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Number Patterns with Multiplication Rules

This lesson introduces number patterns that grow according to a multiplication rule, showing students how to compare terms, discover the constant multiplier, extend a sequence forward, and fill in missing terms in geometric-style number patterns.

What Is a Number Pattern with a Multiplication Rule?

A number pattern is simply a list of numbers that follows a rule. In a number pattern with a multiplication rule, each new term is found by multiplying the term before it by the same fixed number every time. That fixed number is called the multiplier, and once you know it, you can predict any term in the pattern without writing out the whole list.

For example, look at the sequence 2, 6, 18, 54. To move from one term to the next, you multiply by 3 each time. This is different from a pattern like 2, 5, 8, 11, where you add 3 each time instead. Recognizing whether a pattern is built by multiplying or by adding is the first step to describing it correctly.

Each term is found by multiplying the previous term by the same number.

Finding the Multiplication Rule

To find the rule in a multiplicative number pattern, divide any term by the term right before it. If every pair of neighboring terms gives the same quotient, that quotient is the rule. Using the pattern above, \( 6 \div 2 = 3 \), \( 18 \div 6 = 3 \), and \( 54 \div 18 = 3 \), so the rule is "multiply by 3."

This works because a multiplication rule builds the pattern step by step: \( t_2 = t_1 \times r \), \( t_3 = t_2 \times r \), and so on, where \( r \) is the constant multiplier. If you already feel comfortable multiplying quickly in your head, tools like multiplying using mental strategies make it much faster to test whether a rule holds across several terms.

Extending a Number Pattern

Once the rule is known, extending the pattern is straightforward: multiply the last term by the rule to get the next one, and repeat as many times as needed.

Example: The pattern 5, 10, 20, 40 follows the rule "multiply by 2." Find the next two terms.

\( 40 \times 2 = 80 \), then \( 80 \times 2 = 160 \). So the pattern continues 5, 10, 20, 40, 80, 160.

Finding a Missing Term

Sometimes a term is missing from the middle of a pattern. To fill it in, first use the terms you do have to figure out the rule, then apply that rule to the term just before the gap.

Example: Find the missing term in 3, 12, ___, 192.

Since going from 3 to 12 uses \( 3 \times 4 = 12 \), the rule is likely "multiply by 4." Check it against the last term: if the missing term is \( 12 \times 4 = 48 \), then \( 48 \times 4 = 192 \), which matches. So the missing term is 48.

Larger multipliers sometimes mean multiplying bigger numbers, and practicing with multiplying with digits up to 999 helps you check these calculations confidently as terms grow.

Multiplicative Patterns and Geometric Number Patterns

Number patterns that grow by repeated multiplication are closely related to what are often called geometric number patterns, since each term is a fixed multiple of the one before it, just like the terms of a geometric sequence. A simple case is the multiplication table itself: the multiples of 3 form the pattern 3, 6, 9, 12, 15, where each term can also be described by the rule \( t_n = 3n \), with \( n \) being the position of the term in the list.

The graph below shows this idea for the pattern of multiples of 3. Every point lines up because the value of each term is always 3 times its position number.

Breaking a multiplication rule into smaller, friendlier pieces can also make patterns easier to extend by hand. For instance, multiplying by 6 can be thought of as multiplying by 5 and then adding one more group, an idea explored further in converting products to a smaller product and sum.

Multiplicative vs. Additive Patterns

It is worth double-checking which type of rule a pattern uses before extending it. In an additive pattern, the same amount is added each time, such as 4, 8, 12, 16, where 4 is added repeatedly. In a multiplicative pattern, the same amount is multiplied each time, such as 4, 8, 16, 32, where each term doubles. Even though the first two terms of both patterns look similar, the third and fourth terms reveal very different rules, so always test at least two consecutive gaps before deciding on a rule.

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