##
* *Reflection: Continuous Assessment
Words and Abstractions - Section 3: Words and Abstractions #1

I've gone back and forth over the years on the role that a pre-test or diagnostic assessment can play within the first few days of an Algebra 1 course. My current thinking is that I prefer to spend class time setting up class routines, nurturing classroom culture, and demonstrating to students the role of group work, both formally and informally.

Then, assignments like the Number Trick Project and this Words and Abstractions homework assignment play the role of a diagnostic, helping me to see how much each student knows about algebraic representations and vocabulary, as well gauging their dispositions toward the use of the abstract symbols that I've introduced.

For an example of what I can learn from this assignment, take a look at the back of this student's Words and Abstractions assignment.

First, let's think about the "Algebra" column. I notice that this student has a lot of things right. The only outright operation errors are on #'s 20, 27, and 28, where she defaulted to addition rather than using multiplication for the word *product* (#20) and division for the word *quotient* (#27 & 28). Many of my students need some clarification that the word "and" doesn't always mean "addition" - that we have to notice words like *difference*, *product*, and *quotient* before we can interpret what the "and" means. In particular, a lot of kids need clarification on the definition of quotient.

Next, exercise #'s 23 and 24 reveal some common errors with the order of operations. Many students can recite the fact that multiplication comes before addition and subtraction in the order of operations, but they're not sure how to show this in an algebraic expression. So on #23, for example, I know I'll need to show students how to use parentheses to indicate that the addition happens first, *then* we multiply by 3. A quick glance at a stack of student work tells me which particular kids will need to get this message, and I'll pay close attention to who gets it over the next few days. The other common error on #24 is the order of ** 15-x**. We will have to note that "subtract 15 from a number" means that we start with a number,

*, then we subtract 15. Our early dive into the Delta Math assignment (see my other reflection for this lesson) gives kids a chance to see for themselves that the order of terms matters in a subtraction statement, but not so for addition. Eventually, we're able to generalize that addition and multiplication are commutative, but subtraction and division are not. Most students are aware of this; this assignment shows me which of my students can apply that awareness and who will need my help.*

**x**Finally, there is the idea of what I describe to my students are "algebraic traditions". Look at this student's solutions to #21 and #22, for example. It is not wrong to write this expression the way it's written here, but it's not the way that we're going to see it in this class. Kids have to be trained away from writing "* x x 3*" and toward writing "

*". The same goes for that division sign on #'s 17 and 18: it's a fine and useful symbol, but not one you're often going to see in this course. Here's my chance to re-train students to think of any division expression as a fraction.*

**3x**Ok, so now that we've noticed all of that, what about the symbols? It's hard work coming up with novel ways to represent some of these expressions, particularly when we're asked to represent two operations in a particular order. My big takeaway from looking at this student's work is that she certainly brings a lot of mathematical knowledge to the table, but that I'll want to work with her to think about what the operations really mean, rather than just relying on what she's been taught. Look at #29 and #30, for example. That actually seems like a mathematically reasonable way to represent the square of a number! But what does it assume? Is the more important part of squaring a number the use of superscript to write exponent? She's got that part down. But I don't see an indication that she really knows what it means to "square" something. In general, this student doesn't yet understand the different between multiplication and addition, and how each are represented with the squares and dots. There's plenty more to get out of everything she's tried: what else do you notice?

*Continuous Assessment: Student Work: This Assignment as a Pre-Test*

# Words and Abstractions

Lesson 4 of 12

## Objective: SWBAT translate mathematical phrases into symbolic and algebraic representations.

At the end of our last class, and for homework, students commenced their work on translating number tricks from words into symbolic and algebraic representations. For today's opener, I post the second example (see the bottom of this page: Two Example Number Tricks) number trick from our previous class and instruct students to translate it into symbols and algebraic expressions.

As students get to work, I encourage them to look at their prior notes, and if they have room, to record this example on the same notebook page as the previous one. I also encourage them to help each other. Given between 5 and 10 minutes and plenty of firm goading, everyone should be able to get this done.

To see more about what this looks like, see my previous lesson: *How Can an Abstraction Show Me How Things Work? *There is also more to come on this project over the next few lessons.

#### Resources

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At the start of each week, I give students a weekly homework sheet that looks like this: HW Week 2. On the front is a checklist that they can use to make sure that they get everything done each week. As you explore this Algebra 1 course, you'll see that I've shared a variety of weekly homework sheets. On this one is a list of assignments to be completed, as well as a few little tidbits, like talking to an elder about algebra and noting where they keep their textbooks. I try to show the interactivity of these weekly sheets as a way of getting students to buy into these sheets as a useful tool.

