Rules and Patterns

Contributor: Ashley Nail. Lesson ID: 13891

Do you like solving riddles? In this lesson, you will study the relationship between groups of numbers and find the rules that tie them all together. It's like solving a riddle!

categories

Operations and Algebraic Thinking

subject
Math
learning style
Visual
personality style
Lion, Otter, Beaver, Golden Retriever
Grade Level
Intermediate (3-5)
Lesson Type
Quick Query

Lesson Plan - Get It!

Audio:

Jace is going shopping for his family. He sees bags of cookies for $2 each! Every bag has 18 cookies inside. He wants to buy 4 bags of cookies.

  • How much will that cost him?
  • How many cookies will he have?

In order to help Jace, we need to look at rules and patterns in these numbers.

We should look at some other examples first!

Let’s look at the rule for this pattern:

Pattern A: Start at 2 and add 2.

We can use this rule to create a list of terms that can goes on forever.

Pattern A: 2, 4, 6, 8, 10 …

Now, let’s look at another pattern and rule:

Pattern B: Start at 6 and add 6.

Just like Pattern A, we can use this rule to make a list of terms that can go on forever.

Pattern B: 6, 12, 18, 24, 30 …

Let’s arrange these two lists of terms into a table:

  Pattern A   Pattern B
  2   6
  4   12
  6   18
  8   24
  10   30
  ?   ?

 

Using the rules for each pattern, we can easily find the next number in each sequence.

For example, the next term in the Pattern A sequence will be 12. Since we know the rule is to add 2, we will use 10 + 2 to get 12.

The rule for Pattern B is to add 6, so we will use 30 + 6 to get 36.

Let’s add these new terms to the table:

  Pattern A   Pattern B
  2   6
  4   12
  6   18
  8   24
  10   30
  12   36

 

Say we continue to follow the rule for Pattern A to complete more rows of the table. Now, we want to find Pattern B’s missing term.

  Pattern A   Pattern B
  2   6
  4   12
  6   18
  8   24
  10   30
  12   36
  14    
  16    
  18    
  20   ?

 

We could use Pattern B’s rule and continuously add 6 to each term in the sequence. Or we could figure out the relationship between the numbers in Pattern A and Pattern B.

Each set of corresponding numbers in the table can be rewritten as ordered pairs on a coordinate plane. This can help us figure out the relationship between Pattern A and Pattern B.

(2, 6) (4, 12) (6, 18) (8, 24) (12, 36)

*plot points

The first value in the ordered pair is multiplied by 3 to get the second value in the ordered pair.

(2, 6) 2 x 3 = 6

Let’s see if this is true for each ordered pair:

(2, 6) 2 x 3 = 6

(4, 12) 4 x 3 = 12

(6, 18) 6 x 3 = 18

(8, 24) 8 x 3 = 24

(12, 36) 12 x 3 = 36

Now, we can write a new rule describing the relationship between Pattern A and Pattern B!

The terms in Pattern B are 3 times the terms of Pattern A

Let’s use this new rule to find the missing term in Pattern B.

If the term in Pattern A is 20, then to find the corresponding term in Pattern B, we need to multiply by 3.

20 x 3 = 60

The missing term in Pattern B is 60.

  Pattern A   Pattern B
  2   6
  4   12
  6   18
  8   24
  10   30
  12   36
  14    
  16    
  18    
  20   60

 

Using either the rule for Pattern B or the rule for the relationship between Patterns A and B, find the other missing terms.

  • Which rule did you use to complete the table?

Look at the table again, but this time think about the rule you didn’t use.

  • Do you see how both rules will get you the same answers?

When you are ready to practice on your own, click NEXT to visit the Got It? section!

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