Practical and Creative Ideas for Math Teachers

There is Hope - I Think

2008-04-23T11:05:00.000-07:00

Today I read an article in the New York Times concerning education in the USA. As you might expect it does not present a lot of great news. Our local situation is probably better than what is described in this article but we will all have to live with some of these problems if the author is correct.

Here is the link.

Earth Day

2008-04-21T12:59:00.000-07:00

Significant and Insignificant

April 22nd is Earth Day. Since there is no practical way to determine otherwise, let's call it Earth's birthday. The best scientific estimate for the age of the earth is about 4.55 billion years, plus or minus 70 million years. This estimate was calculated from the work of an Iowan, Clair Patterson, in 1953. See the Bill Bryson book, “A Short History of Nearly Everything” (p.149-160) for more information.

From what I can tell, what we would call human beings (homo sapiens) have existed on the earth for about the last 250 000 years. Of course this is a rough estimate but it gives the opportunity to do some interesting math. Large numbers are not easily understood by anyone. In one of my earliest posts (October 30th, 2007), I mentioned that one million seconds is about 11.5 days while one billion seconds is about 31.7 years.

Using the above numbers, 4.55 billion years for the age of the earth and 250 000 years for the time human beings (homo sapiens) have been on earth, there are a few ways to show the relative insignificance of man's time on earth compared to the time that earth has been in existence. One way to do this is to compress the entire age of earth into one 24 hour day. Based on that, how long have human beings existed on earth?

The easiest way to solve this would be to use a proportion:

4 550 000 000 years : 250 000 years = 24 hours : X hours

Solving for X we get about .001318681319 hours

Multiply that by 60 min./1 hour and by 60 sec./ 1 min. (which are forms of one)
and we get about 4.747252747 seconds.

Therefore in the 24 hour history of earth, human beings have been on the scene about the last five seconds. Most people would find that pretty stunning.

Another interesting thing to do with the information is more appealing to students on a visual level. Assume that the average sized man has a wing span (arms stretched out wide), from the tip of the fingernail on the middle finger of one hand to the tip of the fingernail on the middle finger of the other hand, of six feet or 72 inches. Assume that length of 72 inches is all of earth's history. What part of that length would represent the length of human beings time on earth? Again set up a proportion:

4 550 000 000 years : 250 000 years = 72 in. : X in.

Solving for X we get about .003956043956 inches which is very close to .004 inches, or about 1/250 of an inch. In the Bryson book (p.337), one of the authors he cites says, “... a single stroke with a medium-grained nail file [would] eradicate all of human history.”
That is a pretty good visual for students to see that the 1/250th of an inch scraped off one of those middle finger nails would represent all of human beings' time on earth.

Another visual way to look at that would be to imagine that the 4.55 billion years of earth be represented by a 100 yard football field. What would the 250 000 years of human beings be on that field?

4 550 000 000 years : 250 000 years = 100 yards : X yards

Solving for X we get about .005494505495 yards. Multiply by 36 in./1 yard you get about .197802197 inches or about 1/5th of an inch. Imagine looking at a football field from one goal line to the other goal line and realizing that the time of human beings would only be about 1/5th of an inch. That is pretty stunning too.

The very bright student may realize that some of these numerical answers can be obtained from the other numerical answers. Time could be spent on that for those interested. For example, the football field (100 yards) is fifty times as long as the wing span of 6 feet. Multiply 1/250th of an inch times fifty and you would get 1/5th of an inch.

I love math!

Number Tricks

2008-04-16T14:20:00.000-07:00

I have been gone from posting anything for awhile. One of my former students sent me an e-mail saying, "patiently waiting for another blog" so I thought I better get back to posting and that maybe somebody was actually reading them. If you want to comment on a posting directly to my e-mail, send it to me at crxos@yahoo.com.

NUMBER TRICKS

All students enjoy what many people call "Number Tricks." Number tricks are
in two basic types. The first type involves the student starting with their own
chosen number, performing several operations on it, and then ending up with
the same number they started with. The second type involves the student,
again, starting with their own number, performing several operations, but
then everyone in the group ending up with the same number, instead of
their starting number. The students find the first type pretty interesting but
are more fascinated by the second type. At first they are somewhat mystified
that these work. That opens the door for the Algebra teacher to demonstrate
algebraically why they do work and then challenge the students to develop
their own number tricks. I eventually show them why each trick works by showing
three columns that have (1) the direction written out, (2) a numerical example,
and (3) the algebraic expression for that step. I require students to develop an
example of both types of tricks and each trick must have at least six steps.
They show their number tricks just like I show mine to them. The first column is
where they write the direction for that step. In the second column, they write a
numerical example of that step. In the third column they write an algebraic
expression for that step. I encourage them to be as creative as possible. I give
them an example of "dressing up" their directions to allow more creativity. Of
course you cannot do project this until they possess the algebraic skills needed.
This has proven to be an enjoyable project for my students involving creativity
and algebraic knowledge.

(Please excuse the fact that the three columns I wanted to show, separating the direction, the number, and the algebraic expression didn't come out well. The auto editor crams them together so they are a little hard to read. If needed I can e-mail you my document copy which is easier to read.)

Here is an example of the first type of number trick in which everyone ends up
with the same number they started with:

Pick any number: 32, X
Multiply by 20: 640, 20X
Add 1000: 1640, 20X + 1000
Divide by 2: 820, 10X + 500
Subtract 300: 520, 10X + 200
Divide by 5: 104, 2X + 40
Subtract 40: 64, 2X
Divide by 2: 32, X

The teacher should do this trick at least twice (before showing the algebraic expressions)
so the students can see that it works regardless of what number they choose at the beginning.
You might ask them if it matters if they had started with a negative number or a decimal.
If they understand the algebraic expressions they will be able to answer that question.

Here is an example of a number trick in which no matter what number the student chooses
to begin with, they will all end up with the same number - in this case the current year.

Pick any number: 3.14, X
Multiply by 100: 314, 100X
Add 250: 564, 100X + 250
Multiply by 4: 2256, 400X + 1000
Add 1000: 3256, 400X + 2000
Subtract 400 times your
original number: 2000, 2000
Add 8: 2008, 2008

I would have several number tricks of each type ready on that first day in order
to heighten their interest. Make them as fun and interesting as possible. Many times
students would ask how old I was. This was my way of telling them.

An example of "dressing up" the above trick would change the directions to:

Pick any number
Multiply by the number of years in a century
Add the number of pennies in $2.50
Multiply by the number of sides in a quadrilateral.
Add the number of meters in a kilometer.
Subtract twenty squared times your orginal number
Add the value of two to the third power

The steps to any number trick can be "dressed up" in a variety of ways depending
on the age, maturity level, and mathematical knowledge of the class. They could
involve other school subjects also. (Add the number of milliliters on a liter or subtract
the year in which the Declaration of Independence was signed.)

The students may very well surprise you with their creativity in this situation.

Looking into the Past - Really!

2008-03-21T06:52:00.000-07:00

For awhile now I have been reading an absolutely fascinating book. It is, "A Short History of Nearly Everything." It was written by Bill Bryson in 2003. This book is full of extremely interesting information related to science, astronomy, chemistry, biology, geology, anthropology, physics, paleontology, taxonomy, history, mathematics, genetics, and the people who were and are involved in discovering some of the mysteries of the universe. I will eventually post a great deal of information about this book but for now I want to focus on one thing: gamma ray bursts. These are among the biggest and most violent events that take place in the universe.

Today in the New York Times, I read about a gamma ray burst that was briefly visible to the naked eye on Wednesday morning (if you knew where to look). Of course when you look at something like that, you are actually seeing something that happened a long time ago - in this case about seven billion years ago which is about 2.5 billion years before the earth had even formed! Right now the scientific best estimate for the age of the universe is 14 billion years old.

I have been fascinated for a long time by the fact that when we look at the sky, you are seeing light that left the stars, moon, or sun sometime in the past, and sometimes in the ancient, to say the least, past. Actually if you think about the speed of light, whenever we look at anything, we are seeing the light that left that object at some point before we actually experience the sight. In most cases the amount of time is so brief that we simply cannot comprehend it.

However I digress. Here is the link to the Times article on the gamma ray burst.

Here is a link to an opinion article written by Arthur C. Clarke for the Times in 1994. In the article he makes reference to a novel he wrote in 1973, and in a quote from that article, the date of September 11th is used for a monumental event in the history of the planet.

Arthur C. Clarke has died.

2008-03-19T07:07:00.000-07:00

One of the most influential artistic experiences I have ever had in my life was in the spring of 1968. I was in college at UNI at the time. A few college friends and I drove down to Des Moines to see a movie. Yes, there were plenty of movie theaters in the Waterloo-Cedar Falls area so why go to Des Moines? Well the movie we wanted to see was "2001: A Space Odyssey" and it was being shown at the River Hills Theater in Des Moines which, at the time, was the only wide screen movie theater in Iowa and it also had the best (surround) sound system of any movie theater in the state.

I honestly don't remember how I had heard about the movie but probably there had been an article in the newspaper. I do remember hearing enough that I was very curious and felt that it would not be just another movie. I was also looking forward to seeing a movie in the new wide screen (180 degrees) format with the, at the time, new "surround sound."

I convinced some other guys to go down there with me. On the two hour trip to Des Moines I read a very long magazine interview with the director of the movie, Stanley Kubrick. Of course he talked a lot about the making of the movie and I remember the interviewer asking him about the meaning of some of the scenes in the movie. Kubrick's answer was interesting. He basically said to not try to figure it out, but rather just "experience" the movie. In later years Kubrick would say that he himself didn't really know what some of those scenes, and the movie itself, were trying to say. Think about that. Kubrick was the director of the movie and more than any other human being responsible for what people saw and heard on the movie screen. Since then I have read or heard other artists say similar things about their own work. Bob Dylan is a good example. He has often said that his songs "come through him." That implies that they started somewhere else ( location unknowable to the artist), went through him and then emerged into the public consciousness. That makes it a lot easier to understand when an artist claims to not know the meaning of their own work. They aren't trying to be mysterious - they really don't know.

