Thursday's Class

I hope you all are enjoying Raven's Day!

Please get with your partner and check your work from yesterday (pg. 193, 9 - 16). Here are the answers:

9.  23	10.  72	11.  2 · 17
12.  2 · 3 · 7	13.  23 · 32 · 5	14.  5 · 23
15.  2 · 3 · 31	16.  33 · 23

Now with your partner, read the rest of section 4-3 (pgs. 191 and 192). I know that using Prime Factorization to find the Greatest Common Factor (or "GCF") can be very confusing, which is why I'm going to show you another way. First, look at how they find the GCF at the bottom of pg. 191. By breaking down each number (or expression - as in the expressions with the variables) to its prime factors, we are then able to find out which prime factors they have in common. Then, multiply those factors together to find the GCF. For example:

The prime factorization of 18 is 2 x 3 x 3. The prime factorization of 30 is 2 x 3 x 5. Since both prime factorizations contain a 2 and a 3, multiply 2 and 3 together to get 6. We know now that 6 is the GCF of 18 and 30.

There's another way to find the GCF, however, and it's a bit easier as long as you're not dealing with crazy big numbers. Watch the following video with your partner. In this video, the GCF is referred to as the "Greatest Common Divisor" - they are the same thing. Then try a few of the exercise questions after the video. If you have earbuds, please use them:
https://www.khanacademy.org/math/arithmetic/factors-multiples/greatest_common_divisor/v/greatest-common-divisor

Now, with your partner, try pg. 193, 17 - 24. You can use either method, but for 21 - 24, finding the prime factorization first might help. These are very tricky, but try to help each other when solving these. SAVE YOUR WORK and put it somewhere safe. Tomorrow when you check your work I will be walking you through GCF for variable expressions step by step, but for now I'd like you guys to try it after reading the textbook. Please let me know how you do. Are you guys able to comment on these posts?

Wednesday's Class

Hello all my wonderful Math-lings. So sorry I cannot be with you this week. Please check your classwork from yesterday (pg. 188, 1 - 23). The answers are here:

1.  83	2.  r4s2	3.  -7a2b
4.  25a2 or 52a2	5.  95	6.  (-5)4
7.  10,000	8.  64	9.  64
10.  –64	11.  –64	12.  –216
13.  1,000,000	14.  108	15.  –15
16.  31	17.  50	18.  –11
19.  42	20.  73	21.  –212
22.  -9	23.  22	24.  –15x2y

After checking your work, take the next ten minutes to pair up with a partner and discuss what, if anything, went wrong. Help each other out. I know the problems with variables instead of numbers may have been a little confusing, but hopefully now that you know the answers, you'll see that it's just like using regular numbers. NEWSFLASH: I've changed the comments option on our blog so you can comment with questions for me if you are having a hard time with anything. :)

After checking your work, stay with your partner to read through pgs. 190 and half of pg. 191 (read through Example 2). We've already learned about prime factorization by practicing factor trees; now, let's put what we've learned about prime factorization and exponents together to express prime factorization using exponents. Pay close attention to Example 2 - watch how 5 x 5 x 3 x 11 becomes 5² x 3 x 11 because 5 x 5 can be expressed with an exponent.

Then, with your partner, try pg. 193: 9 - 16. Please hold on to your paper. You and your partner will be checking your answers tomorrow and turning the assignment in. Make sure both names are on your paper.

Exponents and ALIENS!!! AHHHH!!!!

Here is some basic info you can refer to when dealing with exponents. Remember, an exponent is that teeny little number floating above and to the right of a number (the base):


This reads "two to the third power." Remember from our example in class, Aliens are VERY fussy about how their powers are used! :) When you see a number with an exponent like the one above, don't multiply the two numbers together, like this: 2 x 3. Your answer will be incorrect. Instead, the base number is multiplied by itself, and the power tells us how many times. The problem above would be solved by multiplying two by itself, three times:

2 x 2 x 2 = 8

And voila! The aliens are happy :)



Image credits:
wclipart.com
solving-math-problems.com

For homework tonight, please solve the following problems:

Write each as a product of the same factor; then, solve.

1)   2⁴

2)   6³

Express each product with an exponent. Then solve.
3)   5 x 5 x 5 x 5 x 5

4)   10 x 10

Prime Factorization

For a quick review of how prime factorization works, please view the info @ the bottom of this blog post (on our other blog):

http://tlsmath6.blogspot.com/2012/09/expressing-prime-factorization-with.html

Midterm Study Guide

Don't be nervous - your midterm will only take one class period to complete. There will only be one or two questions related to each concept. :) Remember, we'll also be reviewing in class for the next week. We'll be doing lots and lots of practice in class so you're ready for next Thursday.

