Remember the trees from the Extend Numerical Patterns Concept?

Previously we worked on looking at the number patterns in nature. Here is one such pattern.

1, 1, 2, 3, 5, 8, 13

This number pattern follows a rule.

Patterns in nature can be visual as well as numerical.

The pattern above is called a Fibonacci Sequence where two of the previous numbers are added together to equal the third number.

Here is a website where you can see the Fibonacci Sequence in nature.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

If you continue to follow the rule for this pattern, what will be the next three numbers in the sequence?

**This Concept is about recognizing and describing visual and/or geometric patterns. You will know how to answer this question by the end of the Concept.**

### Guidance

Visual or geometric patterns are similar to numerical patterns in the sense that the follow a rule. Some geometric patterns are ** repeating patterns.** Repeating patterns have a

**which repeats. Other geometric patterns, much like numerical patterns, change position, increase, or decrease following a pattern rule or formula.**

*pattern unit*Look at the pattern below. What is the pattern unit?

**Let’s begin by naming the shapes.** The shapes follow this order: circle, square, triangle, triangle, circle, square, triangle, triangle, circle, square. To find the pattern unit, we have to find the unit which repeats.

**Where do you begin to notice that the pattern is repeating?**

In this case, the pattern unit is circle, square, triangle, triangle

You can see repeating patterns all around you. The tiling of a floor is a repeated pattern. Wallpaper is often designed in a repeating pattern as well.

**What happens when a visual pattern doesn’t repeat? When it is created by some other rule?**

When this happens, you have to pull out your detective skills once again. You will have to figure out the rule. How did the visual images change? What rule can you write to describe the change?

*Look at the pattern below. What is the pattern rule?*

**We can see that this is not a repeating pattern because no two shapes repeat in the same position. Therefore, we need to look for a pattern rule.** Like with numerical patterns, we compare the shapes to determine how they change. All the shapes are right triangles. In the first figure, the right angle is in the bottom left. In the second shape, the right angle has changed to the top left. In the third shape, the right angle has changed to the top right.

**In this pattern, the right angle is moving clockwise by 90 degrees each time. This is our pattern rule.**

Once you figure out how the pattern unit is repeating or what the rule is to the geometric pattern, you can extend the pattern according to this information.

How many triangles will be in the next step of this pattern?

**How does this geometric pattern change?** To figure this out we have to look at each figure as if it is a step in the pattern. The first step has one triangle. The second step has four triangles. The third step has nine triangles.

We can write the pattern as: 1, 4, 9

**Because this pattern does not change in a systematic way we can’t write a rule for it. We can see that each step increases by two more than the previous step.**

The first step adds 3 to get the second step.

The second step adds 5 to get the third step.

The third step will add two more which is 7.

There are 9 triangles in the third step, \begin{align*}9 + 7 = 16\end{align*}. There will be 16 triangles in the fourth step.

If we know how to describe and extend numerical, visual or geometric patterns, we can also find missing elements in these patterns.

What is the missing number in the pattern?

11, 23, 47, ___, 191

**First we have to find the pattern rule.**

In an increasing number pattern like this one, we start by looking for an addition and/or multiplication relationship between the first two numbers. \begin{align*}11 + 12 = 23\end{align*}, so +12 is a possible pattern rule, but it doesn’t apply to the second number (23) and third number (47), since \begin{align*}23 + 12 = 35\end{align*}, not 47. The third number is 47, which is one more than \begin{align*}23 \times 2\end{align*}, so \begin{align*}(\times \ 2 + 1)\end{align*} is a possible pattern rule.

**To find the missing number (the fourth number), we need to plug 47 (the third number) into the value for \begin{align*}x\end{align*}.**

\begin{align*}y &= 2x + 1\\ y &= 2(47) + 1\\ y &= 94 + 1\\ y &= 95\end{align*}

**To check that we have the right number, we can plug our answer (the fourth number) into the equation to see if we get the fifth number in the pattern.**

\begin{align*}y &= 2x + 1\\ y &= 2(95) + 1\\ y &= 190 + 1\\ y &= 191\end{align*}

191 is the fifth number, so our solution is correct!

**Our answer is 95.**

We can do this visually too. Fill in the missing piece of the pattern.

**First, we examine the figures in the pattern.** The pattern is a pattern of repeating images, so we can look at how the images repeat and figure out the missing image in the pattern.

This is the missing image. We can check our work by examining the pattern. Design faces to the left, then to the right, then to the left, then to the right, then to the left, then to the right.

**This is the correct image to complete the pattern.**

Practice examining visual or geometric patterns. Figure out the rule and write down the next image in the pattern.

#### Example A

**Solution: Right side up smiling face**

#### Example B

**Solution: one arrow to the right**

#### Example C

Continue the pattern with the next two figures.

Triangle, star, triangle, star, square, square, triangle, star, triangle

**Solution: star, square**

Now back to patterns in nature. Here is the original pattern once again.

Remember the trees from the Extend Numerical Patterns Concept?

Previously, we worked on looking at the number patterns in nature. Here is one such pattern.

1, 1, 2, 3, 5, 8, 13

This number pattern follows a rule.

Patterns in nature can be visual as well as numerical.

The pattern above is called a Fibonacci Sequence where two of the previous numbers are added together to equal the third number.

Here is a website where you can see the Fibonacci Sequence in nature.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

If you continue to follow the rule for this pattern, what will be the next three numbers in the sequence?

We can figure out the pattern by following the rule.

We add the two preceding numbers to equal the next value in the sequence.

8 + 13 = 21

13 + 21 = 34

21 + 54 = 75

**The next three values are 21, 34, 75.**

### Vocabulary

- Pattern
- a sequence of number or geometric figures that repeats according to a pattern unit or a rule.

- Geometric Patterns
- visual patterns of geometric figures that follow a rule or repeat according to a pattern unit.

- Ascending Pattern
- a pattern that increases.

- Descending Pattern
- a pattern that decreases.

### Guided Practice

Here is one for you to try on your own.

What would be the next image in the pattern?

**Answer**

If you look at what has happened in each step of the pattern, you will see that the pattern decreased by half.

**The next step in the pattern would show two boxes or two squares.**

### Video Review

Khan Academy Geometric Sequences

### Practice

Directions: Write a rule for each number pattern.

1. 4, 7, 13, 25

2. 216, 196, 176, 156

3. 1, 7, 19, 43

4. 10,000, 1,000, 100, 10

Directions: Now write the next number in each pattern.

5. 4, 7, 13, 25, ____

6. 216, 196, 176, 156, ____

7. 1, 7, 19, 43, ____

8. 10,000, 1,000, 100, 10, ____

Directions: Write down the next image in each pattern based on the description.

9. a hexagon, a pentagon, a square, _______

10. five dots, ten dots, fifteen dots, ______

11. fifteen dots, fourteen dots, twelve dots, _____

12. two triangles, four triangles, three triangles, six triangles, five triangles, _______

13. An octagon, a hexagon, an octagon, a pentagon, an octagon, _____

14. One square, 1.5 squares, 2 squares, 2.5 squares, _____

15. eighteen circles, nine circles, four and a half circles _____