Where does Arthur C. Clarke come into this? He was a science fiction writer among many other things. In 1948 he wrote, "The Sentinel" which in some way was the forerunner to "2001: A Space Odyssey." I won't go into the details but Kubrick and Clarke worked together to bring the movie to the screen. Arthur C. Clarke died on Mar.19, 2008 at the age of 90. He is most remembered for his role in creating the movie but he did many other things. In 1945 he wrote a scientific article that proposed the theory of satellites in geosynchronous orbit around the earth and how valuable that could be. Of course he was way ahead of his time in his thinking. Where would we be today without those communication satellites that almost the entire world relies on these days? This also reminds me of one of Clarke's three famous laws. Law #3 said, "Any sufficiently advanced technology is indistinguishable from magic." I think about that many times when I consider some of the new technology that comes into public usage. How else could you explain using a cell phone to make a call from the middle of a corn field in Iowa to someone in the middle of an oil field in Saudi Arabia? That has to be magic! Here is a link to a Wikipedia article on Clarke if you are curious.

Now back to the trip. We got to the theater and there wasn't a real big crowd. We sat right in the middle of the theater and I was agog at the size of the screen. You had to physically move your head to see from the far left of the screen to the far right. I could see the speakers on the side and behind the audience. To use an expression not known at that time, I was pumped! The lights went down, the sound came up and the movie started. I remember gripping the armrests on each side of me as I saw the first images come on the screen. I felt like I was going to fly off my seat!

I was enthralled the entire time at what I was seeing. The special effects were way ahead of their time ( I didn't see anything comparable until Star Wars in 1978). There were long passages of time in the movie in which there was no dialog, and that was unusual. One scene early in the movie was an encounter between two groups of early humans. The image on the screen was where we saw one of the groups and we knew the other group was behind the camera. We heard that group, behind the camera, from the speakers in the theater behind the audience. That was amazing to me.

The use of classical music was a brilliant touch. With the surround sound and the wide screen it was like the movie had opened its arms and invited the audience into its embrace.

I won't go into detail about the movie. Some people hated it. Some critics hated it. I loved it. Some critics loved it. You might want to do a search for Roger Ebert's review of the movie. The last half hour, in particular the last few minutes, are unforgettable to me. It is pure cinema and I still have no idea what it means (Kubrick always said he didn't know and Clarke never, to my knowledge, gave any explanation) but I "experienced" it and the last scene just gave me such a hopeful feeling about the human race.

I have seen the movie a few times since then on television and on DVD but it isn't even close to the experience I had in 1968 at the River Hills Theater. Watching that movie on a normal TV is like watching a football game on a postage stamp.

I remember walking out of the movie that night thinking I might have experienced something very special in my life. Now, 40 years later I know I did. I have seen many great movies since then, but no other movie has affected me in the way "2001: A Space Odyssey" did. I also remember thinking that one of my goals in life was to see that movie again in the year 2001 at the same theater. I realized I would be very old at that point but I wanted to compare the movie to the reality of 2001. I was so disappointed that Stanley Kubrick died in 1999. Then the River Hills theater was torn down to make way for Wells Fargo Arena. When I went to Wells Fargo Arena in 2006 and 2007 to watch the Spirit Lake girls play basketball in the state tournament, there was a slight tinge of sadness that they were shooting baskets where I had once been sitting and watching and hearing and experiencing my all time favorite movie.

One other thing that relates to parents and teachers. If or when you encounter a young person that is different, and I mean really different, like Arthur C. Clarke or Stanley Kubrick must have been as children, don't dismiss them right away. They can be a nightmare at times because they don't seem to follow the normal rules of life. It's almost like they are aliens from another planet. They might talk about things that seem impossible to a normal way of thinking. Our world badly needs these type of thinkers and dreamers. Some of those dreams may never come to pass, but some of them may end up affecting deeply the people in the world that experience them.

Arthur C. Clarke R.I.P.

NCAA Tourney and Math

2008-03-16T21:03:00.000-07:00

In the past 10-15 years the NCAA Basketball tournament has become a popular item of interest to students, particularly when their favorite team is in the tournament. I found that even kids who weren't normally interested in basketball were talking about it and some of them even would join their classmates (or parents)in picking the outcomes of the games. I wanted to take advantage of this natural interest and come up with some math questions related to the whole thing.

What I ended up doing was pretty simple but turned out to work very well and certainly was interesting to the students. I made a transparency of the full set of brackets with all the teams listed. I showed that to the students and explained the basic idea that a team continued to play as long as they won. (The NCAA actually invites 65 teams into the tournament. Two teams play an early "play-in" game and that narrows the field to 64 teams.) I ask the students to tell me how many total games will then be played, after the "play-in" game, in the tournament to determine a winner. It's the type of question that most, if not all, kids don't already know. Neither do many adults.

When I first thought of this question, I solved it by doing what most people do when they hear this question. I looked at the bracket sheet and counted the first round games -quickly realizing that there had to be 32 games in the first round because there are 64 teams (after the "play-in" game). The next round would be 16 games (because there were 32 winners n the first round), then the third round would have 8 games and the fourth round (regional final) would have four games. Those four games produce the "Final Four." Of course then you have two games in the national semi-finals and then the final game for the championship.

32+16+8+4+2+1 = 63. (I prefer not to count or sometimes even mention the "play-in" game because it isn't usually shown in the brackets that kids see. That way they can count all of the games one by one if they don't see any faster way.) When I first did this question, many years ago and saw the answer "63" I quickly realized that I had not used the most efficient method. When you have 64 teams in a single elimination tournament, 63 teams have to be eliminated in order for the champion to be determined. Each game eliminates one team so 63 games need to be played to eliminate 63 teams. An even better way to look at it is that all teams have to be eliminated except one, so 64-1=63.

Most kids and adults can easily figure out by the original counting method, that the tournament with 64 teams requires 63 games to determine the winner. They usually don't notice or think about the shortcut. Therefore when I present the original question I also present an additional question. Almost every year there are people who say that all of the NCAA (Division 1) teams should participate, regardless of record, just like they do in high school. In a given year I wouldn't always know exactly how many teams were in Division 1 so I would tell the students to assume that there would be 200 teams and to figure out how many games would need to be played. Assuming they didn't know the shortcut mentioned above, they would normally try to solve it just like they had the original question. With 200 teams there would be 100 games, then the second round would have 50 games, the third round 25 games and then they would get stuck. Very few, if any, students or adults, would be able to figure it out from there. What do you do with an odd number of teams?

So the next day when I asked them how they solved the original 64 team question (after they passed their papers in) most would have the correct answer but would not have seen the shortcut. I can't remember any of my 8th graders getting the 200 team question right because they always got stuck with the odd number of teams. At this point, if I had the time, I could suggest two problem solving strategies - solving a simpler problem and finding a pattern along with making a list.

I start with a tournament that has only one team which I know is ridiculous but it serves the strategy. Obviously with only one team, zero games need be played. If there are two teams, you need one game. When you get to three teams, you show them that the only way to do it is to have two teams play and the winner of that game plays the other team for the championship. So three teams requires two games. Four teams (like the "Final Four") needs three games. As this list is being made some kids will see the pattern that every time you add a team you add a game. That is true but if that is all they see then a tournament with 200 teams is going to require making a long list. Eventually somebody usually does see the most efficient method - just subtract one from the number of teams. They can then instantly determine that if the tournament had 200 teams that 199 games would be needed. If the tournament had 35 teams, then 34 games are played.

If you have any other additional time (highly unlikely) you can add in the concept of powers of two. When you have a single elimination tournament, you have to have enough "play-in" games to cut the field down to a number of teams that is a power of two. So a tournament with 35 teams would require three "play-in" games to cut the field to 32 teams. In fact you could ask the students at the beginning why the NCAA field (after the play-in game)is 64 teams. Why not 60 teams or 80 teams or 50 teams?

This was a great problem solving situation that took advantage of the natural interest that students and adults have in the NCAA tournament. One of the things I used to always say to my students is that, "A good mathematician is a lazy one." This particular situation is a perfect example of that.

Iowa Model Core Mathematics Curriculum and Motivation

2008-03-14T04:16:00.000-07:00

The Iowa Department of Education has come out with two documents detailing what it calls the Model Core Mathematics Curriculum. Prior to this Iowa was the only state in the USA that had not established state standards in subjects like mathematics. I will provide a link to the introductory article. In that article is another link to the downloadable curriculum document for high school mathematics. Teachers and parents can read it and draw their own conclusions.

One complaint about mathematics education in the USA has been that the curriculum is "a mile wide and an inch deep," meaning that it has attempted to cover way too many topics and therefore has not been able to cover them in the depth necessary for true understanding and mastery. I believe I first heard that at least as far back as 1983 when the "A Nation at Risk" document came out. I don't disagree with that comment but, if people think that simply decreasing the number of topics addressed in the course of study will solve all of the problems, they will end up being disappointed.

I think it is a first step but what is more important, perhaps even decisive, is the need to find ways to motivate students to do the hard work necessary to learn at a high level. We should work for a higher percentage of parents who put the proper emphasis on education but that is a segment of the population that educators have less control over. What I believe is that math teachers must work hard to determine the most effective ways to teach their students (people other than teachers determine what is to be taught). The effectiveness of a teaching method includes the particular way of having students learn the skill or concept but also must also include motivational ideas that will cause the student to want to spend the time necessary to learn the skill and overcome the obstacles involved in learning difficult things.

Students today have a large number of demands on their time and, getting them to choose learning academic topics over extracurricular activities or jobs or video games or TV or text messaging or phone calls or anything else, requires special attention. I am not saying that students can't do some of those things, some of the time. The ideal thing would be that the student, himself or herself, chooses to put their emphasis on academics and then work in the other activities as their available time permits. It is necessary but not sufficient to just tell them or show them that some particular topic is important to their future. Motivation, motivation, motivation can overcome many obstacles. Motivate daily, motivate weekly, motivate monthly, motivate yearly, and motivate creatively. Lesson planning (daily, unit or chapter) should include motivation.

Success is an excellent motivator but that requires teaching that is correct in how it teaches students to think about a skill or concept so that the success is real and not artificial. When students find that the brain that they have really does work, they get pretty excited and want to do more. Having fun is important to students, but it has to come in the context of learning, otherwise students get the idea that learning is necessarily separate from fun. Having fun is good for teachers too since they may be teaching the same lesson several times a day and/or many times over the course of their career.

