"Rules" for manipulating addition and subtraction number sentences to solve for the variable

If your variable is in an addition number sentence:

Addend1 + Addend2 = Sum (original problem format)
Addend2 + Addend1 = Sum
Sum - Addend1 = Addend2
Sum - Addend2 = Addend1

If your variable is in a subtraction number sentence:

Minuend - Subtrahend = Difference (original problem format)
Minuend - Difference = Subtrahend
Subtrahend + Difference = Minuend
Difference + Subtrahend = Minuend

Example:

1/3 - B = 5/24

let's label each piece....

1/3 (min.) - B (sub.) = 5/24 (diff.)

so, we want to solve for our variable, B. In other works, we want to "isolate" our variable all by itself after the equal sign! So, our subtrahend (which is what the variable B is in our original problem) will follow the equal sign if we subtract our difference form our minuend!

1/3 - 5/24 = B

Now find common denominators and solve!!!

1/3 = 8/24 so 8/24 - 5/24 = 3/24 so B=3/24

Welcome Trimester 2!

Welcome to trimester #2! Off to a fresh start so let's make a good beginning of it! I've purged the homework page of last trimesters assignments and started fresh.

Adding and subtracting mixed #'s - step 1: create common denominators if they are not there! (2) Step 2: Add or subtract (you can add/subtract the whole #'s first and then the fractions or the fractions and then the whole #'s or vice-versa...doesn't matter).
NOTE: You may need to borrow "from the wholes" on some of the subtraction problems (see example below!)

Example: 1 1/2 - 7/8 = _____

step 1 - common denominators: 1 4/8 -7/8 = _____

Ok, so you can't take 7/8 away from 4/8 so we're going to have to borrow some 8ths from our 1 whole...1=8/8 which we can combine with the 4/8 we already have....this would make our number sentence look like this:

12/8 - 7/8= 5/8 (the 12/8 is is the 8/8 + 4/8 from above)

Should be DONE in journal....

By tomorrow (11/22) the following assignments should be DONE in your math journal: (1) Bits & Pieces II: Investigation 1.2 A-D; (2) ACE Questions 16-30 on page 10; Bits & Pieces II: Investigation 2.1 A-F and ACE Question #1 on pg. 24

and

Our Earth Science foldable will be done/finished by Tuesday too!

Percents

- Percent as "out of 100"

- Three equivalencies:
1/4 = .25 x 100 = 25%

- so from fraction to percent just 2 steps:
step 1: numerator divided by denominator = decimal value
step 2: decimal value x 100 = % value
that's it!

Using percents....

strategy 1:
use the definition of percents to make equivalent fractions
18% of people like chocolate ice cream best. How many people out of 200 like chocolate ice cream best?

18/100 = 36/200 so 36 people

strategy 2 (same problem);
multiply decimal equivalent by proposed total of people
0.18 x 200= 36 people

Fraction to decimal

We've been working on converting fractions into their decimal equivalents! We've been specifically looking at the meaning of place value and how to convert fractions with denominators of 10, 100, 1000, etc. into decimals using place value! The denominator determines where the numerator goes using place value...



Example 1:
1/100 = .01 the 1 is in the hundredths place (we have no tenths so the 0 "holds"
the 1 in the correct, hundredths place)

Example 2:
25/1000 = .025 the last digit in the numerator goes in the place value indicated by the denominator, which in this case is the thousandths place...then just back-fill until you get to the decimal. The decimal is read like a whole number but you then add the place value the last digit is in (twenty-five thousandths.....just like the fraction!)

Example 3:
101/100 = 1.01 (this is an improper fraction so we know that we have a value greater than 1 whole - not by very much in this case!)

example 4:
sometimes you have to create and equivalent fraction with a denominator of 10, 100, 1000, etc. before you can use our wonderful base-ten decimal system....if we had 4 fingered hands we've have a base-8 decimal system!!

4/25 = 16/100 (I multiplied the numerator and denominator both by 4)

16/100=0.16

Strategies for finding common denominators (also see video links for more help!)

Finding common denominators - Reasons:
(1) To accurately compare the relative value of 2 fractions (when denominator is same - same sized pieces- the one with the bigger numerator is larger...has more of these same sized pieces!) or
(2) to add & subtract fractions finding common denominators becomes essential because you need to be adding up (or subtracting) the same sized pieces or your answer won't make any sense!

Strategy/Example #1:

1/3 + 2/9 = _____
- Hmmm, well, since the denominator of the 1st fraction is a FACTOR of the second I should be able to use multiplication to create my common denom. because 3x3=9! Don't forget, to get an EQUIVALENT fraction (we don't want to add up a non-equivalent fraction, that would change the actual value of the problem!) we need to multiply the numerator and the denominator by the same #

1 x 3
__ = 3/9 and now we just substitute 3/9 into our original problem for 1/3
3 x 3


3/9 + 2/9 = 5/9 (remember, only add the denominators because that is how many pieces you
are adding - if you added the denominators you'd be changing the size of
your pieces which would, in effect, be making your sum of less value!)

Strategy/example #2:

1/5 + 2/4 = ____ well, neither denominator is a factor of the other so we have to find a
common multiple of both - easiest way to do this is to multiply 2
numbers together!! Here, like this:

1 x 4
___ = 4/20
5 x 4


2 x 5
__ = 10/20
4 x 5


Now, we can substitute our new, fancy equivalent fractions with common denominators in for the old ones and we get

4/20 + 10/20 = 14/20 which reduces to 7/10

Note: you can also use division to reduce fractions in order to find a common denominator - make sure you divide the numerator and the denominator by the same # and that the they are both divisible by that number!