On the back (I always use double-side copies) is a **Record Sheet**. This is place where students can jot some notes about what they learn, notice, or wonder about each day. When I introduce this side of the page, I remind everyone that expect them to "work hard, gather evidence, and demonstrate mastery." The record sheet is a place where they can take notes about how they're doing each of those things. As the year progresses, we will dedicated more and more of our attention to these record sheets, for now, I offer it as a tool but without a great deal of pressure or fanfare; I'd like to see what my students make of these on their own.

#### Resources

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#### Words and Abstractions #1

*30 min*

On the weekly homework sheet, I point out that tonight's homework is to finish an assignment called "Words and Abstractions 1". I say that we're going to get started right away, and that "I hope many of you will have a lot of this assignment complete before you leave class today."

We've already started translating number tricks into symbols and algebraic representations, but many students now need some practice. Many kids buy the idea of abstract representations of verbal phrases, but they lack the confidence or the experience to do it on their own. So today we'll build that.

Additionally, on Part 1 of the Number Trick Project, we left off with a challenge: *How can we represent subtraction if there are not yet any dots from which to subtract? *This assignment will give us a chance to develop an answer to this question as a group.

**Here is how I implement it:**

After distributing the handout to all students, I project the document on the screen at the front of the room. From there, I can provide a few examples for students, but I don't provide too many at first. Usually, this is enough to get everyone started: Words and Abstractions Examples. This is the sort of work that kids must grapple with and figure out on their own.

As class goes on, I might invite students to add their answer to what's on the screen.

The key thing we have to get to is what subtraction looks like. I point out that problem #11 on this handout is the same as the troublesome subtraction step on Part 1 of the project. I propose that we need three symbols: the box for a number that's already been established, the dot that represents 1, and something else for -1. A bunch of random shapes are offered, but the key is that it needs to have a relationship to the 1 dot. An empty dot might work, but I try to avoid that, because it may lead to confusion once we start talking about inequalities. Rather, I move toward the crossed-out dot. My notes look like this: Words and Abstractions Subtraction.

After that, it's pretty much on students to figure out as much as they can. I encourage them to discuss these problems with each other, and I say that tomorrow they'll see the answer key. I tell them that if they can take any approach they'd like: if they want to go one problem at a time, that's great, but if they want to complete the entire algebra or symbols column first, that's fine too. The problems on the front of the handout are expressions with one arithmetic operation, while on the back there are expressions with two or more operations. I tell students that it's ok if the back feels a little more difficult, and I repeat again that I'd like them to compare and discuss their answers with each other.

*expand content*

While students work on Words and Abstractions #1, I try to check in with each of them individually. I make sure that I know each student's name, I keep a record of whether or not they have a binder and a signed syllabus, and I ask them how their doing on both today's work and on Part 1 of the Number Trick Project.

If I move pretty quickly, I can get a moment or two of face time with every student during today's half-hour of work time. This may be at the expense of giving a lot of extra help on today's handout, but it allows me to start to demonstrate another important class structure: the way I refer students to each other for help. If I get to the third table and I see that someone's having trouble with a problem that I've already seen someone else handling successfully, I might ask them to take a look at that problem together.

*expand content*

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- UNIT 1: Number Tricks, Patterns, and Abstractions
- UNIT 2: The Number Line Project
- UNIT 3: Solving Linear Equations
- UNIT 4: Creating Linear Equations
- UNIT 5: Statistics
- UNIT 6: Mini Unit: Patterns, Programs, and Math Without Words
- UNIT 7: Lines
- UNIT 8: Linear and Exponential Functions
- UNIT 9: Systems of Equations
- UNIT 10: Quadratic Functions
- UNIT 11: Functions and Modeling

- LESSON 1: Two Powerful Shapes
- LESSON 2: Number Tricks, Patterns, and How to Succeed in This Class
- LESSON 3: How Can an Abstraction Show Me How Things Work?
- LESSON 4: Words and Abstractions
- LESSON 5: Patterns and Abstractions
- LESSON 6: How to Write a Pattern Rule
- LESSON 7: How to Write a Number Trick
- LESSON 8: Work Period: Patterns and The Number Trick Project
- LESSON 9: Patterns Quiz and Project Work Time
- LESSON 10: What's Wrong With PEMDAS?
- LESSON 11: Problem Set: Number Lines
- LESSON 12: The Parentheses Challenge