This article explores facilitating meaningful mathematics discourse, one of the research-based practices described in *Principles to Actions: Ensuring Mathematical Success for All*. Two tools that can support teachers in strengthening their classroom discourse are discussed in this, another installment in the series.

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### Michael D. Steele

### Rick Stuart and Matt Chedister

While filling three-dimensional letters, students analyzed the relationship between the height of water level and elapsed time.

### Thérèse Cozzo and Joseph Cozzo

This lesson provides an opportunity for students to use mathematical modeling and explore right-triangle trigonometry in the context of protecting battleships.

### Arnulfo Pérez, Bailey Braaten, and Robert MacConnell

A hands-on, project-based modeling unit illustrates how real-world inquiry deepens student engagement with function concepts.

### Arsalan Wares

Determining exact values of trigonometric ratios remains an integral part of the high school mathematics curriculum. Students learn to use 45-45-90° triangles and 30-60-90° triangles to determine exact function values of angles of 30°, 45°, and 60°. Such exact-value ratios can help to determine trigonometric ratios for nonstandard angle measures when trigonometric identities and algebra are used. In this lesson, students apply a geometric approach to determine exact-value trig ratios for angle measures of 22.5°, 67.5°, 15°, and 75°. Some students can extend that approach to other nonstandard angle measures.

### Karine S. Ptak

To maximize classroom time spent on practice and concept attainment, a teaching team discarded traditional warm-up activities and homework assignments.

### Joseph Muller and Ksenija Simic-Muller

What happens with cat populations when they are not controlled? Consider the case of Aoshima Island in Japan. Aoshima Island is called a cat island: Its cat population is 130 and growing; its human population is 13. The cats live in colonies and are fed and cared for by people who live on the islands.

MT's letters to the editor department. Readers comment on published articles and share their mathematical interests.

### Kristen Lew and Juan Pablo Mejía-Ramos

This study examined the genre of undergraduate mathematical proof writing by asking mathematicians and undergraduate students to read 7 partial proofs and identify and discuss uses of mathematical language that were out of the ordinary with respect to what they considered conventional mathematical proof writing. Three main themes emerged: First, mathematicians believed that mathematical language should obey the conventions of academic language, whereas students were either unaware of these conventions or unaware that these conventions applied to proof writing. Second, students did not fully understand the nuances involved in how mathematicians introduce objects in proofs. Third, mathematicians focused on the context of the proof to decide how formal a proof should be, whereas students did not seem to be aware of the importance of this factor.