Sets and Venn Diagrams

Even as an adult, understanding sets; the unions, intersections, complements, empty sets, and universal sets is extremely complicated to understand. Through the course I am taking this semester, I have learned how to use a Venn Diagram to help students better understand sets and the questions that come with them.

The first thing students need to understand is what an Universal Set is; in easiest terms is a set that contains all elements of a problem to be considered.

I took away the Venn Diagram to show that a Universal Set is just all of the numbers all together.

A Union of two sets, denoted by a capital U, is everything in both sets. Now I now as an elementary student, this is difficult to understand (mentally and visually). But, if you explain the same definition with a Venn Diagram, your students will better understand.

For this example, set A={2,5,7} and set B={1,2,5,8}

When asked what the Union of A and B is you would state every single number from both sets like such: AUB= {1,2,5,7,8}

An Intersection of two sets, denoted by a capital upside down U, is anything that appears in both sets; the elements in the middle of both sets. Again, for students reading this or hearing a teacher explains this, would be very complicated to understand, so if you demonstrate a Venn Diagram they can better understand.

For this example, set A={1,2,3,4,5} and set B={1,3,9,12}

When asked what the Intersection of A and B is you would state that the numbers in the middle of the two sets like such: A∩B= {1,3}

A Complement of a set is all the elements in the universal set that are not  in the initial set, and is denoted with a letter c.

In this example, the Complement of  set M would be any numbers outside of the set in yellow, and would be stated as such: Mc={-5,-4,-3,-2,-1,0,1,3,5,6}

Lastly, an Empty Set is a set that has no elements, is often referred to as a Null Set or Disjoint Set, and is notated with {} or a zero with a line through it.

Number Lines

I have found number lines to be one of the most helpful tools in teaching and helping elementary students solve addition and subtraction problems. Working in an elementary school myself, I help a lot of students with their addition and subtraction problems. Adding and subtracting are the most basic of math operations, and I feel that they need to be mastered before moving on to more complicated concepts; adding and subtracting are the base of math. I have also discovered that number lines help clarify how we get negative numbers, I know this to be really hard for students to understand.

I found a great website called funbrain.com that has a great Line Jump game that helps students. Students can chose from multiple games. Once you chose a game, a screen loads up with a number line, and  an airplane above it. The goal is to drop the parachute from the plane onto the correct number on the number line. The games that students can choose from are as follows:

• Easy which is a basic 0 through 10 number line with one digit addition and subtraction facts. The problems, 3+1 for example, show up above the number line. The dot shows up on the first number of the problem, and then the student moves the dot to the left (for subtraction) or the right (for addition). 
• Medium gives the student a number line numbered 0 through 20, and different addition and subtraction problems show up. Once again, you would drop the parachute from the airplane onto the correct number.
• Hard starts introducing students to negative numbers on a number line numbered -8 through 8. Just like the previous settings, you are given different addition and subtraction problems, but this time they start mixing in adding and subtracting of negative numbers.
• Really Hard is almost the same as the Hard game, consisting of the same -8 through 8 number line, but involves more negative problems.
• Super Brain has a -16 to 16 number line, and just like the Really Hard game, involves more of the negative moving problems.

I have found the number line jump game to be extremely helpful in helping the students at my job, as well as my own child. I would even be honest and say that it has helped me myself, I no longer need to use a calculator to figure out basic adding and subtracting problems. 

Properties of Whole Number Addition

Let a,b, and c be any three whole numbers

Closure Property:

a+b is a unique whole number

What this means is that sum of two numbers is always a whole number.

Commutative Property:

a+b=b+a

This means that the sum of two numbers is not effected by the order in which they are added together.

Associative Property:

a+(b+c)=(a+b)+c

Grouping and/or associating numbers in different orders will yield the same answer.

Additive Identity Property:

a+0=a+a=a

The sum of any number and zero will always be the original number.

