unit 1.4  relationships in patterns
what we need to know
 Variables are represented by letters.
 Parts of an expression: variables, constant, coefficients
 Input/output tables
 Definition of a pattern
 Parts of an expression: variables, constant, coefficients
 Input/output tables
 Definition of a pattern
class notes


more information
PATTERNS AND RELATIONS
In math, a pattern is an ordered set of numbers that follow a specific pattern rule. A list of numbers which form a pattern is called a sequence. Each number in a sequence is called a term of the sequence. The first number is the first term of the sequence.
In this example:
 The first term of the sequence is 1.
 This means the pattern rule starts at 1.
 To get the second term, 3 squares are added. This pattern repeats for all the following terms.
 The pattern rule here, then, is: Start at 1. Add 3 squares each time.
The next step, and our goal for this part of the unit, is to come up with is called a relation, and to use such relation to come up with any term in the sequence. A relation is an algebraic expression that results when we relate a variable to the pattern rule (which contains the variable). Let's use the same sequence:
 The first term of the sequence is 1.
 This means the pattern rule starts at 1.
 To get the second term, 3 squares are added. This pattern repeats for all the following terms.
 The pattern rule here, then, is: Start at 1. Add 3 squares each time.
The next step, and our goal for this part of the unit, is to come up with is called a relation, and to use such relation to come up with any term in the sequence. A relation is an algebraic expression that results when we relate a variable to the pattern rule (which contains the variable). Let's use the same sequence:
 1, 2, 3 and 4 are the TERM NUMBERS. Since they are different from each other, they also represent the VARIABLE. Since variables are represented by letters, I'll assign the letter s to the term numbers.
 The PATTERN RULE tells us that, for each TERM of the sequence, we have to add 3 squares, or 3s.
 This means that the relation starts with 3s.
 Next, since the first term number is 1, or s=1, and the relation includes 3s, then for s=1, 3s = 3(1) =3. (Notice how I substituted).
 This means that the first TERM should have 3 squares. But it does not! it only has 1 square, which means that we have to subtract 2.
 Thus, the second part of the relation is  2. The RELATION is, then:
Relation = Term = 3s  2
where s = variable = term number, 2 is the constant
 The PATTERN RULE tells us that, for each TERM of the sequence, we have to add 3 squares, or 3s.
 This means that the relation starts with 3s.
 Next, since the first term number is 1, or s=1, and the relation includes 3s, then for s=1, 3s = 3(1) =3. (Notice how I substituted).
 This means that the first TERM should have 3 squares. But it does not! it only has 1 square, which means that we have to subtract 2.
 Thus, the second part of the relation is  2. The RELATION is, then:
Relation = Term = 3s  2
where s = variable = term number, 2 is the constant
Next lesson, we will use INPUT/OUTPUT tables to determine the relation. The INPUT column are the term or pattern numbers, which themselves are the variable. The OUTPUT column are the terms, which are determined by using the pattern rules, and are the number of squares. In our example, the INPUT/OUTPUT table would look like this:
Examples of relation problems
writing pattern rules
relationship rules for patterns
variables in expressions
videos that may help
interactive online activities
worksheets





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