Metáforas y pensamiento matemático en la formación de educadoras de párvulos / Metaphors and mathematical thinking in the formation of nursery educators

Este trabajo describe el abordaje metafórico y como este se relaciona con el desarrollo del pensamiento matemático, para realizar mejoras en la formación de educadoras de párvulos. Se presenta una experiencia con el cuadrado de Shao Yong, que permite la integración del abordaje metafórico con el desarrollo del pensamiento matemático, donde se estudia la construcción del concepto número con educadoras de párvulos. Este material puede ser considerado como sugerencias de estrategias metodológicas y como lectura base para la discusión y reflexión pedagógica. Se entrega información para docentes innovadores de prácticas de aula y aporta elementos de cambios a la educación y comprensión de la matemática.ABSTRACTThis paper describes the metaphorical approach and how it relates to the development of mathematical thinking, to make improvements in the training of nursery educators. We present an experience with the Shao Yong square, which allows the integration of the metaphorical approach with the development of mathematical thinking, where we study the construction of the concept number with nursery school educators. The proposal highlights methodological strategies and provides as a framework for reading, discussing and pedagogical reflection. Information is provided for innovative classroom teachers of classroom fostering the understanding and teaching of mathematics

A Six-Planet System Around the Star HD 34445

We present a new precision radial velocity dataset that reveals a multi-planet system orbiting the G0V star HD 34445. Our 18-year span consists of 333 precision radial velocity observations, 56 of which were previously published, and 277 which are new data from Keck Observatory, Magellan at Las Campanas Observatory, and the Automated Planet Finder at Lick Observatory. These data indicate the presence of six planet candidates in Keplerian motion about the host star with periods of 1057, 215, 118, 49, 677, and 5700 days, and minimum masses of 0.63, 0.17, 0.1, 0.05, 0.12 and 0.38 Jupiter masses respectively. The HD 34445 planetary system, with its high degree of multiplicity, its long orbital periods, and its induced stellar radial velocity half-amplitudes in the range $2 \,{\rm m\, s^{-1}} \lesssim K \lesssim 5\,{\rm m\, s^{-1}}$ is fundamentally unlike either our own solar system (in which only Jupiter and Saturn induce significant reflex velocities for the Sun), or the Kepler multiple-transiting systems (which tend to have much more compact orbital configurations)Comment: 10 pages, 11 figure

The test case of HD26965: difficulties disentangling weak Doppler signals from stellar activity

We report the discovery of a radial velocity signal that can be interpreted as a planetary-mass candidate orbiting the K dwarf HD26965, with an orbital period of 42.364$\pm$0.015 days, or alternatively, as the presence of residual, uncorrected rotational activity in the data. Observations include data from HIRES, PFS, CHIRON, and HARPS, where 1,111 measurements were made over 16 years. Our best solution for HD26965 $b$ is consistent with a super-Earth that has a minimum mass of 6.92$\pm$0.79 M$_{\oplus}$ orbiting at a distance of 0.215$\pm$0.008 AU from its host star. We have analyzed the correlation between spectral activity indicators and the radial velocities from each instrument, showing moderate correlations that we include in our model. From this analysis, we recover a $\sim$38 day signal, which matches some literature values of the stellar rotation period. However, from independent Mt. Wilson HK data for this star, we find evidence for a significant 42 day signal after subtraction of longer period magnetic cycles, casting doubt on the planetary hypothesis for this period. Although our statistical model strongly suggests that the 42-day signal is Doppler in origin, we conclude that the residual effects of stellar rotation are difficult to fully model and remove from this dataset, highlighting the difficulties to disentangle small planetary signals and photospheric noise, particularly when the orbital periods are close to the rotation period of the star. This study serves as an excellent test case for future works that aim to detect small planets orbiting `Sun-like' stars using radial velocity measurements.Comment: 16 pages, 10 figures, 13 tables, accepted for publication in A

Low Mass Companions for Five Solar-Type Stars from the Magellan Planet Search Program

We report low mass companions orbiting five Solar-type stars that have emerged from the Magellan precision Doppler velocity survey, with minimum (Msini) masses ranging from 1.2 to 25 Mjup. These nearby target stars range from mildly metal-poor to metal-rich, and appear to have low chromospheric activity. The companions to the brightest two of these stars have previously been reported from the CORALIE survey. Four of these companions (HD 48265-b, HD 143361-b, HD 28185-b, HD 111232-b) are low-mass Jupiter-like planets in eccentric intermediate and long-period orbits. On the other hand, the companion to HD 43848 appears to be a long period brown dwarf in a very eccentric orbit.Comment: Accepted for publication on ApJ, 26 pages, 10 figures, 7 table

