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Friday, February 29, 2008

Algebra Period 3 (Friday)

Parallel and Perpendicular Lines 7-8

2 SPECIAL LINES AND THEIR SLOPES

PARALLEL LINES:
2 lines that are parallel to each have the SAME SLOPE!
y = 2x - 10 and y = 2x + 3/4 are parallel because they both have a slope of 2

PERPENDICULAR LINES:

2 lines that are perpendicular to each other have SLOPES that are:
OPPOSITE SIGNS
and
RECIPROCALS
y = 2x - 10 is perpendicular to y = -1/2 x + 3/4


THIS IS ANOTHER TWIST TO OUR MYSTERY LINE PUZZLE!!!
If you know that the mystery line is parallel or perpendicular to another given line
then you know the mystery line's slope!!!


EXAMPLE:
Your mystery line has a point of (2, -5) and
is PARALLEL to line y = 2x + 3/4
So you know the mystery line's slope because it is the same as the given line
( m = 2)
Substitute the slope and the point given on the mystery line and solve for b.


EXAMPLE:
Your mystery line has a point of (2, -5) and
is PERPENDICULAR to line y = 2x + 3/4
So you know the mystery line's slope because it is the opposite sign reciprocal of the given line

Lines that parallel to the y axis in the form x = c, ( such as x = 3) have an "undefined slope"
or NO SLOPE because if you find their slope using
y2 -
y1 / x2 - x1
you will discover
a number/0
which is undefined

On the other had lines parallel to the x axis, in the form y = b ( such as y = -5) have a zero slope becuase using the same
y2 - y1 / x2 - x1
you will discover
0/ a number
which = 0

Thursday, February 28, 2008

Algebra Period 3

Finding the Equation of a Line 7-6

Always remember that y = mx + b and you can substitute and solve for whatever you're missing!
It helps to memorize this little rhyme:
Oh mystery line,
What can you be?
If I could only find you,
y = mx + b

So first I find m
Then I find b.
Now put it all together
And you've found me!
y = mx + b


The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:
1) Do I have the slope (m)? If not, find it by using the slope formula
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope
3) Don't forget to put it all together in one equation at the end.

HERE ARE THE 5 CASES THAT THE BOOK INCLUDES:

First case:You're given the slope and the y intercept (easiest case)
m = 3/2 b = -7/5
Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5

Second case: You're given a point and and the slope and need to find the intercept (b)
(3, 1) m = 2
Plug in the point and the slope and solve for b
1 = 2(3) + b
1 = 6 + b
b = -5
Now put it altogether with the given slope and the intercept you just found:
y = 2x -5

Third case: You're given a point and and the y intercept and need to find the slope
(3, 1) b = 2
Plug in the point and the y intercept and solve for slope
1 = 3m + 2
-1 = 3m
m = -1/3
Now put it altogether with the given intercept and the slope you just found:
y = -1/3 x + 2

Fourth case: You're given 2 points and need to find the slope and the intercept
(1 , 3) and (-2 , -3)
You need to first find the slope:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one of the points and find the intercept b
3 = 2(1) + b
3 = 2 + b
b = 1
Finally, put it all together:
y = 2x + 1

Fifth case: You have a graph of a line and need to determine the equation

Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)
(If the y intercept is not an integer, then follow fourth case completely)
Put the information together in y = mx + b form


Another way to find the equation with 1 point and slope-
Point Slope Form of the equation

You know one point and the slope. This is the same case a the SECOND CASE, but there is a ANOTHER WAY to solve it other using slope intercept form. This is not the most common way. Most people use the slope intercept form for all cases.

This one is very specific, but many students love it! (It is MY personal favorite!!)

Point-slope form of a line: You know one point and the slope. Use the following formula:
y - y1 = m (x - x1)

Using the same example from the second case above: (3 , 1) and m = 2

y - 1 = 2 (x - 3)
Simplify: y - 1 = 2x - 6
y = 2x - 5

If you're trying to link the slope intercept form to the point slope form of the same line:


The point slope version eliminates one step from using the slope intercept form. In the slope intercept form, you plug in the point and slope, solve for b, and then rewrite the equation using the intercept that you found. In point slope form, once you plug in the point and slope, you just simplify and the equation is already done!




