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## Developing Multiplicative Thinking from Additive Reasoning

Using the context of restaurants and ratios to find equivalent fractions can push students' strategies forward.

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## Solutions: Fall Festival Fun

The problem scenario from October 2012 engages students in simple permutations along with finding the probability of an event. Two sixth-grade teachers report on the seasonal, open-ended, authentic learning that their students experienced with a scarecrow-dressing contest.

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## An Appetite for Fractions

Using candy bars as models gives students a taste for learning to represent fractions whose denominators are factors of twelve.

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## Four Ways to Determine Equivalent Ratios

Ratios and proportions are important concepts that occur in real life, but they are difficult to learn and complicated to teach (Lamon 2007). In general, proportional reasoning in the middle grades is an area in need of attention because of its connection to later concepts (Ojose 2015). In this article, we explain the meanings of ratios, proportions, and equivalent ratios and then provide useful methods to determine equivalent ratios using students' examples. Because multiple definitions exist, it is necessary to determine definitions to avoid possible confusion. Johnson (2010) defines a ratio as a pair of positive nonzero real numbers such that there are a units for every b units, written a:b; read a to b; and represented as a fraction, a/b. We follow definitions suggested by the writers of the Common Core (CCSSI 2010): Fraction representation is known as the associated unit rate or the value of a ratio A:B and is found by dividing B into A, where “equivalent ratios have the same unit rate” (McCallum, Zimba, and Daro 2011). Using both contextual and noncontextual proportion tasks, we saw various ways that students found the equivalency of ratios as well as misconceptions they have.

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## Exploring the World Using Proportional Reasoning: If Our Classroom Were the World

Eager to understand their world, students can really engage when population data are introduced in the classroom. The lesson presented in this article was inspired by the book If the World Were a Village: A Book about the World's People (Smith 2011), which presents a great deal of data in a concise form that middle school students typically find interesting, engaging, and, most often, surprising.

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## More Bang for Your Buck

A cartoon involving equivalent fractions is coupled with a full-page activity sheet.

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## Making Sense of Equivalent Ratios

A framework using contexts, language, and representations can support middle school students' justifications.

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## Line 'em up!

Post Script items are designed as rich “grab and go” resources that any teacher can quickly incorporate into their classroom repertoire with little effort and maximum impact. This article shares ideas for using a clothesline number line to build understanding of number relationships across the elementary grades.

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## Problem: Jesse's Train

Each month, elementary school teachers are given a problem along with suggested instructional notes. They are asked to use the problem in their own classrooms and report solutions, strategies, reflections, and misconceptions to the journal audience. This task allows students the opportunity to explore the magnitude of fractions in comparison to different sizes of wholes.

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## Sharing a Birthday Cake

This month, students are presented with a scenario in which two friends must decide how to cut a cake so they each get the same amount. Students will use transformations and their spatial reasoning to determine various ways to cut the cake. Each month, elementary school teachers receive a problem along with suggested instructional notes and are asked to use the problem in their own classrooms and submit solutions, strategies, and reflections to the journal.