Tuesday, February 23, 2016

Decimals: Division

I. Algorithm for Dividing Decimals
Long Division of Decimals
To divide decimals:
1) Set the problem up in long division format.
2) If the divisor is not a whole number move the decimal point in the divisor all the way to the right. Then move the decimal point in the dividend the same number of places to the right.
3) Divide as usual.
4) Position the decimal point in the result directly above the decimal point in the dividend.

Example:
Divide 10 by 0.5

Arrange the problem into the long division format.
       ____
  0.5) 10

Move the decimal point in the divisor all the way to the right making it a whole number. Move the decimal in the dividend the same number of places to the right.
   _____
 5) 100.

Divide 5 into 100.

       2   
 5) 100.
     -10

        20 
 5) 100.
     -10

Example:
Divide 48.65 by 3.5 

Arrange into long division format.
     ______
3.5) 48.65

Move the decimal point in the divisor all the way to the right making it a whole number. Move the decimal in the dividend the same number of places to the right.
    ______
35) 486.5

Divide. Place the decimal point in the answer above the decimal point in the dividend.

        13.9
35) 486.5
     -35
      136
    -105
       315
     -315
         0


II. Conceptual Understanding
1. Decimal divided by whole number
Number Line
Example: 

0.6 ÷ 2

Using the partitive model of division we can translate to "After dividing 0.6 into 2 groups, how much is in each group?"


We can visually see that splitting 0.6 into two groups gives us 0.3 in each group.

0.6 ÷ 2 = 0.3

Base Ten Blocks
Khan: Dividing a decimal by a whole number with fraction models

2. Whole number divided by a decimal
Base ten blocks
Example:

3 ÷ 0.75

Using the quotative model of division we can translate this to " How many 0.75 are contained in 3?


Using base ten blocks we can visually see that there are 4 groups of 0.75 in 3

3 ÷ 0.75 = 4

Number line

3. Decimal divided by a decimal
Example

0.75 ÷ 0.25

Using the quotative model of division  we can translate this as "How many 0.25 are in 0.75?"

The number 0.75 is marked with a red box. We want to know how many 0.25 are in 0.75. We can see there are three jumps of 0.25 so the answer is 3.

0.75 ÷ 0.25 = 3

Putting the problem into the form of a word problem can help with conceptual understanding. "A ribbon is 0.75 centimeters long. If each small piece of ribbon must be exactly 0.25 centimeters in length, how many small pieces can be cut from the ribbon?" When framing the problem this way it becomes immediately clear what the answer is. 
 
4. Reframing The Problem
-The quiz following this video really reinforces this idea


III. Miscellaneous
A. Dividing Whole Numbers That Result In Decimal Quotients
1. Dividend Not an Exact Multiple of the Divisor (Whole Number Remainders)

2. Dividing a Smaller Number by a Larger Number
When dividing a smaller whole number by a larger whole number the result will be a decimal number. 


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Monday, February 22, 2016

Decimals: Multiplication

I. Standard Decimal Multiplication Algorithm  

To multiply decimals:
1) Place the numbers in vertical form lining up the numbers on the right. Do not align the decimal points.

2) Multiply as you would when multiplying whole numbers. Ignore the decimals at this point.

3) Count the number of digits after the decimals in the numbers being multiplied and move the decimal point over that many spaces from the right in the answer.


Example: Multiply 12.26 x 2.5

Place the numbers in vertical form lining them up on the right.

  12.26
x    2.5

Multiply as you would normally multiply whole numbers. At this point ignore the decimals.

     12.26
   x    2.5
      6130
+  24520
    30650

Count the number of digits after the decimals in the numbers being multiplied and move the decimal point over that many spaces from the right of the answer.

     12.26 (2 decimal places)
   x    2.5 (1 decimal place)
      6130
+  24520
   30.650 (3 decimal places starting from the right)


II. Conceptual Understanding
1. Whole number times decimal number
Number Line
Example

4 x 0.3

Thinking in terms of repeated addition we are asking "what is the total of 4 groups of 0.3?" 


