Today's opener serves two purposes. The first is to help students gain confidence with solving inequalities by previewing a problem that will be on Friday's Unit 1 Exam. Second is that it can be used to further emphasize the relationship between guess-and-check and algebraic problem solving strategies.
As students arrive, they get started on this problem (the wording of which is taken from an MCAS exam in Grade 10 Mathematics). After they've worked for a few minutes, I quietly write the notes that appear on the top right in this photo, which is my gentle way of reminding students about what we've been doing for a few days now.
I circulate to see how they're doing, then I call everyone to attention and ask how everyone feels about this problem. It's solution takes just two steps, which most everyone is comfortable with, but the inequality sign and the 1.5 conspire to trip some students up. "If you're not sure what to do, then one thing you can do is guess," I say. "Someone who is not sure: who wants to throw out an answer?" Now a few students give it a try. I'll take what they give me, but if someone says, "ten!" I'll jump on that. "Now ten is a nice place to start," I say. "I think it's easy to multiply by ten, so let's just try it and see what happens." We use it, and see that it yields the inequality 11 > 18, which is clearly not true.
Now comes the most important part. I say, "If I'm using guess and check, then am I mad that my first guess was wrong?" We talk about how, for guess and check to work, you can't sweat a wrong answer. You must just learn from it. If 10 gave us a number that was too low, what should we try next? A few more guesses then move us in the right direction.
But this is inefficient. Usually, before we actually get to an answer, I make the transition to algebra, and we give that a try. When that gives us the solution x > 14.7, we can talk about the words in the problem. What is the first integer that's bigger than 14-and-two-thirds? Will it yield a result that's much bigger than 18 or just a little? Would someone be completely wrong if they said that 20 was the answer? I spend as much time as my students want, then we move on.
After the opener, I review today's agenda, which says that the word of the day is "QUALITY". I tell everyone that they should be nearly done with the Creating Equations Problem Set, and that today I want everyone to focus making sure they're producing their best quality work: both on this problem set, and as they prepare for the exam that's coming up in two days.
I say that this is work period, and that students can work on whatever they see fit, and that I'm going to focus most of my attention on the problem set. Then we set to it.
The optional quiz I reference at the bottom of the agenda will most often wait until tomorrow's lesson; please take a look at that lesson to see it.
In particular, I want to look at #'s 6-8 of the problem set today, and I can count on a good number of students asking for that help.
I am thinking of a number. If I multiply my number by a million and add one, the answer is two. What is my number?
I want to show students that, if this were a smaller number, it would feel much easier. I tell them to cross out the words "a million" and replace them with the number 3, then solve the problem. At this point, almost everyone can do that. When they do, I say, "Ok, now replace the 3 in your equation with a 1,000,000. What happens?" If I'm really lucky, and sometimes I am, a student will crack, "Wait, you were really thinking about the number 0.000001?" to which I'll respond, "Indeed! Have you never thought about one in a million?"
#7 and #8 seem like very different problems if you're only reading the words:
7. A 170-foot piece of rope is cut into two pieces so that one piece is 20 feet longer than the other piece. How long is the shorter piece of rope?
8. Jill makes twice as much as Jack. Together, Jack and Jill’s salaries add up to $159,750. What is Jack’s salary?
I tell students that I'm going to help them with #7, and by doing so, I'm going to try to help with #8. We go through the same process as we've been doing, moving from guess and check to an algebraic solution. After we try a few guesses and use our guessing structure to set up the algebra, the notes end up looking something like this. What's more important is seeing the similarities between the two problems. I introduce the word isomorphic, not because kids must know it, but because it's an awesome word. I tell them that some of the greatest discoveries in science and mathematics have been made when people realize the similarities between different problems. Then we discuss how these problems are related. They're both about two things - two ropes or two salaries - where one is bigger than the other, and we have their sum. By now, I hope that students have enough to continue working.
For any students who are fully up-to-date and confident about what they'll need to do on the exam, I have a set of problems ready to go. I use a page out of K. Elayn Martin-Gay's fantastic Intermediate Algebra textbook. I've always found this series of books to be great resources for problems. Many real-world problems are cited in a way that I can only aspire to.
I use these problems from near the beginning of the Intermediate Algebra book to challenge and engage the students who have shown they're ready by getting this far.