Arithmetic and Problems Solving - Study Guide | MATH 5003, Exams of Mathematics

Download Arithmetic and Problems Solving - Study Guide | MATH 5003 and more Exams Mathematics in PDF only on Docsity! MATH 5003/7003 Arithmetic and Problem Solving 3 hours University of Georgia Oasis Title: ALG & PROB SOLV. Prerequisite: MATH 5002 Brief description: A deep examination of topics in mathematics that are relevant for elementary school teaching. Number theory, algebra and func- tions, including ratio and proportion, probability and statistics. Posing and modifying problems. Course Objectives: To strengthen and deepen knowledge and understand- ing of probability and statistics, elementary number theory and algebra, and how they are used to solve a wide variety of problems. In particular, to strengthen the understanding of and the ability to explain why various pro- cedures and formulas in mathematics work. To strengthen the ability to communicate clearly about mathematics, both orally and in writing. To pro- mote the exploration and explanation of mathematical phenomena. To show that many problems can be solved in a variety of ways. To learn to pose and modify mathematical problems. Topical outline: Division of fractions and decimals: The meaning of division for fractions. Recognizing and writing story problems for fraction division. Understanding the distinction between dividing by 1 2 and dividing in 1 2 . Explaining why the “invert and multiply” procedure for dividing fractions is valid. Explain- ing why the procedure for placement of the decimal point in decimal divi- sion problems is valid. Understanding that division does not always “make smaller.” Ratio and proportion: the meanings of ratio and proportion. Solving ratio problems using only multiplication, division, and logical thinking. Explaining the logic behind setting up proportions by setting two fractions equal to each other and solving these proportions by cross-multiplying. Understanding when problems can’t be solved by a proportion. Optional: the Consumer Price Index. Number Theory: definitions of factors and multiples and concrete problems that use and illustrate these concepts, definitions of greatest common factor 1 and least common multiple and concrete problems that use and illustrate these concepts. Prime numbers. The Sieve of Eratosthenes for producing lists of prime numbers. The trial division method for determining if a num- ber is prime. Factoring counting numbers into products of prime numbers. Optional: the proof that there are infinitely many prime numbers. Lightly: consequences of the irrationality of the square root of 3 for making designs with standard school pattern tile sets. Even and odd: different ways of defin- ing even and their equivalence. Divisibility tests: explaining the divisibility tests for 2, 3, 5, and 9. Algebra and functions: patterns, sequences, formulas, and equations. Writ- ing expressions and equations to go along with a scenario. Evaluating ex- pressions. Solving algebra word problems with Singapore strip diagrams. Using the imagery of a pan balance to understand the technique for solving linear equations in one variable. Arithmetic and geometric sequences. Cre- ating numerical sequences from picture sequences, creating picture sequences from numerical sequences. Describing sequences in words and with formu- las. Determining a specified entry (say, the 100th) in a repeating pattern. Optional: arithmetic and geometric series. Functions and their graphs. Re- lating qualitative descriptions of functions to their graphs. Understanding that the graph of a function is a line exactly when there is a fixed increase in the output for a given fixed increase in the input. Basic descriptive statistics: Lightly: designing investigations and gathering data. Common ways to display data. Three levels of questions about graphs: reading the data, reading between the data, reading beyond the data. The average of a numerical set of data. Understanding the average as “making even” or “leveling out.” The median of a numerical set of data. Showing that different data sets can have the same median but a different average. Showing that different data sets can have the same average but a different median. Understanding that “more than half can be above average.” Percentiles. Understanding the difference between percentile and percent correct. Probability: Basic principles of probability. Simple probability calculations. Using the meaning of fraction multiplication to understand simple probability calculations. Critique of mathematics lessons: students should critique several mathemat- ics lessons that they taught to elementary school children and discuss ways 2