Developing Proportional Reasoning in Intermediate Mathematics

Area(s) of Focus: technology, math
Division(s): Intermediate
Level(s): Grade 7, Grade 8

This project contains a unit for grades 7 and 8 on Proportional Reasoning and a PowerPoint for teachers. The unit includes introductory activities, three-part lessons, mental math strings and assessment strategies.

Developing Proportional Reasoning in Intermediate Mathematics

Proportional reasoning involves the deliberate use of multiplicative relationships to compare quantities and to predict the value of one quantity based on the values of another. Our project focused on how to provide varied intentional opportunities for students to use multiplicative relationships in intermediate grades. In the math curriculum, the development of proportional reasoning is a very challenging concept. In our experience teaching intermediate math, we have found that students seem to struggle more with the concept of proportional reasoning than they do with most other concepts. Although we have varied our instructional practices through the years, the struggle remains.

“All ability to reason using proportional relationships is a complex process that develops over an extended period of time. It takes many varied physical experiences to develop an understanding of what a proportional relationship is and then more time to gain the ability to deal with it abstractly.” (Cordel & Mason, 2000)

Our proposal involved four phases: Research, Planning, Co-teaching/Co-assessing, and Reflecting/Revising/Sharing. During the Research phase, we arranged to meet with curriculum consultants to discuss strategies to develop proportional reasoning. We also investigated technology and conducted a review of the literature in this area. Our goal was to deepen our understanding of the underlying skills and the developmental processes involved in developing proportional reasoning.

During the second phase, we used the information obtained through our research to work together to develop a landscape for proportional reasoning (Fosnot), devise good questions for Three-Part Math Lessons and develop strings for mental math fluency. In our planning, we focused on differentiating for the diverse needs and learning styles of our students as well as the inclusion of technology as a tool in modelling concepts (e.g., the use of virtual manipulatives). Through this project, we found ways for our students to understand proportional reasoning and enable them to make connections between the concepts and in the real world.

Once the planning phase was completed, we planned to co-teach and co-assess student work. We videotaped one lesson each and used our observations to enhance our questioning and sequencing. Important in our project was opportunity to co-assess formative and summative student work, and to co-reflect on student progress and next steps throughout this unit. The opportunities to collaborate with colleagues helped us to develop consistency in our assessment.

The final phase involved reflecting on our learning, revising our unit and sharing our learning with other teachers. Upon completion of the learning cycle, we reviewed our learning and developed next steps for our professional learning. During this reflection, we reviewed our questions, three-part lessons and strings, and revised our unit. We also sought opportunities to share our work with teachers in our respective schools and in York Region. We conducted a workshop at OAME Spring Conference at Humber College.


Team Members

  • Jean Middleton

    York Region District School Board

  • Pamela Coventry

    York Region District School Board

  • Barbara Cadel

    York Region District School Board

  • Katie Dreger

    York Region District School Board

  • Melissa Smith

    York Region District School Board

Professional Learning Goals

Our learning goals were as follows:

    1. Phase #1: Research – Develop an in-depth understanding of proportional reasoning through a literature review and meeting with math consultants.
    2. Phase #2: Planning Student Learning Opportunities – Collaborate with colleagues to develop:
      • A landscape for proportional reasoning
      • Good questions for three-part lessons
      • Strings for mental math
      • Technology integration
      • Scaffolding plan
      • Assessment strategies and a culminating task
      • A unit
    3. Phase #3: Co-teach and Co-Assess
      • Observe our own and each other’s lessons and assess our strengths and work together to develop next steps for our teaching.
      • Collaboratively develop rubrics for student work.
      • Develop consistency in assessment through co-assessment.
    4. Phase #4: Reflect and Revise the Unit/Sharing Our Learning
      • Collectively reflect on our practices and develop strategies to further student learning.
      • Share our learning with colleagues within our school, board and province.

Activities and Resources

1.  Met with consultants.

2. Extensive reading and sharing of reading.

3. Development of student materials.

4. Sharing of developed materials.

5. Development of materials for workshop.

Unexpected Challenges

1.  Funding challenges with school board over the funds being given directly to teachers.

2.  Delays in getting started on project due to funding issues.



Enhancing Student Learning and Development

Student Learning

  1. Students can identify and utilize proportional reasoning in problems and everyday life. (Assessed through culminating activity)
  2. Students will be able to make the connections between different proportional reasoning situations (e.g., fractions, decimals, per cent, ratio, rate). (Assessed through observations and recorded anecdotally, and quizzes)
  3. Students can utilize a variety of strategies including mental math to estimate proportions in everyday life (e.g., 30% of $24.99). (Assessed through observations and recorded anecdotally)
  4. Students can identify proportional reasoning in multiple strands in mathematics, science and social studies.



We will share our work with teachers in our respective schools and in York Region.

We conducted a workshop at the  OAME Spring Conference at Humber College. PowerPoint is enclosed.


Project Evaluation

Project was an overall success. We met our goals.


Resources Used

Van De Walle, John A., Jennifer M. Bay-Williams, LouAnn H. Lovin and Karen S. Karp. Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 6-8 (Volume III) (2nd Edition). Pearson.

Small, Marian. (March 8, 2012). Good Questions: Great Ways to Differentiate Mathematics Instruction, 2nd Edition. Paperback. Canada: Nelson.

Parrish, Sherry. (March 1, 2014). Number Talks Common Core Edition, Grades K-5: Helping Children Build Mental Math and Computation Strategies. Paperback. Math Solutions Publications.

Small, Marian. (February 9, 2012). Making Math Meaningful: to Canadian Students, K-8. Paperback. Canada: Nelson College Indigenous.

Ministry of Education: EduGains, Proportional Reasoning.

Twomey Fosnot, Cathy. (November 5, 1989). Enquiring Teachers, Enquiring Learners: A Constructivist Approach to Teaching. Teachers College Press.

Twomey Fosnot, Cathy. Contexts for Learning Mathematics: Fractions, Decimals and Per Cents.