Tag Archives: instructional strategies

Quadratic Equation Card Sort

I’m so happy I follow @Maths_Masters on Twitter! Such wonderful resources! I must share this one with my algebra teachers.

Quadratic Equations – The Main Ideas.

Better yet, give students the Group 1 through 5 titles from the card sort first and see if they can generate facts for each of them while working in groups of 3 or 4 first. After they have exhausted their knowledge, provide them with the card sort to work out. Finally, have the students, still within their small groups, compare what they came up with with this card sort. While students may not be able to generate all of the facts on their own, nor be able to word their facts as succinctly, the ownership students will experience when they connect what they wrote to the facts on the cards will be immeasurable.

Thank you William Emeny at http://www.greatmathsteachingideas.com/ !

Student Independent Study Routine

Also as a pdf   Student Independent Study Routine

Math is just a different type of subject. It requires a unique balance between memorizing facts, following procedures, problem solving and reasoning. More often than not, these aspects intertwine no matter what age, grade or course. I want to focus on what students can do on their own to help with these skills when you’re not present, i.e. when working at home or in school with a substitute. Learning and practicing independently should be a routine for students. As a teacher, you must teach them how to do this effectively. It’s important to having clear, defined actions that students can practice when you are present, so that they will understand what to do when you’re not. Here are a few suggestions that you can modify to fit your current classroom note taking structure. If you do not have a current note taking structure, maybe this will be motivation to create one. See Helping Students Stay Organized for ideas.

The course binder should be thought of as a record of learning and achievement, as well as information pertaining to the content. Chances are students have made mistakes along the way on assignments or assessments, or have had misconceptions at one point or another. Often teachers get into a routine of teaching, testing, teaching, and more testing. There’s no time set aside for reviewing, reflecting, or reassessing. Reviewing only seems to happen when a subset of skills are passed along to the next chapter. Reassessing may happen on 20-30% of the concepts as a residual effect of the natural sequence of learning mathematics, but never in a capacity that would allow teachers to really determine if true learning has occurred. Lastly, students rarely have the opportunity, or take the opportunity, to reflect on their personal learning and growth on a concept, chapter or unit. This is the area in which students need a great deal of guidance in order to gain the most out of their education.

While in a perfect world it would be wonderful to conference with each and every child after an assessment, to explain, reteach and reassess, or simply have a conversation with the child to gain an understanding of their thought process on particular questions, it’s impossible to do so. It’s imperative to teach students how to assess their own learning, investigate on their own using the tools you have provided them, as well as using outside resources like parents, siblings, and online resources. Before this can occur, you must teach students to value their learning, and essentially fight for understanding in cases when it is not been reached.  It’s only through this desire for understanding that students will be motivated to learn from their mistakes.

When a student takes an assessment of any kind, quick checks, daily assignments, formative assessments like chapter tests, or summative assessments like semester or end of year exams, it’s a snapshot of what the student knows at that point in time. Results are usually recorded in the fashion of points or percentages. Regardless of the type of assessment, for most students 100% performance has not been achieved. There is always room for further understanding. This understanding will only come through review, relearning, and reflecting.

To start this process, students must do the following:

