



Renewing Mathematics Teaching Through Curriculum
RMTC
Summer Workshop Agendas
RMTC is assisting schools in West Michigan who are implementing the CorePlus Mathematics Project (CPMP) Curriculum entitled Contemporary Mathematics in Context. The summer workshops were developed to engage teachers in thinking carefully about the content and teaching of this particular curriculum.
In the summer of 1998, RMTC organized and conducted six oneweek workshops, one for Course 1, three for Course 2 and two for Course 3. The broad goals of the Course 1  3 workshops were:
 to assist teachers in gaining an understanding of the mathematical content in CPMP;
 to model instructional and assessment practices for effective implementation of CPMP;
 to address issues of concern raised by participants, such as classroom management and student achievement;
 to assist participants in understanding the scope and sequence of the threeyear curriculum in the context of the development of mathematical concepts in each course level.
During each of the six workshops, teacherleaders from the RMTC collaborative who are experienced with teaching the curriculum facilitated participant group work on content from the units described in the outlines below, embedding the broad goals of the workshop within the content work. Instructors modeled the teaching approaches they have used with their own students. For evaluation of the mathematical content learned by participants, the Course 1 and Course 2 participants completed pretests and posttests consisting of items from student assessments for the units studied in the workshop.
To read more about the RMTC project Summer Sessions, visit their project website.
Course 1 Agenda
Monday 
8:00  Refreshments 
8:30  Introductions Why reform mathematics programs? Pretest on Course 1 content 
10:30  Overview of the Statistics and Probability Strand Facilitating collaborative groups Modeling of instruction of Unit 1, Patterns in Data Content objectives for Unit 1:
 To use various graphical displays of data to uncover important patterns in the data set and interpret these patterns for a realworld context
 To summarize and interpret sets of data using measures of center and variability
 To compare sets of data using scatterplots and the line y = x, and to interpret these comparisons for the realworld contexts that gave rise to the data

4:45  Synthesis of the day's work Dismissal 
Tuesday 
8:00  Continued work on Patterns in Data 
1:00  Overview of the Algebra and Functions Strand Modeling instruction of Unit 2, Patterns in Change Content objectives for Unit 2
 To begin developing students' sensitivity to the rich variety of situations in which quantities vary in relation to each other
 To develop students' ability to represent relations among variables in several ways  using tables of numerical data, coordinate graphs, symbolic rules, and verbal descriptions  and to interpret data presented in any one of those forms
 To develop students' ability to recognize important patterns of change in single variables and related variables

4:45  Synthesis of the day's work Dismissal 
Wednesday 
8:00  Continued work on Patterns in Change 
10:30  Modeling instruction of Unit 3, Linear Models Content objectives for Unit 3
 To begin developing students' sensitivity to the rich variety of situations in which quantities vary in relation to each other
 To develop students' ability to represent relations among variables in several ways  using tables of numerical data, coordinate graphs, symbolic rules, and verbal descriptions  and to interpret data presented in any one of those forms
 To develop students' ability to recognize important patterns of change in single variables and related variables

4:45  Synthesis of the day's work Dismissal 
Thursday 
8:00  Overview of the Discrete Mathematics Strand Modeling instruction of Unit 4, Graph Models Content objectives for Unit 4
 To use vertexedge graphs to make sense of situations involving relationships among a finite number of elements  for example, conflict and prerequisite relationships
 To gain experience in mathematical modeling by building and using vertexedge graph models to solve problems in a variety of realworld settings
 To develop the skill of algorithmic problem solving: designing, using, and analyzing systematic procedures for solving problems
 To investigate and apply three powerful and widely used graph models: Euler paths, vertex coloring, and critical paths

1:00  Overview of the Geometry and Trigonometry Strand Modeling instruction of Unit 5, Patterns in Space and Visualization Content objectives for Unit 5
 To use geometric shapes and their properties to make sense of situations involving data, change, chance, and discrete structures
 To use visualization to interpret and reason about space and plane situations
 To classify, construct, and sketch models of spaceshapes
 To use plane and spaceshapes to model reallife situations
 To find appropriate measures (perimeter, area, volume) of planeand spaceshapes
 To classify polygons and analyze their properties
 To identify and explain different kinds of symmetry for planeand spaceshapes

4:45  Synthesis of the day's work Dismissal 
Friday 
8:00  Continued work on Patterns in Space and Visualization 
3:00  Synthesis of the day's work Posttest Workshop Evaluation Dismissal 
Course 2, Agenda
Monday 
8:00  Refreshments 
8:30  Introductions Pretest on Course 2 content 
9:30  Modeling of instruction of Unit 1, Matrix Models Content objectives for Unit 1:
 To see the interconnectedness of mathematics through the use of matrices to explore topics in algebra, geometry, statistics, probability, and discrete mathematics
 To use matrices for organizing and displaying data in a variety of realworld settings like brand switching, tracking pollution through an ecosystem, and tournament rankings
 To develop further the skill of mathematical modeling by building matrix models and then operating on them to solve problems
 To learn and apply matrix operations: row sums, scalar multiplication, addition, subtraction, and matrix multiplication
 To use matrices and inverse matrices to answer questions that involve systems of linear equations

4:45  Synthesis of the day's work Dismissal 
Tuesday 
8:00  Modeling instruction of Unit 2, Patterns in Location, Shape, and Size Content objectives for Unit 2
 To use coordinates to model points, lines, and geometric shapes in the plane and to analyze the properties of lines and shapes
 To write systems of linear equations that model reallife situations, to solve systems using linear combinations, and to explain the geometry behind this method
 To use coordinate geometry and programming techniques as a tool to implement computational algorithms, model transformations, and investigate the properties of shapes that are preserved under transformations
 To build and use matrix representations to model polygons, transformations, and computer animations

