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.

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.

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.

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.

CBP Unit Test D

CBP Quiz C Study Guide

Quick reminder on adding or subtracting fractions and mixed numbers

When adding or subtraction fractions with mixed numbers, always change the mixed number into an improper fraction.  We do this by multipling the denominator by the whole number then adding the numerator to create a NEW numerator.  See example below.

Don't forget you have to find a common denominator before you add or subtract.

Finding a total using ratios

On the homework there is a problem where you have to use a ratio to find the yards each person ran, with the total yards ran being 150.  Here is an example of how to do that.

Scott is stronger than Shawn.  They have a goal to lift 500 pounds together.  Scott is 1 1/2 time stronger than Shawn, meaning for every 3 pounds that Scott can lift, Shawn can lift 2.  The ratio of their lifts are 3 to 2, or 3/2 as a fraction.  Here is how to solve who lifts how much weight.


When we add their weights lifted together, we need to reach 500 pounds.  If they were to lift the same about, that would be 250 pounds, or a 1 to 1 ratio.  As it is, for every 3 pounds Scott lifts, Shawn can lift 2.     This is a total of 5 lbs. We have to use an equivalent ratio.  If we multiply this by 100, they would lift 500 pounds.

3 x 100  = 300
2 x 100     200

CBP Quiz B Study Guide

Negative Numbers

We have been working on positive and negative numbers in terms of fractions on a number.  This video may help you and your student to understand this standard a little better.

Go to www.learnzillion.com

enter quick code  LZ1137

If you can't get to the video, here is a review.  

Positive numbers always fall to the right of 0 in ascending order, 1, 2, 3, 4, etc.

Negative numbers always fall to the left of 0, falling from -1, -2, -3, -4.

Positive numbers we are more familiar with.  The larger number is FARTHER from 0.  For example, 4 is farther from 0 and 1 is, so 4 is greater.  We do this by subtracting.  1-0=1  which is the distance from 0.  4-0= 4, which is it's distance from 0.  4 has a greater magnitude therefore 4 is greater.

Negative numbers have a different set of rules.  The number CLOSER to 0 is the great number.  -1 is greater than -4.  0- (-4) = 4, which is farther from 0. 0 - (-1) = 1, which is closer to 0 meaning it that is greater than -1.  

When adding and subtracting negative numbers, if we have a subtraction sign and a negative sign next to each other, we have to use parentheses to separate.  In math, we cannot have two signs touching, they must always be separated by a number.  One way to do this is to change the sign accordingly.  Here is the rule:

+ + = +
+ - = -
- + = -
- - = +

"Like, add, write the common sign
Unlike, subtract, keep the sign of the largerrrrr"

For example, we never write 4 + (+5), because we know it is understand that is is 4 + 5.

We do write this 4 + (- 5 ) really means 4 - 5.

The opposite holds true.  4 - (+ 5) really means 4 - 5.

This is true also.  4 - (-5) really means 4 + 5.

To get even more technical, the - in front of the () is actually a -1, so this means

4 -1(-5).  Order of Operations says to multiply the -1 by the -5, and a negative times a negative is always positive.

Rate Tables

Go to www.learnzillion.com and enter this code: LZ841  This will be helpful.

Let's look at a cost of an item.  If 48 oz juice bottle costs $4.00, what is the price per ounce?  How much would other amounts be?  Let's use a table.

Ounces	48	12	1	24	96	72	480
Price	$4

This means that for every 48 oz I will pay $4.  We can fill in the rest by using equivalent ratios.

Notice that 48 is larger than 4 and on the top row.  This means that no number on the bottom row should be larger than the one on the top.  Now to fill in the blanks, lets use equivalent fractions.

48      12

---- =  -----

4        

I do not know what the bottom number is.  However, I can use factors or fact families to determine what I need to do to get to 12 from 48.  I know:

12 x 4 = 48

4 x 12 = 48

So I also know:

48 ÷ 12 = 4

48 ÷ 4 = 12

SOOOOOO……

48 ÷ 4        12

-----      =   -----

4 ÷ 4           1

So 12 ounces is equal to $1.

Ounces	48	12	1	24	96	72	480
Price	$4	$1

The same principle applies to all of the numbers.

48        1

---- =  -----

4        

What do I need to get to 1 from 48?  Divide the top and bottom by 48!

48 ÷ 48           1                1

-----      =      -----     =      -----

4 ÷ 48           4/48             0.0833

So we would round to $0.08, or 8 cents.

Ounces	48	12	1	24	96	72	480
Price	$4	$1	0.08

The completed table:

Ounces	48	12	1	24	96	72	480	144
Price	$4	$1	$0.08	$2	$8	$6	$40	$12

Unit Rates

Unit rates are a ratio comparing one unit of one item to a quantity of another.  The most common unit rate is miles per hour.  The interstate speed limit is 70 miles per hour which literally means (barring any changes of speed due to accidents, getting off the interstate, bad traffic etc) that you would travel 70 miles in 1 hour.  This is usually represented as a ratio:

70 miles to 1 hour

70 miles : 1 hour

70 miles
1 hour

In class, we have been going over unit rates in terms of chewy gummy worms.  Let's say we have a giant gummy worm that is multi-colored.  Each color represents a segment.  Let's say we have a 4 segment chewy gummy worm and it looks like this:

If we divide this worm for 4 people, each person would get 1 segment, so the unit rate would be 1 segment per person.  If we only had two people, each person would get 2 segments, so the unit rate would be 2 segments per person.  But what if we have 8 people?

