Here's the deal: I'm working on curriculum for my school and Algebra 2 is making my eyes cross. I think the major problem is the state of Virginia is in a transition year between "old" Standards of Learning (SOLs), and "new" ones. This year is supposed to be the year that we're still teaching and assessing the old SOLs, but we're supposed to teach the new ones, too. Those of you that teach Algebra 2 already know that there's an enormous amount of information to cover in a short period of time. To give you context, our school teaches it as a semester-long block course. There's only so much a brain can handle in one day, though!
Here's the first draft of my skills list and structure...I'm not sure what to do about the old vs. new SOLs (my skills list is based on the old SOLs because that is what will be assessed).
Note: Gray items are not included in old or new SOLs but might be necessary for student understanding
Blue items are being taken out of the SOLs starting next year
Red items are new to the SOLs starting this year
Unit 1 Algebra 1 Review/Solving Equations
1 Solve multi-step equations and inequalities
2 Matrix +/-
3 Solve compound inequalities
4 Solve absolute value equations
5 Solve absolute value inequalities
Unit 2 Polynomial Review/Add Depth
6 Factor trinomial a = 1
7 Factor trinomial a > 1
8 Factor special cases (sum/diff of cubes, diff of squares, perfect square trinomials)
9 Factor out GCF first (factor completely)
10 Exponent rules
11 +/- polynomials
12 Multiply polynomials
13 Divide polynomials
Unit 3 Rational Expressions
14 Identify undefined values
15 Simplify rational expressions by factoring and canceling out common factors
16 Multiply and divide fractions
17 Multiply and divide rational expressions
18 Add and subtract fractions
19 Add and subtract rational expressions
20 Simplify complex fractions
21 Solve rational equations
Unit 4 Radicals, Radical Equations and Complex Numbers
22 Simplify numbers under radical
23 Simplify monomials under radical
24 Multiply and divide radicals
25 Add and subtract radicals
26 Nth roots to rational exponents and vice versa
27 Simplify expressions with nth roots and rational exponents
28 Solve radical equations
29 Simplify square roots with negative terms inside radical using i
30 Add and subtract complex numbers
31 Powers of i
32 Multiply complex numbers
Unit 5 Functions (intro)
33 Domain and range of relations (from ordered pairs, mapping, graph, table)
34 Identify relations that are functions and one-to-one
35 Given graph and a value k, find f(k)
36 Given graph, find zeros
37 Given graph and a value k, find where f(x)=k
Unit 6 Linear Functions
38 Slope from graph, equation, points
39 Graph from equation
40 Equation from graph
41 x- and y- intercepts
42 Determine whether lines are parallel, perpendicular, or neither from equation or graph
43 Write equations for parallel and perpendicular lines given line and point off the line
44 Graph linear inequalities
Unit 7 Systems
45 Solve systems of equations by graphing
46 Multiply Matrices using a graphing calculator
47 Inverse matrix method of systems
48 Systems of equations word problems
49 Graph systems of linear inequalities
50 Linear programming max/min problems
Unit 8 Functions (reprise)
51 Function math (addition, subtraction, multiplication, division)
52 Function composition, find a value i.e. f(g(3))
53 Function composition, find the function i.e. f(g(x))
54 Find an inverse function by switching variables
Unit 9 Quadratics
55 Graph from vertex form, identify max/min and zeros
56 Solve by factoring
57 Solve by Quadratic Formula (including complex solutions)
58 Determine roots using the discriminant
59 Write equation for quadratic given roots
60 Quadratic systems
61 Polynomials: relating x-intercept, zeroes and factors
62 End behavior for polynomials
Unit 10 Exponential/Logarithmic functions
63 Exponential growth or decay from function
64 Sketch base graph of exponential/log functions
65 Exponential to log and vice versa
66 Data analysis/curve of best fit for linear, quadratic, exponential and log
Unit 11 Transformations and Parent Functions
67 Graph absolute value functions
68 Horizontal and vertical translations of linear, quadratic, cubic, abs value, exponential and log
69 Reflections and stretching of linear, quadratic, cubic, abs value, exponential and log
70 Combinations of transformations on parent functions
71 Identify parent graphs of parent functions
72 Identify equations of parent functions
Unit 12 Conics
73 Identify a conic from graph
74 Identify a conic from equation
Unit 13 Variations
75 Write equation for direct, inverse and joint variation problems
76 Find the constant of variation
Unit 14 Sequences/Series
77 Write n terms of an arithmetic sequence
78 Find the sum of a finite arithmetic series
79 Write n terms of geometric sequence
80 Find sum of geometric series
81 Use formulas to find nth term
82 Identify sequence/series as arithmetic, geometric or neither
Unit 15 Statistics
83 Determine probabilities associated with areas under the normal crve
84 Compute permutations and combinations
If you made it this far, here's my call for help: Anyone have advice/suggestions for how to make this work and/or a better way to organize the information into cohesive units that seem to occur in a somewhat logical order? There is and will continue to be an emphasis on function families and transformations (as there should be). I find it difficult to express on paper how each function category needs to be a resting place, but they are all connected in the ways that transformations apply. Any ideas?
