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Components of Algebraic Thinking NCTM MATH 2203

Components of Algebraic Thinking NCTM MATH 2203 - The...

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Unformatted text preview: The Navigations series seeks to guide readers through the five . strands of Principles and Srandardrfiar School Mathematics in order to help them translate the Standards and Principles into action and to illustrate the growth and connectedness of content ideas from prekindergarten through grade 12. The Navigations through the algebra curriculum reflect NCTM’s vision of how algebraic concepts should be introduced, how they grow, what to expect of students during and at the end of each grade band, how'to assess what students know, and how selected instructional activities can contribute to learning. Fundamental Components of Algebraic Thinking The Algebra Standard emphasizes relationships among quantities and the ways in which quantities change relative to one another. To ' think algebraically, one must be able to underStand patterns, relations, and functions; represent and analyze mathematical situations and struc— tures using algebraic symbols; use mathematical models to represent: and understand quantitative relationships; and analyze change in vari- ous contexts. Each of these basic components evolves as students grow and mature. Understanding patterns, relatiOns, and functions Young children begin to explore patterns in the world around them through experiences with such things as color, size, shape, design, words, syllables, musical tones, rhythms, movements, and physical objects. They observe, describe, repeat, extend, compare, and create patterns; they sort, classify, and order objects according to various char—_ acteristics; they predict what comes next and identify missing elements in patterns, they learn to distinguish types of patterns, such as growing or repeating patterns . - In the higher elementary grades, children learn to represent patterns numerically, graphically, and symbolically,a as well as verbally. They begin to look for relationships in numerical and geometric patterns and - analyze how patterns grow or change. By using tables, charts, physical- objects, and symbols, students make and explain generalizations about patterns and use relationships in patterns to make predictions. Students in the middle grades explore patterns expressed in tables, graphs, words, or symbols, with an emphasis on patterns that exhibit linear relationships (constant rate of change). They learn to relate sym- bolic and graphical representations and develop an understanding of the significance of slope and y—intercept. They also explore “What if?” questions to investigate how patterns change, and they distinguish lin- ear frorn nonlinear patterns. In high school, students create and use tables, symbols, graphs, and verbal representations to generalize and analyze patterns, relations, and functions with increasing sophistication, and they convert flexibly among various representations. They compare and contrast situations modeled by different types of functions, and they develop an under— Navigating throogh Algebrain PrekindergartennGrade 2 standing of classes of functions, both linear and nonlinear, and their properties. Their understanding expands to include functions of more than one variable, and they learn to perform transformations such as composing and inverting commonly used functions. - Representing and analyzing mathematical situations and structures using algebraic symbols Young children can illustrate mathematical properties (e.g., the com— mutativity of addition) with objects or specific numbers. They use objects, pictures, words, or symbols to represent mathematical ideas and relationships, including the relationship of equality, and to solve problems. When children are encouraged to describe and represent quantities in different ways, they learn to recognize equivalent repre— sentations and expand their ability to use symbols to communicate their ideas. Later in the elementary grades, children investigate, represent, describe, and explain mathematical properties, and they begin to gener— alize relationships and to use them in computing with whole numbers. They develop notions of the idea and usefulness of variables, which they may express with a box, letter, or other symbol to signify the idea of a variable as a placeholder. They also learn to use variables to describe a rule that relates two quantities or to express relationships using equations. During the middle grades, students encounter additional uses of vari— ables as changing quantities in generalized patterns, formulas, identi— ties, expressions of mathematical properties, equations, and inequalities. They explore notions of dependence and independence as variables change in relation to one another, and they develop facility in recogniz— ing the equivalence of mathematical representations, which they can use to transform expressions; to solve problems; and to relate graphical, tabular, and symbolic representations. They also acquire greater facility with linear equations and demonstrate how the values of slope and y— intercept affect the line. ' High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and func— tions, and they recognize and describe the advantages and'disadvantages of various representations for a particular situation.'Such facility with symbols and alternative representations enables them to analyze a math— ematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solution. Using mathematical. models to represent and understand quantitative relationships Very young children learn to use objects or pictures, and, eventually, symbols to enact stories or model situations that involve the addition or subtraction of whole numbersrAs they progress into the upper elemen— tary grades, children begin to realize that mathematics can be used to model numerical and geometric patterns, scientific experiments, and introduction other physical situations, and they discover that mathematical models have the power to predict as well as to describe. As they employ graphs, tables, and equations to represent relationships and use their models to draw conclusions, students compare various models and investigate whether different models of a particular situation yield the same results. Contextualized problems that can be modeled andsolved using vari- ous representations, such as graphs, tables, and equations, engage middle—grades students. With the aid of technology, they learn to use functions to model patterns of change, including situations in which they generate and represent real data. Although the emphasis is on con— texts that are modeled by linear relationships, students also explore examples of nonlinear relationships, and they use their models to develop and test conjectures. High school students develoP skill in identifying essential quantita— tive relationships in a situation and in determining the type of function with which to model the relationship. They use symbolic expressions to represent relationships arising from various contexts, including situa— tions in which they generate and use data. Using their models, students conjecture about relationships, formulate and test hypotheses, and draw conclusions about the situations being modeled. Analyzing change in various contexts From a very early age, children recognize examples of change in their environment and describe change in qualitative terms, such as get— ting taller, colder,.darker, or heavier. By measuring and comparing quantities, children also learn to describe change quantitatively, such as in keeping track of variations in temperature or the growth of a class— room plant or pet. They learn that some changes are predictable but others are not and that often change can be described mathematically. Later in the elementary grades, children represent change in numerical, tabular, or graphical form, and they observe that patterns of change often involve more than one quantity, such as that the length of a spring increases as additional weights are hung from it. Students in the upper _ grades also begin to study differences in patterns of change and to C0m~ pare changes that occur at a constant rate, such as the cost of buying various numbers of pencils at $0.20 each, with changes whose rates increase or decrease, such as the growth of a seedling. Middle—grades students explore many examples of quantities that change and the graphs that represent those changes; they answer ques— tions about the relationShipsrepresented in the graphs and learn to dis— tinguish rate of change from accumulation (total amount of change). By varying parameters such as the rate of change, students can observe the corresponding changes in the graphs, equations, or tables of values of the relationships. High school students deepen this understanding of how mathematical quantities change and, in particular, of the concept of rate of change. They investigate numerous mathematical situations and real—world phenomena to analyze and make sense of changing rela— tionships; interpret change and rates of change fiom graphical and numerical data, and use algebraic techniques and appropriate technol— ogy to develop and evaluate models of dynamic situations. Navigating through Algebra in Prekindergarten-Grade 2 ...
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Components of Algebraic Thinking NCTM MATH 2203 - The...

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