



Renewing Mathematics Teaching Through Curriculum (RMTC)
Summer Workshop Summary
RMTC is assisting middle and high schools in West Michigan who are implementing the CorePlus 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 oneweek in the summer. This summer we are also facilitating a twoday 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 (K5 curriculum), CMP, and CPMP.
How many teachers do you involve in your workshops?
There are a total of 67 teachers attending the RMTC weekone workshops and 20 attending the twoday strand workshop. In addition, 22 teachers are attending CPMP or CMP oneweek 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.
During each of the summer workshops, teacherleaders from the RMTC collaborative who are experienced with teaching the curriculum facilitate 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 model the teaching approaches they have used with their own students. For evaluation of the mathematical content learned by participants, some workshops participants completed pretests and posttests consisting of items from student assessments for the units studied in the workshop.
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.
Renewing Mathematics Teaching Through Curriculum
Summer Workshop Agendas
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, MultipleVariable 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
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 xaxis, translations, or stretches (or combinations of these transformations) of basic functions
 To apply all of the transformations above as they relate to realworld 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 planelinear, planeprojectile, 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 onetoone 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

2Day Algebra Strand Workshop
This 2day 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 K12 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 CorePlus 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.
8:358:45 

Structure of the A.R. Notebooks 
Briefly explain the content of their notebooks. The sections: Investigations, CMP, and CorePlus followed by examples of student work. 
5 min.
8:458:50 

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.
8:509:00 
• 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
9:009:10

• HO examples of student work
• Transp Student Work 
Activity 1Defining 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 2Analyzing 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
9:109:15 
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 
1a1g, 1k* (vocabulary)
4759 
Activity 3
CMP
Investigation 4
Problem 4.1 FollowUp parts 2 and 3a
Problem 4.2
p. 53 #5
Investigations
Changes Over Time
SS#13
SS#14
CorePlus
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
9:1510:10

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
CPMPRates 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
10:1010:25 
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.
10:2511:25 
Chart Paper
Blank transp
Transp grid
• Graphing calculators
HO graphing calc documentation
Transp of questions
HO #7 & 9 (p. 27)
Math Reflections 
CorePlus
Course 4 Unit 2 Rates of Change
Section 1 Investigation 1
Activities 1b (I & IV only), c, 24.
Section 2 Investigation 1
#13
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 changedescribe
Pattern in graphstraight line
Pattern in table
context
meaning of yintercept and of m 
11:2511:55 
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.

11:5512:30 
HO Preservice Teacher’s responses to Test Problem (Optional) 



Moving Straight Ahead
Cover of book
P. 14F,
P. 1a1h, 1j, 24, 1534

Alternative ActivityInv.
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 & FollowUp, parts 13
Extensions
#16 on pages 1415 
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.
1:001:35 
p. 59
Extensions #16
Math Reflections on page 30 
CMP (option)
Investigation 4
Exponential Decay
Problem 4.2 & FollowUp
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, 4851

CMP
Problem 3.1 & FollowUp

This activity must precede the exploration of exponential equations in order that changing parameters makes sense. 
1:352:00
Break to 2:10 

CMP
Exploring Exponential Equations
Problem 4.3 & FollowUp 
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. 5051
Graphing Calculators 
Investigations
Growing Tile Problem
SS#4 (page 4 of 4)
Compare this problem with the Extension problem #16 on page 1415 in 
Have this done while others work on CMP problem 4.2. 

Growing, Growing, Growing
Includes Technology Section
1a1L, 24, 516, 4560

CorePlus
Section 2 Inv.1
Rates of Change for Familiar Functions
Exponential Functions
Activity 910 
Have this done while others work on CMP problem 4.3 & FollowUp. 

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?

