Solving Two Step Equations with Binomials in Numerator
Lesson 18 of 23
Objective: SWBAT determine that a fraction bar acts as a grouping symbol and use this to solve two step equations.
Students enter silently according to the Daily Entrance Routine. Their Do Now assignments are already at their seats. Results from last week’s quiz showed students need to continue improving when simplifying expressions especially when negative coefficients and variable terms are involved. Question 1 exists in the Do Now to check in on student progress when simplifying variable expressions. This type of problem, where a negative number has to be distributed and there is another term in front of it, is especially challenging to students. This video reviews my questioning strategy when reviewing this type of problem.
For question 2 I make sure to review two ways to solve this problem. This second video shows how I use one of the methods to help some students feel comfortable with the idea that the fraction bar represents division as an operation. Once I help struggling students clearing the denominator and breaking this problem into two steps, I walk away and allow them to complete the multiplication step and the final division step on their own.
The third question in the Do Now serves to spiral through a difficult concept students need to continue practicing. It is clear by now that my students this year struggle immensely with fractions and often will skip or freeze when given a problem including fractions. To alleviate this anxiety, I throw as many fraction problems at them as possible to try and get them comfortable with idea that they are not going away. This third video models the way I would review the solution with students. I often help students half of the way through a problem and then leave the rest up to them to finish on their own or explain to the class.
Most likely we will not have time to review the last question whole class, but I do make sure to make time to review answers with students who get to this question. I also offer extra achievement points to students who complete the problem showing correct work by the end of class.
After reviewing questions 1 – 3 in the Do Now I distribute class notes and ask students to put away their Do Nows into the correct section of their binders. Red fonts included in the class notes resource indicate pieces that I write on the board and students are responsible for copying. We always begin with the aim. Then I have a student read it out loud. We begin notes today with a 6 minute activity where students have to find a partner and attempt to solve the equation included in the notes. As students are working I walk around to gauge understanding and pick a pair of students to display their work on the board. These students will be responsible for explaining their through process and for justifying each step in their solution. The pair that I select must be able to defend their answers with mathematical justifications for each step. Their answer does not have to be completely correct, but there must be some correct logic. As a class we fix mistakes and arrive at the correct solution. The pair select must identify multiplying by three as a first step. The following are some guiding questions I may ask as I prepare the pair to present their solution:
- What did you do first? Why? What did you do next?
- How are you making sure the equation stays balanced?
- Is it wrong to “subtract by 6” as a first step? If not, what would be the correct way to do so?
I make sure to review the idea of “getting rid” of a denominator (one of the steps to be written in the notes). By reviewing the idea that we want the variable alone, and therefore with a coefficient of positive 1, I am making sure that students are not making the common mistake of crossing numbers off left and right sides of the equation without mathematical justification. Once the steps have been reviewed, I used them as guiding questions to complete the first problem in the “Task – With Neighbors” worksheet.
Students work with neighbors for 6-7 minutes solving 4 equations and using the steps in the notes. If a pair of students calls me over, I enforce the use of notes by asking, “what is the first step? Look at your notes…” or “what do you need to ‘get rid’ of first? How are you ‘getting rid’ of it?” I am constantly circulating during this time to ensure students are using the vocabulary coefficient, opposite operation, denominator, and numerator. As students work to solve each equation they are in use of MP3 whenever they are asking their partner (or me) to justify a step as well as MP8 while they practice solving using the same pattern of steps.
After 6 – 7 minutes of paired work, students must work independently for at least 10 minutes to complete the remainder of the problems. At this time I circulate less and instead target groups of students to sit with as I remind them of the steps. Each time I make sure NOT to give students answers and instead encourage them to check their work and watch out for mistakes.
Once there are 10 minutes left of class, I write all Task answers on the chalkboard and give students 3 minutes to check them. Then I distribute homework and a half sheet of blank paper. I ask students to write their name on this paper and create a three column chart. In the first column I would like them to write the Task Problem numbers they got correct and to include a note about something they learned today. In the middle section I ask them to write the problem numbers they got incorrect, but understand what mistake was made. I also ask them to include a small note that describes one error they made. In the last column I ask them to copy the problem numbers that baffled them and include one or two questions that describe an idea that most confused them today. I use these sheets to build do nows and other homework assignments so that I can continue to help struggling students understand these concepts. One question I imagine will be asked often is, “why do we have to multiply first to clear the denominator”? The answer for this question is shared during a Do Now sample tomorrow as, “you do not HAVE TO multiply first. When solving equations there are many different steps one could take first. Some are more complicated than others. Therefore, we have to make choices that simplify the problem rather than making it more complicated. By multiplying by the denominator we are able to ‘get rid’ of it and simplify the equation we need to solve.”