Finding the GCF through Prime Factorization

Please read through as a refresher for tomorrow's lesson. We will be spending some more time on GCF.

Let's review how we can find the GCF of each set of numbers by using the prime factorization method. This example does NOT use exponents. I want to give you a visual of how to do it without turning your PF into exponent form juuuuuust in case you forget to use exponents.

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.

Divisibility Rules

An integer is divisible by:

2 if the integer ends in 0,2, 4, 6, 8.

5 if the integer ends in 0 or 5

10 if it ends in 0.

All even numbers end in 0,2,4,6,8,10 and are divisible by 2.

An integer is divisible by 3 if the sum

of its digits is divisible by 3.

An integer is divisible by 9 if the sum

of its digits is divisible by 9.

Example: 924 is divisible by 3

because

9+2+4 = 15.  15 is divisible by 3.

549 is divisible by 9 because 5+4+9 =

18.  18 is divisible by 9.

Chp. 3 Assessment

Hey guys. Just wanted to give you a quick rundown of what's going to be on your Chapter 3 Assessment next Tuesday.

We will  be covering sections 1 - 6:

1 - Rounding/estimating (remember place values and how to round decimals)
2 - Estimating products and quotients (round the numbers first; then, solve using your new numbers)
3 - Mean, Median, Mode & Range
4 - Using formulas (plug in what you know; solve for what you don't know)
5 - Solving Equations by adding/subtracting decimals
6 - Solving equations by multiplying or dividing decimals

Please be sure to bring a calculator. Happy studying!
Oh, and just one more thought from Philosoraptor...

Assignment for Thursday, 11/21

Hello my friends. First, let me thank you for your kind words and support today. You all helped me get through the day and I appreciate it. :)

Tonight for HW, we're going to use the honor system. Pleeeeeease don't use a calculator; you will have a drill tomorrow that will require your awesome brains without one. It will not be graded for correctness, but I'd like to see where we all are in terms of decimal skills.

Try the exercises for multiplying and dividing decimals, as we move into the section for solving equations by multiplying and dividing decimals. This should be review for most of you, but if not, no worries - we will take the time to go over it before we move into the section. Once we're all on the same page and can do it by hand, we'll break out the calculators. Here are the links. Complete several of each and see how you. do.

https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6th-multiplying-decimals/e/multiplying_decimals

https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-arith-operations/cc-5th-dividing-decimals/e/dividing_decimals_1

See you tomorrow!

Equations Game

Howdy!

Today you'll be playing some team games to practice what we've learned about solving equations with decimals. Display the One Step Equation game on the Polyvision board (click this link):http://www.math-play.com/One-Step-Equation-Game.html. Select the 1-player game. Be sure to select the "superstar" level option when playing the game. Please designate a classmate to keep score on the chalkboard.

Divide your class into two teams. Boys vs. girls worked well last time if you'd like to do that again. Take turns going up to the board to answer the questions with the eno pen. Remember to take your calculator with you (or a notebook and pencil).

Team 1 will send a player to the board to complete a problem. Correct answers score 1 point. Incorrect answers score no points. After the first player from Team 1 has gone, Team 2 will send a player to complete a problem. Then Team 1 will send another player, and so on, alternating between your two teams. The team with the most points at the end of the game wins. Have fun!

So far in Chapter 3...

I just wanted to take a second and briefly review what we've learned so far in Chapter 3.

1. Rounding & estimating with decimals
    ---Remember your place-values when being asked to round to a specific number.
    ---The rule is "5 and above, give it a shove; 4 and below, let it go."
    ---Remember that, after we round a number, we drop the digits that came after the digit we've rounded

2. Measures of Central Tendency
     ---Mean, Median, Mode, & Range are all ways to measure/describe/analyze a set of data
     ---Mean: Average // Median: Middle # // Mode: # that appears most often // Range: Highest - lowest
     ---***Take a look at the previous post for some creative math notes!

3. Formulas
    ---To solve a formula, simply plug in what you know (the numbers you are given) and solve for what
         you don't know (the unknown variable).
 
3. Solving Equations with Decimals
     ---Solve these the same way we solve equations with whole numbers - perform the inverse operation
          to solve for the variable.
     ---Now that we've practiced adding/subtracting decimals and have demonstrated that we can complete
         these operations without our calculators, we can use our calculators for assignments which require
         adding and subtracting decimals.

