Reviewing the PUMAS Website

PUMAS (Practical Uses of Math and Science) is a website that provides lesson plans for activities that are rooted in real-world scenarios. I took some time to review some of these activities and found one that would be somewhat useful to me during one of my spring units.

Every year without fail, my students struggle with unit conversions. For some reason, they have a very hard time grasping this abstract concept. As much as I can, I try to use problems that link to their everyday lives; but their are limited activities for this topic. One contributor to the PUMAS website provided an excellent example of a potential oil spill and the ramifications of such a spill (Coastal Threat: A Story in Unit Conversions). To determine how big the effected area of the spill would be, they took the 400,000 gallon capacity of the ship and converted the volume to meters cubed. Assuming that the viscosity of the oil would cause it to form a layer that is 1/100 of an inch thick, they then used unit conversions to calculate the width and length of the potential spill.

The example concludes by having the students actually simulate the spill using a cookie sheet and water. I could also tie this activity in with scale drawings and proportions to have students practice using scale to draw the effected area. Using any of the activities or just using the example itself should help to increase student engagement by allowing me to teach to different learning styles.

Reference: http://pumas.jpl.nasa.gov/examples/layout.asp?Document_Id=02_23_99_1

Multiplicative Property

Word definition: The multiplicative property states that any number multiplied by its reciprocal (one over the number as a fraction) yields a product of one.

Numeric Form: 1/7 x 7 = 1

Algebraic Form: m x 1/m = 1

Proof: m x 1/m = m/1 x 1/m = m/m = 1

Example:

You are making cookies.  The recipe calls for 2 and 3/4 cups of sugar, but you only have a 1/3 cup measuring spoon.  How many of the 1/3 cup spoonfuls will you need to make 2 and 3/4 cups?

Solution:

First, convert 2 and 3/4 to an improper fraction:  11/4 (4 x 2 + 3 to get the numerator.  Keep the same denominator).

Then, divide 11/4 by 1/3:

11/4 ÷ 1/3 = 11/4 x 3/1 = 33/4 = 8 and 1/4 spoonfuls.

You will need to fill the 1/3 cup spoon 8 times and then add 1/4 of the 1/3 cup.

Additive Inverse Property

Word Definition: The additive inverse property states that any number added to its opposite will equal zero. Remember that a number’s opposite is the number that has the same absolute value (see Figure 1).

Figure 1

Numeric Form: From Figure 1, if we add -4 and 4 we get the equation -4 + 4 = 0

Algebraic Form: n + (-n) = 0

Example:

On two consecutive plays, a football team gained five yards on a quick slant pattern, then lost five yards on a sack. What was the net gain for the two plays?

Solution: 5 + (-5) = 0 yards gained

My Mathography

As with any autobiography, this “mathography” starts at the beginning of my math career. My early math years were spent doing many of the same things other students do in their primary years. I learned concepts like my multiplication tables, long division, and other basic math facts. I enjoyed practicing these basic facts and seeing how quickly I could recite them. I mostly enjoyed math because it came easy for me. However, one thing with which I struggled from the beginning was word problems. I just could not identify with Timmy and Mary and why they were buying apples and why they didn’t just get an apple from the refrigerator instead of buying a brand new one. I obviously struggled when it came to finding ways to apply math and thrived when it came to memory, recall, and basic calculations.
I can remember two learning activities in math very vividly, and they both occurred in the second grade. The first activity I can remember was a positive experience for me. It was the first time I had learned long division and double-digit multiplication. I was so proud that I could manipulate such large numbers. I can even remember having my dad give me practice problems on the chalkboard that my grandparents had in the hallway of their old house. The second activity was much more frustrating for me. My teacher gave us a worksheet that had a series of word problems about ice-skating. I hate ice-skating. No other sport has caused greater frustration or embarrassment in my life. Suffice it to say that I could not find a way to identify with these problems. I became so frustrated that I started to cry in the middle of class. Even now, this is an embarrassing memory for me, but it helps me to identify with the frustrations that some of my students feel on a daily basis in the math classroom.
Two teachers greatly impacted my decision to become a math teacher and cultivated my love of math. As I think back, I realize that neither of them used any Earth-shattering teaching techniques. They simply encouraged me to pursue my interest in the subject further. In third grade, Mrs. Arndt first helped me to realize that I had a gift in the subject. She provided all of her students with lots of positive feedback, praise, and encouragement. She also allowed me to work ahead of the rest of my classmates. This suited my personality very well. As I worked to get as far as I could, I found myself enjoying math more and more. The second teacher had more of an influence on my decision to become a teacher. Mr. Kellam was my math teacher for Algebra II, Pre-Calculus, and Calculus. I went to a very small high school, and Mr. Kellam had been around forever. He had taught my mom and all of her siblings, and as of today he is currently teaching my youngest sister in her sophomore year. Mr. Kellam constantly facilitated peer-to-peer instruction. Every example problem was explained by a student. As I grew older, I found myself enjoying this practice more and more. When I went back to visit on one of my breaks, there was no one happier than Mr. Kellam to learn that I had become a math teacher.
I always enjoyed math. However, science was always my favorite subject. It wasn’t that I disliked mathematics; I just saw science as having greater application in the real world. My personality causes me to search for ways to apply what I learned. Physics and chemistry provided me with ways to apply what I had learned in math to the real world. While teaching math pays the bills for me, I will always be more fascinated with science and exploration of the physical world.

