one,

/Getting Headers/
Getting headers,

/Searching/
Searching,

/Reading News/
Reading News,

/Downloading/
Downloading,

/Import NZB/
Import NZB files,

/Filters/
Filters,

/Automation/
Associate a Watch Filter with a Newsgroup,

/Tips & Tricks/
Tips & Tricks,

/FAQ/
FAQ

title: Usenet Explorer User's Manual/Manual


text:
Horticulture/Trees

 * Okame Cherry (Prunus 'Okame')

title: Horticulture/Trees


text:
Change Issues in Curriculum and Instruction/Other Issues Worth Investigating/28 May, 2007

1. Keeping Up and Knowing What to Keep Up With:  How do we know what questions to ask?

2. Who's Responsible?  Who should be held accountable for making change happen?

3. Believing That Change Can Happen:  Becoming Comfortable with Risk

4. Professionalizing the Teaching Profession:  Teacher and Administrator Accountability and Effective Motivation


text: A-level Chemistry/OCR (Salters)/Vacuum filtration


text:
User talk:24.149.178.107

Please stop adding nonsense to Wikibooks. It is considered vandalism. If you would like to experiment, use the sandbox. Thank you.

Xania talk 21:05, 29 May 2007 (UTC)

title: User talk:24.149.178.107


text:
A-level Mathematics

This book covers A-level mathematics with chapters each focusing on the requirements for each examining body.

Table of Contents

 * /OCR/
 * OCR (MEI)
 * /AQA/
 * /Edexcel/
 * /WJEC/
 * /CIE/
 * /CCEA/
 * Advanced A-level Mathematics — greater depth than required by the curriculum, not specific to a particular examining body

title: A-level Mathematics


text:
User talk:71.183.35.13

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks.

Xania talk 22:06, 29 May 2007 (UTC)

title: User talk:71.183.35.13


text:
User talk:83.71.35.206

Please stop adding nonsense to Wikibooks. It is considered vandalism. If you would like to experiment, use the sandbox. Thank you.

Xania talk 22:08, 29 May 2007 (UTC)

Blocked from editing
You have been blocked from editing Wikibooks for one week. The reason for this block is because of the continued addition of misleading information to Wikibook Cookbook articles. I warned you to stop but you didn't listen to my warning and made further misleading changes. After your block has expired you may return but only if you are going to help the Wikibooks community rather than vandalise it.

Xania talk 23:11, 29 May 2007 (UTC)

title: User talk:83.71.35.206


text:
Latin/Spoken Latin

Audio Files
Hi, if you want to learn spoken Latin some of the wide range of lessons on the Latinum podcast are aimed at this: have a look at http://latinum.mypodcast.com

Code Switching Trick
Here is a trick you can use to learn both basic grammar and to get into the hang of speaking in Latin. One of the biggest problems for students who learn Latin is the fact that it is not a word order language. This fact has already been covered in other lessons but the trick presented here is designed to get you past word order thinking and to start producing Latin in your head so it can come out of your mouth or off of your pen. Please note that if you don't have enough vocabulary to produce a real sentence in Latin with out looking up every word you can still use this trick to get yourself used to speaking in this way but using English words that you will slowly replace as your Latin vocabulary grows.

I'd not use this method for more than a day or so. But playing around with it will help a little. Nothing will really work, except for working and studying Latin itself, in Latin.

Of course if you have studied the proper endings for a word then you should use it but if not just keep the English pronouns He, His, Him in you mind. Using these in English it is possible to violate English strict word order and still create an intelligible sentence.

 * He to her its them gives.

Of course this is not good English and we don't know what the pronouns are referring to but we can still understand and that is the whole point. Using pronouns in English we can violate word order and even leave it behind. The interesting thing about Latin is that it isn't just the Pronouns but every word with few exceptions that can do this.

This is what you do. Any word, from any language can be Latinized and used in a pseudo language that will teach you to speak Latin if you stick with it.

Subject of a sentence: do nothing to the word but keep in mind that you can actually drop it from the sentence if everyone already knows what you are talking about.

Possessive: add an `IS` to the end of the word, so instead of car's, you will have caris.

