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

Course Coordinator

Gail Johnston (gjohnsto@iastate.edu)


Catalog Description

MATH 195. Mathematics for Elementary Education I.
(2-2) Cr. 3. F.S. Prereq: Satisfactory performance on placement exam, 2 years high school algebra, 1 year of high school geometry, enrollment in elementary education or early childhood education
Theoretical and hands-on models, mathematical analysis of: elementary students' thinking, standard and non-standard algorithms, and properties related to whole number operations; structure of the decimal system; linear measurement, and two- and three-dimensional geometric shapes and spatial sense; algebra as it relates to elementary curricula. Students in the College of Liberal Arts and Sciences may not count MATH 140, MATH 142, or MATH 195 toward General Education Requirements.

MATH 196. Mathematics for Elementary Education II.
(2-2) Cr. 3. F.S. Prereq: Minimum of C- in MATH 195 and enrollment in elementary education or early childhood education.
Integer, fraction and decimal operations through analysis of properties, theoretical and hands-on models, mathematical analysis of elementary students’ thinking, standard and non-standard algorithms: two- and three-dimensional measurement; probability, statistics, algebra as it relates to elementary curricula/ teaching profession.


Textbook

book coverMath 195 Course Packet (for Math 195)
ISBN: 9780999037591
(new, not used)


book coverMath 196 Course Packet (FOR MATH 196)
ISBN: 9780999037591
(new, not used)


book coverBeckmann (for both Math 195/196)
Mathematics for Elementary Teachers, 3rd ed.
ISBN: 9780999037591 (book)
ISBN: 9781256916741 (activity manual)
Alternatively, you can use the ISU Custom Edition based on the 2nd edition
The book can be used, but the activity manual must be new.


clickerTurning Point RF Clicker
ISBN: 9780108000003



Syllabus for Math 195

The course focuses on the whole number system, numeration, algorithms and interpretations for whole number computation, topics from number theory, algebra, geometric shapes, and measurement.  We will cover portions of chapters 1, 3, 4, 6, 8, 9, 10, 11, & 13 of the text and accompanying activities from the Activity Manual.

Syllabus for Math 196

The course focuses on systems of rational numbers (fractions and decimals) and integers, as well as physical representations, theoretical models and computations, algebraic reasoning, percent and proportional reasoning, surface area and volume, data analysis, and probability. We will cover portions of Chapters 2, 3, 5, 6, 7, 8, 12, 13, 15, 16 and additional material.           


Objectives for Math 195

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.

Day 1

  • Understand course rationale and structure
  • Use manipulatives with a non-base 10 place value numeration system
  • Convert symbols and pictures in Alphabitia
  • Explain the role of zero in place value numeration systems
  • Become familiar with Common Core

Day 2

  • Discover more about our base 10 system by looking at other base systems (where is there trouble with counting, place value, etc)
  • Examine 100 charts, think about patterns and counting by 10s
  • Be able to count in base 5 & 6 (what comes next? what comes before?)
  • Understand 1:1 correspondence, cardinality and subitizing concepts

Day 3

  • Continue to discover more about our base 10 system by looking at other base systems
  • Determine what comes next, what comes before, in a variety of bases
  • Convert between base b and base 10
  • Investigate challenges and solutions to difficulties that arise from bases >10
  • Convert numbers from base10 to Roman Numerals (know symbols)

Day 4

  • Use expanded form and exponential expanded form in any base
  • Describe and draw rough pictures to represent a given positive decimal in 2 ways (set of bundled objects and length models) that fit with and show the structure of the decimal system.
  • (1) Discuss how decimals fill in a number line, (2) label tick marks on number lines, and (3) plot numbers on number lines, in ways that reflects the structure of the decimal system.