So my prescription for the USA to do better in mathematics education (and all education for that matter) is to work to get more positive involvement from parents, find the most effective ways to teach the required content (requires a lot of time and effort on the part of the teacher), and then motivate, motivate, motivate. If you can do only one of those three things, MOTIVATE.

Two Million and Counting

2008-03-10T12:40:00.000-07:00

I recently came across some information about a documentary film with the title,
"Two Million Minutes". It refers to the approximate amount of time in the last four years of a person's high school education. It compares what happens with that time in the USA, India, and China. It probably wouldn't surprise you to know that, from an American perspective, the news is not very good.

Here is a link to the home page of the documentary. On that site you can also click on "BLOG" and you will see a couple of articles from the Wall Street Journal that I think you would find valuable to read. One of the articles is, "Study Finds Sharp Math, Science Skills Help Expand Economy" (from 3-3-08) and the other is "Education Panel Lays Out Truce in Math Wars" (from 3-5-08).

To me, it is a little like global warming. Will we take action before it is too late?
Math (and Science) teachers cannot do this by themselves but they can certainly be in on the beginning of finding solutions.

Forward to the Past?

2008-03-03T02:24:00.000-08:00

One of the controversial methods of attempting to increase achievement in schools is to have single-sex classrooms. This is something that does not get much attention in this area. There was an article on the New York Times website recently that explained the origins of this movement, the studies associated with it, and some of the outcomes so far for students in those type of classes. After reading the article I'm not sure how I feel about it but I do think it is worthy of discussion.

Here is the link.

Exercise to Achieve

2008-02-27T07:28:00.000-08:00

I saw an article in USA Today recently that had a lot to say about the value of regular exercise for helping students learn to their maximum ability. If school administrators think that athletics and/or Physical Education are not important to the academic success, then they have not seen this research. I think it also implies that educators themselves can benefit from their own workouts. For physical and emotional well being as well as mental facility, exercise is very important.

Here is the link:

Evil or Ignorant or Santa Claus?

2008-02-19T12:56:00.000-08:00

EVIL OR IGNORANT OR SANTA CLAUS?

Over my years on this earth I have had occasion to hear or read statements by people which I knew, or later found out, to be untrue. I have said or written things that were untrue. It is unlikely that any human being is 100% truthful.

When people make statements that are untrue, it is my contention that those statements usually come from evil or ignorance. The exceptions to that will be discussed later. For now, let's confine our discussion to statements of fact. Let's say that someone makes the statement that 2+2 = 5. That is obviously not a true statement. Depending on the situation of the person who made that statement, it could have been done with evil intent or out of ignorance. A small child who is just starting to learn the basic facts may make that statement out of ignorance – we are not born knowing basic mathematical facts. However an older student, perhaps looking to cause trouble for a younger sibling, might have evil intentions in trying to convince the young person of this incorrect fact.

This brings up the interesting point of when a false statement is a lie. It seems to me that it is a lie when the statement comes from evil intent, but not when it comes from ignorance. By evil intent, I mean that the person knows that the statement is not true but for whatever reason they choose to portray it as a true statement. Let's use a very politically charged statement from 2003. “Iraq has weapons of mass destruction.” Almost everyone now agrees that was false. Did the people who made that statement know that it was false but, because they wanted to start a war with Iraq, felt they needed to convince the people of the USA that action had to be taken? That could be seen as evil and the statement would be classified as a lie, making whoever said it a liar.

If, however, the statement was honestly believed to be true by whoever said it, then it was made out of ignorance or from erroneous information and therefore not a lie. If the people who said this had “cherry-picked” the evidence, had ignored the counter evidence or established an atmosphere in which counter evidence would not be sought out or presented, then that becomes evil again.

Where does Santa Claus come into this discussion? During the raising of a child, the story of Santa Claus is usually told by parents to their children. I think it is basically harmless and is a good tradition to maintain. Parents are knowingly telling their children a lie, but it is because the myth is a wonderful tradition and fun for everyone. In other words, parents justify the lie on the basis that much more good than bad will occur and it will not do damage to the child.

So maybe, the people who said in 2003, “Iraq has weapons of mass destruction,” were using Santa Claus reasoning. They knew it was a lie, but they calculated that the war in Iraq would do America and the rest of the world much more good than harm and so it was justified in their mind. It makes the statement a lie, but a “benevolent” lie. One of the problems with benevolent liars is that they can't always calculate the damage of their lies, especially in the long run. Also if they employ that tactic often, they risk not being believed when they do tell the truth. The benevolent liar is tacitly saying that the audience for those lies cannot be persuaded to do the right thing if they are told the truth. That could be evil, or at least disrespectful, to those in the audience to which the lies are directed. (Many people will remember Jack Nicholson in “A Few Good Men” telling Tom Cruise, “You can't handle the truth!”)

Benevolent lies may give birth to other lies that are not so benevolent but are necessary to “cover” the original lie. “Oh what a tangled web we weave, when first we practice to deceive,” has always been true. Thank you Sir Walter Scott.

I also believe that those who regularly use Santa Claus reasoning (except when it is actually about Santa Claus!) may start to believe that what they are saying is actually true. Those people can sometimes be dangerous. I think this is where cult leaders are born. Jim Jones (“Jonestown”, in Guyana, was where his followers drank the Kool-Aid that caused them to die) might have been an example of this, but I don't have all of the facts in that situation to say that is more than just my opinion.

Of course opinions are different from facts. Opinions can be considered true or false only to the extent that the opinion is or is not what the person stating the opinion actually believes. I might say that my favorite professional football team was going to win the next Super Bowl. It could be that I truly believe it or it could be simply wishful thinking and not my true belief.

What does this all mean? If I said I knew for sure that would be evil, ignorant, or Santa Claus type of thinking. I think it's best that you decide for yourself what it all means. Let me know what you (truly) think.

Changing the teaching environment

2008-02-07T06:33:00.000-08:00

Yesterday I read an article in USA Today which talked about the positive impact on teaching and learning from the way a new school was built. It was not done "on the cheap" but rather money was spent to do it right with the idea that it would more than payoff in the long run. Schools and teachers are constantly looking for things that help them better educate their kids. The good news is that there are things that work. The bad news is that they are often expensive.

Education has always been underfunded - especially in teaching salaries - but when more funds are spent on the wrong things it leaves the impression that spending more money isn't the answer. Spending more money on things that don't work isn't the answer.

School districts can't build new schools very often. What this article suggests is that when a new school is built it should be done in a way that invests money in a smart way and that it will more than pay off in the long run.

It also suggests to me that teachers can pay more attention to the space they have to work with and find ways to make it safer and more inviting for their students. Research what works and be creative in using limited resources to produce the best environment for you and your students.

Here is the link.

Why are the "MAD ANTS" mad?

2008-01-28T17:24:00.000-08:00

I suppose they have a good reason but, really, why are they mad? I have never thought about the lives of ants and what might disappoint them or discourage them or disgust them, let alone make them mad. Before I go any farther I better explain what has caused me to contemplate this question.

The National Basketball Association (NBA) has their own minor league group of teams. Fourteen teams across the USA play in something called the NBA Development League, commonly abbreviated as the NBA D-League. This league is designed to give basketball players, who are not quite able to make an NBA roster, a chance to develop their skills and to mature into professional basketball players. Since this league does not have a long history, the teams tend to have fairly original nicknames. Often there is a contest in which members of the community are asked to submit their suggestions for what the team nickname should be. One of the new teams in the league this year is located in Des Moines and plays home games at Wells Fargo Arena. They are called the Iowa Energy. That's a nickname designed to take advantage of Iowa's role in ethanol and other alternative sources of energy (I think). The other nicknames in the league are “Thunderbirds”, “Arsenal”, “Toros”, “Jam”, “14ers”, “Wizards”, “Stampede”, “D-Fenders”, “Vipers”, “Skyforce”, “66ers”, and “Flash”,

Recently on Mediacom (cable television) I saw part of an Iowa Energy game in which they played the Ft. Wayne Mad Ants. That nickname caught my attention right away. The logo they have is probably supposed to represent a “Mad Ant” but it wasn't real clear to me. Anyway, this nickname prompted some questions in my mind: Why are those ants mad? Do they live in a state of perpetual madness? Are all ants mad or just some of them? Are there happy ants? Does “mad” mean crazy insane or is it the type of “mad” that means bitterly upset? Is this madness directed to other ants, to human beings or anybody or anything else? Are there ant psychologists available to help these ants recover from their madness?

I have to admit that about the only contact I have with ants is when I crush them with my foot. They might be crawling over a picnic lunch or just meandering down a sidewalk and I admit that my taking of their life is pretty thoughtless on my part. Come to think of it, that might be one of the reasons why they are so mad. If your life could suddenly and unexpectedly be ended by some thoughtless being, thousands of times larger than you, you might be pretty disillusioned too. How many times might you have had one of your family or friends leave home one day and never come back? Why make any plans for the future? Why have children you might not be able to care for? Why even leave your home “hill” when you might never come back? In fact, why even build that hill when some snotty little human being may come along and gleefully destroy it? Can anybody blame ants for having to attend human picnics to get some food for themselves and their families? It really seems that ants might very well consider themselves, not second class world citizens, but maybe fifty-second class. As far as I can tell, they are the Rodney Dangerfield of the world's living beings. They get no respect.

Well, all frivolity (using the term loosely) aside, I did my homework. I wanted to find out how this basketball team from Ft. Wayne Indiana had decided on this particular nickname. As I mentioned before, the officials of the team had conducted a community contest. The name, “Mad Ants” was an easy winner (“Fire”, “Coyotes”, and “Lightning” were the next three in the voting). It was in reference to General “Mad” Anthony Wayne (1745-1796), a US Army General and statesman whose Revolutionary War triumphs and fiery personality had earned him the title, “Mad Anthony”. The town of Fort Wayne Indiana is named after him, as well as many, many other places across the USA including Wayne State College in Nebraska. So, all in all, I have to say that the team did pick a very creative nickname that does reference its community's past as well as reference triumph in battle. However that last part might be not taking hold right now – the Mad Ants are currently in last place in their division in the D-League (something else to be mad at ?).

One final thing that some people might find maddening. All of these teams have their cheerleader dance teams. In Ft. Wayne it is called the “Madame Ants Dance Team.” Ouch.