  1. Have students individually review their assignment/quiz/test, absorb the results, and record in writing their initial feelings and frustrations, as well as identified mistakes or misconceptions. Often times students want to seek your immediate assistance in understanding why they lost points or missed a problem. Teach students to do this step individually first for maximum benefits. As a side note, while I feel strongly about writing across the curriculum, this is not the place for students to feel added pressure and stress with perfecting their spelling, grammar and sentence structure. Allow for lists of comments, ideas or questions to get this aspect of the process accomplished.
  2. Have students to ask their questions pertaining to the particular problems they did not understand and had points deducted. First give other students in class the opportunity to respond to these questions. This is a valuable tool to use in this instance. Allow the student who just reached an understanding of his/her own mistake explain another student’s misconception. Wow! How powerful is that? Not only is the student who asked the question, getting his/her question answered, but the student who just had a major revelation is given the opportunity to verbalize his/her understanding. Relying on students who initially understood the problem to provide an explanation is just as effective. Of course, if this opportunity does not arise, you can certainly take the role of explaining the problem or concept to the class yourself.
  3. After all questions have been answered, students are to formally correct all of the partially or completely missed problems on a new sheet of paper. These corrections would have the problem, including word problems, and problems with graphs and geometric art, and the correct process to achieve the solution. The student should be able to look back at this problem on this formal correction sheet and have all the necessary information to answer it completely.
  4. Below the formal rework of each problem missed, have students summarize their misconception, mistake, or new-found knowledge. Students may want to read what they wrote on Step 1 to reflect on their learning up to this point. They may speak of a simple calculation error, misuse of a formula, an incomplete process, or a total lack of understanding at the time of the assignment/quiz/test. Remind them that this is a form of documentation of their learning. They need to write clearly and completely. This piece is more formal than was done during Step 1. Students should be encouraged to pose further questions in order for them to gain deeper understanding of that particular concept in that problem. These new questions can be brought out in class discussions or one-on-one with the teacher. Flagging these questions with a post-it note may emphasize to students to follow up with the teacher or class.

Understand this is difficult to do on every assignment and task students complete. Targeting a subset of problems, either at the beginning of a unit after base knowledge has been taught, or at the end of a chapter for an overall student evaluation would be more manageable and possibly more effective.

I hesitate to recommend this type of student reflection be tied to a grade of sorts. While students are somewhat motivated to complete assignments because there is a grade attached to it, the purpose of this is to have students examine what it is they know and what information they misunderstood or lacked. This is a discipline. Students have to experience success from of this type of reflection before they buy into the regimen. Teachers should providing non-evaluative, mathematical feedback that is positive and encourages a dialog between student and teacher. If students feel as though this is just another hoop to jump through to get a grade, it becomes a chore and fails as an effective process.

Parent Communication

Year after year, after year, there always seems to be more to do at the beginning of the school year than any human can possibly accomplish. I would like to emphasize one, and only one thing, for the first  3 -4 weeks to accomplish. This would be parent communication. Please make at least one contact home, verbally speaking with a parent or guardian, within the first 3-4 weeks of school. This is a valuable opportunity to make a connection to home; to put your best foot forward; to stress the passion you have for educating their child or student. If at all possible, I would recommend moving those students that are already pushing your buttons to the front of that call line; however, I would only say nice things.

I know…I know….there are some students that are really showing their true colors in the first week of school. Keep in mind…these students are instituting their best defenses during this period. This may just be a front to protect their integrity, their ego, and their pride. These students may be trying to make a place for themselves in the only society they know…school. Let them. However, make a connection home to let their parent(s) or guardians know that you have their best interests in mind. Let them know that you will work with them to insure that “Johnny” or “Susie” have the best year ever. Let them know that the communication lines are always open, in both directions. It’s amazing what positive gains will occur with a simple, casual conversation.

With that said, I would suggest a cheat-sheet of sorts to use when you call parents/guardians the first time. Try to be consistent with the information you give to all parents. As we all know, parents do talk with one another. You’re laughing, but mark my words…. say X, Y and Z to parent A, and G, H, I to parent B, parent A and B talk, …you know what will come your way.

Don’t be afraid to encourage parents/guardians to come into your class for a class period. Some will take you up on it and others will not. When they do, don’t put on a show. Be yourself. Keep your normal routines. Maintain the same expectations. This will paint a real picture for them. For the ones that take you up on this offer, be sure to continue communications home throughout the year, good and bad. Remember, it takes a village to raise a child. With that said, I would also keep a record sheet for each student to record the parental contact you have made and the reasons for doing so, including returning phone calls from parents. Come April or May, this will prove to be a value resource for those students that have been “high-flyers” all year.

