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Aims.The goal of this project is to construct an estimator for the masses of supermassive black holes in active galactic nuclei (AGNs) based on the broad Hαemission line. Methods.We made use of published reverberation mapping data. We remeasured all Hαtime lags from the original data as we find that reverberation measurements are often improved by detrending the light curves. Results.We produced mass estimators that require only the Hαluminosity and the width of the Hαemission line as characterized by either the full width at half maximum or the line dispersion. Conclusions.It is possible, on the basis of a single spectrum covering the Hαemission line, to estimate the mass of the central supermassive black hole in AGNs with all three parameters believed to affect mass measurement – luminosity, line width, and Eddington ratio – taken into account. The typical formal accuracy in such estimates is of order 0.2–0.3 dex relative to the reverberation-based masses. 

We share three suggestions for how teachers can more productively use board work to scaffold joint sense making: (1) make the public record precise; (2) purposefully organize the public record; and (3) take advantage of the public record by referencing it in meaningful ways. 

We present new Very Large Telescope Interferometer (VLTI)/GRAVITY near-infrared interferometric measurements of the angular size of the innermost hot dust continuum for 14 type 1 active galactic nuclei (AGNs). The angular sizes are resolved on scales of ∼0.7 mas and the inferred ring radii range from 0.028 to 1.33 pc, comparable to those reported previously and a factor of 10−20 smaller than the mid-infrared sizes in the literature. Combining our new data with previously published values, we compiled a sample of 25 AGNs with bolometric luminosity ranging from 10⁴²to 10⁴⁷erg s⁻¹, with which we studied the radius-luminosity (R − L) relation for the hot dust structure. Our interferometric measurements of radius are offset by a factor of 2 from the equivalent relation derived through reverberation mapping. Using a simple model to explore the dust structure’s geometry, we conclude that this offset can be explained if the 2 μm emitting surface has a concave shape. Our data show that the slope of the relation is in line with the canonicalR ∝ L^0.5when using an appropriately non-linear correction for bolometric luminosity. In contrast, using optical luminosity or applying a constant bolometric correction to it results in a significant deviation in the slope, suggesting a potential luminosity dependence on the spectral energy distribution. Over four orders of magnitude in luminosity, the intrinsic scatter around theR − Lrelation is 0.2 dex, suggesting a tight correlation between the innermost hot dust structure size and the AGN luminosity. 

Productive use of student mathematical thinking is a critical aspect of effective teaching that is not yet fully understood. We have previously conceptualized the teaching practice of building on student mathematical thinking and the four elements that comprise it. In this paper we begin to unpack this complex practice by looking closely at its third element, Conduct. Based on an analysis of secondary mathematics teachers’ enactments of building, we describe the critical aspects of conducting a whole-class discussion that is focused on making sense of a high-leverage student contribution. 

We share a decomposition of building on MOSTs—a teaching practice that takes advantage of high-leverage instances of student mathematical contributions made during whole-class interaction. This decomposition resulted from an iterative process of teacher-researchers enacting conceptions of the building teaching practice that were refined based on our study of their enactments. We elaborate the four elements of building: (a) Establish the student mathematics of the MOST as the object to be discussed; (b) Grapple Toss that object in a way that positions the class to make sense of it; (c) Conduct a whole-class discussion that supports the students in making sense of the student mathematics of the MOST; and (d) Make Explicit the important mathematical idea from the discussion. We argue for the value of this practice in improving in-the-moment use of high-leverage student mathematical thinking during instruction. 

The more researchers understand the subtleties of teaching practices that productively use student thinking, the better we can support teachers to develop these teaching practices. In this paper, we report the results of an exploration into how secondary mathematics teachers’ use of public records appeared to support or inhibit their efforts to conduct a sense-making discussion around a particular student contribution. We use cognitive load theory to frame two broad ways teachers used public records—manipulating and referencing—to support establishing and maintaining students’ thinking as objects in sense-making discussions. 

Productive use of student mathematical thinking is a critical aspect of effective teaching that is not yet fully understood. We have previously conceptualized the teaching practice of building on student mathematical thinking and the four elements that comprise it. In this paper, we begin to unpack this complex practice by looking closely at its third element, Conduct. Based on an analysis of secondary mathematics teachers' enactments of building, we describe the critical aspects of conducting a whole-class discussion that is focused on making sense of a high-leverage student contribution. 

By using the GRAVITY instrument with the near-infrared (NIR) Very Large Telescope Interferometer (VLTI), the structure of the broad (emission-)line region (BLR) in active galactic nuclei (AGNs) can be spatially resolved, allowing the central black hole (BH) mass to be determined. This work reports new NIR VLTI/GRAVITY interferometric spectra for four type 1 AGNs (Mrk 509, PDS 456, Mrk 1239, and IC 4329A) with resolved broad-line emission. Dynamical modelling of interferometric data constrains the BLR radius and central BH mass measurements for our targets and reveals outflow-dominated BLRs for Mrk 509 and PDS 456. We present an updated radius-luminosity (R-L) relation independent of that derived with reverberation mapping (RM) measurements using all the GRAVITY-observed AGNs. We find our R-L relation to be largely consistent with that derived from RM measurements except at high luminosity, where BLR radii seem to be smaller than predicted. This is consistent with RM-based claims that high Eddington ratio AGNs show consistently smaller BLR sizes. The BH masses of our targets are also consistent with the standardM_BH-σ_*relation. Model-independent photocentre fitting shows spatial offsets between the hot dust continuum and the BLR photocentres (ranging from ∼17 μas to 140 μas) that are generally perpendicular to the alignment of the red- and blueshifted BLR photocentres. These offsets are found to be related to the AGN luminosity and could be caused by asymmetricK-band emission of the hot dust, shifting the dust photocentre. We discuss various possible scenarios that can explain this phenomenon.