Interesting Facts on Factors

Asking a student to find all the factor's of an integer is a relatively simple and common task. Similarly, finding the prime factorization is often asked of our students. It is essential for methods of finding the GCF and LCM that are often taught. However, there are some other very interesting questions surrounding those topics.

• How many factors does an integer have?
• What is the sum of the factors of an integer?
• What is the product of the factors?
• How many odd factors does it have?
• How many even factors?

These questions are all answerable by looking at the prime factorization the integer in question. The following two videos show this process:

This is a great topic to share with your advanced students. It is a way to make prime factorizations more interesting.

My new job is going great. I am learning something new every day. It has been a great experience so far.

One Week Down

As I mentioned before I am started a new job this school year. The Sycamore School is a pre-school through eighh gradet gifted and talented school. I am teaching the fifth and sixth grade pre-algebra and the seventh and eighth grade Geometry classes. It was an amazing first week. I was told that the kids would constantly surprise me with what they know and the question's the ask. I must say they were entirely accurate. Here are some highlights:

• A 7th grade Geometry student was able to prove the square root of two is irrational. I didn't learn this until graduate school and I would imagine many of you have never done it.
• A 5th grader aced the pre-algebra pretest.
• A great dialogue in my 7th grade Geometry class about the definition of skew lines. In general skew lines are not very interesting, but, through their questions we were able to learn a lot about the importance of rigorousness in our definitions.

Professionally it has really challenged me. In my previous teaching assignments I have been able to, how shall I say it...fly by the seat of my pants? Not always, or even most of the time, but if I chose not to spend the time preparing, I could usually come up with very effective lessons on the fly. I am not able to do this at Sycamore. I have to have all my i's dotted and t's crossed as well as spending time considering ways to deepen the discussion beyond what is usually presented in a math class. I love it and am anxious to continue to pursue this challenge.

One of the issues with my new position is it is a long commute and a considerable amount of work. This will make it more difficult to continue to web activities, but I plan on building time into my schedule to continue them as I really enjoy it.

I hope all of you have had a great start to the school year!!!!

Grade Systems Part 4 - Testing

There were many things that bothered me about the structure of my undergraduate education program. One of those was that I was given no information on best practices in testing. Evaluating student performance is a vital component of every classroom. This happens in many ways and has many different names. Formative and summative assessments are two of the more recent names that have been applied to evaluating student learning. There are many things to consider when evaluating students.

• What types of questions should I write?
• Is it OK to use multiple choice? If so, when and how many?
• Should students be asked to explain their answers in a math class?
• How much should each question be worth?
• Should I give partial credit or not?
• How much of the test should be basic skills and how much should require higher order thinking skills?
• Should my tests be summative or formative?

The chair of the board of directors at my new school told a story that I think is important for educators and for this discussion. His son was reading a book that had a genie and the ensuing wishes in it. The son asked, "Dad, what would you ask for if you were given one wish?" The dad said he didn't know and turned the question back on his son. The son responded, " at first I thought I would ask what the meaning of life is, but then I thought maybe that is not the right question. So instead I thought I would wish to know the right questions to ask."

I think a big part of writing tests is about asking the right questions(not just the ones on the test). What are some of the questions that you think need to be answered by someone writing a test?

Intermission: An Lament On Teaching Math

I recently read Paul Lockhart's essay "A Mathematician's Lament" that was published at The Mathematical Association of America Online. It is the best description of the issues that many see in math education that I have come across. On the surface I agree with much of what he discusses. I absolutely agree that discovering math is better than being told it. I believe that the current "ladder" curriculum is weird and disjointed and I agree that most math teachers(me included) are not really qualified to teach math.

But here is the rub and there is really no way to get around this. This essay is idealistic to a point that it is never going to be applicable in any mass educational system. There is one question I have for the author. Why, when I was actually taught by mathematicians, did they teach their class exactly opposite to what you are describing? I understand that I couldn't expect this type of teaching in my K-12 classes and maybe not even in undergraduate, but when I entered graduate school the teaching and lecturing was worse than anything I had experienced up to that point.

With that in mind, I still think it is an excellent read for any math teacher/aspiring math teacher. Found within the idealism are some very good points about teaching math. The most important, though not explicitly mentioned, is that the goal of a math teacher should be to teach students how to learn math, not to teach the math itself.

