In this month's problem, students are presented with a scenario in which two friends have to decide how to cut a cake so they each get the same amount. Students use transformations and spatial reasoning to figure out the various ways to cut the cake. Each month, this section of the Problem Solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics. Find detailed submission guidelines for all departments at http://www.nctm.org/WriteForTCM.

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- Author or Editor: J. Matt Switzer x

### J. Matt Switzer

**NCTM's Principles to Actions:**
*Ensuring Mathematical Success for All* (2014) outlines eight teaching practices for effective teaching and learning of mathematics. One of them, *Use and connect mathematical representations*, involves engaging students in “making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving” (p. 10).

### J. Matt Switzer

In this month's problem, students are presented with a scenario in which two friends are planning a summer vacation. The students must use measurement concepts to compare two lakes to determine what bigger means with respect to the lakes and which of the two lakes is bigger. 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, reflections, and misconceptions to the journal audience.

### J. Matt Switzer

Students are presented with a growing geometric pattern for table configurations at a birthday party. They must verbally describe how to extend the pattern and use their verbal description to generate an expression for the number of tables for different configurations. 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 report solutions, strategies, reflections, and misconceptions to the journal audience.

### J. Matt Switzer

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.

### J. Matt Switzer

Each month, this section of the Problem Solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics. In this month's Problem Solvers Solutions, readers have a window into students' number and operation sense in the early elementary grades. Second and third graders were presented with problem-solving tasks using a hundred chart consisting of two number cards and a challenge card aligned to an addition or subtraction structure. Drawing on the structure of the hundred chart and prior knowledge, students were able to articulate their solution strategies.

### J. Matt Switzer

tudents often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and my students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an inequality, students may lack a deep understanding of the relationship between the inequality and its graph. Hiebert and Carpenter (1992) stated that mathematics is understood “if its mental representation is part of a network of representations” and that the “degree of understanding is determined by the number and strength of the connections” (p. 67). I therefore developed an activity that allows students to explore the graphs of inequalities not presented as lines in slope-intercept form, thereby making connections between pairs of expressions, ordered pairs, and the points on a graph representing equations and inequalities.

### J. Matt Switzer

Knowing both the content and manner in which students learn math in elementary school can help teachers bridge the gap between elementary school and middle school (and then high school) math.

### J. Matt Switzer and Teresa C. Hoppe

In this month's problem, students are presented with a geometric growth pattern for table configurations at a birthday party. Students will verbally describe how to extend the pattern, then use their verbal description to generate an expression for the number of tables for different configurations. Each month, this section of the Problem Solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics.

### J. Matt Switzer, Kelley Buchheister and Barbara Dougherty

This department publishes brief news articles, announcements, and guest editorials on current mathematics education issues that stimulate the interest of TCM readers and cause them to think about an issue or consider a specific viewpoint about some aspect of mathematics education.