Linear Function Machine

What is Linear Function Machine?

This activity allows the user to explore simple linear functions.

The function is determined by looking for patterns in the outputs. The functions available for exploration in the function machine activity have one of the following forms: 

        y = __ * x + __
where the underlines can be any integers between -10 and 10.

The notion of a function is important in higher mathematics such as calculus and in areas which use mathematics such as physics. Any rule that assigns exactly one output to an input or list of inputs can be called a function.

Here are a few examples:

  • Input the number of seconds after the starting gun in a race to get an output of the number of meters the runner has covered.

    Race Chart
    Number of Seconds (input) 1 4 7 8
    Meters covered (output) 5 20 35 40

  • y = x - 6, where x is the place holder (also called a variable) for the input and y is the place holder for the output.

    Function y = x - 6
    X (input) -3 0 7 8
    Y (output) -9 -6 1 2

The rule about only one output each time is crucial and must not be violated. For example, the table below cannot represent a function because 3 has two different outputs:

Not a Function
input 3 2 0 3
output 4 -1 2 -3

The functions produced by this machine are special because they all graph as straight lines. In fact, the slope of the line and the y-intercept of the line are easy to read from the formula: 

          y = (slope) * x + (y-intercept)

The slope tells us the rise over the run, and the intercept tells us where the line crosses the vertical axis when plotted:

Here the red line has intercept = 7 and slope = -1/2, and the green line has intercept = -1 and slope = 4/2.

How Do I Use This Activity?

This activity allows the user to explore simple linear functions.

Controls and Output

As an example, we use function Y = X + 2

  • To find out what the machine will do with a number:

    • Type the number into the X= [____] space (in our example, number 5 is typed in):

    • Press the Enter button:
    • Observe the resulting number in the Y= [____] window (in our example, number 7 is the result):
    • You can input as many numbers as you need to find the rule
  • To check if you guessed the rule right, type numbers in the Y = [___]*X + [___] spaces, and then press the Check formula button. You will see one of the two messages under the picture of the machine: either Correct answer! or Try again!
  • To set a new function machine (that is, a new rule), click once on the Set new function button:
  • This activity automatically keeps track of how successful you are at identifying the function. To view your score press the Show Score button
  • To pause scoring press the Active button and it will turn into a Paused button to indicate that scoring has been paused. To resume scoring press this button again so it reads Active.
  • Every time you press the Check formula button, the score panel is updated. This panel tells you the percentage of your guesses that were correct for the current function, and all the functions.
  • If you want to reset the statistics, you may do so by pressing the Reset button inside of the scoreboard.

Because this activity randomly generates questions, a feature called Seed Random has been implemented that allows multiple users using different computers to work on the same problem. See Seed Random Help for instructions on using this feature.

Description

This activity allows the user to explore simple linear functions; the function is determined by looking for patterns in the outputs. The function is always a simple line of the following form: \{y = m * x + b\} where m, the slope, and b, the y-intercept, can be any integer between -10 and 10. This activity would work well in same ability groups of two to three for about forty-five minutes if you use the and ten to fifteen minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

  • practice students' pattern finding skills
  • introduce basic notions from algebra
  • motivate the idea of a function of one variable
  • practice students' arithmetic skills

Standards Addressed

Grade 6

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences.
    • The student demonstrates algebraic thinking.

Grade 7

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 8

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 6

  • Algebra and Functions

    • 1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results

Grade 7

  • Algebra and Functions

    • 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs
    • 3.0 Students graph and interpret linear and some nonlinear functions
    • 4.0 Students solve simple linear equations and inequalities over the rational numbers

Sixth Grade

  • Expressions and Equations

    • Apply and extend previous understandings of arithmetic to algebraic expressions.
    • Represent and analyze quantitative relationships between dependent and independent variables.

Seventh Grade

  • Expressions and Equations

    • Use properties of operations to generate equivalent expressions.

Eighth Grade

  • Expressions and Equations

    • Understand the connections between proportional relationships, lines, and linear equations.
    • Analyze and solve linear equations and pairs of simultaneous linear equations.

  • Functions

    • Define, evaluate, and compare functions.
    • Use functions to model relationships between quantities.

Algebra

  • Reasoning with Equations and Inequalities

    • Solve equations and inequalities in one variable

Functions

  • Building Functions

    • Build a function that models a relationship between two quantities

  • Interpreting Functions

    • Understand the concept of a function and use function notation

  • Linear, Quadratic, and Exponential Models

    • Construct and compare linear, quadratic, and exponential models and solve problems

Grades 6-8

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols

Grade 6

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will demonstrate an understanding of simple algebraic expressions.

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will demonstrate an understanding of linear relations and fundamental algebraic concepts.

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Introductory Mathematics

  • Algebra

    • COMPETENCY GOAL 4: The learner will understand and use linear relations and functions.
    • COMPETENCY GOAL 5: The learner will understand and use linear relations and functions.

Technical Mathematics I

  • Number and Operations

    • Competency Goal 1: The learner will apply various strategies to solve problems.

8th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

Elementary Algebra

  • Elementary Algebra

    • Standard EA-3: The student will demonstrate through the mathematical processes an understanding of relationships and functions.
    • Standard EA-4: The student will demonstrate through the mathematical processes an understanding of the procedures for writing and solving linear equations and inequalities.
    • Standard EA-5: The student will demonstrate through the mathematical processes an understanding of the graphs and characteristics of linear equations and inequalities.

Grade 6

  • Patterns, Relationships, and Algebraic Thinking

    • 4. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes.

Algebra I

  • Foundation for Functions

    • 1. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.
    • 2. The student uses the properties and attributes of functions.
    • 3. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations.
    • 4. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

  • Linear Functions

    • 5. The student understands that linear functions can be represented in different ways and translates among their various representations.
    • 6. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations.

Secondary

  • Algebra II

    • AII.08 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators.
    • AII.13 The student will solve practical problems, using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems.
    • AII.8
    • AII.13

Be Prepared to

  • answer the question "How do I figure out what the numbers are supposed to be? I don't know how to figure it out."
  • discuss pattern recognition, input, output, etc...