There are many great songs to help students in remembering the properties, the Properties of Addition song I have discovered goes along to a very popular song by a group called One Direction entitled What Makes You Beautiful that a lot of children have heard, and is extremely catchy.

The two properties I have found to be most difficult for students are the Commutative and Associative Properties.I found the Turn Around Addition Song (a teacher at my job showed me this) to help students better understand how the Commutative Property works:

I also found another great song for the Associative Property to help students out:

A lot of teachers have told me that they use songs for teaching their students the properties for addition, subtraction, multiplication, and division. The songs stick in their students heads, and they are constantly singing them when they go to solve their problems, and the students are getting it and getting their homework and tests correct.

Repeated Subtraction Model to Solve Division Problems

If I had been taught to do division this way in my own schooling, I don't think that I would dislike it as much as I do. You can also set up the problem as sixteen minus 4, and keep subtracting four until you get zero:

16

 -4 

12

-4

8

-4

4

-4

0

One you get to zero, you go back and count how many times you took away four, which is four times, and you've got your answer: 16/4=4.

This method works wonderfully with all types of division problems, and greatly decreases frustration and confusion that the method of long hand division creates. The example below is a division problem that does not divide evenly:

55/6=?

55

-6

49

-6

43

-6

37

-6

31

-6

25

-6

19

-6

13

-6

7

-6

1

For this problem, you cannot get down to zero because 6 does not go into 55 evenly, you are left with a remainder of 1. Now that you know your remainder, you go back through and count how many times you subtracted 6, which is 9 times. Thus your final answer to 55/6=9R1.

Understanding Absolute Value

I have found that using a number line is the quickest, and smartest way for finding the absolute value of a given number. Below are two images for finding the absolute value of a positive number, and negative.

When broken down, and shown with a number, I myself was better able to find the absolute value of any number put in front of me. If elementary children are shown this way, I think that they will have a better time understanding absolute value.

Coming across a problem like│5│=?  This problem is asking the number inside the bars distance from zero. It is important to remember that the answer will always be positive. The first thing you do when you come across an absolute value problem, after reading and analyzing, is to set up a number line. Second, you locate and label the number that is inside of the bars. Last, you count how many you are away from zero and have your answer. I discovered a great song to help students in calculating their answers.

Sometimes you may come across a problem that looks like this:

-│7│=? or -│-7│=?

What do you do? This can be extremely confusing for students because they have been taught that the answer will always be positive. In these special cases however, the negative is placed on the outside of the bars (it is not actually part of the absolute value, its on the sidelines if you will). You would go about and use the number to find your answer, and in the end would carry the negative over, thus getting -7.

Problem Solving Strategies: Understanding, and using the Guess and Check method.

Once upon a time when I was in grade school, the guess and check method could keep me up all night trying to “guess” the correct answer, and “checking” my guess to see if it fits. In today’s math classes, teachers are teaching students a much more quicker and efficient way to guess and check. Below I will demonstrate the process.

Example Problem: Mr. Young has 32 18-cent stamps and 27-cent stamps all told. The stamps are worth $7.65. How many of each kind of stamp does he have?

Now instead of blindly guessing numbers, teachers are teaching students to start guesses with multiples of ten. Watch below:

*Note: if you guess 20 stamps you have to do 12 for the 27-cent (32-20=12)

Guess #1) 18 X 20 + 27 X 12=$6.84

This first guess is too low for our $7.65 total, so now let’s take and use 10:

Guess #2) 18 X 10 + 27 X 22 =$7.74

Now this is an extremely close guess. For you next guess, you’re going to want to increase by just one:

*Note: because you are guessing 11 don’t forget to minus 11 from 32 to get 21 for the 27-cent to be multiplied by.

Guess #3) 18 X 11 + 27 X 21 =$7.65

Tadah, we just learned how to do a guess and check problem in less than a minute!

I really wish I could have learned this technique back in my day. Children these days are so much more fortunate in the methods they learn for solving math problems.