4 x 0.3 = 1.2

2. Decimal number times whole number
Number Line
Example 

0.3 x 4 

Thinking of this problem in terms of scaling we are asking what is three tenths of four. To solve using the number line, we first divide the number line into tenths. 4 ÷ 10 = 0.4 so each tenth of the number 4 is 0.4 which is how the number line is partitioned. Finally we count the parts. We need three of the tenth sized parts (0.3 of the whole)


0.3 x 4 = 1.2

3. Decimal number times decimal number
Number Line
Example

0.4 x 0.3

We can word this problem as asking " what is four tenths of three tenths?" But how do we figure that distance from 0 to 0.3? I asked myself "what is one tenth of 0.3?" That would be 0.03. So I need to make 4 jumps of 0.03 to find my answer. 
0.4 x 0.3 = 0.12    





Practice
K5 Learning: Multiplication of Decimals


Reference
Master Math: Basic Math and Pre-Algebra
Math.com: Multiplying Decimals




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Tuesday, February 16, 2016

Decimals: Addition & Subtraction

I. Standard Algorithm

To add or subtract numbers with decimals:
1) Place the numbers in vertical form with the decimal points aligned.
2) Add or subtract each column of numbers the same as you would add or subtract whole numbers.
3) Place the decimal point in the answer directly below the decimal point in the terms.

Example: Add 8.53 + 15.2

Write the numbers in vertical form with the decimals aligned. You can add zeros to fill empty spaces.

     8.53
+ 15.20

 Add as you would normally add whole numbers and place the decimal in the answer directly below the decimal point in the terms.

     8.53
+ 15.20
   23.73

Example: 95.2 - 23.15

Write the numbers in vertical form with the decimals aligned. You can add zeros to fill empty spaces.

   95.20
-  23.15

 Subtract as you would normally subtract whole numbers and place the decimal in the answer directly below the decimal point in the terms.

        1 10
   95.20
-  23.15
   72.05

II. Conceptual Understanding
Base Ten Blocks







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Monday, February 15, 2016

Decimals

I. Intro
Decimals are numbers that are written with a decimal point. The number to the left of a decimal point is a whole number (or integer) and the number to the right is a fractional number (number between 0 and 1). This fractional part uses base 10 place value.

Hence, writing numbers with decimals (decimal notation) is another way of writing fractions and mixed numbers.

Reading Decimal Numbers
There are two ways to read a decimal number. 

1. Digit-by-Digit 
The first way is to simply read the whole number followed by "point", then to read the digits in the fractional part separately. It is a more casual way but more common way to read decimals. For example, we read 85.64 as eighty-five point six-four. 

2. Place Value Reading
The second way is to read the whole number part followed by "and", then to read the fractional part in the same way as we read whole numbers but followed by the place value of the last digit. For example, we can also read 85.64 as eighty-five and sixty-four hundredths. 

This method can be confusing so let's work through it.
12.1 - Twelve and one tenth
12.12 - Twelve and twelve hundredths
12.123 - Twelve and one hundred twenty three thousandths
12.1234 - Twelve and one thousand two hundred thirty four ten thousandths
12.12345 - Twelve and twelve thousand three hundred forty five hundred thousandths
12.123456 - Twelve and one hundred twenty-three thousand four hundred fifty-six millionths


II. Conceptual Models
Number Line
We can use the number line to help conceptually make sense of decimals. Here we are asked to find 0.6 on the number line. To do so, we divide 0 to 1 into ten parts. Next, we move from 0 to the right counting six places till we get to 0.6.


Area Model
Decimals can also be graphically represented with the use of an area model. The whole block is divided into 100 parts with the 35 blue colored blocks representing the number. 

