Solving Equations with Fractions
Lesson 16 of 23
Objective: SWBAT solve two step equations with fraction constants and fraction solutions.
(5 mins) Students enter silently according to the Daily Entrance Routine. There are Do Now assignments at their desks. This worksheet includes addition and subtraction problems with rational numbers. Students will use the problems in the Do Now for a foldable study aid at the end of this section of class.
(6 – 7 mins) We review the correct answers for each problem as I justify the process for calculating the answers. For example, for problem #1 I may explain that I am “combining two negative rational numbers. This means my answer will be negative and I must add the absolute values of the numbers”. This is the type of language and vocabulary I expect students to use when justifying the process for calculating these answers. Each problem is used as an example for the rules of adding and subtracting rational numbers. These examples will be written under the appropriate section of the foldable.
Create Study Tool
(10 – 12 mins) Once we have reviewed the answers I give students the template for the foldable study aid. I explain to students that this tool will remind them about the rules when adding and subtracting rational numbers. They are to use this tool during the task today. Students must take each of the problems in the do now and write them into the appropriate section for the rule followed when solving for the answer. There are two examples per rule. Students must fold the paper in half, lengthwise, and cut along the solid line in the center to create two different sections (one for addition and one for subtraction). If students finish these basic steps with time left, they may decorate the outside of their foldable.
Students are asked to copy the aim which is written on the white board at the front of the class (KWBAT solve two step equations with fraction coefficients, constants and/or fraction solutions. ), as well as the definition of equation written on the black board. It’s important for students to understand that an equation is made up of two expressions separated by an equal sign. Many students are struggling to understand this concept which affects their ability to correctly isolate the variable terms as well as their ability to correctly complete check steps.
For example, in an equation such as 3x + 3/2 = 9 students often divide the variable term by 3 to isolate the variable and divide the constant on the right side of the equation by three. If the student does not also divide the improper fraction 3/2 by 3, they are making a mathematical error in solving the equation. Understanding that both sides of the equation are equivalent quantities allows me to explain these types of errors to students. I have attached a sample of the work used to illustrate this error.
I use the class notes’ examples to show students how to use their study aid to solve the equations. For example, 5/6 – m = –11/12, I ask student to identify the variable terms. This is the term we want to isolate first. Since a positive 5/6 makes up the expression, we must first cancel out +5/6 with –5/6 on both sides. This results in the expression
–11/12 – 5/6
on the opposite side. We will follow the rule “when subtracting from a negative, the answer will remain negative and you must add the absolute values”. In order to add 11/12 and 5/6 we must write an equivalent fraction for 5/6 with a common denominator of 12:
– m = –11/12 – 5/6
– m = –11/12 – 10/12
– m = – [ 11/12 + 10/12 ]
– m = – [ 21/12 ]
– m = – 1 9/12 = – 1 ¾
The last step is to isolate the variable:
– m = – 1 ¾
(divide by –1 on both sides)
m = 1 ¾
Students work in pairs to solve the first 5 equations from the homework sheet by using their rules sheet. After 6-7 minutes, students are assigned to one of the first five equations and are asked to show the work on a piece of white paper, justifying each step with a rule or a step used to solve. They are asked to use my solution to the example from the class notes as a guide for the type of information that needs to be on their chart paper. As I walk around to help students with the homework questions, I guide them through the steps (isolating the variable term, and then the variable) as well as guiding them through the use of the rules (“are you adding or subtracting the numbers?”). Sometimes students make the mistake of using double signs when canceling out constants or coefficients to solve equations. This causes them to be confused about whether the number are being added or subtracted. If this happens I ask them to think back to the balance scale. In an equation like #1 in the HW, -2 + (1/2)x = -6/7, there are two negative chips being combined with half of a quantity, x. To begin isolating the quantity, x, we would need to cancel out two negative chips with two positives. Thus, the singular sign to use is positive and the quantity of positives is 2. There is no need to say, or write, “adding positive two”, or erroneously stating, “adding negative two”. These are the most common, confusing ideas I see students encounter in these types of equations. By breaking down these problems into smaller steps, students are using MP7 as they take note of the numbers’ signs and determine which rules to follow when solving..
Once there are 10 minutes of class left, students are asked to tape their paper posters on the chalk board. I will number each poster with a different number. Every student receives a post it. They are asked to write their name at the top of the post it and divide it into two equal sections with a vertical line. The left side can be labeled with a + sign and the right side with a – sign. Students will use the post it to give one comment about a positive attribute of each poster and on thing the student creator needs to improve. For example, a student may write, “great job labeling each step in your solution” for a positive comment and/or “you did not include a sign or operation for each step. Are you adding or subtracting 2 in your first step?”
Like the previous lesson, this is again an opportunity for students to use MP3 as they asses and give each other feedback on their work. Students are randomly assigned numbered poster/papers and are asked to take one minute to view the poster, 3 minutes to write the feedback, and 2 minutes to stick it on the paper. When they return to their desk they are to pack up their belongings. Once students are packed up at their seats, I will call them to line up and hand them the poster and post its given by their peers. They are to go home and make any necessary improvements to the poster for HW. These posters will be displayed on a new bulletin board the next day.