20804218Mathematics for Elementary Education: Arithmetic Models
Course Information
Description
A first course in mathematics needed for teaching K-8. Emphasis will be on the student communicating how and why standard and alternative arithmetic algorithms work Content will focus on problem solving strategies and word problems involving whole numbers, fractions, decimals, integers and percents.
Total Credits
3
Course Competencies
  1. Apply a variety of appropriate problem-solving strategies
    Criteria
    Apply Polya's four-step guideline for solving problems
    Communicate the advantages and disadvantages of different problem-solving strategies

  2. Identify different sets of numbers in correlation with the number line
    Criteria
    Determine if a given number is counting, whole, integer, rational, and/or real
    Identify counting numbers and place them on a number line
    Identify whole numbers and place them on a number line
    Identify integers and place them on a number line
    Identify rational numbers and place them on a number line
    Identify real numbers and place them on a number line

  3. Represent place value using different strategies
    Criteria
    Represent decimal numbers using bundled objects
    Represent decimal numbers using powers of ten
    Represent a decimal number using appropriate rounding techniques
    Compare numbers to determine which is largest by using a number line

  4. Communicate how to use fractions to describe parts of objects, collections, and quantities
    Criteria
    Determine if fractions are equivalent by changing to decimals
    Determine if fractions are equivalent by changing denominators
    Compare the magnitude of different fractions by converting to decimal numbers
    Compare the magnitude of different fractions by using common denominators
    Compare the magnitude of different fractions by using common numerators
    Compare the magnitude
    Express improper fractions as mixed numbers
    Express mixed numbers as improper fractions
    Reduce fractions to lowest terms
    Plot proper fractions, improper fractions, and mixed numbers on a number line

  5. Communicate how to use percents to describe parts of objects, collections, and quantities
    Criteria
    Interpret the definition of percent
    Convert fractions into percents
    Convert decimals into percents
    Convert percents into fractions
    Convert percents into decimals
    Solve percent problems for base
    Solve percent problems for amount
    Solve percent problems for percentage
    Translate a verbally stated application involving percentages into an equivalent computation
    Determine if a given problem that involves combining percentages can be solved by adding the percentages
    Calculate percent increase and percent decrease in several different ways
    Calculate quantities from a given percent increase or decrease
    Distinguish between percent increase/decrease and percent of
    Solve problems involving percent increase ore decrease

  6. Apply different procedures used to add and subtract whole numbers, decimal numbers, and integers
    Criteria
    Apply the regrouping algorithm for adding and subtracting
    Show the regrouping algorithm for addition using manipulatives
    Show the regrouping algorithm for subtration using manipulatives
    Apply the regrouping algorithm to other bases such as time and weight when adding and subtracting
    Use addition or subtraction to translate story problems
    Use subtraction when comparing quantities
    Use number line to add or subtract
    Use manipulatives or a picture diagram to demonstrate part-part-whole problems, take-away problems, or comparison problems
    Write story problems that involve addition or subtraction of negative numbers
    Change a subtraction problem to a related addition problem involving the additive inverse
    Relate subtraction facts to their corresponding addition facts via the four-fact (number-fact) families
    Write a story problem for a given addition or subtraction problem
    Describe how the associative and commutative properties of addition aid in mental math
    Identify where the commmutative or associative properties of addition have been used

  7. Apply different procedures used to add and subtract fractions
    Criteria
    Describe how to add and subtract fractions by changing to a common denominator
    Demonstrate why fractions must have common denominators in order to add and subtract
    Write story problems for adding and subtracting fractions
    Use pictures, manipulatives or number lines to explain the logic of writing improper fractions as mixed numbers
    Use the regrouping algorithm to borrow when subtracting fractions

  8. Apply different procedures used to multiply whole numbers
    Criteria
    Use pictures or manipulatives to describe why multiplication applies to a problem exhibiting equal groups
    Explain why multiplying by 10 moves each digit one place to the left
    Explain why multiplying by powers of 10 moves each digit to the left an equal number of places as the exponent
    Explain why we can multiply to find the area of a rectangle by drawing rectangles filled with unit squares
    Explain the commutatitive property of multiplication by subdividing rectangles or arrays of objects in two different ways
    Use the commutatitive property of multiplication to write a related problem
    Explain why we can multiply to find the volume of a box by subdividing it into unit cubes
    Explain the associaitive property of multiplication by subdividing boxes or groups of groups of objects in two different ways
    Explain the distributive property by illustrating the total number of objects in a subdivided array in two different ways
    Use the associative and commutative properties of multiplication in problems
    Use the distributive property in problems
    Show how the associative, commutative and distributive properties of multiplication can aid in mental math
    Use subdivision to demonstrate the logic of the partial-products alogorithm
    Use partial products to demonstrate the logic of placing a zero in the one's column of the second line in the standard multiplication alogorithm and extend this concept to any other lines needed when multiplying by larger numbers
    Subdivide arrays containing groups of grooups and show that the portions correspond to the lines in the standard multiplication algorithm
    Compare how place value is handled in the partial-products and standard algorithms
    Validate the partial-products algorithm by writing equations that use expanded forms and the distributive property

