School:Mathematics/Print Version
Welcome to the Wikiversity School of Mathematics, a collection of lesson plans and related materials to assist in the learning of mathematics.
Pre-university level course areas
editReadings
editSee also
editExternal Links
editKronos 400 Free printable multiplication & division worksheets. Free lessons, online activities, worksheets, lessons & assessments in all subjects. [1]
- This section of the School of Mathematics focuses on providing teaching tools for parents and educators, whether in traditional or home-schooling environments. A parallel project is underway to teach students directly, whether children or adults, at this Wikibooks page.
Welcome to the Wikiversity Division of Primary School Mathematics, part of the School of Mathematics. The face of teaching mathematics is in constant flux. This course, therefore, is an attempt to stay in step with the current pedagogue(s) used to teach mathematics to the primary grades.
While educators should find this course helpful in learning or reacquainting themselves with some of the methodologies currently used, this course attempts to use as much lay language as possible to also be helpful to parents who, when looking over the shoulders of their children, are sometimes baffled by the "New Math". Those who wish to home-school their child(ren) may find the most value from this course; attempts will be made to cover a number of pedagogically related topics in depth; many core components (not necessarily math related) of learning will be discussed and links to more in-depth coverage of those topics will also be included. Also, if you have trouble with the material, links will be found on each page to other more basic and/or remedial sources in the Wiki community to help bring you up to speed. By providing links "in both directions" as described above, it is hoped that this course might serve as a starting point for those wishing to gain more insight into the teaching of mathematics to the primary grades.
What This Course is Not
editThis course assumes that the reader has reached a certain level of competency with the topics it covers. As such, this course is not designed to teach mathematics to the reader. For a very elementary introduction to mathematics, for the child or for the adult who considers themselves mathematically handicapped, you may want to read a book in development in Wikibooks, Primary Mathematics.
This course is meant to give the reader insight into how and why mathematical core understandings and skills are taught the way they are. A basic mathematics background should be all that is needed to find value in this course. Oftentimes parents (and even teachers) only understand a topic in the way they were taught it, but modern math curriculums are typically constructed such that topics are covered in many different ways in order to, among other things, accommodate the different learning styles of students.
There are many ways to teach any given mathematical understanding, and the depth of students' mathematical understandings are enhanced when they have explored it from multiple perspectives. For example, the Theory of Multiple Intelligences is but one perspective that teachers employ when leveraging student's individual strengths.
Although the idea that there are many learning styles is not well supported by the research in this area so far, this course recognizes that there are many teaching styles. As such, this course is not necessarily intended to be a math curriculum, although it could serve as part of the foundation of one. For example, this course will not teach the reader in step by step fashion how to multiply two large numbers, but the various building blocks and algorithms that students can be introduced to in the process of learning this skill will be explained.
There are many very good texts/curriculums/approaches out there; there are even "schools" of thought that suggest that math is best taught without textbooks if the teacher is skilled enough. This course could never pretend to cover the range of curricula out there. Emphasis will be on content knowledge and how it can be taught to the primary grade student. Nonetheless, activities will be suggested in this course that support the concept or skill being presented, which may serve to present the reader with ideas for their own course content.
Connections
editOne of the overarching ideas that should be highly leveraged in the teaching of mathematics in the primary grades involves the connections students make in their learning. All of our mathematical understandings are intertwined. For this reason, different skills such as multiplication and concepts like those learned from the study of geometry should not be explored by students in a vacuum. Rather, they should be taught together, in such a way that they reinforce each other. Internal links, when found in this course, will purposefully be placed there to emphasize the need for an ever-present awareness of the connectedness of mathematical understandings. Teachers should prefer not to "teach", they should prefer to "guide".
So how do teachers not teach? They do this by inspiring students to inspire their own learning. The most inspiring inquiries are often born of students' own interests. Primary and even upper school teachers recognize that understanding mathematics in the abstract is not the goal of most students. They need to see connections to the real world to inspire their learning. Teachers should prefer to use real world problems that require the need for mathematical models (see below). Students should then be encouraged to make connections by looking for patterns, exploring extremes, and forming and testing conjectures. At the same time teachers should be aware that all mathematics, from basic numeracy upwards is an abstraction and that some ideas are better explained as abstractions or generalizations from what is already known, rather than by constructing tedious and implausible 'real world' examples. After all, how often do carpenters really use trigonometry to measure the height of a flagpole?