I will continue my grading systems series this weekend. I wanted to comment on the essay while it was still fresh in my mind.

Grading Systems Part 3a - Homework

This is part three to a series of posts on grading. Part one dealt with percentages and part two dealt with grade weighting. As I have mentioned before, I know that I do not have all the answers on these topics. My goal is to present some of my ideas and then to hear some thoughts of other teachers. So please feel free to let me know what you think in the comments.

I have actually used five different methods of dealing with homework. Let me give a quick recap of each of them.

• During my student teaching I taught Pre-calculus. The teacher I worked with assigned homework everyday. The interesting thing is he never picked it up or even checked to see if it was completed. We went over it and by the questions the students asked the clearly did it. The students entire grade came from tests.
• Also during student teaching I worked with a different teacher who taught Pre-algebra. He also assigned homework everyday and never checked the assignments. What he did was give a homework quiz everyday. These quizzes usually involved problems on the homework as well as problems not on the homework.
• During my first two years of teaching I generally checked the homework for completion. I would occasionally give homework quizzes as well. I usually allowed students to turn in homework late for half credit.
• While in graduate school I taught a class that was comparable to Algebra 1/Algebra 2. We were asked to pick up and grade all of their homework assignments(we met everyday). I always tried to have them graded before the next class and I picked up the homework after going over it. The grade was out of 10 points. I would give them 5 points for completion and randomly pick 5 questions to grade.
• Last year I used a combination of grading the homework and giving homework quizzes. I had more trouble grading the assignments in a timely manner because I had a 100+ students versus 20.

In general I think all of these systems had value. My goal with homework is very clear and two-fold.

1. It gives the students to practice problems on their own. Practicing is an essential part of learning math.
2. It is used as a catalyst for reteaching the material the next day. Every homework system needs to foster discussion of the problems the next day. This gives a chance to answer questions, review the material and occasionally teach to students who missed the previous day.

Which of the systems best accomplishes the afore mentioned goals? I am not sure...I want to think about it a little more and talk with the other teachers about my school before I decide what I am going to use in my class this year. I will revisit this post next week. Until then I would love to hear how you deal with homework in you class.

Grading Systems Part 2 - Weighting

From my first year of teaching I have always placed weights on the different categories of graded work in my classroom. For example this year, though not by choice, my weights looked like this:

70% - Tests

20% - Quizzes

10% - Everything else

I was not a fan of this weighting system because I thought it devalued responsibility (we also allowed retests). But I believe a system that weights the grade categories is the most appropriate. As I was going through school I was always amazed at how "fly by the seat of the their pants" teachers were with their grading system. What I mean is they seemed to randomly assign points and then they would add up the points and divide by the total to see what the grade was. The problem with this method is that from grading period to grading period the grade represents something different. Also it is much more likely that one category will dominate the grade. Often this leads to homework being overvalued, which I think is a problem.

Here is a few things I believe:

• The majority of the grade should be knowledge/skill based. The grade should be an accurate representation of what a student knows.
• Homework and responsibility should play a minor roll in a students grade. The grade should also represent a student's probability of success at higher levels. Hard work and responsibility are key to success(especially in math classes) as students reach harder classes.
• Behavior should play no role in a student's grade.

So the question is, what is the appropriate percentages?

Grading Systems Part 1 - Logic and Percentages

This is part one of a series of posts on grading students. Last year during a professional development seminar my former principal starting discussing perceived inequities in grading systems. He rambled on about an F being worth 50% and 5 point grading scales. I attempted to follow what he was talking about, but failed to understand its relevance to my(or any) classroom. Recently, I ran into an article describing exactly what he was talking about. I finally am starting to grasp the issue that lead to the faulty conclusions discussed in that article and by my principal.


The general problem is when teachers arbitrarily assign a letter grade to an assignment and then try to convert it to a percentage. Thus what happens is a D is worth 60% and an F a 0%. The article is correct, this does not make sense. The problem is the use of a letter grade as the initial grading system and then trying to convert this into a number. This problem does not include the more common practice of converting a numerical grading system to a letter grade. Thus for most math classrooms this article should not apply. Math lends itself to figure all grades in terms of percentages. Then when grades are reported letter grades are assigned to those percentages. Here is the statement I am referencing from the article.