  9. Apply different procedures used to multiply fractions, decimals and negative numbers
    Criteria
    Use pictures and manipulatives to solve story problems involving fraction multiplication
    Use simple story problems and pictures to show why we multiply both the numberators and denominators when multiplying fractions
    Use estimation to explain where to place the decimal point in a multiplication problem
    Show how decimal multiplication is used in area problems
    Write story problems for decimal multiplication
    Use a story to demonstrate why multiplication by a positive number and a negative number will result in a negative number
    Use a story to demonstrate why multiplication by a negative number and a negative number will result in a positive number
    Use commutative, associative and distributive properties to explain the rules for multiplying negative numbers
    Rewrite numbers using standard and scientific notation
    Use scientific notation to solve problelms

  10. Apply different procedures used to divide whole numbers
    Criteria
    Use the "How many groups" intrepretation and the "How many in each group" intrepretation to write two different stroy problems for the same division fact
    Determine whether a division story problem should be answered exactly or with a decimal, mixed number or whole number/remainder format based on the context of the problem
    Explain why division by zero is not defined
    Use division to solve problems
    Explain how to use the scaffold method to carry out division
    Compare the scaffold method to the standard long division algorithm
    Explain why zeros are sometimes necessary as place holders in the long division algorithm
    Show how long division can be used to create a decimal when dividing two whole numbers
    Write a story problem tinvolving division of whole numbers that results in a decimal answer
    Describe how a fraction can be represented as a division problem
    Explain how long division is used to convert a fraction into a decimal
    Explain why a quotient involving a whole number and remainder can be described as a mixed number
    Write a story problem that requires converting an improper fraction into a mixed number

  11. Apply different procedures used to divide fractions, decimals and negative numbers
    Criteria
    Use the "How many groups" intrepretation and the "How many in each group" intrepretation to write two different story problems for the same division fact for fractions
    Solve fraction division story problems with the aid of pictures
    Validate the "invert and multiply" procedure of dividing fractions
    Use the "How many groups" intrepretation and the "How many in each group" intrepretation to write two different story problems for the same division fact for decimals
    Use estimation to determine the location of the decimal point in a decimal division problem
    Explain in several different ways why we move the decimal points the way we do when we divide decimals
    Use a story to demonstrate why division of a negative number by a positive number will result in a negative number
    Use a story to demonstrate why division of a positive number by a negative number will result in a negative number
    Use commutative, associative and distributive properties to explain the rules for dividing a negative number by another negative number

  12. Use ratios and proportions to solve problems
    Criteria
    Explain the reasoning behind the use of ratio tables to aid in solving ratio problems
    Explain the reasoning behind the use of strip diagrams to aid in solving ratio problems
    Explain how to use ratios to represent the amount of the first quantity per unit amount of the second quantity
    Explain how you can use multiplication and division to solve simple ratio problems
    Explain the reasoning behind the use of cross-multiplication to aid in solving proportions
    Identify problems that should not be solved with a proportion

  13. Apply different procedures to solve problems involving factors and multiples
    Criteria
    Explain the meaning of the terms "factors" and "multiples"
    Explain different ways to find the factors of a given counting number
    Write story problems that require the use of factors to solve problems
    Write story problems that require the use of multiples to solve problems
    Explain the meaning of GCF and LCM
    Explain how to use the definitions to determine GCFs and LCMs
    Explain how to use the slide mehtod to determine GCFs and LCMs
    Write story problems that require the use of GCFs to solve problems
    Write story problems that require the use of LCMs to solve problems
    Apply GCFs and LCMs to fraction arithmetic

  14. Apply different number theory procedures to solve problems
    Criteria
    Explain the difference between a prime number and a composite number
    Explain how to use the sieve of Eratosthenes to produce a list of prime numbers
    Explain why you only need to divide by prime numbers when determining if a counting number is prime
    Explain when to stop dividing by prime numbers when determining if a counting number is prime
    Describe the difference between an even and an odd number
    Explain why we should examine the ones place to determine if a number is even or odd
    Explain why the divisibility tests work for finding the factors 2, 3, 4, 5, 6, 9 and 10
    Explain why rational numbers are those that may be written as a terminating or repeating decimal
    Explain how to write a terminating or repeating decimal as a fractioin
    Explain why 0.99999... = 1
    Prove that various square roots, such as the square root of 2 and the square root of 3 are irrational

This Outline is under development.