Number Patterns: Rules, Terms and Tables | Grade 5 Math | LevelKite

Start with $2$ . Add $5$ . Add $5$ again, and again, and again. You get $2$ , $7$ , $12$ , $17$ , $22$ .

You just followed a rule. In this lesson, a rule is an instruction that tells you how to get from one number to the next. You use the same rule every time. Here the rule is "add 5". The number you begin with is the starting number. Here it is $2$ .

The list you made is a number pattern: a list of numbers, kept in order, that follows a rule. Each number in the pattern is called a term. A term's position is its place in the list: 1st, 2nd, 3rd, and so on. In this pattern, the 1st term is $2$ and the 4th term is $17$ .

Building the pattern with the starting number 2 and the rule "add 5"

Position	How you get the term	Term
1st	Write the starting number.	$2$
2nd	Add $5$ to the 1st term: $2 + 5 = 7$	$7$
3rd	Add $5$ to the 2nd term: $7 + 5 = 12$	$12$
4th	Add $5$ to the 3rd term: $12 + 5 = 17$	$17$
5th	Add $5$ to the 4th term: $17 + 5 = 22$	$22$

Look at how the table works. The starting number goes in first. It is the 1st term, and you do not need the rule to get it.

After the 1st term, every term comes from the term just before it. You use the rule on the newest term to get the next one. Five terms take four uses of the rule, because the starting number already gives you the 1st term.

These steps work for every rule in this lesson. Some rules tell you to add a number. Others tell you to multiply by a number.

Worked example

One rule, one pattern

The starting number is $5$ and the rule is "multiply by 3". What are the first five terms of the pattern?

Write the starting number first. The starting number is the 1st term, so you do not use the rule yet.
$\text{1st term: } 5$
Use the rule on the 1st term to get the 2nd term. The rule says to multiply by 3, so multiply 5 by 3.
$5 \times 3 = 15$
Keep using the rule on the newest term, because each term after the 1st comes from the term just before it. Do this three more times to reach five terms.
$15 \times 3 = 45 \quad 45 \times 3 = 135 \quad 135 \times 3 = 405$
List the terms in the order you found them, so each term keeps its position.
$5,\; 15,\; 45,\; 135,\; 405$

Answer

The first five terms of the pattern are $5$ , $15$ , $45$ , $135$ and $405$ .

Sometimes a problem gives you two rules and asks for two patterns. Each pattern has its own starting number and its own rule. Give the patterns names, such as Pattern A and Pattern B, so you can tell them apart.

Build one pattern at a time. Finish Pattern A, then build Pattern B. Use each rule only on its own pattern: Pattern A's rule works on Pattern A's terms, and Pattern B's rule works on Pattern B's terms.

Keep each list in the order you found the terms. Then you can always tell which term is 1st, 2nd, 3rd, and so on in each pattern.

Worked example

Two rules, two patterns

Pattern A has the starting number $0$ and the rule "add 3". Pattern B has the starting number $0$ and the rule "add 6". Write the first five terms of each pattern.

Build Pattern A first. Write its starting number, 0, as the 1st term. Then use the rule "add 3" on the newest term four times, because five terms take four uses of the rule.
$0 + 3 = 3 \quad 3 + 3 = 6 \quad 6 + 3 = 9 \quad 9 + 3 = 12$
List Pattern A's terms in the order you found them and label the list. The label keeps these terms apart from Pattern B's terms.
$\text{Pattern A: } 0,\; 3,\; 6,\; 9,\; 12$
Now build Pattern B. It also starts at 0, but it follows its own rule, so do not use "add 3" here. Write 0 as the 1st term. Then use "add 6" on the newest term four times.
$0 + 6 = 6 \quad 6 + 6 = 12 \quad 12 + 6 = 18 \quad 18 + 6 = 24$
List Pattern B's terms in order and label this list too, so the two patterns stay apart.
$\text{Pattern B: } 0,\; 6,\; 12,\; 18,\; 24$

Answer

Pattern A is $0$ , $3$ , $6$ , $9$ , $12$ . Pattern B is $0$ , $6$ , $12$ , $18$ , $24$ .

A table keeps two patterns lined up. Make one column for each pattern. A column goes down the table. Make one row for each position. A row goes across the table.

Each pattern's terms go down its own column in order. So the 1st terms share the 1st row, the 2nd terms share the 2nd row, and so on. Terms in the same position sit side by side.

This is a little like naming a point on a coordinate plane, where two numbers stay together in a fixed order. Here each row holds two terms in a fixed order: Pattern A's term, then Pattern B's term.

The table below records Pattern A and Pattern B from the last example. Read across the 4th row. It holds $9$ and $18$ . These are the 4th term of Pattern A and the 4th term of Pattern B.

Pattern A (start at 0, add 3) and Pattern B (start at 0, add 6)

Position	Pattern A (add 3)	Pattern B (add 6)
1st	$0$	$0$
2nd	$3$	$6$
3rd	$6$	$12$
4th	$9$	$18$
5th	$12$	$24$

Key concept

A rule builds a pattern, and a table lines patterns up

A starting number and a rule build a number pattern one term at a time. The starting number is the 1st term. You use the rule on each term to get the next term, and you keep the terms in order.

When you record two patterns in a table with one column for each pattern and one row for each position, terms in the same position sit side by side in the same row.

Worked example

Two patterns in one table

Pattern A has the starting number $7$ and the rule "add 10". Pattern B has the starting number $3$ and the rule "multiply by 2". Complete a table that shows the first five terms of each pattern.

Set up the table. Give each pattern its own column and each position its own row, from 1st to 5th. This layout puts terms in the same position in the same row.
Fill in the 1st row with the two starting numbers, because each starting number is the 1st term of its pattern.
$\text{Pattern A: } 7 \qquad \text{Pattern B: } 3$
Work down Pattern A's column. The term just before each new term is the one above it, so use "add 10" on the term above.
$7 + 10 = 17 \quad 17 + 10 = 27 \quad 27 + 10 = 37 \quad 37 + 10 = 47$
Work down Pattern B's column the same way, but use its own rule, "multiply by 2", because each rule works only on its own pattern.
$3 \times 2 = 6 \quad 6 \times 2 = 12 \quad 12 \times 2 = 24 \quad 24 \times 2 = 48$
Check a row. Count down each column to the 3rd term: 27 in Pattern A and 12 in Pattern B. Both sit in the 3rd row, so terms in the same position are side by side.

Answer

The completed table shows Pattern A as $7$ , $17$ , $27$ , $37$ , $47$ and Pattern B as $3$ , $6$ , $12$ , $24$ , $48$ , with terms in the same position in the same row.

The completed table: Pattern A (start at 7, add 10) and Pattern B (start at 3, multiply by 2)

Position	Pattern A (add 10)	Pattern B (multiply by 2)
1st	$7$	$3$
2nd	$17$	$6$
3rd	$27$	$12$
4th	$37$	$24$
5th	$47$	$48$

Takeaway: A starting number and a rule build a number pattern. Write the starting number as the 1st term. Then use the rule on the newest term, again and again, and keep the terms in order.

With two patterns, give each one its own starting number, its own rule and its own column. Keep one row for each position, so terms in the same position sit side by side.

Next, you will compare the terms that share a row and describe how the two patterns are related.