Wednesday, December 6, 2017

Decimal Operations

The next unit is over decimals.  Here are some things to remember:

Addition-

  1. line up the decimals
  2. look for words like added together, gave, found, total, etc.
Subtraction-
  1. line up the decimals
  2. look for words like difference, how much farther, what is left, pays, owes, etc.
Multiplication-
  1. line up the farthest RIGHT digit in your number
  2. look for words like per (as in price PER pound), area, product, total, how many times, altogether, twice, multiplies, in all.
Division-
  1. put the 1st number inside the "house", the second number outside.  If there is a decimal in the outside number, move it right until it hits the "door".  For however many times you moved it outside, move it inside the same amount, then put it on top of the "house"
  2. look for words like quotient, average, amount per, per, ratio, divided by, out of

Here is one last thing for multiplication and division:

Now look at the statement and question.
Find the word “each”.
If “each” is in the statement -- multiply.
If “each” is in the question -- divide.

Thursday, November 16, 2017

Dividing Fractions

Dividing fractions is EXACTLY like multiplying fractions with one extra step

COPY CHANGE FLIP

Let's look at a few fraction division problems.








Notice that I COPY the 7/8, CHANGE the divide sign to X, then FLIP the 2nd fraction to be 8/1.  Then multiply like normal.












Here is another example







Here is one more:









This works for all fraction dividing problems including whole numbers and mixed numbers.  With whole numbers, put them over 1.  For example, 12 would be 12/1.  For mixed numbers, change them to improper fractions first.  for 4 1/3 would become 8/3.  Then copy, change and flip as necessary.

Some common mistakes include flipping both numbers, not changing the sign, or not changing whole numbers/mixed numbers appropriately.  BE SURE TO

COPY  CHANGE  THEN FLIP.

Wednesday, November 15, 2017

Multiplying Fractions

When multiplying fractions, we just multiply the numerators together and then the denominators together.  Let's look at an example:

1  *   1  =   1
3       4       12

Here is another example:

2   *   5   =   10  or 5
3        6        18      9

What if you are multiplying a fraction and a whole number?

12 *   1
          4

Every whole number has a denominator of 1, so....

12 *   1   =   12
 1       4         4

Since the fraction bar means divide, we have 12/4 which is 3.  When multiplying fractions, the product will never be larger than the sum.

We can use a brownie pan method to show the same thing.  Here is a blank, square brownie pan.


Let's say that I am selling brownies.  I have only 2/3 of the pan left.  The red section is gone.


Let's say I have a customer who wants to buy 1/4 of what I have left, or 1/4 of 2/3.  I would have to cut my 2/3 into 4th's.  It would look like this.


So now I need to color in 1/4 of what I have here.  The green shows what I would sell.



This represents 2/12 of the whole pan, or 1/6.  Mathematically:

1    *    2   =    2
4          3         12



Modeling Multiplying Fractions

2/3 of 7/10 means 2/3 x 7/10.  The of can be replaced with a multiplication sign.  We begin by drawing a square.




Split the box according to the denominator of the SECOND factor.


Mark off the amount that makes the second factor true.  In this case we marked out 3/10 of the whole because we only have 7/10 of the whole left.

Then split the WHOLE box by the denominator of the FIRST factor.




I labeled the top 1/3 to show that each vertical bar represents 1/3 and the left side 1/10 to show that each horizontal bar represents 1/10.




















I then shaded 2 out of every 3 boxes left over in green.  (sorry the picture is blurry).

The total green boxes represents my numerator.  The TOTAL boxes of the whole becomes my denominator, not just the part leftover.






This shows mathematically why that works.

Tuesday, November 14, 2017

Subtracting/Adding Fractions

We went over adding and subtracting fractions in class.  Here are the 2 ways we discussed.  This also works for adding fractions.

Separate the whole numbers first.  Here, we have 11 - 4 = 7.

Then subtract your fractional parts (we had to find a common denominator first).

Since 1/8 came first, we have to subtract the 1/4 or 2/8 from it.  This give us -1/8.  Now we subtract our two parts

7-1/8.  When the denominator and numerator are the same, we know that it is equal to 1, so 8/8=1




In this method, we changed our mixed numbers into improper fractions.  From there we had to find a common denominator.   Then regular subtraction rules apply.  Once we find our improper fraction, we have to change it back to a mixed number.