Measure of Central Tendency: Mean, Median, & Mode

Aloha friends,

Measure of central tendency, what an intimidating & scary phrase! It's probably best to put that word aside. Let's say, in the closet. We're going to put that phrase into the closet, along with a lot of other words we might come across I imagine. So from this point forward, if I'm putting a word in the "closet" then it's probably not necessary to remember at this point in our learning. The measure of central tendency breaks down to the mean, median, and mode of data. More intimidating words, at least to me they are...let's break those bad boys down to the bare essentials. 

The mean is another word for the average. To the find the average of any numbers, we must add all of the numbers together and divide that number by how many numbers were added together. For instance, what's the average of 5, 5, 10, 15, 5, 10?

1) First, add all of the numbers...  5 +5 + 10 + 15 + 5 + 10 = 50

2) Second, divide the number by how many numbers were added together (6).

50/6  =    8.3333

3) The average number is 8.333. 

The median is the literally the middle number when all of the numbers are lined up in numerical order. 

So using our past numbers: 5, 5, 10, 15, 5, 10 we're going to find the median.

1) List the numbers in numerical order, smallest to largest or largest to smallest (smallest to largest makes more sense to me personally). 5, 5, 5, 10, 10, 15

2) Find the middle number, if the middle number can't be found because there is an even number of values then averaging the two middle ones gives you the median. For instance, add 5 + 10 = 15 ... divide by 15/2 = 7.5. The median is 7.5. Our math teacher says it's easy to remember median by thinking of medium. 

The mode is "data value that occurs most often." Looking over our data set: 5, 5, 5, 10, 10, 15 we can see that 5 occurs the most often so the mode is 5. Our math teacher says it's easy to remember mode by thinking of most. 

Carpe diem, 

Cass

* Links to other helpful websites about the measure of central tendency:
1) Kids Math Games -
http://www.kidsmathgamesonline.com/numbers/meanmedianmode.html
2) Measure of Central Tendency Rap Song -
http://pinterest.com/pin/122582421080180899/

Stem-and-Leaf Plot

Morning friends, have you had a chance to go outside today? Have you seen any living flowers  lately? I hope so, I personally consider nature to be my church. I feel most alive and at peace when I'm engulfed in it.

My Blog for today is about stem-and-leaf plots. When I hear stem-and-leaf I think of flowers. Stem-and-leaf plots are valuable tools when quickly organizing and analyzing data. They are wonderful visual tools. They are particularly valuable when looking at line plots and histograms (graphs with intervals that are easier to read and analyze with specific numbers).

An example of a stem-and-leaf plot is below:

1     1 1 2 5 8 9
2     0 1 2 2 4 7
3     4 8 9 9
4     1 2 3 7 9

The numbers on the left (1, 2, 3, and 4) represent the tens place. The numbers on the right represent the ones place and represents how many are in that category. For instance, the 1's have six numbers: 11, 11, 12, 15, 18, 19.

You can introduce stem-and-leaf plots as early as third grade probably.

Carpe diem,
Cass

Circle Graphs

Good morning friends

I want to talk about circle graphs today. Circle graphs are fun to introduce because most teachers introduce them by comparing them to a pizza and what kid (and adult) doesn't like pizza?
Circle graphs are more "complex" than bar graphs. You "can turn the numbers into percents and then sketch a circle graph" or you "can turn them into degrees and make the graph with a protractor."

The grade students begin to use protractors is a good age to begin creating circle graphs using degrees. It's good practice of circle graphs, geometry, and measurements and data.

A standard protractor (fairly inexpensive, less than $5.00)

Bar graphs and circle graphs have different advantages and advantages. One student may prefer a circle graph and another might prefer a bar graph to depict the same data. It's not a matter of what's right or wrong, but preference...and understanding the pros and cons of each. This is a perfect example of why it's important to support more than one way of thinking, not to push one method (your preferred method). One activity you could possibly do to compare and contrast circle & bar graphs is to have two sets of data (one ideally better suited for a bar graph and one ideally better suited for a circle graph). Instruct the students create a bar and circle graph of each sets of data and choose which one helps them answer a set of questions better. Then you can have them analyze their results and explain why they came up with those results. After there can be a class discussion over the two types of graphs. Critical thinking is necessary in all subjects and unfortunately math is sometimes forgotten.

Carpe diem,

Cass

* Links to other helpful websites about bar circle graphs (or pie charts/graphs):
1) Pie Chart -
http://www.mathsisfun.com/data/pie-charts.html

Bar Graph vs. Histogram Graph

Good day friends! Today's Blog is all about graphs. A graph is a "picture that displays data and [is] used to tell a story." Bar graphs and circle graphs are the most common graphs elementary aged students will see in school. First I'm going to compare and contrast bar graphs and histogram graphs because at first glance they appear identical.

Bar graphs are "used to represent situations where the data are categories." I discussed categorical data in my past blog (please refer to it if you have questions). You can arrange bar graphs "randomly, alphabetically, or by popularity." Don't be confused, bar graphs (or categorical data) is not absent of numbers but the categories or choices aren't numerical. If you had your first grade class choose their favorite holiday and 12 people chose Christmas and 3 chose Halloween then you can see that numbers are attached to the categories.

