Summer Workshop Plans

RMTC Summer Workshop Agenda

RMTC is assisting middle and high schools in West Michigan who are implementing the Core-Plus Mathematics Project (CPMP) and the Connected Mathematics Project (CMP) curricula. The summer workshops engage teachers in thinking carefully about the content and teaching of these particular curricula.

How long do your workshops last?

Grade level specific implementation workshops are held for one-week in the summer. This summer we are also facilitating a two-day institute where teachers will build understanding of the development of algebraic ideas from kindergarten to high school by exploring activities from Investigations in Number, Data, and Space (K-5 curriculum), CMP, and CPMP.

How many teachers do you involve in your workshops?

There are a total of 67 teachers attending the RMTC week-one workshops and 20 attending the two-day strand workshop. In addition, 22 teachers are attending CPMP or CMP one-week workshops organized by the curriculum projects.

What are your major goals for your workshops in terms of content and pedagogy? (Just one or two paragraphs)

The broad goals of the workshops are:

• to assist teachers in gaining an understanding of the mathematical content in CPMP and CMP;

• to model instructional and assessment practices for effective implementation of CPMP and CMP;

• 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 topics in the curricula,

• to assist teachers in becoming more reflective about their teaching.

Are you offering professional development on specific curricula that you are hoping will be used in the classroom? If so, which curricula are you using?

Yes, our project is using specific curricula. See first three questions above.

CPMP Course 3 Agenda

Monday
7:30	Refreshments
8:00	Introductions Participant Issues and Concerns Reflections on Student Learning Overview of Course 3
11:00	Content Pretest for evaluation of teachers mathematical learning 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 multiple-variable relations (including trigonometric relations) where one equation relates more than two variables To develop the ability to solve multiple-variable 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 Synthesis of Unit 1
4:00	Dismissal
Tuesday
8:00	Modeling of 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 Synthesis of Unit 2
2:00	Assessment - Participants do a group test
3:00	Issues and concerns discussion
4:00	Dismissal
Wednesday
8:00	Modeling instruction of Unit 3, Symbol Sense and Algebraic Reasoning Content objectives for Unit 3 To develop a more formal understanding of function and function notation To reason about algebraic expressions by applying the basic algebraic properties of commutativity, associativity, identity, inverse, and distributivity To develop greater facility with algebraic operations with polynomials, including adding, subtracting, multiplying, factoring, and solving To solve linear and quadratic equations and inequalities by reasoning with their symbolic form To prove important mathematical patterns by writing algebraic expressions, equations, and inequalities in equivalent forms and applying algebraic reasoning Synthesis of Unit 3
2:00	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 relating to triangles, lines, and quadrilaterals to prove the correctness of related geometric statements or provide counterexamples
4:00	Dismissal
Thursday
8:00	Continued work on Unit 4 Synthesis of Unit 4
1:00	Assessment - individual test
2:00	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
Friday
8:00	Continued work on Unit 5 Synthesis of Unit 5
1:00	Content Posttest completed by participants
2:00	Overview 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 x-axis, translations, or stretches (or combinations of these transformations) of basic functions To apply all of the transformations above as they relate to real-world situations Synthesis of Unit 6
2:30	Issues and Concerns
3:00	Synthesis of the week's work Dismissal
Course 4 Objectives
Monday
7:30	Refreshments
8:00	Introductions Participant Issues and Concerns Reflections on Student Learning Overview of Course 4
10:00	Content Pretest for evaluation of teachers mathematical learning Modeling of instruction of Unit 1, Rates of Change Content objectives for Unit 1 To develop student ability to estimate the rate of change for a variety of quantities using tables of numerical data, graphical representations, and symbolic rules and to develop student ability to relate the rate of change in a quantity to the graph of that quantity To develop student ability to recognize that many nonlinear functions "look" linear when zoomed in on at a point and thus the rate of change at a point for a nonlinear function can be approximated with the rate of change for a linear function To develop student ability to estimate the net change in a quantity whose rate function is given in graphical, tabular, and symbolic forms using systematic approximations to its rate of change function and geometric considerations To develop student ability to estimate and calculate areas, adapting the approximation method to the estimation of areas or using integrals in conjunction with a calculator or computer integration tool
4:00	Dismissal
Tuesday
8:00	Continued work on Rates of Change Synthesis of Unit 1
3:00	Pedagogical Issues
4:00	Dismissal
Wednesday
8:00	Modeling of instruction of Unit 2: Modeling Motion Content Objectives for Unit 2: To describe and use the concept of vector in mathematical, scientific, and everyday situations To represent vectors geometrically and to operate on them using this representation To describe, represent, and use vector components synthetically and analytically To use vector concepts to represent parametrically plane-linear, plane-projectile, and plane circular motions To use parametric models of motions to answer questions concerning the motions (linear, projectile and circular)
4:00	Synthesis of Unit 2 Dismissal
Thursday
8:00	Modeling of instruction of Unit 3: Counting Models Content Objectives for Unit 3 To develop the skill of careful counting in a variety of contexts To understand and apply a variety of counting techniques, such as the Multiplication Principle of Counting, tree diagrams, systematic lists, and combinatorial reasoning To identify, understand, and solve combinatorial problems involving combinations, permutations, selections, and arrangements of indistinguishable objects To understand and apply the Binomial Theorem and Pascal’s triangle To develop the ability to prove statements using combinatorial reasoning and the Principle of Mathematical Induction Synthesis of Unit 3
3:30	Reflective Writing
4:00	Dismissal
Friday
8:00	Modeling of instruction of Unit 4: Composite, Inverse and Logarithmic Functions Content Objectives for Unit 4 To explore, understand, and represent composition of functions in geometric, algebraic, and numeric contexts To explore, understand, and represent inverse relationships in geometric, algebraic, and numeric settings To use function composition as a tool to analyze the graph of a function To understand and discuss how the inverses of functions that are not one-to-one can become functions by appropriate restrictions on the domain of the function To produce and use inverse functions for y = ax2 and y = a(bx) To understand the inverse relationship between logarithms and exponentials and to use it to simplify complex computations and solve exponential equations To linearize bivariate data by transforming one or both variables To use linearizing as a tool in finding an appropriate model for bivariate data Synthesis of Unit 4
1:00	Posttest completion
2:00	Issues of Implementation: Political Understanding the full CPMP program Evaluation of mathematics programs

