Private: Learning Math: Patterns, Functions, and Algebra
Patterns in Context Part C: Different Uses of Variables (25 minutes)
Session 2, Part C
In the toothpick problem, we used a variable to describe the relationship between the number of toothpicks and the number of triangles in the pattern. The use of variables is the most familiar part of working algebraically. We have also seen in the Eric the Sheep problem that algebraic thinking does not always require using variables. The concept of variable, however, is an important one in algebra. In fact, there are many meanings for variables and how they are used in mathematics.
Mathematician Zal Usiskin has outlined four conceptions of algebra based on different uses of variable:
Conception 1: Algebra as generalized arithmetic. Here, variables are indeterminates— they do not have specific values, but allow you to analyze operations like multiplication and addition.
Example: The sum of 2 even numbers is even: 2a + 2b = 2(a + b).
Example: Any number times zero is zero: 0 x n = 0
Conception 2: Algebra as a study of procedures for solving certain kinds of problems. Here, the variables are unknowns, and you want to solve for them.
Example: When 4 is added to 9 times a certain number, the sum is 40. Find the number. We represent this as 4 + 9n = 40, and n is the unknownsolution.
Example: You get paid $10 per hour and earned $30 in tips. In total, you made $380 last week. How many hours did you work? Here, the unknownis the number of hours worked.
Conception 3: Algebra as the study of relationships among quantities. Here the variables really vary, and you look at how changes in one variable affect the others.
Example: In a rectangle, area is length times width: A = L x W.
Example: What happens to the value of 1/x as x gets larger and larger?
Example: In Part B, what happens to the number of toothpicks as the number of triangles increases?
Conception 4: Algebra as the study of structures. This conception of algebra explores the nature of numbers and operations, and we will explore this conception in greater detail in Session 9.
It is clear from these descriptions that variables can be used in different ways and for different purposes. The relative importance given to these multiple uses of variables affects the purposes for which algebra is used as well.
Come up with at least two other examples of “generalized arithmetic.” Write your examples in words, symbols, or both. Note 9
Think of as many ways as you can to solve the equation 4 + 9n = 40; that is, to find a number so that 4 added to 9 times that number is 40.
Describe at least two other situations where variables represent relationships between quantities.
We’ve been using variables to describe patterns concisely, and some would claim that this is a move toward algebra. But what is algebra?
Groups: Discuss this question in small groups, then share answers with the whole group.
Reflect on this statement from mathematician Zal Usiskin that describes what algebra is: “Algebra is not easily defined.” In the first session, we clearly used algebraic thinking to describe the pattern in the Eric the Sheep problem, but we didn’t use an equation with variables. When we focus on the concept of variable, we can see that variables have many facets, as illustrated in the following examples:
- A = LW
- 40 = 5x
- sin x = cos x * tan x
- 1 = n * (1/n)
- y = kx
Each of these has a different feel. Can you explain the differences?
If you get stuck, here’s a way to sort through these examples:
- Example 1 is a formula.
- Example 2 is an equation or open sentence to solve.
- Example 3 is an identity that is true for any value of x, other than when cos x = 0.
- Example 4 is a property that is true for all n not equal to 0.
- Example 5 is an equation representing a direct variation function, where it is implied that k is a constant and y and x are variables.
Each of these has a purpose in the study of algebra, and to quote again from Usiskin:
“Purposes for algebra are determined by, or are related to, different conceptions of algebra, which correlate with the different relative importance given to various uses of variables.”
Also, consider this quote by Bob Davis: “Algebra is the way we talk about what numbers do when we don’t know what the numbers are. ”
Now go on to Usiskin’s four conceptions of algebra. Read through the conceptions as described in the course materials.
Groups: Discuss the first three conceptions (saving the fourth for the last session) and the examples contained in the descriptions. Group members can help each other understand the different representations in the examples — for instance, the representation of “even numbers” as “2a” and “2b.”
Groups: Work in small groups on Problems C1-C3. Spend some time sharing responses. Some people may need help in using symbols to describe their work to explain what the symbols are showing.
Generalized arithmetic allows you to state things like the commutative property of addition: a + b = b + a. Or the property that when you add zero to a number, you get the same number: 0 + n = n. Or the property that the product of two square numbers is a square number: x2 * y2 = (x * y)2.
The answer is n = 4. One method is the patented “guess-and-check.” Others include systematic testing of values of n starting with n = 1, or “undoing” the operations on the left side – if we have to multiply n by 9, then add 4, a way to find n would be to subtract 4, then divide by 9.
Some situations include the relationship between the length of a side of a square and the square’s area (A = s2), the relationship between Fahrenheit and Celsius temperature (F = 1.8C + 32), and the relationship between distance, rate, and time (D = r * t). In all of these, the variables vary and represent relationships between two or more quantities.
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.