##
* *Reflection: Relevance
Creating your own Systems of Equations - Section 3: Closure

In this lesson, my students chose the first three scenarios far more than the others. No one chose the Grass Eaters scenario. The Bake Sale, Nickels and Dimes, and Cell Phone problems were modeled and solved quite well. In the cell phone problem, some students did not really state which was the better plan. They simply solved their system and wrote the coordinates of the solution. A couple of groups did not interpret the meaning of the solution with respect to the problem, which was required (see student work phone plan).

The Nickels and Dimes coin problem was set up and solved well. At the end, the students wrote the coordinate pairs and what coin each variable represents, but many added a solution with variables switched (see student work nickels dimes). I asked the students why they did this, and one of them said that it did not matter what variable came first here because their is no x and y.

So, I need to make things clearer to this group without getting too deep in theory. I indicated that the common definition of an ordered pair, which is generally accepted, satisfies the property (a, b) = (x, y) if a = x and b = y, and that we could state the order of the coordinates, at the beginning so that we remain consistent until the end. I talked with the students about the fact that in this case they began and should continue with the model (n, d).

**Teacher**: Can you tell me when could it be important to state the variables in a particular order?

**Students**: {No answer}

**Teacher**: *Well, you solved this system by what method?*

**Students**: Substitution, Oh….when I have to graph.

**Teacher***: *Yes, why would the order matter then?

**Students**: Because the first variable is like the x and the second the y and they fall in different places on the graph.

**Teacher**: Yep, we generally use the first coordinates as our x, indicating horizontal movement, and the second as our y for vertical movement.

I believe that the reason for these scenarios being chosen most was the relevance aspect they have. Personal and real-world relevance is motivating to students because they relate the subject matter to the world around them. The scenarios like the **Cell Phone Plan** and the **Bake Sale** are topics that students not only had previously encountered in other lessons, but probably have had personal experiences with.

In the classroom I find it very important to link material to the real world. I can usually tell when too much time is being spent on abstract ideas by observing student behavior. Many students become unmotivated and start to daze off, or just stop participating. Any time you stop and tell a story, or ask a question involving a real-world situation, students become more attentive.

Through listening to students' comments and discussions as they went around the scenarios, I quickly found that they identified with these cases and were more willing to come up with a situation involving a system of equations. Besides, similar situations to these particular scenarios had been modeled before. Students did not even try the Grass Eaters problem. Maybe this scenario could be replaced, or the idea could be portrayed better by adding signs near the animals indicating what they are eating or how much food the buckets contain. This may trigger some "equation" ideas in students' minds. Replacing it by a more relevant situation to the students is probably better.

Another reason why students may have chosen these cases is that they lend themselves to the use of equations in Standard form. Students have seen some of these types of problems before and these may have seemed familiar to them. In the phone problem, students wrote equations in slope intercept form easily, indicating the cost per minute as slope and a fixed fee as the y intercept but failed to interpret the solution. The coin problem was the next best. Most students obtained both equations correctly and performed the solving quite well.

*Relevance: Successful Scenarios in the lesson*

# Creating your own Systems of Equations

Lesson 10 of 13

## Objective: SWBAT create and solve systems of equations based on their own formulated real world circumstance

## Big Idea: Designing problems and applying learned skills in different real world contexts fosters higher order thinking.

*65 minutes*

#### Launch (Warm up)

*15 min*

The warm up exercises (Warmup Creating Systems) in this Launch section access students' prior knowledge and give me a good picture of student readiness for the activity. I already have a pretty good idea of those students that have struggled more than others throughout the unit.

Today, I want to pair these students up with those that have done well. At first, I tell students to complete the warm up individually. While they are working, I walk around monitoring and helping where necessary and I distribute one card piece (Pair up cards) to each student. The cards were cut out before class and they are ready to be handed to students during the warm up. I distribute the cards strategically so students who may have difficulty with the main activity, are paired with more advanced students to work together.

Once students are finished with the warm-up exercises, I review each question by calling on volunteers to share an answer.

* *

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#### Application

*35 min*

Once the warm up is done, I ask students to find their partner by finding the Pair up cards that matches their's. I then take the six Scenarios for creating a word problem and post them in separate areas around the room, preferably on whiteboard.

Before asking students to get to work, I project the Burgers image on the board in order to model what each pair of students in class is suppose to do during the activity. The image shows hamburgers and cheeseburgers. I leave the image up for a while and tell the class that I am going to think of a situation which lends itself to writing a system of two linear equations, in order to find the solution to my problem. I then project the model system of equations problem. I call on someone to read the word problem and then ask if they can write the two equations on the board for us. The desired equations are:

**b + c = 24**

**2.5b + 3c = 68**

I expect that all of my students will be able to solve this system at this point in the unit. If we're not pressed with time, I may ask the class to solve it and call on someone to write the solution on the board.

I then ask students to stand and go around with their partners, looking at the 6 scenarios posted around the room. I ask that they choose 4 out of the 6 scenarios and use their imagination to...

**Create a problem situation from the image****Write the two equations in the system****Solve the system by any method**

Some students may want to create their own situation, not a situation in any of the posts. I always allow students to create one of their own if they wish. I also vary the number of problems for students to solve if time runs out.

The students should do all their work on blank sheets of paper (see reflection), one set per group, indicating the scenarios chosen. In order to avoid chaos, I may ask students to choose the images they want to work with and work at their desks. Moving around is always a good idea to help student thinking. I'm flexible enough to allow some groups to sit and work on the floor if they want to. As the class is working, I walk around giving help and guidance, yet allowing each group to create and solve their own problem. Calculators always come in handy here because the values students choose will many times give decimal results when computing results.

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#### Closure

*15 min*

Once groups are done with creating their own word problems and solving their systems, I call on volunteers for each of the six scenarios posted, to write and solve their system under the corresponding image. Once all six systems are solved on the board, I call on each group and ask that they state their word problem to the whole class, guide us through the problem explaining what method they used and what the solution to their system equations means with respect to the problem situation.

I ask students to pose any questions after each presentation and allow some time for the students to respond to these. All students should hand in their work containing the four scenarios and solved systems, before leaving the class.

Student sample work (see *Successful Scenarios in the Lesson* reflection for more about how students worked on these tasks):

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The Creating Systems of Equations homework assignment involves some research so I ask that students choose one out of the two problem given. All work should be shown.

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: Sharing Equations (Day 1)
- LESSON 2: Sharing Equations (Day 2)
- LESSON 3: Catching Up
- LESSON 4: On Substitution (A continuation of "Catching Up")
- LESSON 5: Solving Systems by Addition
- LESSON 6: Elimination to Solving Systems (1)
- LESSON 7: Elimination to Solving Systems (2)
- LESSON 8: Systems and Parallel Lines
- LESSON 9: Counterfeit Coins
- LESSON 10: Creating your own Systems of Equations
- LESSON 11: Going Mobile?
- LESSON 12: Systems Assessment 1: Scavenger Hunt
- LESSON 13: Systems Assessment 2: Mastery Test