Basketball, Algebra, and Probabilities

This article is an attempt to illustrate some humanistic aspects of mathematics in context, in particular, sports and scoring (basketball). The intriguing and dynamic illustrations demonstrate innovative and creative ways of integrating basketball snapshots into the pedagogy of a high school or college-level mathematics-in-context course. I have used this activity with several mathematics education students in a mathematics-in-context class as they worked in groups of five. I include here a presentation and a discussion of their explorations and analyses

Dramathizing Functions: Building Connections between Mathematics and Arts

This article focuses on connections between mathematics and performance arts (drama). More specifically we offer an exposition of a segment of college algebra mathematics (an introduction to functions), with an approach primarily emphasizing the aesthetic aspects of mathematical learning, teaching, and performing

Regional odontodysplasia of the deciduous and permanent teeth associated with eruption disorders : a case report

Regional odontodysplasia (RO) is an unusual, non-hereditary anomaly of the dental hard tissues with characteristic clinical, radiographic and histological findings. Clinically, RO affects the primary and permanent dentition in the maxilla and mandible or both jaws. Radiographically, there is a lack of contrast between the enamel dentin, both of which are less radiopaque than unaffected counterparts. Additionally, enamel and dentin layers are thin, giving the teeth a ?ghost-like? appearance. Histologically, areas of hypocalcified enamel are visible and enamel prisms appear irregular in direction. Coronal dentin is fibrous, consisting of clefts and a reduced number of dentinal tubules; radicular dentin is generally more normal in structure and calcification. The RO etiology is uncertain; numerous factors have been suggested and considered as local trauma, irradiation, hypophosphatasia, hypocalcemia, hyperpyrexia. The treatment of RO has given rise to controversy. These cases require a continuous and multidisciplinary approach. Most clinicians advocate extracting the affected teeth as soon as possible and inserting a prosthetic replacement. Other clinicians prefer restorative procedures, if possible, to protect the affected erupted teeth. A case of RO in an 8 year-old male whose chief complaint was the absence of eruption of permanent teeth is presented. Clinical, radiographic and histological findings are described

What are the Odds at the Russia 2018 FIFA World Cup?

In this note we explore the group stage competition format in a standard FIFA World Cup Soccer Championship group phase. The group stage involves thirty-two teams divided into eight groups of four teams each, based on a draw that takes the national teams’ seeding and geographical location into consideration. Each of the four teams in a given group is scheduled to play one match against every other team in the same group. Upon the completion of six games in each of the eight groups (for a total of 48 games), the top two highest scoring teams (the winner and the runner-up) advance to the knockout stage. In this note we focus on the forty different ways (sequential configurations or states) that a group stage in an arbitrary group can result at the end of the group stage upon the completion of the six games in a typical World Cup Championship. We generate simulations for these configurations on spreadsheets. We use the Russia 2018 FIFA World Cup as an example, along with other relevant historical data, to compare and contrast theoretical versus actual configurations and their probabilities

Pattern Blocks Art

Pattern blocks are versatile manipulatives facilitating connections that can be made among various strands of mathematics such as number sense, algebra, geometry and measurement, spatial reasoning, probability and trigonometry. This note focuses on an artistic interpretation of the pattern blocks with primary focus on convex polygons made with pattern blocks, and describes five mathematically rich activities using them

A case study on the investigation of reasoning skills in geometry

The aim of this study is to evaluate the reasoning skills in geometry-related subjects of six 8th Grade students. The study data were obtained at the end of the 2011-2012 spring period in a public elementary school. The study uses a case study with qualitative research techniques to investigate how students use reasoning skills. In this study, six geometry problems were used to collect the study data. The students were asked to think aloud when solving the  problems so as to be better able to explain their thoughts. From the data obtained, it was identified that the processes involved when demonstrating reasoning skills showed a number of differences.Keywords: geometry, reasoning, reasoning skills, teaching geometr

Eccrine porocarcinoma of the head: An important differential diagnosis in the elderly patient

Background: Eccrine porocarcinoma is a rare malignant tumor of the sweat gland, characterized by a broad spectrum of clinicopathologic presentations. Surprisingly, unlike its benign counterpart eccrine poroma, eccrine porocarcinoma is seldom found in areas with a high density of eccrine sweat glands, like the palms or soles. Instead, eccrine porocarcinoma frequently occurs on the lower extremities, trunk and abdomen, but also on the head, resembling various other skin tumors, as illustrated in the patients described herein. Observations: We report 5 cases of eccrine porocarcinoma of the head. All patients were initially diagnosed as having epidermal or melanocytic skin tumors. Only after histopathologic examination were they classified as eccrine porocarcinoma, showing features of epithelial tumors with abortive ductal differentiation. Characteristic clinical, histopathologic and immunohistochemical findings of eccrine porocarcinomas are illustrated. Conclusion: Eccrine porocarcinomas are potentially fatal adnexal malignancies, in which extensive metastatic dissemination may occur. Porocarcinomas are commonly overlooked, or misinterpreted as squamous or basal cell carcinomas as well as other common malignant and even benign skin tumors. Knowledge of the clinical pattern and histologic findings, therefore, is crucial for an early therapeutic intervention, which can reduce the risk of tumor recurrence and serious complications. Copyright (c) 2008 S. Karger AG, Basel