This 2-day workshop focused on the development of algebraic ideas in the Connected Mathematics Project (CMP) middle school program, with a special emphasis on how these ideas fit into a K-12 approach to learning algebra. In addition to the focus on CMP, we looked at the Investigations in Number, Data, and Space curriculum for elementary school and the Core-Plus Mathematics Project curriculum for high school. Discussion revolved around several aspects of algebraic reasoning, but focused mainly on the rate of change as a unifying theme. The workshop was designed to provide middle school and high school mathematics teachers with the opportunity to learn about and reflect upon the development of a particular mathematical topic throughout the grades.

Introduction

Activity	Description	Time	Materials
Introduction of participants	Have participants introduce each other. Instruct participants to tell of partners experience with Standards based curricula, level and number of years as well as school district.	10 min. 8:35-8:45
Structure of the A.R. Notebooks	Briefly explain the content of their notebooks. The sections: Investigations, CMP, and Core-Plus followed by examples of student work.	5 min. 8:45-8:50
Brainstorm Activity 1 What is algebra?	Have participants discuss their ideas in small groups and then share with the whole group. Record their ideas. Explain that this is a preliminary attempt to define a complex term. What does the word "algebra" bring to mind?	10 min. 8:50-9:00	• Chart paper
Activity 2 Beginning ideas of representing change over time.	Have participants examine examples of Investigations student’s representations elevator trips. Students were asked to construct graphs for themselves that represented imaginary trips on an elevator. Focus Questions: How do visualize the elevator trip? Is it clear where the trip started? The order of changes? Can you interpret the trip?	10 min 9:00-9:10	• HO examples of student work • Transp Student Work

Activity 1--Defining the meaning of algebra.

Have participants discuss their ideas in small groups and then share with the whole group. Record their ideas. Explain that this is a preliminary attempt to define a complex term.

• modeling
• pattern finding
• describing, and using patterns
• describing and using functions

There are ways of thinking that are considered the foundation on which to build algebraic thinking.

• Predicting
• generalizing
• validating
• extend thinking beyond specific values to values not yet found

Activity 2-Analyzing young childrens representations of change over time

Focus Questions:

How do these children visualize the elevator trip?
Is it clear where the trip started?

The order of changes?

Can you interpret the trip?