...oh...and I'm going to be teaching one section of deaf students and one section of blind students...in case that makes a difference
**edit: I've added links to the old and new Virgina SOLs for Algebra 2 if anyone's interested**
multiplying radical expressions calculator
Tuesday, August 31, 2010
Biology and the Laws of Marriage
After waiting a bit to hear if there was any argument in regards to what he said grandpa continued his talk at the meeting.
“First of all we must admit that those who favor decisions in regards to the rights of homosexuals are predominantly those who have accepted the theory that homosexuality behavior is a genetically determined trait rather than a learned trait. How many would favor homosexual marriage if sexual behavior were not genetically determined?
And there are societies that show tolerance towards the behavior of homosexuality. But let’s not confuse behavior with state of being. Before we decide to give certain rights to certain individuals who are members of a certain collective we need to make the determination that to do so is beneficial to society as well as it is beneficial to the individuals.
“I brought up the question in an earlier talk of, what would happen in a single gender collective in regards to sexual behavior? Well, we do have collectives of single gender in this society already. Those collectives are known as prisons. And we already know what happens in them in regards to sexual activities. And I challenge anyone to declare that only homosexuals are sexually active in prisons.
If DNA is a valid means to determine biological traits in people and if biological traits are an aspect of the principle of determination rather than free will that can mean only one thing. There must be a gene that controls the sexual orientation of a person and that must be a common gene in each of us.
I have but one question in regards to this. If this is true then two persons possessing the same DNA must be of the same sexual orientation. So, explain to me how there can be identical twins and one of those twins is a heterosexual while the other is a homosexual?
Wikipedia has an interesting entry entitled “Biology and Sexual Orientation”. It describes the many ways that studies have been done in regards to sexual orientation. It seems as if the best that these studies could conclude is and I quote;
“No simple, single cause for sexual orientation has been conclusively demonstrated, but research suggests that it is by a combination of genetic, hormonal, and environmental influences,[1…”.
In another area of this entry in regards to the study of twins it had this to say and I quote,
“Nonetheless, it is possible to conclude that, given the difference in sexuality in so many sets of identical twins (who are genetically identical), sexual orientation cannot be purely caused by genetics.”
In closing I remind everyone that rights belong to individuals not groups. Thus if we are to determine whether or not the rights of an individual is being denied to any individual by law then we must address the individual.
“ Do the laws on marriage prevent a homosexual from getting married and knowing all of the benefits that there are in marriage? The answer is no. There are no laws that prevent that. The sexual preference of the parties to be married is not a requirement of the State for the purpose of marriage. The only requirement is that the person that one is to be married to is a person of the opposite gender. And this applies to all persons.
Is it true that persons of certain sexual orientation can marry anyone that they want to while persons of a different sexual orientation cannot? The answer is no. Once a law has been enacted in regards to a certain issue no one has the right to do as they want in regards to that issue. The law applies to every single individual thus all are abiding by the law or not abiding by the law. The only way for a person to do what they want is for there to be no laws in regards to what he wants. Think about what this means the next time you hear someone say that there oughta be a law…”
With these words said, grandpa stepped down from the podium and sat next to grandma to await the words of the next speaker.
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“First of all we must admit that those who favor decisions in regards to the rights of homosexuals are predominantly those who have accepted the theory that homosexuality behavior is a genetically determined trait rather than a learned trait. How many would favor homosexual marriage if sexual behavior were not genetically determined?
And there are societies that show tolerance towards the behavior of homosexuality. But let’s not confuse behavior with state of being. Before we decide to give certain rights to certain individuals who are members of a certain collective we need to make the determination that to do so is beneficial to society as well as it is beneficial to the individuals.
