In this lesson you will learn how to describe, extend, and make generalizations about numeric patterns. Working with number patterns is an very useful skill for solving many types of problem.

Identifying a pattern when you look at individual examples helps you to generalize and find a broader solution to a problem.

Probably without even knowing it, you have been observing and creating patterns ever since you were a very small child. You probably made repeating patterns with shapes, such the one below with triangles, circles, and squares.

When you got just a little bit older, you probably learned skip-counting, which is nothing more than applying a pattern to counting.

Skip-count by 2’s: 2, 4, 6, 8, 10, 12, 14, 16…

Skip-count by 5’s: 5, 10, 15, 20, 25, 30, 35, 40 …

Skip-count by 10’s: 10, 20, 30, 40, 50, 60, 70…

All of these are patterns, or mathematical rules.

A fun part of math is creating and playing with patterns. Math is organized with rules to follow. If you know what the rule is, you can create a pattern. The rule is often organized on a function table as shown in the examples below.

| This table shows the rule "add 5 to a number". The "x" column is the input side. The "y" column is the output side. The numbers that are side by side go together. In the first row, start with 0 and add 5. The result is 5. Start with 1 in row two, add 5. The result is 6, and so on. |

| This table shows the rule "multiply a number by two". In the first row, start with 0 and multiply it by 2. The result is 0. Start with 1 in row two, and multiply it by two. The result is 2. In row three, start with 2 and multiply by 2. The result is 4. And so on. |

| This table shows the rule "subtract 3 from a number". Although it is common for a function table to begin with 0 on the X side, it does not have to. The X numbers usually start with the smallest and get larger, but the first number can be any number. The numbers can also increase by more than one. This table begins with 6, and each X number goes up by two. In row one, 6 - 3 = an output in the Y column of 3, and so on. |

| Sometimes you are given a table with a rule, and asked to apply the rule to the numbers in the X column in order to complete the numbers in the Y column. What would the Y numbers be for this function table? Click the boxes to show or hide the answers. |

How did you do filling in the missing information on the Y side? Click in the boxes to check your answers . Since the rule is X +9 = Y, you add 9 to each of the X numbers to get the corresponding Y number.

| Let's try another one. What would the Y numbers be for this function table? Click the boxes to show or hide the answers. |

How did you do filling in the missing information on the Y side? Check your answers by clicking the the boxes. Since the rule is X -7 = Y, you should have subtracted 7 from each of the X numbers to get the corresponding Y number.

| Sometimes you might be given parts of both sides of the table, and you need to fill in what's missing by using what you know. See if you can fill in what's missing from this function table. Click the boxes to show or hide the answers. |

How did you do filling in the
missing information? Since the rule
for this function table is "multiply by 5,"
the first missing number is 15, since
3 x 5 = 15. For the next missing number,
you have to think "what number times 5
would give me 25?" You can also think of
it as division: "25 divided by 5 = what
number?" since division is the *inverse*, or
opposite of multiplication. The final
missing number is 5 x 7, or 35.

| Sometimes, you may be given the function table without the rule. You have to analyze what you see happening with the numbers and decide what the rule is for that particular function table. You will often be asked to use the rule you discover to complete missing numbers in the function table, and/or continue the pattern by extending the function table. What do you think the rule is for this table? What are the missing numbers? Click the boxes to show or hide the answers. |

How did you do? Were you able to figure out the pattern? This table shows "multiply a number by 3." You can use the filled in numbers to figure out the rule by asking, "what is the relationship between the X numbers and their corresponding Y numbers?" Look for patterns. Once you have determined the pattern, it is a simple matter to fill in the missing numbers. Here, 21 (3 x 7) and 33 (3 x 11) were missing. If you were asked to extend the table, you would get the pair: 13, 39.

So when it comes to number patterns, remember these things:

- Look for a relationship between the input “X” side, and the output “Y” side.
- Check the pattern against every row. It has to be true for the whole table, or it’s not the rule.
- Use the rule, and the numbers you know, to complete or extend the pattern.

- Number Patterns with Function Tables - 3 Page Worksheet