Indirect object: add `i` to the end. This will sound funny but will work. If a word end in an ee kind of sound just pronounce it twice.

Direct object: add `EM` to the end of a word. thus you get things like flowerem, spacem, computerem, carem, personem and so on.

If you need to use a preposition just use the prepositions you have learned and don't change the word (just use the subject form as the preposition tell you the grammar of the word)

Thus you could say a sentence like;
 * boy girli manis flowerem gives

The above sentence is not English, nor Latin but it is what is called Code Switching and anyone who is on the way to fluency in a second language does it automatically. That is why is Japan people go around saying things like

 * Car WA Boy OH Hit imasu.

We don't understand this sentence but it again is what may happen when two languages mix in the brain. Code Switcing is an important step to really becoming efficient in a new language. It allows you to move past only translation and actually make words come out of your mouth.

Note; when you practice this trick don't worry about the plurals till you really study them.

Anyone who has studied Latin even a little probably knows the beginnings of the conjugation for
 * amo, amas, amat

What you should take note of is the fact that where the second person in Latin is governed by an S at the end and the third by a T it is reversed in English.

 * I love, thou lovest, he loves

So this is what you do to start adding Verbs to your new Latin Code Switching hack; flip what you would normally say for HE from S to T at the end and start using S for the You form.

 * I love, you loves, he lovet

Yes, It looks strange but it is to help you learn to code switch. Again, don't use plurals till you really learn them and have already gotten used to using the singulars here. When you learn the correct Latin Vocabulary then replace the English words for the correct Latin words with correct Latin endings for that particular word.

As a final step before really forcing yourself into creating all Latin sentences, start using either an `M` or an `I` after verbs that are in the first person

 * I lovem, I lovei

This will be important when you learn the perfect tense. Then start dropping the I, we, you, them pronouns all together. You will find that you don't need them since the verb tells you who is doing the action of the verb. Still use He and she as the difference from Male and Female is more important in Latin then in English.

Your resulting sentences should be something like this when spoken

 * boy girli manis flowerem givet.

Don't write in this code. This should be used only for spoken exercises. Get a friend and both of you learn this code and just speak back and forth. Keep it simple. Don't try anything too complicated till you actually learn it. Try to make it sound nice as well, Latin was a fluid, beautiful language that gave use the romance of French and Italian so giving yourself a little accent can't hurt.

Another tip is to put the verb at the end of the sentence. The reason is that this creates a spoken period. Because we don't have word order to tell us when the sentence ends anymore, by putting the verb, which every sentence must have, at the end you are also telling the other person that the sentence is coming to an end. To make a question you can still put the verb at the end and just make it sound like a question or use a prober Latin question creator when you learn it.

Here is an example of a conversation in this code

 * computerem uses?
 * yes. use wants?
 * dog momis computerem broket.
 * really, terrible ist.
 * mom around house dogem chaset
 * funny soundt
 * say, girlis from computer class numberem gets?
 * no, she tall guyem liket
 * dark hairem she liket?
 * Maybe, not knowi
 * heris numberem get tryi

Yes, silly conversation, but this is just to give you an example of how to start breaking word order grammar and get into the mindset of the inflexive Latin grammar. Once you learn the proper endings start applying them to your English words. When you learn the proper vocabulary then switch to using the Latin words. When you learn a new Verb ending for past tense, infinitive and what not then start practicing with this new ending till you got it down, and not just with Latin words but with English Code Switching. You will be able to speak intelligible, near fluent Latin in less than five months.

Once you actually do get some real Latin vocabulary under your belt you should start paying attention to the grave marks. Macrons can help you to tell the difference in words in their written form but really doesn't help with speaking. Find a book or article in the net that shows the little accent markes that are sometimes called graves and sometimes called Breves. They look like just a little slash over a vowel. é é à and the like. These tell you what part of the word to sort of choke up on. Try saying the word Spaghetti. You automatically want to choke up or stress the E sound in the middle. You actually say this `spaghétti`. Not all words have a stressed sound in them but many do and it can give you a great way to really sound Latin. All in all they are really helpful.

