Clear the Fractions!
Lesson 17 of 23
Objective: SWBAT solve equations with fractional constants and coefficients by using a common multiple.
Students enter silently according to the Daily Entrance Routine. There are Do Now assignments at their desks. This worksheet includes four expressions which students must simplify by using the distributive property. The first problem is the most basic, and as I walk around the room while students are working I am checking to make sure students understand that the factor on the outside of the parentheses must be multiplied by each of the terms on the inside. Students sometimes forget to multiply by the second term, which is most often the constant. This is a great way to spiral back to the few who are still making this mistake. The second problem adds complexity with the negative signs in use. I am checking to see if students use the correct rules for multiplying negative integers (for ex: a negative times a negative equals a positive). In the third example I begin to check students’ understanding and skills with distribution of fractions. I mainly want to know which students can mentally calculate ½ x 2x = 1x or x, and which students still need to write it out on paper to calculate it. During this time, I can also check in with students who may be confused about the use of fractions. I am still battling “fraction intimidation” with a small group of students who seem to freeze up when they see a fraction in use. I coach them through this particular problem by asking, “what is half of 2x?” and “what is half of +8”? By stating the question in this form I am helping students understand the relationship between fractions and multiplying with whole numbers. I am also hoping that some students will begin to see that when multiplying ½ by 2 (reciprocals) they are “leaving x alone” because the coefficient results in a positive 1. Students who begin to notice this fact through problems 3 and 4 will have a better understanding of today’s lesson since it will require them to multiply fractions in an equation by an LCD to clear the denominators.
I ask students to take out a sheet of lined paper, fill out a heading at the top, and copy the aim off the board “KWBAT distribute ______________ to clear fractions”. I also ask them to set up their paper using Cornell Notes style by drawing a vertical line down the paper creating a “Topics” section which takes up a third of the paper and a “Notes” section taking up the rest of the space. I use the SMARTboard Notebook program to display this on the board and write the notes along with students. The first topic we write down to review is “Solving Equations: Behind the Scenes”. I ask students to write the equation 2x – 2 = 8 and we review the solve steps (add two on both sides, divide by two on both sides, x = 5). Then I ask students to recopy the equation and substitute the x with our answer, 5. I explain that I will be showing students what is happening “behind the scenes” when we solve and I follow the same solve steps (add two on both sides, divide by two on both sides, 5 = 5). I’ve noticed a common mistake when solving two step equations is to divide or multiply only one of the terms in a binomial expression to solve an equation. For example, in the equation 2x – 2 = 8, some students choose to divide by two first, but only do so to 2x and 8, erroneously missing the constant –2. Students can divide first, but multiplication and division must always be applied to every term in an equation. It is important to address this common error today because students will be using LCDs to clear the fraction in an equation. Since we will be multiplying to do this, students must understand that they must multiply every term in the equation by that common multiple, even if the term is not a fraction itself.
I ask students to re-write the statement 5 = 5 into their notes. Then I ask if the equation is still true when I multiply 5(3) = 5. The answer is no, 15 does not equal 5, and then I ask “what would I have to do to the other side of the equation to make it true”? Students reply, “multiply by 5”:
5(3) = 5(3)
15 = 15
Students must then write 3 + 2 = 5 on the board and I ask if this is a true statement. Students respond yes and I follow it up by asking if this is a true statement:
3(3) + 2 = 5(3)
Students respond no, and I ask, why? I multiplied numbers on both sides by 3, why didn’t I get equivalent expressions? If one student cannot give me the answer right away I ask them to conference with neighbors until someone can explain that I would need to multiply 2 by 3 as well:
3(3) + 2 = 5(3)
3(3) + 2(3) = 5(3)
9 + 6 = 15
I ask students to keep these ideas in mind as we transition to the next topic in the notes, “Solving Equations with Fractions: Using LCD”. I ask students to review the homework together, asking any neighbors around their seat which question they thought was the most difficult of the first four. Then I take a quick vote by a show of hands and we write that equation into our notes section. The most likely choice will be an equation including three fractions with different denominators, or #4 in the homework. Students must copy this equation next to the new topic we will review. I ask them to tell me the LCD of all fractions in the equation and to multiply every term in the equation by that multiple. I guide them through the work by showing each step and identifying vocabulary words like LCD, the multiple, term, expression, equivalent, etc. A narrated sample of the work is included in the resource for this section.
Students receive the task sheet and are instructed to work in pairs or independently to solve. They are free to choose the method used in class yesterday to solve or the new method utilizing the LCD to clear fractions in equations. Students must use this new method on at least 4 of the problems in their task. I use a random name generator on the SMARTboard to send 6 students into booths at the side of the room. This will free up more space in the main part of the class. Based on work shown in the Do Now, I will be asking some students to work with me at the front of the room and also welcoming others to join us. All work and answers for the first four problems will be generated on the SMARTboard and students will be asked to check their answers after I complete each problem on the board.
When working with a target number of students at the front, I begin the first problem by modeling the first step and ask for help on the following step. In the rest of the problems, I cold-call different students to guide me through the steps or to help another student identify a step. Another option, given sufficient time, is also to have students come to the front of the board and write the steps themselves. This is also a good opportunity to discuss better strategies. In problems like #8, it may be better not to use the LCD method to clear the fractions since the constants can be cleared using opposite operations.
Time has been a challenge this year in my class as I now have 60 minutes each day for lessons. Today we review homework at the end of class because I knew the lesson itself would require more time for students to ask questions. Students are asked to take out their homework. Worked out answers are displayed on the board and students are responsible for checking their own work and their answers. They can also ask their partners for help in figuring out their mistakes. If their partner cannot determine the error or help, students can raise their hands and I will come over to help. Students are also encouraged to use this time to ask me questions about the lesson itself. Essentially I am using this last ten minutes of class to close loops and answer any questions students are still struggling within this algebraic topic.
During this time I ask students to focus on their work with fractions and to make sure they are showing all steps in an equation, including the opposite operations performed on opposite sides of the equation. This attention to the details and the steps are examples of MP6.