Learning Math: Patterns, Functions, and Algebra
Linear Functions and Slope Part D: Putting It Together (35 minutes)
Session 5, Part D
Linear functions have come up in many situations so far in this session. In this section, we’ll consolidate some of our ideas about linear functions and look for connections between them.
Below are several input/output tables. For each table:
- Find a closed form rule for taking an input and finding the correct output
- Find a recursive rule for going from one output to the next
- Graph the pairs of numbers in the table
- Determine if the rule describes a linear function or not. If it is a linear function, find the slope.
Tip: The recursive rule here includes n, the input number.
What are the characteristics of a linear function? How can you tell that a function is linear if you are given:
- a closed form rule for a function?
- a recursive rule for a function?
- a description of a situation?
- a table?
- a graph?
The point of this section is to make connections between the different situations in which we’ve seen linear functions.
Take 15-20 minutes to work on Problems D1-D7. Then think about the connections in the different representations, paying particular attention to instances where linear functions are different from other kinds of functions.
In Problem D6, the recursive rule for the function y = 1/x can be quite challenging. The easiest way to describe it is to use your input in the rule.
Some people consider a rule recursive only if it truly depends on previous outputs, with no reference to the input.
- Closed forms for linear functions look like y = ax + b, where a and b are some numbers, x is the independent variable, and y is the dependent variable.
- Recursive rules for linear functions add a constant value from one output to the next. This constant is the same as the value of a in the formula y = ax + b.
- Graphs of linear functions look like lines. The slopes of the lines are the same as the difference between successive outputs, and the same as the value of a in the y = ax + b formula.
Groups: Discuss the above statements.
The following solutions refer to the input variable as “x” and the output variable as “y.”
The closed-form rule is y = x – 1.
The recursive rule is yn = yn-1 + 1, since the outputs grow by 1 each time.
This is a linear function, according to its graph, and the slope is 1.
The closed-form rule is y = x2.
The recursive rule is harder to formulate for this one: it is yn = yn-1 + (2n – 1). The key here is finding the pattern in the differences between each term. This is not a linear function.
The closed-form rule is y = 2x + 1.
The recursive rule is yn = yn-1 + 2. Outputs grow by 2 each time.
This is a linear function, and the slope is 2.
The closed-form rule is y = -x + 10, or y = 10 – x (both are the same).
The recursive rule is yn = yn-1 – 1. Outputs drop by 1 each time.
This is a linear function, and the slope is -1.
The closed-form rule is y = 5x.
The recursive rule is yn = yn-1 + 5. Outputs grow by 5 each time.
This is a linear function, and the slope is 5.
The closed-form rule is y = 1 / x.
The recursive rule is very difficult. Two possible answers are 1 / yn = 1 / yn-1 + 1, and yn= yn-1 + 1 / (n)(n-1).
This is not a linear function. Notice that the rate of change is not constant.
The closed-form rule is y = -7.
The recursive rule is yn = yn-1, because every term is the same as the last.
This is a linear function, according to the graph, and the slope is 0 (which means it is a horizontal line).
If there is a closed-form rule for a function, and the function is linear, it will be in the form y = Mx + B, where M and B can be any real number — positive, negative, or 0. Note Problems D5 and D7, in which one of the two values is 0.
If there is a recursive rule given, it should be in the form yn = yn-1 + M, where M is the slope of the line.
If a situation is described, it should involve a constant rate of change, such as a constant speed of a car, the constant slope of a ramp, or the constant price of gasoline per gallon.
If a table is given, the rate of change (change in output, divided by change in input) should always be the same number. If inputs are a sequence of numbers (like 1, 2, 3, 4, 5), the outputs should also form a sequence (3, 5, 7, 9, 11; 5, 10, 15, 20, 25).
If a graph is given, it should be a straight line (a linear function).
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.