A Unifying Theme

2003 ◽  
Vol 8 (8) ◽  
pp. 387
Keyword(s):  
Middle Grades ◽  
Proportional Reasoning ◽  
Additive Models ◽  
Classroom Teaching ◽  
Central Component ◽  
Active Engagement ◽  
Mathematics Students ◽  
Linear Relationships

Roportionality is a central component of middle-grades mathematics. Students encounter contexts and concepts involving ratios, proportions, fractions, decimals, percents, similarity, rates, linear relationships, and related situations when counting and additive models are no longer appropriate or efficient methods to apply. Experience has shown us that it takes time and active engagement to develop the thinking needed to conceptualize these areas. Such understanding lays an important foundation for later mathematics. To provide you with classroom-based ideas and examples and to stimulate your own thinking about proportional reasoning, the Editorial Panel brings you this focus issue. The authors share activities and insights for classroom teaching on a variety of proportional reasoning topics.

Research into Practice: Similarity in the Middle Grades

1988 ◽  
Vol 35 (9) ◽  
pp. 32-35 ◽  
Author(s):  
Glenda Lappan ◽  
Ruhama Even
Keyword(s):  
United States ◽  
Middle Grades ◽  
Proportional Reasoning ◽  
International Study ◽  
Indirect Measurement ◽  
Grade 8 ◽  
Second International

Simila rity is an important topic in geometry, basic to understanding the geometry of indirect measurement, proportional reasoning, scale drawing and modeling, and the nature of growing. When United States’ teachers were asked to rate the importance of this topic for the Second International Study of Mathematics, they rated similarity of plane figures as being important for all students in grade 8.

Improving Middle School Teachers' Reasoning about Proportional Reasoning

2003 ◽  
Vol 8 (8) ◽  
pp. 398-403
Author(s):  
Charles S. Thompson ◽  
William S. Bush
Keyword(s):  
Professional Development ◽  
Middle School ◽  
Middle School Teachers ◽  
Middle Grades ◽  
Proportional Reasoning ◽  
Development Project ◽  
School Teachers

Article describes a professional development project to increase teachers' understanding of proportional reasoning, the thinking patterns associated with proportional reasoning, and the applications of proportional reasoning across the middle-grades curriculum.

Pattern-block frenzy

2012 ◽  
Vol 19 (2) ◽  
pp. 116-121
Author(s):  
Katie L. Anderson
Keyword(s):  
Proportional Reasoning ◽  
Sixth Graders ◽  
Virtual Manipulatives ◽  
Mathematical Tasks ◽  
Active Engagement ◽  
Classroom Discussions ◽  
Success Stories ◽  
Use Of Technology ◽  
Creative Solutions ◽  
Effective Use

Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K–grade 6 classrooms. This article describes a set of lessons where sixth graders use virtual pattern blocks to develop proportional reasoning. Students' work with the virtual manipulatives reveals a variety of creative solutions and promotes active engagement. The author suggests that technology is most effective when coupled with worthwhile mathematical tasks and rich classroom discussions.

Four Ways to Determine Equivalent Ratios

2018 ◽  
Vol 24 (1) ◽  
pp. 48-52 ◽  
Author(s):  
Ricardo Martinez ◽  
Ji Yeong
Keyword(s):  
Common Core ◽  
Middle Grades ◽  
Proportional Reasoning ◽  
Real Life ◽  
Real Numbers ◽  
The Common

Ratios and proportions are important concepts that occur in real life, but they are difficult to learn and complicated to teach (Lamon 2007). In general, proportional reasoning in the middle grades is an area in need of attention because of its connection to later concepts (Ojose 2015). In this article, we explain the meanings of ratios, proportions, and equivalent ratios and then provide useful methods to determine equivalent ratios using students' examples. Because multiple definitions exist, it is necessary to determine definitions to avoid possible confusion. Johnson (2010) defines a ratio as a pair of positive nonzero real numbers such that there are a units for every b units, written a:b; read a to b; and represented as a fraction, a/b. We follow definitions suggested by the writers of the Common Core (CCSSI 2010): Fraction representation is known as the associated unit rate or the value of a ratio A:B and is found by dividing B into A, where “equivalent ratios have the same unit rate” (McCallum, Zimba, and Daro 2011). Using both contextual and noncontextual proportion tasks, we saw various ways that students found the equivalency of ratios as well as misconceptions they have.

Take Time for Action: Proportional Reasoning

2000 ◽  
Vol 5 (5) ◽  
pp. 310-313
Author(s):  
Jane Lincoln Miller ◽  
James T. Fey
Keyword(s):  
Direct Instruction ◽  
Middle Grades ◽  
Collaborative Work ◽  
Proportional Reasoning ◽  
New Approaches ◽  
Mathematics Program ◽  
Authentic Problems

Developing facility with proportional reasoning should be “one of the hallmarks of the middle grades mathematics program” (NCTM 1998, 213). Such reasoning has long been a problem for students, however, because of the complexity of thinking that it requires. Several standards-based curriculumreform projects have explored new approaches to developing students' proportional reasoning concepts and skills. Instead of offering direct instruction on standard algorithms for checking equivalence of ratios or solving proportion equations, these new approaches encourage students to build understanding and strategies for proportional reasoning through guided collaborative work on authentic problems.

