## Learning Math: Patterns, Functions, and Algebra

# 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

### In This Session

**Part A:** Doing and Undoing

**Part B:** Undoing Algorithms

**Part C:** Function Machines

**Part D:** Number Games

**Part E:** Other Kinds of Functions

**Homework**

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.

**Note 1**

### Learning Objectives

In this session, we will explore algorithms and functions. We will:

- Understand the importance of doing and undoing in mathematics
- Determine when a process can or cannot be undone
- Use function machines to picture and undo algorithms
- Understand that functions produce unique outputs

### Key Terms

**Previously Introduced:**

**Variable: **The term variable can have different meanings:

- An indeterminate, a symbol used to represent generalized arithmetic (such as 0 x a = 0)
- An unknown, used to represent a particular number (such as a + 3 = 7)
- A relationship between quantities (such as a + b = 10)
- A term used to help in understanding the study of algebraic structures (such as a + b = b + a)

**New in This Session:**

**Algorithm: **An algorithm is a recipe or a description of a mechanical set of steps for performing some task.

**Function: **A function is any relationship between inputs and outputs in which each input leads to exactly one output. It is possible for a function to have more than one input that yields the same output.

**Function Machine: **A function machine is a way of visualizing functions and their inputs and outputs.

**Network: **A network of function machines is made when two or more function machines are connected so that the output from one function machine becomes the input for the next.

**Iterate: **To iterate a function means to repeat its algorithm, using the previous output as the next input.

**Whole Number: **A whole number is a counting number or zero. The whole numbers can be written as the series 0, 1, 2, 3, … . Negative numbers and fractions are not included in the set of whole numbers.

**Factors: **The factors of a whole number are the numbers that divide evenly into it. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

**Prime Number: **A prime number is a whole number with exactly two factors: 1 and the number itself. For example, the numbers 2, 7, 13, 19, and 31 are prime. The numbers 0, 1, 12, and 100 are not prime. In particular, 1 is not a prime because it does not have exactly two factors (1 is the only factor).

**Polygon: **A polygon is a two-dimensional figure made up of sides of any length. A triangle, rectangle, pentagon, and octagon are all polygons. Note that the sides of a polygon do not all have to be the same length.

### Notes

**Note 1**

In this session, we will explore algorithms and functions. We will use function machines to illustrate function as algorithm, to picture operations as machines, and to provide a visual image of inputs and outputs.

When people begin to move from arithmetic to algebra, they start thinking about properties of operations — specifically on “what undoes what.” Function machines are used both to build algorithms and to provide experience in thinking about how to “undo,” or create inverses of, those algorithms.

We will also focus on the concept of uniqueness, looking at non-numerical examples to illustrate the uniqueness of functions. A key idea in the study of functions is that inputs must give unique outputs, but outputs may not have unique inputs. If the latter is the case, these functions cannot be undone.

By the end of the session, we will see that the same function can be expressed by different rules or algorithms, something that we have seen informally in Sessions 1 and 2.

**Review**

**Groups:** Discuss any questions about the homework. Look at the descriptions of the rules found for Problem H1, and, if it has not already been done, write the rule using symbols. Discuss the notation in x^{2} + 1, especially reviewing the meaning of the exponent. This session contains a brief problem on the function y = x^{4}, and later, Session 7 contains extensive work with exponential functions, so it’s good to preview some of these ideas.

Another interesting question on the homework is how we can know that 102 wouldn’t appear in the “output” column of the table (as long as the “inputs” remained integers). One explanation: The input of 10 gives 101, and the input of 11 gives 122. Since the function seems to be only increasing, it has skipped 102. Since each number in the table is one more than a perfect square, another good explanation would be that 101 is not a perfect square, so 102 cannot appear in the table.

**Groups:** Consider sharing solutions and representations for the frog in the well problem, as well.