Thinking with Mathematical Models

Activity	Description	Time	Materials
Overview of CMP Units	Ask participant to share the ways in which the concept of variable is developed in CMP, particularly in the unit Variables and Patterns. [See overview on page 1a.]	5 min 9:10-9:15	Transp CMP Algebra Units
Lead in to CMP units.	Variables and Patterns develops students’ ability to explore a variety of situations where changes occur. Students develop three ways of representing a changing situation: narrative description as well as a data table and a graph, both of which show changes in the two variables. Ask experienced participants to share their experiences in teaching this unit.   Come back and add questions.	Still 9:15
Thinking with Math Models	In this unit, students explore the advantages of using algebraic models, in the form of graphs and equations, to describe situations.	Still 9:15	1a-1g, 1k* (vocabulary) 47-59
Activity 3 CMP Investigation 4 Problem 4.1 Follow-Up parts 2 and 3a Problem 4.2 p. 53 #5 Investigations Changes Over Time SS#13 SS#14   Core-Plus Investigation 1 1a	Launch Show the transparency of dropped beanbags. Ask for an interpretation. CMP Participants will be asked to match a story to the graphs of a bus and a car that leave from the same location. Have those exploring CMP write a story for each of the 6 graphs. Exchange stories and have others match story to graph. Discuss any for which there is not agreement or are a surprise. Investigations Investigations: Make a poster displaying their work. Ask them to explain how the Investigations activity relates to the CMP problem.	55 minutes 9:15-10:10	Launch Transp of bean bags dropped Thinking with Mathematical Models 6 index cards per group. Blank transparencies Transp of Problem 4.2 Changes Over Time Chart paper Transp SS#13 Transp SS#14 CPMP--Rates of Change Transp graphs Prob 1a
Summary/ Discussion	Questions: What differences do you notice among the graphs representing different situations? When the graph becomes a straight line, what does that mean? What is similar and what is different about graphs A and D (Investigations, SS#13)? How can the graphs be interpreted? What are the issues involved in interpreting the graphs? In what ways do students have to think about the graphs?	Same 10:10 ending   Break 10:10-10:25	Applications p. 53 # 5

  Moving Straight Ahead

Activity	Discussion	Time	Materials
Intro to Unit	In Moving Straight Ahead, students study linear functions and relationships. Learning to recognize a linear situation from its context, a table, a graph, or an equation is at the heart of this unit. The idea of dependent and independent variable and the concept of rate, that is, how one variable changes with respect to the other are key ideas developed in this unit.	10:25
CMP Investigation 2 Walking to Win Problem 2.4 Problem 2.5 Applications #7 & 9 (page 27)	Launch: The only information available is the rate at which the two brothers walk and that Henri gets a 45 meter head start. Have those working on this task make posters showing their equations, graphs and tables. Prompts: Encourage the use of multiple representations, that is, symbolically as well as using tables and graphs. They will be asked to argue for their conclusion, that is, the length or lengths for the race. They will also be asked to explain their strategy Ask how they choose their range of values.	60 min. 10:25-11:25	Chart Paper Blank transp Transp grid • Graphing calculators HO graphing calc documentation Transp of questions HO #7 & 9 (p. 27) Math Reflections
Core-Plus Course 4 Unit 2 Rates of Change Section 1 Investigation 1 Activities 1b (I & IV only), c, 2-4.   Section 2 Investigation 1 #1-3 Investigations Growing Tile Problem SS#4 (page 1 of 4) See questions on page 3.	Questions/Discussion: In what ways are the activities in the three programs similar? In what ways are they different? What do students have to know in order to understand velocity and acceleration as they relate to linear functions? Questions: How can you determine whether a situation is linear by examining a table of data or an equation? How does changing one of the quantities in a situation affect the table, the graph, or the equation? What are the important characteristics of linear functions? Of functions in general? Y=mx+b the rate of change--describe Pattern in graph-straight line Pattern in table context meaning of y-intercept and of m	11:25-11:55	Grid Chart Paper Transp Grid paper - one per group Square Tile
CMP Related Task (optional) Examine preservice teachers responses	(Optional) Examine student work (preservice teachers). Explain their thinking and describe commonly held misconceptions about linear pattern of change. Talk about the misconceptions held by preservice teachers after completion of Thinking with Mathematical Models and Moving Straight Ahead.	11:55-12:30	HO Preservice Teacher’s responses to Test Problem (Optional)
			Moving Straight Ahead Cover of book P. 14F, P. 1a-1h, 1j, 2-4, 15-34
Alternative Activity-Inv. Investigations Growing tiling pattern SS#4 (page 1 of 4)	Have experienced CMP teachers do the Growing Tile problem (linear growth) from Patterns of Change (Grade 5). Have them present to the class. Prompts: In what way is this problem similar to the Walking problem? How is it different? How does it help develop algebraic reasoning?		Chart Paper

