Tuesday, 2/19

First of all, let me thank all of you for the super-awesome, very thoughtful cards I received from each of you. They were such a blessing to read. I can't wait to get back to school!

Today is a game day :) Visit http://www.sheppardsoftware.com/math.htm and try the games under "Fractions". Equivalent Fractions Matching, Reduce Fractions Shoot and Equivalent Fractions Mathman (this one uses the keyboard from my computer) are great ones to start with. They involve finding equivalent fractions and simplifying fractions. You may use a calculator. Please take turns.

Have fun!

Thursday 2/14

Happy Valentine's Day! Did anyone else get snow last night??? :)

Today we move along in 4-4 to "Simplifying Fractions" - pgs. 196 - 197. To simplify a fraction:

1) Find the GCF of the numerator and denominator (don't worry, you can just use the regular ol' listing method for now to find the GCF)

2) Divide the numerator and denominator by the GCF for your simplified fraction

*Skip over Example 3 on pg. 197 - this requires you to understand polynomials & GCF, which we need to travel back to*

Try pg. 198, 7 - 12 and 28 - 31, 40, 42, and 48. 

Enjoy your long weekend!

Wednesday 2/13

Well, folks... after much debate... I think we're going to save 4-3 for when I return. For now let's move ahead with some easier stuff, and then we'll come back to polynomials.

Today's assignment:

1) Read "Finding Equivalent Fractions" on pg. 196.
Notice that, in order to find an equivalent fraction, all you have to do is multiply or divide both the top and bottom numbers by the same number. 

For example:
If we need to find an equivalent fraction to 2/3, we can multiply both the top and bottom numbers by the same thing to get an equivalent fraction. Let's use "4."
If we multiply the top number (2) by 4, we get 8. If we multiply the bottom number (3) by 4, we get 12. So our new fraction is 8/12. This means that 2/3 = 8/12 (they are equivalent fractions).
Here's a visual from imoamal.org - notice that the same amount of "pie" is covered by both fractions, which means they are equivalent:


2) Watch the following video on KA:
https://www.khanacademy.org/math/arithmetic/fractions/Equivalent_fractions/v/equivalent-fractions

3) Complete a, b, and c under Example 1 on pg. 196 AND 1 - 6 on pg. 198.

Happy Math-ing.

PS - Roscoe says, "Let it Snow!"

Another Tuesday Post

If you're reading this it means you've already gone through the other blog post for today and checked your answers for Friday's questions along with the step-by-step pictures.

P.S. - Roscoe says hello:

Now that we (hopefully) have a good understanding of how to find the GCF of two numbers by using prime factorization, let's take a look at polynomials.

First, let me explain what a polynomial is. A polynomial is an algebraic expression with more than one term. We did a lot with polynomials back when we were simplifying expressions. 3ab is a polynomial - it says "3 times a times b" because they're all right next to each other. 6a³b and 4a²b are also polynomials. 

So... what does this have to do with GCF, you might ask? We're going to be finding the GCF for polynomials, using prime factorization, just like we did with regular numbers.

Let's take those two polynomials from above (in red) and first find the prime factorization of each. We do this by breaking down any regular numbers in the polynomial into its prime factor. Then, for all the variables, just write them out (take them out of exponential form):

It's kinda like take a puzzle apart. Now, the second polynomial:

 Now we can compare the two prime factorizations:

We have a few matches. One 2 on both sides, 2 a's on both sides, and a b on both sides. Multiply your matches together, and now we have the GCF:

The GCF = 2a²b

This example is also given in your textbooks on pg. 191. 

Please watch the following video from Khan Academy only until 2:31:

https://www.khanacademy.org/math/algebra/polynomials/multiplying_polynomials/v/factor-polynomials-using-the-gcf 

As usual, if you have any questions please leave me a comment. Using this information try pg. 193, 21 - 24 and 37 - 40. Please turn in what you have to your sub. 

Tuesday 2/12

Hello all. Just a quick note about showing work - please remember to show your work by hand. While a few of you got all of the answers correct for the blog questions from Friday, I didn't see any of your work, and was not able to tell if you found those answers by using the prime factorization method or the listing method.

Right now, it's important that you're using the prime factorization method specifically so that we can move on to the next step.

