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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

Thursday, February 14, 2008

Math 6 Honors Periods 6 & 7 (Thursday)

Products With One or More Negative Factors 11-4 & 11-5

The product of a positive integer and a negative integer is a negative integer.
3 X -2 = -6
Multiplication is just repeated addition so this is really -2 + -2 + -2 = -6

The product of ZERO and any integer is always ZERO!!


4(-5X3) = 4(-15) = -60

-12(3 X 5) = -180

[25 - (6 X -2)](-3) = -111


The product of -1 and any integer equals the opposite of that integer.

-1 X 7 = -1

-1 X -8 = 8

The product of two negative integers is a positive integer

-3 X 4 X -2 = 24

-5 X 2 X -3 X -4 = -120

For a product with NO zero factors:
1. If the number of negative factors is ODD, the product is negative.

2. If the number of negative factors is EVEN, the product is positive.

-6 X -3 = 18

8 X -2 X -1 = 18

-2 X - 7 X 11 X 0 = 0

What happens in life -- happens in math:

Something good happens to someone good---- that's good

Something bad happens to someone good --- that's bad

Something good happens to someone bad--- that's bad

Something bad happens to someone bad ( really bad of course)-- that's good!!

Otherwise know as

positive X positive = positive

negative X positive = negative

positive X negative = negative

negative X negative = positive



Wednesday, February 13, 2008

Algebra Period 3

We won't be covering Chapter 7-1 in class.
This is simple Pre-Algebra!
I have included a review below:

Review of x y Coordinate Plane Graphing from Pre-Algebra (ch 7-1 in your book)

Cartesian plane:
Named after French mathematician Descartes.
plane: a two dimensional (across and up/down) flat surface that extends infinitely in all directions.

quadrant: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals

The axes are NOT part of any quadrant. A point on the x-axis or the y-axis is not in a quadrant since it is on the boundary between quadrants.

coordinate - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)

"ordinate" means in order

So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first

Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y ) format

origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.

x comes before y in the alphabet so the order is (x, y) ....
always go right or left first, then up or down

the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)

NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x
, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)

EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down

BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?

If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT

If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN

Sunday, February 10, 2008

Pre Algebra Periods 1, 2, & 4

Chapter 4-9: Scientific notation
You've had this since 6th grade!
You restate very big or very small numbers using powers of 10 in

exponential form--Move the decimal so the number fits in this range:

less than 10 AND greater than or equal to 1


Count the number of places you moved the decimal and make that your exponent

Very big numbers - exponent is positive
Very small numbers (decimals) - exponent is negative (just like a fraction!)

Remember that STANDARD notation is what you expect (the normal number)

When you multiply or divide scientific notations, use the power rules!
Just be careful that if your answer does not fit the scientific notation range, that you restate it.


ORDERING SCIENTIFIC NOTATION NUMBERS:

As long as numbers are in scientific notation,

they are easy to put in order from least to greatest!

1) If they are all different powers, simply order them by powers
2) If they have the same power, simply order them using your decimal ordering skills.

EXAMPLE 1: Order 3.7 x 108, 4.3 x 10-2, 9.3 x 105, and 8.7 x 10-5

8.7 x 10-5, 4.3 x 10-2, 9.3 x 105, 3.7 x 108



EXAMPLE 2: 3.7 x 108, 4.3 x 108, 9.3 x 108, and 8.7 x 108

3.7 x 108, 4.3 x 108, 8.7 x 108, 9.3 x 108

Algebra Period 3

Reviewed final checklist of how to factor:

1. Look for a GCF of all terms
2. Binomials - look for difference of two squares
both perfect squares - double hug - one pos, one neg - square roots of both terms
2. Trinomials - look for Trinomial Square (factors as a binomial squared)
first and last must be perfect squares - middle must be double the product of the two square roots
SINGLE hug - square roots of both terms - sign is middle sign
3. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle

4. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor
5. Trinomial with "a" coefficient - Use XBox - multiply first to last to get new product - then find factors that multiply to that new produce and either add or subtract to the middle term (use trinomial rules above) - replace middle term with these two factors and place appropriate signs so they will add to the original middle term - proceed as if you have factoring by grouping (see 6 below)
6. 4 term polynomial - factor by grouping - pair of the first 2 terms and then the second 2 terms by placing parentheses around them - make sure you always have a plus sign between the 2 pairs ( you may need to double check) - factor out the GCF of each pair - if it factors, there should now be a new GCF - factor that out in front parentheses and place what ever is left in the second parentheses


REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!