Struggle is Good

2008-01-24T20:54:00.000-08:00

STRUGGLE IS GOOD

The above statement might seem a little surprising but allow me to explain. If you know me, you know the explanation will be pretty long! In the January 21, 2008 edition of Sports Illustrated (Brett Favre, in the snow, on the cover), there is an article on basketball coach Rick Majerus. He has been a highly successful college basketball coach over the years, known primarily for taking the University of Utah to the national championship game in 1998 against Kentucky and his great sense of humor. Because of his health and other reasons, Majerus was out of coaching (doing analysis on TV instead) for the past few years. He got back into coaching this season with the Saint Louis University basketball team. The article is an interesting one even if you aren't a sports fan. Here is a link to it.

I don't agree or advocate all of the things that Coach Majerus does with his teams (which doesn't mean that those things are wrong for him) but he made an interesting statement in the magazine article. He said, “Parents today want to take all the pain, all the heartache and sadness, out of their kids' lives. All the things that make you a better person.” That particular statement reminded me of one of the philosophical underpinnings of my teaching career. This was something that developed over several years of teaching and seeing how kids react to certain situations and also having former students come back to talk to me after moving on to high school, college, or adulthood. In essence, one of the things they most appreciated about my class was that it was demanding and it caused them to develop better study habits and to have a greater sense of accomplishment when they succeeded. In other words, the students felt more pride in getting a good grade in a demanding course than getting an excellent grade in a less demanding course.

Actually this can be a little more complicated. All kids like to get good grades, and like even more to get excellent grades. Of course part of that happiness is that the good or excellent grade also makes their parents happy. However that is temporary for the student if they conclude that the excellent grade had required little, if any, effort on their part and that almost everyone else in the class – perhaps some who had worked even less than they had – had received the same excellent grade.

As I developed my classes and my expectations I felt it was important to do things the right way, to have my students understand things in ways that would best enable them to learn more in the future, and to encourage them to be detailed and careful with their work. If a student wasn't used to that or couldn't rise to that level of expectation, they would not do well in my class. I would count things wrong, or partially wrong, if they made a slight error. In some cases approximate answers are correct, but in most situations in math, precision in thinking and in answers is necessary. I eventually became much more concerned about how they were thinking about something (being well aware that there wasn't necessarily just one correct way of thinking about something), rather than the answer they eventually got. Sometimes the most trouble I had was with the parents who wanted their kids to do well and didn't always see the value of that attention to detail and precision.

It wasn't unusual that a student in my class would get an average grade (which is a “C”, believe it or not) when they had previously received good or excellent grades. That was not my intention. My intention was to do things the right way, which wasn't always the easiest way. Fortunately I had the backing of the administration who had confidence in me, and eventually I got the backing of my former students who could testify to the effectiveness of what had happened in my class. I did have to fight this battle throughout my teaching career but I felt strongly that I was doing the right thing and enough people agreed with me that it did work out well.

As any teacher knows, middle school students can be pretty simple and direct in how they explain things, Their most frequent explanation of me to their younger siblings who were about to start my class was something to the effect, “He is strict but you will learn a lot.”

Many years ago, I watched the movie, “Wall Street” and the character played by Michael Douglas famously said, “Greed is good.” A few days after that I realized that I had been using a similar philosophy in my teaching - “Struggle is good.” Beyond the learning of mathematical skills and concepts, there was value in requiring students to work to their maximum, or at least to work very hard in order to get the grade they desired. It would bring out the best in them and it would not deceive them (or their parents) into thinking that they really knew more than they did. At first they would resist or be uncomfortable, but I really left them no choice. They had to meet my standards, not the other way around.

Now I need to add something to this philosophy. If the instructor cannot “deliver the goods”, in terms of effective teaching, then the students will not continue to work. They will either conclude that the teacher doesn't know what they are doing or that they themselves don't have the ability to be successful. The teacher has the same responsibility as the student – to work very hard and do their best. I want to stress that setting high expectations for kids requires that we set high expectations for ourselves, whether we be parents or teachers.

What I advocate for teachers and parents is to carefully put struggle into their children's experiences. Don't make the struggle artificial - make it true to the value and importance of what they are doing. Teach them. Don't train them. Set high standards, but not impossible ones. If you are teaching a twelve year old how to pitch and your standard is that they never throw a wild pitch then I would submit that an expectation like that is impossible to reach. If you expect a child (or an adult) to reach an impossible level, they will soon quit and may never be willing to trust your judgment again. The wise teacher or parent can also model the value of struggle by working at something difficult in their own life. Therefore it can be “do as I do” and not just “do as I say.”

Find ways to motivate the kids to want to do what you require because they understand the value of what you want them to learn, but also because you want them to understand the value of struggle and the importance of overcoming the obstacles in their life. Motivation of students was always something that I considered to be a weakness of mine, but I always recognized its importance and I tried to constantly get better at it.

Any adult knows that adversity is a regular part of life and someone who has never had to deal with adversity before, may not be able to overcome it. Effective teaching should not just result in the learning of specific concepts and skills but also result in confidence gained from being required to meet high standards and overcome obstacles and difficulties in their experience. This is done for two reasons – it is the right thing to do (which many would say is enough by itself) and it promotes in kids the confidence and ability in themselves that they can overcome the adversity that will take place in their future.

So I say, “Struggle is good.” Now you know why I think that is true. Do you agree?

Hair-raising Experience

2008-01-16T12:37:00.000-08:00

THE STREAMLINED LOOK

Earlier this year the Spirit Lake boys basketball team beat Sheldon in a home game. As they were preparing to play Sheldon again, this time at Sheldon, I mentioned to (Spirit Lake) Coach Walz that the last time the Spirit Lake boys (varsity) basketball team had won at Sheldon was in 1997. I remember that game well. We had lost 7 games in a row at that point of the season and had lost our best player, Todd Gengerke, to knee surgery the day before the game. The team decided to dedicate the game to him. Sheldon was 14-2 going into the game and was ranked #9 in the state at the time. However our guys played with great purpose and excellent focus throughout the game. It was close all the way. We shot eight for eight from the free throw line in the last minute and a half and ended up winning 55-53. We asked the Sheldon people for the game ball and took it back to Spirit Lake. The team, managers, and coaching staff went out to the hospital that night and presented the game ball, with all of our signatures, to Todd. It was a great moment for all of us, and we asked the photographer and sports reporter for the paper, Mike Early, to take a picture of that presentation. I have that picture framed and in my house.

After I told Coach Walz that story, he asked me to relate that to the current team on the night before they played at Sheldon. I said I would do that and as I thought about it, it seemed to me that I needed to add something that would give the players some extra motivation since they didn't have a teammate in the hospital having knee surgery. I told them that it would be the first time in a long time that we had beaten Sheldon twice which should be motivation enough but, if they did win, I would have my barber cut off my hair – military style. Yes, it was a moment of temporary insanity on my part but I have kept my word. Spirit Lake beat Sheldon on Tuesday night 52-47 in overtime. The morning after the game I went in and got my “induction cut.” I'm not going to be needing my comb for a long time – not that I miss that.

I stopped by Wal Mart and picked up a stocking hat and a baseball hat. Hopefully that will stave off pneumonia during this Iowa winter. After my trip to the barber I stopped by the library and did a little research. I found out that the previous time Spirit Lake had beaten Sheldon twice in the same season was in 1993. Now that they have done it again in 2008, I won't forget what year it took place. Spirit Lake is now 10-3 and is having a very fine season. They have a big game this Friday against LeMars. Spirit Lake is in fourth place right now, just behind LeMars. I am going to see the team this afternoon at their practice and let them see what their win last night accomplished. And I am going to tell them that against LeMars, they are on their own, since I don't have any more hair I am willing to have removed.

HALFWAY NUMBERS

2008-01-13T15:24:00.000-08:00

Back on Nov. 27, Nov. 29, and Dec. 3rd,on this blog,I posted some information on Mental Math which included philosophy and numerous mental math questions. I would urge you to read that if you have time. The questions represented a wide variety of mental math situations and math content. If time does not permit you to do much mental math, with students or children, but you want to do something rather than nothing, then "halfway" numbers might be the solution. I thought of this by accident but it proved to be a very good concept and a source of good mental computation opportunities. It is something that can be tailored to almost any age group and/or ability. A teacher or parent that does just one or two of these a day can help their students develop some confidence and ability in mental math. Below is an explanation of the concept and techniques.

HALFWAY NUMBERS

Halfway numbers present some very good mental math opportunities for middle school students. Halfway numbers are numbers that are halfway between two other numbers. For example, students could be asked to find the number halfway between 48 and 64. When I first presented this type of question to my students, I had in mind two distinct methods that could be used. Obviously the halfway number would be the average of 48 and 64. (I call this "The Average Method".) So the student would have to add 48 and 64 in their head. A good mental math strategy for that could be to add 40 and 60 (100) and then add 8 and 4 (12) and then add 100 and 12 getting 112. That is called "Front End Addition". (There are other good mental math strategies for adding 48 and 64 but that is not my focus here.) 112 must then be divided by 2. A good mental math strategy is to "distribute" the division. 100/2 is 50. 12/2 is 6. 50 +6=56. Therefore the number halfway between 48 and 64 is 56. The other common strategy for halfway numbers I call "The Difference Method." The (positive) difference between 48 and 64 is found first. The student could mentally subtract 48 from 64 by first subtracting 40 from 64 getting 24 and then subtracting the remaining 8. 24-8 could be done by 24-10+2 (subtracting 8 is the same as subtracting 10 and adding 2.) 24-10+2=14+2=16. So the difference between 64 and 48 is 16. The halfway number is half of that difference (half of 16 is 8) away from 48 and from 64. So the student could add 8 to 48 or subtract 8 from 64. Either way you end up at the halfway number of 56. (By the way, although it is not my focus here, I did want to mention another method for finding the difference between 48 and 64. In general, one of the best ways to subtract in your head is to figure out what you have to add to the smaller number to get to the larger number. In this situation it would be natural to think that you would add 2 to 48 to get to 50 and then add 14 more to 50 to get 64. Thus in total you would add 2+14 or 16 to 48 to get 64 and therefore the difference between 48 and 64 is 16. The concept of “adding on” is a great way for people to do mental math subtraction.)

