Fractions on a Clock Face.

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One great way to get students to see that they can list several names for one fraction is by exploring what we could name each section of a clock face that has had the hands placed on different numbers. We always leave one hand at 12, and our clocks have equal length arms so we never know know whether we are looking at an hour hand or a minute hand.

Starting with the bottom clock, we see that one hand is on the 12 and one hand is on the 2. Most students first see this as 2 out of 12 hours, and they write the fraction 2/12. Many also see that this could be 10 out of 60 minutes, or 10/60. From that point, many notice that neither of these fractions is stated in its lowest terms. Most then "chop-chop" 2/12 (divide by 2/2) to get the fraction 1/6. Many then can say that 16 2/3% of the clock face has been "covered up" or "rotated through".

The next picture up shows one hand on the 12 and one on the 8. This shows 8/12 or 40/60. Again, neither fraction is in lowest terms. Many students will just see this as 2/3, as 4/12 looks like 1/3, but to prove it, one could divide 40/60 by 10/10 to get 4/6. Then one could divide 4/6 by 2/2 to get 2/3. We have mathematical PROOF! Most students now see that 8/12 is the same as 2/3, and that is the same as 66 2/3%. SWEET!

The goal in all of this is to be able to add fractions with unlike denominators. The third picture up illustrates the addition of 1/4 + 2/3. 1/4 is seen to be equivalent to 3/12 and 2/3 was proven to be equivalent to 8/12. So, we get 3/12 + 8/12 = 11/12.

We might also get 25% + 66 2/3% = 91 2/3% but 91 2/3 / 100 is not in its lowest terms, and most middle school teachers would simply freak-out if the answer is presented in a percent. So, even though the % is correct, I would emphasize finding the answer as a fraction in its lowest terms.

One last note, please remember that any fraction can be stated as another equivalent fraction simply by multiplying or dividing the fraction by the number 1, and any fraction where the numerator and denominator are the same (N/N) equals 1.

So, 32/48 divided by 16/16 , still has the value of 32/48, but it is more universally recognized as 2/3. 400/500 = 4/5 not because "you can drop the zeros", but because 400/500 divided by 100/100 = 4/5!

Similarly, 3/4 = 9/12, because 3/4 X 3/3 = 9/12. 5/6 = 10/12, because 5/6 X 2/2 =10/12!

I hope you are not confused, but if you are, please send me your comments :0}

Fractions Are Instructions to Divide! Sir!

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Just how do you go about solving a question like: What is 3/12 of 36 penguins? There seems to be an almost endless list of strategies that don't work well. However, there is one strategy that works VERY well.

In my class, I have the kids repeat, military style, "Fractions are instructions to divide by the denominator and then multiply by the numerator! Sir!"

Almost all kids can remember this"direct order". Many do not, however, know what needs dividing up, or into how many groups. In the problem above, the 36 penguins need to be divided into 12 even groups, and that puts 3 penguins in each group. 3 would be a fine answer to the question, "What is 1/12 of 36 penguins?", but is not a good answer for 3/12. This is where the "and multiply by the numerator" comes into play. We need to take into account 3 of the 12 equal groups of penguins, or 3 X 3 = 9 penguins.

The work also shows an array of 36 "penguins", and it would be quite correct to say that 3/12 could be understood as 3 out of every 12. This is a great model for small numbers, but I would not want to split up 3600 penguins into an array!

So, "Fractions are an instruction to divide by the denominator and then multiply the answer by the numerator! Sir!"

Fraction Equivalent Strips

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Please take a look at the fraction strips above. The goal of the above was to match about 20 landmark fractions to their equivalent percents. The goal was not rely on memory, but to strategically think about the placement of these fractions. For example, most students immediately know that 1/2 = 50%. They can use that knowledge to "discover" that 1/4 = 25%, and then to find that 1/8 = 12.5%. Even the more difficult fractions like 1/6 become easy to place when there are landmark percents already present. 16 2/3 % is easy to place if 10% and 20% are already listed.

Fraction Action in Fifth Grade

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This story about fractional numbers is told from the bottom picture up.