Mead, Median, Mode & Range!

Key

R.I.P., flip-flop.

As many of you know, my favorite flip-flop died today. Here is a memorial picture:

Goodbye, flip-flop. You will be missed.

...Anyway.

Everyone seemed to do a really nice job with identifying equations today. Let's keep the momentum going and read pages 86 & 87. This activity lab shows you how to model equations with rectangles.

Please read through all of the examples. Your assignment is to choose any one of the problems from 7-12 or 16-21 and model/solve that equation. Make it big and creative as I will be choosing one to post on the blog. Have a crazyawesomefantastic weekend!

Thursday 10/3

HAPPY THURSDAY!

Please watch the following videos to prepare for our lesson tomorrow:

https://www.khanacademy.org/math/algebra/introduction-to-algebra/algebra_why/v/one-step-equation-intuition

https://www.khanacademy.org/math/algebra/introduction-to-algebra/algebra_why/v/adding-and-subtracting-the-same-thing-from-both-sides

Monday, 9/30

Ok, guys. I know the combining like terms thing is a little stressful, so let's take some time to go over how it works. 

1) Remember that, no matter what you're doing in math, you must always follow PEMDAS. 

2) Keep the distributive property in mind. Check out this graphic from scimathmn.org:

(The main thing is to remember that whatever's outside the parenthesis gets multiplied by what's inside. 

3) Remember that, with addition and subtraction, you may online combine like terms (the stuff with the same variables). 

Example:

5(3x +2) - 6x +4   ---> First step: Distribute the 5 by multiplying:

15x + 10 - 6x +4   ---> Next step: Combine like terms (they're highlighted in purple):

9x + 10 + 4

= 9x + 14

b

As you complete the following problems, if you have any questions, leave a comment below. Here is your assignment:

1) 6 + x - 4x + 3

2) 3(w + 3) + 4w

3) -3z + 8(z + y)

See you tomorrow!

Thursday, 9/26

Howdy, y'all. Please watch the following video. It will show you a bit more about combining like terms. Aaaaaand have a really wonderful evening! See you tomorrow!

https://www.khanacademy.org/math/algebra/introduction-to-algebra/manipulating-expressions/v/combining-like-terms

If for some reason the link doesn't work, please visit Khanacademy.org and search for the video called "Combining Like Terms." You can also do the exercise on Khan Academy and practice online. 

8/29 - Did you find it???

Hello all and welcome to the Pre-Algebra blog!

If you've found the blog and would like some extra credit, complete the following problems to turn in on Tuesday:

If x = 3 and y = 5:

1) 3x + y
2) 2xy
3) (9+y) - x

Don't be scurred of rational numbers!

Hey guys. I know a lot of you are scratching your heads trying to remember how we graph rational numbers.  Look back at your old test if you have it - many of you graphed them correctly on the test. Here is a video that might help you understand the "why" to graphing rational numbers (after you watch, continue to read below please):
http://www.showme.com/sh/?h=X4EovCa

For those of you having a hard time, let's review:
- A rational number is any number that can be expressed as a fraction (which is, pretty much, any number!)
- All of the rational numbers we've graphed have been LESS than one (between zero and one)

Just so you know, BOTH rational numbers on your final will be "tenths". Just to make things a little easier.

Here's a quick rundown:

If I need to graph 7/10, I know my number lands somewhere between zero and one. Make sure your number line goes from zero to one:

Now, because I'm dealing with tenths, that means I need to divide my number line into ten different sections. It's like saying I'm dividing a giant cake into ten pieces, or cutting a pizza into ten different slices. Just place your lines so that there are ten little sections between 0 and 1:

Now that we have a graph of "tenths" - we can graph our fraction. 7/10 is "seven tenths". So we count seven places from zero to reach our destination (it's like finding the "seventh piece of cake"):

What if you have a decimal? Either turn it into a fraction or say it aloud so you have an understanding of where it goes. For example, 0.3  is "three tenths". Applying what we learned above, we can graph 0.3:

Does this help?

Final Review Questions

Hello mathlings. Your final review questions are as follows. We will be working through these problems and reviewing the concepts for the week leading up to finals.