2:503:10 

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. 1A1k & 1p (TG)
p. 218 
CMP
Introduction to Quadratic Relationships
Problem 1.1 & FollowUp 
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
8:309:00 
p. 5 
CMP
Reading a Graph
Problem 1.2 & FollowUp 
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 introducedparabola. 
30 min
9:009:30 
P. 79 
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
9:3010:00
Break to 10:15 
P. 1011 
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 
10:1510:35 
4b (TG)
211
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 48
Checkpoint p. 27




CMP
Investigation 2
Quadratic Expressions
Problem 2.1 & FollowUp (parts 1 & 2) 
CMPInvestigation 2
Rate of change in area as side lengths change.
Equivalence of quadratic expressions. 
25 min
10:3511:00 
1940
(will not cover 2830)
Graph paper 
CMP
Changing One Dimension
Problem 2.2 part B 
Bring in the terminology on page 23.

25 min
11:0011:25 

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
11:2512:15
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 14 



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
1:001:30 


Investigation 5
Painted Cubes



CMP
Painted Cubes
Problem 5.1 & 5.2 
Make a table with data from the painted cube problem.. 
60 min
1:302:30 
p. 7184 
Project 
Compare and generalize the patterns. 

Chart paper 



p. 70R, 7174
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
 yintercept
 xintercept(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
RMTC Grade 7 Workshop
June 2125, 1999
Monday, June 21, 1999 
8:3010:30 a.m. 
Welcome
Discuss mathematical emphasis of Variables & Patterns
Variables & Patterns Investigation 1:
"Variables and Coordinate Graphs"
Mathematical and ProblemSolving 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

10:4512:30 p.m. 
Variables & Patterns Investigation 2: "Graphing Change"
Mathematical and ProblemSolving 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

1:303:00 p.m. 
Variables & Patterns Investigation 3:
"Analyzing Graphs and Tables"
Mathematical and ProblemSolving 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 
8:3010:30 a.m. 
Variables & Patterns Investigation 4: "Patterns and Rules"
Mathematical and ProblemSolving 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 ProblemSolving 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 straightline (linear) function by examining graphs

10:4512:30 p.m. 
Discuss mathematical emphasis of Stretching & Shrinking
Stretching & Shrinking Investigation 1: "Enlarging Figures"
Mathematical and ProblemSolving Goals:
 To make enlargements of simple figures with a rubberband stretcher
 To describe in an intuitive way what the word similar means
 To consider relationships between lengths and between areas in simple, similar figures

1:303:00 p.m. 
Stretching & Shrinking Investigation 2: "Similar Figures"
Mathematical and ProblemSolving 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 
8:3010:30 a.m. 
Assessment Discussion/Practice
Stretching & Shrinking Investigation 3:
"Patterns of Similar Figures"
Mathematical and ProblemSolving 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 reptile)
 To find reptiles 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

10:4512:30 p.m. 
Stretching & Shrinking Investigation 4: "Using Similarity"
Mathematical and ProblemSolving 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

1:303:00 p.m. 
Stretching & Shrinking Investigation 5: "Similar Triangles"
Mathematical and ProblemSolving 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 realworld problems
 To collect data, analyze it, and draw reasoned conclusions from it

Thursday, June 24, 1999 
8:3010:30 a.m. 
Discuss mathematical emphasis of Comparing & Scaling
Comparing & Scaling Investigation 1: "Making Comparisons"
Mathematical and ProblemSolving 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

10:4512:30 p.m. 
Comparing & Scaling Investigation 2:
"Comparing by Finding Percents"
Mathematical and ProblemSolving 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

1:303:00 p.m. 
Comparing & Scaling Investigation 3:
"Comparing by Using Ratios"
Mathematical and ProblemSolving 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 
8:3010:30 a.m. 
Comparing & Scaling Investigation 4:
"Comparing by Finding Rates"
Mathematical and ProblemSolving 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 straightline graph
 To find the missing value in a proportion

10:4512:30 p.m. 
Comparing & Scaling Investigation 5:
"Estimating Populations and Population Densities"
Mathematical and ProblemSolving Goals:
 To use geometric scaling to estimate population counts
 To apply proportional reasoning to situations in which capturetagrecapture 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)

1:303:00 p.m. 
Comparing & Scaling Investigation 6: "Choosing Strategies"
Mathematical and ProblemSolving 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.