Pre Algebra Periods 1, 2, & 4

Probability and Odds 6-4

Outcomes-- possible results of any action

an event-- any outcome or group of outcomes.

Probability of an even is expressed either as a ratio or as a percent (%)

P(event) = # of favorable outcomes/ # of possible outcomes
The Probability of finding an 's' in Mississippi
P(s) = 4/11 = .363636... = 36 4/11%

Odds

favorable outcomes/ unfavorable outcomes

DO NOT WRITE AS A PERCENT

odd of getting a 's' in Mississippi becomes 4/7 or 4:7 or 4 to 7


The odds of UCLA winning the NCAA 3:1 3 favorable/1 unfavorable

changing that to probability it becomes 3/4 or 75%


probability -- the denominator is TOTAL outcomes

odds-- the denominator is unfavorable outcomes.

Wednesday, February 27, 2008

Sunday, February 24, 2008

Pre Algebra Periods 1, 2, & 4

Similar Figures & Scale Drawings 6-3

Proportions are used a lot in Geometry


Two figures are similar if they are proportional

(1) they are exactly the same shape but different sizes

(2) each corresponding side of one of the figures is the same ratio of every other corresponding side.

(3) every corresponding angle is exactly the same measurement ( congruent)

All circles and all squares are similar.

Every square has 90° angles and each side will be in the same ratio to any other square.

Every circle has a measurement of 360° and every circle has the same ratio between its circumference and its diameter. That is C/d = π.

This is actually true for every regular polygon.

What is a regular polygon? All sides of the polygon are equal and all angles are equal!! Or.. a polygon with congruent sides and congruent angles. For example:

Equilateral triangle

Square

A good example of a regular octagon would be a stop sign.

When you name similar figures, you should go in the same order of their corresponding parts.
The symbol for SIMILAR looks like 1 of the 2 wavy lines that means approximately equal The symbol ~ means “is similar to”


SCALE DRAWING PROBLEMS ARE PERFECT TO SOLVE WITH PROPORTIONS!

For example a map says that 3 cm: 1000 miles
You measure that you need to go 5.4 cm on the map
How many miles is that?

Set up a proportion with the labels

cm/mi = cm/mi

3/100 = 5.4/x

Use cross products to solve

3x = 1000(5.4) but don’t multiply out yet

divide both sides by 3 so

x = 1000(5.4)/3 use your divisibility rules to simplify first

x = 1000(1.8) because 5.4/3 = 1.8

x = 1800 miles. Make sure your answer seems reasonable. That is, you know if 3cm is 1000 miles , then 6 cm would represent 2000 miles so your answer should be less than 2000 miles.

Indirect measurement

Person 5 ft tall casts a 4 ft shadow. A tree casts a 10 foot shadow. How tall is the tree. Set up proportions but make sure your labels match

5/4 = x/ 10 5(10) = 4x again don’t multiply out first

divide both sides by 4 so 5(10)/4 = x 25/2 = x so the tree must be 12.5 ft tall

set up

actual height of person/shadow = actual height of tree/shadow

or

person shadow/ tree shadow = person’s height/ tree’s height


You will arrive at the same answer!!

Saturday, February 23, 2008

ISTE Standard-Related Goals

Through this course, I have gained greater insight in how to use many technological tools in new ways and to integrate them effectively into my classroom. While I was very comfortable with using several of the tools prior to this class, I believe I have grown tremendously in my knowledge of technological tools and expanded my abilities significantly as well. Now, focusing upon the elements contained in Standards I and V of the National Educational Technology Standards for Teachers I see how those key multimedia tools will support my growth and movement across the rubric to becoming a level four on all of the performance indicators.


I. Technology operations and concepts

A. Demonstrate introductory knowledge, skills, and understanding of concepts, related to technology (as described in the ISTE National Educational Technology Standards for Students).

