Sequence and Number Pattern

Number Sequence:

In teaching number sequences, it is important to start with concrete examples using blocks or other manipulatives.

It is easiest to start by showing the growth of a simple repeating pattern.

Simple repeating pattern.

Show how it grows by adding successive identical units of repeat.

Growing pattern.

Counting the number of blocks gives the sequence 2, 4, 6… Students can see that each time a unit of repeat is added, the total number of blocks increases by 2. Also, the total number of blocks is twice the number of units of repeat.

Identical units of repeat:

Repeating patterns can be used to introduce students to many concepts in the early mathematics curriculum, especially multiplication. However, students need to be able to find the unit of repeat in each pattern.

The idea can be introduced using 'trains' of interlocking cubes arranged in a row.

A train of six cubes.

A train of twelve cubes.

These trains can be made using the following units of repeat.

A unit of repeat for the first train.

A unit of repeat for the second train.

The first unit is repeated three times, and the second one is repeated four times.

Students can often copy or extend such patterns without being able to find a unit of repeat. For example, students often see the first pattern as 'alternating white and green' rather than 'white-green repeated'.

If students cannot readily identify units of repeat in one-dimensional patterns, then they may not be able to grasp the multiplication that is implicit in such patterns.

Number Patterns:

Using Patterns to Solve Problems

When solving problems that involve greatest common factors, you can use patterns to help. 

Let's look at an example:

Name the next value in the series. 4, 7, 10, 13, ______

First, you need to identify the pattern by looking at the numbers.

4 + 3 = 7; 7 + 3 =10, 10 + 3 = 13

The pattern consists of adding 3 to the previous number, so the next number will be the sum of 13 and 3.

13 + 3 = 16

The answer is 16.

Although the first example involved addition, any of the four operations can be used when trying to identify the pattern. 

Let's look at an example of a pattern that uses multiplication.

Name the next value in the series. 2, 6, 18, 54, _____

Notice that the next number is found by multiplying the current number by 3. Based on that pattern, the next number will be the product of 54 and 3.

 54×3=162

The next number in the series is 162.

Example 1: 

Earlier, you were given a problem about Coach Gerald.

His team is in a tournament in which teams are knocked out in each round. He wants his team to understand the number of games that they will have to win in order to win the tournament, so he provides his team with part of the pattern that the winners of each round experience.

32, 16, 8, 4, ______

What is the next number in the pattern that Coach Gerald wrote for his players?

First, figure out what changes between the given values.

32 - 16 = 16; 16 - 8 = 8; 8 - 4 = 4

Next, look at the changes and determine the pattern. 

Subtract half of the previous number from the current number to calculate the next number.

Then, find the next number.

4 - 2 = 2

The answer is 2.

Example 2:

What is the next number in this sequence?

1, 1, 2, 3, 5, 8, 13, 21, _____

First, figure out what changes between the given values.

1 + 0 = 1; 1 + 1 = 2; 2 + 1 = 3; 3 + 2 = 5; 5 + 3 = 8; 8 + 5 = 13; 13 + 8 = 21

Next, look at the changes and determine the pattern.

The next number is found by finding the sum of the current number and the number before it.

Then, determine the next number.

21 + 13 = 34

The next number is 34.

Example 3:

Name the next value in the series.

3, 7, 15, 31, _____

First, figure out what changes between the given values.

3 + 4 = 7; 7 + 8 = 15; 15 + 16 = 31

Next, look at the changes and determine the pattern.

Add 2 times the previous number that was added

Then, determine the next number.

31 + 2(16) = 31 + 32 = 63

The answer is 63.

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