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!