Jackie McCarthy's Portfolio Page

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Based on Jamie Woodcock's Models: an Instructional Tool, I will be integrating instruction with models into my current 7th grade mathematics classroom. My students will soon encounter the use of variables in math, so instruction with models will be integrated with instruction for understanding when I am teaching my students to solve simple linear equations.


Instructional Problem
Seventh grade mathematics is where students first encounter significant instruction in the use of variables in mathematics. Shortly after being introduced to the concept of a variable, seventh grade students must learn to use inverse operations to solve simple linear equations. This is a significant skill in forming the foundation of algebra proficiency, and must be strong for success in later mathematics courses.

Nature of What is to Be Learned
Students are learning a new set of skills to prepare them to work with a new type of mathematics. This new concept is the first of many abstract mathematical topics students will encounter. To successfully understand variables and solving linear equations, students must understand the following:

  • Addition and subtraction as inverse operations
  • Multiplication and division as inverse operations
  • Variables represent a quantity that can change
  • A solution to an equation is a specific value of a variable that makes the equation statement true

About the Learners
Models instruction will be integrated into several 7th grade math classes in the Troy City School District in Troy, New York. The class that will be analyzed and reported specifically for this project has 22 students. The students in this class represent many different demographic areas, and 7 of the class members have limited English proficiency.

Instructional Context

Curricular Context:Solving linear equations is preceded in the curriculum by translating and evaluating simple algebraic expressions

School Contextual Factors: The initial learning goal may have to be slightly altered to accommodate the specific learners of this context. Many students need additional vocabulary instruction due to the language barrier for the ESL students. So far in this school year, students have consistently needed reminders of previous material before covering new topics. So in the context of this lesson, students will need a reminder of variables and algebraic expressions before equations and inverse operations can be considered. To accommodate the time that will be taken for the reminder of earlier material, instruction and learning goals for the main topic will be altered. Broad, conceptual understanding of variables and inverse operations will be more likely developed over time throughout the year, whereas this specific lesson will accomplish an understanding sufficient to solve linear equations.

New Instructional Goals
The ideal goal is for students to develop a broad, conceptual understanding of variables and inverse operations and to use that understanding to solve linear equations. Taking into account the context of the school and the specific learners of the class, the new goal is for students to successfully solve linear equations using inverse operations.


NOTE: Since my project is a case study of an existing mini-course, these objectives are specific to the lesson I will be using in my classroom. These are NOT overall course objectives or overall unit objectives.
Assessment Situation: Students will be taking a traditional mathematics test at the end of the unit. Traditional assessment is used to provide students with distributed practice for the state assessment at the end of the year. Some problems on the test will require students to write equations from written descriptions (i.e. "the product of 3 and a number is 15"). Other problems will be explicit examples (i.e. "Solve for x: x-7 = 11")

Behavior to be Observed: Students will solve simple linear equations that can be solved in two steps.

Tools and Constraints: Students must show inversion of the given operations- they can not simply provide an answer. Students are allowed and encouraged to show a formal "check" for their solution.

Learned Capability Verb: Students will accurately SOLVE two-step linear equations in one variable.

Object of Learned Capability: Two-step linear equations :


Assessment Situation: Also a traditional assessment, as with objective 1. Students will have to identify inverse operations for the given operations in equations, and will demonstrate this by showing work on the test questions.

Behavior to be Observed: Students will identify and use inverse operations in the problem-solving process.

Tools and Constraints: Accurate work must be shown in distinct problem-solving steps. Strings of unequal quantities joined by equal signs will not be given credit.

Learned Capability Verb: Students will USE inverse operations in the problem-solving process.

Object of Learned Capability: Inverse operations


Prerequisite Knowledge: Objective 1
Objective 1 is conceptually larger than objective 2, so an understanding of inverse operations (a large component of objective 2) is part of the prerequisite knowledge for this task. As far as content is concerned, an understanding of solving one-step linear equations is also needed in order to understand two-step equations and their solution processes. Students must also have an understanding of the order of operations, since this must be reversed in the equation solving process. As was observed during lessons about one-step equations, self-confidence and a positive attitude towards math are important prerequisites for this concept. I had the opportunity to observe two students in my class during the lessons on one-step equations, both with low confidence. I spoke with each of them outside of class about how they should give it a shot, and I told them I believed that they could do it if they really focused. One student improved his attitude in class and began to participate more, and now has a better understanding of one-step equations. The other maintained his poor attitude and, as a result, still has a limited understanding of one-step equations. This demonstrated to me first-hand the importance of going into a lesson such as this with a positive attitude and self-confidence.

Prerequisite Knowledge: Objective 2
For objective 2, students must have an understanding of how inverse operations work in the context of one-step linear equations. Similarly to objective 1, students must understand order of operations in order to reverse the process. Self-efficacy is also important to this objective, as this objective serves as a prerequisite for the first objective.

NOTE: Prerequisite information is very similar for both objectives since this is a case study taking place over the course of only one or two lessons, as opposed to an entire unit or course.


For the final results of my project, click here