Equations with Variables on Both Sides
Lesson 3 of 20
Objective: To solve equations with variables and integers on both sides.
Students enter the class silently according to the Daily Entrance Routine. Do Now assignments are at their desk and include a series of pan balance problems. Everyone is coming back from Thanksgiving break and it is best to start with a concrete representation of equations so that students can refresh their knowledge of solving equations. There are two problems on the front of the sheet, representing two different equations. The first equation 3x = 9 is represented in the picture of 3 boxes with an unknown amount of coins inside each and an equivalent expression on the opposite side of the = sign. I made 12 of these cube boxes using attached nets (see resources). I will have these boxes available in class, with the given amount of coins inside each. These will be set up at booths on the side of the room so that I can send student who struggle with the pictured representations. These concrete representations are helpful for review as well as for students still struggling to understand complex relationships within equations. I will be using pictorial and abstract representations in today’s class, thus using the Concrete-Pictorial-Abstract approach. For more information on this approach click here for a great summary provided by an online resource.
Students can earn up to two achievement points for completing both sides of the worksheet correctly within 6 minutes. The set of extra problems on the back of the sheet serves multiple purposes: it keeps students who finish early busy, it allows me to travel to several students to check in about the basic concepts of solving equations before diving into today’s topic and it introduces the topic using a pictorial representation before diving into abstract representations in equations.
I reserve 3-4 minutes to review the last problem. The illustration depicts the equation 4A + 5 = A + 14. In this example, I model with the boxes I made that we can begin by taking away 5 marbles, or chips from each side of the equation. Then I take away 1 A from each side of the equation. I select one other student to show the same steps on the opposite side of the room for more students to see.
I ask students to organize their Do Now into their binder and get ready for class notes which are distributed. At the beginning I provide the definition for a “pan balance” or “balance scale” as reference and to use this concept to transition into the next topic. I have students write the steps for solving equations which we reviewed in the last Do Now problem:
Step 1: Isolate the variable terms on each side of the equation by using opposite operations (+/–). Remember to keep the equation balanced!
Step 2: Isolate the variables on the opposite side of remaining constants by using opposite operations (+/–). Remember to keep the equation balanced!
We use these steps to review the solution to the equation in the notes which yields a negative answer. I ask students to share their opinion with their neighbor about the order of these steps. Do we have to isolate the variable terms on one side of the equation first, or can we start by isolating the constants on one side of the equation first? The goal is to get students to identify the more basic goal of isolating variables and constants on opposite sides of the equation so that we can figure out how many units the variable is worth. This is also a good opportunity to practice MP4 as we create real world examples for the equation we are solving:
2x + 6 = 2 + x
2x + 4 = x
x + 4 = 0
x = –4
Ex 1: Ms. Chavira owes the same amount of money at Bloomingdales and Nike.com. She has 6 dollars in the bank. Kaya owes the same amount as Ms. Chavira to Nike.com only and she has 2 dollars in the bank. If they both pay off their debts they will have the same amount of money. How much do they each owe Nike.com?
I can check to see if students can apply this real-world situation to the equation by identifying expressions and symbols aligning to the context of the situation. Which expression summarizes the first sentence in this problem? What constant represents the amount of money Ms. Chavira/Kaya have in the bank? What part of the equation represents the quantity obtained when Ms. Chavira pays off her debt?
I set up sticky notes labeled 1 – 24 on the blackboard before class. By now students are getting curious about those numbers. At the beginning of the task section I ask students to organize their notes, close their binders, clear their desks, and be ready to move. Then, I ask them to stand up on row at a time, select a sticky note off the board and return to their seat. Students with ODD numbered sticky notes will remain seated on the left side of the room at a table to themselves. Students have the option of moving to a different table even if they were already seated on the left side. Students with even numbers must choose a partner with an odd number or have the option of working independently on the right side of the room. As we near the winter break and as students prepare to take interim assessments for the next two days, I know that I need to provide as many chances to move and to work with different peers as possible so that students don’t lag and get bored.
Students work in pairs or independently to solve the equations on their worksheet. They are split into two levels of difficulty for which students can receive achievement points if scored correctly and within a given amount of time. I will be walking around with counter chips and “x boxes” to concretely represent any equations I can and help students who are struggling. The most common error to look out for with these types of equations is when students incorrectly combine terms on opposite sides of the number line. For example, #3 reads:
n + 8 = –4 + 4n
Some students may incorrectly combine the n's on opposite sides of the equation like this:
5n + 8 = –4
In these cases, having the algebra balance to concretely show how this action unbalances the equation can be very useful for delivering this concept.
Students will also be notified that today’s classwork is being graded for effort. A rubric which details how they will be graded will be distributed. This rubric includes three categories: completion (student must complete 10 problems), work shown (student must show the steps we discussed during the notes), and partner evaluations (student must listen to their partner and remain on task). Students who choose to work independently will have a “partner” score from me based on how well they remained on task and how well they listened to my feedback.
Students are asked to complete the same rubric and grade themselves. This will inform both student and teacher about perceptions of understanding the directions of a task; both can then problem solve to improve learning of the lesson.
After completing the rubric, students are to complete their exit ticket and turn it in at the end of class.