Plane Figurate Number Proofs Without Words Explained With Pattern Blocks

This article focuses on an artistic interpretation of pattern block designs with primary focus on the connection between pattern blocks and plane figurate numbers. Through this interpretation, it tells the story behind a handful of proofs without words (PWWs) that are inspired by such pattern block designs

Numerical modeling of porous media using a 3D Darcy-Forchheimer approach

LAUREA MAGISTRALEModellare la porosità nella fluidodinamica computazionale (CFD) in genere non consente una rappresentazione esplicita delle strutture dei pori. Tipicamente si utilizzano modelli macroscopici studiati per replicare effetti equivalenti sul flusso, quali salti di pressione e deflessioni del flusso. Un metodo per modellare i mezzi porosi nella CFD è il modello Darcy-Forchheimer, il quale introduce un termine con segno negativo nell'equazione di bilancio della quantità di moto. Questo termine è definito da un tensore che consente di rappresentare il comportamento anisotropo del mezzo poroso. Questa tesi presenta l'implementazione del modello Darcy-Forchheimer con una formulazione tensoriale completa in OpenFOAMv2306, colmando una lacuna nelle attuali capacità di modellazione. Il nuovo modello offre un'ampia flessibilità applicativa, indipendente dal tipo di solutore, e consente di catturare con precisione gli effetti anisotropi nei mezzi porosi. Il modello sviluppato ha riprodotto con successo il comportamento previsto e le distribuzioni di carico sullo schermo poroso nei casi di validazione. Tuttavia, se confrontato con altre strategie di simulazione, come l'approccio Pressure Velocity Jump (PVJ), sono emerse alcune discrepanze, che indicano la necessità di coefficienti di scala e di ulteriori indagini. Nel complesso, questa tesi introduce un modello più versatile e accurato per la simulazione del flusso attraverso i mezzi porosi, con potenziali applicazioni in vari campi dell'ingegneria.Porous modeling in computational fluid dynamics (CFD) typically does not allow for an explicit representation of pore structures. Instead, it relies on macroscopic models designed to replicate equivalent effects on the flow, such as pressure jumps and flow deflections. One method for modeling porous media in CFD is the Darcy-Forchheimer model, which introduces a sink term in the momentum balance equation. This sink term is defined by a tensor, enabling the representation of anisotropic behavior in the porous medium. This thesis presents the implementation of the Darcy-Forchheimer model with a full tensorial formulation in OpenFOAMv2306, addressing a gap in current modeling capabilities. The new model offers broad application flexibility, independent of solver type, and enables the accurate capture of anisotropic effects in porous media. The developed model successfully reproduced the expected behavior and load distributions on the porous screen in validation cases. However, when compared with other simulation strategies, such as the Pressure Velocity Jump (PVJ) approach, some discrepancies emerged, indicating the need for scaling coefficients and further investigation. Overall, this thesis introduces a more versatile and accurate model for simulating flow through porous media, with potential applications across various engineering fields

Preservice Teachers’ Mapping Structures Acting on Representational Quantities

In this article, I write about my research on five preservice secondary teachers’ (PST) understanding and sense making of representational quantities associated with magnetic color cubes and tiles. Data came from individual interviews during which I asked PST problems guided by five main tasks: prime and composite numbers, summation of counting numbers, odd numbers, even numbers, and polynomial expressions in x and y. My work drew upon an analysis framework (Behr et. al, 1994) supported by a unit coordination construct (Steffe, 1988) associated with linear and areal quantities inherent in the nature of figures produced by these PST. Linear quantities can be thought of as generated via linear measurement units (e.g., inches, centimeters, units) whereas areal quantities are generated via areal measurement units (e.g., square inches, square centimeters, square units, etc.) I used thematic analysis supported by constant comparison and retrospective analysis to explain my theories and hypotheses concerning PST’s representational quantities. I developed a data analysis framework which I named “Relational Notation” to describe these PST’s understanding of linear and areal units. PST also treated the quantitative multiplication and addition operations as some kind of functions, mappings, when expressing the area of their growing rectangles made of magnetic color cubes and tiles as sums and products. Their behavior necessitated the existence of another component for my data analysis framework which I called “Mapping Structures”