Final Review Assignment/Guide

Complete these problems on the following pages:

CHP. 4 Concepts (pg. 230)

Exponents:

13, 14, 15, 63

Prime Factorization:

28

GCF:

34, 38

Simplifying Fractions:

39, 44

Graphing Rational Numbers:

45, 46

Chp. 5 Concepts (pg. 286)

LCM:

1, 3

Comparing and ordering fractions:

5, 6, 11

Converting fractions to decimals and decimals to fractions:

13, 16, 19, 21

Operations with fractions:

25, 26, 38, 31, 33, 36

Equations with fractions:

43, 44, 46

Final Study Guide

'ello, everyone! Here are the concepts that will be included on your Pre-Algebra Final. The jist of it is:
Exponents, GCM, Fractions, LCM, Equations. In more detail:

Chp. 4: Sections 2, 4, 6, 7, 8

- Exponents - How they work & what to do if you must add, subtract, multiply, or divide them

- Greatest Common Factor using Prime Factorization

- Simplifying Fractions

- Rational Numbers (you will need to graph two rational numbers, just like your Chp. 4 test)

Chp. 5: Sections 1, 2, 3, 4, 7, 8

- Finding the Least Common Multiple (use Prime Factorization for large numbers)

- Fractions:
   - Comparing/ordering fractions
   - Turning a fraction into a decimal (and a decimal into a fraction)
   - Adding & subtracting fractions
   - Mixed numbers: How to turn them into improper fractions, how to add & subtract them
   - Multiplying and Dividing fractions

- Solving equations by adding, subtracting, or multiplying fractions

Your test will not be super long but it will include all of these concepts using at least one question for each. And remember, don't stressed - we have plenty of time for review!

5-8 Solving Equations by Multiplying Fractions

Okay. Last section. You can do it!!! :)

We are now solving equations in which we must use multiplication or division. This is no different than what we've done in the past:

3x = 33 ----- To solve, we do the opposite of what's happening to x. Since we're multiplying x by 3,
                      the OPPOSITE would be to divide both sides by 3.

3x  =  33
 3        3

We now solve both sides and are left with x = 11.

What we're doing today is no different - we're just going to see some pesky fractions.

Take a look at Example 1 for this section:

5a = 1/7

Since we're multiplying a by 5, we do the OPPOSITE to both sides (Divide by 5). Here's the thing, though - how do you divide 1/7 by 5??? Going back to what we learned about dividing when there are fractions involved, we know we can multiply instead by the reciprocal (or "inverse"). Remember?

So all we have to do is turn that right side into a multiplication problem:

a = 1/7 x 1/5
a = 1/35

Now jump ahead to see Example 3. I know it looks like a hot mess, but pay close attention to what they're doing - they're remembering their negative rules, and then simplifying by using common factors before solving. I know we haven't touched on simplifying BEFORE solving, but it's there as an option for you if you'd like to try it. Otherwise, be SURE to simplify your answer after you've solved.

Example 4 shows us that, as always, when dealing with mixed numbers in an equation, it's easier to convert them to improper fractions first, and then solve.

Let's try a few:

Pg. 274, 1 - 20 EVENS ONLY

5-7 Solving Equations by Adding or Subtracting with Fractions

Well guys, home stretch. I am SO CRAZY PROUD of all you've accomplished this year. It's time to put what we learned near the beginning of the year and what you've just recently learned together. 

*cue Star Wars theme music*

This section is taking us back to solving equations. Remember those days? A few things I want you to recall:

1) Solving an equation means finding out what the variable is.  
2) In order to do this, we need to get the variable all alone. 
3) We get the variable by itself by doing the OPPOSITE of what's happening to it TO BOTH SIDES.

A quick example just to jog your memory:

3 + y = 10 --- What's the value of y?

To find out, we do the opposite of what's happening to y to both sides. Since the equation is ADDING 3 to y, we now must do the opposite - SUBTRACT 3. (And make sure you do this to both sides!) It's like balancing a scale.

3 + y - 3 = 10 - 3  ---- Now we solve both sides .

y = 7

Equations with fractions work exactly the same way. You just need to remember to apply what you know about fractions when you solve. 