What to expect:
- We'll be covering things we've been tested on so far this year (Chapters 1 - 3, excluding inequalities and sections 3-7 and 3-8)
- You may use your calculator on every part EXCEPT integers (adding, subtracting, multiplying and dividing with negative numbers) - you will take this portion of the test after you're finished with everything else
- Any formulas you'll need, such as A = lw, will be included on your test
- Pllllleeeeaaaase show work whenever possible

Resources to study:
- Previous blog posts
- Khan Academy videos
- Be sure to look back @ all 3 tests we've taken (I'll  be handing back any old tests I have tomorrow)
- Quick Checks in your online textbook
- The Chapter Review @ the end of each chapter (we'll be using these a lot in class)

Things you'll need to know by chapter:

Chp. 1
Variable expressions: Knowing how to express a numerical value using a variable. For example, "twenty-five less than x" would be expressed by x - 25.
Order of Operations: PEMDAS
Evaluating a variable expression: Just substitute the variable. For example, 3x + 4 for x = 5 would look like this, after substituting 5 for x: 3(5) + 4. Then we'd use order of operations: 15 + 4 = 9.
Integers: Everyone's favorite! (hehe just kidding, I think?) Take a look at old blog posts to refresh your memory. There's lots of info on here about how to solve problems with negative numbers.
- Absolute value
- Comparing integers (for example, what's greater, -7 or -9?)
- Adding and subtracting integers
- Multiplying and dividing integers
Number patterns: Figuring out what the "rule" is for a pattern, and then finding the next few numbers. For example, for the set 0, 6, 12, 18... we can conclude that the rule is "start w/ zero and add 6", and that the next two numbers will be 24 and 30.
The coordinate plane: You'll need to know how to graph a set of coordnates, such as (3, 5)

Chp. 2
Distributive Property: Know how to distribute by multiplying what's on the outside of the parentheses to the items inside the parentheses separately. For example, 3(6 + 2) is distributed as 3(6) + 3(2), which gives us 18 + 6 = 24.
Simplifying variable expressions: Remember to:
1) Change any subtracting to adding a negative
2) Combine like terms
For example: To simplify 2x + 3y - 2y + 7, we first rewrite the expression without subtraction:
2x + 3y + -2y + 7
Then, combine like terms:
2x + y + 7
Solving Equations: This is the big one! Remember to do the opposite to both sides in order to get the variable by itself! Check out previous examples in the blog. I think most of you have a pretty good handle on solving equations, but we'll be practicing a lot in class too.

Chp. 3
Rounding: You will only be asked to solve one problem that has anything to do with rounding. For example, to round 73.57 to the nearest tenth, that means we can't have any numbers after the tenths place. Look to the right of the tenths place; "5 and above, give it a shove - 4 or below, let it go." So... would the 5 that is currently in the tenths place get a "shove", or would it stay the same? Since the number next to it is a 7 ("5 or above" category), we give the 5 a shove up to 6. Our new number is 73.6.
Measures of Central Tendency: This means mean, median, mode (and range)
Mean: Add 'em all up and then divide by however many numbers you have
Median: Line 'em up in order from least to greatest - then identify the middle number (OR, if there are two middle numbers, the median is the mean of those two middle numbers)
Mode: The number that appears most often
Range: The difference between the greatest and least values in the data set
Formulas: You don't need to memorize any formulas. You just need to be able to plug in the right numbers into a formula. For example, if a rectangle has a length of 5 inches and a width of 2 inches, and the formula for the Area is length x width, you'll  need to plug 5 x 2 into the formula and solve (we'd get 10 inches squared).
Solving Equations w/ Decimals: Same exact method for solving equations, only a little more complex because we're using decimals instead of whole numbers. Just remember, that calculator is your friend. :)

A quick guide to inequalities

Make sure you know your symbols:
< - less than
> - greater than
< - less than or equal to
> - greater than or equal to

To graph an inequality, use an open dot for > and <. Use a closed dot for > and <.

Use the same rules for solving equations to solve an inquality (get the variable by itself by using the inverse operation) - just be sure that you're not using an equals sign (use the inequality symbol instead).

*VERY IMPORTANT*: When multiplying or dividing each side of an inequality by a negative number, reverse the direction of the inequality symbol!!!

Example:

-3x < 12    ---> We need to divide both sides by -3 in order to get x by itself.

-3x < 12   
-3      -3

x > -4       ---> Be sure to FLIP the inequality sign!!!

The Very Mathtastic, Super-Quick Guide to Equations

To solve an equation:

1. Remember, your mission is to get the variable all by itself. .:Cue Mission Impossible theme:.

2. To do this, use the inverse operation. Ask yourself, "What's happening to the variable?" and then
    do the opposite of that to both sides. Remember that addition and subtraction are opposites, and
    multiplication and division are opposites.

3. Your equation will now tell you what the variable equals.

Examples:

6 + y = 17  ---> What's happening to the variable? We're adding 6 to it. The inverse operation
                          would be to subtract 6. DON'T FORGET - YOU MUST DO THE SAME
                          TO BOTH SIDES.

6 + y - 6 = 17 - 6  ---> Solve both sides of the equation.

y = 11

--------------------------------------------------------------------------------------------------------
-2 = a - 10  ---> What's happening to the variable? We're subtracting 10 from it. The inverse
                          operation would be to add 10.

-2 + 10 = a - 10 + 10  ---> Solve both sides.

8 = a
---------------------------------------------------------------------------------------------------------
3x = -15   ---> Since we're multiplying the variable by 3, the inverse operation would be to divide
                        by 3.

3x = -15   ---> Remember, the fraction bar means "divided by."
3         3

x = -5
---------------------------------------------------------------------------------------------------------
m = 3   ---> Since we're dividing the variable by 9, the inverse operation would be to multiply by 9.
9

9 x m = 3 x 9   ---> Don't forget, a 9 on top and on bottom cancel each other out.
      9

m = 27
--------------------------------------------------------------------------------------------------------
Well, folks, there you have it. Just make sure you:
1. Use the inverse operation
2. Do the same thing to both sides
3. Get the variable by itself
4. Check your answer by plugging that value into the original equation (in place of the variable)