One class period when I was doing halfway numbers with my students, one of them, named Gary, used a different method. He took half of each number and then added those together. He did get the halfway number. In my example above, that would mean taking half of 48 (24) and half of 64 (32). Then add 24 and 32. You get, of course, 56. I have to admit that at first my reaction was that it was a lucky coincidence. After school that day I looked at it algebraically, and it turns out that "The Gary Method" does work all of the time. You just have to show that a/2 + b/2 = (a+b)/2. As you can see in this example, "The Gary Method" might be the easiest way to find a halfway number between 48 and 64.

Finally, one day, when I was explaining "The Difference Method" I used a number line diagram. I drew a number line putting a tick mark on the left labeled 48 and a tick mark on the right, labeled 64. I put a tick mark halfway between 48 and 64 and said we needed to find what that number was. I drew a little arc from 48 toward 64. The arc represented a jump of two to 50. Then I drew a similar arc from 64 to 62. So now the halfway number is halfway between 50 and 62. Some students can see right then that the halfway number has to be 56. If others don't see that you can continue moving toward the halfway number equal amounts until the halfway number is obvious. I ended up calling this "The Number Line Method." They do need to be able to do it in their head, of course. Some students really like this method. They "see" what is going on a little better.

Give them some easy examples at first and let them try to find the answer their own way. Students will naturally gravitate to one or two of the above methods. You can gradually introduce the other methods and then show how each method could be the best way to do certain halfway number questions. Who knows? One of your students may discover an altogether different method that also works.

By the way, when students are first asked to do any type of mental math question, they will tend to try to do the paper and pencil method in their head. I do everything I can to discourage that. Pencil and paper methods are good when you use pencil and paper. When doing mental math we want to use strategies that are appropriate for mental math. Thinking this way and using the strategies of mental math can be very empowering to students who previously assumed that they were not good at math. They may realize that they are good at math when they are taught and allowed to use their brain in a natural way.

One-on-One With Obama

2008-01-08T03:10:00.000-08:00

Recently I read an interesting article in Sports Illustrated in the "Point After" section, the last page of the magazine. That section used to be written by Rick Riley, who I really liked, but he has moved on to another position. Anyway this column is written by S. L. Price and talks about a one-on-one basketball game he had with Barack Obama, during a campaign trip at the Spencer, Iowa, YMCA. I have played basketball at that "Y", years ago, and I have become an Obama supporter. You can also find out what basketball had to do with the budding relationship between Barack and his future wife, Michelle. It's a good "read."

Here is the link.

Caucus Night and Math

2008-01-04T02:09:00.000-08:00

On Thursday night I attended my first caucus. It was a fairly interesting experience.
First I will tell you who I supported and why. I consider myself an independent although I tend to favor the Democrats. I have voted for some Republicans over the years, like Senator Grassley, and various other candidates in more local elections.

As I looked at this year's slate of Presidential candidates, both Republican and Democrat, I saw several good choices. I am old enough, or should I say experienced enough, to know that no one is going to be perfect. There is no candidate that I agree with on everything but my first priority was to look for someone who might be able to unite the country. Unless that happens we are going to continue to have gridlock. For example I agree with Hillary Clinton on a lot of issues but she is easily the most polarizing candidate that has a reasonable chance to win. If she were to win the Presidency, her best intentions and best programs would never come to fruition because of the intense hatred so many of her enemies have for her and her husband. Her ideas would be rejected by many just because they came from her, not necessarily because they were bad ideas.

Of the remaining "viable" candidates, I felt that Obama, McCain, Richardson, Huckabee, and Edwards (in that order) were the most likely to be able to work with the other party in a constructive way and unite the country. That is the most important thing in my opinion because the challenges we face, both foreign and domestic, are going to require a united effort. Immigration and health care are good examples. There is basically unanimous agreement that these are problems that need to be addressed but nothing has happened in the last seven years because of gridlock.

Barack Obama does lack experience in certain areas but I feel he has the most unifying message (which he first outlined at the Democratic convention in 2004)and I think he has the potential to be a great President. He has brought a lot of young people back into the political process and seems to give the most hope for uniting the country and starting a groundswell of momentum to make things better WITHOUT denigrating the other side. I also have the feeling that if Obama became President that it would be a positive in the eyes of most of the rest of the world. That would help us in defeating the actual threats to the freedom of our country and our allies.

Now on to the caucus on Thursday night. My precinct in Dickinson County had 121 people that showed up to express their preference. We were told that for a candidate to be "viable" that they would need to have at least 15 percent of that total. A little mental math on my part (move the decimal point one place to the left, take half and add it on) told me that would be 18 people. We then moved to our separate groups. We in the Obama group had way more than 18; Clinton and Edwards also had more than 18. Biden had about six or seven and Richardson had three. Those people had to reorganize and they received "pitches" from the other three groups. Some went to Edwards, but most came to Obama. In the end, Obama had 60 voters, Clinton had 32 and Edwards had 29.

Our particular precinct, based on population, had been assigned seven delegates to the state convention, so those seven had to be divided among the three viable candidates. So of course they took the number of voters in each group divided by the total number of voters (121) and then multiplied by seven. Edwards got about 1.68, Clinton about 1.85 and Obama about 3.47. By the rules they had to round to the nearest whole number (no partial delegates!) so Edwards and Clinton received two delegates each and Obama got three delegates. Those were the three numbers reported to the state Democratic party and those are the numbers used in determining the percentages that are reported on television and in the newspaper.
As an Obama supporter, I was a little disappointed since we had almost twice as many votes as each of the other two but only got one more delegate.

From what I had heard I was expecting the process to last from one to two hours but it was actually over for the voters in about half an hour. I'm glad I participated and it was interesting to see some of the people I know and who they favored. I especially respect the people who originally stood up for Biden and Richardson, knowing that they would be unlikely to reach that 15% level. (Early on, a few months ago, I had favored Richardson but I ended up feeling that Obama would be more likely to unite and inspire the country as a whole. Their policy differences are slight.)

Barack Obama does remind me a little of John Kennedy (once again showing my age). He is young, an inspiring speaker, relativley inexperienced on the national stage, has to overcome a possible prejudice (Kennedy as a Catholic and Obama as an African-American) and overall gives people hope for a new era in politics and a new era for America. Ladies and gentlemen, I think we are about to experience a significant historical moment in American history.

Helping the Boys

2008-01-02T10:33:00.000-08:00

For anyone who has taught and/or parented a middle school/high school boy, you know that organization, planning, and logical thinking are often in short supply. I came across an article that relates to that with some suggestions for ways to help.

Here is the link.

Improving math instruction

2007-12-26T09:09:00.000-08:00

I recently saw an interesting article in the Washington Post on some elementary math teachers incorporating more algebraic concepts in their teaching. The article also talked about some developments in the areas of educating and training math teachers to enable them to serve their students more effectively.

Here is the link.

Mom's Recipe

2007-12-20T20:10:00.000-08:00

My mom is a great cook as well as a great person. I thought I would let you know about her recipe for Cavatelli. It is extremely tasty and I have never known anyone who doesn't like it. Combine it with a salad, garlic bread, and beverage of your choice and you have a great meal! This recipe serves about six - unless me and my brothers are there in which it serves about three. Anyway, if you try it I know you'll like it.

Cavatelli

1 lb. Ground Beef
1 lb. Italian Sausage
1onion chopped
1 Green Pepper chopped
1 garlic bud
Saute all together

Add:
2 cans chopped tomatoes
16 oz. can tomato sauce
6 oz. can tomato paste
1 tsp sugar
1/2 tsp salt
1/2 tsp pepper
1 tsp basil
1 tsp oregano
1 tbls. parmesan cheese

Simmer for 2 hours

Prepare 3 kinds of pasta (shells,penne,etc.) 1 lb. Total

Mix with sauce, put in baking dish, bake at 350 until hot and bubbly. Top with shredded mozarella cheese. Heat until cheese melts.

ENJOY!!

I couldn't resist.

2007-12-17T19:55:00.000-08:00

Because of my recent posts on mental math, the article linked below had a very interesting title. The story really isn't about the type of mental math that I have been talking about, but it does report on a pretty surprising test result. The students at Duke University that participated in this study probably hope that no one ever finds out that they were involved!

Here is the link.

Using Math to Save Money and Make Decisions

2007-12-13T05:59:00.000-08:00

The following article contains a lot of useful information that people can use to help make some of the most common money decisions that they have in life. It would also be something that high school teachers can use with senior students, or college teachers with their students, because they will be making these types of decisions in their near future. The article also provides links to other websites that have more detailed information.

Here is the link.

"EXTREME" Mental Math

2007-12-12T07:40:00.000-08:00

Some of my recent posts have been on the subject of mental math. Well have you ever spent any time trying to mentally figure out the 13th root of a randomly selected 200 digit number? (For those that don't know the 13th root of a number would be a number that when taken to the 13th power would equal the given number - for example 2 is the 13th root of 8192 because two to the thirteenth power is 8192.) I recently read a story of a young man who did just that in a little over one minute!

Here is the link.

Basketball Math

2007-12-08T09:51:00.000-08:00

BASKETBALL COMBINED
FIELD GOAL PERCENTAGE

During my thirty-three year math teaching career, I spent thirty of those years coaching basketball. One of the things that I found interesting was that when the three point shot became part of the game, it changed the concept of what was a good field goal percentage (meaning what percent of the field goals attempted were successful). Most high school coaches were happy if their team made at least 45% of their field goal attempts when all shots made were worth two points. The advent of the three point shot changed that because the team scored three points for a made shot from behind the arc as opposed to two points for a made shot from inside the arc. So making a three point shot was naturally more valuable to a team than making a two point shot.

The made three point shot is worth 1.5 times a much as a made two point shot so the three point percentage should be multiplied by 1.5 to fairly compare it to the percentage from two point range. So if a team or player shot 40% from three point range that would be equal to shooting 60% from two point range. As an example: making four out of ten (40%) from three point range would score 12 points (in ten shots) and making six out of ten (60%) from two point range would also score 12 points (in ten shots). Therefore if a coach would be happy with 45% shooting from two point range they would have to be happy with 30% shooting from three point range.

In the reality of basketball there might be some slight differences in those two situations. Three point shooters are less likely to be fouled than two point shooters. Also missed three point shots are not as often rebounded for baskets as are the missed two point shots. (One thing that some people consider to be negative about the three point rule is that the “in between” shot [between the lane and the three point arc] has almost completely disappeared from the game at the high school and collegiate level.) However for the balance of this discussion I am not going to consider these realities.