MagAO Imaging of Long-period Objects (MILO). I. A Benchmark M Dwarf Companion Exciting a Massive Planet around the Sun-like Star HD 7449

We present high-contrast Magellan adaptive optics (MagAO) images of HD 7449, a Sun-like star with one planet and a long-term radial velocity (RV) trend. We unambiguously detect the source of the long-term trend from 0.6-2.15 \microns ~at a separation of \about 0\fasec 54. We use the object's colors and spectral energy distribution to show that it is most likely an M4-M5 dwarf (mass \about 0.1-0.2 \msun) at the same distance as the primary and is therefore likely bound. We also present new RVs measured with the Magellan/MIKE and PFS spectrometers and compile these with archival data from CORALIE and HARPS. We use a new Markov chain Monte Carlo procedure to constrain both the mass ($> 0.17$ \msun ~at 99$\%$ confidence) and semimajor axis (\about 18 AU) of the M dwarf companion (HD 7449B). We also refine the parameters of the known massive planet (HD 7449Ab), finding that its minimum mass is $1.09^{+0.52}_{-0.19}$ \mj, its semimajor axis is $2.33^{+0.01}_{-0.02}$ AU, and its eccentricity is $0.8^{+0.08}_{-0.06}$. We use N-body simulations to constrain the eccentricity of HD 7449B to $\lesssim$ 0.5. The M dwarf may be inducing Kozai oscillations on the planet, explaining its high eccentricity. If this is the case and its orbit was initially circular, the mass of the planet would need to be $\lesssim$ 1.5 \mj. This demonstrates that strong constraints on known planets can be made using direct observations of otherwise undetectable long-period companions.Comment: Corrected planet mass error (7.8 Mj --> 1.09 Mj, in agreement with previous studies

Two Jupiter-Mass Planets Orbiting HD 154672 and HD 205739

We report the detection of the first two planets from the N2K Doppler planet search program at the Magellan telescopes. The first planet has a mass of M sin i = 4.96 M_Jup and is orbiting the G3 IV star HD154672 with an orbital period of 163.9 days. The second planet is orbiting the F7 V star HD205739 with an orbital period of 279.8 days and has a mass of M sin i = 1.37 M_Jup. Both planets are in eccentric orbits, with eccentricities e = 0.61 and e = 0.27, respectively. Both stars are metal rich and appear to be chromospherically inactive, based on inspection of their Ca II H and K lines. Finally, the best Keplerian model fit to HD205739b shows a trend of 0.0649 m/s/day, suggesting the presence of an additional outer body in that system.Comment: 16 pages, 5 figures, accepted for publication on A

DOPPLER MONITORING OF THE WASP-47 MULTIPLANET SYSTEM

We present precise Doppler observations of WASP-47, a transiting planetary system featuring a hot Jupiter with both inner and outer planetary companions. This system has an unusual architecture and also provides a rare opportunity to measure planet masses in two different ways: the Doppler method, and the analysis of transit-timing variations (TTV). Based on the new Doppler data, obtained with the Planet Finder Spectrograph on the Magellan/Clay 6.5 m telescope, the mass of the hot Jupiter is 370 ± 29[subscript ⊕]. This is consistent with the previous Doppler determination as well as the TTV determination. For the inner planet WASP-47e, the Doppler data lead to a mass of 12.2 ± 3.7[subscript ⊕], in agreement with the TTV-based upper limit of <22 M[subscript ⊕] (95% confidence). For the outer planet WASP-47d, the Doppler mass constraint of 10.4 ± 8.4[subscript ⊕] is consistent with the TTV-based measurement of $15.2[+6.7 over -7.6] M[subscript ⊕].United States. National Aeronautics and Space Administration (Origins Program Grant NNX11AG85G

Doppler monitoring of the WASP-47 multiplanet system

We present precise Doppler observations of WASP-47, a transiting planetary system featuring a hot Jupiter with both inner and outer planetary companions. This system has an unusual architecture and also provides a rare opportunity to measure planet masses in two different ways: the Doppler method, and the analysis of transit-timing variations (TTV). Based on the new Doppler data, obtained with the Planet Finder Spectrograph on the Magellan/Clay 6.5 m telescope, the mass of the hot Jupiter is 370+/- 29M⊕. This is consistent with the previous Doppler determination as well as the TTV determination. For the inner planet WASP-47e, the Doppler data lead to a mass of 12.2 +/-3.7 M⊕ in agreement with the TTV-based upper limit of <22 M⊕ (95% confidence). For the outer planet WASP-47d, the Doppler mass constraint of 10.4 +/- 8.4 M⊕ is consistent with the TTV-based measurement of 15.2 -7.6 to +6.7 M