Category Archives: Solving Equations

This category would contain all types of equations from simple one and two step all the way through solving quadratic and radical equations.

Solving Quadratic Equations

Here I go again…talking equations. Must be the Libra in me!

As a follow-up to my last post, Simplifying Radical Expressions, this instructional piece transitions to quadratic equations in which simplifying radicals is necessary. Solving quadratic equations is the precursor to graphing quadratic equations and studying the nature of parabolas. This unit hinges on students bringing all of their knowledge of solving equations and factoring, while also acting as a springboard to understanding everything there is to know about parabolas.

The instructional piece below is intended to help you teach your students to solve quadratic equations. Inside the Word document are links to previously posted material that you may find helpful. The pdf version is there in case you’re unable to open the Word file. As always, I look forward to your comments, questions and suggestions. I also encourage you to post instructional strategies you have found successful in your classroom.

Solving Quadratic Equations Word File

Solving Quadratic Equations Pdf

Revised File

There was an error on the second page. The value of x on both of the equations should have been 4. This has been corrected and the new file has been attached.

Being Strategic in Solving Equations PartII

I want to thank my new online editor and dear friend for finding this. : )

Being Strategic in Solving Equations Part II

I want to thank you for taking time to visit. I see that the number of visitors to my blog increase substantially through out the last few months. I hope this is a sign that what I have been posting is relevant to you as an educator. Regardless, it has been a motivating factor for me, in that I want to stay current on the concepts you are teaching your students. Please feel free to post questions and suggestions. I would like to design my posts around your needs as educators.

In this post, I continue our discussion of being strategic and flexible in solving multistep equations. I’ve really enjoyed writing this piece. I have attached it as a pdf due to formatting issues. Math type does not format well when I copy and past.

ENJOY!

Being Strategic in Solving Equations PartII

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!

Error Analysis

Interestingly enough, as a teacher you’re concerned with students getting correct answers when it comes to math problems. However, a student’s wrong answer can tell you a great deal about what they know and understand.

 Have you ever graded a test, multiple choice or single answer, and found a large portion of students submitted the same wrong answer? If it’s a multiple choice question, hopefully all choices were designed with common student errors in mind. By doing so, you create the opportunity to group students post-test into groups according to the mistake they made.

 This next instructional strategy I’m about to explain is not my own. In fact, I’m not certain who initially came up with this idea, but it is one that I, and fellow teacher s, have used and found very informative. What you do before and after this strategy is up to you as the teacher. This idea simply gives you an idea of how to collect data about what students know and understand.

 Let’s start with a simple concept like solving one-step equations. Provide students with a questions such as x – (-3) = -7. We know the answer to be -10, because subtracting 3 from both sides is the correct step in isolating the variable for this problem. We see this equation to be x + 3 = -7 by use of the definition of subtraction that states a – b = a + (-b). Only after we “clean up” the question are we able to see what operation we need to undo during the solving process. This particular problem is a quite difficult for 7th and 8th grade students who are just learning how to solve equations. Common mistakes are to add 3 to both sides, producing a -4 solution. Other mistakes are to correctly perform the operation to solve only to apply integer operations incorrectly. If students did indeed subtract 3 from both sides, they could come up with -4 or 10 as incorrect solutions. If we play the role of the student when designing question, we can perform error analysis during classroom instruction.

 Using this particular example, we arrived at solutions of -10, -4, and 10. The value of 4 could be justified as an incorrect response if students did not apply integer rules correctly when solving the simpler, yet equivalent, equation x + 3 = -7. Provide students the original question, x – (-3) = -7, and have them solve this equation and show their work. Have the potential answers of -10, 10, -4, and 4 posted around the room inconspicuously. Once students have found an answer, and before you have disclosed the correct solution, have students get up and stand by their response. Give them one or two minutes to discuss as a group how they found their common answer. Next, have one member from each answer group explain to the whole class how they arrived at their answer. After all students have listened to all of the groups’ responses, they must then decide to agree with their original group, or decide to join a new group. They must provide an explanation to why they are choosing to join a new group by stating the mistake they made in their solving process that arrived at a new answer.

 This can be applied to any multistep problem, on in which common misconceptions or error may occur. There may be anywhere from 2 to 5 possible responses students may have, depending on the error they made. Do your best not to make any judgment calls prior to students forming groups and making their final decision on an answer based on each group’s explanation. This provides students the opportunity to evaluate their steps and problem solving process against others in the class. This is a level of analyzing that most of the time does not happen when we, as teachers, simply mark a student’s paper for incorrect responses. Once students become accustomed to this type of discussion and error analysis, they will become leaders in their own education.

 I will caution you on stopping the discussion at disclosing the correct solution for the problem. I encourage you to provide students the reason as to how they could have arrived at the incorrect solutions where predetermined.  You will find students with incorrect answers that do not fall into the anticipated incorrect responses. Take this opportunity to allow the student to perform their own error analysis on their work and have them explain their mistake.

 This is a powerful tool that creates analytical students and better test takers. Of course, it’s important for students to understand how to arrive at correct solutions for problems. However, students can gain a great deal more information from learning what they did incorrectly. It’s important for students to know when they got a question incorrect, but to maximize learning; students must understand how they got a question incorrect.

Please share ways in which you have utilized, or will utilize, this stategy.

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.

 SolvingEquationsDoUndoMethod

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

http://nlvm.usu.edu/en/nav/frames_asid_201_g_4_t_2.html?open=instructions

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

            http://www.mathsnet.net/algebra/balance.html

            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

            http://www.explorelearning.com/

            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.

Solving Equations Ideas…Coming Soon!

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