How do you start a unit on Systems of Linear Equations, or Simultaneous Equations (depending on how your curriculum or school refers to this concept)? Do you front-load the vocabulary? Do you start with the visual solution, i.e. graph? Or do you start with whatever example is in the book for that chapter? I start with a Give Me Five!
Students have a good understanding of what it means to be tardy to class, and running laps in Phys. Ed. I attach a problem to that understanding and ask a simple question.
Tardies have been a growing problem in P.E. lately, so the teachers have created their own system to try and quell the problem. Mr. Crunch usually starts class with 2 laps around the gym as a warm-up, but now he will be adding an additional lap for every tardy student. Mrs. Yogger usually starts with 5 laps as a warm-up, but now will add 1/2 lap for every tardy student.
Which class would you rather be in this semester?
It seems simple enough, but as students start to discuss in their groups, they come up with a lot of, "Well, it depends..." Argumentation among students quickly ensues. Once the groups are fully entrenched in their response to the question, I tell them they must back their response up with a Give Me Five.
Students must justify, prove or back up their answer using five representations of the problem and their answer. Writing an equation for each teacher happens quickly. Then the groups decide if they make two tables or one table and while making that decision they are debating the efficiency of the choices. The graph spurs conversations on how far to go on the x and y axis, and then what scale to use to best represent the problem and solution. The picture is a challenge for some students, but if they can draw a picture that makes sense, they definitely understand the math. Then the written explanation makes connections among the representations and deepens the learning.
By the time the students finish the five representations, each student knows and understands that the point (6,8) is the most important point in their response. They can clearly explain that in the table, the point (6,8) is where the number of tardy students and laps are the same, therefore the points before and after that depend on if you like running more or running less. Students can clearly demonstrate understanding that finding the point (6,8) on the graph allows them to show where students would run more or run less. There are so many teachable moments throughout this activity and you'll want to stop class use them all, but I suggest letting a lot of them go. Allow for mistakes. Allow students to make tables that stop at five tardies. Allow students to make two graphs even though one is what you expect. Respond to the mistakes later with respect and a learning focus. It pays off throughout the unit.
At this point students have an understanding of solving systems of linear equations using a table and graph. You may want to practice with different equations for a day or two. Eventually students will ask, "Is their an easier way to solve for this point than building a table or graph every time?"
Yes! Now there is a desire to learn!