Monthly Archives: July 2010

Multiplying Monomials

Check out the file below for instructional notes for teaching the multiplication of monomials. These notes include the multiplication of like bases, the application of scientific notation and the concept of a power raised to a power. These notes are designed to enhance the understanding of the algebraic rules students often times confuse. The central theme should be to emphasize the meaning of a power over and over with students instead of restating algebraic versions of rules. I hope this helps.



Algebra Alien

As I’m cleaning up the last 5 years worth of files from two computers and two thumb drives, I want to share a really fun end-of-the-year activity that was a HUGE hit! I have included the directions/rubric and student sample work.

The premise of this activity is that students research their own notebook for the BEST sample problem that meets the criteria to showcase on their alien. The alien is of their own creation and depicts the scope and depth of the algebra learned throughout the year. Students are allowed to get as creative as they wish. When all students have competed their alien, have them post them all around the room and have the class do a walking gallery. With post-it notes and pencil, have students examine the difficulty level of the problems on their peer’s alien and make comments. Students can bring these comments back to their desk to share out with the whole class or they can post them beside the alien. This provides yet another type of review of the content for the students. You’ll be amazed at your student’s creativity, as well as how on-mark they will be with showcasing the BEST problems from the year.

 Directions and Rubric  algebra-alien


First Days of School

I have here another version of a powerpoint that I have used the first couple of days of school to share with my students my classroom rules and expectations. With this presentation, I have students complete a “Get to Know You” activity.


Number Sense Foldable

In my last post here I left discussing the importance of creating strong number sense skills for students. I provided instructional notes and a foldable example. I have attached below a Smart Notebook file that walks through creating this with students. If you do not have this software, you can save this file and then open it using the Smart Notebook Express Web Application. This is a free viewer/editor and is easy to use. The Number Sense file that is attached here is posted to the Smart Exchange site that is designed for educators to share files.

Smart Notebook Express Web Application

Number Sense (Notebook File)

Solving Equations – Do/Undo Method

I dearly love to teach solving equations. I could spend the entire year on just this one concept. Wait a minute…isn’t solving equations, of any type, 80-90% of the algebra curriculum already? Well, that might explain my love of algebra.

 I do love solving equations and I look forward to teaching this at the beginning of the school year. An algebra student typically comes in with equation solving knowledge. However, we have all experienced students in our classrooms who had not mastered the prerequisite desired of a beginning algebra student. In this segment I want to talk about the ways that we can vary the instruction in the beginning so that students master the concept of solving equations, even if they did not come to you class with the desired prior knowledge. 

 There are many methods of teaching equation solving, including buying expensive manipulative materials such as equation boards and balance beams. I personally have not experienced any product out there that does a comprehensive job of teaching students to solve equations. That’s not to say that I do not believe in using manipulatives during the instruction process. Everything that you’d need to teach students to solve equations is within your classroom or school. Balance beams from your science teachers may be helpful in addition to white boards/laminated papers, markers/color pencils and two color counters or even simple object such as textbooks and binders. Using manipulatives will allow your tactile and visual learners to grasp what is going on during the solving process. Lastly, I cannot discount the wonderful websites out there that have applets that provide students reinforcement of understanding the solving process. I will share those with you in a bit.

 I will start with your most challenging algebra student, those who do not come to your class with the desired equation solving ability. Start with a review of basic integer operations as well as the definition of subtraction. For this method it will be essential for students to recognize situations such as x – (-2) is really x + 2 and x + (-3) is x – 3. To evaluate an expression it is essential to follow the proper order of operations. To solve an equation is to undo the order of the operations that were done to the unknown value. To do this we must reverse the order of operations. While this method does not take care of all the types of equations students will be solving algebra, this will assist in students understand the concept of opposite operations while maintaining equality. It works nicely to use a “Do/Undo” table. When given an equation, students fill in the left side of a table that explains what has been done to the variable. Then they simply fill in the right hand side of how to undo the operations that were done initially. See the attached file for examples and explanations to this method.


