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 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!