The key is communication home. I certainly do not want to disillusion you into thinking that every parent or guardian will be responsive to you making calls home. Some will and some won’t. The key is to do what you should on your end….and that’s making sure that the line is open. Sometimes, it’s just one-sided.

Best of luck!

#1 Teacher’s Resource

I am, and will forever be, a lover of learning. I was the first of my family to get my bachelor degree, and continued on to get my masters. This of course came with a hefty price tag, but completely worth it. My master of mathematics education program was entirely online and I enjoyed it thoroughly. I embrace the online world as a means of learning and exchanging of knowledge. Many students today have not known of a time without computers; therefore, learning online is just as natural as learning via textbooks and videos of previous generations.

When my friend introduced me to Khan Academy, the learner inside of me got very excited. Free! Online! Learning! Thousands of instructional videos that allow me to review previous learned topics or to branch out and learn something completely off the charts, like Galvanic Cells in chemistry. Math.  Science. Humanities. Test Prep. It’s all just a click away. Oh, and Mr. Gates himself is part of this amazing project.

This blog is about me sharing information, strategies, lesson plans, and resources with teachers across all subjects, but particularly mathematics. So, what are the implications for the classroom with regard to Khan Academy? Let me list the ways…

  1. Teacher learning/reviewing resource – Teachers can see instructions and examples of content they will teach their students. This is very helpful for cases where teachers are required to teach a grade or a subject they have not taught before.
  2. Student learning/reviewing resource – Students can access content or lessons they feel they need assistance in mastering. This could be on the whim of the student, or as the teacher, you could assign students to review particular videos and content for purposes of understanding material taught in the classroom.
  3. Parent learning/reviewing resource – Parents can learn in along with their child. Parents usually want to help their child understand all of their subjects, but feel too many years have passed to do so effectively. Directing them to particular videos pertaining to material being taught in the classroom will provide them an opportunity to learn along with their child. Moreover, parents can benefit from this resource to broaden their own knowledge base.
Since most school systems and teachers utilize websites to communicate with students and parents, as well as fellow teachers, it would be very beneficial to create links to these videos on a weekly or unit basis. Doing so in an organized fashion according to your curriculum’s pacing guide will provide this “one-stop-shopping” experience for students, parents and teachers alike.
I truly hope you find this as exciting as I do. Work with your peer/subject teachers to organize this information into a form that works for you, and all those involved.

Maintaining Momentum

This is the hardest time of the school year! The time between Thanksgiving and the winter break/holidays are so stressful. If you’re like most, you’re behind in your curriculum pacing and you need to start preparing your students for semester exams. This coupled with all of the interruptions this time of year brings, like plays, performances and celebrations, albeit as nice as they are, you’re struggling to keep the focus on learning. I’ve been there!

This is just a small list of ideas to help keep the pace on track and help you make it to the end of the semester. Please feel free to add any ideas and suggestions that may have worked for you in the past.

  • This is that time of year where working in groups may not feel like it’s a good idea, but you might want to incorporate some into you lessons/activities. Students are social beings. They have a ton of built up energy and they need to release it. Forcing students to work independently, because their behavior has not warranted otherwise, may be a recipe for disaster and may increase the number of classroom interruptions. Allowing students to work in 2’s, or 3’s if necessary, may help students release some social energy while working on an assignment/project. Be sure the activity students are working on is relevant and meaningful; otherwise, students may become a major disruption.
  • Think hands-on! Allow students to participate in kinesthetic activities that involve color and different textures. Allow them to be creative in displaying their understanding of the topic being learned. This is not to say let students “color” their assignment, but if students can have some creative license in doing their work, they will appreciate the freedom. Most of all, display student work! Showing off their successes will motivate them more than you know.
  • Allow your stronger students act as facilitators in small group activities while you work with students who may need some remediation prior to the semester exam. This should be an activity that takes minimal instruction and revolves around review material. The students in the class should be able to work independently on an activity while you work with individual students, but you have assigned students to use as instructors when students in class have questions. Be careful how you advertise this. I have picked students based on grade and attitude, but announced to the class that these are the “go-to” people for their section. Of course, clear classroom rules and procedures MUST be in place prior to this type of activity.
  • Open up time for tutoring sessions, either in the morning, during a “free period”, or in the afternoon.  While providing incentives for attending tutoring sessions is inappropriate, because getting to tutoring may not be feasible for all students, students will feel rewarded getting help in a smaller setting. This may not feel like it would impact the classroom instruction and help with pacing, but it does in a huge way. If you can get those 2-5 students caught up during tutoring, they will be able to follow you during the remaining instructional periods. They will appreciate understanding what they did not know before, and be motivated to keep up with you in class. Draw them in during class to answer questions or to summarize a concept if you feel their ready. This is the time to reel them back into the class!

