Division

So what comes after Multiplication... You got it, Division! On Thursday in class we began talking about Division....

Quotient: the result of division (the answer)

Dividend: the number divided

Divisor: does the dividing


Division of Whole Numbers:
-For any whole numbers r and s, with s cannot = 0, the quotient of r divided by s, written r/s, is the whole number k, if it exists, such that r = s x k.


-Sharing (Partitive) Concept

Suppose you have 9 basketballs, which you want to divide equally among 3 people. How many basketballs would each person receive?

Solution: The answer can be determined by separating the basketballs into 3 equivalent sets. The following figure shows 9 balls divided into 3 groups and illustrates 9/3. The divisor 3 indicates the number of people.

Person 1

Person 2

Person 3

 © CKSinfo.com

This method is easy for kids to understand, because you can actually see the total number of basketballs being divided evenly.

-Measurement (Subtractive) Concept
Suppose you have 10 basketballs and want to give 3 tennis balls to as many people as possible. How many people would receive tennis balls?

Solution: The answer can be determined by subtracting, or measuring off, as many sets of 2 as possible. The following group of 10 basketballs shows the result of this measuring process and illustrates 10/2. The divisor 2 is the number of balls in each group, and the quotient 5 is the number of groups.









 © CKSinfo.com

This method of measurement (subtractive) concept is a different way to look at dividing. It allows the student to use something they are more comfortable with. Another perspective helps them understand division in a different way.

Multiplication

Now that we've went over Addition and Subtraction, let's talk about Multiplication!

Multiplication: Repeated Addition

Multiplication of Whole Numbers
-For any whole numbers r and s, the product of r and s is the sum with s occuring r times.
   
            r x s = s + s + s . . . + s
                        r times


One way of representing multiplication is with a rectangular array. Base 5, 10, and 100 pieces are commonly used to make rectangular arrays. Singular pieces called units are also used. The pieces can be set up according to the numbers in the problem.

©LEARNING THINGS

I remember using base pieces throughout Elementary School, except ours were green. This method really helped me learn, because it was hands on. We could actually see what we were adding, subtracting, multiplying and dividing.



Number Properties
Closure Property for Multiplication: states that the product of any two whole numbers is a whole number.
-For any whole numbers a and b,
  a x b is a unique whole number

Identity Property for Multiplication: the number 1
-For any whole number b
  1 x b = b x 1 = b

- When multiplied by another number, it leaves the identity for the number unchanged.
For example,
1 x 4 = 4      1 x 50 = 50     1 x 0 = 0

Commutative Property for Multiplication: states that in any product of two numbers, the numbers may be interchanged (commuted) without affecting the product.
-For any whole numbers a and b,
  a x b = b x a

For example,
342 x 26 = 26 x 342

Associative Property of Multiplication: states that in any product of three numbers, the middle number may be associated with and multiplied by either of the two end numbers.
- For any whole numbers a,b, and c,
   a x (b x c) = (a x b) x c

Distributive Property of Multiplication: when multiplying a sume of two numbers by a third number, we can add the two numbers and then multiply the third number, or we can multiply each number of the sum by the third number and then add the two products.
- For any whole numbers a,b, and c
   a x (b + c) = a x b + a x c

-For example, to compute 35 x (10 + 2), we can compute 35 x 12, or we can add 35 x 10 to 35 x 2. This property is called the distributive property for multiplication over addition.
   35 x 12 = 35 x (10 + 2) = (35 x 10) + (35 x 2)



A times table is a very helpful tool when kids are learning multiplication facts. Teachers often use activities such as timed worksheets. Students may be given one minute to do as many multiplication problems as they can. Here the perfect squares are highlighted. Once kids become more familiar with these multiples and factors, they will  be able to do more challenging problems much more efficiently.

Table from :Vaughn Aubuchon

In Elementary School we were required to memorize our perfect squares, factors, and multiples. Although this seemed difficult at the time, it was well worth it in the future. Today I know my multiplication facts, because of this table. 

Video by: King Yakko

Models for Subtraction Algorithms

Here we go with subtraction! When I was in Elementary School there was something about subtraction that I didn't like... Maybe this was because we became so comfortable with addition we didn't want to try something different.  In this class we have gone over several different methods to do subtraction.

Examples

The definition of subtraction says that we can compute the difference 17-5 by determining the missing addend.

Missing addend: finding the number that must be added to 5 to give 17.

Cashiers often use this approach when making change. It is simpler to pay back the change by counting up rather than down. For example, Jimmy bought a candy bar for 85 cents with a one dollar bill. The cashier then counts 90 by adding a nickel.  Then a dime reaches $1.