In our EDC notebooks, we have been tracking equivalent fractions. We started on day 1 with the fraction 1/16. All students were able to understand that there were 16 pieces in the top row and if we were to shade one of those boxes it would equal 1/16 of the whole. On day 2, most students immediately realized that when we added another 16th, we created an area equal in size to the the 8ths row immediately below. When we got to 4/16, also 2/8, students saw that this was equal to 1/4 of the whole. It was at that point that students started to fill in percentages for the fractions that we had created. Most students knew that 1/4 = 25%, and they concluded that 1/8 was half as large so it must = 12.5%. A few students then realized that 1/16 must equal half of 12.5% or 6.25%. This knowledge of fractions and their equivalent percents will be a key component in adding, subtracting and comparing fractional pieces. Most students know several landmark percents like 25%, 50%, and 75%, and they can easily shade an object like a circle graph correctly with those amounts. Soon, students learn to represent other amounts like 30% (3/10) with accuracy based on their knowledge of the landmarks mentioned. Putting fractional amounts into percentages just makes life easier.

The third picture up comes from our current investigation, and it shows how students come to see "out of" statements as fractions. One out of three is the fraction 1/3, which seems easy enough but is VERY important conceptually.

The next two pictures are the absolute most important concepts that we will cover in this unit. Students shade in portions of a 10 X 10 grid and report the percent (parts out of 100) that it took to cover the portion. One picture shows one fourth of a grid shaded (25 blocks and 25%). the shading is a bit "artistic" , and it probably led to the misconception about shading in 1/8 of the grid, which should be 12 1/2 blocks or 12.5% . The percent was listed correctly, but the grid does not have 12 1/2 blocks shaded. Still, I can tell that the idea of 1/8 being half of 1/4 is understood.

The top picture is the Holy Grail of understanding fractions. All of the fractions listed are considered landmark (IMPORTANT) fractions in our number system. Understanding the percentages that go with each of these is a very critical conceptual step.

So, know the fraction. Be The fraction!

Know the percent. Be the percent!

Revisions: Another Great Example!

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Doing revisions correctly takes a bunch more work than just stating a new answer. Only by clearly stating why a prior answer was wrong and providing new evidence (NEW WORK), can an instructor be sure that the concept involved has really been mastered.

In this example, every distractor (answer choice) is examined. Clear reasons are stated as to why there is only one correct answer. This is enough to convince the person in charge (ME) that the concept of what a square number really is is now understood.

PTB Prime-Time-Battle

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Prime-Time-Battle is a really fun EDC game that we play daily. This games gets students to try and find the prime factorization of numbers, as all numbers can be expressed as the product of prime numbers (OK, not 1 or 2). Besides building automaticity with multiplication facts, PTB prepares kids for much of the learning that they will see in many advanced courses. In advanced math courses, many times LCM and GCF are found through comparing prime factors. It also seems to just be a bunch of fun, as many kids see it as a a chance to compete in friendly math endeavor!

Have I Got A Story For You!

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To leave a comment, you have to click on the word "comment" in the comment counter at the bottom of the blog :-} I smell candy!

All of these samples of student work show how creative and efficient students can be when asked to solve basic multiplication and division problems. All of these students used very efficient strategies to carry out work that is usually a very scripted set of steps (Traditional algorithms allow for very limited flexibility in mathematical thinking).

Equally important, the students were able to create their own word problems that could be answered by solving the numerical problems. This is always a very tricky situation for students of this age, (in truth, it is for students of any age) as students are so used to solving numerical problems that are written for them. Many start by imitating problems that they have read in math books. However, when students start to create problems that are really relevant to their lives, they gain a very deep insight into what multiplication and division really are.

These are works in progress, and there are some very simple errors, especially in the division problems (usually because of faulty subtraction), but the kids are showing a pretty deep understanding of what the answers really mean. In most math books, the remainders of division problems are just written as a fraction or a decimal amount, but in the real world, remainders mean something. Remainders can be valuable or inconsequential, and contrary to popular opinion, not all remainders can be split into equal pieces.....kids, puppies, and crunchy potato chips included!

In photo number 1, there is a subtraction error. 835-690 = ?

In photo number 2, one factor pair, 26 X 20, was left out.

In photo number 3, the remaining $17 could, and probably should, become 50 cents extra per person, although giving the money to charity is admirable :-}

In photo number 4, 3135-2300 = ?