Hello-
IPR's (interim progress report cards) are coming out this Thursday....meaning I have to submit my grades NO later than Wednesday. Soooooo, if you have any missing or late assignments to turn in this would be the weekend to get 'er done!

Science: Mass, Volume, and Density

Density, the measure of mass per unit volume, can be found by simply dividing the mass of an object by it's volume.
M/V=D

If you HAVE the density and volume and you want to figure out the MASS of an object, take your density and multiply it by your volume.
D x V = M

And if your looking for the VOLUME and you have the mass and density, just divide the mass by the density.

M/D = V

Remember, volume is the amount of SPACE something takes up, mass is a measure for how much matter is in an object and density is a measure of mass per volume. I also like to think of my garbage can after I forget to take out the garbage the previous week - the bin is the same size (volume) but I've been forced to increase the amount of matter (mass) in that same space thus increasing it's DENSITY!

oh, yeah...units we've been using in class:
* the gram(g) is a unit of MASS
* cubic centimeters (cm^3) or milliliters (mL) are units of VOLUME
* and grams per milliliter (g/mL) and grams per cubic centimeter (g/cm^3) is for DESNSITY

Inv.1.3 (9/28 Homework)

Investigation 1.3 (purple worksheet) Tips/Hints: Use Soleil's Strategy for finding the value of a fractional portion!

Steps

1) Identify the fraction represented if it's not given
2) Identify the value of the whole or the total
3) Divide your whole by the denominator of the fraction you "have"
4) Multiply the quotient from step 3 by your numerator
Example: Billy bought a bag of 20 cookies and ate 3/4 of it.

20 /4 = 5

3 x 5 = 15 cookies eaten by Billy

*Note on division: When you divide, the divisor (in this case 4) chops the dividend (in this case 20) into 4 groups and gives you the value of 1 of these groups (the quotient or answer)....OR you could say you can create 5 groups of 4 out of 20....OR if you had to SHARE 20 things among 4 people each person gets 5. Grouping or Sharing - whichever makes more sense to YOU!

Now, some of you will read the example problem and say "of" means multiply, which it mostly does....but this would require the students to divide the fraction and then multiply by the fractions decimal equivalent (i.e.: 0.75 x 20 = 15). While this way is more efficient, we haven't studied place value yet so I'm not ready to explain WHY this works to the students yet.

Hint for tonight's Green worksheet homework - Like mentioned in class today, you can always find common multiples of 2 numbers by multiplying them by EACH OTHER
Example:
If I wanted to find common multiples of 4 and 5....4x5=20
20 is thus a common multiple of BOTH! To find more common multiples I would then just count by 20's. So additional common multiples would be 40, 60, 80, etc.

Factor Game Rules

These are the rules to the Factor Game - if your sixth grader can explain them to you without this all the better! Let them teach you...you never learn something better than when you have to teach it!
1) Player A chooses a # on the game board and circles it.
2) Using a different color (or "circling" it with a square works too), Player B circles all of the factors of the # A chose (can't circle the number itself even though it is a factor because it has already been taken - a number can't be circled twice!) You get the value of the numbers you circle as points.
3) Player B now circles a new number and Player A circles all of the factors of the number that are not already circled.
4) The players take turns choosing numbers and circling factors.
5) IF a player circles a number that has no factors left because they have already been circled that player DOES NOT get the points for that number and then loses their next turn. You have to pick a number that has factors your opponent can circle!
6) THE GAME ENDS when you cannot do the latter, i.e. when there are no numbers left with uncircled factors.
7) Total up the points and see who wins! Winner does tacky victory dance!

Hola!
Switching gears...going to be working on identifying factors and multiples for about a week.
Example:

3 x 4 = 12
3 and 4 are both factors of 12
12 is a multiple of both 3 and 4.

It's all about perspective...if you're looking for factors of a PARTICULAR number just think of all of the whole numbers you can multiply together to GET that number. Those are the factors (hint: factors ALWAYS come in pairs)

On the other hand, if you have a particular number and want to find the multiples of that number, just start multiplying that number by ANY WHOLE NUMBER and the product will be a multiple of that number. See example above!

"Hidden" Multiplication

Ok, we're going to begin weening you off of using "x" to represent multiplication in preparation for working with variables in algebra.
Examples:

* We can represent 5 x (5+3) by just getting rid of the "x" multiplication symbol and snuggling the 5 right up on the addition problem in the parenthesis. So it would look like this: 5(5+3)
or five multiplied by the sum of three and 5. Thus, the multiplication symbol is "hidden."

With "x" and then without
5x5 = (5)(5)
5 x (4-2) = 5(4-2)
5x(10) = (5)(10)

Remember, the order of operations still applies so in the second example in the columns you would have to find the difference BEFORE multiplying.
Good luck!

Everything up to 9/13 and HELPFUL EXAMPLES/HINTS

* Exponent practice worksheet (it's blue!) Remember, the shrimpy little number (the power or exponent) floating off to the right is telling you how many times to MULTIPLY the big number (the base) by ITSELF!!!

Example:



* Magnifying Me Biography
* Signed Parent contact information from the pink "Mr. White Things to know" worksheet
* Order of operations vocabulary in math journal
* Order of operations practice worksheet (it's tan and has an A and B side)
Remember to focus on the OPERATIONS between the numbers and to rewrite the number sentence after solving each component on the PEMDAS list until it's all done!
Example:


See how after doing each step in the order of operations the number sentence is re-written...after the "P" step, the (10-2), the number sentence is rewritten with everything the same except for the 8, which was inserted into the problem because it's the difference inside the parenthesis! Doing this will prevent errors!!!

Good Luck!