One specific goal I have set for myself to help me move from level three to level four on the ISTE rubric for standard 1A is to use standards set for students to plan, design and implement technology-based activities in my classroom. I will do this by creating projects such as the Data Analysis Project, refined through this course. Assessments such as this project take current classroom math concepts and embed them into projects which are also designed to teach new technologies to students. In this example, students learn how to use Excel and PowerPoint as mediums for presenting the results of their data gathering. I realize some students will come into the class with knowledge of how to implement these programs; however, many students will need to be taught how to use Excel, PPT and how to integrate them together. Time in the computer lab for creating this project is essential. Moreover, as per the ISTE NETS for 5-8 graders I see the importance of including in my instruction data gathering, pattern examination, and information application for decision making using digital tools and resources. One digital tool that I found extremely exciting from this class was using filamentality to create hotlists and webquests as an alternative means of exploration for my students. With projects like these, I feel confident that I will be able to meet level four status on the ISTE Standards rubric. I am comfortable with my abilities to meet some of the level four requirements, such as solving routine hardware and software problems and proficiency with the use of common input and output devices. However, I recognize that I need to expand my abilities to fully meet this level of proficiency across the board.

B. Demonstrate continue growth in technology, knowledge and skills to stay abreast of current and emerging technologies.

I am definitely at a level three as I continually make an effort to learn about “new advances in technology by attending conferences, communicating with colleagues about the use of technology in education, and engaging in hands-on exploration of variety of technology tools.” My goal is to remain at the forefront of any and all new and emerging technology that will benefit my teaching as well as my students’ learning. I plan to achieve this goal by continuing to attend as many conferences, learning forums, and district-sponsored technology support courses as possible. More importantly, I want to continue attending Southern California’s CUE (Computer Using Educators) Conference every spring. At this conference I have continually been introduced to new ways of using technology to improve my students’ learning. Signing up for technology classes provided by my district during school as well as attending those offered after school hours is another way I will stay abreast of such new technologies. At such various learning forums, my immediate goals include become more proficient at using a blog as a classroom tool, learning moodle, and podcasting. I also intend on continuing my education in web design, becoming more proficient at using DreamWeaver. The information and knowledge I gained from Lynell Burmark in her book Visual Literacy Learn to See. See to Learn about using color, font, music, and images, has transformed my opinion of how to create both an instructional PowerPoint presentation and an effective classroom website.


V. Productivity and Professional Practice

A. Use technology resources to engage in ongoing professional development and lifelong learning.

I rank myself somewhere between level three and four; I am constantly looking for new classes or conferences to attend, always striving to acquire new skills and learn new technology. I will continue to belong to CUE and ISTE and as a result of EDUC 6662, I have recently joined several listservs—including Middle-L and Math Forum’s mathedcc. In addition, I am a member of both the National Middle School Association and NCTM, receiving online journals about subjects and topics that are relevant to middle school teachers. My goal is to expand my professional development repertoire and explore new venues in depth. During the next six months, I intend to enroll and participate in an ISTE Webinar that focuses on classroom blogging, learning with second life, or podcasting as an educational tool.

B. Continually evaluate and reflect on professional practice to make informed decisions regarding the use of technology in support of student learning.

I rank myself once again at level three. Being able to evaluate technology for its benefits in my classroom so that it supports my students in learning, and make decisions accordingly is a skill that I want to develop further. My goal is to engage in continuous assessment of the effectiveness of the technological tools implemented in my classroom to insure that the most meaningful learning is achieved. I feel that this is a critical component of my teaching because I do utilize a great number of technological tools within my classroom. I definitely evaluate the benefits of using various technological tools and resources in my classroom.

C. Apply technology to increase productivity

For the most part, I rank myself at a level three on this performance indicator. I am aware of technology resources available to me and I have begun using many of those outlined in our course. Throughout this class, I have participated in discussion boards, increasing my technological knowledge and productivity. The Week Two discussion dealt with finding relevant resources on a variety of professional development topics, such as assessments, teaching with standards, and classroom management. Not only did I gain tremendous awareness of what is available with my topic of assessment but I now have a huge reservoir of resources provided by my colleagues at Walden. My goal is not only to follow the discussions on some of the listservs but to post questions on either the Middle-L or mathedcc listservs. I have not participated in that manner and have been intimidated by this type of communication. However, I realize that I need to experiment with this venue. During the next few weeks, I plan to respond to someone’s posted question and pose a question of my own.

D. Use technology to communicate and collaborate with peers, parents, and the larger community in order to nurture student learning.

