Case Study of Models in Math Instruction

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Welcome to the final page for my case study of the Models: an Instructional Tool mini-course!


During the Summer 2011 semester, I took a class that had a strong focus on using models to improve math and science instruction at the secondary level. Throughout that course, I learned about the importance of realistic problems in the math classroom, and learned about different types of models that can be incorporated into math instruction. When selecting a topic for this project, I noticed a former student created an entire mini-course on using models in math (see Models: an Instructional Tool). Using my knowledge from the Summer course and the information provided in the previous mini-course on this website, I wanted to use models in my own classroom to see how it affected my students' interest and performance.


In order to incorporate modeling into my instruction and have time to analyze the results after an in-class assessment, I decided to take one day to focus on a specific topic in my class using modeling. For my 7th grade math class, I found a modeling activity to use to reinforce the topic of solving two-step linear equations (such as 3x + 4 = 13). For the development and design behind my case-study, see my portfolio page here. For the final product, I have included on this page a formal lesson plan with the incorporated modeling, the assessment data from the unit test given after this material was covered, and a brief reflection about the success of modeling instruction.


Topic: Linear Equations in Two Steps

Instructional Context: Follows the unit on solving one-step equations and inequalities. In addition to two-step equations, basic operations with positive and negative integers are included in this unit. By the end of the unit, two-step equation examples with negative integers can be used.

Time Frame: One 43 minute class period after two-step equations have been introduced (prior lesson includes an informal pre-assessment)

Behavioral objectives: (see my portfolio for detailed objectives)

  • Students will identify inverse operations as part of the equation solving process
  • Students will correctly solve linear equations in two steps

NYS Learning Standards:

  • 7.PS.11- Work in collaboration with others to solve problems
  • 7.CN.1- Understand and make connections among multiple representations of the same mathematical idea
  • 7.R.6- Use representations to explore problem situations

Materials: Modeling activity sheet

Introduction: 5 minutes
As a reminder of inverse operations and the equation solving process, put a warm-up problem on the board for students to complete. The example I used is x+5=9 (with directions "solve the equation to find the value of x").

Main Activity: 30 minutes
Distribute modeling activity sheet to students. Split students into partners or groups of 3, depending on class size. Go through the first example as a class. Then students work in their small groups to complete the modeling activity. Circulate throughout the room, checking in with student groups to make sure they are progressing through the activity. Clarify issues as students encounter misconceptions.

Conclusion: 8 minutes
As a whole class group, summarize what was learned through the modeling activity. Provide a wrap-up word problem for students to complete as a "ticket out the door". (My example was "Aliyah had $24 to spend on seven pencils. After buying them she had $10. How much did each pencil cost?")

Follow Up & Assessment: This lesson can be followed by standard homework assignments. Because of the department-wide curriculum at my school, my particular class took a traditional assessment following the unit.


The activity seemed to go well. I had to explain the first problem to every group, but after that problem was explained, they were able to make their way through the rest of the activity. Some groups finished the entire sheet, so I provided them with a new example- "A triangle is worth one more point in a game than a square. Two squares and three triangles are worth 13 points. How many points is one triangle worth? One square?" This problem elevates the level of difficulty to something in the Integrated Algebra curriculum- solving systems of equations with two variables. One group that had finished the sheet excitedly got through this example fairly quickly. I provided another systems of equations question that used actual equations with two variables, and this was more challenging for them. Only two students correctly solved this example. However, students were not expected to answer those supplemental examples, and the rest of the group did well with the original activity, so I think it was successful. Students were very clearly more engaged than when traditional lecture is used as the means of instruction.

After analyzing assessment data, I can more confidently say that completing this activity was worth while. Three of my classes took the test, but only one of them used this modeling activity. Of those three classes, the one that used the modeling activity had a test average that was higher than either of the other two classes. I think the fact that this model tied linear equations to concrete examples had an effect on my students' performance on the exam.