Exploring Math Ratios, Proportions & Similarity

By Jerry Burkhart

Reviewed by Mickie Gibbs

This installment of the Advanced Common Core Math Explorations focuses on areas that are given great importance in the middle grades: ratios, proportions, and similarity. Ten explorations are presented, giving students an in-depth look at topics from scale models and percentages to the Pythagorean Theorem and the Golden Ratio.

Deep, rich and open ended

Each exploration is divided into three stages, with each stage divided into problems, all centered around the same scenario. Each stage is a little more challenging than the one before it.

Most explorations include an “Algebra Connections” page, giving teachers the opportunity to connect their students’ discoveries to their future study of algebra.

Conversation Starters and Classroom Vignettes are also included. These give teachers ideas for questioning and examples of how classroom conversations might go. Each problem has detailed solutions. There are two tables showing the exact Common Core alignment for each exploration.

The focus is conceptual understanding & discovery

Many explorations are cross-curricular. The first exploration deals with what is a common project in science classes: developing a scale model of the solar system. My favorite exploration laid a foundation for the study of trigonometric functions. Inverse variation is explored in one situation, and another looks at the difficult-to-teach concepts of accuracy and precision.

Explorations are deep, rich, and open-ended. They are excellent DOK (Depth Of Knowledge) 3 experiences. The focus is conceptual understanding and discovery, not rules and rote practice. Students – and teachers – are encouraged to be prepared for what will be a productive struggle. Answers will not come quickly or easily, but students and teachers are reminded that is not a bad thing.

Students and teachers are encouraged to be prepared for what will be a productive struggle.

There is masterful use of representations such as tables, double number lines, and graphs in the book. Students go back and forth between various representations. Formulas are developed, presented, and used when appropriate and after students have strengthened their conceptual understanding of the math behind the topic.

When I received the book and saw it was directed primarily at gifted students, I was a little disappointed. I teach collaborative pre-algebra classes containing students with learning disabilities. After looking at each exploration, however, I believe at least Stage 1 is accessible to all students.

While the explorations definitely have high ceilings, each one also has a low threshold. Many problems give a “Testing the Waters” suggestion, a way to simplify the problem and make it even more accessible (as well as “Diving Deeper” suggestions for those who need even more challenge).

As an eighth grade teacher whose standards come after the major development of ratio and proportionality, I was also concerned I wouldn’t have much use for the explorations. But there were problems throughout the book I can use to both review concepts with my students and connect them to new learning, such as slope (plus the Pythagorean Theorem exploration!).

Realizing the power of Common Core

I was taught math by lecture and example and tons and tons of rote practice. I taught that way for the first 15 years or so of my career. Common Core has stretched me as a teacher and forced me to look for rich problems that do more than demonstrate an algorithm.

I am learning to enjoy the “productive struggle” that comes when students work on real, conceptual understanding of topics in math. Advanced Common Core Math Explorations: Ratios, Proportions, & Similarity, Grades 5-8 provides such problems and gives teachers the support they need in using the problems in their classrooms.

Mickie Gibbs is a National Board Certified, eighth grade Pre-Algebra and Algebra 1 teacher in Arab, Alabama who has been teaching for 20 years. She sponsors her school’s Algebra 1 competitive team and National Junior Beta Club. Every morning before school, she can be found facilitating “Homework Help” for students who need a little extra time or assistance with their work. She loves teaching math and seeks to make it an enjoyable experience for her students. Website: Mrs. Gibbs Flips Algebra 1

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