If you finished your work Friday, it will be passed back to you with comments I've made. If not, please take it out. Everyone should compare their work from Friday to the answers on today's blog. Let's take a look at how we can find the GCF of each set of numbers by using the prime factorization method:

1)  16, 60

First, find the prime factorization of each:

Next, identify the matches on each side. A "match" is when there's a number on one side (at the bottom of the tree) that has an EXACT twin on the other side. The two numbers team up to create a "match."

We have underlined our matches. There's a "2" on one side, and another "2" on the other side. This gives us our first match: 2. Our second match is also 2, since there is a second two on both sides. As shown in the picture below, the remaining numbers have no matches:

Now, list our matches. We have one match of "2" and another match of "2". We multiply our matches together:

The GCF = 4.


Let's look @ #2 from yesterday's blog:

2)  36, 81

Step 1: Find the prime factorization of both numbers. Then, find your matches:

The 1st 3 on the left side matches with the 1st 3 on the right side. This gives us our first match of "3." The next 3 on the left side matches with the second 3 on the right side. This gives us another match of "3." None of the other numbers have matches. So, our matches are 3 and 3. Now, we multiply our matches to get a GCF of 9:

3)  15, 30

Here's one big diagram that puts everything together (click the pic to make it bigger):

4)  32, 64

We had five 2s on the left side, which all had matches on the right side. So our matches are 2, 2, 2, 2, and 2:

5)  10, 13 --- Don't forget that every number has a factor of "1":

That was a tricky one :). Now we need more practice. Your classwork today will be collected. Please hand in Friday's work as well if you have not already done so. I will be checking for work shown - this is the most important part, so I can help you when you make a mistake.
Let's try the following five problems. Same directions as Friday - find the GCF of each pair of numbers using the prime factorization method. 

1)   18, 24
2)   36, 54
3)   75, 100
4)   24, 36
5)   30, 90

Friday's Class

Hmmm... I wonder if you'll be reading this on Friday, or Monday? Since we're supposed to get some .:snow:.! :)

Okay. Let's get down to business with these GCF things.

Here are the answers from yesterday's assignment (pg. 193, 17 - 24). I know they were very difficult so please don't worry if you got them wrong. I wanted to let you try them first though after reading the textbook:

		17.  5
18.  7	19.  25	20.  3
21.  7c	22.  3y2	23.  6c3
24.  2mn

Before we can really tackle finding the GCF for those complicated variable expressions, we need to master the "hard" way of finding a GCF.

As we learned yesterday, we can find the GCF of two numbers two different ways. First, by listing the factors, and identifying the biggest one they have in common. 

Example:

Find the GCF of 30 and 18. 

If we list the factors of 30, we get: 1, 2, 3, 5, 6, 10, 15, and 30. 

If we list the factors of 18, we get: 1, 2, 3, 6, 9, and 18. 

When we compare the two lists, we see that the greatest common factor shared by both of them is 6. 

This method is easiest with relatively small numbers. 

Let's look at the other way. 

To find the GCF of two numbers by using prime factorization, first do factor trees for the two numbers. 

                     30                              18

                  2  x  15                      2  x  9   

               2  x  3  x  5                2  x  3  x  3

The prime factorizations for both numbers are, as usual, @ the bottom of the tree. 

Now here's where it gets tricky. After this step, identify all of the common factors @ the bottom of the trees. The thing is, each common (or "shared") factor has to have a match!

We see that both sides have one 2. So we can select 2 as a common factor. Only one side has a 5, so we cannot use the 5 as a common factor. Both sides have one 3, so we can select 3 as a common factor. Now, look on the side for 18. We have an extra 3 hanging around at the bottom. The thing is, that 3 doesn't have a match on the other side. No twin, no bff, no match. So the extra 3 is not a "common" factor. 

Follow me so far? We've identified the common factors at the bottom of each tree to be 2 and 3. Now we multiply those common factors together: 2 x 3. We get 6. Now we know that 6 is the GCF of 30 and 18. 

Let's stop here and practice the method of finding the GCF using prime factorization without worrying about polynomials, variables, exponents, etc. We're going to break it down step-by-step. 

Try the following 5 problems individually using the prime factorization method. Find the GCF:

1)  16, 60

2)  36, 81

3)  15, 30

4)  32, 64

5)  10, 13 ---> This is a tricky one! Can anyone figure it out? :)

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