Linear Equations: A Subtle Note, and Choosing Your Own Adventure
Lesson 3 of 12
Objective: SWBAT develop their knowledge of linear equations by solving equations at the level of their choice.
For the third day this week, class opens with a series of patterns problems. Yesterday, students were given a pattern rule followed by a series of questions. Today they're given the first five terms in a pattern. I ask them to write a pattern rule, and then to find the 100th term in the pattern, both of which are review questions designed to help students gain confidence with these skills.
The third question is of a type that was introduced on Monday: given a the value of the term, can you write an equation and solve it to determine what number term that would be? The algebra looks like this:
t(n) = 4n + 11
t(n) = 799
4n + 11 = 799
Before we answer this third question, I ask students to make estimates. We've already seen that the 100th term is 411. So where might you estimate that 799 would fall?
Like yesterday, the fourth question moves us into inequalities. When will the numbers in this pattern pass 200? The most fun point to explore here is the idea that when we solve the inequality
4n + 11 > 200
n > 47.25
We have to interpret the value of n to be the first integer that is greater than 47.25. Even though rounding to the nearest integer gives us 47, the 47th term is still less than 200. It's the 48th term that is the first to exceed 200. Also, some students find it interesting to note how you can't really have a 47.25th term in a pattern. Patterns are discrete functions, as we will see later in the course.
All of this continues to lay groundwork for creating and solving linear equalities, which is coming up soon!
Today's class notes give me space to solve a few example equations at whatever level my students request. I ask them to look at their handouts from yesterday and their homework, and I elicit a few examples that will comprise the first part of today's notes. I love to put these few minutes of class in the hands of my students, and their attention is high because they get to ask for exactly what they need.
After a few equations at whatever level they choose, I say I'm going to talk about Level 4 for a few minutes. I write these three equations on the board:
-x/9 + 13 = 20
5x-10 = 45
I say that all of these are two step equations. The first one is useful to everyone; most of my students still get a little stuck when they're confronted with division. But we note that there are two steps here: we have to undo the addition with subtraction, then undo the division with multiplication. We pay close attention to the positive and negative signs here.
When I work through the second one, kids are driving, because they know what to do on an equation like this.
Then, I write the word subtle above the work, and say that I'd like to share a subtle hint. I say, "I know that many of you know how to use the Distributive Property to solve this equation, but I'd like you to see a different way to think about this. This is subtle," I add, "Which means that you'll have to pay close attention to see what I'm talking about here."
I ask for someone to read the second equation aloud. Invariably, someone says, "five x minus 10 equals 45." I say that's true, but I'd like to think of it like this: "take a number, multiply by five, then subtract 10, and you'll get 45." I look around to check that students are with me. I might repeat that this is "subtle, so watch closely here."
I ask for a volunteer to read the third equation like I just read the second. Sometimes, someone gets it, other times, I have to say it. Either way, my goal is to read it like this: "take a number, subtract 10, then multiply by 5, and you'll get 45." I ask students to compare these two equations. We see that they're the same, but the operations happen - thanks to those parentheses - in a different order. "So we'll have to undo these operations in the reverse order." When things are really going well, this is where a student will ask if this will change the solution. I express my gratitude that someone asked that question, and I say that it's a great habit to ask questions like this.
I show how to solve the third equation by undoing the multiplication with division then undoing the subtraction with addition, and that you don't have to distribute if you do it this way. I tell students that if they can understand this, what a great tool this will be moving forward. "Even though you can distribute," I say, "I think it's easier to do it like this."
It's always satisfying to see how satisfied students are with this distinction, and many are excited to be able to solve this equation as I've demonstrated. I'll be able to see how well the knowledge hold for any students taking the Level 4 quiz today.
Today's class ends with the first check-in quiz on solving linear equations. I use Kuta Software to prepare quizzes at each level from 3 (one-step) to 6 (multiple steps, requiring distributive property). Whatever level students choose, their quiz will consist of 10 equations at that level.
When I grade these, I will check how many solutions are correct, I write that number at the top of the page, and I record it in my grade book for SLT 1.1. I outline what I'm saying on the last slide of today's class notes. More importantly, I grade all of these quizzes on Mathematical Practice #1, and I repeat my talking points from yesterday's class:
- Showing some work is a way of demonstrating perseverance. On simple problems it shows how you've made sense of this kind of problem. On the more difficult problems it shows that you didn't quit when things got hard. Along these lines, tomorrow's quiz will also be graded on Mathematical Practice #1.
- As we move on, you must get used to showing these steps. A lot of people may be able to solve a one-step equation in your head, and some of you have proven that you can solve two-step equations without showing any work. What happens when the problems get more complicated, however? You're going to have to be able to go step by step.
When I distribute these quizzes, it's up to kids to choose their level, but if they're uncertain, I make suggestions based on what I saw yesterday.
Some students who choose Level 5 decide they'd rather get help than count this for a grade. That's great: I say, "Show me whatever steps you know, and I look forward to helping you learn this." This is "quiz", and it's graded as such - but it's definitely formative, it's definitely practice, and it's definitely meant to be a chance for everyone learn to something. I call it a quiz, but really it's an opportunity to learn.