Their argument: Other letter grades — A, B, C and D — are broken down in increments of 10 from 60 to 100, but there is a 59-point spread between D and F, a gap that can often make it mathematically impossible for some failing students to ever catch up.

The problem with this is they are incorrectly viewing the problem. There are two sets, passing and failing. Failing is from 0-60 and passing is from 60-100. Now the other letter grades give the student, parent, school and college exactly how well they are passing. Now if they want to make an argument that failing should be 0-50 and passing from 50-100 that is fine, but the argument that 50% should be a minimum failing number is weird and doesn't make sense. If you want to do that then work off of a 4 point grading system.


However, I am not sure I logically agree with a 0 to 4 scale. My problem is that it doesn't distinguish well between a student that is almost passing and one that isn't close. This is an important distinction to make. Not because it matters on their report card, but from a teaching perspective. When I grade something I need to be able to quickly determine the skill level of each student. I am digressing though because, as I said before, this doesn't apply to a math classroom. Math lends itself to total points and percentages.


As you can tell I disagree with the ideas put forth in this article. They are born out of a desire to pass more students and are disguised as a self esteem booster. It always amuses me when schools attempt to dictate grading systems to their teachers. In a classroom it is always possible to manipulate the system so the grades look like you think they should. I think in general schools spend way to much time looking a the grading and numbers involved with failing students. At my last school our failure numbers were monitored. If they were too high then you were going to here about it. Accordingly many teachers lowered standards.

In my mind the best way to grade a student is using point totals and percentages. Letter grades are only useful as communication tools to parents, students and colleges. All assignments should have number values associated with them. A much more difficult question is, how do I determine what number represents a minimum competency in my classroom? This question is not asked nearly often enough.

Eighteen Days and Counting!

As I mentioned earlier I am starting a new school this fall. The first day for students is August 18th. This will be a completely new experience for me as I will be teaching middle school instead of high school. Also my new gig is at a gifted and talented school private school, which I suspect will be a much different teaching environment from the public schools I have taught at in the past.

Anyway, it is time for me to start considering the norms that I want to establish in my classroom. There are many different things that I need to consider.

• Discipline
• Homework
• Presentation Style
• Organization of Students Work
• Group work
• Seating Chart and Desk Arrangement
• Communication with Parents
• Website
• Grading Structure
• Retesting Policy

I am sure there are others but those are the ones that immediately came to mind. I am going to spend some time the next couple of weeks discussing these topics. Some of them (Discipline, Grading, Retesting) I will delay until I learn more about how the other teachers at my school handle them. I have always thought it was important to try and follow the norms at my school as much as possible.

Per usual I am very interested in what others think of these topics. Check back the next couple of weeks and let me know how you handle these in your classroom. Most of what I do in my room has been stolen from those around me.

Graphing Basics

My plan tomorrow is to work on the discussion for graphing linear equations. I am always amazed at how many of my students are able to graph equations one day and then look at me blank faced the next. When I teach students how to graph from the beginning I emphasize its meaning. Thus I think it is appropriate to spend a considerable amount of time having students graph using T-Charts. Other forms of graphing are important for various reasons, but it is very important for students to view a line as a collection of solutions. In my opinion, the best way to accomplish this is to work with T-charts. Eventually, I teach the intercept method and obviously the slope-intercept method, but conceptually I think it is essential that students have a concrete understanding of the relationship between solutions and a graph.

What are some of your best tips for introducing graphs? I will be talking about slope later on, so I am just looking for ideas about how to graph using T-charts.

Google Knol

Google has jumped head first into the content creation business. There new platform is called Google Knol. It is very similar to squidoo, which I have discussed before and is what most of the sites listed at the right were created with. I decided to created a "knol" on euchre to try it out. The process of creating a "knol" is much easier than creating a squidoo. However, there are a lot less options. If you are a teacher looking to publish something for your class, knol would be a pretty easy place to do it. It will be very interesting to see how the knol project goes. Will google rank its own pages better in its search engine? I haven't decided if I will contribute any math content there or not.

