# Standards for Mathematical Content

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Developing Proportional Reasoning

This multimedia overview offers an introduction to students’ development of proportional thinking and its relationship to cross-multiplication. It describes recommended instructional strategies for solving problems related to ratio, rate, and proportion, including buildup and unit ratio strategies, and illustrates examples of real-world context for such problems. This recommended practice contains three tools to support the teaching and learning of ratio, rate, and proportion.

*SOURCE:* Doing What Works

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Cross Multiply? Not So Fast

Eighth-grade co-teachers lead a class through activities involving ratios. Students use concrete materials, build-up strategies, and cross-multiplication to solve the real-world problems found in *Ratio: Warm Up Activity*, *Build-Up Strategies Worksheet*, and *Build a Cafe: PowerPoint and Report*.

**FEATURING:** James Ro and Jackie Price, Howard County Public Schools (MD)

*SOURCE:* Doing What Works

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Building on Intuitive Understanding

Thomas P. Carpenter discusses the importance of building children’s conceptual understanding of fractions based on intuitive ideas about sharing and apportioning objects. He demonstrates examples of children’s responses to challenging problems that help them expand and articulate their thinking.

**FEATURING:** Thomas P. Carpenter, University of Wisconsin - Madison

*SOURCE:* Doing What Works

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A Learning Trajectory for Fractions

Mathematics coaches describe what they’ve observed about how children deepen their understanding of fractions. They unpack the skills required to perform fraction operations using a technique called “tip of the iceberg.”

**FEATURING:** Renee Sherry, Ken Jensen, and Kim Pippenger, Math Coaches, Tollgate Elementary School (CO)

*SOURCE: *Doing What Works

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Ways to Measure One and a Half Cups

A fourth-grade teacher asks students to help solve his problem of measuring 1-1/2 cups without either a 1-cup or a 1/2-cup measuring cup, and instead with the use of many other size measuring cups. Students write number sentences to describe the results of their measurement exploration.

**FEATURING:** Keith Phelps, Fourth-Grade Teacher, Eliza Hart Spalding School of Math and Technology (ID)

*SOURCE: *Doing What Works

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Solving a Real-World Fraction Division Problem

A fifth-grade teacher poses a fraction division problem and challenges students to find different approaches to a solution. One student uses a number line; others use build-up strategies.

**FEATURING: **Brett Mosely, Fifth-Grade Teacher, Eliza Hart Spalding School of Math and Technology (ID)

*SOURCE:* Doing What Works

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