Learning Math: Patterns, Functions, and Algebra
Algebraic Structure Part A: Comparing Operations (10 minutes)
Session 9, Part A
Students begin to move from arithmetic to algebra (in the structural sense) when they start thinking about properties of operations rather than properties of numbers. This happens quite early. For example, “missing addend” problems (such as 4 + ? = 9) can be solved one at a time, each as a special case, by any number of techniques (counting up, counting back, even subtracting). But when your students start saying things like “subtraction is the opposite of addition” or “subtraction undoes addition,” they are starting to realize a structural relationship between two operations rather than a collection of relationships between pairs of numbers. Note 2
We took an initial look at algebra from a structural approach when we examined the concept of doing and undoing in Session 3. At that point, we were looking at relationships between operations with a focus on undoing, or inverting, operations. Another hallmark of a move to algebra as structure is a focus on comparing algorithms. For example, consider the following two algorithms:
|•||Take a number|
|•||Double your answer|
|•||Take a number|
|•||Add 2 to your answer|
Explain why algorithms A and B will always give the same output if you give them the same input.
In what sense are algorithms A and B “the same?” In what sense are they different? Do you think algebra students would classify them as the same or different? Would you classify them as equivalent?
In this session, we’ll be exploring a structural approach to algebra. When we begin to think about properties of operations rather than of numbers, we’re moving from arithmetic to algebra, in a structural sense. You may recall that in Session 3 we looked at algorithms through the lens of “doing and undoing.” A focus on “what undoes what” often marks the beginning of reasoning about operations. In this session, we’ll continue to examine algorithms, as well as other systems that represent a structural approach to algebra.
Spend some time thinking through the following exercise.
Groups: You may want to put this on an overhead.
One hallmark of a move to algebra as structure is a focus on undoing, or inverting, processes; another is comparing them.
A simple case of this is the following:
Is adding 3 to 2 the same as adding 2 to 3?
Is subtracting 3 from 2 the same as subtracting 2 from 3?
Thinking about properties rather than individual numbers indicates a move to a structural point of view. Keep this in mind as you work on Problems A1 and A2.
Follow both algorithms using N as the input number:
Algorithm A: N -> N + 1 -> 2(N + 1) = 2N + 2
Algorithm B: N -> 2N -> 2N + 2
Each algorithm produces the same output, as long as you’re willing to trust that the distributive property is always true! If you don’t trust or remember the distributive property, remember that 2(N + 1) is the same as (N + 1) + (N + 1).
Most algebra students would classify them as identical, since they determine identical expressions. But the steps are different — it’s like giving two sets of directions to the same place. Certain problems have many different algorithms of solution, some much more difficult than others.
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.