I rank myself somewhere between level three and four. I am aware of the wide variety of technology recourses to communicate and collaborate with colleagues and I am making use of several of these resources in my classroom. My goal is to use them greater to help facilitate communication with peers, parents, and others. Students and parents use my websites to verify daily homework assignments, review requirements for long term projects, participate in online quizzes, and check current grades. My webpages are updated continuously to support the changes in my classroom. My goal is to increase the time between each communication; although I usually send out a NewsFlash twice a month, I intend to add at least once a week. The positive feedback I received from two parents this week further supports the need to communicate with parents even with these little flashes on a regular basis. In addition, during EDUC 6662, I have begun adding notes from the class lessons on my recently established classroom blog. Again, the goal is to increase the addition of notes from class. My ultimate goal is to have students submitting their notes and comments about mathematical content covered during the class period. I also communicate with parents, tutors, and students via email. I encourage students to email me with specific math questions, answering them on a daily basis. My goal includes learning how to collaborate with peers and the larger community using other technology resources such as listservs and videoconference-ing.

References

Burmark, L. (2002). Visual literacy: Learn to see, see to learn. Alexandria, VA: Association for Supervision and Curriculum Development.

ISTE’s Educational Technology Standards for Students Retrived Feb. 18, 2008, from http://www.iste.org/Content/NavigationMenu/NETS/For_Students/NETS_S.htm

ISTE Webinar Series Retrieved Feb. 20, 1008 from http://www.iste.org/Content/NavigationMenu/Professional_Development/Webinar_Series1/ISTE_Webinar_Series1.htm

Laureate Education, Inc. (Executive Producer). (2002). Multimedia tools: how to research, plan, and communicate with technology. Program three, “multimedia design, part one: text and images.” [Videotape]. Baltimore: Canter & Associates, Canter & Associates, Inc. (Producer)

Laureate Education, Inc. (Executive Producer). (2002). Multimedia tools: how to research, plan, and communicate with technology. Program four, “multimedia design, part two: color and music.” [Videotape]. Baltimore: Canter & Associates, Canter & Associates, Inc. (Producer)

Leu, D. J., Leu, D. D., & Coiro, J. (2004). Teaching with the Internet K–12: New literacies for new times (4th ed.). Norwood, MA: Christopher-Gordon.

Friday, February 22, 2008

Pre Algebra Periods 1, 2, & 4

Proportion Problems:
Remember it's all about the labels
Set up your problem first with your labels on either side!!

A lion has 12 heartbeats in 16 seconds. and h heartbeats in i minutes.

Oops, what do we need to do first?

Change the 1 minute into seconds so we are using the same labels!!

heartbeats/ seconds so 12/16 = h/60

Solve using cross products or 12(60) = 16h

Don't multiply out just yet. Get that variable alone first so divide both sides by 16

12(60)/16 = h Now simplify before you multiply--

That is, think what number goes into both 12 and 16?
4... so your equation becomes 3(60)/4 = h
but wait... you can simplify again because 4 goes into 60 -- how many times?
15 times.
So you get 3(15) = h or h = 45
You aren't done yet...
Your answer is 45 heartbeats.

Algebra Period 3 (Friday)

Slope 7-4

You can think of slope of a line as the slope of a ski mountain.

When you are climbing up-- it's positive
When you're sliding down, it's negative.

(Okay, okay-- you must be looking at the mountain from left to right!!)

The steeper the mountain, the higher the slope.
A slope of 5 would be an expert slope because it is much steeper
than a slope of 2 ( an intermediate's slope)

"bunny slopes' for beginners will be the lower numbers--
generally fractional slopes ( like 1/2 or 2/3).

A good benchmark is a slope of 1 or -1. Notice it is a 45 degree angle.

You can also think of slope as rise/run which is read ' rise over run'

Rise is how tall the mountain is ( the y- axis)
Run is how wide the mountain is ( the x value)

A 1000 foot high mountain ( the rise) is very steep-- if it is only 200 feet wide ( the run)
It's slope is 1000/200 = 5/1

Another mountian that is also 1000 feet high is not very steep if it is 2000 feet wide.
It's slope is 1000/2000 = 1/2

You can think of slope as a calculation:

Risexxx= wChange in y values= sDifference in y valuess= swy2 - y1
____ ss=s_____________ =s_______________ s= x_____
Run sssssswChange in x value xxx Difference in x value ssw x2 - x1

To calculate slope you need 2 coordinates. It doesn't matter which one you start with!! Just be consistent!!. If you start with the y value of one point, make sure you start with the same x!!

