Some New Videos

There are certain topics that really interest me when teaching different math classes.  Recently my Algebra 2 class was working on some absolute value equations and inequalities and I made a couple of videos on the topic.  I thoroughly enjoy teaching the students this topic. The first video explains how to turn an absolute value equation into a piece-wise function:

The second and third video are a couple of examples.  The examples are $$|x+5|-|x-2|=8$$ and $$|3x-2|+2x=8.$$  There are certainly other methods for solving these problems, but creating a piece-wise function is effective and also useful when trying to graph functions of a similar nature.

Example 1

Example 2

Google Does It Again

Google has never made sense to me. It seems like such a paradoxical organization. Clearly they are generating vast amounts of revenue, but it seems like all of their best products are completely free. Adding to that list is the ability graph functions through a simple Google search. Follow the link for a demonstration (sorry for the sarcasm):

Graph of the Sine Function

You can change the scale easily and it has an extremely fine trace function. Now, they just need to design a way to find roots and points of intersection and it would be really amazing! You can also graph multiple functions at once:

Multiple Graphs

It is against my nature to think a company the size of Google could possibly be "good", but man they are generous with their technology.

Power Point Who?

I have never really been a fan of power point. I used it at times, but always felt like it should be able to do more than it did and that it should be easier make dynamic than it was. At the beginning of this school year, our head of middle school gave the staff a presentation. I was blown away. For the past four months I have been under the false impression that what she did took a lot of skill and hard work. Actually, though she utilized it well, all it really took was a web address. Prezi

Saturday, I finally took the time to look at the website, thinking it would take a big block of time like Christmas break for me to become competent using Prezi. Nope, I spent an hour playing around and learning the software. Maybe two hours later, I was able to produce the following video. IMO this is the best video I have created:



You can also view the Prezi itself:

Quadrilaterals

What do you think?

UPDATE: Looks like I can embed it as well.

To Knol or Not to Knol

I have recently forayed into a new avenue for putting my material on the web. I have decided to give Google's version of Squidoo a try. If you have not heard about it before they are referred to as "Knol". I believe in Google language Knol = Unit of Knowledge. In general, the Knol platform seems more inclined towards academics and might be a better platform for what I have been doing. Here are my first two on Geometry Proofs. I am still adding to both of them.

Knol on Angle Theorems

Knol on Example Proofs

Happy Teaching/Learning!

Two Examples of Proofs

I uploaded two more videos to YouTube today. They are both examples of how to use angle theorems to prove things. Here they are:

Four More Geometry Proofs

I completed 4 more Geometry proofs. My plan is to prove most of the theorems I go through with my Geometry class for YouTube and then I will also be adding some harder proofs as well. Tell me what you think and if there are any proofs you would like to me to add.

Alternate Interior Angle Theorem



Same-Side Interior Angle Theorem



Triangle Angle Theorem



Remote Interior Angle Theorem

New Squidoo Lenses Published

I have just finished two new Squidoo lenses. Squidoo is a pretty cool site that allows anyone to create a webpage about any topic. Check out my lenses and feel free to give it a thumbs up if you like what you see.

Writing Linear Equations

Graphing Linear Equations

Geometry Videos

Hello! Been awhile since I have posted, but I plan on sharing some of what I have been doing with my online videos and websites. Essentially, my new job has been wonderful, but also time consuming. Finally, I am to the point that I can start uploading new videos in an organized way and finish some of the work I started with the Algebra videos and website. My current goal is to focus my new video creation on Geometry videos, filling in Algebra videos when I have time and a need for my websites. Here are two Geometry videos I created today!

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?