Learning Math: Patterns, Functions, and Algebra
Algebraic Structure Part B: Guess My Rule (20 minutes)
Session 9, Part B
One way to compare algorithms is to play the “Guess My Rule” game. Here’s how it goes:
|•||Someone makes up a mystery algorithm (like “take a number, double it, and add 2 to the answer”). He or she writes it down or draws a picture of it (with machines, for example) and keeps it secret.|
|•||The algorithm writer then takes “requests.” Other people suggest inputs and request outputs. Everyone records the results.|
|•||Other people try to guess the algorithm with as few requests as possible. If someone says, “I’ve got it,” that person writes down his or her guess of the algorithm, either in words or pictures.|
|•||If the guess is the same as the mystery algorithm, the round is over. If (as often happens) the guessed algorithm looks different from the mystery algorithm, the guessers have to either prove that they will always produce the same result, or they must find an input where the algorithms produce different outputs. For example, a match for “take a number, double it, and add 2” might be “take a number, add 1, and double the answer.”|
Describe some strategies for playing “Guess My Rule.”
When you play “Guess My Rule,” you often come up with an algorithm that acts the same as the mystery algorithm but contains different steps. In what sense are these algorithms the same? In what sense are they different? Do you think they represent the same function?
Groups: You may organize a participatory game of “Guess My Rule.”
One strategy is to pick consecutive inputs, trying to find the pattern to the output. This will help decide if the rule is linear, quadratic, exponential, inverse, or cyclic. Guessing numbers that are in proportion may help as well; if you’ve tried 10, see what happens with 20.
These algorithms produce the same final result, but may have completely different intermediate steps. One important concern is that the final result is always identical, not just for a few values of the input. Graphically, the functions should be identical, even though they may be created from different steps.
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.