I hope this sparks some ideas of things you can do in your classroom. Please share!

Being Strategic in Solving Equations

This post is also available as the following pdf Being Strategic in Solving Equations

In a previous post under Solving Equations, I shared with you the Do/Undo Method for basic equations. In this post, I’d like to explore more advanced equation that have a variety of starting points for solving. It’s important to not require students to be so routine and procedural when solving multistep equations. I do say that with a little hesitation, because we’ve all had those students, usually not the strongest mathematics students, who thrive on procedures, steps, and routines. It is perfectly fine for students to utilize a procedure or follow steps, so long as they understand the purpose behind them. This involves students having flexibility in number sense and solving basic equations.

Here is an interesting opening problem/demonstration that you may want to use prior to starting a unit on solving equations with the variable on both sides of the equals.

      Start with a basic two step equation, say 3x + 5 = 8. Inform students to simply watch and evaluate your mathematical reasoning. Don’t allow for students to call out any observations until you give the signal. Begin by subtracting 3x from both sides to end up with 5 = 8 – 3x. Then subtract 5 from both sides to get 0 = 3 – 3x. Then subtract 3 from both sides to get -3 = -3x. Finally, divide both sides by -3 to get x = 1. You could keep this going for some time if you’d like. However, the purpose of this demonstration is to show students that so long as they are mathematically correct in their solving process and in their integer operations, they are free to take as many steps as they’d like to determine the value of their variable. I’ve often called it “Taking the Scenic Route” to my students, which is a polite way of saying, “You didn’t have to do that many steps to solve the problem”. It’s okay to take the scenic route once and a while. In fact, there are things that can be learned by doing so; however, it’s not the most efficient way to solve a problem.

Let’s look at a multistep equation with the variable on both sides.

3x – 7 = 8 – 2x

At quick glance, there’s four ways to begin this problem without the use of the commutative property. I personally like to get one representation of the variable as my first step, and then proceed with solving the two step equation. I teach this method first so that students see the connection to the previously learned material. Before I combine my variable terms, I need to decide which term to move and what I desire to be the value of the combined terms. If I remove the 3x , I must subtract 3x  from both sides of the equation, producing a -5x on the right-hand side.

 3x – 7 = 8 – 2x

– 3x              – 3x

        – 7 = 8 – 5x                                                                 

If I remove the  -2x, negative 2x through the definition of subtraction, I must add 2x to both side of the equation, producing a 5x on the left-hand side.

    3x – 7 = 8 – 2x

+ 2x               + 2x

    5x – 7 = 8                                                                  

Notice the differences between these two resulting equations. Each term in the second equation is simply the opposite of each of the terms in the first equation. This is a characteristic that you can share with your students as you work out problems. As these equations become more and more complex, and the number of methods of solution increase, you need to give students the tools to assess their own understanding. If not, you will find yourself doing the same problem over and over again, wasting instruction time.

