# 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 |

46 |

12 | Tests & Review |

1 | Basic Skills Test |

59 |

## OBJECTIVES

**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.