Learning Math: Patterns, Functions, and Algebra
Classroom Case Studies, Grades 3-5 Part A: Classroom Video (30 minutes)
Session 10: 3-5, Part A
To begin the exploration of what algebraic thinking looks like in a classroom at your grade level, watch a video segment of a teacher who has taken the “Patterns, Functions, and Algebra” course and has adapted the mathematics to her own teaching situation. When viewing the video, keep the following three questions in mind:
|a.||What fundamental algebraic ideas (content) is the teacher trying to teach? Think back to the big ideas of the previous sessions: patterns, functions, linearity, proportional reasoning, nonlinear functions, and algebraic structure.|
|b.||What mathematical thinking tools (process) does the teacher expect students to demonstrate? Think back to the processes you identified in the first session: problem-solving skills, representation skills, and reasoning skills.|
|c.||How do students demonstrate their knowledge of the intended content? What does the teacher do to elicit student thinking?|
In this video segment, Liza Jones introduces her students to the process of undoing. She begins by asking her students to figure out the “recipe” or formula she is using to turn an In into an Out. She then asks them to undo the “recipe” by finding the In when given the Out.
You can find this segment on the session video, approximately 33 minutes and 48 seconds after the Annenberg Media logo.
Reflect on questions (a), (b), and (c) above.
How does Ms. Jones incorporate the concept of doing and undoing into the model of function machines?
How do the students think about undoing the recipe they constructed?
How could Ms. Jones have used recursive thinking in this lesson?
Before examining specific problems at your grade level with an eye toward algebraic thinking, we’ll watch another teacher — one who has also taken the course — teaching in her classroom. The purpose is not to be critical of the teacher’s methods or teaching style. Instead, look closely at how the teacher brings out algebraic ideas in teaching the topic at hand, as well as how the teacher extends the lesson and asks questions that elicit algebraic thinking.
Review the meaning of algebraic ideas (that is, the content of algebra) and mathematical thinking tools (the processes used in analyzing problems). Keep in mind questions (a), (b), and (c) as you watch the video.
Groups: Work in small groups on Problems A1-A4. Share answers to Problem A4 especially, because recursive thinking is not necessarily something that third-grade teachers often consider in their teaching. The vocabulary (recursive) is not the important part here. The method of thinking recursively, however, is important for teachers to consider as students make sense of patterns.
Reflection on the three questions should include the ideas described below.
|a.||The fundamental algebraic idea (content) in this video is the notion of function and inverse function as doing and undoing.|
|b.||The teacher expects students to write a function rule to represent a relationship shown in a table of values and then find a rule for the inverse function.|
|c.||Students demonstrate their knowledge of the intended content by writing function rules and by describing inverse functions as reversing a recipe by reversing steps and using “opposite” operations.|
Ms. Jones asks students to think of a function as a recipe for “doing,” which then leads to inverse functions as “undoing” the same recipe.
The students describe undoing as reversing the steps and using opposite operations.
Ms. Jones could have used recursive thinking with the function machine that doubles and adds one. This function could be described recursively by saying, “The Out number for In = 1 is 3. The Out number for any other In is the previous Out + 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.