This table is essential for students to be successful with solving equations. It breaks down the steps into smaller parts, more manageable for students who are not the strongest algebra student. As students master solving basic multistep equations, other strategies for solving equations can be introduced. The underlying theme in solving equations is that what is done to one side of the equation must be done on the other side as well. This will call for the use of the distributive property in many cases. I will look at more sophisticated methods of solutions in further posts. If there is one thing that I can leave you with on this post, I would stress to students that there is more than one way to solve an equation. Flexibility and strategy are key elements in solving equations.

 Online Resources

National Library of Virtual Manipulatives

A wonderful resource and an easy to use app; however, I’m not such a fan of “throwing away” blocks from both sides to show division. Division to me is a regrouping. There’s no opportunity to show that regrouping aspect here.

 MathsNet: Interactive Algebra


            This is a great app for students to use at home when practicing. I find it very easy to use and following the natural solving process, even with the variability as to what step you take first. What I like most about this app is that it takes on the direction the student wishes to take. The one drawback is that there is an assumption that what is done to one side is done to the other side of the equals sign by simply typing in one command, such as +6x or ÷2. I’d personally like to see students enter this information under particular terms on each side of the equals sign to reinforce the properties of equality.

 Explore Learning


            This site has wonderful apps for students to manipulate across the pre-algebra curriculum, including the basic one and two step equations. They also have integer applets that can be utilized beforehand to sharpen those skills. The equation apps follow similar strategies and the integer practice problems. The drawback to this is that it costs. Check with your site technology person and see if your school has a site license for this product and if not, check into seeing what it would cost for both your science and math teachers to use with your students.

The Easy Way to Factor Trinomials

As a student back in the 9th grade, I remember my first encounter with this thing called F-O-I-L. We were learning how to multiply two binomials using this method. I seriously thought it was a joke and that my algebra teacher was playing a trick on us. I recall looking around at my peers in class, hoping someone would ask the first question, or better yet, burst out laughing like I was wanting to do. I did catch on. It was just a set of procedures. I could follow directions. The problem came when I needed to reverse this procedure and factor a trinomial to get back to the two binomials that were multiplied. I was ready to throw in the towel. I couldn’t make sense of the procedures. Actually, I wouldn’t even say it was a procedure per se, but rather a lot of steps to run steps to see which set of numbers did the trick. Apparently, I didn’t have the patience for trial and error. This was certainly a turning point for me as a student. I remember at asking “Why” for the first time in my math education experience.

 This brings me up to my first year teaching. I was very excited to get to teach algebra and I certainly felt prepared to do so with my wonderful experience as a student teacher. I remembered prepping for the lesson on factoring trinomials. My students had mastered multiplying two binomials as we recited, “First, outer, inner, last”, aloud in class. The first example, of course, didn’t go so smoothly. Lots of wrinkled foreheads! It was most telling by the fact that the fifth example received the same reaction as the first. I had hit a brick wall. I didn’t know any other way to teach factoring trinomials. Unfortunately I, as well as my students, suffered through several years of me teaching the trial and error method of factoring, or reverse FOIL as some refer to it as, before I found the easiest way to teach this concept. I did not come up with this way of teaching. In fact, since learning this method, I have found it in current textbooks. I first experienced this through a workshop at the 2009 South Carolina Council of Teachers of Mathematics conference in Columbia, South Carolina. Unfortunately, I do not recall the name of the presenter who facilitated this workshop.

 To set the wheels in motion for this easier factoring method, I begin with NOT teaching the FOIL method for multiplying two binomials. Instead I have students apply the distributive property twice, pulling each term of the first binomial through the second binomial. I take this opportunity to utilize the different pen tools of my Tablet PC when teaching this method. You can do the same by using different color chalk, whiteboard markers, or smartboard markers to get the same effect.

 I have included an instructional sheet that is quite comprehensive and explains this method of factoring. I not only provide worked out examples, but also guiding questions and tips in teaching factoring trinomials to students. I’m very proud of this piece and will certainly work it in to the book I’m currently writing. I look forward to your feedback on this method of factoring as well as on the explanation I have included in the attached document.