As you can see, there is a ton of input to manage, but the concepts are coming together!

Double-Trouble or Split-It!

Recently we have added a "game" to our EDC notebooks. Double-Trouble and Split-It are games that are designed to get students to strategically think about doubling and splitting numbers. These skills are crucial if a student is to build great mental number sense.

For example:

When I ask kids to split 1,700, many kids balk, because they see 17 groups of 100 as an odd number of hundreds. Most kids are not seeing this as 8 1/2 groups of 100. However, most students can readily replace 1,700 with 1,600 + 100, and they can easily split these numbers to get to 850! This still may not seem important, but since the product of 17 X 10 is so easily calculated, it seems logical that 17 X 5 (half as big as 17 X 10) should be easily found if 1,700 can be easily split.

In a similar way, being able to double numbers with ease leads to being able to double small factor pairs into larger factor pairs. If 7 X 2 = 14, then 7 X 4 is twice as big. So, 7 X 4 = 28. Then, 7 X 8 = 56, and 7 X 16 = 112...likewise, 17 X 2 = 34, 17 X 4 = 68...17 X 4 = 68, 17 X 40 = 680, and 17 X 80 = 1,360 ...all done through doubling.

When doubling a number, many students find it easiest to double the largest place value first and then work through the smaller place values.

Doubling a number like 1, 486 might look like 1,000 + 1,000 = 2,000...400 +400 = 800 ...80 +80 = 160...6 + 6 = 12...added together, the sum is 2,972.

These are VERY important concepts! Please practice. It's the kind of thing you can do in the car.

Division and Multiplication A Match Made in Heaven

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This student shows why a great understanding of basic multiplication is so vital to solving division problems. By knowing the simple factor pairs shown (15 X 1 =15 15 X 2 =30...) and knowing the relationship of these to larger landmark factor pairs (15 X 10=150 15 X 20=300), this division problem becomes an absolute snap to complete correctly. I highly advise students to write the first several multiples, and a few larger landmark multiples, of the divisor when solving almost any division problem.

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Boxes of Markers is a student sheet designed to introduce the idea of portioning items evenly as a form of division. Many students simply skip count by the number of students , in this case 23, in order to find out how many groups of 23 can be made by multiples of 70 markers. It might look like 23, 46, 69;at which point the kids realize that they have run out of markers. Most can see that each "skip count" represents a student with a marker. So, three skip counts would equal 3 markers per student.

The work above is from a student that immediately recognized this as a division problem that could be solved using easy multiples of 23 until the number of markers was exhausted. This really is the goal of this type of exercise, as it makes division and multiplication forever seen as related activities. Once the student that did this work adds sentences that explain the numerical answers, the work will be at standard to say the least!

Revisions Are Vital to Mastery!

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Students in our class are able to complete "revisions" of tasks to insure that concepts that were missed are cleared up or at least made more clear. Students do have to prove that they understand why they were wrong, and they have to show new work that proves that they really have reached an understanding of the concept.

In the top photo, the student explains what went wrong in the problem in the bottom photo. In this case there was a fairly major misconception about the formula used to compute the amount of money to subtract ($1,000 X the date or $25,000) from the original balance ($712,000). This misconception made the problem impossible for the student to solve. However, it would not be correct to assume that the student in question did not know how to subtract. Therefore, this thorough revision provides equitable remediation. Revisions are hard work, but they are well worth the effort!

It's Not Your Father's Math Anymore!

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The work above shows some more interesting strategies for subtracting and for showing possible combinations. On a recent test, students were asked to subtract $25,000 from $712,000. Many students used the traditional algorithm with mixed results. Today, the class decided that decomposing the numbers, less the 0s, and using positive and negative values made solving the problem easier. Most students realize that using negative values is way easier than it first sounds. Look at the middle photo.

The possible combinations problem involved making sandwiches with three major ingredients, a bread (Wheat or Rye), a meat (turkey, ham, or chicken), and a cheese (Swiss or American). Most students as comfortable using a tree diagram to show the combinations, but many students fail to understand exactly what a combination is. In this case, a combination is a type of bread with one type of meat and one type of cheese. Once that became clear, students could easily see (via their tree diagrams) that each bread type could have six possible combinations of ingredients. Look at the top photo.

Why Do We Do the Things We Do?

