Common Core State Standards for MATHEMATICS
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
2.1 the Algebra of Sets
Chapter 2 I Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations between sets, developing an “algebra of sets” that strongly resembles aspects of the algebra of sentential logic. In addition, as we discussed in chapter 1, a fundamental goal in mathematics is crafting articulate, thorough, convincing, and insightful arguments for the truth of mathematical statements. We continue the development of theorem-proving and proof-writing skills in the context of basic set theory. After exploring the algebra of sets, we study two number systems denoted Zn and U(n) that are closely related to the integers. Our approach is based on a widely used strategy of mathematicians: we work with specific examples and look for general patterns. This study leads to the definition of modified addition and multiplication operations on certain finite subsets of the integers. We isolate key axioms, or properties, that are satisfied by these and many other number systems and then examine number systems that share the “group” properties of the integers. Finally, we consider an application of this mathematics to check digit schemes, which have become increasingly important for the success of business and telecommunications in our technologically based society. Through the study of these topics, we engage in a thorough introduction to abstract algebra from the perspective of the mathematician— working with specific examples to identify key abstract properties common to diverse and interesting mathematical systems. 2.1 The Algebra of Sets Intuitively, a set is a “collection” of objects known as “elements.” But in the early 1900’s, a radical transformation occurred in mathematicians’ understanding of sets when the British philosopher Bertrand Russell identified a fundamental paradox inherent in this intuitive notion of a set (this paradox is discussed in exercises 66–70 at the end of this section). -
Db Math Marco Zennaro Ermanno Pietrosemoli Goals
dB Math Marco Zennaro Ermanno Pietrosemoli Goals ‣ Electromagnetic waves carry power measured in milliwatts. ‣ Decibels (dB) use a relative logarithmic relationship to reduce multiplication to simple addition. ‣ You can simplify common radio calculations by using dBm instead of mW, and dB to represent variations of power. ‣ It is simpler to solve radio calculations in your head by using dB. 2 Power ‣ Any electromagnetic wave carries energy - we can feel that when we enjoy (or suffer from) the warmth of the sun. The amount of energy received in a certain amount of time is called power. ‣ The electric field is measured in V/m (volts per meter), the power contained within it is proportional to the square of the electric field: 2 P ~ E ‣ The unit of power is the watt (W). For wireless work, the milliwatt (mW) is usually a more convenient unit. 3 Gain and Loss ‣ If the amplitude of an electromagnetic wave increases, its power increases. This increase in power is called a gain. ‣ If the amplitude decreases, its power decreases. This decrease in power is called a loss. ‣ When designing communication links, you try to maximize the gains while minimizing any losses. 4 Intro to dB ‣ Decibels are a relative measurement unit unlike the absolute measurement of milliwatts. ‣ The decibel (dB) is 10 times the decimal logarithm of the ratio between two values of a variable. The calculation of decibels uses a logarithm to allow very large or very small relations to be represented with a conveniently small number. ‣ On the logarithmic scale, the reference cannot be zero because the log of zero does not exist! 5 Why do we use dB? ‣ Power does not fade in a linear manner, but inversely as the square of the distance. -
Feasibility Study for Teaching Geometry and Other Topics Using Three-Dimensional Printers
Feasibility Study For Teaching Geometry and Other Topics Using Three-Dimensional Printers Elizabeth Ann Slavkovsky A Thesis in the Field of Mathematics for Teaching for the Degree of Master of Liberal Arts in Extension Studies Harvard University October 2012 Abstract Since 2003, 3D printer technology has shown explosive growth, and has become significantly less expensive and more available. 3D printers at a hobbyist level are available for as little as $550, putting them in reach of individuals and schools. In addition, there are many “pay by the part” 3D printing services available to anyone who can design in three dimensions. 