Many speakers of Latin will change a C to a CH if it is followed by an E or I vowel sound. You don't have to do this and many people that teach Classical Latin say it is wrong but if you are speaking Latin today it really is by definition Not Classical Latin. Doing this will get you saying the word in a very European kind of pronunciation that really seems very natural and makes some words easier to say. Again, you don't have to do it and really shouldn't if it makes your teacher angry. Similar to changing the C to CH is the use of changing a G sound to a J sound if again followed by an E or an I sound.
 * Thus Ge-nu becomes `jen-u as in the modern genuflect.
 * Civitas- Kee vee tas becomes Chee vee tas which eventually came to be See vee tas- so that we can say civics and citizen.

The last point of concern is to choose to use the letter V as a v,v,v,v,v,v,v sound or a W,W,W,W,W,W sound. Once you pick it is best to stay consistent and not change from one word to another. If you don't like the way it is going with one and want to change to the other then change all words to the new way of saying them.

Sentence patterns
Once you have played around with the language a bit you will want to start putting things in order the way they should be in Latin. Now, it is true that Latin does not have a specifically set word order, but there is a usual order that sentences tend to go in. This does not mean that they HAVE TO go in this order, it just is the common way. The word order, it should be remembered, has zero relation to the grammatical meaning of a word. Regardless of word order the sentence will mean the exact same thing. However the normal word order goes something like this.

Nominative Dative Accusative Ablatives/Adverbs Verbs.

Put into modern English language grammatical terms, this translates into something like this.

Subject Indirect Object Direct Object Compliments verbs.

Please note, the verb is usually last. It does not have to be, but just usually is. The subject (nominative) is usually first but does not have to be and an interesting feature of Latin is that the Subject can be dropped anytime it is understood. Many sentences in Latin actually wind up being Direct Object Verb.

The reason most teacher teach the declention list in a certain order N-G-D-Ac-Ab is because of this normal Latin word order even if they do not realize it.

Most Latin sentences will not contain every case. Latin has a number of sentence patterns and these patterns determine what cases are needed and what those cases mean.

The most basic sentence pattern in Latin is Nominative Esse. For this sentence pattern you basically put your noun in the nominative case and put the correct form of the "be" verb Esse after it.

Examples:
 * Homo sum. Mulier es. Umbella est. Pueri sumus. Puellae estis. Furnaces sunt.

Make sure you match singulars with singular verbs and plurals with plural verbs.

The next sentence pattern in complexity is Nominative Nominative Esse. This basically says that you are equating the first thing with the second thing. This pattern is also used to place the qualities of an adjective on a noun. All that really matters is that they are both in the nominative. Make sure they match in gender.

 * Pater homo est. Amicae mulieres sunt. Magister bonevolus est. Magistra irata est. Nauta bonus est. Manus dextra est.

These last two sentence patterns were pretty easy and straight forward, you just have to remember not to put the "be" verb in the middle but even if you do it is still correct Latin. It just sounds better to put the verb at the end. The next sentence pattern isn't so hard but since we don't really have it in English it can throw some.

Nominative Dative Esse. This is where you state that a thing is "for" a person. "The book is for him" and the like. English doesn't really have a Dative case anymore so the idea is a bit out there but datives come up all the time in Latin so it is time to start getting used them.

 * Liber puero est. Mulier homini est. Homo mulieri est. (Nominative dropped) Mihi est. Tibi est.

At this point in time it is appropriate to mention the Genitive case. You will notice even though most of your books state that the Genitive case is the possessor case similar to English "His" "Her" "Its" "Their", it was not used for that purpose above. In Latin the Genitive case is a quasi-adjective that states an internal quality of the noun. Take a look at the difference.

 * Puer mulieri est. Puer irae est.

I can say "the boy is for the woman / He is the woman's son" but I can't say "the boy is for anger." That makes no sense. But I can say "He is a boy of anger." This gives you the impression that the boy is just angry all the time. Datives used with the Esse verb tend to be people, not objects. Genitives used with the esse verb tend to be objects but can often be people. Here is the difference.

 * Filius homini est.

The very point of this sentence is to tell you that the Son is for the Man.

 * Filius hominis nauta est.

