##
* *Reflection: Problem-based Approaches
Solving Systems by Graphing - Section 4: Homework 5.3 and How to Do It

I think it's very important to start a unit on systems with problem solving. Students have to see why this tool exists, and they can only get to that idea by seeing it in context. On the other hand, there are all sorts of important algebra skills to practice during this unit, and old-fashioned practice of those skills will play a central role to the rest of this unit. Now that we've spent nearly four classes with a focus primarily on problem solving, I can start to make the transition.

There are many benefits to starting the unit with problem solving. As I've mentioned, it's helpful for students to see the context for why the tool exists. Many of my students are inclined to express their distaste for word problems, and I'm trying to undo that. A side benefit to all this is that when we do "finally" get to just practicing the algebra, kids are hungry for it. I can't tell you how many times I've heard kids say, "finally, real math!" when they get a practice worksheet. I don't get into it with kids about the definition of "real math" here. But it's always amused me a bit at how, by starting with context and then moving toward practice, students express their excitement at having the opportunity to drill a few skills, so why not use that to my advantage? At the very least, it makes the job of classroom management that much easier.

*Why start systems with problem solving? Glad you asked.*

*Problem-based Approaches: Why start systems with problem solving? Glad you asked.*

# Solving Systems by Graphing

Lesson 4 of 20

## Objective: SWBAT solve a system of equations by graphing, and interpret the intersection of two graphs in terms of a context.

## Big Idea: As students gain confidence in graphing - and interpreting the intersection points on - systems of equations, we continue to see how this skill relates to guess and check.

*43 minutes*

#### Opener: At Odds!

*10 min*

Today's opener is on the first slide of today's lesson notes. As was the case on Monday and Tuesday, and will continue to be throughout the unit, we're going start the problem today before finishing it tomorrow.

I give kids about five minutes to get started, then I ask for volunteers to share their work. The problem is similar in structure to the "Ways to Make 50 Cents" problem from earlier in the week. It's not too hard to come up with a few ways to have three odd numbers that sum to 25, but that's not the point of the problem. The idea is that we have to get organized in a way that makes it obvious that the list is complete. So when a student shares their work, I use the document camera to display the work to the class, and then we work together to organize it.

Sometimes the student presenting their work wants to be in charge of organizing what's here, and other other times another student will volunteer to lead us in this task. This time, we have two decisions to make. First, how is each set of three numbers organized: from least to greatest or greatest to least? Second, I ask the same question about the list of sets. For both decisions, it doesn't matter which option we choose, as long as we pick one and stick to it. (In this case, the problem is easier if the decision is "least to greatest" for each, but that's an understanding I allow kids to come to on their own. Try it for yourself to see why.)

With those decisions made, the class can work together to revise the student work they're seeing now. Then I ask if everyone can imagine a few sets of three numbers that might be missing, and everyone is excited to keep going. But we're not going to do that right now: I say that I'll expect a complete solution to this problem at the start of tomorrow's class.

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To help transition to the next part of class, I return homework #5.1 and #5.2.

Today the guiding question is whether or not we solve each of these by graphing. Some students were successful in creating graphs for these problems yesterday. Now, to get there, we'll take a few steps. I want students to be clear on, what a great guess and check solution looks like, so I show an assignment that earned a 3 and one that earned a 4 on the learning target

**5.1: I can use guess and check to solve problems with two or more unknowns.**

First I show the "3" which translates to "proficiency" but not quite "mastery" (or an 85 on the 0-100 scale). As you can see, this work is beautiful. It is organized and well-presented. But it's also *too much*. "To master guess and check," I say, "you have to use your guesses to move efficiently toward a solution." As an example, I show everyone the "4" (you can also look at today's slides to see how I present this work to the class). In this example, the student uses each guess to move much closer to the correct solution, and even better than that, she notes a pattern that emerges in the work. To conclude, "You can see that both of these are great looking solutions. The purpose of algebra is to see how *efficiently* you can solve a problem, and we're going to continue to see ways to do that."

Of course, the "3" solution is very useful in terms of the next learning target:

**5.2: I can solve a system of equations by graphing, and I can interpret the intersection of two graphs in terms of a context.**

because it clearly shows a set of points that could be graphed to represent one part of the problem. That leads into the next section of today's lesson.

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#### Key Elements of a Graph

*13 min*

**Debrief From Yesterday**

Our work continues from yesterday's lesson in the computer lab. I give students a little more time to complete their graphs of the Ducks and Cows problem from Monday, the Chloe and Zeke problem from Tuesday, and the problems on homework assignments 5.1 and 5.2.