Modern educators realize that students gain true "ownership" of their understandings through, inasmuch as it is possible, making connections on their own - by way of their own work and explorations. "Telling them how to do it" does not respect or celebrate their abilities. On the other hand, letting students explore concepts, learning from their mistakes as they go, ultimately leads to much stronger mathematical understandings. It also leads them to form learning habits that make their future explorations more efficient and successful. Connecting with a child's personal interests is also important. However, it is unrealistic to expect students to construct their own understandings of many areas in Mathematics. For example, calculus eluded mathematicians for centuries before it was created by Sir Isaac Newton. More appropriately for the primary school level, the long division algorithm is not something that students can be expected to figure out for themselves.
Manipulatives and Models
editOne very important component of the contemporary methods used to teach math is the use of manipulatives (such as toys) and models (visual representations) to give added dimension to students' understandings. In each chapter of this course, the reader will find various examples of models used to teach different mathematical understandings. Often, they will find that these models serve to make connections to material covered in other sections. Keep in mind that teachers in the classroom tend to be very creative and resourceful. Often, the models found in this course have many possible permutations, and can come in various shapes, sizes, and guises. They are purposely presented here in simple forms to facilitate, for the reader, their identification.
In this course, the word model is occasionally used in a subtly different context as well - one that is commonly used by teachers of any discipline. We like to model good mathematical habits just as we like to see any kind of good behavior modeled. Occasionally, teachers should model problems by doing them as if they are a student.
Course Contents
editNote: a copy of some of the following pages initially was created from Wikibooks, but the content will start to diverge to take the two different audiences into account. See Wikiversity:Import.
Numbers and Operations
editPre-Algebra
edit- Negative numbers
- Powers, roots, and exponents
- Fractions
- Working with fractions
- Percentages
- Factors and Primes
- Method for Factoring
- Proportions
- Introduction to variables
Measurement and Geometry
edit- Polygons and Area
- Angles, see also: Angles
- Solids
Advanced topics
editLinks commonly found in this course
editMove on to High School Mathematics
External links
edit- http://hub.mspnet.org/index.cfm/13083
- http://math.berkeley.edu/~wu/
- https://mathnuggets.com
- http://archive.aft.org/pubs-reports/american_educator/fall99/amed1.pdf
- http://www.teachrkids.com/
- http://nlvm.usu.edu/en/NAV/vlibrary.html
- http://archive.aft.org/pubs-reports/american_educator/issues/fall2005/aharoni.htm
- http://www.math.jhu.edu/~wsw/papers/PAPERS/ED/ee.pdf
- http://www.air.org/expertise/index/?fa=viewContent&content_id=598
- http://www.cs.nyu.edu/faculty/siegel/siegel.pdf
- http://www.helpinghandeducation.com/
Welcome to the Wikiversity Division of High School Mathematics, part of the School of Mathematics. This division focuses on high school mathematics. Progression should continue through each section. Each section may be instructed or learned thoroughly per year, or can be tailored to a custom study plan.
Introduction
editThese are the topics which are normally taught in high school. It should tend to be inclusive. Topics which are only present in some curriculae and topics that are not normally taught at the high school level but which high school student are capable of learning should be put under supplementary materials.
- Brief overview over each topic
- Study plan
Building basic mathematical knowledge
editFurther algebra
edit- Quadratic functions
- Polynomials
- Factorization of polynomials
- Linear Algebra and Matrices
Functions
edit- Trigonometry
- Exponential and Logarithm
- Complex numbers
Others
edit- Sets
- Series and Sequences
- Permutation and Combination
- Probability
- Statistic
- Vectors
- Geometry
Calculus
editSupplementary
edit- Graph Theory
- Group Theory
- Logic
- Number Theory
- Mathematical errors
Country-specific courses
edit- Advanced Level Mathematics - United Kingdom (Under Construction)
Extensions
edit- High school mathematics extensions contains numerous topics not covered in the standard high school curriculum.
Links to other languages in other Wikimedia projects
edit- 日本語高等学校数学 ＠Wikibooks - Japanese
External links
editAlgebra/Function graphing Algebra/Systems of simultaneous equations Algebra/Solving equations Algebra/Solving linear inequalities Calculus/Functions Algebra/Quadratic functions Algebra/Trigonometry Algebra/Polynomials Algebra/Factorization of polynomials
Introducing Limits
editLet's consider the following expression:
As n gets larger and larger, the fraction gets closer and closer to 1.