3D graphics programs are also widely available; where 10 years ago few could afford the technology to design in three dimensions, now anyone with a computer can download Google SketchUp or Blender for free. Many jobs now require more 3D skills, including medical, mining, video game design, and countless other fields. Because of this, the 3D printer has found its way into the classroom, particularly for STEM (science, technology, engineering, and math) programs in all grade levels. However, most of these programs focus mainly on the design and engineering possibilities for students. This thesis project was to explore the difficulty and benefits of the technology in the mathematics classroom. For this thesis project we researched the technology available and purchased a hobby-level 3D printer to see how well it might work for someone without extensive technology background. We sent designed parts away. In addition, we tried out Google SketchUp, Blender, Mathematica, and other programs for designing parts. We came up with several lessons and demos around the printer design. -
The Five Fundamental Operations of Mathematics: Addition, Subtraction
The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms Kenneth A. Ribet UC Berkeley Trinity University March 31, 2008 Kenneth A. Ribet Five fundamental operations This talk is about counting, and it’s about solving equations. Counting is a very familiar activity in mathematics. Many universities teach sophomore-level courses on discrete mathematics that turn out to be mostly about counting. For example, we ask our students to find the number of different ways of constituting a bag of a dozen lollipops if there are 5 different flavors. (The answer is 1820, I think.) Kenneth A. Ribet Five fundamental operations Solving equations is even more of a flagship activity for mathematicians. At a mathematics conference at Sundance, Robert Redford told a group of my colleagues “I hope you solve all your equations”! The kind of equations that I like to solve are Diophantine equations. Diophantus of Alexandria (third century AD) was Robert Redford’s kind of mathematician. This “father of algebra” focused on the solution to algebraic equations, especially in contexts where the solutions are constrained to be whole numbers or fractions. Kenneth A. Ribet Five fundamental operations Here’s a typical example. Consider the equation y 2 = x3 + 1. In an algebra or high school class, we might graph this equation in the plane; there’s little challenge. But what if we ask for solutions in integers (i.e., whole numbers)? It is relatively easy to discover the solutions (0; ±1), (−1; 0) and (2; ±3), and Diophantus might have asked if there are any more. -
Elementary Math Program
North Shore Schools Elementary Mathematics Curriculum Our math program, Math in Focus, is based upon Singapore Math, which places an emphasis on developing proficiency with problem solving while fostering conceptual understanding and fundamental skills. Online resources are available at ThinkCentral using the username and password provided by the classroom teacher. The mathematics curriculum is aligned with the Common Core Learning Standards (CCLS). There are eight Mathematical Practices that set an expectation of understanding of mathematics and are the same for grades K-12. Mathematical Practices 1. Make sense of problems and persevere in 4. Model with mathematics. solving them. 5. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the 7. Look for and make use of structure. reasoning of others. 8. Look for and express regularity in repeated reasoning. The content at each grade level focuses on specific critical areas. Kindergarten Representing and Comparing Whole Numbers Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away. -
Math for Elementary Teachers
Math for Elementary Teachers Math 203 #76908 Name __________________________________________ Spring 2020 Santiago Canyon College, Math and Science Division Monday 10:30 am – 1:25 pm (with LAB) Wednesday 10:30 am – 12:35 pm Instructor: Anne Hauscarriague E-mail: [email protected] Office: Home Phone: 714-628-4919 Website: www.sccollege.edu/ahauscarriague (Grades will be posted here after each exam) Office Hours: Mon: 2:00 – 3:00 Tues/Thurs: 9:30 – 10:30 Wed: 2:00 – 4:00 MSC/CraniumCafe Hours: Mon/Wed: 9:30 – 10:30, Wed: 4:00 – 4:30 Math 203 Student Learning Outcomes: As a result of completing Mathematics 203, the student will be able to: 1. Analyze the structure and properties of rational and real number systems including their decimal representation and illustrate the use of a representation of these numbers including the number line model. 2. Evaluate the equivalence of numeric algorithms and explain the advantages and disadvantages of equivalent algorithms. 