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. 

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.