Histogram graphs "can be used to summarize numerical data that are on an interval scale, either discrete or continuous." Histograms are good graphs to use for ranges of data, data that's best read if in order. A histogram must be in order

You can begin to introduce graphs to children as young as first grade but the depth to their understanding may take a few years to develop. For instance, first graders can be taught to make simple bar graphs or line plots and they can analyze data from the two types of graphs. However a histogram has a little more depth to it and the vocabulary one might use to explain a histogram would probably be over the heads of most first graders. I would suggest introducing histograms in second or third grade.

*A wonderful book to use to begin teaching about graphs to young children is Lemonade for Sale by Stuart J. Murphy and Illustrated by Tricia Tusa.*

Carpe diem,

Cass

* Links to other helpful websites about bar graphs &   histograms:
1) Creating Bar Graphs - http://www.readingrockets.org/article/43814/
2) Histograms - http://www.mathsisfun.com/data/histograms.html
3) Bar Graphs -
http://www.mathsisfun.com/data/bar-graphs.html

Data & Measurements

Hello friends!

Today we'll dive into data. I'm going to talk about categorical data and numerical data.

• Categorical data is data that isn't numbers but categories. An example of categorical data might be if you ask your kindergarten or first grade students what their favorite color is: red, orange, yellow, green, blue, etc.--If one student answered "purple"--purple would fall into a category (colors), purple is not a number. 
• Numerical data is centered around numbers, data collected from numerical variables.For example, how tall are all of the students in your classroom? You can have them measure themselves in inches or centimeters, both are numbers (numerical data). 

Both categorical data and numerical data can be taught as early as first grade. My children began to learn how to measure in the first grade. They began to learn how to measure with a ruler but you can teach your students to measure using a finger, a button, a paper clip, a cheerio, marshmallows, etc. For example, how many marshamallows long is the paper?

In the first grade it's important to introduce measurement but why we measure and how to measure accurately comes with age and practice, so patience is key when teaching data measurement to young children.

Activity:
*You could have your first graders practice measuring for a week using different objects, including a ruler.*
On the last day of the data measurement unit you can have your students:
1) Color a picture of an animal (normal paper size).

2) Pass out (x) number of cheerios (m&m's, mini marshmallows, skittles, etc.) to each student. The number is based on how many on average it will take to measure the perimeter of the animal plus 5-10 more.

3) After the students finish coloring, instruct them to measure the perimeter of their animal with their measuring tool (teacher's choice) and record the number at the bottom of the page.
Total Number: ___

4) Have them tally their results on the board.
Giraffe Perimeter:
8   9   10   11   12   13   14

5) Discuss how the number with the most tallies is probably the most accurate but double check it with the class, pointing out things that may have led to different data results. For example: misshaped cheerios (smaller or bigger ones).

6) After discussing the results as a class, instruct the students that they can eat their measuring tools (cheerios) if they want.

Let me know how it goes if you try this activity.

Carpe diem,

Cass

* Links to other helpful websites about measurement:
1) Article about measurement for prekindergarten-kindergarten teachers/or parents -
Measuring Experiences for Young Children

Introductions

Hello Friends,

Welcome to my first Blog post in my thirty years of living on planet Earth. My name is Cassandra and I'm happy to say that I'll be blogging about math. I have no idea what I'm doing when it comes to Blogs but I do have a small idea of what I'm doing when it comes to math, and more importantly I'm hopeful I can help make math easier for you. 

Shamu (above)

First a little background information on me. I was born in Daytona Beach, Florida where my father was a race car driver. My mother was the head whale trainer at Sea World, and yes I spent my weekends growing up feeding Shamu--the original Shamu. I was a professional surfer into my early twenties but a hungry shark ate my left leg for lunch off the North Shore of Kauai in December 2007. After I lost my tasty left leg I chose to go to school for teaching. I love community service and it's the closest thing to it in my opinion. I leave the classroom filled with love and happiness, like a deflated balloon filled up over the school day--it must be the right profession for me.

Math was a subject I feared until college. It wasn't until college that I believed it wasn't the subject (mathematics) or me (my brain) but probably how the subject was taught to me. It's  a simple subject, not much to fear at all. The importance lies with how you teach people and what works for one person may not work for another. I believe children learn by doing and by making connections to their daily lives. Simply learning to find the answer to a problem won't cut it. My blogs will teach basic elementary math and will give you ideas on how to make meaningful connections to the math we ask our students to learn. 

I'm going to end each of my Blogs with "Carpe diem," translating to "seize the day" -- I hope you continue on with me friends. My first official math Blog will be about data and measurements, stay tuned.

"In this world, nothing can be said to be certain except death and taxes." - Benjamin Franklin

Carpe diem,

Cass

*One link that may be helpful for teaching in Arizona or for learning where your child/children should be academically for their grade level is:
1) Common Core State Standards - http://www.corestandards.org/Math