Math 195/196 – Mathematics for Elementary Education I/II

• Perform and understand notation for set operations
• Draw, interpret and use Venn Diagrams to represent information, perform set operations and solve word problems
• Know and understand the basic meaning of multiplication
• Write and solve repeated addition, array, area, ordered pair, and multiplicative comparison situations for story problems.
• Use arrays, organized lists, tree diagrams, strip diagrams and number lines to exhibit multiplicative structure
• Interpret an expression for a product in terms of number of groups and number of items in a group
• Use and identify the commutative and associative properties of multiplication
• Write an expression that uses the operations and properties to describe a given picture/diagram.
• Draw a picture/diagram to represent a given expression.
• Know and apply order of operations to evaluate expressions
• Identify and use the distributive property in a context and/or a set of equations
• Write a story problem and draw an area diagram/array to correspond to a given expression involving the distributive property.
• Use the distributive property to lighten the load for learning single-digit multiplication facts
• Write equations and draw an area diagram/array corresponding to a method of calculation
• Write/identify “how many groups” and “how many in each group” types of division problems
• Write the corresponding multiplication (and addition) equations for division problems with and without remainders
• Write or identify story problems in which the quotient is rounded up or down.
• Explain the validity (or lack thereof) of 0/a, a/0 and 0/0 types of division problems.
• Write the corresponding multiplication (and addition) equations for division problems with and without remainders
• Understand and write equations (including generalized division algorithm form) that correspond to student-generated methods of division.
• Explain scaffold division algorithm using “how many groups” analogy, including context situations
• Identify, explain, and correct student errors in standard and non standard division algorithms
• Explain standard division algorithm using “how many in each group” analogy, including how interpretation of number changes within the method
• Identify, explain, and correct student errors in standard and non standard division algorithms
• Use a geometry (area) argument to describe factors and multiples.
• Write, identify and solve word problems involving multiples and factors
• Use and understand terms: factor, multiples, prime, composite.
• Use the Sieve of Eratosthenes to find prime numbers, and explain why this works
• Use and understand rationale for an efficient method for determining if a number is prime

Unit 3

• Use a factor tree to find the prime factorization of a number
• Explain and use Number of Factors Theorem and observations from factor table activity
• Find the GCF and LCM of two or more numbers, using both the list method and the prime factorization method.
• Identify, solve and write story problems involving GCF and LCM
• Define even and odd numbers
• Use algebra and pictures to prove conjectures about even or odd numbers (numeric examples are not enough to prove something is true)
• Know and explain the rationale behind divisibility tests for 2, 3, 4, 5, 9, and 10 using:
  • manipulative diagrams,
  • place value arguments, and
  • division algorithm proofs
• Understand that a specific example is not a proof of all cases
• Know, use and explain rationale for divisibility tests for 4, 5, 9 and 10
• Construct combined divisibility tests
• Know vocabulary: variable, expression, equation, formula
• Write expressions that correspond to a design or pattern, and reflect the meaning of multiplication
• Write numeric and algebraic equations that correspond to sums of different types of numbers
• Use algebraic, geometric and numerical methods to justify formulas for adding sequences of numbers.
• Flexibly use the formulas for sums of consecutive numbers and the sums of odd numbers to solve a variety of problems.
• Formulate and flexibly use equations and/or geometric diagrams arising from a scenario.
• Write equations to represent related quantities; write story problems to represent such equations
• Distinguish between situations such as “three times as much as” and “three times more than”
• Solve equations using number sense: annotated pictures
• Solve equations algebraically and show correspondence to pan balance.
• Connect features of a linear function with features of corresponding graph (intercept, slope) and table of values
• Determine if a given sequence is arithmetic, geometric, or neither
• Write successive terms and formula for arithmetic sequences
• Given a series of figures or table of values, write the formula that arises from an arithmetic sequence
• Know geometry terms from text 10.1 and 10.2
• Know and use the parallel postulate
• Understand why the formula for the sum of the measures of the angles in a polygon makes sense
• Compute the sum of the measures of the angles in a polygon, as well as the individual measures of angles in a regular polygon
• Solve angle problems involving parallel lines and/or polygons