As n approaches infinity, the expression will evaluate to fractions where the difference between them and 1 becomes negligible. The expression itself approaches 1. As mathematicians would say, the limit of the expression as goes to infinity is 1, or in symbols: .
An interesting sequence is As gets bigger (in symbols , we have smaller values of , for
and so on. Clearly, can't be smaller than zero (for if we have that is less than zero). Then we may say that . Continuing with this sequence, we might want to study what happens when gets near to zero, and later what happens with negative values of going near to zero. Usually, the letter is reserved for integer values, so we are going to redefine our sequence as . If we take a sequence of values of , say
We see that the respective values of grows indefinitely, for
In this case, we might say that the limit of , in words, the limit of as goes to zero from right (as the sequence of values of goes to zero from the left in a graphic) diverge (or tends to infinity, or is unbounded, but we never say that it is infinity or equals infinity). Other case happens if we study sequences of values of such that every element of the sequence is lower than zero, the sequence is increasing but never exceeding zero. One example of such sequence is:
- , with
- , with
- , with
- , with
- , with
The values of decrease without bounds. The we say that , or that tends to minus infinity when goes to zero from left.
Basic Limits
editFor some limits (if the function is continuous at and near the limit), the variable can be replaced with its value directly:
For example,
and
(with not equal to 0)
Others are somewhat more complicated:
Note that in this limit, one may not immediately set equal to because this would result in the expression evaluating to
which is an undefined expression. However, one may reduce the expression by separating the terms into separate fractions (in this case, and ), which can be evaluated directly.
Right and Left Hand Limits
editSometimes, we want to calculate the limit of a function as a variable approaches a certain value from only one side; that is, from the left or right side. This is denoted, respectively, by or .
If the left-hand and right-hand limits do not both exist, or are not equal to each other, then the limit does not exist.
The following limit does not exist:
It doesn't because the left and right handed limits are unequal.
Note that if the function is undefined at the point where we are trying to find the limit, it doesn't mean that the limit of the function at that point does not exist; for an example, let's evaluate at .
Left-hand limit | Right-hand limit |
---|---|
Therefore:
Some Formal Definitions and Properties
editUntil now, limits have been discussed informally but it shouldn't be all intuition, for we need to be sure of certain assertions. For an example, the limit
We have seen that the function decreases as increases, but how do we guarantee that there isn't a value , say
such that is never smaller than ? If there is such , we might want to say that the limit is , not zero, and we can't test every single possible value of (for there are infinite possibilities). We must then find a mathematical way of proving that there isn't such , but for that we need to define formally what a limit is.
Right Limits, Left Limits, Limits and Continuity
editLet be a real valued function. We say that
if for every there is a such that, for every between and
where is the absolute value of .
This is the formal definition of convergence from the left. It means that for each possible error bigger than zero, we are able to find a interval such that for all in that interval, the distance between the value of the function and the constant is less than the error.
TODO: Graphics illustrating this.
In an analogous fashion, we say that
if for every there is a such that, for every between and , .
And to finish the necessary definitions,
if
and
- .
Example:
This is a assertion that must be proved. First, lets study the behavior of near zero;
where the arrow pointing right means implies. So, define the function
If , then . We have show how to find the delta in the definition of limit, showing that the limit of as x tends to zero is zero.
In fact, for any real number ,
Lets see how to construct a suitable function .
- , then
So,
implying that makes for any .
Functions with the property that
are called continuous, and arise very naturally in the physical sciences; Beware that, against the intuition of most people, not every function is continuous.
Property of Limits
editProperty one: If , then
for any constant .
Proof: Construct the function for . Then
So
Then the limit of is , for the delta function of is
QED.
TODO: Demonstrate main properties of limit (unicity, etc)
L'Hôpital's Rule
editL'Hôpital's Rule is used when a limit approaches an indeterminate form. The two main indeterminate forms are and . Other indeterminate forms can be algebrically manipulated, such as .
L'Hôpital's Rule states if a limit of approaches an intederminate form as approaches , then:
Example:
Both the numerator and the denominator approach zero as approaches zero, therefore the limit is in indeterminate form, so l'Hôpital's rule can be applied to this limit. (note: you can also use the Sandwich Theorem.)
