A 2D version of Cavalieri's Principle is productive for the teaching of area. In this manuscript, we consider an area-preserving transformation, “segment-skewing,” which provides alternative justification methods for area formulas, conceptual insights into statements about area, and foreshadows transitions about area in calculus via the Riemann integral.

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### Nicholas H. Wasserman, Keith Weber, Timothy Fukawa-Connelly and Juan Pablo Mejía-Ramos

The Asked & Answered department shares excerpts from discussion threads on the online MyNCTM community. In this issue, featured threads highlight responses to members' questions regarding 1st grade number sense, multiplication and division of fractions, issues of definition and precision related to circles, and the value of rationalizing denominators.

### John K. Lannin, Delinda van Garderen and Jessica Kamuru

This manuscript discusses two important ideas for developing student foundational understanding of the number line: (a) student views of the number sequence, and (b) recognizing units on the number line. Various student strategies and activities are included.

### Amy Noelle Parks

Children experience joy in well-designed mathematics classrooms. This article describes five research-based practices for bringing joy into PreK-Grade 2 math lessons.

### Randall E. Groth, Jennifer A. Bergner and Jathan W. Austin

Normative discourse about probability requires shared meanings for disciplinary vocabulary. Previous research indicates that students’ meanings for probability vocabulary often differ from those of mathematicians, creating a need to attend to developing students’ use of language. Current standards documents conflict in their recommendations about how this should occur. In the present study, we conducted microgenetic research to examine the vocabulary use of four students before, during, and after lessons from a cycle of design-based research attending to probability vocabulary. In characterizing students’ normative and nonnormative uses of language, we draw implications for the design of curriculum, standards, and further research. Specifically, we illustrate the importance of attending to incrementality, multidimensionality, polysemy, interrelatedness, and heterogeneity to foster students’ probability vocabulary development.

### Gerhard Sonnert, Melissa D. Barnett and Philip M. Sadler

Students’ attitudes toward mathematics and the strength of their mathematics preparation typically go hand in hand such that their specific effects are difficult to disentangle. Employing the method of propensity weighting of a continuous variable, we built hierarchical linear models in which mathematics attitudes and preparation are uncorrelated. Data used came from a national survey of U.S. college students taking introductory calculus (*N* = 5,676). A 1-standard-deviation increase in mathematics preparation predicted a 4.72-point higher college calculus grade, whereas a 1-standard-deviation increase in mathematics attitudes resulted in a 3.15-point gain. Thus, the effect of mathematics preparation was about 1.5 times that of mathematics attitudes. The two variables did not interact, nor was there any interaction between gender and these variables.

### Debasmita Basu, Nicole Panorkou, Michelle Zhu, Pankaj Lal and Bharath K. Samanthula

We provide an example from our integrated math and science curriculum where students explore the mathematical relationships underlying various science phenomena. We present the tasks we designed for exploring the covariation relationships that underlie the concept of gravity and discuss the generalizations students made as they interacted with those tasks.

### S. Asli Özgün-Koca and Matt Enlow

In this month's Growing Problem Solvers, we focused on supporting students' understanding of congruence and similarity through rigid motions and transformations. Initial understandings of congruence and similarity begin in first grade as students work with shapes in different perspectives and orientations and reflect on similarities and differences.

### Alyson E. Lischka, Kyle M. Prince and Samuel D. Reed

Encouraging students to persevere in problem solving can be accomplished using extended tasks where students solve a problem over an extended time. This article presents a structure for use of extended tasks and examples of student thinking that can emerge through such tasks. Considerations for implementation are provided.

### Ed Dickey

A personal reflection by Ed Dickey on the influence and legacy of NCTM's journals.