Unit 4

• Understand/use the definition of circle and sphere and solve related distance problems
• Know definitions of quadrilaterals: rhombus, parallelogram, trapezoid, rectangle, square
• Compare and contrast characteristics of quadrilaterals and show these relationships with Venn diagrams
• Know, justify and use formula for number of diagonals in a polygon
• Find the midpoint and slope of specific and general (using algebra) line segments
• Prove a figure is a parallelogram by using general coordinates and computing slopes
• Describe why measuring length with a ruler can be challenging for some students.
• Explain why we multiply or divide when we convert units
• Know assigned units in both the US customary & the metric systems
• Convert among/between US customary and metric system units using dimensional analysis
• Know terms related to different types of polyhedral and their parts
• Visualize a named or general polyhedron to determine its characteristics, such as number of vertices, faces and edges
• Know how Euler’s formula relates the number of vertices, faces and edges of any convex polyhedron
• Draw a pattern for a named polyhedron, cylinder or cone
• Name the polyhedron, cylinder or cone associated with a given pattern or net
• Know definition of a platonic solid
• Know names & characteristics of all platonic solids
• Understand why only a certain number of platonic solids exist

Objectives for Math 196

Unit 1

• Understand the definition of fraction and the role of numerator and denominator
• Given the fractional amount of a quantity, represent the unit/ whole or another fractional amount for:
  • Area models (pattern blocks, rectangular models)
  • Discrete models
  • Length models (number line, bar models)
• From the wording of a problem, identify the “whole” or “unit” for each given fraction, and understand how the “whole” may change in a problem
• Use fractions to compare quantities
• Draw pictures to depict equivalent fractions and explain their relationship to the calculation used to find them.
• Connect procedures for finding common denominators and simplifying fractions to the meaning of fractions
• Solve problems involving fractions with unlike denominators
• Use multiple methods to find a fraction between two numbers
• Find decimal equivalency of a fraction
• Flexibly understand meaning of numerator, denominator in a way that broadens the definition of a fraction
• Use and explain the reasoning behind several methods of comparing fractions
• Know/use the following methods to solve percent problems (finding %, part (or portion), and whole):
  • pictures
  • percent equations
    • equivalent fractions
  • percent tables
• Know/use the following methods to solve percent problems (finding %, part, or whole):
  • pictures
  • percent equations (using decimal equivalents or % notation)
    • equivalent fractions
  • percent tables
• Solve percent increase or decrease problems using
  • equations,
  • percent tables*, and
  • strip diagrams*
• Recognize the “whole” when solving percent increase or percent decrease problems
• Understand nuances of language used in percent problems
  • When comparing two quantities to each other
    • % of
    • % as much/many as
  • When comparing a difference in two quantities to one of the quantities
    • % increase/% more than/% profit,
    • % decrease/% less than/% loss/% discount
  • When comparing percentages directly:
    • percentage point increase
• Explain the logic behind the procedure for converting between mixed numbers and improper fractions and draw a picture to motivate that understanding.
• Use several methods to add/subtract mixed numbers and identify student errors involved in these operations.
• Use expanded form to express decimals as a sum of fractions
• Analyze story problems involving fractions to determine whether or not they can be solved by adding the given fractions directly.
• Identify/write fraction story problems of compare, add-to, take-away and part-part-whole problem types.
• Understand and model the addition /subtraction of fractions as combining/separating like parts that refer to the same whole.
• Model integer addition as “combining sets” with +/- manipulatives and corresponding equations
• Model integer subtraction as “take away” with +/- manipulatives and corresponding equations
• Use number lines to model integer addition (as combining sets) and integer subtraction (as take away)
• Explain and model subtraction as comparison using a vertical number line.