Read through the section as a class and pay close attention to the examples. Notice how the first step is to do the opposite; THEN, your "fractions stuff" comes into play. In Example 1, for instance, in order to subtract 1/4 from 1/3, we need to first convert the fractions into the same "family" (same denominator) and then solve. 

Stop @ the end of each example and try the Quick Checks together as a class. The last example may look tricky but you can always transform the mixed numbers into improper fractions. If you go through the section and are ready to move on to practice, try  pg. 270, 8 - 19. 

5-4 Part 2: Dividing Fractions

Now then - on to dividing with fractions. 

I'm about to tell you the most well-kept secret about fractions: You don't have to. 

?!

It's true. 

Dividing by a fraction is very simple once you learn the two main steps. 

Step 1:

Change the problem to a MULTIPLICATION problem instead. For example: 

2/3 ÷ 1/4 becomes 2/3 x 1/4. The most important step is next. 

Step 2: 

"Flip" the second number (or, in other words, find the reciprocal of the divisor - a reciprocal is the inverse - the number gets flipped around). It's that simple. So for our example above, we would "flip" 1/4 and turn it into 4/1. Our math problem is now this:

2/3 x 4/1

Now we can use what we learned yesterday to multiply the fractions. First, the tops:

2 x 4 = 8

Then, the bottoms:

3 x 1 = 3

Our new fraction is 8/3. We must simplify:

8/3 = 2 2/3 (two and two-thirds)

It's that simple. Just change it to multiplication and make SURE you flip that second number. If you run into mixed numbers, same thing from yesterday applies - just transform the mixed number into an improper fraction first. 

Let's try pg. 255, 18 - 28. 

5-4 Part 1: Multiplying Fractions

Hey guys! I miss you all. Lymie says hello :)

When I told Roscoe you guys were working on fractions, he gave me the stink eye:

:) Anyway. Onto more fraction stuff.

Fortunately, it's quite easy to multiply fractions. All you need to do is multiply the tops, multiply the bottoms, and then simplify.
The example @ the bottom of pg. 252 sums it up quite well:

3/7 x 4/5 --- First, multiply the tops. 3 x 4 = 12, so our new numerator is 12.
              ---- Next, multiply the bottoms. 7 x 5 = 35, so our new denominator is 35.
              ---- If we put our new numbers together, our fraction is 12/35.

Since 12/35 is already in simplest form, 12/35 is our answer. You just have to remember to simplify after you've multiplied if the fraction is not already in simplest form.

Sometimes, to make life a little easier, you can simplify fractions BEFORE you multiply. Read through example 2 on pg. 253 as a class to see how it works.

If you encounter a mixed number, simply transform it into an improper fraction and then multiply. This can be seen in example 3 on pg. 253. Please read through this example as a class as well - you won't be solving too many word problems for fractions in the near future, but this is actually one of those times when, if you said, "Are we EVER going to use this outside of school?!?!" I can say, "Most likely." :) Fractions are everywhere.

I have a feeling some of you might be totally fine with this concept but will get stuck when you see negative numbers. REMEMBER your negative rules! :) Work on pg. 255, 1 - 13 today. If you finish early you can try the quick checks.

Tomorrow I will post the rest of 5-4 (Dividing Fractions). It's actually pretty easy!

5-3 Continued (Adding & Subtracting Mixed Numbers)

Fractions are pesky, aren't they? They're always asking us to change them around.

Read through the rest of p. 248 and 249 as a class.

Let's sum up what we learned. Sometimes, when adding and subtracting with mixed numbers, I have to think of myself as a secret, government-appointed scientist who has to solve an important calculation before an asteroid destroys the earth, just 'cause fractions irritate me that much. But - if you use the super-secret fraction-action method, you'll be fine.

1) Convert your mixed numbers into improper fractions. Here's a diagram from mrmelsonmath624.blogspot.com to remind you how we do this:


If you don't remember how to convert mixed numbers to improper fractions, you'll need to practice that before you move on. Since I can't accurately display mixed numbers from this keyboard, I'd like you to look at pg. 250, #15.

Once you convert those mixed numbers to improper fractions, you should have:
81/8 + 15/4.

2) Now we proceed as with regular fractions. Since we can't add them 'til they've got the same bottom number, we find the LCM of the denominators. The LCM of 8 and 4 is 8.

