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9.21: Number Patterns

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Remember the skateboard park? Well after the grand opening, kids continued to go to the skateboard park every day. The first day, there were 32 kids. The next day there were 30 kids. The third day, there were 28 kids.

As part of the research of the skateboard park, Taylor took on keeping track of attendance for the first week. She wrote these numbers down and noticed a pattern.

Do you notice a pattern? If this trend for attendance continues, how many students will attend on the fourth day?

Guidance

When solving problems that involve greatest common factors, we can use patterns to help us. The strategy “look for a pattern” is just that.

What pattern can be seen in the numbers that we are working with? How does the pattern appear?

There are 280 girls and 260 boys playing on soccer teams this fall. If each team has the same number of girls and the same number of boys, what is the greatest number of teams that can be formed?

To solve this, we need to find the prime factors of 280 and 260. Then, we can figure out the greatest common factor which is the largest number that divides into both groups. Once we have this factor, we will know the number of teams. The greatest common factor is also the number of teams that can be formed.

We start by factoring 280.

Next, we factor 260

If we look at what is common here, we can see that 25 and two 2's are common.

5 $\times$ 2 $\times$ 2 $=$ 20

There are 20 possible teams.

By looking for patterns, we could use 10 as a factor. Right in the beginning, we have 10 as one of the factors, then we just had to find any other factors. This gave us our answer.

Example A

Name the next value in the series. 3, 7, 15, 31, _____

Solution: 63

Example B

Name the next value in the series.

12, 8, 4, _____

Solution: 0

Example C

Name the next value in the series.

2.5, 5, 10, 20, 40, _____

Solution: 80

Here is the original problem once again.

After the grand opening, kids continued to go to the skateboard park every day. The first day, there were 32 kids. The next day there were 30 kids. The third day, there were 28 kids.

As part of the research of the skateboard park, Taylor took on keeping track of attendance for the first week. She wrote these numbers down and noticed a pattern.

Do you notice a pattern? If this trend for attendance continues, how many students will attend on the fourth day?

To notice a pattern, let's write out the numbers of kids who went to the skateboard park on the first three days.

$32, 30, 28$

If you look at these values, you can see that the number of kids in attendance decreased by two every day.

Based on this pattern, we could predict that 26 kids will go to the skateboard park on the fourth day.

However, attendance is a tricky thing to predict and it could change at any moment!

Vocabulary

There aren't any new vocabulary words for this Concept.

Guided Practice

Here is one for you to try on your own.

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

What is the next number in this sequence?

If we were going to solve this problem, we would need to look for a pattern in the numbers. A Venn diagram wouldn’t really help us here-we have one set of data and we aren’t comparing anything. We are looking to figure out the next number.

How can we figure this out?

We can look for different ways to get the numbers using different operations. Were these numbers multiplied?

$1 \times 1 & = 1 \\1 \times 2 & = 2\\ 2 \times 3 & = 6$

That’s right, it doesn’t work. You have to use a different operation.

Do you see a pattern?

The pattern here is to find the sum of the two previous numbers. That sum is the next number in the pattern.

Let’s see if this works.

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

$1 + 1 & = 2 \\1 + 2 & = 3 \\2 + 3 & = 5 \\5 + 3 & = 8 \\5 + 8 & = 13 \\8 + 13 & = 21 \\13 + 21 & = 34$

Video Review

Here is a video for review.

Practice

Directions: Figure out the pattern in each of the following problems. Then write in the next number in each pattern.

1. 2, 4, 6, 8, 10 ____

2. 20, 17, 14, 11, ____

3. 4, 8, 16, 32, ____

4. 200, 100, 50 ____

5. 120, 60, 30, 15, ____

6. 22, 33, 44, 55, 66, ____

7. 4, 12, 36, _____

8. 5, 10, 6, 12, 8, 15, 11, _____

9. 6, 4, 8, 5, 10, 6, 12, _____

10. 5, 11, 6, 13, 7, 15, 8, _____

11. 6, 12, 18, 24, 30, 36, _____

12. 100, 75, 100, 75, 50, 75, _____

13. 14, 28, 35, _____

14. 200, 50, 12.5, _____

15. 621, 207, 69, _____

Basic

Oct 29, 2012

Sep 10, 2014