Here, we have to cut each segment in half, meaning each person would get 1/2 segment.  The unit rate now would be 1/2 segment per person.  This can be done a different way, but the ultimate goal would be to reach the 1/2 segment per person ratio (unit rate).

You can do this without drawing a picture.  Just divide the amount of segments by the number of people.  Below are two examples.

If we have an 8 segment worm divided among 6 people, you can set it up like a fraction (see picture).  We have 1 6 inside of 8, so we pull it out.  That leaves us with 2/6 left, which reduces to 1/3.  So the unit rate would be 1 1/3 segments per person.  What if the you have more people than segments?

Take 8 segments and divide it by 12 people.  Basically, simplify the fraction 8/12 by dividing both the numerator and the denominator by the GCF (4).  This give you 2/3.  The unit rate would be 2/3 of a segment per person.

The key when doing unit rates by fractions is to make what you want to be the 1 on the bottom (in this case we wanted PER PERSON).  Then simplify the fraction or divide.

Equivalent Fractions

We can find equivalent fractions by multiplying the numerator and denominator by a common factor.  This gives us multiples of the numerator and denominator.  We can continue to do this until we have the required number of equivalent fractions or until we reach the fraction we need.  The best idea is to stick with prime numbers or numbers that are easy to multiply by.  See the example below.

Ratios part 2

As we move along with ratios/proportions, we start to look at equivalent ratios.  Finding equivalent ratios is exactly the same as equivalent fractions (fractions are ratios in the sense of part to whole).  We can find equivalent ratios in a number of different ways.

Lets start with halving and doubling.  Any ratio can be halved (if both numbers are even), or doubled.  Take a look at this ratio.  The easiest way to find the simplest fraction is to divide the denominator (bottom number) and the numerator (top number) by the GCF.

4 boys to 8 girls.  This means for every 4 boys in a class, there are 8 girls.  We can simplify this by halving both numbers.

2 boys to 4 girls is equivalent.   We can even half this again to get a unit rate (we will cover this more later)

1 boy to 2 girls = 2 boys to 4 girls = 4 boys to 8 girls.

Alternatively, we could find equivalent ratios by doubling each number in the ratio.

4 boys to 8 girls is equal to 8 boys to 16 girls.

These two ways are, in my opinion, the easiest way to find equivalent ratios.  You can also multiply each number in the ratio by the same number and find larger equivalent ratios.


What about finding equivalent ratios when given a fraction?  Let's take a look.  In class we have been using $300 to describe a fundraising goal.  Let's say over the course of a few days that 3/5 of the goal has been raised.  We can set this up like an equation

3      
--- =  --------
5         300

We have to determine what number 5 has to be multiplied by to get to 300.  We can do this by dividing 300 by 5, which gives us 60.  So we would know that 5 * 60 equals 300, so 3 * 60 would give use what 3/5 of 300 is.

3 * 60        180
---       =   --------
5 * 60        300

We can do this with any fraction and ratio.

Ratios

Ratios are comparing two things together.  Let's take comparing boys to girls in a classroom.  There are 27 students in class. Of the 27 students, 15 are boys and 12 are girls.  These ratios may be represented as:

15 boys to 12 girls
For every 15 boys, there are 12 girls (however, only used when comparing larger quantities through scaling)

15 boys : 12 girls

15 boys
12 girls

These ratios all compare part to part.

It is important to keep the same order in which was given in the problem.  Notice I said:

"Let's take comparing boys to girls in a classroom".

I kept the same order because that was what was given to me in the problem.

You can compare ratios in terms of part to whole as well.

15 boys out of 27 students
12 girls out of 27 students

In this case, we used part of the classroom (boys or girls) and compared it to the total amount of students.  Since we know that there are only 2 genders, if I said 15 boys out of 27 students we could subtract the part FROM THE whole to find the amount of the other part.

 27 total students
-15 boys
12 girls.

These are the basics to ratios.

Prime Time Unit Test D

Order of Operations

We have reviewed Order of Operations in class and I have previously written about it in a blog but I have decided to dedicate one post to it prior to this test.

Order of Operations is the standard, accepted method for evaluating expressions and equations.  Order of Operations goes like this:

Parentheses
Exponents
Multiplication or
Division (whichever comes first, left to right)
Addition or
Subtraction (whichever comes first, left to right)

Let's look at an example. (remember that the ^ means an exponent)

2^3 + 9 (6 - 4) ÷ 2

The parentheses would have to be expressed first so we would rewrite the expression changing only what was inside the parentheses.

2^3 + 9 (2) ÷ 2.

Now, the (2) means to multiply, so we would not do that next.  We would, however, express the exponent.
2^3 = 2 x 2 x 2 = 8, so when we rewrite the expression it would be

8 + 9 (2) ÷ 2

Multiplication would come next, so we would multiply 9 and 2

8 + 18 ÷ 2

Division would come next and 18 divided by 2 would have use rewrite the expression

8 + 9

With addition left, we add 8 + 9 and our answer would be

17.

Step by step would look like this:

2^3 + 9 (6 - 4) ÷ 2
    8 + 9 (2) ÷ 2
       8 + 18 ÷ 2
         8 + 9
            17

I call this in class the upside down pyramid and I prefer students doing it this way because it shows each and every step and it is organized.

Use this link to find order of operation problems.

http://www.math-drills.com/orderofoperations/ooo_integers_foursteps_positive_pemdas_all.html