Everyday Math – PLS on the Parkway

My wife and I are currently taking an on-site PLS course entitled Purposeful Learning Through Multiple Intelligences. If you’re not familiar with the how the on-site courses work, they meet from five until nine on Friday, eight until five on Saturday, and eight until five on Sunday. Needless to say, you get a lot done in a weekend but the days can be exhausting. We found a site not too far from our house; it was only a driving distance of about fifteen or twenty miles. This was no problem. The problem was The Parkway. Now, if you know anything about Pittsburgh, you know that we don’t have what you might call logical roadway systems. The Parkway is I-376 which runs through the middle of Pittsburgh and out to the Penna Turnpike. The road itself is not bad, but it’s construction season here in the ‘Burgh (construction season usually runs from about the beginning of February to the end of January), and when our PennDot crews work on The Parkway they will shut anywhere from one to all lanes. This would not be a problem except that just about everyone uses this road to get around. Needless to say, traffic is a NIGHTMARE! Anyways, back to the story. We were less than thrilled to have to drive this route three times over the weekend, so we thought it may be a good idea to look into staying in a hotel. After all, the class was being held at the Holiday Inn. My job was to figure out if this option would be cost-effective when factoring in gas, food, and the cost of the hotel room.

Option 1: Stay at the Hotel, Avoid the Headaches of Traffic

I started by looking up the cost of the room on holidayinn.com.  The lowest rate they had was $124 per night.  I wasn’t off to a great start.  For two nights that would come to $248 (124 x 2 = 248).  But I thought maybe once I factored in the price of gas, it would even out.  I assumed that the price of gas would at least stay around the same price it was when I did this calculation, $3.65 per gallon (a risky assumption to make these days).  My wife’s Ford Focus gets about twenty-five miles per gallon for a trip like this where there is both city and highway driving.  The distance from our house to the hotel was 22 miles.  This would mean that if we stayed at the hotel we would be making this trip twice (once there and once back) and would travel a total of 44 miles for the weekend (22 x 2 = 44).  This would use up 1.76 gallons in the Focus (44 miles / 25 miles per gallon = 1.76 gallons).  At $3.65 per gallon, our total for gas if we stayed at the hotel would be $6.42 (1.76 x 3.65 = 6.42), and our grand total so far for the hotel weekend would be $254.42 (we’ll round that up to $255).  Next, I had to factor in meals.  I budgeted about $50 each day ($10 for breakfast and $20 each for lunch and dinner) for both of us.  This would amount to $120 for food for the weekend since we would only be eating dinner on Friday (50 x 2 + 20 = 100 + 20 = 120).  Our grand total for the entire weekend if we stayed in the hotel was $345 (120 + 255 = 345).  Ouch!

Option 2: Make the Drive, Stay at Home

Next I wouldn’t to see what we would spend if we did not stay in the hotel.  Obviously we would spend more money on gas, but we could save money on lodging and food (not to mention we would be able to sleep in our own bed).  I assumed the same values for gas mileage and price, except now we would be making the 22 mile trip six times (there and back each day).  We would travel 132 miles (22 x 6 = 132, three times more than what we would in the other options) and would use 5.3 gallons of gas (132 / 25 = 5.3).  This would give us a gas price of $19.27 (I rounded that to $20).  That’s not too bad if you can stand the traffic.  Next I calculated our food expenses.  Since we could eat breakfast and dinner at home, we would only be spending $20 for lunch on Saturday and Sunday.  This gives a total of $40 spent on food for the weekend (20 x 2 = 40).  That brought our grand total for not staying at the hotel to $60 (20 + 40 = 60), a difference of $305 from the other option (365 – 60 = 305).  Needless to say, we decided to brave the traffic and make the drive.  I’m glad I did the math, though.  I may have been tempted to take the easy way out otherwise.

Vocabulary

Today’s “Word of the day” is:

in·te·ger /ˈɪntɪdʒər/ Pronunciation Key – Show Spelled Pronunciatio[[in-ti-jer] – noun

1. Mathematics. One of the positive or negative numbers 1, 2, 3, etc., or zero

To put that in “math speak,” an integer is basically the set of all whole numbers and their opposites.

Opposites are numbers who have the same absolute value, or distance from zero on a number line. For instance…

…in this diagram, points B and E are opposites because they are 2.5 units away from zero. -2.5 is the opposite of 2.5.

Here are some neat websites to help you explore this topic further:

http://math.com/school/subject1/lessons/S1U1L10GL.html

http://www.mathleague.com/help/integers/integers.htm

http://amby.com/educate/math/integer.html

References:

http://dictionary.reference.com/browse/integer

Revision to “About Me” Post

I realize that I forgot to add my two learning goals for this course.  My first goal is to learn new strategies for helping my students to think mathematically.  I’m always trying to get them to find the meaning behind what we are learning and apply it to their everyday lives.  I’m hoping that this course will help me do that.  Secondly, I would like to gain some new ideas for activities that I can use in my classroom to increase active engagement.

A Little Bit About Me

Welcome, everyone. I’m setting up this blog as an assignment for an online class I am taking. However, once the class is done, I may update it to keep my family in the Midwest updated on my life here in western Pennsylvania. I am a seventh grade Algebra teacher in a suburban school district. I’ve been happily married for two years (still in the honeymoon phase, and we don’t plan on exiting it). My wife and I both attended Grove City College, and we love to get back there as often as we can. Having grown up in the Midwest, I’m also a huge basketball nut (you don’t see many of those where I live now). During the winter, I even do a little bit of coaching. I also enjoy golfing and spending time outdoors. I closely follow the Chicago Cubs and Bears, Notre Dame football, and anything having to do with college basketball (I loathe the NBA). Some of my favorite TV shows include The Office, Amazing Race, Survivor, Biggest Loser, and the Big Bang Theory.

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