“I brought up the question in an earlier talk of, what would happen in a single gender collective in regards to sexual behavior? Well, we do have collectives of single gender in this society already. Those collectives are known as prisons. And we already know what happens in them in regards to sexual activities. And I challenge anyone to declare that only homosexuals are sexually active in prisons.
If DNA is a valid means to determine biological traits in people and if biological traits are an aspect of the principle of determination rather than free will that can mean only one thing. There must be a gene that controls the sexual orientation of a person and that must be a common gene in each of us.
I have but one question in regards to this. If this is true then two persons possessing the same DNA must be of the same sexual orientation. So, explain to me how there can be identical twins and one of those twins is a heterosexual while the other is a homosexual?
Wikipedia has an interesting entry entitled “Biology and Sexual Orientation”. It describes the many ways that studies have been done in regards to sexual orientation. It seems as if the best that these studies could conclude is and I quote;
“No simple, single cause for sexual orientation has been conclusively demonstrated, but research suggests that it is by a combination of genetic, hormonal, and environmental influences,[1…”.
In another area of this entry in regards to the study of twins it had this to say and I quote,
“Nonetheless, it is possible to conclude that, given the difference in sexuality in so many sets of identical twins (who are genetically identical), sexual orientation cannot be purely caused by genetics.”
In closing I remind everyone that rights belong to individuals not groups. Thus if we are to determine whether or not the rights of an individual is being denied to any individual by law then we must address the individual.
“ Do the laws on marriage prevent a homosexual from getting married and knowing all of the benefits that there are in marriage? The answer is no. There are no laws that prevent that. The sexual preference of the parties to be married is not a requirement of the State for the purpose of marriage. The only requirement is that the person that one is to be married to is a person of the opposite gender. And this applies to all persons.
Is it true that persons of certain sexual orientation can marry anyone that they want to while persons of a different sexual orientation cannot? The answer is no. Once a law has been enacted in regards to a certain issue no one has the right to do as they want in regards to that issue. The law applies to every single individual thus all are abiding by the law or not abiding by the law. The only way for a person to do what they want is for there to be no laws in regards to what he wants. Think about what this means the next time you hear someone say that there oughta be a law…”
With these words said, grandpa stepped down from the podium and sat next to grandma to await the words of the next speaker.
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Calculus with TI89 calculator
I love my TI-89 calculator - but I hate the manual that came with it.
Dont get me wrong, maybe it is just that I have a really old version and it has been improved since my version was released - or maybe the manual is supposed to be a "reference" book instead of a user guide.
Anyway, I'm doing Calculus at University and so I decided to spend some time going through the manual and summarising the functions that I use the most.
Remembering the function names
All of the functions shown below can be accessed from the "Catalog" menu or the "Math" menu. The "Math" menu is much better as it groups all of the functions by their usage. (EG Matrix, Calculus etc)
Unfortunately the "Math" menu does not provide the (limited) help that the "catalog" menu provides. After using the commands for a while, you quickly get used to the available parameters.
C! alculus
Differenciation
Use the F3 function key (option 1) or 2nd + 8 keys
d(x^2+2x,x) - Differenciate the function x^2+2x
d(x^2+2x,x,2) - Get the second differencial of the function
Integrals
Use the F3 function key (option 2) or 2nd + 7 keys
S(2x+2,x) - Get the integral of 2x+2 dx
S(2x+2,x,1,5) - Get the area under the function between 1 and 5
Local Minimum and Maxima (turning points)
fMin(x^2,x) - Get the min candidates of the function x^2
fMax(x^2,x) - Get the max candidates of the function x^2
Limits
limit(1/x^2,x,0) - Find the limit of 1/x^2 as x approaches 0
limit(1/x,x,0,1) - Find the limit of 1/x as x approaches 0 from above
limit(1/x,x,0,-1) - find the limit of 1/x as x approaches 0 from below
Algebra
Intercepts
To find the x intercepts, use the zeros function.
zeros((x+2)^2-2 , x) - List where the function crosses the x axis
To find where the function crosses the y axis, we use the "with" operator to set x equal to zero. The with operator is a vertical bar (on the button above the EE button)
(x+2)^2+2 | x=0
Solving Inequalities
solve( abs(2x+2)+1 < 5 )
Solving Quadratics
"factor" and "expand" are opposites of each other.
