Today's opener follows the same sequence as yesterday's. This is a chance for students to begin to sure up some of their knowledge of patterns, and my hope is that students are feeling comfortable with problems like these.
I focus most of my attention on the fourth problem, which, just like yesterday, helps us move into inequalities. In order to answer this question, we'll solve the inequality:
8n - 48 > 300
We'll interpret the solution as the first integer greater than 43.5, and I'll show students that
t(43) = 8(43) - 48 = 296
t(44) = 8(44) - 48 = 304
in order to help make sense of what's going on here.
The problem with inequalities is the same as the problem with fractions and negative numbers. For some reason, their utility is lost on kids and they feel much more complicated and difficult to deal with than they really are.
I focus a lot of attention on dispelling the villainy from all three of these importantly useful and usefully important mathematical ideas, by trying to show students the use and facility of them as tools. I've been doing the same with function notation so far this week.
Today's notes follow a structure similar to yesterday's: I ask students to share with the class and me examples of equations on which they need some help. These equations might come from the homework, or from yesterday's Check-In Quiz, which I returned to students while they got started on the opener.
After we go over a few of these, I put up a fairly simple two-step equation, like
4x - 21 = 39
and I ask students to solve it. I let them tell me the steps, and although some students are still getting comfortable with this skill, at the whole-class level, this is pretty straightforward, and I have an easy time eliciting the two steps necessary to solve it.
When it's done, I tell everyone to watch closely, and I erase the equals sign from the original equation, replacing it with a greater than sign. I say, "Ok, keep watching!" Then, I move line-by-line through our solution, and at each step, including the answer, I make the same change. At the end, we have x > 15.
"The equation originally said what value of x will make four times x minus 21 equal to 39?" I say. "Now the inequality says what values of x will make four times x minus 21 greater than 39? The solution to the equation was that x had to be 15. Now, in this inequality, any value for x that is greater than 15 will make it true." I ask for a number bigger than 15 to plug in for x, and we see that, indeed, 4x - 21 comes out greater than 39.
I draw a number line on the board. "Every number greater than 15 will make this inequality true," I say. "To represent that, I want to draw a line that shows all numbers greater than 15." I draw the line. "Now, does x = 15 make this inequality true?" We decide that it wouldn't, and I draw the open circle at 15.
One more time, I change the original problem, so that the greater than sign becomes a greater than or equals sign. Once again, this is the only change I make as I go through. "Now does 15 work?" I say. It does. I fill in the circle. "That's basically all you need to know about inequalities." For the moment, I leave out the detail about multiplying or dividing through by a negative number. That's an exploration of its own. To begin, my real goal is to show students that they should not be intimidated by inequalities.
My students tell me that they've seen how to graph inequalities on the number line before, so we briefly review that with a few examples. Today's Check-In Quiz will serve as a pre-assessment of what they really can do with one-dimensional graphs like these.
Just like yesterday, the second half of today's class is dedicated to a Check-In Quiz on SLT 1.1. Thus far, our focus has been only on the equations part of this Student Learning Target, but it's important to note that the SLT also calls on students to be able to solve inequalities. As it was during the first half of today's class, the purpose of this quiz is to ease the transition from equations to inequalities.
Today's quiz is another Kuta Software-generated worksheet, consisting a few parts. It includes four Level 4 (two step) linear equations to be solved, and students are prompted to show where each solution falls on a number line. Following that are some number lines representing inequalities, and students are prompted to write each of these in algebraic terms. Then, there are four Level 4 inequalities to be solved, also with an instruction to plot each solution on its own number line.
When I grade these, I will once again show students how many they got right and wrong, but the most important grade I will record will be their score on Mathematical Practice #1, for which I'm assessing the extent to which students showed a record of the efforts for solving each equation.
At this point, about two-thirds of my students are feeling comfortable solving two-step equations. For these, and even those who are well beyond needing the practice solving two-step equations, this Check-In is appropriately scaffolded for them to shore up their basic understandings of inequalities and the related representations on a number line. For those students who still lack confidence solving two-step equations, I tell them to show me whatever they can on the first four equations, then to try plotting the inequalities. If they finish that, I circulate and try to help them see the similarities between equations and inequalities before asking that they just try to solve the first one, and so on.