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This edition of my blog emphasizes two things that are hard for most 5th graders and adults to do well. The first is subtraction (be honest adults, you know it's easy to make mistakes), and the second is metacognition, or thinking about your thinking.

This task asked that students solve a fairly easy subtraction task two ways and then to give a rationale for the method that they prefer. Many kids wrote that they preferred one method over another, "because it was easier for them to do" or because, "that's the way that they did it last year in class". However, some students were able to make statements that relate to the value of the numbers, the distance between the numbers, a preference for addition over subtraction, a need for a visual strategy, or a like or dislike of negative numbers!

If you are an adult, try to answer that same question. I bet it will not be easy :-}

Rounding Using A Number Line

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Today we worked rounding numbers using a number line to provide "visual evidence" that conclusively indicates which end of the number line the number is closer to. This process helps students see the relative distance of numbers, and leads students to really think about place value. This, in my mind, is much more valuable than the traditional method of looking at digits and playing the "4 or less is down and 5 or more is up" game.

Counting Puzzles Stimulate Great Thinking

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These pictures show two recent student sheets that we have been working on in class. Both are designed to stimulate thinking about counting patterns. Often, students fall back on methods of solving these questions that are far from efficient. For example, a student might simply list all of the multiples of 25 in order to get to 300 and then state, after counting the written numbers, that it takes 12 people counting by 25 to get to 300. However, many students quickly latch onto the idea that it is much easier just to figure out how many 25s are in 100 and then triple the amount to get to 300.

On the "What's in Between" page, students have to sort through a multitude of important concepts in order to find the answers to these puzzles. For example, students must be able to find the mid-point of two numbers in order to find a reference point to know if a number is a number is closer to the numbers on the ends of the number line. Some students get really confused with larger numbers like 7,900 and 8,100. These same students would also have no trouble finding what comes exactly in the middle of 79 and 81. Sometimes, "ignoring the zeros" can be a great strategy. Students also have to be flexible and efficient when they are forced to find numbers that are multiples of two different numbers. If a puzzle said that the mystery numbers were said if you count by 125s and were also multiples of 500, then only certain numbers would qualify. I would try to think about multiples of 500 first, as that means only numbers that end in two or three zeros would qualify. See if you can find the puzzle with answers that are not quite correct.

Basic Operations Voted Most Efficient Week 2!

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Our first homework sheet proved to be of great use, as many simple and very efficient strategies emerged!

Problem number one was a subtraction problem, and the most popular strategy was the use of an open number line to add up from $267 to $323. The class then gave a go at "straight subtraction" which uses positive and negative values. Many in the class found this to be even easier to navigate, and most thought that it was faster too. Fast and accurate; you cannot beat that!

The second problem shows a very efficient execution of left to right adding. Adding the greatest place value first can help reduce large errors in calculations. I'd much rather be off by a dollar than a million dollars!

The final problem was almost universally solved by using a "generic rectangle" to multiply 16 by 12. The partial products are usually accurate, easy to calculate, fast, and easy to add up. Some students used multiplication clusters like 16 X 10 and 16 x 2 to solve the problem, and that is another fast and accurate strategy that is great for this factor combination!

These students ROCK!

Homework With Meaning!

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These two homework problems may seem too easy for a 5th grade student, but on examination of the thinking displayed, it is clear that a deep understanding of important concepts, like place value, are well understood by this student.

The first problem was quickly identified as a "subtraction" problem, but it was solved using simple addition. This young mathematician added up from 267 to 323 on an open number line. This is kind of like what folks did in the good old days before cash registers had built in electronic calculators. They would count back the change using landmarks along the way. This student also used landmarks (easily recognized and "easy to work with numbers"), as she first "jumped from 267 to 270. This allowed her to easily add on to 270 in order to get to 300. From 300, the jump to 323 was a piece of cake. Finally, all of the jumps were totalled, and the distance between $267 and $323 was found correctly.

Also of note are the sentence restating the prompt (question) and the matching equation, These also signify a real sense of math understanding.

On the second problem, the student added from left to right, and to me that is great! Adding the largest place values first makes it less likely to make a mistake of great magnitude. Using the traditional algorithm makes it more likely to make a mistake in the larger place values, and that's a real drawback to sticking with traditional algorithmic thinking, unless you REALLY understand the method well.