3. Analyze multiple approaches to solving problems from elementary to advanced levels of mathematics, using concepts and tools from sets, logic, functions, number theory and patterns. Prerequisite: Successful completion of Math 080 (grade of C or better) or qualifying profile from the Math placement process. This class has previous math knowledge as a prerequisite and it is expected that you are comfortable with algebra and geometry. If you need review work, some resources are: School Zone Math 6th Grade Deluxe Edition, (Grade 5 is also a good review of basic arithmetic skills); Schaum’s Outline series Elementary Mathematics by Barnett Rich; www.math.com; www.mathtv.com; and/or www.KhanAcademy.com. -
Niobrara County School District #1 Curriculum Guide
NIOBRARA COUNTY SCHOOL DISTRICT #1 CURRICULUM GUIDE th SUBJECT: Math Algebra II TIMELINE: 4 quarter Domain: Student Friendly Level of Resource Academic Standard: Learning Objective Thinking Correlation/Exemplar Vocabulary [Assessment] [Mathematical Practices] Domain: Perform operations on matrices and use matrices in applications. Standards: N-VM.6.Use matrices to I can represent and Application Pearson Alg. II Textbook: Matrix represent and manipulated data, e.g., manipulate data to represent --Lesson 12-2 p. 772 (Matrices) to represent payoffs or incidence data. M --Concept Byte 12-2 p. relationships in a network. 780 Data --Lesson 12-5 p. 801 [Assessment]: --Concept Byte 12-2 p. 780 [Mathematical Practices]: Niobrara County School District #1, Fall 2012 Page 1 NIOBRARA COUNTY SCHOOL DISTRICT #1 CURRICULUM GUIDE th SUBJECT: Math Algebra II TIMELINE: 4 quarter Domain: Student Friendly Level of Resource Academic Standard: Learning Objective Thinking Correlation/Exemplar Vocabulary [Assessment] [Mathematical Practices] Domain: Perform operations on matrices and use matrices in applications. Standards: N-VM.7. Multiply matrices I can multiply matrices by Application Pearson Alg. II Textbook: Scalars by scalars to produce new matrices, scalars to produce new --Lesson 12-2 p. 772 e.g., as when all of the payoffs in a matrices. M --Lesson 12-5 p. 801 game are doubled. [Assessment]: [Mathematical Practices]: Domain: Perform operations on matrices and use matrices in applications. Standards:N-VM.8.Add, subtract and I can perform operations on Knowledge Pearson Alg. II Common Matrix multiply matrices of appropriate matrices and reiterate the Core Textbook: Dimensions dimensions. limitations on matrix --Lesson 12-1p. 764 [Assessment]: dimensions. -
Variables in Mathematics Education
Variables in Mathematics Education Susanna S. Epp DePaul University, Department of Mathematical Sciences, Chicago, IL 60614, USA http://www.springer.com/lncs Abstract. This paper suggests that consistently referring to variables as placeholders is an effective countermeasure for addressing a number of the difficulties students’ encounter in learning mathematics. The sug- gestion is supported by examples discussing ways in which variables are used to express unknown quantities, define functions and express other universal statements, and serve as generic elements in mathematical dis- course. In addition, making greater use of the term “dummy variable” and phrasing statements both with and without variables may help stu- dents avoid mistakes that result from misinterpreting the scope of a bound variable. Keywords: variable, bound variable, mathematics education, placeholder. 1 Introduction Variables are of critical importance in mathematics. For instance, Felix Klein wrote in 1908 that “one may well declare that real mathematics begins with operations with letters,”[3] and Alfred Tarski wrote in 1941 that “the invention of variables constitutes a turning point in the history of mathematics.”[5] In 1911, A. N. Whitehead expressly linked the concepts of variables and quantification to their expressions in informal English when he wrote: “The ideas of ‘any’ and ‘some’ are introduced to algebra by the use of letters. it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics.”[6] There is a question, however, about how to describe the use of variables in mathematics instruction and even what word to use for them. -
An Introduction to Mathematical Modelling