Walking to Win

Have participants consider the following questions:

• Did you find one representation more helpful than others? If so, why?
• What would the graph look like if someone walked 3 meters per second? 5 meters per second? 1/2 meter per second? How would each of these walking rates affect the table? the equation?
• How are the representations alike? How are they different?
• What patterns do you see in the graphs? in the tables? in the equations?
• How does the rate at which each person walks affect the graphs? the tables? the equations?
• How do you find the distance traveled in a given number of hours that exceeded the time recorded using the table? the graph? the equation?
• What is the reasoning involved when solving the problem by writing and solving equations?
• The equations y = __ + x and y = 2.5x model the distance Emile and Henri walk during the race. What are other ways to represent the same relationships.

Growing, Growing, Growing

Activity	Description	Time	Materials
CMP Investigation 1 Exponential Growth Problem 1.2 & Follow-Up, parts 1-3 Extensions #16 on pages 14-15	Examine only the plan proposed in Problem 1.2. Launch: If a ruba is worth one of our pennies, do you think the peasant’s plan is a good one for her? Questions: What are the patterns in the table? In the graph? Where will the graph intersect the x and y axis? What do these points (or lack of existence of these point) mean? What is the relationship between consecutive entries in your table?	35 min. 1:00-1:35	p. 5-9   Extensions #16 Math Reflections on page 30
CMP (option) Investigation 4 Exponential Decay Problem 4.2 & Follow-Up CPMP Find average velocity, approx. Velocity at differnt points.	Optional   Discuss rate of decay and the decay factor. Compare patterns in tables, graphs and equations in decay problems to those from the rubas problems.	Probably omit 40 min.	p. 45, 48-51
CMP Problem 3.1 & Follow-Up	This activity must precede the exploration of exponential equations in order that changing parameters makes sense.	1:35-2:00 Break to 2:10
CMP Exploring Exponential Equations Problem 4.3 & Follow-Up	Discuss the effects of changing the value of b. Then focus on the role of a. Need to look at If experienced CMP teachers have finished their assignment have them join the groups exploring these problems and ask probing questions?	40 min. 2:10- 2:50	P. 50-51 Graphing Calculators
Investigations Growing Tile Problem SS#4 (page 4 of 4)   Compare this problem with the Extension problem #16 on page 14-15 in	Have this done while others work on CMP problem 4.2.		Growing, Growing, Growing Includes Technology Section 1a-1L, 2-4, 5-16, 45-60
Core-Plus Section 2 Inv.1 Rates of Change for Familiar Functions Exponential Functions Activity 9-10	Have this done while others work on CMP problem 4.3 & Follow-Up.		Math Reflections on page 60
Summary	If extra time, go back to exponential decay problems.   Discuss patterns in tables, graphs and symbolic form of exponential relationship. What does a represent? b represent? What are similarities among the graphs? What differences did you notice? How can you predict the shape of the graph for the equation y = b^x when given a specific value for x? What happens as a increases when b is greater than 1? Less than 1?	2:50-3:10

