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

This course targets the mathematics subject matter specialization standard of the Iowa State Teacher Education Standards. It is designed to help you understand the central concepts, tools of inquiry, and structure of mathematics and prepare you to create learning experiences that make these elements meaningful for elementary students. At the end of this course, you will have both content and process knowledge. You will have experienced what it means to think mathematically, understand the value of conceptual insight, and appreciate how mathematical knowledge is constructed in an exploratory manner.  You will be familiar with the Common Core’s Standards of Mathematical Practice and use them to solve problems mathematically.

Numbers and the Base-Ten System – Unit 1

• Use manipulatives with a non-base 10 place value numeration system
• Explain what it means for the base-ten system to use place value.  Discuss what problem the development of the base-ten system solved.
• Describe base-ten units and explain how adjacent place values are related in the base-ten system.
• Explore the Alphabitia numeration system in an effort to uncover the key features of our own numeration system
  • role of a zero symbol
  • how place value changes as we move right or left one place
  • how the number of symbols used is related to the base of the system
• Discover more about our base 10 system by looking at other base systems
• Examine how elementary students can utilize 100s charts
  • Patterns
  • Counting by 10s
  • Addition/Subtraction
• Count in base other bases (what comes next? what comes before?)
  • How does this parallel the struggles/insights of our future students (when counting in base 10?)
• Understand 1:1 correspondence, cardinality and subitizing concepts
• Convert between base b and base 10
• Use exponential expanded form to represent numbers (sometimes moved to a different unit)
• Use a number line (that reflects the structure of the decimal system) to model and explain rounding decisions (sometimes moved to a different unit)
• Describe and make math drawings to represent a given counting number in terms of bundled objects in a way that fits with the base-ten representation for that number of objects. (sometimes moved to a different unit)

Addition and Subtraction – Unit 1

• Categorize +, – story problems
  • Put Together/Take Apart, Add To, Take From, and Compare
  • Result unknown, start unknown or change unknown (for add to & take away types only)
  • Discrete or continuous quantities
• Write /critique exemplars of each +/- story problem type
  • Clear cut, simple and age appropriate
  • When appropriate, use tenses and/or adverbs to clearly indicate order of actions (via words like “then”, “next”, “more”, “now”, “yesterday”, “this morning”, etc).
  • Write equations and make math drawings (strip diagrams) to represent the problems.
  • Recognize that keywords alone are not effective for solving problems and that problems whose keywords indicate the opposite operation of a solution are difficult.
• Know/demonstrate the relationship between + and –
• Explain how to use a number line to add and subtract numbers (non-negative numbers)
• Recognize/demonstrate /use associative, commutative prop. of addition
  • Give examples to show how to use the associative property of addition to make problems easier to do mentally including the make-a-ten strategy.
  • Give examples to show how to use the commutative property of addition to make problems easier to do mentally including the “count on from the larger addend” methods.
• Describe how to use subtraction problems as unknown add in problems, explain how this can help young children with basic subtraction facts, and explain how this can be applied to other mental math problems.
• Write correct equations to go along with a mental method of addition or subtraction (learning paths for addition and subtraction). Identify where the commutative or associative properties of addition have been used in calculations.
• Add/subtract in different bases, staying in base.  Understand and relate this to base 10.
  • What struggles do students face in base 10?
  • When is it appropriate to introduce the common algorithms?
  • How do these algorithms relate to what students already know/do with manipulativs?
• Know/demonstrate ways to lighten the load for learning single digit addition facts (addition chart)
• Use the equal sign correctly to write an equation or sequence of equations modeling a mental math strategy
  • Students will understand the research-based importance of the equal sign.
• Analyze methods of addition and subtraction other than the common algorithms.  Use multiple bases to understand the role of place value in these methods.
  • Lattice
  • Partial Sums
  • Other variations of subtraction algorithm
• Explain the common (standard) addition and subtraction algorithms in terms of bundled objects/place values, paying special attention to regrouping.
• Make sense of and explain why alternative addition and subtraction algorithms give correct answers.
• Become familiar with a variety of solution methods.