Unit 2

• Apply the meaning of multiplication (as defined by author) to write and analyze fraction story problems
• Model proper fraction multiplication using:
  • Area models
    • fraction circles
    • rectangular area model (make the connection between it and the standard algorithm for multiplying fractions)
  • Length model/number line (if time)
  • Discrete model (if time)
• Use the following models to show improper fraction multiplication
  • fraction circles
  • rectangular area (showing connection to std alg)
  • rectangular area (showing connection to FOIL)
  • number line
  • discrete (if time)
• Model decimal multiplication with base 10 manipulatives and/or decimal squares
• Explain why the procedure for multiplying decimals makes sense mathematically
• Know and justify exponent rules for
  • multiplication
  • division
  • raising to a power
• Represent numbers with powers and scientific notation
• Use manipulatives to model multiplying negative numbers, when possible
• Understand and justify the rules for multiplying negative numbers:
  • For (-) (-)=(+), explain with:
    • Pattern method
    • Substitution for Zero method (using the distributive property)
• Interpret remainders from “How many groups?” fraction division problems correctly
• Write “How many groups?” fraction division story problems and see connection to whole number division problems
• Draw pictures to solve “How many groups?” fraction division problems
• Understand the connection between invert and multiply procedure and solving “how many groups/” fraction division problems using a picture
• Identify and write “How many in one group?” fraction division story problems, (and recognize them as ratio problems also)
• Understand the connection between the invert and multiply procedure and solving “how many in one group?” fraction division problems using a picture
• Distinguish fraction division story problems from fraction multiplication story problems (distinguish dividing in half from dividing by one half.)
• Use double number lines to solve “how many in 1 group?” fraction division problems and explain how this is related to the “invert and multiply” algorithm.
• Write story problems involving decimals using both perspectives of division;
• Justify why we can divide two decimals by shifting the decimal point of both the dividend and divisor the same number of decimal points by:
  • Using equivalent fractions/ratios
  • Using a picture
• Solve ratio and proportion problems in multiple ways
  • ratio table
  • double number line
  • cross multiplying with a proportion
  • equivalent fractions
• Use strip diagrams to solve ratio problems and explain their use
• Identify situations that can’t be solved with a proportion
• Solve problems involving inversely proportional quantities using logical reasoning

Unit 3

• Be familiar with how the various number systems (counting numbers, the whole numbers, the integers, the rational numbers, and the real numbers) are related
• Give examples of problems that can be formulated within a number system but require a larger number system to solve.
• Convert fractions to decimals using long division and understand what the resulting reminders tell you about the fraction’s decimal representation and about whether it will terminate or repeat
• Justify why the prime factorization of the denominator of a simplified fraction allows us to determine if a fraction will terminate or repeat.
• Given a terminating or repeating decimal, use algebraic equations to write it as a fraction (Know the value of 0.999… )
• Use a decimal square to “see” a fraction as a decimal
• Solve story problems
  • using strip diagrams,
  • using algebraic equations, and
  • identifying connections between the two methods.
• Justify the formula for the area of a rectangle
• Use moving and additivity principles to solve area problems
• Understand a variety of ways to develop the formula for the area of a triangle
• Given any base of a triangle, draw its corresponding altitude and use it to compute area
• Use appropriate units for area
• Understand a method of developing the formula for the area of a parallelogram
• Understand several ways to develop the formula for the area of a trapezoid
• Investigate the relationship between the circumference and diameter of a circle
• Develop a rationale for the formula of the area of a circle
• Develop a proof for the Pythagorean Theorem and use the theorem to solve problems
• Determine if a square of given area can be made on a geoboard and explain why or why not.
• Know that while perimeter and area are not directly related, knowing the perimeter gives insight into the maximum area of a shape.
• Draw a net for a solid and use it to find the solid’s surface area
• Understand the rationale for and use the formula for the volume of prisms and cylinders
• Understand how volume can be calculated using repeated fillings of water.
• Relate formulas for the volume of a prism to the volume of a pyramid that has the same base and the same height, or of a cone that has the same base and height as a cylinder
• Find volume of various solids, including pyramids and cones
• Use the scale factor and internal factor methods of solving problems about lengths in similar shapes, and understand conceptual differences between the two methods
• Solve problems involving area and volume for similar shapes.

Unit 4

• Understand how sampling design affects the data that is collected
• Display both categorical and numerical data in a variety of appropriate ways.
• Recognize mean, median and mode as measures of center and identify which is most appropriate to use for a given data set
• View the mean of numerical data in two ways:
  • as the “leveling out” of data values and see how this corresponds to the way we calculate it
  • as the “balance point” for data when looking at dot plots and histograms
• Solve problems involving the mean and median
• Solve problems involving GPA
• Compute a weighted course percent grade
• Understand percentile language
• Create and interpret box plots
• Understand how volume can be calculated using repeated fillings of water.
• Relate formulas for the volume of a prism to the volume of a pyramid that has the same base and the same height, or of a cone that has the same base and height as a cylinder
• Find volume of various solids, including, prisms, pyramids, cylinders and cones
• Display both categorical and numerical data in a variety of appropriate ways.
• If two outcomes of an experiment/situation are equally likely, then their probabilities are equal
• Given a set of equally likely outcomes to an experiment, the probability of a given event, E, occurring is the number of outcomes satisfying E divided by the total number of outcomes. For equally likely outcomes ONLY: P (2H) = # of ways 2H can occur/# of outcomes possible