An Introduction to Mathematical Modelling Glenn Marion, Bioinformatics and Statistics Scotland Given 2008 by Daniel Lawson and Glenn Marion 2008 Contents 1 Introduction 1 1.1 Whatismathematicalmodelling?. .......... 1 1.2 Whatobjectivescanmodellingachieve? . ............ 1 1.3 Classificationsofmodels . ......... 1 1.4 Stagesofmodelling............................... ....... 2 2 Building models 4 2.1 Gettingstarted .................................. ...... 4 2.2 Systemsanalysis ................................. ...... 4 2.2.1 Makingassumptions ............................. .... 4 2.2.2 Flowdiagrams .................................. 6 2.3 Choosingmathematicalequations. ........... 7 2.3.1 Equationsfromtheliterature . ........ 7 2.3.2 Analogiesfromphysics. ...... 8 2.3.3 Dataexploration ............................... .... 8 2.4 Solvingequations................................ ....... 9 2.4.1 Analytically.................................. .... 9 2.4.2 Numerically................................... 10 3 Studying models 12 3.1 Dimensionlessform............................... ....... 12 3.2 Asymptoticbehaviour ............................. ....... 12 3.3 Sensitivityanalysis . ......... 14 3.4 Modellingmodeloutput . ....... 16 4 Testing models 18 4.1 Testingtheassumptions . ........ 18 4.2 Modelstructure.................................. ...... 18 i 4.3 Predictionofpreviouslyunuseddata . ............ 18 4.3.1 Reasonsforpredictionerrors . ........ 20 4.4 Estimatingmodelparameters . ......... 20 4.5 Comparingtwomodelsforthesamesystem . ......... -
Complex Eigenvalues
Quantitative Understanding in Biology Module III: Linear Difference Equations Lecture II: Complex Eigenvalues Introduction In the previous section, we considered the generalized two- variable system of linear difference equations, and showed that the eigenvalues of these systems are given by the equation… ± 2 4 = 1,2 2 √ − …where b = (a11 + a22), c=(a22 ∙ a11 – a12 ∙ a21), and ai,j are the elements of the 2 x 2 matrix that define the linear system. Inspecting this relation shows that it is possible for the eigenvalues of such a system to be complex numbers. Because our plants and seeds model was our first look at these systems, it was carefully constructed to avoid this complication [exercise: can you show that http://www.xkcd.org/179 this model cannot produce complex eigenvalues]. However, many systems of biological interest do have complex eigenvalues, so it is important that we understand how to deal with and interpret them. We’ll begin with a review of the basic algebra of complex numbers, and then consider their meaning as eigenvalues of dynamical systems. A Review of Complex Numbers You may recall that complex numbers can be represented with the notation a+bi, where a is the real part of the complex number, and b is the imaginary part. The symbol i denotes 1 (recall i2 = -1, i3 = -i and i4 = +1). Hence, complex numbers can be thought of as points on a complex plane, which has real √− and imaginary axes. In some disciplines, such as electrical engineering, you may see 1 represented by the symbol j. √− This geometric model of a complex number suggests an alternate, but equivalent, representation. -
Example 10.8 Testing the Steady-State Approximation. ⊕ a B C C a B C P + → → + → D Dt K K K K K K [ ]
Example 10.8 Testing the steady-state approximation. Å The steady-state approximation contains an apparent contradiction: we set the time derivative of the concentration of some species (a reaction intermediate) equal to zero — implying that it is a constant — and then derive a formula showing how it changes with time. Actually, there is no contradiction since all that is required it that the rate of change of the "steady" species be small compared to the rate of reaction (as measured by the rate of disappearance of the reactant or appearance of the product). But exactly when (in a practical sense) is this approximation appropriate? It is often applied as a matter of convenience and justified ex post facto — that is, if the resulting rate law fits the data then the approximation is considered justified. But as this example demonstrates, such reasoning is dangerous and possible erroneous. We examine the mechanism A + B ¾¾1® C C ¾¾2® A + B (10.12) 3 C® P (Note that the second reaction is the reverse of the first, so we have a reversible second-order reaction followed by an irreversible first-order reaction.) The rate constants are k1 for the forward reaction of the first step, k2 for the reverse of the first step, and k3 for the second step. This mechanism is readily solved for with the steady-state approximation to give d[A] k1k3 = - ke[A][B] with ke = (10.13) dt k2 + k3 (TEXT Eq. (10.38)). With initial concentrations of A and B equal, hence [A] = [B] for all times, this equation integrates to 1 1 = + ket (10.14) [A] A0 where A0 is the initial concentration (equal to 1 in work to follow). -
Differentiation Rules (Differential Calculus)
Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below.