30+ and Counting!!!

My math teaching website is coming along nicely. I have added over thirty resources and have begun to write some of the articles. I am very excited about the site and the relationship this blog is playing in its creation. I know two things for certain about it:

1) I have some good ideas about how to teach Algebra 1.

2) I have SOME good ideas about how to teach Algebra 1.

I definitely know that I don't have all the ideas about teaching Algebra 1. Thus I am very thankful for the contributions of some of the readers of this site. I believe collaboration with other math teachers is the best form of professional development and I appreciate those that are engaging in that with me. It is my goal this blog turns into a community of math teachers discussing very practical means of teaching math.

What's the Function of a Function?

While watching a 53 year old golfer winning the Brittish Open I decided to tackle the discussion on functions. I feel pretty comfortable with how to teach about functions. What has me struggling a little bit is the "why". I understand the value of functions, especially as they relate to Physics. As well, I understand the importance of one-to-one and many-to-one concepts in Abstract Algebra and the importance of domain and range. But my question is:

Why is it important to teach an Algebra 1 student if a relation is a function or not?

Any takers?

Should Teachers Avoid Social Networking Sites?

From 2005-2007 I went to Indiana University as a full-time math graduate student. In the timeline of my life this was 3 years after graduating from the University of Evansville. Thus, I essentially missed out on the Facebook revolution that took over college campuses. However, the majority of my classmates at IU were fresh out of college and were all on Facebook. Accordingl they coerced me into setting me up with an account. During grad school I pretty well ignored it except to use it as a means to identify students that I tutored at IU.

Fast forward to the last two weeks. I figure if I am going to create these resources I might as well try to market them to as many people as possible. Through my research on how to internet market, I was told sites like MySpace and Facebook were usually reccomended as one option. Thus I dove into Facebook and have been an active user the last two weeks. I am very glad I did because I have reconnected with an old college roomate and a friend from Brazil who studied at UE for a semester. Her whole family was at UE that semester and they were some of my favorite people I met at Evansville.

The question I have is what is the appropriate behavior of a teacher on these sites? I am of the inclinationt that I would not accept "friend requests" from students that are currently at my school and would definitely be careful what is posted on my page. It is an interesting question and one that more and more new teachers and schools are going to have to deal with...Let me know what you think.

And and Or

The latest article I have been writing for my website deals with combined inequalities(or compound inequalities). One of the difficulties in teaching this topic is helping students understand conceptually the difference between "and" and "or". I try to accomplish this by developing real world examples. For example I give them the statements such as:

"The sky is blue and the grass is green."

"Peyton Manning is a Colt and a running back."

"Barrack Obama is running for president or he is a math teacher."

I then ask them which of the following are true and why. The goal is to get the students to understand that both statements must be true for an "and" inequality to be true. And that at least one must be true for the "or" inequality to be true.

What are other ways to help students understand combined inequalities?

What Does It Mean to Be Equal?

I am currently working on the Algebra 1 standard "Equations and Inequalities" for my new math teaching website At the core of teaching Algebra is an understanding of equality. One of the major obstacles for students is realizing the differences between an equaiton and an expression. I usually try to make the connection by explaining that an equation is a comparison of two expressions. I emphasize that most equations are actually questions. What value of the variable(s) make the equation true?

The above is more of a conceptual understanding of equations. But just as importantly students must learn how to apply the properties of equations (and inequalities). I introduce them by setting up an equation of two numbers. For instance, 20 = 20. I then add a number to one side and ask them the question, "what must I do to the other side so the equation is still true?" I then use this general method to try and generate the properties of equations.

What are some ways that you introduce and reinforce the properties of equations to your students?

What is Rational About an Exponent?

I have been helping a friend paint his house this week so I have not been able to write as much content for my new website as I had hoped. However, I was able to work on an article about a method of introducing rational exponents. It is not complete, nor completely edited (my editor a.k.a my wife, has been working insane hours this week) but it will give you an idea about the content I hope to create. Let me know what you think.

Associate With Me?

Eventually, I hope my math teaching website will include a comment section so I can get feed back from teachers on how they teach each topic. Unfortunately, that involves a much more dynamic type of website construction than I am currently employing. Luckily, I am helping a friend paint his house this week and he happens to do web design as a career so I should be able to pull it off eventually. For now, however, I am going to use this blog to get some feedback on different topics.