Special Slopes:

Horizontal lines in the form of y = k have slopes of ZERO ( they are FLAT)

Vertical lines in the form of x = h have NO SLOPE or UNDEFINED because the denominator is zero.

Wednesday, February 20, 2008

Algebra Period 3 (Wednesday)

Linear Equations and Their Graphs 7-3

Equations whose graphs are lines are linear equations.
Must have only 1st powers only, no products or variables and no variables in the denominator.
Examples of linear equations:

y = 2x + 1 y - 3x = -2 5y = -4 9x - 15y = 7

Examples of Nonlinear Equations

y = x2 - 4 x2 + y2 = 16 y = 2/x xy = 3

Since two points determine a line, plotting 2 points is sufficient for graphing linear equations. We should, HOWEVER, use a third point as a check!!

Graph the equation 2y - 4 = 4x

solve for y first

y = 2x + 2
find three solutions by setting up a T-chart and choosing values for x

if x = 0 y = 2(0) + 2 y = 2

if x = 1 y = 2(1) + 2 y = 4

if x = -2 y = 2(-2) = 2 y = -2

Plot the points ( 0,2) ( 1, 4) and ( -2, -2) AND DRAW the line containing those points.

Every point on the line is a solution to the linear equation!!

We can graph a linear equation by finding any two points that belong to the graph. Often the easiest points are where the graph crosses the axes.

The x- intercept of a line is the x-coordinate of the point where the line intercepts the x-axis.

The y- intercept of a line is the y-coordinate of the point where the line intercept the y-axis.

Graph 4x + 3y = 12 using intercepts

To find the x-intercept, let y = 0
then 4x + 3(0) = 12
4x = 12
x = 3 the x intercept is 3. WE plot the point (3,0)

To find the y-intercept, let x = 0
then 4(0) + 3y = 12
y = 4 the y intercept is 4 We plot the point (0,4)

Connect the two points and draw the line containing them. Make sure to always include arrows on your lines.

The standard form of a linear equation in two variables is
Ax + By = C, where A, B, and C are constants and
A and B are not BOTH zero.

5x + 7y = 35 3x -2y = 6 are examples of equations in standard form

y = 3 will be a line parallel to the x-axis written in standard form
0X + 1Y = 3 you can see that for any value of x y = 3. Thus, any ordered pair (x,3) such as (0,3), or (4,3) or even (-20, 3) would be a solution.
The line is parallel to the x-axis with y=intercept 3.


x = -5 will be a line parallel to the y-axis.

Math 6 Honors Periods 6 & 7 (Wednesday- Friday)

AstroCampers-
Take lots of pictures and email some to me!! I can't wait to see all those great pictures. Let's post them here!! I want to hear all about your adventures at AstroCamp. This blog is a great way to let us know what a great time you had, what you did, what the weather was like, how good the food was, who your cabin mates were-- and if they snored... etc. Give us all the details!!

We definitely missed you but had fun practicing squares and approximating square roots, learning base 5 and the binary system-- just ask any one of the students who were in class Wednesday about what 1001001 represents in binary.

Pre-Algebra Student Entry (Tues. & Wed.)

Ratio and unit rates 6-1

A Ratio compares two quantities by division.
Ratio= fraction with MEANING
Labels are CRITICAL
16:14 What does it mean?
16 girls: 14 boys Has meaning

Rate compares quanities with two different units.
Rate=Ratio with 2 different lables.
I drive 180 miles in 3 hours
180 miles/ 3 hours

A unit rate has a denominator of 1.
EX:
60/1 MPH

Order of numbers in ratio are VERY IMPORTANT!


Proportions 6-2

A proportion is two equal ratios.
EX:
4/3= y/21
Solve for the variable! (y)

Propotions are all about setting up the labels!
3 oz. = $1.65 How much is 5 oz.?
Set up the problem FIRST! DONT FORGET THE LABLES!