Multiplication and Division – Unit 2

• Know and understand the basic meaning of multiplication as a groups of b
  • Understand the purpose of order in helping students make sense of multiplication in problem solving and mental math strategies
  • Explain why multiplication solves a problem by exhibiting or describing equal groups and reinterpreting the problem as asking how many units are equal to M groups of N units. Write a corresponding multiplication expression (M*N) or equation (M*N=P)
• 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
• Describe multiplication by 10 as moving digits one place to the left. Utilize bundled objects to help explain this occurrence.
• Write expressions that correspond to a design or pattern, and reflect the meaning of multiplication
• Explain why we can multiply to find the area of a rectangle by describing rectangles as partitioned into groups of 1 unit by one unit squares.
• State the commutative property of months placation and explain why it is true (for counting numbers) by partitioning rectangles or arrays in two different ways.
• State the associative property of multiplication and explain why it is true for counting numbers by partitioning boxes two different ways, or by organizing groups of groups of objects in two different ways.
• Give examples of how to use the associative and commutative properties of multiplication and problem and recognize when these properties have been used.
• Know and apply order of operations to evaluate expressions.  Understand why the common pneumonic device PEMDAS leads to misunderstanding.
• State the distributive property and explain why it is true for counting numbers by describing the total number of objects in a partitioned array in two different ways. Use simple situations to explain or illustrate the distributive property.   Write sequences of equations to model mental math strategies illustrated by these arrays.
• Identify and use the distributive property in a context and/or a set of equations
• Use equations and decompose arrays when describing how basic multiplication facts are related to other basic facts via properties of arithmetic.
• Given a mental method of calculation, write a string of equations that correspond to the method and that show which properties of arithmetic were used.
• Write/identify “how many groups” and “how many units in 1 group” types of division problems.
• Write the corresponding multiplication (and addition) equations for division problems with and without remainders.  Include units for each.
• Relate a multiplication problem to an array and partition the array so that the pieces correspond to the lines in the partial products method. Use the decomposition to explain the validity of the partial products method. Show the portions of the array that correspond to the lines in the common method.
• Solve a multiplication problem by writing equations that use expanded forms in the distributive property. Relate the equations of the lines in the partial products method. Use the relationship to explain why the partial products method calculates the correct answer to the multiplication problem.
• Explain the standard algorithm for multiplication in terms of the definition of multiplication, place value, and the properties of arithmetic.
• Explain the validity (or lack thereof) of 0/a, a/0 and 0/0 types of division problems.
• 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

Number Theory – Unit 2 and Unit 3

• 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 a factor tree to find the prime factorization of a number
• Explain and use Number of Factors Theorem to determine the number of factors a number has (and what those factors are) using a number’s prime factorization.
• 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
• Construct combined divisibility tests
• Use algebraic, geometric and numerical methods to justify formulas for adding sequences of numbers.
• Use the Sieve of Eratosthenes, and explain why it produces a list of prime numbers (if time).
• Use trial division method to determine if a given counting number is prime.  Explain why you have to divide only by prime numbers and at what point you can stop (if time).
• Flexibly use the formulas for sums of consecutive numbers and the sums of odd numbers to solve a variety of problems.
• Write numeric and algebraic equations that correspond to sums of different types of numbers
• Formulate and flexibly use equations and/or geometric diagrams arising from a scenario.
• Draw, interpret and use Venn Diagrams to represent information, and solve word problems

Algebra – Unit 3

• Know vocabulary: variable, expression, equation, formula
• 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

Geometry – Unit 3 and Unit 4

• Find the midpoint and slope of specific and general (using algebra) line segments
• Use the fact that a straight line forms a 180 degree angle to explain why the angles opposite each other, which are formed when two lines meet, are equal.
• Know how to show informally that the sum of the angles of a triangle is 180 degrees.
• Know and use the parallel postulate to prove theat the sum of the angles in every triangle is 180 degrees.
• 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
• 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 (if time)

Measurement – Unit 4

• 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
• Recognize that in some cases one object can be larger than another object with respect to one attribute, but smaller with respect to a different attribute.