1:00  Programming Animation on graphics calculators 
4:45  Synthesis of the day's work Dismissal 
Wednesday 
8:00  Modeling instruction of Unit 3, Patterns of Association Content objectives for Unit 3
 To describe the association between two variables by interpreting a scatterplot
 To interpret correlation coefficients
 To understand that just because two variables are correlated, it does not mean that one directly causes the other
 To know when it is appropriate to make predictions from the least squares regression line or its equation
 To understand the effects of outliers on correlation coefficients, on the equation of the least squares regression line, and on the interpretations of correlation coefficients and regression line in realworld

1:00  Modeling instruction of Unit 4, Power Models Content objectives for Unit 4 
4:45  Synthesis of the day's work Dismissal 
Thursday 
8:00  Continued work on Unit 4  Quadratic Models 
10:15  Modeling instruction of Unit 5, Network Optimization Content objectives for Unit 5
 To gain experience in mathematical modeling by building and using vertexedge graph models to solve problems in a variety of realworld settings
 To develop the skill of algorithmic problem solving by designing, using, and analyzing systematic procedures for solving problems
 To optimize networks in different ways and in different contexts by finding minimal spanning trees, shortest paths, Hamiltonian paths, and by analyzing the Traveling Salesperson Problem

4:45  Synthesis of the day's work Dismissal 
Friday 
8:00  Unit 8, Capstone  Investigations 2  7
 To review the major objectives of the course in the context of one large project
 To synthesize mathematical concepts from Course 2

11:00  Posttest Workshop Evaluation 
1:00  Capstone Group Presentations 
4:00  Dismissal 
Course 3, Agenda
Monday 
8:00  Refreshments 
8:30  Introductions 
10:30  Overview of the algebra strand in Course 3 Modeling of instruction of Unit 1, Multiple Variable Models Content objectives for Unit 1:
 To develop an understanding of, and the ability to solve, problems involving multiplevariable relations (including trigonometric relations) where one equation relates more than two variables
 To develop the ability to solve multiplevariable equations for one variable in terms of the other variables
 To model situations with systems of equations and inequalities where two or more output variables are related to the same input variables, and to apply those systems to solve problems

3:15  Modeling of instruction of Unit 3, Symbol Sense and Algebraic Reasoning 
4:45  Synthesis of the day's work Dismissal 
Tuesday 
8:00  Continued work on Symbol Sense and Algebraic Reasoning Content objectives for Unit 6 
10:15  Modeling instruction of Unit 6, Families of Functions Content objectives for Unit 6
 To describe the table and graph patterns expected in linear, direct power, inverse power, exponential, sine, cosine, absolute value, and square root models given the corresponding algebraic rules in function form
 To identify a function as a variation of a basic family of functions
 To recognize how the patterns in graphs, tables, and rules of functions relate to the functions' transformed graphs, tables, and rules
 To write function rules which are reflections across the xaxis, translations, or stretches (or combinations of these transformations) of basic functions
 To apply all of the transformation above as they relate to realworld situations

4:45  Synthesis of the day's work Dismissal 
Wednesday 
8:00  Modeling instruction of Unit 2, Modeling Public Opinion Content objectives for Unit 2
 To measure and analyze public opinion through a mathematical analysis of voting and surveys
 To use and analyze a variety of election analysis methods, particularly those based on preferential voting
 To understand and apply basic ideas related to the design and interpretation of surveys, such as background information, random sampling, and bias
 To construct simulated sampling distributions of sample proportions, and to use sampling distributions to identify which proportions are likely to be found in a sample of a given size
 To construct and interpret margin of error and confidence intervals for population proportions
 To critically analyze surveys and elections in everyday life and as reported in the media

2:15  Modeling instruction of Unit 4, Shapes and Geometric Reasoning Content objectives for Unit 4
 To recognize the differences between, as well as the complementary nature of, inductive and deductive reasoning
 To develop some facility in producing deductive arguments in geometric situations
 To know and be able to use the relations among the angles formed when two lines intersect
 To know and be able to use the necessary and sufficient conditions for two lines to be parallel
 To know and be able to use triangle similarity and congruence theorems
 To know and be able to use the necessary and sufficient conditions for quadrilaterals to be (special) parallelograms
 To use a variety of conditions among triangles, lines, and quadrilaterals to prove the correctness of related geometric statements or provide counterexamples

4:45  Synthesis of the day's work Dismissal 
Thursday 
8:00  Continued work on Shapes and Geometric Reasoning 
1:00  Modeling instruction of Unit 5, Patterns in Variation Content objectives for Unit 5
 To understand the standard deviation as a measure of variability in a distribution
 To understand the normal distribution as a model of variability
 To understand and be able to use the number of standard deviations from the mean as a measure of position of a value in a distribution
 To understand the construction, interpretation, and theory of control charts
 To understand and apply the Addition Rule for mutually exclusive events

4:45  Synthesis of the day's work Dismissal 
Friday 
8:00  Continued work on Patterns in Variation 
10:15  Modeling instruction of Unit 7, Discrete Models of Change Content objectives for Unit 5
 To use iteration and recursion as tools to represent, analyze, and solve problems involving sequential change
 To formalize and consolidate previous study of NOWNEXT equations, particularly through the use of subscript notation and the introduction of recursion equations
 To understand and apply arithmetic and geometric sequences, and series
 To understand and apply the method of finite differences
 To explore function iteration and, in the process, informally introduce function composition
 To investigate and apply recursion equations, particularly combined recursion equations of the form A_{n} = r A_{n  1} + b
 To review linear, exponential, and polynomial models from a recursive perspective

4:00  Synthesis of the day's work Workshop Evaluation Dismissal 