 = .85 + __ + __

I think kids would like to learn subtraction this way, because it's just like addition. Once or twice a week I take Madaline to preschool and it never fails, she always finds either a dime or a quarter in the back seat. She gets so excited and says, "I'm putting this in my piggy bank!" Kids love money, even if it is just a coin or two. I think learning addition and subtraction with money would keep their interest and make learning fun.

Taking away concept

Suppose that Emily has 4 blocks and gives away 2. How many blocks will she have left?

  

1.       Start with the 4 blocks and then take away 2.

Center for Theoretical and Computational Materials Science, NIST

                                        

2.       The take away concept shows that 4-2=2.



Center for Theoretical and Computational Materials Science, NIST

      The take away concept should be easy for kids to understand, especially for ones who don't like sharing! It is an easy method to explain and understand.


Comparison Concept

              

                                 

     

 Picture by Christine Pearson

Suppose that you have 1 pencil and someone else has 3 pencils. How many more pencils does the other person have than you?  Compare your collection to the other to determine the difference.  Here you can see that they have two more pencils than you.

The comparison concept is a nice visual way to see subtraction. However, it is not very useful when using large numbers.

Missing Addend Concept

Suppose that you have 6 stamps and you need to mail 10 letters. How many more stamps are needed? In this case we can count up from 6 to 10 to find the missing addend.

+


+


+







                     +                 







                     +                    = 10

 

All clip art in Discovery Education's Clip Art Gallery created by Mark A. Hicks, illustrator.

The missing addend concept is the most popular method for subtraction. When I was in Elementary School I don't remember being taught this. It makes sense on why it is so comfortable to use, because it is addition.




Video by: Harry Kindgergarten

Models For Addition Alogrithms

On Tuesday in class, we learned about Addition Alogrithms...

Alogrithm: step-by-step procedure for computing
1. adding digits
2. regrouping or "carrying"

Examples
Partial Sums
In this method, the digits for each place value are added, and the partial sums are recorded before there is any regrouping. This method is easy for kids, because it allows them to look at the problem in a much simpler way. It is less overwhelming for them to work with smaller numbers.


1.   +345              2.               +345    = 3 hundreds + 4 tens + 5
        278                                278    = 2 hundreds + 7 tens + 8
          13                                             5 hundreds + 11 tens + 13
        11                          Regrouping:   6 hundreds + 2 tens + 3
        5                                               = 623
        623

I was never taught this way of adding in Elementary School, Junior High, or High School. I can't believe that over all those years I never saw a teacher add this way. Not even once! It would have been really helpful in my early years of math, because I would have been able to actually see what I was doing. In the partial sums method, it is obvious exactly what is being added. You can clearly see that there is three hundereds, four tens, and five ones in 345. If I become an Elementary teacher, I will make sure my students learn this method.



Left to Right Addition

First Step                  Second Step                   Third Step
  +897                           +897                             +897
    537                              537                               537
  13                                132                                1324
                                        4                                    43

To add 897 and 537 from left to right, first the 8 and 5 are added in the hundreds colum. In the second step, 9 and 3 are added in the tens colum. Because regrouping (carrying) is necessary, 3 in the hundreds colum is scratched out and replaced by the 4. In the third step, the units digits are added. Again regrouping is necessary, so 2 in the tens column is scratced out and replaced by 3.

This method is comfortable for kids, because they learn to read from left to right, some find it natural to add in this direction as well. I thought this was interesting. This is another method that I have never seen before. The first time doing this in class was kind of awkward. It was weird for me to add this way. It felt backwards. However, I think that if I had learned this when I was younger it would have worked very well, with little or no confusion.


Associative Property for Addition
For any whole numbers a,b, and c,
a+(b+c) = (a+b) +c

In any sume of three numbers, the middle number may be added to either of the two end numbers.

6 + 7 = 6 + (4 + 3) = (6 + 4) + 3 =10 + 3 = 13


Commutative Property for Addition
a+b = b+a

When two numbers are added the numbers may be interchanged without affecting the sum.

26 + 37 + 4 = 26 + 4 + 37 = (26 + 4) = 37 = 30 + 37

The numbers 26, 37,and 4 are arranged more conveniently on the right side of the following equation than on the left, because 26 + 4 = 30 and it is easy to compute 30 +37.
Breaking down problems helps kids understand and see how the problem is actually being solved.

video from xoax.net

This video is a great visual explanation of addition. It specifically shows kids how adding works step by step. The different colored markers help to see what is changing in each problem.

So that's all for addition... Can't wait to see the new things we will learn about subtraction. I'm sure there will be something I've never heard of!

Numeration Systems

This is My Blog for Math for Elementary Teachers...

In this blog I will be emphasizing:
                           interesting topics 
                           unique methods
                           helpful hints
                           fun games

Language deals with letters as math deals with numbers. Without numbers there would be no math. So let's get started by recognizing the history of numbers and how they came around....

