Grades 6–8, Ratios and Proportional Relationships

teaching math

Proportionality is a cornerstone topic in middle school mathematics as emphasized by the Common Core State Standards for Mathematics. We encounter problems involving multiplicative comparison and proportional relationships in our daily life such as comparing prices in the grocery store, reading a map scale, determining a percent discount, or modifying the amount of ingredients in a recipe. As such, students build their reasoning with proportional relationships skills by comparing quantities, looking at the relative ways numbers change, and thinking about proportional relationships through multiple representations (e.g., tape diagrams, double number lines, ratio tables, graphs, equations, and verbal descriptions). In this WSMI professional development experience, teachers develop essential understanding related to ratios and proportional relationships that are prevalent in grades 6–8 mathematics. By experiencing the big ideas in reasoning with proportional relationships, teachers deepen and broaden their mathematical understanding and pedagogical practices of one of the most challenging topics for students. Through hands-on activities teachers learn how to engage their students through meaningful tasks that enhance student understanding of and reasoning with ratios and proportional relationships.


  • Broaden teachers’ mathematical understanding and pedagogical practices of ratios and proportional relationships.
  • Study meaningful tasks that will increase students’ understanding of ratios and proportional relationships.
  • Critically examine student misconceptions when reasoning with ratios and proportional relationships in order to make informed instructional decisions.
  • Study the mathematical progression that students pass through as they become proficient at reasoning with ratios and proportional relationships.

Grades 6–8, Expressions and Equations

WSMI Expressions and Equations

In grades 6–8 students build on their understanding of order of operations and properties of operations to reason about numerical and algebraic expressions and equations. By providing students with multiple representations such as words, tables, graphs, and equations, they begin to analyze and model mathematical relationships. Through contextual situations students make sense of solving one-variable linear equations and inequalities as well as systems of linear equations. Finally, students extend their understanding of ratios and proportional relationships in order to make connections between unit rates and slopes of linear equations. 

This module focuses on developing the knowledge of teachers aligned with the Common Core State Standards for Mathematics. Specifically, participants will study the development of algebraic thinking in grades 6–8 drawing from CCSSM domains of Expressions and Equations and Functions. Emphasis will be placed on focus standards from each domain that are essential in the progression of algebraic reasoning across the grades. In addition, participants will deepen their understanding of the multiple representations and extend their knowledge of meaningful contexts that are fundamental to these standards.


  • Participants will become familiar with the progression of the Common Core State Standards for expressions, equations and functions across grades 68.
  • Participants will develop an understanding of the role of using multiple representations and contextual situations to support student algebraic learning.
  • Participants will understand, represent, and explain the important concepts in the CCSSM including writing, interpreting, and using expressions and equations; applying operations with rational numbers to work with expressions and linear equations; modeling relationships between quantities with linear equations and functions; and solving linear equations and systems of linear equations through real world contexts.

Grades 6–8, Statistics 

The focus of this module is centered on the development of key statistical concepts that are part of the CCSSM domain of Statistics and Probability in Grades 6 to 8.  Investigations are developed to help illustrate standards 1 to 5 under the Statistics and Probability Domain in Grade 6, standards 1 to 4 under the Statistics and Probability Domain in Grade 7, and standards 1 and 4 under the Statistics and Probability domain in Grade 8.    A number of the standards in Grades 6 to 8 are new topics for students and these standards need time to be developed as they extend into the high school Conceptual Category of Statistics and Probability.  Some of the key concepts are:  statistical questions, histograms, mean absolute deviation (Grade 6), use a random sample to draw an inference about a population, measuring the difference between two means (Grade 7) and understanding association between two quantitative variables and two categorical variables (Grade 8)


  • Broaden teachers’ understanding and pedagogical practices of statistical relationships.
  • Investigate key statistical concepts through meaningful tasks and investigations.
  • Study the progression of key statistical concepts from Grade 6 through Grade 8.
  • Provide investigations that illustrate how classroom teachers can empower their students to become data collectors and analyzers.