Frogs, Fleas, and Painted Cubes

Activity	Discussion	Time	Materials
Launch	A technology section is needed in the handout.		Frogs, Fleas, and Painted Cubes Cover of book p. 1A-1k & 1p (TG) p. 2-18
CMP Introduction to Quadratic Relationships Problem 1.1 & Follow-Up	Investigation 1 Discuss the shape of the graph and what it tells about the area enclosed by a fixed perimeter. Begin to observe characteristics of a graph that relate to maximum area. Note relationship between area and dimensions of the rectangle.	30 min 8:30-9:00	p. 5-
CMP Reading a Graph Problem 1.2 & Follow-Up	Discuss the information that can be read from the graph of length and area data for a fixed perimeter. Give a point and ask for a question that could be asked and answered with that information. Terminology introduced--parabola.	30 min 9:00-9:30	P. 7-9
CMP Problem 1.3	Focus on the patterns in the tables, graphs and equations and the information that each representation reveals. Discuss which form is the most useful, for example, for predicting a maximum.	30 min 9:30-10:00 Break to 10:15	P. 10-11
Investigations Growing Tile Patterns , SS#4 (page 2 of 4) and (page 3 of 4)	Summarize/Discussion of CMP Problems 1.1 to 1.3	10:15-10:35	4b (TG) 2-11 ACE Questions Connections #9 p. 16 Extensions #11, 12 p. 17 Math Reflections p. 18
CPMP Section 2 Inv 1 Rates of Change for Familiar Functions Quadratic Functions Activity 4-8 Checkpoint p. 27
CMP Investigation 2 Quadratic Expressions Problem 2.1 & Follow-Up (parts 1 & 2)	CMP--Investigation 2 Rate of change in area as side lengths change. Equivalence of quadratic expressions.	25 min 10:35-11:00	19-40 (will not cover 28-30) Graph paper
CMP Changing One Dimension Problem 2.2 part B	Bring in the terminology on page 23.	25 min 11:00-11:25
CMP Changing Both Dimensions Problem 2.3 parts A & B	Suggest #7 on page 27 as a challenge for anyone who finishes early.	50 mn 11:25-12:15 Lunch	ACE Connections #3, 32, 33, 34, Extensions 37, 39?, 41 & 43 part a, 44
	Discussion/Summary -- Problem 34 is good to discuss because its context relates to rate of growth. At some point discuss the pattern of change in the linear factors and then the pattern of change in their product. Why is this?
CPMP Section 2 Inv 2 The Linear Connection Activities 1-4
Ritzler Video	Show as an example of algebraic reasoning and of students analyzing the effect of changing constant values and coefficients on the graphical representation.	30 min 1:00-1:30
	Investigation 5 Painted Cubes
CMP Painted Cubes Problem 5.1 & 5.2	Make a table with data from the painted cube problem..	60 min 1:30-2:30	p. 71-84
Project	Compare and generalize the patterns.		Chart paper
			p. 70R, 71-74 Math Reflections p. 84

Comparison of Functions with focus on patterns in change

Pattern in table
Pattern in graph
Pattern in the symbolic form
Pattern in the rate of change
Essential characteristics

• Min or Max
• Symmetry
• y-intercept
• x-intercept(s)
• additive and multiplicative components?

Equivalent expressions
Effect of changes in symbolic form (change of parameters) on rate of change and on graphical representation.
Context of the situation