solve( x^2+4x+2=0 , x) - Find the values of x that make the equation true
factor( x^2+3x+2 ) - Find the factors [ eg (x+1)(x+2) ] of the equation
expand( (x+1)(x+2) ) - Convert the factors into an equation
Matrices and Vectors
Use ; to sepe! rate rows and commas to seperate columns
[1,2,3;4,5,6] x [3,2;4,5;6,7] - Find the multiple of two matrices
ref( [1,2,3;4,5,6] ) - Convert to row echelon form
rref( [1,2,3;4,5,6] ) - Convert to reduced row echelon form
Transpose a matrix
Use the catalog button to get access to the letter T used in the following example
[1,2,3;4,5,6]T - Transpose the matrix
Inverse of a matrix
Note : The matrix needs to be square
[1,2;3,4]^-1
Determinant
det( [1,2,3;4,5,6;7,8,9] ) - Return the determinant of a matrix
Identity Matrix
Here is a quick way to enter an identity matrix. Saves entering all the 0's and 1's
identity(3) - Returns a 3 x 3 identity matrix
Normal (length) of a vector
norm( [1,2,3] ) - returns the norm of a vector
dotP( [1,2,3] , [4,! 5,6] ) - Return the dot product of two vectors
Showing your working
We are often asked to show our working in our exams/tests. The easiest way to work with matrices is to use the rowSwap and mrowAdd functions. It is also a good idea to use the "Ans" key (2nd (-) button) to save re-entering the matrix for each step.
You need to remember that the "last" value entered is the row that will be replaced
rowSwap( Ans , 1 , 2) - Swap row 1 and 2
mrowAdd( 3, Ans , 1 , 2) - Row 2 is replaced with row 2 + 3 times row 1
Subtraction of a row can be done by using -1 as the first element in the mrowAdd function
Conclusion
There are many more functions that the TI89 is capable of. But these are the ones that I have found most useful in Stage 1 Mathematics. As I proceed through the year, I will no doubt find even easier ways to do things and will post them as I find them.
limit calculator step by step
Math online tutoring
i used to be a Math whiz when i was young. seriously, i was an excellent student in Math. i often competed in inter-school quiz bees as our school's representative. my parents were very pleased because they didn't need to hire a tutor for me. most students my age then had difficulty understanding the lessons in Math, and i do recall having to teach my classmates in their assignments. it was harder for students back then to actually get the help they need in comprehending the lessons.
free online tutoring math
good thing K-12 students can now get online tutoring so their lives will be a lot better when dealing with the much-dreaded subject. 5th grade Math has become easier, and so is 4th grade Math. 5th grade Math may be tougher than it looks, but with the proper online tutoring tool, students can excel in this subject, too. before they know it, adding fractions will just be a piece of cake.
online tutoring is not only great for 4th and 5th graders, but also for K-12 students who take up Algebra 2. yes, a lot of people may find Algebra 2 difficult but good thing help is now available to those who find it challenging. i know formula for volume and graphing linear equations can seem daunting, but there is hope now that online tutoring is within one's fingertips.
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Pan balance problems to teach algebraic reasoning
Today I have a "goodie" for you all: a free download of some pan balance (or scales) problems where children solve for the unknown:
Just right click on the link and "save" it to your own computer: Balance Problems (a PDF). This lesson is also included in my book Math Mammoth Multiplication 2 and in Math Mammoth Grade 4 Complete Worktext (A part).
The problems look kind of like this:
These can help children avoid the common misconception that equality or the equal sign "=' is an an operation. It is not; it is a relationship.
You see: many students view "=" as "find the answer operator", so that "3 + 4 = ?" means "Find what 3 + 4 is," and "3 + 4 = 7" means that when you add 3 and 4, you get 7. To students with this operator-view of equality, a sentence like "11 = 4 + 7" or "9 + 5 = 2 × 7;" makes no sense.
You might also find these resources useful:
Balance word problems from Math Kangaroo
Algebraic Reasoning Game - a weighing scales game that practices algebraic rea! soning
Interactive Pan Balance with Shapes
A Balanced Equation Model from Absorb Mathematics
An interactive animation illustrating solving the equation 4x + 6 = x - 3. Drag the green handles to balance each side. Click the arrow button to reset the animation. On the right side, you'll see links to similar animations of equation solving using a balance.
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Have questions about math problems
Have questions about math problems, get help on this website. You will learn the toughest questions with the easiest way to solve it!!
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easy math problems
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