One day while I was watching a game I started to wonder if there was a way to take all of the field goal attempts and field goals made and come up with one percentage which reflected the different value of the shots. In other words it would not be realistic to simply add the two point shots made and the three point shots made and divide by the total number of field goals attempted. Here is why: let's say a team made ten of twenty from two point range (50% and scoring 20 points) and made five of twenty from three point range (25% and scoring 15 points). Just adding them normally that would be 15 made shots in 40 attempts which is 37.5%. However the team actually scored 35 points with those 40 shots and if they had made those same 15 out of 40 shots under the old rules (no shot worth three points) they would have only scored 30 points.

After some experimentation, thinking, and calculating, I finally came up with a way to actually figure out a combined field goal percentage method that accurately reflected the value of a three point shot as well as the two point shot. What it actually does is to give you a field goal percentage that would accurately reflect how many points you scored with your field goal attempts if all the attempts were from two point range. The original formula I had wasn't “user friendly” so I made an adjustment that was mathematically equal but much easier to use. Here is the “friendly” formula:

[Total points scored from all field goals] divided by [Twice the total number of all field goal attempts] Convert the resulting decimal to a percent.

In the above example the team scored 35 points with 40 shots so it would be 35 divided by (2x40) or 35 / 80 which is .4375 or 43.75%. That is higher than the 37.5 % which would be the normal way and that reflects those five made three pointers scored 15 points not ten points.

The interested reader might want to comment on why this formula works. I think I know but I would be anxious to hear what you have to say about it.

Here is one additional challenge. Can someone come up with a formula that also takes into account the free throws made (worth one point of course) and the free throws attempted along with all the field goals made and attempted? In other words, one percentage that would reflect one point shots, two point shots and three point shots. It is possible that no one in the world has ever done that, maybe because they had many other better things to do! As I am typing this, I do have an idea on what the formula would be but I would like to hear what other people think. If you figure it out or find that someone else has figured it out, please let everyone know by posting a comment on this blog.

One other way that basketball coaches and/or fans could look at this information is to simply compute what I would call the “PPS” for their team. “PPS” stands for “Points Per Shot.” Take the total number of points scored by a team and divide by the total number of shots (free throws, two point shots, and three point shots) that the team took during the game. Here is an example: A team makes 12 of 18 free throws scoring 12 points, 20 of 45 two point shots scoring 40 points, and 5 of 17 three point shots scoring 15 points. Therefore the team scored a total of 67 points and took a total of 80 shots. The PPS would be 67 / 80 which is .8375. So the team averaged .8375 points for each shot. That decimal should not be converted to a percent.

Let me know what you think, if you are still awake and make it this far!

Encouraging news

2007-12-05T10:57:00.000-08:00

This is an article from Thomas Friedman, the author of "The World is Flat" and the winner of three Pulitzer prizes. I am encouraged by this information.

Here is the link.

"USE YOUR BRAIN, NOT YOUR CALCULATOR." part 3

2007-12-03T19:35:00.000-08:00

Here is a final group of mental math questions. In the near future, I will talk more in depth about the concept and use of "halfway numbers."

Mental Math

Mental Math # 11
8.What is two-thirds of four-fifths?
9.40 is what percent of 1O? Ten is what percent of forty?
10.How many thirds in 60?
11.8.4 /1000
12.163 cm is how many meters?
13.What is 15% of $12?

Mental Math #12
16.What is three-fourths of three-fourths?
17.One and one-half minus one-fourth?
18.How many quarts in 12 gallons?
19.If marbles are $1.20 per dozen, what is the cost of 3 marbles?
20.9 is what percent of 6? 6 is what percent of 9?
21.What is halfway between 490 and 522?
22.A trip lasted from 7:50 am to 2:05 pm. How long was the trip?
23.Which is better in percentage - making 8 out of 14 free throws or making 12 out of 25 free throws?

Mental Math #13
24.4.3+9.8
25.202-39
26.Round 68.561 to the nearest tenth.
27.Five minus one and seven-eights
28.9.6 / 1000
29.How many pints in one gallon?
30.What is four to the third power in factored form? Standard form?
31.What would be a 15% tip on a restaurant bill of $21.50?

Mental Math # 14
32.218 to the zero power equals what number?
33.What is one-half of seven eights?
34.What is 40% of 40?
35.Round 4.088 to the nearest tenth.
36.Eleven and one-fourth minus two and one-half.
37. 981-208.
38.A trip lasted from 9:45 am to 2:20 pm. How long is the trip?
39.Approximate a 15% tip on a restaurant bill of $26.49.

Mental Math #15
40.4 is what percent of 20?
41.20 is what percent of 4?
42.Two-fifths minus one third.
43.How many quarts are in two and one-half gallons?
44.$4.85 (10 000).
45.Two fifths minus one third.

Math and Science Education Collaborative Initiative

2007-12-02T19:25:00.000-08:00

In the December 2nd edition of the Des Moines Register there is an article on the Iowa Board of Regents' Mathematics and Science Education Collaborative Initiative, a joint effort of the three state universities, to strengthen math and science education in the state. I cannot give you a link to the article but it is on the desmoinesregister.com website right now. Go to that site and click on "opinion" and look for the article by Benjamin Allen (President of UNI).

Another way to find out the information of this very important program would be at this link

When you are on that page you can click on "news release" and "funding proposal".

If possible read the introductory article in the Register first.

I think this is an exciting program that has the potential to make a big difference in math and science education and the recruitment of many new and talented teachers.

"USE YOUR BRAIN, NOT YOUR CALCULATOR.", part 2

2007-11-29T05:52:00.000-08:00

Here are some more Mental Math questions. In a future post I will talk more about the concept of halfway numbers.

Mental Math Questions

Mental Math #6
1.Six minus two and three-fourths
2.Estimate the product of 57 and 81.
3.What is 50% of 73?
4.108 + 999
5.How many fourths in six and one-half?
6.1.7 / 100
7.What is the decimal equivalent of five-sixths?
8.What number is halfway between 138 and 190?

Mental Math #7
9.What is two-thirds of one fifth?
10.How many quarts in 5 gallons?
11.What is 25% of 46?
12.What is 2001 / 1000 ?
13.Four to the third power is what number?
14.11 feet is how many inches?
15.10 is what percent of 5?
16.Apples are 12 cents each and oranges are 13 cents each. What is the total cost of three of each?

Mental Math #8
17.Seven-eights minus three-fourths
18.What is 10% of 156?
19.Write two to the fourth power in factored form and in standard form.
20.How many thirds in eight and two-thirds?
21.What is one-fourth of three-fourths?
22.Find a number halfway between 88 and 106.

Mental Math #9
23.Estimate 82 times 66.
24.How many cups in 3 pints?
25.10.96 / 10
26.Five-sixths minus one-third. (answer in simplest form)
27.What is three-fifths of forty? .
28.Eight to the zero power, divided by eight.
29.If you left at 8:45 am and arrived at 11: 19 am, how long (time) was the trip?
30.Find a number halfway between 102 and 74.

Mental Math # 10
31.Bananas cost 10 cents each and pears cost 12 cents each. What is the total cost of three bananas and four pears?
32.What is 25% of 80?
33.Seven-eights times two.
34.Eleven-sixteenths minus one-half.
35.30 is two fifths of what number?
36.What number is halfway between 390 and 518?

"USE YOUR BRAIN, NOT YOUR CALCULATOR."

2007-11-27T15:05:00.000-08:00

MENTAL MATH

The ability to do math in your head is a skill that I don't believe we spend enough time on with students. If you think about it, when presented with a mathematical situation, most people would first try to do it in their head. For example, if a recipe serves 12 people and calls for 3/4 cups of flour, how much flour would you need if you wanted to serve 24 people? If an answer can't be found mentally, a person would likely try a calculator next. If a calculator weren't available, pencil and paper would be the next option. In this day and age, I believe kids (and adults) use calculators "reflexively". That means they use it before thinking whether or not the question could be answered more quickly just using their brain. I found over the space of a few months of working on Mental Math in class, I can get most kids to not use the calculator reflexively (at least in my room). This must be constantly reinforced in the regular classroom activity also. My constant refrain was "Use your brain, not your calculator."

When you begin doing Mental Math, many kids will want to use paper and pencil methods in their head. This must be discouraged or you won't get anywhere. For example, let's add 66+48. You DON'T want them adding 6+8, getting 14, putting down the 4 (in their mind), carrying the 1. Then they add 6+4, get 10 add the 1, get 11, put it next to the 4 (in their mind) and end up with 114. That is very inefficient for mental math. They could, however, add 70+50, getting 120 and then subtract 4 and subtract 2 (or just subtract 4+2), getting 114. Or they could add 60+40, getting 100 and then add 6+8, getting 14 and then adding 100+14. Or take 4 off of the 48, leaving 44 and add that to 66, getting 70, and then adding 40 and then 4 (or adding 44 all at once). Early in their mental math experiences let them come up with their own strategies, but also show them some efficient strategies that you know they can use in other situations.

The Mental Math questions I will eventually show (I used with 8th graders) are based on the work of Larry Leutzinger of Area Education Agency 7 in Cedar Falls, Iowa. I modified some of the questions and then added some of my own. He advocated using 5 questions a day. For me it is easier to do 8 questions every 2 to 3 days. Each teacher has to find what works best for him or her. I feel it is invaluable for middle school students to be forced to "use you brain, not your calculator." You then should see less "reflexive" use of the calculator.

My classroom procedure for this was the following: Students get out their Mental Math answer sheet, and stand up next to their desk. I give the question orally and then repeat it once. They think about while they are standing, then sit down to write their answer (all they can write is an answer), then stand back up. The standing and sitting is helpful for several reasons. First they get to move around a little. Second, you can observe who comes up with an answer quickly and who takes a while. Third, when everybody is back up, I can go on to the next question. Sometimes a question is difficult and I will give them a little longer. In some cases I will feel they have had enough time and I will move on to the next question even if some have not written an answer. I normally do not grade their performance on mental math. I do all of the questions this way. Then we go over each question in class. I have already prepared a transparency showing what I think is(are) the best way(s) to do the question and then see if they have any unique alternative methods. I usually ask for a show of hands as to who got each question right. Then at the end I might ask who got 8 out of 8,7 out of 8, and so on. The whole purpose is for students to develop the ability to think and reason mathematically and to be confident in using their brain to do math mentally. One other positive benefit to doing mental math this way is that it will increase the listening skills of the students since they do not see the questions, they hear them.