Now the limit is in a usable form, which equals 1.
If the limit resulting from applying l'Hôpital's Rule is still one of the two mentioned indeterminates, we may apply the rule again (to the limit obtained), and again and again until a usable form is encountered.
Where does it come from
editTo obtain l'Hôpital's Rule for a limit of which approaches as approaches , we simply decompose both and in terms of their Taylor expansion (centered around ). The independent terms of both expansions must be (because both and approached ), so if we divide both the and by (or, equivalently, find their derivatives), our limit will stop being indeterminate.
It could be the case that the Taylor expansions of both the numerator and the denominator have a as coefficient of the term, thus yielding an indeterminate. This is the same case mentioned above where the trick was to repeat the process until a suitable limit was found.
The case of a limit which approaches can be transformed to the case above by exchanging with , which obviously approaches .
Prerequisites
editIn order to understand and calculate derivatives, one must understand the following topics:
Introduction to Differentiation
editDifferentiation is the process of finding the unique slope of a line at any point on that line, should such a slope exist. As we shall show, there are commonplace functions that fail this requirement at some point.
Generally, a slope can be found by taking the change in y-coordinates divided by the change in x-coordinates or:
where m is the slope.
Now what happens when we attempt to take the slope of a point on a line, or how fast the y's are changing in respect to the x's at a single point? The formula fails when or
In order to do this, we have to undergo differentiation.
Definition of a Derivative
editThe following formula will calculate the slope of a line at any point:
where indicates the derivative, and h is the difference between a point h units away from x and x
Given , to find the derivative of f(a) (where a is any x coordinate within the domain of f(x)), use the definition of derivative.
Here's an exercise to try:
Find , given that . The solution is below.
Therefore, the slope of 4x^2 at x=2 is 16.
This is a somewhat tedious process when bigger functions are involved. Take for example:
Using the definition of derivative, your equation looks like this:
Have fun solving that algebraically!
Luckily, this is where rules for derivatives come in.
Where Derivation Fails
editTwo examples of commonplace functions either have ambiguous derivatives or none at some point. The absolute value function has an ambiguous derivative at x=0 For |x|, |x| = -x and the derivative of |x| is -1 if x < 0. |x| = x and the derivative of |x| is 1 if x > 0. But if x=0, then |x|, x, and -x are all zero. No matter how close a number is to zero, the derivative of |x| is 1 for x > 0 but -1 for any x less than zero. Because one cannot determine whether the derivative of |x| is -1, 1, or something else at x=0, one has an ambiguous derivative at x=0... and thus none.
Another function that most off us know well is 1/x. For f(x) = 1/x
implies
No derivative of 1/x can exist at x=0.
Derivative Rules
editIn the following table, f(x) and g(x) are functions whereas "a" is a constant number. f'(x) and g'(c) are the derivatives of the functions f(x) and g(x), respectively.
Rule Name | Original Function | Derivative |
---|---|---|
Chain Rule | f(g(x)) | f'(g(x))•g'(x) |
constant function derivative | a | 0 |
Addition Rule | f(x)+g(x) | f'(x)+g'(x) |
??? | x | 1 |
Scalar Rule | a•f(x) | a•f'(x) |
Power Rule | (f(x))^{a} | a•(f(x))^{a-1}ˆ•f'(x) |
Product Rule | f(x)•g(x) | f(x)•g'(x) + g(x)•f'(x) |
Quotient Rule | f(x)/g(x) | (g(x)•f'(x) - f(x)•g'(x)) / (g(x))^{2} |
derivative of trigonometric function | sin(x) | cos(x) |
derivative of trigonometric function | cos(x) | -sin(x) |
derivative of trigonometric function | tan(x) | sec^{2}(x) |
derivative of trigonometric function | ln(x) | 1/x |
derivative of trigonometric function | e^{x} | e^{x} |
See http://en.wikibooks.org/wiki/High_school_mathematics_extensions for more High School Mathematics info and http://en.wikibooks.org/wiki/Wikiversity:School_of_Mathematics#WIKIVERSITY_SCHOOL_OF_MATHEMATICS for more links and courses.
Print version created by: writerhawk 23:34, 30 May 2006 (UTC)