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Text: Modern Mathematics for Elementary School Teachers, Billstein, et. al., 6th Edition.

Materials required: You are to bring a ruler, pair of scissors, set of crayons or colored pencils to each class. You also need a folder or loose-leaf notebook for saving and organizing your work because you may be asked to discuss an overview of all your work at any time during the quarter.

Course Project: A project investigating different internet searches and giving a critique of and preparing a lesson plan. Further details will be given in class.

Recommended Reading: Will be announced in class.

Conference: Attend the CMC Mathematics Conference if Fall Qtr in Palm Springs

One of our goals this quarter is to help you become more proficient in seeing mathematics as a process. Contrary to common opinion that mathematics involves only exact answers, mathematics is a procedure we apply every day in our lives.

Any adventure into mathematics is challenging at times; however, it is always exciting and rewarding for those who persevere. It is recommended you budget your time so as to be able spend a minimum of two hours concentrated study time per hour of class time to pass this course.

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Overview of the Course, Mat 391

In Mat 391, we take a brief look at how the counting numbers (the natural numbers) were adapted in some of the more significant historical numeration systems from BC to 800-900AD. By 900 AD, zero was added to the developed natural number system generating the whole number system. Adding the negative numbers,Neg= {-n | -n+n=0, for all whole numbers n}, to the set of whole numbers (circa: 1600 AD) led to the establishment of the integers, and the integers, together with all the ratios, a/b, of all integers a and b0rational numbers, established formed the rational number system.

After discovering that the rational numbers are equivalently the set of all infinite repeating decimals, we complete our study of our development of the real number system by adding the set of all infinite non-repeating decimals. These latter numbers are more commonly called the irrational numbers because they cannot be expressed as the ratio of two integers. High school algebra students extend the real numbers to the complex numbers by incorporating the imaginary number, i (the square root of negative 1), while even higher mathematical systems are studied in college. Businesses use higher mathematics to conduct banking and other international trade. Therefore, it is necessary that elementary students have a thorough comprehension of the real number system to be successful later in the real world.

In Mat 391 we learn about the properties of the real number system, such as, closure and the existence or non-existence of identities or inverses for the four basic operations of +, -, *, and . We learn which of these operations are associative, commutative, or have a distributive relationship one over another. We cover concepts such as fewer than,"<", or more than, ">" and discover how the concept of place value is applied to give us the basic algorithms we use every day to simplify number calculations. We also show how today's technology, especially hand-held calculators, are affecting our use and teaching of these algorithms in accordance with the current NCTM Standards.

Summary of Topics Week by Week

Week 1.	Chapter 3.1 Numeration Systems and Introduction to Bases

Egyptian:	Numerals include; surprised man (one million), lotus flower (one thousand), scroll(one hundred), heelbone(ten), and a vertical mark, |, represents one.
	System is additive: has no base designation, little order, no place value

Babylonian:	Numerals: tens (< ) and arrowhead-like one,
	System is additive: similar to a base sixty system; has a fixed order, but there is a problem in that there is no numeral zero to act as a placeholder. For example, does the numeral I I, represent (one) one + (one) one = two or does it represent (one) sixty + (one) one = sixty-one?

Mayan:

Numerals - football = zero, dot = one, bar = five;

System is additive: almost base twenty, and does have a zero numeral place holder. Numeral script is vertical counting from the bottom up. The bottom level goes from zero to five and the next level up covers the twenties from one*twenty to seventeen*twenty. A dot or bar numeral in the third level up stands for one*eighteen*twenty or five* eighteen*twenty. From there, the levels increase vertically as 18*20*n , n>2.

Roman:

Numerals - I, V, X, L, C, D, M, MM, MMM, (a bar over a numeral represents multiplication by a thousand, starting with four thousand).

System is essentially additive with a subtractive feature. One always adds the value of a smaller numeral to the value of a larger if it immediately follows the larger numeral. But subtract the value of certain smaller numerals if they appear immediately in front of a larger numeral. Numerals can be repeated consecutively a max number of three times. There is also a multiplicative feature from four thousand on.

Week 2.	Chapter 3.2 Definitions of Addition and Subtraction Chapter 3.3 Definitions of Multiplication and Division

Week 3.	Chapter 3.4 Algorithms for Addition & Subtraction Chapter 3.5 Algorithms for Multiplication and Division

It is encouraged that students be able to convert directly from a given base to a multiple of that base by the beginning of the third week . For example, to convert 101nine directly to base three utilizing the rules of exponents, know that is also in base ten.

Recognize that this value in base ten is directly written as 10001three in base three. Students are to be able to explain like processes in order to exhibit that they thoroughly understand the grouping concept behind our base ten place value system.. Students are to be able to use the various models given to explain these equivalencies although students need not be held responsible for the formal proofs.

Week 4.	Chapter 4.1 Integer Operations: Addition and Subtraction Chapter 4.2 Integer Operations: Multiplication and Division ­ emphasis is given to the concept of division
	Assessment 1 considered.
	By the end of week 4, students are to be able to explain:
-all four operations, - the difference - the equivalencies	addition, subtraction, multiplication and division and to know and be able to use the various models of these operations in both Chapter 3 and Chapter 4. between the direction concept of "inverse" and the operation of "subtraction" -of integer additive inverses:-(-(ab) )= (-a)(-b) = ab= ­-( -a)b=a( -b)=(-a)b,and that a--b=a+(­b) and ­ (a ± b)=­a ± (­b).
	For example, students should be able to explain the steps of the Charged Field Model.

Week 5.	Chapter 4.3 Integer Divisibility (the divides relation) Chapter 4.4 Integer Primes & Composites ­ GCD's & LCM's
	Note: The page 208 Brain Teaser ­ The Elephant and the Ant, is an interesting way to emphasize how division by zero can be misused.

Week 6.	Chapter 5.1 Rationals Chapter 5.2 The Rational System: Addition & Subtraction

Week 7.	Chapter 5.3 The Rational System: Multiplication & Division

Week 8.	Chapter 6.1 Introduction to Exponents and Decimals
	Assessment 2 considered.

Week 9.	Chapter 6.3 Real Numbers

Week 10.	Chapter 7.5, Percents Quarter Review

Week 11.	Final Exam

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