Thursday, February 7, 2008

Math 6 Honors Periods 6 & 7 (Thursday)

Subtracting Integers 11-3

Isn't it true that if you want to have a little less negativity, you just need to add a little positiveness!! What works in life-- works in math as well.


Rule

For all integers a and b, a- b = a + (-b)

Instead of subtracting-- ADD THE OPPOSITE

4 - (-5) = 4 + +5 = 9

We used Algebra tiles to prove this using Zero Pairs

+1 - 1 = 0 ( ZERO PAIR)

-3 - (-8) = - 3 + +8 = 5

5 - (-5) = 5 + + 5 = 10

-6 - -6 = -6 + + 6 = 0

Wednesday, February 6, 2008

Algebra Period 3

Using Equations that Factor 6-9

Word Problems

Problem Solving Guidelines

Phase 1: Understanding the problem
Ask yourself:

What am I trying to find?
What data am I given?
Have I ever solved a similar problem?

Phase 2: Develop and carryout a PLAN

Ask yourself:

What strategies might I use to solve the problem?
How can I correctly carry out the strategies I selected?

Phase 3: Find the ANSWER and CHECK

Ask Yourself:

Does the proposed solution check?
What is the answer to the problem?
Does the answer seem reasonable?
Have a stated the answer clearly? ( labeled??)


Practice translating some of these word problems:

The product of one more than a number and one less than the number is 8.
Find the number.

Start with a let statement. It can be as simple as
Let x = the number.

the product... means it will be a multiplication problem
one more than a number x + 1
one less than the number x - 1

so (x +1)(x- 1) = 8

in order to solve you must use FOIL or the box method
but just looking at ( x + 1) (x -1) you know that is the difference of 2 squares... so
x2 - 1 = 8 move the eight to the other side setting the equation equal to 0

x2 - 9 = 0 and now it is also the difference of two squares or
(x-3)(x+3) = 0

so x - 3 = 0 and x + 3 = 0 so x = + 3 OR x = -3



The square of a number minus twice the number is 48. Find the number

Let x = the number

the square of a number x2
minus twice the number -2x

so x2 -2x = 48

again it becones x2 -2x - 48 = 0

factor and you get (x -8)(x+6) = 0 Using the principle of Zero products
x = 8 or x = -6

Pre Algebra Periods 1, 2, & 4

Dividing Powers with Like Bases

m8
___ = m 8 - 5 = m3
m5


m3
__ = m 3 - 5 = m-2
m5


1
__

m2




56
_____ = 5 6 - 8 = 5-2 =
58


1
__
52



1
__

25

b3
__
b9

written without a fraction bar is

b3-9 = b -6

Any number raised to the ZERO POWER is equal to 1

PROOF:

an
____
an

= 1 but it also equals an-n or a0
for every a , except a cannot equal zero!!

Math 6 Honors Periods 6 & 7 (Tuesday & Wednesday)

Negative Numbers 11-1

On a horizontal number line we use negative numbers for the coordinates of points to the left of zero. We denote the number called ‘negative four’ by the symbol -4. The graphs of 4 and -4 are the same distance from 0—but in opposite directions. Thus they are opposites. -4 is the opposite of 4.

The opposite of 0 is 0

Absolute Value is a distance concept. Absolute value is the distance of a number from 0 on a number line. The absolute value of -4 is 4. l-4l = 4

The absolute value of 5 is 5. l5l= 5

Counting numbers: 1, 2, 3, 4, ….

Whole numbers 0, 1, 2, 3, 4….

Integers are whole numbers and their opposites.

…-4, -3, -2, -1, 0, 1, 2, 3, 4….

The opposite of 0 is 0.

The integer 0 is neither positive nor negative.

The farther we go to the right on a number line--- the bigger the number. We can compare two integers by looking at their position on a number line.

if x <0>

Adding Integers 11-2

Rules: The sum of two positive integers is a positive integer.

The sum of two negative integers is a negative integer.

So- if the two numbers have the same sign, use their sign and just add the numbers.

-15 + -13 = - 28 -10 + -4 = -14

Rules: The sum of a positive integer and a negative integer is :

POSITIVE… IF the positive number has a greater absolute value

NEGATIVE… IF the negative number has a greater absolute value

ZERO… IF both numbers have the same absolute value

30 + -16 … ask yourself the all important question… “WHO WINS? in this case the positive and then ask “BY HOW MUCH?” take the difference so 30 + -16 = 14

14 + - 52… “WHO WINS?” the negative… “BY HOW MUCH?”

