Mathematics for Elementary School Teaching: Arithmetic and Problem Solving, Exams of Mathematics

Download Mathematics for Elementary School Teaching: Arithmetic and Problem Solving and more Exams Mathematics in PDF only on Docsity! MATH 5001/7001 Arithmetic and Problem Solving 3 hours University of Georgia Syllabus as of Spring 2005 Oasis Title: ARITH & PROB SOLV. Brief description: A deep examination of topics in mathematics that are relevant for elementary school teaching. Problem solving. Number systems: whole numbers, integers, rational numbers (fractions) and real numbers (dec- imals) and the relationships between these systems. Understanding multipli- cation and division, including why standard computational algorithms work. Properties of arithmetic. Applications of elementary mathematics. Course Objectives: To strengthen and deepen knowledge and understand- ing of arithmetic, how it is used to solve a wide variety of problems, and how it leads to algebra. In particular, to strengthen the understanding of and the ability to explain why various procedures from arithmetic work. To strengthen the ability to communicate clearly about mathematics, both orally and in writing. To promote the exploration and explanation of mathe- matical phenomena. To show that many problems can be solved in a variety of ways. Topical Outline: Problem solving: Polya’s principles. Writing explanations. Numbers: The natural numbers, the whole numbers, the rational numbers (fractions), and the real numbers (decimals). The decimal system and place value. Representing decimals with bundled objects. Representing decimals on a number line. Comparing sizes of decimals. Finding decimals in between decimals. Rounding decimals. The meaning of fractions. The importance of the whole associated with a fraction. Improper fractions. Equivalent fractions. Simplest form of a fraction. Fractions as numbers on number lines. Comparing sizes of fractions: by giving them common denominators, by converting to decimals, and by cross-multiplying. Using other reasoning to compare sizes of fractions. Solving fraction problems with the aid of pictures. Percent. Benchmark percentages and their common fraction equivalents. Solving percentage problems with the aid of pictures. Solving percentage problems numerically. 1 Addition and subtraction: Interpretations of addition and subtraction. The relationship between addition and subtraction. Explaining why the standard algorithms for adding and subtracting whole numbers and decimals work. Using regrouping in situations other than base 10, for example in calculating elapsed time by replacing 1 hour with 60 minutes. Adding and subtracting fractions. Explaining why we add and subtract fractions the way we do. The importance of the whole when adding and subtracting fractions, espe- cially in story problems. Recognizing and writing story problems for fraction addition and subtraction. Recognizing story problems that are not solved by fraction addition or subtraction. Mixed numbers. Understanding when percentages should and should not be added. Calculating percent increase and decrease with the aid of pictures. Calculating percent increase and de- crease numerically. Percent of versus percent increase or decrease. The commutative and associative properties of addition and their use in mental arithmetic. Using properties of addition to aid the learning of basic addition facts. Other (mental) methods for adding and subtracting: rounding and compensating, subtracting by adding on. Writing equations that correspond to a mental method of calculation (to demonstrate the connection between mental arithmetic and algebra). Multiplication: The meaning of multiplication. Ways of showing multiplica- tive structure: with groups, with arrays, and with tree diagrams. Using the meaning of multiplication to explain why various problems can be solved by multiplying. Explaining why multiplication by 10 is easy in the decimal sys- tem. Why the commutative and associative properties of multiplication and the distributive properties make sense and how to illustrate them with arrays, areas of rectangles, and volumes of boxes. Using properties of arithmetic in solving arithmetic problems mentally. Writing equations that correspond to a mental method of calculation (to demonstrate the connection between mental arithmetic and algebra). Using properties of arithmetic to aid in the learning of basic multiplication facts. The distributive property and FOIL. Using multiplication to estimate how many. The partial products multiplica- tion algorithm. Using pictures and the distributive property to explain why the standard and partial products procedures for multiplying whole numbers are valid. Explaining why non-standard strategies for multiplying can be correct or incorrect, such as explaining why 23 × 23 6= 20 × 20 + 3 × 3 and explaining why 32 × 28 = 30 × 30 − 2 × 2. The meaning of multi- plication for fractions. Recognizing and writing story problems for fraction 2