III. Place Value
Place value with decimal numbers can be thought of as an expansion of the place value explanation used with whole numbers. Numbers are written using digits 0 through 9 with the position of the digit in the number determining the value of the digit. Each place has a value 10 times the place to its right. For example, the number 364.314 in expanded form is:

3 x 100 + 6 x 10 + 4 x 1 + 3 x 1/10 + 1 x 1/100 + 4 x 1/1,000

or

3 Hundreds + 6 Tens + 4 Ones + 3 Tenths + 1 Hundredths + 4 Thousandths

Here is an image showing place values of both whole and decimal number parts.


Lets focus on the digits to the right of the decimal. Each of these place units refer to a whole divided into equal parts. So the tenths place is a whole divided into ten equal parts, the hundredths is a whole divided into a hundred equal parts, the thousandths is a whole divided into a thousand equal parts, and so on.

Here are examples of equivalent forms of decimals and fractions for the place value units to the right of the decimal point in the chart above:

Tenths: 0.1 = 1/10
Hundredths: 0.01 = 1/100
Thousandths: 0.001 = 1/1,000
Ten Thousandths: 0.0001 = 1/10,000
Hundred Thousandths: 0.00001 = 1/100,000
Millionths: 0.000001 = 1/1,000,000

10 Times Relationship
The following graphic helps drive home the point that the relationship between any two adjacent places is that the place on the left is worth ten times as much as the place on the right, and the place on the right is worth one tenth as much as the place on the left:
Face Value x Place Value
Another important idea in our base ten system that extends to decimal numbers is that the quantity represented by a digit is the product of its face value and its place value. The face value is the value of the digit without regard to its position. In the number 73.6, the quantity represented by the 7, for example, is its face value 7 multiplied by its place value 10, which is 7 x 10 = 70. The face value 3 is multiplied by it's place value 1, which is 3 x 1 = 3  The face value of the 6 is multiplied by it's place value of 0.1 (or 1/10), which is 6 x 0.1 = 0.6. 

The result is 7 x 10 + 3 x 1 + 6 x 0.1 = 73.6

Regrouping With Decimals

IV. Types of Decimals
Decimals can be divided into different categories.

Terminating decimals: Terminating decimals mean it does not reoccur and end after a finite number of decimal places. For example: 543.534234

Non-terminating decimals: It means that the decimal numbers have infinite digits after the decimal point. For example, 54543.23774632439473747... 
The non-terminating decimal numbers can be further divided into 2 types:

1) Recurring (Repeating) decimal numbers: In recurring decimal numbers, digits repeat after a fixed interval. For example, 94346.374374374...

2) Non-recurring decimal numbers: In non-recurring decimal numbers, digits never repeat after a fixed interval. For example 743.872367346...

V. Conversions 
A. Converting Terminating Decimals to Fractions
Place the number to the right of the decimal point in the numerator. Next, place the number 1 in the denominator and then add as many zeroes as the numerator has digits to the right of the decimal. Reduce if necessary

Example:

5.025 = 5 .025/1 = 5 25/1000 = 5 1/40

B. Converting Repeating Decimals to Fractions
See:
Cliffnotes: Changing Infinite Repeating Decimals to Fractions
Note: Non-recurring, non-terminating decimals can not be converted to a fraction.

C. Converting Fractions to Decimals
Convert fractions to decimals by dividing the numerator by the denominator.

Example:
                                            .5 
1/2 = 2) 1    = 2) 1.0   = 2) 1.0
                                           1.0
                                           0 0

D. Converting Decimals to Percentages & Converting Percentages to Decimals
See Percentages


Practice:
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Tuesday, February 9, 2016

Multiplying Fractions (And Scaling)

I. Standard Algorithms
A. Multiplying Fractions Algorithm 
To multiply fractions:
1. Multiply the numerators to get the new numerator.
2. Multiply the denominator to get the new denominator.
3. Simplify the result if necessary.

Example:


B. Multiplying Mixed Numbers
To multiply mixed fraction:
1. convert to improper Fractions
2. Follow the general algorithm for multiplying fractions (above). 
3. Convert the results back to mixed fractions.