38 so 14 + -52 = -38


(-2 + 3) + - 6 you can work this 2 ways

(-2 + 3) + - 6 = 1 + -6 = -5 or

using all the properties that work for whole numbers

Commutative and Associative properties of addition

can change expression to (-2 + -6) + 3 or -8 + 3 = -5 you still arrive at the same solution.

Thursday, January 31, 2008

Class Notes for the 2nd Semester

We will begin posting class notes from our class lessons starting 2nd semester. Look for this information starting February 4, 2008. Students who would like to earn extra credit by posting their own notes should email me-- or post right here. Looking forward to posting great class notes.

Thursday, December 27, 2007

Math 6 Honors Periods 6 & 7

We are almost finished with our study of Geometry!! All that remains is a quick study of congruent figures. We will study geometric constructions at a later date in 2008. Students already have the study guide for the Geometric Quest II, which is scheduled for Friday, January 11th. The solutions to the study guide will be posted online after 1/1/2008.

You will be able to find valuable notes for the week on this site!!-- Starting in 2008, check this blog often. Students will be contributing as well-- for extra credit!!


Have a wonderful rest of the winter break. See you in 2008!!

Pre Algebra Periods 1, 2, & 4

We are in the midst of our study of fractions-- adding, subtracting, multiplying, dividing -- understanding the GCF as well as the LCM. Start to memorize those important fractions families-- 1/5, 1/6/ 1/8 1/9, and 1/11 families. You already know the 1/2, 1/3 and 1/4 families!! Know the fraction, decimal and percent equivalences!! See the 'Incognito' Sheet on our main web page!! It has all the important equivalences-- and if you are going to memorize them-- you want the correct information!!

You will be able to find valuable notes for the week on this site!!-- Starting in 2008, check this blog often. Students will be contributing as well-- for extra credit!!


Have a wonderful rest of the winter break. See you in 2008!!

Algebra Period 3

We finally completed our study of Chapter 5- Polynomials. Most algebra students did great- our mean score on our Sections 5-5 through 5-11 Quest was 90.4%. Way to go, students!! Look forward to the upcoming Power Project-- once we are back in school after New Years!!

Starting in 2008, you will be able to find notes from our class for each week. Make sure to check out this blog often. Students will be able to contribute as well and earn extra credit.



Have a wonderful rest of the winter break!! See you in 2008

Friday, December 21, 2007

Teaching With Learning Styles and Multiple Intelligences

“The bottom line is that learning is a complex process and students learn in various ways. The teacher who acknowledges and actively responds to these truths will facilitate learning success for more learners.” (Guild, 1998)

Right from the start, this course has made a significant impact upon me and my teaching practices. Although I have been aware of Howard Gardner’s theory of multiply intelligence for years; Silver, Strong and Perini’s approach to integrating learning style and multiple intelligences in So Each May Learn enabled me to synthesize the steps necessary and implement changes in my lessons to reach more students. Their belief that “depth of learning comes as students process and think more intently about the content from various perspectives and in many lights” (Silver, et. al., 2000) has truly opened my eyes to the importance of incorporating these elements into my students’ daily lives both inside of my classroom and beyond.

In both the planning and implementing of my unit lessons, I am beginning to focus on each of the learning styles. I now attempt to make sure that each style is addressed by developing lessons and activities around them. By connecting the models as described in So Each May Learn, I am actively analyzing my curriculum to identify my areas of strength and those areas in which I need to focus and develop. However, none of this will benefit my students unless I take the time to truly discover each of their learning styles and dominant intelligences.

Therefore, I have come up with a plan, a sort of New Years Resolution (if you will) that lays out how I will integrate LS/MI into my classroom and more importantly, my students’ perspectives of themselves. I intent to use the three weeks left in the first semester (after we return from winter break) to introduce these concepts in depth to my almost 200 students. Upon returning to school, my students will complete both a learning-style survey and a multiple intelligences checklist. ISTE recommends Walter McKenzie’s Multiple Intelligences & Instructional Technology for not only lesson plans and planning materials but also as having an excellent MI survey. I also need to thank Denise Bakkum, a colleague in this course for providing a great website with an online survey for students. http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks3/ict/multiple_int/questions/questions.cfm

After spending time learning about their multiple intelligences and learning styles, both my students and I will have a greater understanding of their varied ways of learning. Also, my students will understand more about me as I share with them my own learning/teaching style. Then, with the start of the second semester, I will be able to implement various teaching and learning aspects, with the knowledge and understanding of which ones will allow each individual student to be successful.