3) Convert the fractions to new fractions that have 8 as the denominator (thankfully, we only need to convert one of them):

15/4 = ?/8    ---> We know we can multiply 4 by 2 to get 8, so we'll do the same thing to the top:

15/4 = 30/8

4) Now, add your new fractions: 81/8 + 30/8 = 111/30.

5) Finally, simplify. This will most likely result in a mixed number. Remember, to convert (or simplify) an improper fraction to a mixed number, we divide the top by the bottom, and use the remainder as the top of the fraction and the original denominator for the bottom. Here's a diagram from ricksmath.com:



In our case, we divide: 111 ÷ 30.

111 ÷ 30 = 3 remainder 21. So, for our mixed number, our whole number is 3, the top of our fraction is 21, and the bottom is 30:

3 and 21/30 (again, sorry I can't display mixed numbers properly on here). But wait! Our mission is not done. Our answer is not in simplified form. Our whole number stays the same, but the fraction can be simplified. Our final answer is:

3 and 7/10.

I know this is a lot to take in. It takes practice and lots of remembering what you've already learned about fractions, but you can do it!

Are you ready to try p. 250, 15 - 18?

I'll be posting more tomorrow on 5-4. It'll be there when you're ready. Also, please take a look around the room - there's a fractions poster on the Math board that will remind you of some fraction principles.

Adding and Subtracting Fractions

The very first thing I'd like you to do for this section is to read through 5-3 in your textbook and STOP at Section 2 (Adding and subtracting with mixed numbers). Don't worry about notes or quickchecks or anything - just simply read this part of 5-3 silently to yourself.

Now, let's sum everything up:

Fractions can be added and subtracted as long as they have the same denominator (bottom number). Kind of like being in the same "family" - everyone has the same last name.

In the first section, we learned that we can easily add and subtract fractions with the same denominator simply by adding or subtracting the numbers on top. Basically, solve the problem with the TOP numbers and leave the bottom number the same (and don't forget to simplify if necessary!)

Example:
 3/5 and 1/5 have the same denominator, so I can easily add them by just adding the top numbers and keeping the denominator the same:

3/5 + 1/5 = 4/5.

The tricky part comes in when you have two fractions with different denominators. Before we can add or subtract them, we have to give them the same denominator (put them into the same "family"). So, how do we do this?

1) Find the LCM of the two denominators. The example in the book shows us 2/3 + 1/5. Since our denominators here are 3 and 5, our LCM = 15.

2) Now we need to change our fractions to new fractions that have a denominator of 15. Recall that to find any equivalent fraction, we must do the same thing to the top that we did to the bottom. So, 2/3 equals SOMETHING over 15, and 1/5 also equals SOMETHING OVER 15:

2/3 = ?/15
1/5 = ?/15

To find out what goes where the question marks are, figure out how we got from the old denominators to the new denominators. For the first one (2/3), how did we get to 15? We know that we can multiply 3 by 5 to get 15. Now, do the same thing to the top: 2 x 5 = 10. Our new fraction is 10/15.

Let's figure out the second fraction (1/5). How did we get from a denominator of 5 to a denominator of 15? we know we can multiply 5 by 3 to get 15. Now, do the same thing to the top: 1 x 3 = 3. Our new fraction is 3/15.

3) Now our new fractions can finally be added because they have the same denominator!

10/15 + 3/15 = 13/15. ---> This is already in simplest form, and so it is our answer.

To sum up the summary (lol!):

- Fractions need to have the same denominator in order to be added or subtracted.
- If they already have the same denominator, just add or subtract the top numbers, leave the bottom the same, and simplify if necessary.
- If they do NOT have the same denominator:
1) Find the LCM of the denominators.
2) Convert the fractions to equivalent fractions with the LCM as your common denominator.
3) Solve.

Try p.250, 5 - 12 individually. Check as a class. When you're ready to continue, the next post will address the rest of 5-3.

P.S.: To L-Dubs - thank you SO much for my Lyme Disease plushie! It's seriously the cutest thing ever. Made me smile. His name is Lymie. I will post pics of Lymie helping me work soon. :)

Review and Repeating Decimals

Hi guys. Let's review what we already know about converting decimals to fractions. Remember this step-by-step guide when you get stuck.

