Portfolio for Models: A Case Study for Instructional Tools


Models as an Instructional Tool


The following lessons have been designed to give students the opportunity to physically manipulate objects to understand the concepts of adding, subtracting, and multiplying integers. The students can make their own integer tiles by cutting out two different color squares (or any shape). Each tile will represent either one positive or one negative. The students will engage in these hands on activities and begin to generate their own methods for adding, subtracting, and multiplying integers. More importantly, students will be able to explain what they are doing and why it is a mathematically correct procedure.


The intent of this course is to provide students with an opportunity to use models in the mathematics classroom. Students in middle school mathematics will learn to use these models as representations for mathematical concepts. Integers is a specifically difficult topic for many students. Instead of learning “rules” for adding, subtracting, and multiplying integers, students will develop mathematical reasoning. For professional development purposes, participants in this course can examine different ways to implement models. The mini-course will use models in the classroom to aid in student-centered, discovery, and exploratory learning. As a result of this mini-course, students and participants will further their understanding of mathematics through the use of models.

For information regarding models and an overview of how to use models within the classroom, please visit Jamie Woodcock’s page: Models: an Instructional Tool

Needs Assessment

Instructional Problem

How to use Models (or manipulatives) with integers in a middle school mathematics classroom.

What is to be learned

Students of this case study will be working with integer models in a classroom setting.

About the learners

This course is aimed at middle school mathematics students who are interested in models (or manipulatives) and technology in the classroom. Educators can use this course and apply/adapt the lessons to fit their specific classroom needs.

Instructional content

Students often have difficulty understanding the rationale for mathematical procedures. The tools and assessments will aid in the development of the concept and provide alternative methods to solving integer problems.


The middle school student will develop mathematical strategies to solve different operations of integers.

Performance Objectives

Students will be able to:

  1. Use models as representations to solve addition, subtraction, and multiplication problems
  2. Describe how and why the rules of integers work
  3. Use mental mathematics to solve integer problems

Task Analysis


The lessons provided in the case study will allow students to use models to represent integer problems. Students will be able to answer the “how” and “why” questions of mathematics. By introducing models in the classroom, students will learn to accept and understand mathematical procedures, without the use of memorizing procedures.

Learning Outcomes

At the end of this unit, students will be able to:

  1. Understand how to add/subtract/multiply/divide integers
  2. Develop procedures for add/subtract/multiply/dividing integers
  3. Perform mental mathematics to solve integer problems
  4. Develop critical thinking by generating methods for using tiles

Prerequisite skills

  1. Basic understanding of the definition of integers
  2. Add, subtract, multiply, and divide whole numbers
  3. Commutative property of multiplication and addition

Supportive Prerequisites

  1. Basic computer proficiency
  2. Basic ability to navigate the World Wide Web
  3. Willingness to use mathematical models (representations)
  4. Willingness to learn new mathematical procedures


  1. Adding integers with different signs using models
  2. Adding integers with the same sign using models
  3. Subtracting integers using models
  4. Multiplying integers using models

File:Christine's Curriculum Map.pdf

End of Unit Reflection