EDC and More!

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Today we spent time completing another day of our EDC curriculum, and we began to diagnose the needs of students. It was great to see that many students do remember the strategies that they have learned over the last five years. It was so great to see students paying such close attention to detail on our fourth day of EDC. I wrote, in black pen, some of the important concepts that we discussed during today's session on one student's awesome work :-}

Simple Beginnings Have Powerful Meaning

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These are pictures of the first couple of days work in a our calendar math series, "Every Day Counts". All of these examples show student work that is remarkably complete and detailed. The concepts may seem simple, but in reality all of this work involved very deep conversations about place value and other important concepts that relate to number sense.

A big "Booooyyyaaahhhh" goes out to the three students that produced this work.

2009-2010 School Year Approaches!

Well if you are reading this, you have found my "famous" math blog. This year, I plan to include, dare I say this, almost daily pictures of the work that is being produced in our math classes. I think that by doing this, students and parents will get a very clear picture of just what is really going on in class. There will be way less writing by me (stop clapping), and way more for you to DISCUSS.

To do this, I really need parents to sign and turn in all of the Internet permission slips. I will never post a child's name. I might call Suzy "Student S." :-}, or I might just leave it at "This student". You guys please tell me what you are OK with. I will never embarrass a student by leaving a negative comment about work.

Oh, just another note about the photos of work that I attach. What you will see first is a small "thumb nail" image at the top of the blog. By clicking on the thumb nail, you will see the full size piece of work. Sometimes, I forget to select the correct size for the image, and the work will be too large to view. If I do this, please let me know.

I am blessed and honored to have your kids in my class. They WILL create remarkable work!

Line Plots, Cats, and 5th Graders

If you were presented with this data about cat height, what could state about what was typical of the cats in this group. Could you find the mean, the mode, the range, and the median? Could you tell what fraction and percent of the cats were taller than 5.5 inches but less than 8 inches? (Click on the picture if you need a full-size view.)

I just measured my cat and she is 10 inches tall at the shoulder. She is a smallish seven-year-old female cat. Does that make you think anything about what kind of sample this might have been?

If you have a cat, could you measure their height and reply? We'll stick to inches since the chart was created in that unit:-}

Spinner Games and Probability

If the above picture was a spinner game, and you were given the choice of "two scoring options" to choose (you would get a point if you landed on a number that met the requirements of either of your two options chosen), what would you choose?

A. a prime number
B. an odd number
C. a multiple of 5
D. an even number

Many students would look at choice A, and state that it has a 4:8 or 50% probability, as 4 of the 8 numbers are prime.

B would have a 5:8 or 62.5% probability, as there are 5 odd numbers

C would have a 4:8 or 50% probability, as there are 4 multiples of 5

D would have a 3:8 or 37.5% probability, as there are 3 even numbers


Man students would just look for the highest probabilities. Perhaps, they would choose A and B, but if you study those choices closely, you may see that several sections would be possible "losers". So, what two options would you choose?

Hints for Taking the FCAT Test!

1. Your teachers, principal, family, friends, and classmates have faith in you!

2. You have all the tools you need in order to be successful. I've taught a bunch of kids, and you guys have the goods!

3. Do not stress out. This is a time to show off.

4. If a question is too difficult, skip it! It is far better to miss one item than the 23 items that come after it :-}

5. Read each question carefully. Be sure to read with "a voice" that you can hear in your head. Rereading with a voice has helped me solve so many problems over the years.

6. Make some notes to back up your mental math strategies. If you show some work, your chances of being correct go up a bunch. I have seen that most errors occur through rushed simple operations like adding or subtracting. So, don not rush (you know who I am talking to ;-} )However, please remember that you do not have time to "write a book" about each problem (again, you know who I am talking to ;-} ) .

7. Look at your final answer and see if it makes sense. Elephants do not weight 17 pounds!

PEACE!

Seven Things.............

Seven Things You Don’t Really Need To Know About Me!

(You can click on the images at the top for a more detailed and spectacular view of these rare artifacts! The ones that are embedded do not show all the available glorious details.)

Well, there was this time when I had to have Mrs. Shannon come rescue DA at the ER at 12:00 midnight, because I had a kidney stone! Oh, I already emailed that huh?