1) NAME the decimal (use your place value chart to find out what it "says").
    Example: .075 would be "seventy-five hundredths" since the last digit ends in the hundredths place
                   .6 would be "six tenths" because the number ends in the tenths place
                   1.457 would be "one and four hundred fifty-seven thousandths" because the last digit ends in the
                             thousandths place

2) Write your fraction. Let's use .075 as our example. Since we know .075 is "seventy-five hundredths", we write our fraction out as 75/100.

3) Reduce to simplest terms. 75/100 simplified would be 3/4 (remember, this is where we use the GCF to simplify)

Once you're able to turn decimals into fractions, all we need to do is use a little algebra to figure out how to turn those pesky repeating decimals into fractions. I'm not a big fan of the way your textbook explains it, so let me see if I can explain it in a simpler way.

Let's take the following number as an example (the same one in your textbook) - remember to click on the pictures to make them bigger:

1) Step one: Ignore the repeating symbol (pretend it's not even there). Move the decimal to the END of the number until it's a whole number.

2) Count how many times you moved your decimal to the right. In our case, it was TWO times.

3) Raise the number 10 to THAT power (the number of times you moved the decimal). In our case, we moved it TWO times, so we raise 10 to the second power:

4) Here comes the weird part. Subtract 1 from whatever number you got when you did step 3. In our case, we got 100. 100 - 1 = 99.

5) Now set up an equation that says "99x equals 72" (we are bringing back that "72" from when we moved the decimal in step 1).

6) Use what you already know about algebraic equations to solve for "x" by dividing both sides by 99.

7) We now have a fraction on the right side. Use the GCF (in this case, 9) to divide down and simplify your fraction.

So our answer is 8/11.



Here are the first two quick checks for this concept written out using our steps (sorry the second one is a little blurry):

I know it's tough but if you keep going through these steps and examples I know you'll get the hang of it. Try the third Quick Check as a class and let me know how it goes!

Monday, 4/15

Happy Monday everyone. Please check your answers to Friday's assignment (pg. 244, 1 - 12).

Please let me know if you guys had any trouble with these. If not, today you can  move on to more of 5-2. Read through the first half of pg. 243 as a class. You'll be learning how to turn a decimal into a fraction. Check back here once you've read through the first example (example 4).

In order to change a decimal to a fraction, you must first think back to when you learned about place value and be able to say what a decimal is correctly. Here is a place value chart from mathtutorvista.com to remind you:



Let's do the first Quick Check together.

1) 1.75 --- We would call this "one and seventy-five hundredths"
     Now we write the number out as a fraction using the fraction we just said:
     1 75/100 (sorry, I can't write fractions the normal way here on the blog)
     --- Since 75/100 is not in simplest form, simplify. Our new number is:
     1 3/4 (One and three fourths)

Here's another example:

Rewrite .6 as a fraction.

--- We would read .6 as "six tenths" - which gives us the fraction 6/10. Now simplify. We can write our
      fraction as 3/5.



We aren't going to move on to repeating decimals until tomorrow. For today, practice what you've just learned. Complete pg. 244, 16 - 27.

Friday, 11/12

Happy Frrrrriiiidaaayyyyyyyyyyyy!!!!!

Here are the answers to yesterday's questions (p. 760, 1 - 18):

1) 30
2) 40
3) 24xy
4) 15t²
5) >
6) < 
7) = 
8) > (Remember, we're dealing with negatives for this one)

For today's class, I'd like you to read through pgs. 241 and 242 as a class. Please try the quick checks along the way on your laptop after you've read each Example. You'll see through the first example that, in order to turn a fraction into a decimal, just divide the top by the bottom.*Remember that any whole number can be written with a decimal. In the first example, to turn 5/8 into a decimal, we had to divide 5 by 8. In order to do so, rewrite "5" as "5.0", carry the decimal up, and then begin to divide. 

We are still not yet using calculators for Chapter 5.

The second example shows how to indicate that you have a repeating decimal. For example, if you divide a fraction to transform it into a decimal and you find yourself continuing to divide - over and over again, getting the same number - that's called a "repeating" decimal and can be indicated with a bar over the numbers that repeat. 

The third example will show you that in order to compare a mixed group of fractions and decimals, turn each fraction into a decimal and then compare. 