This sentence uses a Genitive because the reason that the sentence exists is not to tell us that the Son is for the Man but that the Son is a Sailor. The fact that he is the Man's Son is just an extra idea that we threw on but could easily take off and still have the meaning of the sentence preserved. Filius nauta est. Filius himinis nauta est.

Basically, for grammar, we will treat the genitive like an indeclinable adjective. It can go anyplace an adjective can go.

The next sentence pattern we will look at is Nominative Adverb est. You basically state a nominative and give extra information about it using an adverb, or a preposition plus case noun.

 * Petasus illic est. Petasus in mensa est.

With Intransitive and Transitive verbs we can use adverbs, prepositional phrases and Ablatives by themselves for this.

 * Bene video. In via sto. Placido quiesco.

This means that you already know another sentence pattern. Nominative Adverb Intransative Verb.

Once we jump into the world of transitive verbs you will have a bit more in the sentence to keep track of. A transitive verb allows an Actor (subject nominative) to act (transitive verb) on an object (Accusative). The sentence pattern goes like this: Nominative Accusative Transative Verb.

In English we use word order to separate these two because unless we use a pronoun like him, the subject and object of the sentence will look the same. Not in Latin, or at least not often in Latin. In Latin, Subjects sound and thus look different than Objects. Keep in mind that word order could be any order but, so that you can see and get used to what you are seeing, I will stick to Nominative Accusative Verb. In a real Latin sentence, multiple adjective/genitives, adverbs/ablatives and or datives may be in a sentence to extend and enhance the meaning.

 * Puella puerum videt. See, even with Puer and Puella being next to each other and them being similar in pronunciation, you can still easily determine who is the subject and who is the object.

 * Pueri puellas cum intentissime observant. No mystery here as to what is going on I hope.

 * Next we will look at giving verbs. Their sentence pattern goes like this: Nominative Dative Accusative Verb. Nominative gives Accsative to Dative.  If you can understand that then this sentence pattern is easy.

 * Puella puero rosam dat. Puer puellae cephalalgiam dat. Mulier hominibus litteras misit.

You will notice I hope that I used the verb Dat twice above. The Dative does indeed take its name from its frequent use with this verb. Also I used the verb Misit. Any verb that means "give" "send" "grant" and the like can logically use this construction.

Next we look at Making verbs. This is where you say that you turned one thing into another thing. This can be a noun into another noun or a noun into an adjective. Nominative Accusative Accusative Verb.

 * Coquus similaginem crustulum fecit. Coquus crustulum dulce fecit.

The last of the most basic sentence patterns that you will need to remember and use in everyday speech is Nominative Ab Ablative Passive Verb. In this sentence the Nominative Case is not the actor of the verb but instead receives the action that the verb gives, just like a passive verb in English. Logically, there can be no Accusative in this kind of sentence. The person that is the actor does not have to be stated but if he or she is it will be using the Ablative case after the preposition Ab.

 * Ab familia mea laudar. Circenses ab omnibus videntur.

Those sentence patterns listed above are the most basic and most frequent sentence patterns that you will encounter and have to use is you become a Latin speaker (and you should be trying to become a Latin speaker). These sentence patterns are your base and onto them you will eventually add additional sub patterns that you will use. The important thing is to master these simple patterns first and then move on to more complex structures. Once you know the pattern, it becomes just a matter of vocabulary replacement. Move one word out and put in the desired next word. You should know how to do this through repetition so much that you do not have to think about it to parse or translate. The goal is to understand in Latin and not have to need to translate into English to get it.

title: Latin/Spoken Latin


text:
A-level Chemistry/OCR (Salters)/Reactions

Classes of chemical reaction
Reactions can be classified by numerous criteria.

 * Addition reactions
 * Condensation reactions
 * Elimination reactions
 * Substitution reactions

Organic reactions

 * Nitrile hydrolysis
 * Reactions of arenes


text:
A-level Chemistry/OCR (Salters)/Froth flotation

Exam-style description of froth flotation

 * grains are given water repellent coating
 * air and detergent cause the solution to froth
 * ore grains are / metal is concentrated in the froth
 * ore grains are scooped off with the froth


text:
User talk:Bbi5291

Hello, welcome to Wikibooks!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be a good idea to add this page to your "watchlist" so that you can see when any new information is posted there. You can do that by clicking the tab labeled "watch" at the top of the page.