Numeration Systems

On Thursday January 27th, we learned that there were several other ways to write numbers other than the way we do today. Most of the ways are extensive and can be confusing at times, which is why we no longer use them.

Numerals: written symbols for numbers
Numeration System: a locally organized collection of numerals

The earliest numeration system appears to have grown from tallying.

© Annenberg Foundation 2011

Each tally stroke represents one. When there are a total of five tallies per set, a diagonal is used. Tallies are very easy to use. I remember using them all throughout elementary school and even in high school when we would play games and keep score.

Base: the number of objects used in the grouping process
Base-ten numeration system: when grouping is done by 10s

Ancient Numeration Systems

• Egyptian Numeration

The Ancient Egyptian Numeration System used pictures symbols called hieroglyphics. This was a base-ten numeration system. Each symbol represented a power of ten.

Made by Fleur de Roos, Ilse De Waele, Vanessa Heyndrickx
September 1998 

Shown above a stick represents numbers 1-9. A heel bone represents 10, a coiled rope represents 100, and a lotus flower represents 1,000. The symbols continue with a pointing finger representing 10,000, a frog (sometimes written as tadpole) representing 100,000 and finally a Pharaoh's kneeling slave representing 1,000,000.

This number system is an example of an additive numeration system, because each power of the base is repeated as many times as needed.

Can you imagine drawing all of these pictures for every single number you used?! I can't imagine how many pieces of paper one algebraic expression would use.

• Roman Numeration

Today we see Roman Numerals on clocks, buildings, gravestones, in books and in sports events like the Super Bowl. (GO Packers!!!) The Romans also used base-ten in their modified additive numeration system. Unlike the Egyptian Numeration System, the Roman Numeration System has symbols for powers of the base, such as 5, 50, and 500.


© 2000-2011 Nicholas Academy

The Romans wrote their numerals so that the numbers they stood for were in decreasing order from left to right (Bennett, Burton, Nelson). When a Roman Numeral is placed in front of a numeral for a larger number its position indicates subtraction. This is shown, for example, in the Roman Numeral for the number 9. It would be written as IX. Showing that one is subtracted from ten.

Roman Numerals are over all simple to understand, however, I think it is one of the numeration systems that has been forgotten. Kids today may not even know what they mean. This chart is very helpful in showing what each numeral stands for. Also unlike the other systems the Roman Numerals do not take long to write.

• Babylonian Numeration

The Babylonians formed a base-sixty numeration system. Their basic symbols for 1 through 59 were additively formed by repeating symbols as shown below:


 ©2005-2011 Red Gate Software

The example shown above is a number much larger than 59. To write numbers larger than 59 the Babylonians used their basic symbols for 1-59 and the concept of place value.

Place Value: a power of the base

The Babylonian place values were 1, 60, 60² , 60³ , etc. The symbols have different values depending on the position or location of the symbol. Above the (1*60²) makes 3600 and the (2*60) makes 120. Add the two together the total is 3720. Add this to the 36 and the number 3756 is the outcome. This system was very particular, therefore, the smallest mistake could change the number entirely.

The Babylonian System seems very confusing to me. I can easily see why it is no longer used. Writing out the number 100 would take some time and thinking. The fact that the placement of each symbol matters can be very confusing as well. The example problem shown above helped me understand the concepts of Babylonian numbers. It also showed the shorthand way of writing numbers and problems.

• Mayan Numeration

The Mayans used a modified base-twenty numeration system. Unlike the previous systems, the Mayan's included a symbol for zero.


J J O'Connor and E F Robertson

As shown numbers greater than 19, place values were used. The Mayans wrote their numerals vertically with one numeral above another. The powers of the base increase from the bottom to the top. The numeral in the bottom represents the number of units. The numeral in the second position represents the number of 20s.

The Mayan Numeration System is very clear and easy to comprehend in the chart above. It makes sense and was one of the best systems during its time. The Mayans were the only group to include the number zero. That says something right there.

• Hindu-Arabic Numeration

Today, most of the world uses the Hindu-Arabic Numeration System. This positional numeration system was invented by the Hindus and the Arabs who transmitted it to Europe. It is a base-ten numeration system, in which place value is determined by the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit in a numeral has a name that indicates its position (Bennett, Burton, Nelson).

In English the number names for whole numbers from 1-20 are all single words. The names for numbers 21-99 are compound number names that are hyphenated. Except for the numbers 30, 40, 50, etc...

Obviously this is the shortest system of them all. It has been very successful for many years and that is why it is the one we use today.




© Puentes

Aren't you glad we don't use Hieroglyphics, or the Babylonian Number System!? Could you imagine doing algebra and calculus problems with all of those symbols?! Problems are hard enough with our number system today. I think it's interesting how numbers have developed over time, and I am very thankful they have =)