Day 5

  • Continue to
    • Describe how decimals fill in a number line
    • label tick marks on number lines, and
    • plot numbers on number lines in a way that reflects the structure of the decimal system.
  • Show how to zoom in on portions of number lines to see the portions in greater detail
  • Use number line to model and explain rounding decisions
  • Use decimal representations and  number line to compare decimals

Day 6

  • Categorize +, - story problems
    • Part-part-whole, add to, take away, or 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
    • Age appropriate
    • Use tenses and/or adverbs to clearly indicate order of actions, & when total occurred (via words like “then”,  “next”,  “more”, “now”, or “yesterday”, “this morning”, etc).
  • Understand/demonstrate the relationship between + and -

Day 7

  • Draw strip diagrams for addition/sub problems
  • Know/recognize/demonstrate /use associative, commutative prop of +
  • Know/recognize/demonstrate learning paths for addition (CA 3F)
  • Add in different bases, staying in base

Day 8

  • Know/demonstrate ways to lighten the load for learning single digit addition facts; with equations and on addition chart

Day 9

  • Know/recognize/demonstrate, and for Level 3, write equations for, procedures children use to perform single-digit subtraction.
  • Subtract in different bases, staying in base
  • Write correct equations to correspond to a student’s solution strategy

Day 11

  • Write equations that correspond to a method of calculation for addition and subtraction problems.
  • Use number lines to find the difference between two numbers
  • Analyze methods of addition and subtraction other than the common algorithms.

Day 12

  • Add/subtract multi-digit numbers in base b
  • Use the lattice algorithm to add in base b or 10
  • Use the partial sums algorithm to add in base b or 10

Day 13

  • Explain the common (standard) addition and subtraction algorithms in terms of bundled objects, paying special attention to regrouping.
  • Make sense of and explain why alternative addition and subtraction algorithms give correct answers.

Day 14

  • Describe a set
    • in words,
    • by listing the elements
  • Perform and understand notation for set operations
  • Draw, interpret and use Venn Diagrams to represent information, perform set operations and solve word problems

Day 15

  • Know and understand the basic meaning of multiplication
  • Write and solve array, repeated addition, area, ordered pair, and multiplicative comparison types of story problems.
  • Use arrays, organized lists, tree diagrams, strip diagrams and number lines to exhibit multiplicative structure.

Day 16

  • 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 which uses the operations and properties to describe a given picture/diagram.
  • Draw a picture/diagram to represent a given expression.
  • Write equations that correspond to a method of calculation

Day 17

  • Identify and use the distributive property in a context and/or equations.
  • For a given expression involving the distributive property, write a word problem and draw a diagram to correspond to the expression.
  • Use the distributive property to lighten the load for learning single-digit multiplication facts

Day 18

  • Write equations that correspond to a method of calculation
  • Understand  the relationship between quantities such as 38x19 and 40x20 using rectangular (area) arrays and equations.

Day 19

  • Know, write equations for and draw a rectangular (area) array for the standard and partial products multiplication algorithms.
  • Solve multi-digit non-base 10 multiplication problems—working in the given base.
  • Solve a multiplication problem (with either whole numbers or binomials) using the lattice method.

Day 20

  • Write/identify “how many groups” and “how many in each group” types of division problems
  • 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.
  • Understand what properties hold for division
  • Write the corresponding multiplication (and addition) equations for division problems with and without remainders

Day 21

  • Use scaffold method of division and interpret process in terms of “how many groups” and types of manipulatives.
  • Explain steps in a student created division solution and determine its validity.
  • Understand that division problems must specify evenly distributed (equal) amounts

Day 22

  • 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

Day 24

  • Write, identify and solve word problems involving multiples and factors
  • Use and understand terms and notation: factor, divisor, multiples, divisibility and  notation.
  • Use the Sieve of Eratosthenes to find prime numbers, and explain why this works
  • Know and defend an efficient method for determining when a number is prime
  • Use a geometry (area) argument to describe factors and multiples

Day 25

  • List all the factors of a number.
    • Explain the Fundamental Theorem of Arithmetic
    • Use factor trees to find the prime factorization of a number.
    • Explain and use Number of Factors theorem/observations from factor table

Day 26

  • 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

Day 27

  • Define even and odd numbers
  • Use algebra and pictures to prove conjectures about even or odd numbers (examples are not enough to prove something is true)

Day 28

  • Explain the rationale behind divisibility tests for 2, 3,  4,  5,  8,  9, 10 using manipulative diagrams and place value arguments, or division algorithm proofs, and understand that a specific example is not a proof of all cases.
  • Use divisibility rules correctly to determine if a given number is divisible by 2, 3, 4, 5 ,8, 9 or 10,  and to construct a divisibility test for other numbers, where appropriate.