For example, in my Algebra classes, the first unit of the second semester is on graphing and linear equations. While I do include many interesting and fun activities in this unit that encompass many styles and intelligences, I now see the benefit of allowing students to chose those geared to their dominant learning styles and intelligences…rather than requiring all students to complete every aspect (as was the case with my previous attempts to bring variety into my classroom).

MY PLAN

While continuing with my regular lessons and reviews in the three weeks leading up to the end of the first semester, I will weave in the following information as I prepare to fully integrate LS/MI into my classroom. (Probably not on consecutive days, but rather spread throughout the three weeks).

Step 1

Anticipatory Set-

I wanted to create a really easy but fun way to introduce the concepts of learning styles and multiple intelligences to my students that would catch their attention, but not require too much work on my part. What I came up with, was using an activity we have recently done from a workbook called FACEing Algebra, where students solve equations and chose from two possible answers. Each option corresponds to a different version of a facial feature that they must draw onto a blank head. Link to sample worksheet: http://www.mathcounts4ever.com/prealgebra/faceingmathinequality11.pdf

The end products of this activity are some really adorable and easily recognizable faces. I thought it would be great to use four of these faces to represent the cast of characters from figure 6.7 (p. 90) of the course text. For example, the mastery and understanding learners (and their statements) would be:

Samuel T.

“I will often make a list of my next day’s activities so I can be ready. Then I can check them off when I get them done, which usually happens. I don’t mind class projects, as long as the teacher gives us an exact set of directions as to what is due and when. Usually, I turn those projects a few days early to make sure I have them done. Teachers like my work, although they say that I need to be more flexible and realize that there isn’t always a right and a wrong answer. I am not exactly sure what they mean by that. I come to school to learn, and so I like it when the teacher shows me exactly what to do and what the answers are. I know I have mastered the material when I get a test or project back and everything on it is 100 percent right.”




Nancy T.

“I like learning about ideas and their history and the reasons that people believe in them. The part of a class that I like best is when we get a chance to really think through a topic, usually on paper but sometimes out loud in discussion. I remember my mom saying that as a little kid I was always asking ‘Why?’ I guess that hasn’t changed much. If people give me a chance to compare choices and make my own decisions, I usually make the right one. I think school is a great place to find out all sorts of things. If, after a long discussion or an assignment, I have been able to look at all the different viewpoints and start to understand them, then I feel like I haven’t wasted my time. For this reason, I guess I like essay tests the best because they give me some time to really express my opinions and prove my ideas”


As each picture is displayed, I will read the corresponding descriptive quote (attempting to change my voice for each style). Students will then be given a simple chart, similar to the one found in the course text along with this cast of characters, where they are asked to rank the four characters according to their similarity. My hope is that with this fun introduction to learning styles, my students will become excited to further discover their own unique learning profile.




Step 2

I will then formally introduce the concepts of learning styles and multiple intelligences to my students. Using various sources, including Strong et. al. examples to explain these ideas.


Step 3

Next, I will ask my students to speculate what they think my dominant learning style and intelligences are. Then I will actually display both of my test results and analysis. I will then discuss what I learned about myself from discovering this information, saying something like, “This is who I am and how I learn.” I will then ask for feedback and further development of the importance of recognizing our own individual learning profiles… “What do you think? Is this me? I know what works for me… Wouldn’t it be great to discover how you learn best?”


Steps 4/5

In the following days I will take the students to the computer lab where they will complete both the MI indicator and a learning style profile test.

Each student will receive an individualized chart of his/her results.


Steps 6/7

I plan to present class results as a whole. Perhaps, even analyzing the information with various charts, graphs, etc. (Our current unit).

We will also discuss when is it great to work with people who think the same way and when do you like variety of styles/intelligences in a group.


With a plan in place to bring the concepts of learning styles and multiple intelligences to the forefront of my classroom environment, I know that my students (as well as my teaching) will benefit immensely. I look forward to the new year, and see it as a fresh start in getting to know all of my students better.


Resources

Guild, P. B., & Chock-Eng, S. (1998). Multiple intelligence, learning styles, brain-based education: Where do the messages overlap? Schools in the Middle, 7(4), 38–40.

Iste site http://www.iste.org/source/orders/excerpts/multi2.pdf

Multiple Intelligences image from http://www.newhorizons.org/strategies/arts/cabc/oddleifson3_2.gif

Silver, H. F., Strong, R. W., & Perini, M. J. (2000). So each may learn: Integrating learning styles and multiple intelligences. Alexandria, VA: Association for Supervision and Curriculum Development.