After reviewing the learning target

**5.2: I can solve a system of equations by graphing, and I can interpret the intersection of two graphs in terms of a context.**

I ask students what to describe the "minimum" amount of information a graph should show if we're using it to solve a problem. Then, I use this to summarize yesterday's most important ideas. This photo shows what I describe to students as the "Key Elements of a Graph". We have to see a sketch of both lines, with x- and y-intercepts, if possible, and the intersection of the two lines. The intersection point should be labeled, and we should be able to interpret the meaning of that intersection. The purpose of this mini-lesson is to make this process *feel* approachable to kids. A lot my students have the impression that creating a graph is hard work; I want them to see that sketching a solution can be quick, and that a lot information can be embedded in that sketch.

**Practice Time**

Depending on how the rest of the lesson has progressed, there may be some time to practice here. If I can see that it's useful, I may set aside another class period tomorrow to this sort of practice, but there's also benefit in moving on and letting kids loop repeatedly back to working with graphs as we continue to see new problems. As was the case at the end of yesterday's class, there are lots of possibilities here. If anyone needs to finish the algebraic analysis for Ducks & Cows or either of the Zeke & Chloe problems, they can. Otherwise, they might get at some of the homework problems. The spiders and ants problem from homework 5.2 is a great option here.

The second problem on Homework 5.2 is fun because it has three variables, so what happens in this case? We might look at a 3D graph (now that computer software has made such a graph accessible), or we recognize that there there must be a different way, which paves the way for substitution next week. At some point in the next week, I'll make sure to share a 3D graph with my students, but I won't force it. I've found that the "wow factor" of such a graph is directly proportional to the extend to which kids have asked on their own to see it.

*expand content*

For tonight homework, students will practice graphing systems of equations. I use Kuta Infinite Algebra to create the practice assignments that I'll use in the next few weeks. Textbook problems work fine too. I like Kuta because it's customizable, and I can create leveled worksheets to meet my students exactly where they are. For this assignment, I provide eight systems of equations, the first six of which give two linear equations in slope-intercept form. The settings in the software allow me to make sure that every solution consists of a pair of integers between -10 and 10.

I distribute the handout and graph paper. For graph paper, I choose the smallest grid squares I can find. I tell students to fold their graph paper into quarters, which will give them space to graph each of the eight problems separately. Then I use the document camera to show everyone what I expect on the first problem.

With my work projected at the front of the room, I'm modeling. "Here is what your work can look like," I say. Students can see how I start by carefully drawing and marking my axes, and then how I use the y-intercept and slope to plot points for each line. Then I connect the dots. This is background knowledge that I want everyone to have already, but I know that some lack confidence and an understanding of what craftsmanship looks like.

As I describe what I'm doing, I want kids to feel like they get it, which for the most part they do, but it's always about confidence. I say, "The y-intercept of this equation is -6, so you know where that point goes," and students have the chance to say to themselves, "Yeah! I do know that!" The same goes for slope. Soon, both lines are graphed and we can see the intersection. I label it, and then I ask if anyone has any questions. This year, every time I've done this, kids are excited to try the next one on their own.

Moving forward, we'll spend a lot of time looking at problems in class, and then homework will be used as a tool to help students practice the algebra.

*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: Solving Problems in Two or More Unknowns
- LESSON 2: Organizing a List and Guess & Check
- LESSON 3: What's In an Intersection?
- LESSON 4: Solving Systems by Graphing
- LESSON 5: Mastery Session: MP1, 5.1, and 5.2
- LESSON 6: From Guess and Check to Graphing Systems
- LESSON 7: Graphing Lines and Getting Stuff Done
- LESSON 8: Linear Equation Drills (Computer Lab)
- LESSON 9: Three Ways to Solve a Problem
- LESSON 10: From Graphing to Substitution
- LESSON 11: REALLY BIG IDEAS! Graphing vs. Substitution
- LESSON 12: An Intersection Point, and What Happens Around It
- LESSON 13: Algebraic Nuts and Bolts: Solving Systems
- LESSON 14: The Most Important Algebraic Idea of the Year
- LESSON 15: How Far Can Substitution Take You?
- LESSON 16: Sweet Problems with Two Variables
- LESSON 17: Many Lines Through One Point
- LESSON 18: Solving Systems by Elimination
- LESSON 19: Three Days of Review, Problem Solving, and Achieving Mastery
- LESSON 20: Unit 5: Systems of Equations, Two Day Exam