It is necessary to do Mental Math on a regular basis or you will lose whatever gains you have made. It takes awhile for students to gain confidence in their ability to do math in their head. Sometimes the reason that they are not successful is that the methods they are using are inefficient for mental work. You must show them a variety of ways to do math mentally, which will give them tools they can use, but it also builds their desire to think more creatively on their own. I have also seen the “spillover” of their increased confidence into the other areas of their math class. Of course if they aren't taught methods that will work, then the opposite consequences take place.

Here are some examples of Mental Math questions. In upcoming posts I will supply more questions.

MENTAL MATH QUESTIONS

Mental Math #1
1.58+16
2.How many inches are in 6 feet?
3.What is 50% of 54?
4.Mary has five times as much money as Bob. If Bob has 40 cents, how much does Mary have?
5.Five and one-third yards is how many feet?
6.A class has 24 students. Three-fourths of the class were participating in athletics. How many students in that class are participating in athletics?

Mental Math #2
7.Estimate the product of 196 and 52.
8.48+62
9.62 times .5
10.What is 3/8 of 40?
11.What is 2/3 of 21?
12.If a quart of milk costs 75 cents, what would be the cost of a gallon of milk?
13.20 feet is how many yards?
14.A trip takes two hours and forty minutes. If you need to be at your destination by 4:15 pm, you need to leave no later than what time?

Mental Math #3
15.27/2
16..939 (100)
17.66+48
18.Six and one-half minus two and one-fourth
19.If a dozen oranges cost $1.80, what would be the cost of six oranges?
20.165cm is equal to how many meters?
21.What is 75% of 32?
22.You make a purchase for $1. 72. If you pay with a $5 bill, how much change should you get back?

Mental Math #4
23.If a pound of candy costs $2.80, what would 4 oz. of that candy cost?
24.What is 10 squared divided by 5?
25.How many eights of an inch are in four inches?
26.You make a purchase for $5.73. You pay with a $10 bill. How much money should you get back?
27.Joe has three fourths as much money as Alice. If Alice has $30, how much does Joe have?
28.How many tenths in ten and one-half?
29.Two thirds minus one half.
30.The average classroom at SLMS has 22 students. If there are 20 classrooms in the school, how many students are in the school?

Mental Math #5
31.What is the reciprocal of 3?
32.A shirt costs $9.35 to make. How much would it cost to make 10 000 of those shirts?
33.How many inches are in 8 feet?
34.Write a quick estimate of the product of 55 and 47.
35. A special breakfast meal at McDonalds costs $2.50? What would five of those meals cost?
36. Three fourths plus one third.
37. A sweater normally costs $60 but is on sale for 20% off. What is the sale price?
38. What number is halfway between 16 and 30?

More on the Kindle

2007-11-24T12:57:00.000-08:00

For those of you interested in more information on the Kindle, I found a detailed article on the Newsweek website. It puts this device in greater perspective and describes some of the experiences the author has had in using it.

Here is the link.

Reading Innovation

2007-11-21T10:26:00.000-08:00

I recently heard and read about one of the most fascinating innovations in the history of mass communication. That's just my opinion of course. My opinion also is that the current price of this device will not allow it to become commonplace. When and if it does come down in price, I see a huge impact on education and society in general. It is exciting to be living during a time of technological innovation in which things like this can come into the marketplace.

Here is the link.

Wave of the Future?

2007-11-20T05:32:00.000-08:00

I came across this article in the New York Times. To me it suggests one of the possibilities for high school education that might seem to be impossible for most schools but with the proper motivation it may very well be possible. It is already being done in a few places with great success. When people know what can be done and the value of it, they may very well find a way to get it done.

Here is the link.

Calculators

2007-11-19T05:22:00.000-08:00

CALCULATOR USAGE ON TESTS AND QUIZZES

In some math classes, a teacher often faces a situation in which he or she is giving a quiz or test and which the teacher wants the student to use a calculator on part of the test but not on the other part of the test. I found a good way to handle this. You can do it in either order but I had the students place their calculators on the table in the front of my room and they took a marker out of a box on that front table. (I would put enough markers in the box so that each student could get one.) When they start the test they do all the questions on the test that I want them to do without a calculator. I would group those questions together or designate them in some way so that the student knew which ones they were. They had to do their work, if needed, with the marker, and write the answer(s) with the marker. When they had completed those questions, then they would put their test face down, take the marker back up to the front table, put it back in the box, and then pick up their calculator to finish the rest of the test.

When working with the calculator they show their work with pencil and write their answer(s) with pencil. Because their answers to the non-calculator questions are written with a marker they can't use their calculator to redo the question and change their answer (when they have their calculator, the marker is back in the box). Ink pens can be used instead of markers as long as students use a pencil with their calculator. This method works very well. Students should know ahead of time what kinds of questions they will be doing without the calculator (ideally when you are first teaching the skill). Also the day before the first time I use this method, I tell them about it and show them where they will put their calculator and where the markers will be and so on.

E-commerce at Risk?

2007-11-17T07:26:00.000-08:00

I came across an article that points out a potential flaw in the security of online transactions. It also illustrates one of the current uses of mathematics that is designed to protect those transactions. I do plan on continuing to buy things online until I hear otherwise. However I always told my students that, when someone says that a code or system is totally safe, that it isn't necessarily true. What man puts together, man can take apart and sometimes man can make an honest mistake.

Here is the link.

Movie Review #1

2007-11-16T03:08:00.000-08:00

ACROSS THE UNIVERSE

What in the world is a movie review, an amateur one at that, doing on “Practical and Creative Ideas for Math Teachers?” Well I just thought it would be something interesting and teachers are people too, right?

Being a long time fan of the Beatles I thought I should see this movie. I had read good and bad things about it. It turned out to be very enjoyable – but it did not “change my world.” The basic story is a very common one – lonely boy meets lonely girl, they fall in love, they fall out of love, they get back together. The End.

The various plot points are illustrated by about thirty Beatle songs, sung mostly by a group of young, talented actors. In many cases the arrangements are slightly different from the Beatle originals but still recognizable. Some lyrics are given more emphasis, some less, in service to the plot. A series of stunning visuals accompany the songs, again in moving the story forward, or at least sideways. The creativity of director Julie Taymor is commendable, as well as her audacity to try to use music that is sacred to many people in a way that some may not like. I did like and admire what she did. She shot for the moon, only made it into orbit, but certainly didn't stay on the ground. How many of us have made it into orbit?

The movie is actually a series of music videos, strung together by the love story of the plot. Bono has a good part singing, “I Am the Walrus”. Joe Cocker is involved in an enjoyable take on “Come Together.” Other segments built around, “Strawberry Fields Together”, “I Want You (She's So Heavy)”, Hey Jude”, and “Let it Be” were also highlights. I also liked the multiple Selma Hayeks, but not necessarily the setting of the song, “Happiness is a Warm Gun.” The actors playing a Janis Joplin type character (“Sadie”) and a Jimi Hendrix type character (“Jo Jo”) were very good. I do wish the writers would have done more with the title song, “Across the Universe”, one of my all time favorite John Lennon compositions.

Watching this movie reminded me once again of the genius of the Beatles and showed the timelessness of their music. In fact there were many other Beatle songs that could have been used, but there is only so much celluloid. One thing I did wonder about – did they write the story and then pick which songs would go with that story, or the other way around?

I read something the other day (“oh boy”) about the fact that a lot of tweens, after watching the movie, are going out to buy the soundtrack album. New Beatle fans are being created every day. This movie is like the popcorn that many people ate while watching it – very tasty, very enjoyable but, in the end, not nutritious. But not everything has to be nutritious. I did enjoy this movie and am glad I went to see it.

Numerous other professional reviews of “Across the Universe” can be found on the movie review sites on the web.

Bush v. Gore part 2

2007-11-15T06:22:00.000-08:00

A VERY CLOSE ELECTION
SOLUTION PLAN

Florida

Bush – 2 912 790 votes, Gore – 2 912 253 votes.
Total is 5 825 043 votes. Bush won by 537 votes. Using a proportion: 5 825 043 votes / 100yd. = 537 votes / X yd. Therefore:
5 825 043 X = 100 (537) Solving for X, we get
X = .009218816 yd.
.009218816 yd. multiplied by 36in./yd = .331877376 in.
4/16 = .25, 5/16 = .3125. 6/16 = 3/8 = .375

Therefore if all the votes in Florida were represented by a 100 yard distance and rounding to the nearest sixteenth of an inch, Mr. Bush won by about 5/16 of an inch.

Using the same information but with the 84 ft. basketball court and using a proportion, Mr. Bush won by .007743805 ft. or about 3/32 of an inch.

USA

Taking the same approach as above, Mr. Gore would have won on the football field by about 19 3/16 inches. On the basketball court Mr. Gore wins by about 5 12/32 inches.

An interesting way to use one of your answers to get another answer is as follows:

In Florida we calculated that Mr. Bush would have won by
about .331877376 in on a football field. We could use that answer to figure out the answer to a basketball court.
(100 yd. = 300 ft.) Again using a proportion:
300 ft./84 ft. = .331877376 in./ X in. Solving for X we get X = .092925665 in. which is virtually identical to the answer we got the other way (.092925666 in.)

If you want more information about this solution plan, write a comment at the bottom of this post.

Brain Development

2007-11-13T05:50:00.000-08:00

I saw an interesting article this morning on the New York Times website that I thought you might find interesting. It is self explanatory.
Here is the link:

It is exciting to think that as time goes on we will be able to find out more about how the brain actually works. That will enable us to figure out better ways of addressing the problems that teachers and parents observe in their children.

Three Answers

2007-11-11T13:52:00.000-08:00

RIGHT, WRONG, NOT WRONG

When students give answers to math questions they sometimes will give an answer that is not what the teacher really wants but it is equal to the correct answer or is written in a different way. For me, those answers fell into the "Not Wrong" category. That told the students two things. First and most important to them, their answer was not counted wrong. Second, there was something else I wanted for that answer. They got the benefit of not counting the answer wrong but they knew there was a different answer or a different way of writing the answer that I wanted them to understand.

