MATH 195: Mathematics for Elementary Education I
(2-2) Cr. 3. F.S.
Prereq: Satisfactory performance on placement assessment, 2 years high school algebra, 1 year of high school geometry, enrollment in elementary education or early childhood education
Whole number operations through analysis of properties, theoretical and hands-on models, mathematical analysis of elementary students’ thinking; standard and non-standard algorithms; structure of the decimal system; linear measurement; two- and three-dimensional measurement, shapes and spatial sense; number theory; algebra as it relates to elementary curricula/teaching profession. Students in the College of Liberal Arts and Sciences may not count 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; continuation of two- and three-dimensional measurement, shapes and spatial sense; probability and statistics; proportional reasoning; algebra as it relates to elementary curricula/teaching profession.
Math 195 Course Packet (for Math 195)
Math 196 Course Packet (for Math 196)
Mathematics for Elementary Teachers, 5th ed. (for both Math 195/196)
ISBN: 9780134423401 (e-book, 180 day subscription)
ISBN: 1323785280 (custom book)
Mathematics for Elementary Teachers, 5th ed. (for both Math 195/196)
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. 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
- Counting by 10s
- 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.
- 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.
Objectives for Math 196
Fractions and Problem Solving
- 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
- Given a fraction, use our definition of a fraction, math drawings and number lines to explain why multiplying the numerator and denominator by the same counting number produces an equal fraction. In the process, attend to the number of parts and the size of the parts.
- Connect procedures for finding common denominators and simplifying fractions to the meaning of fractions. Make math drawings to justify reasoning.
- Solve problems involving fractions with unlike denominators
- Use multiple methods to find a fraction between two numbers
- Find decimal equivalency of a fraction
- Use and explain the reasoning behind several methods of comparing fractions.
- Compare fractions by comparing them to benchmarks such as 1/2 and one. Compare fractions by reasoning about the number of parts and the sizes of the parts.
- Compare fractions that have the same numerator and use math drawings and our definition of fraction to explain the rationale for this method of comparison.
- Recognize that reasoning about fraction comparison requires the fractions to have the same size whole.
- Know how to reason with the following to solve basic percent problems: working with equivalent fractions; using a percent table; going through 1%; going through one; and using math drawings, benchmark, fractions, and mental calculation if appropriate.
- Solve percent increase or decrease problems using percent tables and strip diagrams.
- Understand nuances of language used in percent problems
- Explain the logic behind the procedure for converting between mixed numbers and improper fractions and draw a picture to motivate that understanding.
Addition and Subtraction of Fractions and Integers (Unit 1)
- Use several methods to add/subtract mixed numbers and identify student errors involved in these operations.
- 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 different 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.
- Model integer subtraction as “take away” with +/- manipulatives.
- Use number lines to model integer addition (as combining sets) and integer subtraction (as take away).
Ratio and Proportional Relationships (Units 1 and 2)
- Solve ratio and proportion problems in multiple ways. Using a ratio table, explain what it means for quantities to be a certain ratio from the multiple batches perspective. Using a ratio table and strip diagram as support, explain what it means for quantities to be in a certain ratio from the variable parts perspective.
- Compare the qualities of quantities in different ratios by reasoning about tables, for example, compared to walking speeds or two different mixtures of paints or juices.
- Use strip diagrams to solve ratio problems and explain their use
- Recognize the common era of making an additive comparison in a ratio situation.
- Identify situations that can’t be solved with a proportion
- Recognize and give examples of inversely proportional relationships and distinguish them from proportional relationships. Represent inversely proportional and proportional relationships with tables and graphs, and reason about quantities to find entries in the tables.
- Solve problems involving inversely proportional quantities using logical reasoning
Making Sense of Fraction, Decimal, and Integer Multiplication and Division (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)
- Area models
- 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, area, and/or decimal squares
- Explain why we put the decimal point where we do when we multiply decimals.
- Explain why the procedure for multiplying decimals makes sense mathematically
- Know and justify exponent rules for
- raising to a power
- Represent numbers with powers and scientific notation.
- Explain why we add exponents when we multiply two powers of the same number.
- Explain why it makes sense to define a number to the power of 0 to be 1.
- Use manipulatives to model multiplying negative numbers, when possible. Understand and justify the rules for multiplying negative numbers with the pattern method.
- Write or identify story problems in which the quotient is rounded up or down.
- 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)
- 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 (if time)
- Use a decimal square to “see” a fraction as a decimal (if time)
- Distinguish fraction division story problems from fraction multiplication story problems (distinguish dividing in half from dividing by one half.)
- 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
Area of Shapes (Unit 3)
- Justify the formula for the area of a rectangle
- Explain why we calculate perimeters of polygons the way we do. Discuss misconceptions with perimeter calculations. Recognize that perimeter does not determine area.
- 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 several methods 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
- No how the number Pi is defined and use the definition to explain circumference formulas. Explain that Pi is a constant of proportionality in the relationship between the diameter and circumference of any circle.
- Develop a proof for the Pythagorean Theorem and use the theorem to solve problems
- Know that while perimeter and area are not directly related, knowing the perimeter gives insight into the maximum area of a shape.
Solid Shapes and Their Volume and Surface Area (Unit 3)
- 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
- Name the polyhedron, cylinder or cone associated with a given pattern or net
- 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
- Explain why the 1/3 in the volume formula for pyramids and cones is plausible.
- Discuss the distinction between the volume, the surface area, and the height of the solid shape.
Algebra, Probability and Statistics (Unit 4)
- Solve algebraic story problems
- using strip diagrams,
- using algebraic equations, and
- identifying connections between the two methods.
- 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
- Describe how to view the mean as leveling out, and explain why this way of viewing the mean agrees with the way we calculate the mean (by adding and then dividing).
- Use the leveling-out view of the mean to solve problems about mean. Also use the standard way of calculating the mean to solve problems about the mean.
- Solve problems involving GPA
- Compute a weighted course percent grade
- Understand percentile language (use percentiles and percent correctly)
- Given a data set, determine the median, the first and third quartiles, and the interquartile range. Use medians and interquartile ranges to discuss and compare data sets.
- Create and interpret box plots to compare data sets.
- Given a data set, determine the mean and the mean absolute deviation (MAD). Use means and MADs to discuss and compare data sets.
- 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
- Recognize that an empirical probability is likely to be close to the theoretical probability when a chance process has occurred many times.