Learning Math: Patterns, Functions, and Algebra
Patterns in Context Part E: Summing Up (20 minutes)
Session 2, Part E
At the beginning of this session, we learned that finding, describing, explaining, and predicting with patterns are some of the most important skills in mathematics. We had a chance to practice these skills throughout the session as we explored several patterns.
We have used many different kinds of descriptions for patterns. Two kinds of descriptions, in particular, are very important:
A recursive description tells you how to get from one term to the next. In the table in Part A, for instance, some people might say, “Each output is 4 more than the previous one.” That’s a recursive description. In the pattern of Part D, if you said, “To get from staircase n to staircase n + 1, add n + 1 blocks,” you were using a recursive description.
Recursive descriptions are most useful when the instruction to get from one step to the next works no matter where you start in the pattern. But you need to know the output from one or more of the previous steps to find the next one.
A closed-form description tells you how to get from any input to any output, without needing to know any previous outputs. In the table in Part A, the instruction “take the input, multiply it by 4, and add 2” or “the output for n is 4n + 2” are closed-form descriptions. In the pattern of Part D, if you said, “Staircase n has n(n + 1) / 2 blocks,” you were using a closed-form description.
Problem E1: Write and Reflect
Give an example from your work so far that has been a recursive description of a pattern. Did you ever describe a pattern with a closed-form description? Can you describe the use of the variable in your description?
Problem E2: Write and Reflect
In your opinion, why is the study of patterns important, and how does it connect to algebraic thinking?
Review the “Big Four” of patterns: finding, describing, explaining, and predicting with patterns. Do you think you have practiced all four of these skills? When? What was the most enjoyable? Most difficult? Do you practice these processes with your students?
Read about recursive and closed-form descriptions.
Groups: Discuss the difference between closed form and recursive form. Take five minutes to jot down answers to Problems E1 and E2, then spend some time sharing a few of the answers.
Recursive descriptions include things like “Take the previous number of blocks and add n” and “Take the previous number of toothpicks and add 4.” Closed-form descriptions include things like “The number of toothpicks is given by the formula T = 4n + 2” and “The number of blocks in the nth staircase is given by the formula n * (n + 1) / 2.”
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.