Once you have mastered the Quick Checks, you can move on to your Classwork Assignment:
pg. 244, 1 - 12 

(If you don't finish in class, please finish for Homework)

Have a fantastic weekend!

Thursday, 4/11

Hello everyone! I hope you are all enjoying this crazy awesome weather. If the font for these posts is too small to see from your seat, leave a comment and I'll change the font size for next time (or your can use the "zoom" tool back on my computer).

For today's lesson I'd like to review what we covered in 5-1.

First, we learned how to find the Least Common Multiple (LCM) of two numbers, either using the listing method or using prime factorization. Let's review the prime factorization method:

First, find the prime factorization of the two numbers:

               8                   18
           2 x 4               2 x 9
        2  x 2 x 2         2 x 3 x 3

Our prime factorizations are at the bottom. Now, we need to multiply all of the bottom "tree" numbers together, using the matches ONLY ONCE. Our only match between the two tree bottoms is a pair of twos (in red), so we use it only once, and then bring all the other numbers down to multiply:

2 x 2 x 2 x 3 x 3 = 72

The LCM of 8 and 18 is 72.

Let's try it with three numbers. Find the LCM of 6, 14, and 28 using prime factorization:

First, find the prime factorization of all three numbers (I'm going to skip the whole tree and go straight to the answers we'd have at the bottom):

      6                       14                      28
   2 x 3                   2 x 7               2 x 2 x 7      

We have a matching 2 for all three trees, so we use it once. We also have a match of 7 between 14 and 28, so we use it only once. Then, we multiply those with all of the leftover, non-matching numbers:

2 x 7 x 3 x 2 = 84. The LCM of 6, 14, and 28 is 84.

Okay, everyone take a two-minute brain break by using Squidward hands to greet the people next to you and ask them how their day has been.

Now. What on earth does all this LCM stuff have to do with fractions? Remember, in ancient times (a couple of days ago) when we had to compare fractions, that we used the LCM to turn fractions with different denominators into fractions with the same denominators. (Because you can't compare two fractions that have different denominators.) Kinda like giving them the same last name, or putting them into the same "family."

In order to compare fractions with unlike denominators, you must:

1) Find the LCM of the denominators (now called the Least Common Denominator).
2) Convert the original fractions into equivalent fractions with the Least Common Denominator.
3) Compare the numerators, and voila!

Let's walk through an example using these steps:

Which is greater, 2/3 or 7/8? (Sorry about the sideways fractions!)

1) Find the LCM of the denominators:
          3                   8
          3               2 x 2 x 2 
We have no matches, so we multiply everything together: 3 x 2 x 2 x 2 = 24. This is now the Least Common Denominator we're looking for. 

2) Convert the original fractions into equivalent fractions with the Least Common Denominator.
That means we need to change 2/3 and 7/8 into fractions that have 24 at the bottom.

2/3 = ?/24 --- First, figure out how we can get from 3 to 24. Once we realize we multiply by 8, we must do the same thing to the numerator. Our new fraction is 16/24.

7/8 = ?/24 --- How do we get from 8 to 24? We multiply by 3. Do the same thing to the top for our new fraction: 21/24.

3) Now we can finally compare the two fractions by simply looking at the numerator. Which is bigger, 16/24 or 21/24?
We know that 21/24 is bigger. Therefore, 7/8 is greater than 2/3. 

And now, Mathlings, we are fully equipped to compare fractions and to list them in order from least to greatest, even if they have different denominators. Let's do one more exercise to practice all this stuff:

Classwork: pg. 760, 1 - 8. Please use the prime factorization method for finding the LCM. We will check answers tomorrow (or, if everyone finishes in class, check answers together with Mrs. Croskey).

Roscoe says, "Have a superawesometerrific Thursday!"

Wednesday 4/10

Greetings, Mathlings. Since you have a short class today because of testing,  take the opportunity to share some things about yourself with Mrs. Croskey! I suggest a round of the favorites game, Have You Ever, and maybe even a little silly story time. Tomorrow we get back into Chapter 5. Have fun!!!

Weekend Assignment 3/1

Howdy, y'all.

Please watch the following video from Khan Academy to prepare for Monday's lesson. (Don't forget to click "full screen" so you can see! Your drill question will be:

What is a rational number?

Then, you will need to provide an example of a rational number.

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? :)