Good luck! Mattb112885 (talk) 23:30, 29 May 2007 (UTC)

title: User talk:Bbi5291


text:
User talk:Reimer8829

Hello, welcome to Wikibooks!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be a good idea to add this page to your "watchlist" so that you can see when any new information is posted there. You can do that by clicking the tab labeled "watch" at the top of the page.

Good luck! Mattb112885 (talk) 23:31, 29 May 2007 (UTC)

title: User talk:Reimer8829


text:
User talk:Lohman 2572

Hello, welcome to Wikibooks!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be a good idea to add this page to your "watchlist" so that you can see when any new information is posted there. You can do that by clicking the tab labeled "watch" at the top of the page.

Good luck! Mattb112885 (talk) 23:32, 29 May 2007 (UTC)

title: User talk:Lohman 2572


text:
User talk:Walker5199

Hello, welcome to Wikibooks!

You will find more resources in Community Portal. If you want to ask a question, visit the Study help desk, the Staff lounge, IRC channel or ask me personally on my talk page. For site news, see the Bulletin board. It might be a good idea to add this page to your "watchlist" so that you can see when any new information is posted there. You can do that by clicking the tab labeled "watch" at the top of the page.

Good luck! Mattb112885 (talk) 23:34, 29 May 2007 (UTC)

title: User talk:Walker5199


text:
Algebra/Roots and Radicals

What are Roots?
Roots are the inverse operation for exponents. An expression with roots is called a radical expression. It's easy, although perhaps tedious, to compute exponents given a root. For instance 7*7*7*7 = 49*49 = 2401. So, we know the fourth root of 2401 is 7, and the square root of 2401 is 49. What is the third root of 2401? This article gives a formula for determining the answer, while this article gives a detailed explanation of roots.

Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. When we graph functions we will see that expressions that use exponentiation use curves instead of lines. We will see using algebra that not all of these expression are functions, that knowing when an expression is a relation or a function can allow us to make certain types of assumptions, and we can use these assumptions to build mental models for topics that would otherwise be impossible to understand.

For now we will deal with roots by turning them back into exponents.

If a root is defined as the nth root of X, it is represented as $$\sqrt[n]{x} = r$$. We get rid of the root by raising our answer to the nth power, i.e. $$r^n = x.\!\,$$  Problems Using Roots

Square root
If you take the square root of a number, the result is a number which when squared gives the first number. This can be written symbolically as:

$$\sqrt{x} = y \mbox{ if } y^2 = x \,$$. In the series of real numbers, $$y^2\geq 0$$ regardless of the value of $$y$$. As such, we cannot define the $$\sqrt{x}$$ when $$x<0$$.

Examples:
 * $$\sqrt 9 = 3$$ since $$3^2 = 3\cdot3 = 9.$$
 * If $$x=25$$ then $$\sqrt{x-16}=\sqrt{25-16}=\sqrt{9}=3$$.
 * If $$x=7$$ then $$\sqrt{x-16}$$ is undefined because $$\sqrt{x-16}=\sqrt{7-16}=\sqrt{-9}$$, but there is no number $$y$$ so that $$y^2=-9$$. Notice that the answer is not $$-3$$ since $$-3\cdot-3=9$$ and not $$-9$$.

You may notice or discover that there is a solution to square roots of negative numbers. This will be discussed in the future chapter of Complex Numbers, which will require learning intermediate concepts.

Cube roots
Roots do not have to be square. One can also take the cube root of a number ($$\sqrt[3]{ } $$ ). The cube root is the number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. For example, the cube root of 8 is 2 because $$2 \cdot 2 \cdot 2 = 8$$, or:

 * $$\sqrt[3]{8} = 2.$$

Other roots
There are an infinite number of possible roots all in the form of $$\sqrt[n]{a}$$ which corresponds to $$a^\frac{1}{n}$$, when expressed using exponents. If $$\sqrt[n]{a} = {b}$$ then $$ b^{n} = a \,$$.