Day 29

  • Know terms: variable, expression, equation, formula
  • Write expressions that correspond to a design or pattern, and reflect the meaning of multiplication
  • Write equations that correspond to sums of different types of numbers

Day 30

  • Write equations to correspond to sums of different types of numbers.
  • Understand and justify formulas for adding sequences of numbers algebraically, computationally and geometrically.
  • Flexibly use the formulas for sum of odd numbers and the sum of consecutive numbers to solve a variety of problems.
  • Develop a series of geometric drawings to represent a scenario (could be either a situation or a sequence of numbers). Use the geometric drawings to develop/justify a formula to represent the scenario

Day 31

  • Write equations to represent related quantities; write story problems to represent such equations
  • Distinguish between “three times as much as” and “three times more than”, etc.
  • Practice problem solving skills using charts, algebra, etc. in an organized format.

Day 32

  • Solve equations using number sense: pictures and written descriptions
  • Solve equations algebraically and show correspondence to pan balance

Day 34

  • Visualize lines and planes, disjoint regions
  • Know geometry terms from text

Day 35

  • 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 interior angles in a polygon, as well as the individual measures of angles in a regular polygon
  • Solve angle problems involving polygons

Day 36

  • 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
  • Understand how formulas for area of rectangle and parallelogram are derived and why they make sense
  • Know and use formulas for area of rectangles and parallelograms

Day 37

  • Draw the three altitudes for all types of triangles
  • Know and justify the formula for the area of a triangle
  • Find the midpoint of specific and general (using algebra) line segments
  • “Prove” a figure is a parallelogram/other polygon by using slope or lengths of sides.

Day 38

  • Describe why measuring length with a ruler can be challenging for some students.
  • Know assigned units in both the US customary & the metric systems
  • Convert among/between US customary and metric system units using dimensional analysis

Day 39

  • 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

Day 40

  • Know definition of a platonic solid
  • Know names & characteristics of all platonic solids
  • Understand why only a certain number of platonic solids exist
  • Draw a pattern for a named polyhedron, cylinder or cone
  • Name the polyhedron, cylinder or cone associated with a given pattern or net

Objectives for Math 196

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. Therefore, an emphasis will be placed on communication of mathematical ideas as you work actively in small and large group settings to discover mathematical concepts.  At the same time, this is a content course and you are expected to learn the mathematical terminology and concepts. Topics covered will include fractions, decimals, integers, percents, algebraic reasoning, geometry, elementary statistics and probability concepts.

Day 1

  • Understand the definition of fraction, role of the numerator and the denominator.
  • Realize the importance of the unit in understanding /solving a problem about fractions.
  • Given the fractional amount of a quantity using a discrete model, pattern blocks, number line or other model, express the unit or another fractional amount.

Day 2

  • Use pictures and numberlines to motivate understanding of converting between mixed numbers and improper fractions.
  • Identify  the “whole” or unit, based on the wording of a problem
  • Represent partitioning the whole (unit) in a variety of ways  (fraction of parts, fraction of area, discrete/set model, area model)
  • Understand that the meaning of equal parts may be tied to a specific attribute of an object, quantity or collection as defined by a problem
  • Use fractions to compare quantities

Day 3

  • Flexibly understand meaning of numerator, denominator
  • Find decimal equivalents of fractions by long division.
  • Approximate locations of fractions on number lines, and explain the reasoning using meaning of numerator and denominator                       
  • Draw pictures to depict equivalent fractions       
  • Use equivalent fractions/decimals to find a fraction between two numbers

Day 4

  • Connect procedures for finding common denominators and simplifying fractions to text’s definition of fractions
  • Correctly identify that finding equivalent fractions involves multiplying by a form of ONE, not by a factor
  • Understand the relationship between numbers that do not have the same denominators and to demonstrate this on a number line.