Instructional Piece FactoringTrinomialstheEasyWay

Word Document Version Factoring Trinomials the Easy Way

Teaching and Assessing Number Sense

It is important for algebra students to know the characteristics of the different sets of real numbers, including natural numbers, whole numbers, integers, rationals and irrationals. They should be able to determine the smallest subset the value belongs, as well as all subsequent sets. Students should understand and be able to interpret the Venn diagram of the real number system.

I have included two documents here. One is for instructional purposes and may even be used for student notes. The other is a test that I have used with algebra students. The test goes a bit further than the instructional piece that I have included in that I also assess students ability to evaluate and operate using real numbers. Students will need to be able to apply the order of operations and integer rules to evaluate algebraic expressions.

Instructional Piece on the Real Number System RealNumberSystem

Algebra Test NumberSenseTest 

Instructional Strategies:

  1. Have students create a foldable to record this information. See instruction sheet Instructional Strategy Number Sense Foldable
  2. Create a large Venn Diagram on the whiteboard, provide students with notecards with a real number, one at a time have them place their card in the appropriate spot within the Venn.
  3. Create a large number line within the classroom. Provide each student with a real number written on a notecard and a clothes pin. Have students place their card appropriately on thenumber line. Once everyone has  finished, have each child call out their number, what set(s) the number belongs and why they placed it where they did on the number line.

Additional Binder Item

Last time I spoke of how to help students stay organized through the use of a binder and folder system. In addition to the items discussed for the binder, I create one extra item over the summer months to give the students at the beginning of the school year. I take a basic sheet protector, usually found in boxes of 50 or 100, and attach a pair of sticky velcro circles between the two plastic sheets. It needs to be at the top and in the center of the sheet. The purpose of this item is for students to keep their foldable or other content frames created in class. The velcro keeps these items from falling out. Many of these items are difficult to punch holes through and store in the notes section of their binder. In addition, keeping them easily accessible within the sheet protector will allow students to use these items as they were intended, as well as keeping them in nice condition. You can have students store this in the front binder, in notes section or in the back of their binder, whichever you prefer.

Here are examples of these supply items you will need to create these for your students.

Sheet Protectors


Helping Students Stay Organized

The following are two organizational tools that I have students create for my class. One is an overall semester binder. The letter attached is what I provide to my students to place inside their binder as their first sheet. This explains how to divide the binder into 5 sections and what belongs in each section. It follows a typical Explicit/Direct Lesson Plan (EDI format) of Do Now (Warm Up), Class Notes, Assignments, Quizzes/Tests, and Vocabulary/Standardized Test Practice. It usually takes about the first nine weeks before students are able to go on auto pilot on this procedure.

Binder Composition

It’s important to emphasize throughout the transitions during instruction where students should be in their binder. For instance, at the beginning of class students should be in the front of their binders for their warm up, or do now, problems. However, during your instruction you’ll want students to record notes from the lesson, so they will need to be in their “Notes” section within their binder. If students choose to have loose leaf paper out of their binder to do these activities that would be fine. Be sure that students have ample time at the end of class to place these papers in the appropriate sections to insure the organizational structure is maintained.

Within the Assignment section of the binder, I provide students an Assignment Log. This too has been included here, but in two different formats. This has evolved over the years at the request of my students. Initially I had intended for students to utilize this as a means of creating a table of contents to what was stored in the assignment section of their binder. This included the assignment number, date, description and points earned. Over time, students requested that they be allowed to record their other grades for projects, quizzes, tests and the like. I approved and thus modified this to be the Grade Log Sheet. By doing this students can calculate their overall grade in the class faster since I have always done a point system. Students kept a running sum of points earned and points possible. With each grade they were able to see the changes in their overall average for the class. In addition, they were able to see the impact not completing an assignment did on their average in the class. Providing students almost instant access to their grade will instill a sense of ownership and motivation to achieve on the next assignment, quiz, test or project.

