Elimination to Solving Systems (2)
Lesson 7 of 13
Objective: SWBAT solve more systems of equations by Elimination.
At the start of the class, I project the work of two students solving a system, on the whiteboard:
I ask the class to analyze both students' work, and be prepared to explain both strategies used. After a couple of minutes I call on volunteers and ask that each student stand and explain a strategy. I expect students to offer explanations like:
- Student A decided to cancel the x variables and chose a fraction, 4/3, to multiply the first equation by, to yield -4x, then added the equations.
- Student B also canceled the x variable, but by multiplying the second equation by 3/4 to yield 3x, and then added.
Then, I will ask the class to finish solving each system. I will call two volunteers to write their work on the board. The two solutions show students that the Multiplication Property of Equality can be multiplied to either equation to eliminate the variable x. The answer is the same either way.
In this section of the lesson, I want students to realize that they can simultaneously multiply each equation by a different number. So I tell the class to turn to an elbow partner and come up with a strategy to solve this same system, Launch Elimination, but totally avoiding fractions.
Giving as little information possible, I walk around listening to students discuss their strategies. The idea is to let students struggle and figure it out themselves. A pair of students is bound to come up with multiplying each equation by an appropriate value so that adding the equations afterwards, cancels the variable. I ask a student to come up to the show their work on the board, and explain it to the class. I don't expect students to make the ideal choice of numbers to multiply by, every time. Students may take a little time to become skilled at selecting the most efficient multipliers, one that will keep the products small and minimize the arithmetic needed for the rest of the solution.
A student should then summarize the strategy used and explain when it is useful. The student should indicate that a multiple of the two coefficients of the variable to cancel, must be sought, and each equation multiplied by its appropriate factor. This strategy is useful when no two coefficients of the same variable are multiples of each other.
Before proceeding to the following activity, I make certain that students understood from the recent board example, that both equations can be multiplied simultaneously by a different factor to obtain an equivalent system, and that although this can always be done, it is a strategy used when no two coefficients are multiples of one another.
I randomly break up the class into groups of threes, and give each group one sheet of paper preferably a colored sheet with no lines. (With larger classes, groups should of fours is ok, but no larger; adding more stations is an alternative) I choose a "secretary" for each group, which will do the writing on their colored paper. I try and always choose a student who works slowly for this job.
With help, I arrange the desks into 7 stations and ask that each group stand at a station. At each station I place a 4x6 Index Card with a question facing down on the desk, and I write the station number on the back. Students should not to turn it over until we begin.
Resource Note: Print the Index Cards and cut and paste them on each index card. Then write the station number on the back of each.
Each group will have to answer the question on their index card, writing their response on a sheet of paper. When I say it's time, the students will move to the next station and so forth. I give groups 5 minutes at each station. When 4 minutes have past, I give a one minute warning. As students are responding to the questions, I walk through listening to their discussions and encouraging all members to participate. I try to keep an eye open for those students who I think will simply "tag along" and motivate them to become engaged. I will ask a student "What do you think?" or "Can you answer this one for the group?"
Sometimes none or few of the groups are ready to go on the next station. I then give them more time to finish the problem. When we are done I ask students to sit at whatever station they are at.
To close the lesson, I ask each group to draw an emoticon face by each question on their paper indicating their level of mastery. Students have closed lessons similarly in the past, so they know what to do.
I may project the faces on the board so they could refer to them. Sometimes there is only one student that had problems with a particular question, while the rest of the group knew it well. If this case, and similar cases, I ask that the name of the student be written in parenthesis with the corresponding face beside it. I take these papers and go over them to determine if additional practice is needed for certain types of question, or if I need to re-teach any concept before going on to the next lesson.
I keep this extension of our lessons on solving systems of linear equations to give to more advanced students. Yet, my goal is that eventually, every student is able to apply their skills to solving these systems. Since solving systems in 3 variables is not part of our learning standards, I don't include them in any assessment toward their grades.