The only exception is 0. $$\sqrt[0]{a}$$ is undefined, as it corresponds to $$a^\frac{1}{0}$$, resulting in a division by zero. Even if you attempt to discover the 0th root of 1, you will not make progress as practically any number to the power of zero equals 1, leaving only an undefined result.

Irrational numbers
If you square root a whole number which is not itself the square of a rational number the answer will have an infinite number of decimal places. Such a number is described as irrational and is defined as a number which cannot be written as a rational number: $$\frac{a}{b}$$, where a and b are integers.

However, using a calculator you can approximate the square root of a non-square number:

$$\sqrt{3} \approx 1.73205080757 $$

The result of taking the square root is written with the approximately equal sign $$\approx$$ because the result is an irrational value which cannot be written in decimal notation exactly. Writing the square root of 3 or any other non-square number as $$\sqrt{3}$$ is the simplest way to represent the exact value.

Proof
Taking $$\sqrt{3}$$ as an example.

Suppose $$\sqrt{3}$$ is rational and $$\sqrt{3}$$ = $$ \frac{a}{b} $$ where a and b are integers and relatively prime.

$$ 3 = \frac{a^2}{b^2} $$

$$ a^2 = 3b^2 $$

This implies that 3 is a factor of $$a^2$$. Since a is an integer and 3 is prime, 3 is a factor of a. Let a = 3k where k is an integer.

$$ a^2 = 3b^2 $$

$$ (3k)^2 = 3b^2 $$

$$ 9k^2 = 3b^2 $$

$$ b^2 = 3k^2 $$

Similarly, 3 is a factor of b, which contradicts the first statement that a and b are relatively prime. Therefore $$\sqrt{3}$$ cannot be rational. Therefore $$\sqrt{3}$$ is irrational.

Irrational numbers also appear when attempting to take cube roots or other roots. However, they are not restricted to roots, and may also appear in other mathematical constants (e.g. π, e, φ, etc.).

title: Algebra/Roots and Radicals


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User:Kfasimpaur/sandbox2

= Glossary proposal =

This is a proposal for organizing glossaries for Wikijunior books. Please comment, make suggestions, etc.

Purpose
The purpose of a glossary in a Wikijunior book is to give simple, child-appropriate definitions of potentially unfamiliar words in the context in which they are used in the book. This differs from a dictionary in that only the pertinent definition is given. For example, in a book about the human body, the word "organ" would be included as a glossary word. The definition would be "a part of the body made up of cells doing a similar job" (not "a musical instrument with a keyboard").

Visual representation
Glossary links should be designated in a way that is visually distinct from other links (as external links are, for example). Perhaps a small icon after the link would work for this.

Use
Optimally, all key terms that are potentially unfamiliar to children in the target age group should be included in the glossary.

Question: Should only the first use of a word be glossary-linked (as would be done in an ebook) or should every entry be linked? One problem with the former is that it assumes a linear reading. The problem with the latter option is that there may be so many links that is distracting to (children) readers.

Technical implementation
Depending on others' input, I am thinking of implementing this in a way similar to the way citations are done in Wikipedia. For example, to add the word "organ" to the glossary, the markup could be something like:

 * The stomach is an organ that is a part of the digestive system.

Then the glossary would be dynamically generated including the entry:

 * organ - a part of the body made up of cells doing a similar job

Question: Would the glossary be generated at the bottom of each module, or should there be a consolidated glossary for the whole book? From a programming standpoint, the first option is probably easier? From a pedagogical standpoint (and in a print format especially), I'd prefer the latter.

Espohagus - a tube made of muscle that goes from the mouth to the stomach; food that is swallowed goes down this tube

blah blah

title: User:Kfasimpaur/sandbox2


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User:Martian.knight

'''Talk goes to the talk page. | Send me an email.'''

For now, see my page at Wikipedia.

title: User:Martian.knight


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User talk:Martian.knight

Please leave new messages at the bottom of my page, and responses under the same heading they were started.Click here to leave me a new message. I will typically answer on your talk page to ensure you receive the communication.

title: User talk:Martian.knight


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User:Dark Pegasus

I'm a Java developer interested in learning about non-java development.

title: User:Dark Pegasus