Todays topic is: Associative and Commutative Properties

I am interested in your thoughts on how you teach students about these properties. Obviously, the properties themselves are pretty simple to illustrate, but is it important to make students name the illustrations or is it more important for them to just understand the properties but not know their names?

The Latest Project

As I have talked about before, this past year I began creating single page websites called "lenses". One of the features of Squidoo, the website where I created these, is that it has a stats page. On this stats page I can see every word/phrase typed into a search engine that is used to find my site. From looking at these search terms I came to find out that a lot of teachers were finding my websites looking for resources and ideas on how to teach a topic. This has led me to create a new website designed for teachers. If you visit this website right now you will see that it is far from complete. I am daily adding resources and contributing to the discussions on the different topics. My goal is to finish Algebra 1 by the end of the summer.

Please take a look at the site and let me know what you think. Thanks.

http://www.how2teachmath.com

The Slope of My Life

As a line changes over time so has my life recently. For various reasons that I may blog about in the future, I began looking for a new job in April of this year. I found three schools that I was very interested in and were very interested in me. They were the Sycamore School, Herron School and the International School of Indiana. I was excited about all three schools and believe each would have been a very good place to work.

While in middle school I competed in a great math program called MATHCOUNTS. As an 8th grader I was fortunate enough to finish 3rd at the state competition. This allowed me to compete for Indiana at the national competition. The coach of that team was a teacher from Terre Haute named Bob Fischer. From that trip and other experiences with MATHCOUNTS I have a tremendous amount of respect for him as a coach. This is relevant because the job at Sycamore gave me the opportunity to assist him in coaching their MATHCOUNTS team. While at Franklin I found out I really enjoyed coaching academic teams, so the opportunity to work with someone with his experience and record was very appealing to me.

Thus, with this in mind, as well as other considerations I have accepted a position at Sycamore School in Indianapolis. There are some huge changes for me as this will be a middle school position (though I will be teaching Geometry and Algebra 2) and it is a private school. My other two jobs have been at public high schools. As well Sycamore is considered a "gifted and talented" school. This will also be a big change for me because I have always taught regular/low level classes. This is the change I am most comfortable with because I have always desired to teach high level classes.

I decided this year to break my summer into two parts. The first part was reliving my glory years on the golf course by preparing for and then competing in the Indiana State Amateur Tournament. It went very well as I made the cut(top 60 out of about 500 that attempt to qualify), made a hole-in-one and finished 51st. Unfortunately, I played very poorly in the 4th round or I would have finished even higher. The second part of my summer is going to be focused on creating online math resources. This past semester I worked on creating Squidoo pages on Algebra 1 topics(See top right corner). I am going to continue that work, but I am also creating a website more geared for teachers. I will post more on that tomorrow.

How Do You Factor That?

I recently finished another squidoo lens on working with polynomials. One of the goals of the lens is to teach students how to factor quadratic expressions. I was taught the guess and check method to factor quadratics and have always held the belief this was the best method to factor quadratic expressions. One of my fundamental theories on math education is the importance of teaching the why along with the how. The guess and check method may not always be the easiest "how" but I have always thought it was the best "why".

I will confess I have not worked much with the other more algorithmic methods of factoring quadratic expressions. If you teach/use this method, I would love to hear why you think it is the best way to teach the concept?

What Math Should I Teach?

I have recently finished two more Squidoo pages on Quadratic Equations and Square Roots. Both of these, as have the others lenses linked to the right, force me to face the question "What Math Should I Teach?" In my mind, the Squidoo pages and the acccompanying videos I have created are designed for a regular 8th grade/freshman Algebra 1 class. But as I have learned from teaching those classes the curriculum can vary widely from class to class, school to school and state to state.