Monday, June 21, 1999
8:30-10:30 a.m.	Welcome Discuss mathematical emphasis of Variables & Patterns Variables & Patterns Investigation 1: "Variables and Coordinate Graphs" Mathematical and Problem-Solving Goals: To collect data from an experiment and then make a table and a graph to organize and represent the data To search for explanations for patterns and variations in data To understand that a variable is a quantity that changes and to recognize variables in the real world To understand that in order to make a graph that shows the relationship between two variables, you need to identify the two variables, choose an axis for each, and select an appropriate scale for each axis To interpret information given in a graph
10:45-12:30 p.m.	Variables & Patterns Investigation 2: "Graphing Change" Mathematical and Problem-Solving Goals: To make sense of data given in the form of a table or a graph To read a narrative of a situation that changes over time and make a table and graph that represent these changes To read data given in a table and make a graph from the table To read data given in a graph and make a table from the graph To compare tables, graphs, and narratives and understand the advantages and disadvantages of each form of representation
1:30-3:00 p.m.	Variables & Patterns Investigation 3: "Analyzing Graphs and Tables" Mathematical and Problem-Solving Goals: To change the form of representation of data from tables to graphs and vice versa To search for patterns of change To describe situations that change in predictable ways with rules in words for predicting the change To compare forms of representation of data
Tuesday, June 22, 1999
8:30-10:30 a.m.	Variables & Patterns Investigation 4: "Patterns and Rules" Mathematical and Problem-Solving Goals: To understand the relationship between rate, time, and distance To represent information regarding rates in tables and graphs and to use tables and graphs to compare rates To search for patterns of predictable change To learn to express in words and symbols situations that change in predictable ways Variables & Patterns Investigation 5: "Using a Graphing Calculator" Mathematical and Problem-Solving Goals: To use a rule to generate a table or graph on the graphing calculator To use a graphing calculator to compare the table and graphs of various rules; in particular, to decide whether a given rule defines a straight-line (linear) function by examining graphs
10:45-12:30 p.m.	Discuss mathematical emphasis of Stretching & Shrinking Stretching & Shrinking Investigation 1: "Enlarging Figures" Mathematical and Problem-Solving Goals: To make enlargements of simple figures with a rubber-band stretcher To describe in an intuitive way what the word similar means To consider relationships between lengths and between areas in simple, similar figures
1:30-3:00 p.m.	Stretching & Shrinking Investigation 2: "Similar Figures" Mathematical and Problem-Solving Goals: To review locating points in a coordinate system To graph figures using algebraic rules To predict how figures on a coordinate system are affected by a given rule To learn that corresponding angles of similar figures are equal and that corresponding sides grow by the same factor To compare lengths and angles in similar and nonsimilar figures informally To experiment with examples and counterexamples of similar shapes
Wednesday, June 23, 1999
8:30-10:30 a.m.	Assessment Discussion/Practice Stretching & Shrinking Investigation 3: "Patterns of Similar Figures" Mathematical and Problem-Solving Goals: To recognize similar figures and to be able to tell why they are similar To understand that any two similar figures are related by a scale factor, which is the ratio of their corresponding size To build a larger, similar shape from copies of a basic shape (a rep-tile) To find rep-tiles by dividing a large shape into smaller, similar shapes To understand that the sides and perimeters of similar figures grow by a scale factor and that the areas grow by the square of the scale factor To find a missing measurement in a pair of similar figures To recognize that triangles with equal corresponding angles are similar
10:45-12:30 p.m.	Stretching & Shrinking Investigation 4: "Using Similarity" Mathematical and Problem-Solving Goals: To use the definition of similarity to recognize when figures are similar To determine the scale factor between two similar figures To use the scale factor between similar figures to find the lengths of corresponding sides To find a missing measurement in a pair of similar figures To use the relationship between scale factor and area to find the area of a figure that is similar to a figure of a known area To solve problems that involve scaling up and down
1:30-3:00 p.m.	Stretching & Shrinking Investigation 5: "Similar Triangles" Mathematical and Problem-Solving Goals: To recognize similar figures in the real world To find a missing measurement in a pair of similar figures To apply what has been learned about similar figures to solve real-world problems To collect data, analyze it, and draw reasoned conclusions from it
Thursday, June 24, 1999
8:30-10:30 a.m.	Discuss mathematical emphasis of Comparing & Scaling Comparing & Scaling Investigation 1: "Making Comparisons" Mathematical and Problem-Solving Goals: To explore several ways to make comparisons To begin to understand how to determine when comparisons can be made using multiplication or division versus addition or subtraction To begin to develop ways to use ratios, fractions, rates, and unit rates to answer questions involving proportional reasoning
10:45-12:30 p.m.	Comparing & Scaling Investigation 2: "Comparing by Finding Percents" Mathematical and Problem-Solving Goals: To further develop the ability to make sensible comparisons of data using ratios, fractions, and decimal rates, with a focus on percents To develop the ability to make judgments about rounding data to estimate ratio comparisons To observe what is common about situations that call for a certain type of ratio comparison
1:30-3:00 p.m.	Comparing & Scaling Investigation 3: "Comparing by Using Ratios" Mathematical and Problem-Solving Goals: To recognize situations in which ratios are a useful form of comparison To form, label, and interpret ratios from numbers given or implied in a situation To explore several informal strategies for solving scaling problems involving ratios (which is equivalent to solving proportions)
Friday, June 25, 1999
8:30-10:30 a.m.	Comparing & Scaling Investigation 4: "Comparing by Finding Rates" Mathematical and Problem-Solving Goals: To find unit rates To represent data in tables and graphs To look for patterns in tables in order to make predictions beyond the tables To connect unit rates with the rule describing a situation To begin to recognize that constant growth in a table will give a straight-line graph To find the missing value in a proportion
10:45-12:30 p.m.	Comparing & Scaling Investigation 5: "Estimating Populations and Population Densities" Mathematical and Problem-Solving Goals: To use geometric scaling to estimate population counts To apply proportional reasoning to situations in which capture-tag-recapture methods are appropriate for estimating population counts To use ratios and scaling up or down (finding equivalent ratios) to find the missing value in a proportion To use rates to describe population and traffic density (space per person or car)
1:30-3:00 p.m.	Comparing & Scaling Investigation 6: "Choosing Strategies" Mathematical and Problem-Solving Goals: To select and apply appropriate strategies to make comparisons To review when ratio and difference strategies are useful in solving problems To use proportional reasoning to fairly apportion available space so that the group is representative of the larger community

Note: Pedagogical/implementation issues were discussed throughout the week in the context of the particular unit being investigated. Specific time was allotted to discuss the Launch/Explore/Summary method and different ways of managing homework and assessment.