Well then, there was this fall from my road bike that peeled me like a tater! Oh yeah, I emailed and sent pictures.

OK, I’ve got it. When I was born, I cost exactly $196.80, and I can prove it!



Now, if you knew that, you must work for the CIA or something.

Keeping things chronologically oriented, I have always been better at math than I was at more cultured things like the fine arts, and again, I can prove it.


Thanks to my dear old mom’s careful interrogation and subsequent documentation, I know that the picture (both are from first grade) contains one black bear, one deer under a tree, one raccoon hiding in a hole in a log, one bunny, one bird flying someplace, and a partridge in a pear tree! You try and figure it out.


I have never acted in a play unlike Ms. Lipsky, but my family did kind of resemble a TV family that traveled on a big groovy bus, and again, I have the evidence!

Hyde Grove Elementary Patrol Boy on the loose!
Nice Lamb-Chops Pops!

I often spent part of the summer with my grand mom and granddad in Vidalia, Georgia. They had the biggest house in town, and granddad always had the latest Caddy, except for the time that he bought a Rolls Royce! They were pure southern gentry right down to their bigoted cores. Granddad built almost all of the roads in S.E. GA. (well, he owned the company, but his job seemed to be riding around in his 30 foot long land-yacht checking on the crew and admiring or criticizing their work). Grand mom stayed home, made homemade candies, crocheted quilts, told the maids what to do, played bridge, played the stock market and counted her money. Being the son of a preacher man that marched with Dr. King, opened the first integrated church in Montgomery, Alabama, and started an integrated summer camp in Pensacola, Florida, I did my dead-level best to set my grandparents on the right path. When I was seven, I asked my grand mom why the very nice maid was not eating lunch with us; after all, she made the lunch. I got the stare and then was told that “the help” eats on the back porch. Without a word, I got up with my food and started to stroll away from the table. Grand mom asked where I was headed. Needless to say, I had a very nice lunch on the porch. The scene of the crime is pictured below. The Chamber of Commerce used their house on their brochure cover.

Outlaw biker days! From 6 to 15-years-old, I do not remember a day that I did not ride a motorcycle. At 15, you were legally allowed to pilot a motorcycle that was “producing less than five brake horse power”. Somehow, I convinced my dad that my 120 MPH Honda café racer only made 4-5 horsepower. He was not, and never will be, mechanically inclined! However, the joke would be on me as, on December 30, 1979 I left planet earth after my motorcycle rear-ended a turning car. I was taken to St. Vincent’s ER and “pronounced”—“called”—and about to be toe-tagged when an older ER nurse said that my vitals did not show due to a build up of fluid around my heart and in my lungs. She “bagged me” with a salt and ammonia mixture and saved my life. I spent the next five days in ICU. My folks say that I kept asking the same question over and over again, “Has the Gator Bowl started?” (that was the year that Clemson beat Ohio State and Woody Hayes punched Clemson linebacker Charlie Bauman in the mouth ending Hayes career). I also complained about bugs running up and down my IV drip (there were no bugs). When I left the hospital, I had blurred vision for a while and I tended to repeat myself, I tended to repeat myself. I still love cycles, and I have owned about 20, but it took me 15 years to get on one after the wreck. I do not ride today, as the costs outweigh the benefits. My 15th birthday card was a bit prophetic?

Number VI: I really could run a 4.45 forty coming out of high school!

Final bit of totally useless knowledge: My whole family calls me by a different name than my CCE family. I was known to the world by my middle name, Rives, right up until I enrolled at UNF. It’s pronounced Reeeeevs. However, most of the time when people read my name it turned into Rivers, Riiiiiiiiiives, Rivas, Rico, boy with the blonde hair, or “did they misspell this?”. Being shy and unwanting of extra attention, I cringed every time I got a new teacher, because they would butcher my name and then argue that they were right, because “the silent e makes the vowel long”. Combine the middle name with the equally challenging last name, Ruark, and you simply cannot say them both without sounding like you have a mouth full of marbles. So finally, at UNF when a prof started to say “Ruh, uh Rii, uh Ria”, I broke in and said, “It’s Tom. Teee Ohhh M.” Oh, Ma Bell still can’t get with the program either. To wit,



Peace!