Indiana, my home state, has created a list of standards for us to follow. The Algebra 1 standards are the governing list of concepts for me to cover in an Algebra 1 class. And I use them as such. However, even with those as a guide there are many choices to make on the depth and breadth that each topic is covered. For instance, I am currently teaching basic trigonometry concepts to a high school Geometry class. The class is what FCHS refers to as a "lab" class, which means the students background puts them in a category of needing extra help to pass Geometry. Thus I am left to ask the question of how in depth do I teach the concepts to them. The Indiana Geometry standards state:

G5.4 Define and use the trigonometric functions (sine, cosine, tangent, cotangent, cosecant, secant) in terms of angles of right triangles.

G5.5 Know and use the relationship sin2(x)+cos2(x) = 1.

With this in mind I look at the two days it has taken the students to begin to grasp just the basic 3 trigonometric functions and I wonder if I can possibly cover both of those. Especially in light of the fact that I still need to cover quadrilaterals and circles before the end of the year(under 8 weeks left). I have more to say on this, and will at a later time, but I would love to hear any opinions that other teachers have on how they decide what they are going to cover.

Exponent Properties

I recently finished my squidoo lens on exponent properties. I have enjoyed the process of creating these single page sites. It has allowed me to accomplish two important things:


1) The actual building of the site is very simple and easy to do. I had to work a little bit to figure out the html need to create exponents, but overall the process of creating a website on squidoo is pretty painless.


2) Squidoo is well recognized by Google. Within the first week of publishing the exponents lens I am average 15+ visits a day from people searching on google. Considering how specific the topic is that is pretty cool. It is nice to know that people are finding and benefitting from the work I am doing. Yesterday nearly 100 people visited the four lenses I have on squidoo.


The lens I am creating currently is on simplifying square roots. It should be done some time next week.

MIMIO Demo's Part 2

This is the second video I have created showing the different functions of the MIMIO I use in my math classroom. The technology is truly amazing and I think has a very positive impact on the classroom. In this video you will learn to make videos yourself, write on top of other programs (word, powerpoint, internet explorer, sketchpad, etc...) and use the different backgrounds and pictures MIMIO provides. My favorite part so far has been the ease with which I can bring my own picture into the MIMIO. Not only is it easy to do, it is very easy to save the picture and access whenever you want. I use the x-y grids and number lines I created often. If you have any questions or want me to create a video on a specific function in MIMIO let me know.

MIMIO Demo's Part 1

As I said below, I was recently fortunate enough to have a MIMIO Interactive placed in my classroom. After less than two months I can't imagine teaching without it. You might ask why I like it so much. Here are just a few reasons.

• Not having to worry about dry erase markers and the mess they make


• Having multiple pages of work to go back to if necessary


• Being able to easily access coordinate grids and number lines


• Recording my work as I do a problem(or a video on how to use a MIMIO!)


• The straight line feature(I was never very good at drawing straight lines)


• Printing out notes for a student who has trouble concentrating during class


• There are many others reasons including some I probably haven't figured out yet

The following video shows some of the features of the MIMIO notebook. It is designed for someone who has never used a MIMIO before. The MIMIO is capable of much more and I will be adding videos in the coming weeks showing these things.

To MIMIO or not to MIMIO

One of the goals of this blog is to talk about some of the teaching practices I am using in my classroom. Currently the new exciting thing I am doing is using a MIMIO for all my direct instruction. The MIMIO is a new technology similar to a SMART board used in many classrooms around the country. The term that is used to describe both is "interactive white board."

Differences Between the Two

I have not used a SMART board so I know very little about them, but I believe they are functionally the same as a MIMIO. Meaning they can accomplish the same thing. The big difference is that the SMART technology is a white board in and of itself. The MIMIO is designed to be installed on an existing white board. Because of this the MIMIO requires much less hardward and thus imys much less expensive. As well the MIMIO is much more portable around a school than a SMART board. Conversely, I would imagine that the MIMIO is more likely to have technical problems. As well the MIMIO requires you to calibrate it before use. I would assume the SMART board would not require this. Whatever the case is, I don't have a preference or reccomendation between the two, but I can say that I have been 100% satisfied with the MIMIO so far.

An Example of a MIMIO

I have yet to create a video about a MIMIO. This was the best I could find on youtube:




I plan on creating some videos showing how I do different things on MIMIO. I plan on posting a series of MIMIO tutorials in the next couple of months.

I will end this post with another video I have created for the tutorials I am creating. They can be accessed in the top right corner of this blog.