There are answers that are "Right." There are answers that are "Wrong." There are answers that are "Not Wrong." Here is an example:
Let's say the student is asked to solve P=2L + 2W for L.
For me the "Right" answer is L=(P-2W)/2.
A "Wrong" answer could be L=P-W.
A "Not Wrong" answer could be L= -W + .5P.
Some teachers might very well consider that that last answer is just as "Right" as my first answer and that is fine. My point here is that there are situations in which the student may give an answer in a way that really isn't wrong but is not in the form that the teacher might want, in that situation, at that time. In the above example if the teacher had specifically told the students that there were to be no decimals or negative signs in a formula, then that "Not Wrong" answer would be "Wrong."
My experience with most students is that when they hear that an answer is "Not Wrong", they will attempt to understand the different type of answer that the teacher wants and try to do it that way next time.

There are other situations in which there are multiple right answers. A simple example would be the following:
What is the probability that a card randomly drawn from a standard deck is a diamond?
"Right" answers could be 13/52, 1/4, 25%, or .25. Of course if the directions had stated that the answer had to be a fraction in simplest form, then the only "Right" answer would be ¼.

The concept of "Not Wrong" might also apply to the method that a student uses to find an answer. It is good for students to know that there are multiple ways of solving a problem. However the teacher might want the student to use the most efficient method and/or a method that can be applied to many other problems. For example a student might use the Guess and Check method to successfully solve a certain problem. The teacher might call that method "Not Wrong" if they had wanted the student to use a more efficient strategy.

As you can imagine the concept of "Not Wrong" is in the eye of the beholder and the situation. What is "Not Wrong" to one teacher might be "Wrong" to another teacher and "Right" to another teacher. The concept of calling some answers or methods, “Not Wrong” can be an effective teaching tool that shows respect for student thinking but also teaches that there is something else they need to learn.

Bush v. Gore

2007-11-01T19:37:00.000-07:00

A VERY CLOSE ELECTION

The 2000 Presidential Election was one of the closest in history. To get some perspective on how close that election was, I came up with the following questions that I believed the students would find interesting, as well as using some good mathematics. This makes for a very good extra credit situation although it has lost some of its interest to today's students. It does suggest an approach that teachers could take with a similar situation in the future. The problem uses the visual images of a basketball court and a football field. This type of familiar image will help students understand the closeness of the vote. Older students and adults will also enjoy this.

In the state of Florida in 2000, the official “certified” voting results showed that George Bush got 2 912 790 votes and Al Gore received 2 912 253 votes. Imagine that the combined votes of these two candidates equaled the length of a football field, from goal line to goal line (100 yards). If the two candidates had received exactly the same number of votes, then they would each have 50 yards. However they didn't receive the same number of votes. Based on the vote totals above, Mr. Bush won Florida by what distance? Round your answer to the nearest sixteenth of an inch.

Using the same vote totals from above, assume the combined vote total is the length of a high school basketball court (84 feet). In this case, Mr. Bush won Florida by what distance? Round this answer to the nearest thirty-second of an inch.

In the total USA vote in 2000, Mr. Gore received
50 996 582 votes and Mr. Bush received 50 456 062 votes. Assuming the combined vote total of the two candidates is a football field, as above, Mr. Gore won by what distance? Round to the nearest sixteenth of an inch. Assuming the combined vote total is the length of the high school basketball court, as above, Mr. Gore won by what distance? Round to the nearest thirty-second of an inch.

The answers and my solution plan will appear in a future post.

Fun with Time

2007-10-30T14:12:00.000-07:00

MILLION SECONDS,

BILLION SECONDS

Another idea that I often used to show the stunning power of mathematics was fairly simple. How long does it take a person to become one million seconds old? The answers I usually got from students were way off (that's what makes it interesting). We fairly soon could work out together that a person reaches one million seconds in a little over 11.5 days. So then I asked them to figure out how long it would take for a person to reach one billion seconds old. I might ask them if they thought they had already reached a billion seconds. Most thought they had since they had reached one million seconds in less that two weeks of life. Of course I wouldn't be using this problem if it came out the way most kids thought it would. Either later in that period or the next day, the students were stunned to find out that they would not become one billion seconds old until they were approximately 31.7 years old. Again most adults are amazed by that answer also.

This problem can be used in a variety of ways. One extension I had them do was to figure out on which day they would reach one billion seconds. That is not easy because of leap days and so forth, but also not that interesting to most of my students because it was so far away. For my 7^th and 8^th graders it was more interesting to figure out when they would be 5 000 days old. I had them assume that they had been born on noon of their birthday.

5 000 days is about 13.7 years so it was between their 13^th and 14^th birthday. I told them that if they came to me on their 5 000 day old birthday (and could prove to me that it was the right day) I would give them a little present. If they had already passed that day, they had to tell me what day it was and if they were correct then they got the present. The present was something small and inexpensive and the same for everyone that year. If the 5 000 day old birthday happened in the summer they could see me just before school got out or the first day of school the next year.

The creative teacher can find other uses for this situation, depending on the age and ability of their students.

Adults can figure out when they will reach or did reach one billion seconds. Those of us who are a little older can figure out when we will be two billion seconds old. Maybe you don't want to do that! Some of the students liked to use this information with their parents birthdays. Don't forget that problems like this can show the power and fun of doing math. Things like this help kids to think in a more mature and logical way.

60x60x24 =86 400 seconds in one day

1 000 000 / 86 400 = about 11.57 days

1 000 000 000 / 86400 = about 11 574 days

11 574 / 365.25 = about 31.7 years

Tissue Paper Problem

2007-10-30T10:48:00.000-07:00

TISSUE PAPER PROBLEM

I often used this on the first day of school to let my students know that this class was going to be special. I used an ordinary piece of tissue paper (Kleenex). When you pull it out of the box, it is two-ply. That is there are two sheets together. I took those apart so I had only one sheet. I told the students that this one piece of tissue was so thin that 1024 of them stacked on top of each other, and then compressed down as far as possible, would be exactly one inch high. That is a way of telling them that the thickness or height of the one sheet was 1/1024 of an inch. I then took that one piece of tissue and folded in in half. I asked them how thick it was now compared to before I folded it. They could easily see that it was now twice as thick or, if you thought of the thickness as its height, it was now twice as high. I again folded it in half and asked them how thick or how high it was. They knew it was now four times as thick or high compared to before the folding. I continued folding and asking the same question. If you have ever tried this you know you can only fold it seven or eight times before it becomes too thick to continue. I did show them that it was ¼ of an inch high after 8 folds. I then asked them how thick or how high it would be if it could be folded in half a total of twenty times. I would say to them that even though we can't actually do that many folds that we could figure it out mathematically. I took some random answers from the students which were usually pretty far off. I then gave them another hint: after ten folds it would be one inch high. Of course that caused many of them to jump to the erroneous conclusion that after 20 folds it would be two inches high. I made sure that they knew that answer was wrong but I did not explain any further (that it would be two inches high after the 11^th fold). Their homework for the next day was to figure out the thickness or height of that one piece of tissue paper after it had been folded in half 20 times. When they came in the next day, I might have one or two kids who had figured it out. At any rate, they were all amazed that after 20 folds it would be 85 feet, four inches high. I find that most adults are amazed by that too.

When I first started using this activity, I was teaching on the third floor of a rather old building. The height of the building was about 48 feet. The building had a gym whose floor was below ground about 15 feet. It really stunned the students that a person could be down on their hands and knees on that gym floor and start folding that tissue paper and that after 20 folds it would be much higher than the top of the three story school building.

Most years I gave the students tissue paper to fold along with me. Sometimes that didn't work well because they didn't always hear the information I was giving them while they were folding. I would suggest that you try it with them folding their own tissue and if you don't like it, then just have them observe you.

(One ply of tissue paper is 1/1024 inch thick.)

0 folds – one thickness, 1 fold – two thicknesses,

2 folds – four thicknesses,3 folds- eight thicknesses,

4 folds – 16 thicknesses, 5 folds – 32 thicknesses,

6 folds – 64 thicknesses, 7 folds – 128 thicknesses,

8 folds- 256 thicknesses, 9 folds – 512 thicknesses,

10 folds – 1024 thicknesses, therefore one inch high,

11 folds – two inches high, 12 folds – four inches high,

13 folds – eight inches high, 14 folds – 16 inches high,

15 folds – 32 inches high, 16 folds – 64 inches high,

17 folds – 128 inches high, 18 folds – 256 inches high,

19 folds – 512 inches high, 20 folds – 1024 inches high.

1024 inches divided by 12 is 85 feet 4 inches.

Introduction

2007-10-30T09:35:00.000-07:00

PRACTICAL and CREATIVE

IDEAS

FOR MATH TEACHERS

INTRODUCTION

Welcome to my blog! On here you will find a potpourri of interesting and novel ways to increase your effectiveness and your enjoyment of teaching. Everything I mention here has been used successfully in my own classroom but I know that doesn't guarantee that it will work for you. Some teaching ideas that worked very well for other people did not work well for me. Either way, reading and thinking about these things may cause you to come up with some creative ideas of your own that will be very effective for you.

I was a math teacher in the Spirit Lake, Iowa, school system from 1970-2003. My basic teaching assignment was 8^th grade Pre-algebra and Algebra 1 for 8^th graders. I also taught some 7^th grade math and some 9^th grade General Math. In many cases I found the textbook approach to certain topics to be ineffective as well as not much fun. Over the years I worked to develop better ways to teach those topics and ways of managing the classroom that solved some of the typical problems associated with math teaching or teaching in general. I also came up with several excellent projects that challenged my students and enabled them to use the math they were learning in some practical and creative ways.

Much of the content of this blog can also be used by elementary and high school math teachers and many ideas are applicable to other subjects besides math. You may want to tell your friends who teach other subjects, at any level, about some of the things you find here, especially the classroom management topics. Some of the projects I will mention are very good for inter-disciplinary work and you may find it enjoyable and valuable to work with some of your colleagues on those projects.

Please come back to this site often. I plan to post many ideas which could be very helpful to you. Tell you friends and colleagues about this site. Leave comments for me on any questions or problems you might like me to address. I look forward to sharing ideas with everyone in order to make teaching more effective and more enjoyable!