Improper Applications of Proportional Reasoning

2003 ◽  
Vol 9 (4) ◽  
pp. 204-209
Author(s):  
Wim Van Dooren, ◽  
Dirk De Bock ◽  
Lieven Verschaffel ◽  
Dirk Janssens
Keyword(s):  
Mathematics Education ◽  
Middle Grades ◽  
Proportional Reasoning ◽  
Deep Understanding ◽  
School Mathematics ◽  
Method Of Solution

One of the major goals of elementary and middle-grades mathematics education is for students to obtain a deep understanding of the proportional model in a variety of forms and applications. However, the reinforcement of proportionality at numerous occasions in school mathematics, along with the teaching of some standardized methods for solving proportionality problems, appear to lead to a resistant tendency in some students and adults to see and apply proportions everywhere. This same application occurs in situations where another method of solution is appropriate. Along with mastering the proportional scheme, its misuse seems to appear, as well. This overgeneralization of proportion has many faces: It has been found at different age levels and in a variety of mathematical domains, such as elementary arithmetic (Cramer, Post, and Currier 1993), algebra (Matz 1982), geometry (De Bock, Verschaffel, and Janssens 1998, 2002) and probability (Van Dooren et al. 2002).

scholarly journals The Development and Implementation of a Human-Caused Wildland Fire Occurrence Prediction System for the Province of Ontario, Canada

2020 ◽  
Author(s):  
Douglas G. Woolford ◽  
David L. Martell ◽  
Colin McFayden ◽  
Jordan Evens ◽  
Aaron Stacey ◽  
...  
Keyword(s):  
Fire Management ◽  
Wildland Fire ◽  
Fire Suppression ◽  
Generalized Additive Models ◽  
Temporal Trends ◽  
Additive Models ◽  
Fire Occurrence ◽  
Fine Scale ◽  
High Resolution Data ◽  
Linear Relationships

We describe the development and implementation of an operational human-caused wildland fire occurrence prediction (FOP) system in the Province of Ontario, Canada. A suite of supervised statistical learning models was developed using more than 50 years of high-resolution data over a 73.8 million hectare study area, partitioned into Ontario’s Northwest and Northeast Fire Management Regions. A stratified modelling approach accounts for different seasonal baselines regionally and for a set of communities in the far north. Response-dependent sampling and modelling techniques using logistic Generalized Additive Models are used to develop a fine-scale, spatio-temporal FOP system with models that include non-linear relationships with key predictors. These predictors include inter and intra-annual temporal trends, spatial trends, ecological variables, fuel moisture measures, human land use characteristics and a novel measure of human activity. The system produces fine-scale, spatially explicit maps of daily probabilistic human-caused FOP based on locally observed conditions along with point and interval predictions for the expected number of fires in each region. A simulation-based approach for generating the prediction intervals is described. Daily predictions were made available to fire management practitioners through a custom dashboard and integrated into daily regional planning to support detection and fire suppression preparedness needs.

Assessing Proportional Thinking

2003 ◽  
Vol 9 (3) ◽  
pp. 166-172
Author(s):  
George W. Bright ◽  
Jeane M. Joyner ◽  
Charles Wallis
Keyword(s):  
Teaching And Learning ◽  
Middle Grades ◽  
Proportional Reasoning ◽  
Making Sense

Proportional thinking is an important part of mathematics in the middle grades and “connects many of the mathematics topics studied in grades 6–8” (NCTM 2000, p. 217). Partly in response to this need, NCTM's 2002 Yearbook titled Making Sense of Fractions, Ratios, and Proportions (Bright and Litwiller 2002; Litwiller and Bright 2002) addressed proportional reasoning across the grades but with special emphasis on the teaching and learning of this important area in the middle grades.

scholarly journals Analyzing the Influence of The Secondary to University Core Courses Alignment

2021 ◽  
Vol 3 (2) ◽  
pp. 21-32
Author(s):  
Flamur Bidaj ◽  
Anila Paparisto
Keyword(s):  
High School ◽  
Student Success ◽  
Core Curriculum ◽  
The Other ◽  
Classroom Teaching ◽  
First Year ◽  
Academic Factors ◽  
Linear Relationships ◽  
Core Courses ◽  
Transition From High School

The student success  in the first year, is influenced, among the other things, even by academic factors: college readiness, core  curriculum in high school, cognitive, etc.  The alignment analysis of the some core courses between university and high school, is the main objective of this article. The qualitative method and  student questionnaires, are used to carry out this analysis. The results obtained indicate the influence of curriculum alignment on classroom teaching and student success for three core courses: Mathematics, Physics and Chemistry,  on the first year. Using the regress analyze,  some linear  relationships are found, either for two classroom teaching and student success indicators as well. Based on these results, we  emphasize the necessity for a greater student support during the transition from high school to university, in order to foster student success. This study was conducted in engineering study field, but it can be used in the other fields as well.

Dilations and the Equation of a Line

2016 ◽  
Vol 21 (7) ◽  
pp. 406-414
Author(s):  
David A. Yopp
Keyword(s):  
Proportional Reasoning ◽  
Similar Triangle ◽  
Linear Relationships ◽  
Transformational Geometry

Track students' understanding of proportional reasoning by combining transformational geometry, similar-triangle reasoning, and linear relationships.

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