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Mathematical Structures and Concepts for Elementar

Catalog Description

Designed for elementary education majors. Others admitted only with Mathematics Department
approval. Emphasis is placed on structure, meaning, relationships, and types of thinking in elementary
mathematics. Problem solving, sets, numeration systems, integers, logic, rational numbers, ratio and
proportion, and percents are among the topics considered.

Course Objectives:

See attached

Student Evaluation (Type of Grading)

Chapter tests, workbook activities, presentations, and a comprehensive final exam.

Days Topic
5 Problem Solving
4 Sets
2 Functions
3 Logic
3 Numeration Systems
5 Whole Number Operations
3 Integers
3 Number Theory
1 Clock Arithmetic
3 Rational Numbers
1 Integer Exponents
3 Decimals
2 Real Numbers, Radicals & Exponents
2 Ratio and Proportion
3 Percents
1 Algebraic Expressions
2 Algebraic Equation


12 Tests & Review
1 Basic Skills Test


1. Problem Solving
 Use pattern finding techniques to determine any term of a sequence.
 Use pattern finding techniques or formulas to find the general term of a sequence.
 Classify sequences as arithmetic, geometric, or neither.
 Use a variety of problem-solving strategies in solving problems.
 Describe the steps of the problem-solving process.

2. Sets
 Shade a Venn diagram to illustrate set unions and intersections and complements.
 Distinguish between finite sets and infinite sets.
 Recognize and diagram set relations.
 Define a set and the elements or members of a set.
 Define a well-defined set, and be able to distinguish between sets that are well-defined and those that are not well-defined.
 Define and give examples of equal or identical sets.
 Define the universal set.
 Determine the Cartesian product of two sets.
 Distinguish between subsets and proper subsets.
 Define the null set, or the empty set.
 Determine the subsets and proper subsets given any universal set.
 Define the complement of a set, and be able to correctly find the complement of any set given any universal set.
 Define equivalent sets, and be able to recognize them.
 Define the cardinal number of a set, and be able to recognize them.
 Define the intersection of two sets, and be able to determine the intersection given any two sets.
 Define disjoint sets, and be able to recognize them.
 Define the union of two sets, and be able to determine the union given any two sets.
 Correctly use Venn diagrams to find the solutions to work problems.
 Define the set difference and be able to correctly find the set difference of any given sets.

3. Functions
 Define a relation and a function.
 Evaluate functions using function notation.
 Write rules or phrases that could describe a relation or function.
 Determine domains and ranges of functions.
 Define the composition of two functions and be able to determine the composition.

4. Logic
 Classify conjunctions, disjunctions, conditionals and biconditionals as true or false.
 Write the converse, inverse, and contrapositive of a conditional statement.
 Determine whether a valid conclusion can be made on the basis of a conditional and a simple statement.
 Draw Euler circles to determine if an argument is valid or invalid.
 Define quatifers and their negations.

5. Numeration Systems
 Identify the characteristics of the Hindu-Arabic numeration system.
 Write a numeral using different forms of expanded notation.
 Convert numbers from decimal notation to non-decimal notation and vice versa.
 Perform addition, subtraction, multiplication and division in non-decimal bases.
 Take any numeral expressed in any other base and convert it to an equivalent numeral in base ten (base 2, 5, or 12).
Convert any numeral expressed in base ten to an equivalent numeral in any other base (base 2, 5, or 12).

6. Whole Number Operations
 Illustrate addition and subtraction situations using set models and number line models.
 Recognize statements of the properties of the whole numbers under addition.
 Illustrate multiplication and division situations using set and array models.
 Recognize statements of the properties of the whole numbers under multiplication.
 Relate operations with multiples of 10 to “basic fact” results.
 Demonstrate an understanding of the algorithms by using some form of expanded notation or modeling.
 Simplify multiplication expressions involving exponents.
 Use and describe mental math techniques.
 Estimate results by using rounding of front-end digit strategy and grouping to nice numbers strategy.
 Use the scratch addition algorithm.
 Use the lattice multiplication algorithm.

7. Integers
 Form the set integers, and be able to distinguish between positive integers, negative integers, and 0.
 Use a set model and number line to illustrate operations with integers when appropriate.
 Perform calculations with integers.
 Use the chip model for addition, subtraction, and multiplication of integers.
 Use the order of operations to simplify expressions.
 Factor integer expressions.
 Recognize statements of the properties of integers under addition and multiplication.

8. Number Theory
 Distinguish between numbers that are prime numbers and those that are composite numbers.
 Find the greatest common divisor of any two or three numbers.
 Find the least common multiple of any two or three numbers.
 Determine whether a given number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10 or 11.
 Write the prime factorization of a number.
 List the factors of a number.
 Define and use the Euclidian algorithm.
 Define and use factor, divisor, and multiple with the “divides” symbol.

9. Clock Arithmetic
 Perform the operations of addition, subtraction, multiplication, and division in clock arithmetic and in any given module system.
 Define modular congruence and use the definition to determine if statements are true or false.

10. Rational Numbers
 Find fractions that are equivalent to a given fraction.
 Change a given fraction to an equivalent fraction in lowest terms.
 Add and subtract fractions with like denominators and unlike denominators.
 Multiply fractions and mixed numbers.
 Divide fractions and mixed numbers.
 Determine whether any two given rational numbers are equivalent.
 Order fractions.
 Illustrate and use the denseness property.

11. Ratio & Proportion
 Write ratios to represent real-life situations.
 Solve problems involving proportions.

12. Integer Exponents
 Apply the properties of exponents to rational numbers

13. Decimals
 Distinguish between rational and irrational numbers.
 Determine if a rational number is terminating or repeating.
 Convert from decimal notation to scientific notation.
 Convert from scientific notation to decimal notation.
 Order repeating decimals.
 Round decimals.
 Convert a repeating decimal to a fraction.

14. Real Numbers, Radicals and Exponents
 Form the set of real numbers and distinguish between rational numbers and irrational numbers.
 Define the principal square root and the nth root.
 Change expressions from radical form to rational exponent form.
 Change expressions from rational exponent form to radical form.
 Solve problems with rational exponents and/or radicals.

15. Percents
 Define percent
 Change from percent notation to fraction or decimal notation.
 Change from fraction of decimal notation to percent notation.
 Solve percent application including simple and compound interest.

16. Translate English sentences to algebraic expressions.

17. Solve algebraic equations and applications.

18. Graph linear equations.