II. Conceptual Understanding
1. Whole number times fraction (Fraction times whole number)
Number Line
To grasp the concept, we start with multiplying a whole number times a fraction.

Example 1:

4 x 2/3

We could think of this problem in terms of repeated addition where we are asking, 'what is the total of 4 groups of 2/3'? To demonstrate this, the following number line shows each number 1 through 4 divided into three equal parts as reflected by the denominator in 2/3. Starting at zero, we move to the right 2 one third units (represented by the numerator) 4 times. This leaves us at 8/3 or 2 2/3.



solving using the rules of multiplying fractions:

4 x 2/3 = 4/1 x 2/3 = 8/3 = 2 2/3

Alternatively, instead of thinking of the problem as 4 groups of 2/3, we could also have thought of it as asking 'what is 2/3 of 4'.

Example 2:

2/3 x 4

This number line shows 4 divided into three equal parts (4 ÷ 3 = 4/3 = 1 1/3). Starting at zero, we move to the right 2 of those 4/3 units ending at 8/3 = 2 2/3.
*Note: Dividing 4 into 3 parts equals 1.333.. which equals 1 1/3. It would have been much clearer to mark the first jump as 1 1/3 then the second as 2 2/3. Instead I marked each jump as 4/3 as a movement of 4/3. Just realize that  4/3 + 4/3 + 4/3 = 12/3 = 4.

2. Fraction times fraction
Number Line
Example 3:

3/4 x 1/2

We can think of this problem as asking 'what is 3/4 of a 1/2' or, since we are using the number line, we can read it as '3/4 of the distance to 1/2.' We first draw a number line which shows 1/2 of 1.

Next, we are going to figure what is 3/4 of this 1/2. To do this, we divide the 1/2 into four equal sections as indicated in the denominator of the first fraction. We can now visualize the distance, but we don't know the actual length of this distance in terms of the whole from 0 to 1. To figure this, divide the other 1/2 of the line also into four parts. Now, the entire whole from 0 to 1 has been divided into 8 parts. Finally, starting at 0 we move three of these units to the right ending on 3/8.


Also see: 
Khan: Multiplying 2 Fractions: Number Line


III. Multiplication as Scaling
As examples 2 and 3 above illustrate, the idea of multiplication as repeated addition doesn't work very well as a concept when multiplying fractions. It's at this point that some educators introduce the idea of multiplication as scaling. 

To scale a number is to make it larger or smaller by a certain factor. The concept is taught in geometry where an object is enlarged or reduced by a certain magnitude but still retain it's shape. Conceptually, repeated addition only allows you to think of multiplication as an operation that makes things bigger. The benefit of thinking of multiplication as scaling is that you can conceptually also see multiplication as a process of making a number smaller.

A good definition I came across is from the book "Putting Essential Understanding of Multiplication and Division into Practice"

“multiplication is a scalar process involving two quantities, with one quantity -the multiplier – serving as a scaling factor and specifying how the operation resizes, or rescales, the other quantity – the multiplicative unit. The rescaled result is the product of the multiplication.”

As the above definition indicates, one number of the factors in a multiplication problem is viewed as a scaling factor and the other number is seen as the number being scaled up or down. Some tutorials I've seen refer to the first number as the scaling factor and some refer to the second number as the scaling factor. Ultimately it doesn't matter given the commutative property off multiplication but I will stick to what is generally taught when multiplying whole numbers. That being the first factor is the multiplier (scaling factor) and the second number is the number being multiplied (the multiplicand) 

Example 2 (above) 

2/3 x 4 

The 2/3 is the scaling factor. Since it is less than 1 we know the answer will the less than 4. The multiplicand, 4, is being scaled down. 

Example 3 (above)

3/4 x 1/2

Again, the scaling factor, 3/4, is less than 1 so we know the answer will be less than 1/2.


Scaling
Multiplication as Scaling
Does the order of multiplicand and multiplier matter in a multiplication equation?
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