Day 5

  • Use and explain the reasoning behind the following methods of comparing fractions:
    • Decimal equivalence
    • Common denominator
    • Common numerator
    • Cross-multiplying
    • Comparing to a benchmark

Day 6

  • Solve basic percent problems (finding %, part, whole)
  • Know/use the following methods to calculate percents
    • percent tables
    • equivalent fractions
    • pictures
    • decimal equivalents
    • write equations to solve percent problems, & correctly interpret your answer

Day 7

  • Solve % story problems in a variety of ways

Day 8

  • Use fraction-decimal –percent equivalencies to compute
  • Identify student errors in adding and subtracting mixed numbers

Day 9

  • Identify and write fraction story problems of compare, add-to, take-away and part-part-whole problem types.
  • Express a decimal as a sum of fractions
  • Determine the fraction of a figure shaded

Day 11

  • Use fraction circles to help understand/explain the standard algorithm for adding and subtracting fractions
  • Model addition of integers as combining sets using (+) (-) manipulatives.

Day 12

  • Model subtraction as take away using (+) (-) manipulatives
  • Explain and model subtraction as comparison using a vertical number.

Day 13

  • Write and interpret fraction multiplication story problems in a way that is consistent with the meaning of multiplication
  • Use fraction circle and rectangular area models for proper fraction multiplication and make the connection between these models and the standard algorithm for multiplying fractions

Day 14

  • Use fraction circle and rectangular area models for improper fraction multiplication and make the connection between these models and the standard algorithm for multiplying fractions
  • Use number line and discrete (optional) models for multiplying improper fractions
  • Demonstrate the connection between multiplying improper fractions using the distributive property (“each with each” or FOIL) and an rectangular area model showing the partial products

Day 15

  • Understand why the procedure for multiplying decimals works.
  • Model decimal multiplication using base 10 manipulatives and decimal squares
  • Explain/use exponent rules and scientific notation

Day 16

  • Understand and justify the rules for multiplying negative numbers
  • Understand  rationales for why (-)(-) = (+):
    • Pattern method
    • Concept of zero method

Days 17 & 18

  • Write/recognize fraction division story problems from the “how many groups” perspective
  • Draw pictures to solve fraction division problems
  • Understand why we ‘invert and multiply” to divide fractions from the “how many groups?” perspective.
  • Understand and solve fraction division problems using the “divide the numerators” method
  • Interpret remainders from a “how many groups” division stoy problem

Day 19

  • Write/recognize division story problems from the “how many in one group?” perspective.
  • 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 fraction division problems and relate to the invert and multiply procedure

Day 20

  • Write division story problems involving decimals using both the “How many groups?” and “How many in one group?” perspectives
  • Use decimal squares and manipulatives to model both the “How many groups?” and “How many in one group?” perspectives of decimal division
  • Justify why we can divide two decimals by shifting the decimal point of both the dividend and divisor the same number of decimal places
  • Recognizing fraction division problems as ratio problems.

Day 21

  • Solve ratio & proportion problems in multiple ways
  • Use strip diagrams to solve ratio problems and explain their use. 
  • Use ratio tables to solve problems

Day 22

  • Solve percent increase or decrease problems using percent tables, diagrams, and symbolic means. 
  • Understand the difference between percentage point increase/decrease and percent increase/decrease

Day 24

  • Explore the remainders and length of repetend, resulting from long division in converting a fraction to a  decimal
  • Use an argument about remainders to identify when a fraction’s decimal representation will terminate (remainder = 0) and when it will repeat (remainder repeats)
  • Justify why the prime factorization of the denominator of a simplified fraction allows us to determine if a fraction will terminate or repeat.
  • Use algebra and a system of equations to write a fraction for a repeating decimal
  • Know the value of 0.999…

Day 25

  • Explore multiple ways to solve story problems including algebraic equations and strip diagrams. 

