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.
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.
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.
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.
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.
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.
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.
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.
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!
