# Patterns: Describing patterns using tables and solving variables

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##### Intros
###### Lessons
1. Introduction to Describing Patterns using Tables and Solving Variables:
2. What is a function machine and what is a function table?
3. What are two-step rules?
4. How do we write number pattern rules as formulas with variables?
5. Solving the formula for one-step rules
6. Solving the formula for two-step rules with consecutive inputs
7. Solving for formula for two-step rules with random inputs
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##### Examples
###### Lessons
1. Solve for the Function Table's Missing Variables
Use the rule to complete the function table:

2. Solving for Function Table Rules
Write the rule for the function table
• Write the one-step rule as a formula with a variable

3. Using Two-Step Rules to Complete Function Tables
Use the two-step rule to complete the function table.

4. Solving for Two-Step Rules in Function Tables
Write the two-step rule for the function table.
1. $output = (m) input \pm b$
2. $output = (m) input \pm b$
3. $output = (m) input \pm b$
4. $output = (m) input \pm b$
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##### Practice
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###### Topic Notes

In this lesson, we will learn:

• How to describe number patterns using a function table (input output table)
• How to write formulas with variables for function tables and solve for variables
• The steps for solving the rule (one-step and two-step) or formula for a function table

Notes:

• We can think of the relationship between numbers in a pattern as a machine
• The machine takes the number you give it (the “input”), applies a function (the “rule” or math operations), and gives you a resulting number (the “output”)

• The input output table (or function table) keeps track of these inputs and outputs
• Unlike the number sequence, order is not necessary for a function table
• Ex. for the number sequence/pattern “start at 1 and add 3 each time” it would be:

• Ex. but for the function table with a rule of “add 3” it could be:

• It is also possible to have two-step rules for function tables
• The first step is to either multiply or divide (× or ÷)
• The second step is to either add or subtract (+ or –)
• Instead of writing “input” and “output” in the function table, variables can be written instead
• Variables are symbols (letters) that represent values that can change (“varying”)
• Variables can be used to write a formula for the function table using the format:
• $(output variable) = (multiplier/divisor) x (input variable) \pm (addend/subtrahend)$
• Or more commonly written as $y = m x + b$
• To solve for the variables in function tables:
• If solving for an output: plug the input value into the formula
• If solving for an input: plug the output value in and solve backwards (algebra)
• If you are given a complete function table and asked to solve for the formula:
• Check horizontally across input/output for one-step rules
• If it is not a one-step rule:
• If the inputs are consecutive, the multiplier m (in formula $y = m x + b$) is the difference between outputs
• If the inputs are random, the formula can be either found by:
• (#1) trial and error
• OR (#2) using two pairs of input/output and m is the ratio of $\large\frac{\Delta y}{\Delta x}$