Now let’s compare these two equations on another level. As you’re illustrating to students how to solve an equation like this, it’s important to engage students in this type of conversation and planning before jumping in and doing steps procedurally. Generally speaking, students are more likely to have success in solving equations when negative values are minimal. The more students have to keep up with negative signs as they solve an equation, the more likely they will make a mistake. This is not researched based, although I believe the research is there to be found, but rather this is my 10 years of classroom experience talking. Moreover, if students understand they have a choice in what term they move, and the ramifications of that move, then students are more likely to choose the step that results in the positive value. With that said, in many cases it is impossible to combine variable terms and avoid negative results, simply by the nature of integer addition and subtraction rules.

When solving these multistep equations during instruction or guided practice with students, good questions to pose are the following:

  • “What are your options?”
  • “Do you want to combine your variable terms first or your constant terms?”
  • “What will happen if you subtract/add ____ from both sides of the equals?”
  • “If you do that first step, do you end up with a positive or a negative term?”

Repeatedly asking students these questions throughout instruction will actually cause students to start to internalize them. Soon they will be asking themselves these questions as they work independently. You are simply providing them the research tools they need to investigate future problems for themselves. This is a good thing.

I want to caution you on one thing. I would stay away from creating the habit of ending up with the variable term always on the left side, or the right side. Many misconceptions can be developed from doing so, and can have a negative impact on student performance in later topics and in later courses.

I’m going to stop here for now and let you digest these ideas. I will come back another time and talk about different scenarios that provide us a bit more strategic opportunity. Stay tuned!

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

4-Corners Linear Equation Poster


This is a culminating activity that ties together students finding the slope from a graph, from two points and from an equation. This activity can be differentiated based on the level of students you teach. I used this for an honors algebra class, but varied the equation based on the level of the student. I gave slope intercept form of equations to struggling students and equations that were in no particular form to students who needed a challenge, so they must arrange into either standard form or slope intercept form. Once students have their equation, they must complete all four sections of this poster, of which can be done nicely on a 11 by 14 sheet of white paper. They can start in any section they wish, but they must find the necessary information to graph the equation of the line. This can be done first by creating a table of values for x and evaluating the equation to find the corresponding y-values and plotting these points. It can also be done by solving the equation for y, if not done so already, to create the slope intercept form. From this students can identify the slope and the y-intercept and graph the line. Finally, students can determine the x and y-intercepts from the equation and plot these two points to graph the line. Moreover, students can calculate the slope between these two intercepts to verify the value of the slope of the line.

What students should take away from this activity is that they have multiple ways to access the information they need to graph the equation of the line, regardless what they are given. As a teacher, you want to teach students flexibility in their solving methods and to be able to come at a problem from many different angles. To increase this flexibility you can have students do this activity again without giving them the equation, but rather one of the four corners of the poster. For instance, provide them with only the slope and the y-intercept and have them find the equation of the line, in multiple forms if you desire, the graph of the line, the x and y-intercepts and the table. Students will find it more difficult to go from just the slope and the y-intercept to a table, but if they understand what slope is they will quickly figure out what needs to be done. If the slope is 2/3 and the y-intercept is (-2, 4), then they have a starting point to work from, increasing the y-values by 2 and the x-values by 3. Additionally, students should also see they can decrease the y-values by 2 while also decreasing the x-values by 3 to create new order pairs for their table. This is the flexibility that is essential for students to fully understand working with linear equations.

Summertime Planning

It’s July 1st and half of summer break is just about over. Depressing thought for many, but also a stressful time for math teachers that may have a new grade level to teach or a new curriculum to implement this upcoming school year. Summer is a time for us to regain our sense of saneness, but it’s also a time for us to get organized for the next round of students. This may mean making changes in lesson plan format, classroom rules and procedures and activities/projects you want your students to complete. Maybe you’re cleaning out those files and have run across a really cool resource or activity that you forgot about or you’re revamping your organizational method for keeping such papers/resources. Share ideas on what you’re doing to prepare for this next school year.

Sharing What’s Worked

I will be sharing with you some of the rules and procedures that have worked throughout the three distinctly different schools over the last 10 years. My plan is to upload documents in .pdf format. Stay tuned to gain a jumpstart on next school year!

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