Lesson: I & We: Systems of Equations
Printed example/practice problems
Colored Whiteboard/overhead markers
Computer with internet access
Algebra 9.0 Students solve a system of two linear equations in two variables algebraically
and are able to interpret the answer graphically. Students are able to solve a
system of two linear inequalities in two variables and to sketch the solution sets.
Present review problems to students to remind them of their skills in solving one and two step equations. You can do this by presenting problems on the overhead, or, for more fun, you can play this game briefly. (The only downside of this game is that shooting the basket has nothing to do with math skills, but it is just frustrating enough to keep things interesting for a short game. The upside is that you can play without time restrictions, so you can discuss any difficult problems.)
Remind students that we are talking about individuals and collectives in this project. There is a math skill that has to do with how the individual affects the collective. In fact, thinking in these terms will help us master this complex skill. We know that an equation (function) represents a line that can be graphed. So, if I had y = 2 +4x, I could plug in value for x (say, 1) and that would tell me the value of y (6). If I graphed that point and two others, I would have a line. This we know. So, we can say that the value of the individual (the first variable) is what will determine the fate of the collective (the rest of the equation). Let’s use this thinking to figure out where TWO lines intersect. Of course, one way is to just plot all the points on both lines forever until we run into their intersection. But it is easy to learn the identity of an individual, and that will tell us about the collective.
So, if we have two lines and we want to know where they intersect, they might be
y= 2 + 4x
and y= 10 + 3x
So, we know that one thing that y IS (EQUALS) is (2 + 4x). It’s like saying that this “family” of terms is the same as the individual. So, we can replace y with its value (“family”), like in all substitution.
(2 + 4x) = 10 +3x
Oh! Wait! We know how to solve that for x! Let’s combine like terms!
2 + 1x = 10
Wam! Bam! x=8
Since we now know the identity of that individual, we can put in back in the collective and find the identity of the other individual.
y= 2 + 4x
y= 2 +4(8)
y= 2 + 32
y = 34
Now we know that these two individuals, x=8 and y=34 bring their “families” or “communities” together.
(I have also taught this- because it has so many steps and it is hard for students to remember how to start – by saying that y can’t go to the birthday party, but it wants to know what the present will be. So, it sends its family. They find out the present (x) and come home and tell y. Since the story is so short, students memorize it and it guides them up to the last step, which is then obvious.)
If you’re more comfortable or want a more visual lesson, follow the instructions here:
(they call the “family” a “blob”)
Assume one page will be guided practice and one will be independent practice.
Prepare index cards with the answers to the first page, one answer per card. Number them! Mix them up and hand them out to the class. Tell them not to let anyone see their card until we have finished the problem they have and then they will tell us whether we were right or wrong. They can’t cheat when their problem comes up!
Go about putting problems on the board and calling on students to name an individual who stands alone, name their family, place their family into the “community” (or “party”, the other equation) and then find the other variable (“present”). Call on a student to say whether it is correct or not, based on the card they are holding.
Point at that if we don’t have an individual on his own, we need to isolate him so that we can clearly see him and his family. We just rearrange one of the equations so that it isolates him, and then we do our normal steps.
If working on an overhead, you may want to color code the “individual” and the “family” to make them stand out when they are being moved and manipulated. Students benefit from using colored pencils to do the same.
Distribute the next page of exercises, and colored pencils is applicable. Allow 5 – 10 minutes of practice, then display the correct answers to the first 3-5 problems, depending on where students are. Ask students to check their work and let you know if they are having trouble. Circulate and check in with students.
Exit Ticket/Have students solve this problem on a piece of paper and check percentage of correct answers:
y= 4 +3x