Label A ________ = _______ Label A
Label B wwwwwwwwwwwwww Label B


Lorenzo L. Per:2

Tuesday, February 19, 2008

Math 6 Honors Periods 6 & 7 (Tuesday)

Solving Equations 11-7

Now that we have learned about negative integers, we can solve an equation such as
x + 7 = 2

subtract 7 from both sides
x + 7 = 2
-7 = -7

ask yourself 'Who Wins? and set up a side bar with the winner on top.

Then take the difference. So in this case we know that -7 wins-- so our answer will end up being negative. We put 7 on top and take the difference 7 -2 = 5
x = -5

What happens if we have 3x = 15. First looking at the problem, notice that x is being multiplied by 3-- so we need to use the inverse to isolate the variable. So we will divide by 3

3x /3 = 15/3 or x = 15

But what happens if we have 1/2 x + 3 = 0. First we need to separate the variable from the addition or subtraction operation so we use the inverse of + 3 and subtract 3 from both sides
1/2 x + 3 - 3 = 0 - 3 or 1/2x = -3
Now, we need to multiply both sides by 2/1 ( the reciprocal of 1/2)
(2/1)1/2x = -3(2/1) or x = -6

Monday, February 18, 2008

Algebra Period 3 (Tuesday)

Graphing Linear Equations 7-2

How do you determine whether a given number is a solution?
.. as the commercial states... 'Plug it in, plug it in, plug it in.!'

Determine whether (3,7) is a solution of y = 2x + 1
y= 2x + 1
7 ? 2(3) + 1 substituted 3 for x and 7 for y

7 = 7 The equation is TRUE!!

Determine whether ( -2, 3) is a solution of 2y = 4x -8

2y = 4x - 8
2(3) ? 4(-2) - 8

6 ≠ 16 The equation is FALSE!!


How do you find a solution to an equation yourself?

Plug in for x and FIND y!!

Find 3 solutions of y -3x = -2

Solving for y first makes it easier to substitute for x and compute. so

change y - 3x = -2 to y = 3x - 2
We can substitute values for x and find corresponding y values.
We choose any value for x-- start with the easiest. x = 0

y = 3x - 2 substitute zero in for x

y = 3(0) - 2

y = -2 So the ordered pair (0,-2) is a solution.

Choose another number-- how about x = 2

y= 3(2) -2
y = 4 so the ordered pair ( 2, 4) is a solution

Choose another number This time let's use x = -2

y = 3(-2) - 2
y = -6 -2 which becomes y = -6 + -2
y = -8 so the ordered pair ( -2, -8) is a solution

How can you graph a linear equation? Make an x/y table of values and then graph the coordinates. You only really need 2 points-- because 2 points determine one line.
HOWEVER, always make a table of 3 ordered pairs to make sure no errors have occurred in your calculations. All 3 points will be on 1 line.

I always try to use x = 0 and solve for y and then set y = 0 and solve for x. It is usually easy. Then pick another value for x and solve for y. If this doesn't work well ( if you get a fraction as an answer-- and that isn't easy to graph) then try setting x = 1, then x = 2, then x = 3.

The graph of an equation is a drawing that represents the solution set. So 'to graph an equation' means to draw the graph of its solution set. You MUST connect the points to show the solution set. Make sure your line has arrows at both ends!!

Friday, February 15, 2008

Math 6 Honors Periods 6 & 7 (Friday)

Quotients of Integers

One of last night's homework problems could have been
-7 X 4 = -28 if that is true then,

-28 ÷ 4 = -7

The rules that apply for multiplication also apply for division.

The quotient of two positive integers is positive

The quotient of two negative integers is positive

The quotient of a positive integer and a negative integers is negative.


Notice: the quotient of two integers need not be an integer. For example, there is no integer n such that 10/4 = n because there is no integer n such that n X 4 = 10

Actually the quotient 10/-4 = -2.5

Remember that we never divide by 0. 10/0 is UNDEFINED

Let n be a positive integer and –n be its opposite

n ÷ -1 = -n

-n ÷ -1 = n

-n ÷ -n = 1

-n ÷ n = -1

n2 ÷ n = n