Assignment Sheet2


The second organizational tool I have my students create is an Assignment Folder. This does not house all of the student’s assignments. That is what the assignment section of their binder is for. This is simply to house only the assignments associated with the particular unit we are studying. This could be the assignments that are practice for the up and coming quiz or test. I have had students keep a simple three-prong folder as a means of making it easier for students to transport their study materials from school to home and back. Taking home a binder and textbook home almost daily is quite a challenge, especially if your school does not allow backpacks.


Within this folder should be all of their practice problem from their assignments, with the corrected solutions to the ones they missed. For every question they got incorrect they were responsible for re-working the problem show how to arrive at the correct solution, the one I gave them in class. This put the responsibility on the students to some of the error analysis work in understanding, and correcting, their mistakes. I also encouraged students to create 5-10 question self-quizzes daily from this folder as a means of keeping their skills sharp. They were allowed to keep current notes in the pocket of their folder to help them in their studying, including any instructional aids such as foldables or content maps. It is important to remind students and provide sufficient time after each unit to place these items in the appropriate places within their binder. It’s one thing to say to stay organized a certain way, but if reminders and time in class to do so is not provided, chances are the binders and folders will become an ultimate failure.

I would pick up their folders for a folder check the day of the test or quiz. I would assess their completion of their assignments and corrections and assign points accordingly. Each assignment ranged from 5-10 points depending on their complexity and each unit had 4-8 assignments, so this could equate to hefty points if not completed.

In the world of math if students do not practice a variety of problems within each unit, they will not be proficient. With that said, if there is no plan for you as a teacher to assess student work, students probably won’t do it. Therefore, it’s important to have a plan and a procedure in place for students to be aware of in advance with regard to a folder grade. You as a teacher will see on a daily basis who did their assignments and who did not. You may agree that so long as students have all of the work completed, checked and corrected by the time the folders are collected you will take full credit in terms of the points for their grade. Others may keep track on a daily basis and assign a check, check minus and 0 in terms of completion per student and make the necessary phone calls  home to inform parents. Other consequences may be to keep students in from any free time they may have, including lunch in order for their work to be completed. Morning or after school tutoring sessions may be required as well. It’s important to understand that not every child will complete every question on ever assignment. As the teacher you need to determine where your procedure will be if it is not done to your specification and what time limit, if any, will you allow for it to be completed. Many administrators will not allow you to fail a child based on not completing homework, if he/she is passing test and quizzes with good marks. The goal is to assess what students know and are able to do. Homework is simply the formative assessments that take a snap shot of what the students know at particular intervals.

Classroom Procedure Posters

*These posters were created in Microsoft Publisher, but were then converted to .pdf format for uploading purposes. I would have like to have kept them in their original format for your editing purposes, but this format did not support that. Luckily these can be used as is by most if you agree with the procedures for your classroom.

The following are the procedures that I have had in my class for over 8 years. With any type of rules or procedures they can be modified to suit the students that you teach and the school you are at. However, I found that these were the guidelines that I needed in my classroom in order to function as a teacher. To put it bluntly, these are the requirements I must have in place to make me happy as a teacher. With the exception of minor changes in design and wording, these have been my procedures throughout 8 years of teaching which span across 3 distinctively different school environments. Simply put, they work!.

I spend the first three days going over these procedures and having students act out what they look like, as well as what they don’t look like. I also spend every opportunity the first few months reiterating them as the occasion arises. I repeat this in a review format when we come back from winter break and again after spring break. I have taught 7th & 8th grade students. They are mature and intelligent individuals, but without the review of my classroom expectations, they have a tendency to try to “rule the roost” so to speak. This is very much in line with the philosophy of Harry Wong. Here are a few sites that will give you information on Harry Wong and his effective teaching methods:

I consider the following the top 5 classroom procedures that happen on a daily basis.






These procedures are need once and a while. I do go over these, but usually on day 2 or 3. Generally speaking you probably aren’t going to have an assembly the first day or two of school. If you do, your administration staff is just cruel. : ) In addition, incorporating these in with the major 5 procedures from above can be overload for your students.





Just a note:

I usually print these out on a color printer and offset each one on different color paper. I laminate them and place on a well-seen bulletin board in the classroom.

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