Day 26

  • Determine if a given sequence is arithmetic, geometric, or neither
  • Write successive terms and formula for arithmetic or geometric sequences
  • Given a figure or table of values, write the expression that arises from either an arithmetic  or geometric sequence
  • Connect features of a linear function with features of corresponding graph and table (intercept, slope)

Day 27

  • Use Additivity  and Moving Principles to find the area of a variety of geometric figures
  • Know and apply formula for area of rectangle & parallelogram
  • Understand a variety of ways to develop the formula for the area of a triangle
  • Use appropriate units for area

Day 28

  • Develop Pick’s formula and use it to find the area of figures whose vertices lie on lattice points
  • See connections between Pick’s formula and graph
  • See how the domain of a function affects how the graph is drawn
  • Justify the formula for the area of a trapezoid

Day 29

  • Understand the meaning of π
  • Develop a rationale for the formula of the area of a circle

Day 30

  • Use dimensional analysis and alternate methods to find the area of an irregular shape
  • Estimate the area of an irregular figure by finding an upper and lower bound of the area.

Day 31

  • Know that while perimeter and area are not always directly related, perimeter can give insight into the maximum area of a shape.
  • Consider the historical perspectives of the Pythagorean Theorem
  • Describe/explain the Pythagorean Theorem proof covered in class
  • Apply the Pythagorean Theorem to solve problems

Day 32

  • Determine if a square of given area can be made on a geoboard and provide a convincing explanation

Day 33

  • Find surface area of various solids including: polyhedra, prisms, pyramids, cylinders and cones

Day 34

  • Find surface area and volume of polyhedra and other solids
  • Understand how volume can be calculated using repeated fillings of water
  • Relate formulas for the volume of a pyramid to the volume of a prism that has the same base and the same height, (or for a cone that has the same base and height as a cylinder, or for a sphere that has the same diameter as thdiameter and height of a

Day 35

  • Create and interpret common data displays
  • Consider different types of questions that we can ask about display of data
  • Recognize common errors and misleading practices associated with displays
  • Write questions for the three levels of questions related to graphs/displays

Day 36

  • Recognize mean, median and mode as measures of center and identify which is most appropriate to use for a given data set
  • View mean as “leveling out” and see how this corresponds to the way we calculate it
  • Use the compensation method for finding the mean
  • Solve problems about the mean
  • Create data sets with different means and medians

Day 37

  • Compute a course percent grade when given weighted components.
  • Solve problems involving GPA.
  • Report and interpret percentiles
  • Understand why a “measure of spread” is important in adequately describing a data set
  • Create and interpret graphical displays showing the spread of  data
    • Box-Plots

Day 38

  • Compute measures of spread:
    • MAD (Mean Absolute Deviation)
    • standard deviation
  • Use standard deviation and mean to answer questions about a data set
  • Understand the basic idea of a normal distribution
  • Compute z-scores and use to compare performance

Day 39

  • Understand what is meant by theoretical and experimental probability
  • Use a tree diagram to represent possible outcomes and probabilities associated with the outcomes,  with and without replacement

Day 40

  • Use a tree diagram to represent and solve a variety of conditional  probability problems

 


Old Exams

(none provided)


Official Math Department Policies

The Math Department Class Policies page describes the official policies that all instructors have to follow. It covers rules on make-up exams, cheating, student behavior, etc.


Students With Disabilities

If you have a documented disability and require accommodations, you should obtain a Student Academic Accommodation Request (SAAR) from the Disability Resources office (Student Services Building, Room 1076, 294-6624 or TDD 294-6335, disabilityresources@iastate.edu or accommodations@iastate.edu). Please contact your instructor early in the semester so that your learning needs may be appropriately met.

More information about disability resources in the Mathematics Department can be found at http://www.math.iastate.edu/Undergrad/AccommodationPol.html.