Inductive Patterns

Learning Objectives

• Recognize patterns in sequences of numbers and geometrical/real world objects.
• Apply inductive reasoning to predict the value of unknown terms in a sequence.

Some people define mathematics as the study of patterns. Patterns abound in our world—in nature, in the behaviors of people, and in technology. The twist of a seashell and the swirl of buds across the face of a sunflower both make clear patterns.

Although such patterns may be easy to see, we often need mathematics to help describe and explore them. It wasn’t until the growth patterns of organisms like flowers and snails were measured mathematically that biologists realized that their regular spirals are designed for efficiency—they pack the most amount of material into the smallest amount of space.

The mathematical study of pattern is a powerful analytical tool for making sense of our world.

Let’s take a close look at a visual pattern and see how mathematics can help us understand and make use of it.

Mario is a clothing designer. He wants to decorate belts with a triangular pattern made out of strips of leather. Before he starts making belts, he wants to know how many strips he will need for different belt lengths. Mario will start by counting the number of strips of leather needed for the first few stages of the pattern.

To make `1` triangle, he uses `3` strips, one for each side. Mario creates a second triangle alongside the first one by adding two more leather strips. The right side of the first triangle acts as the left side of the second triangle. So he takes `5` strips to make a `2`-triangle design. To add a third triangle, Mario again adds two more strips to one side of an existing triangle, for a total of `7` strips. The fourth triangle takes two more strips, bringing the count to `9` strips.

At this point, Mario realizes that he doesn’t need to keep counting—once he makes the first triangle, each additional triangle takes two more strips of leather. He has found the mathematical pattern of his designs.

Mario can use words to describe the growing amount of strips he needs to make more triangles. He can also describe it as a mathematical sequenceAn ordered list of numbers or objects.—an ordered list of numbers. The sequence of numbers that describes the number of strips needed for the first four triangles of the belt design is `3,5,7,9`.

Mario’s mathematical sequence: `3,5,7,9`

The numbers that make up a sequence are called termsA value in a sequence--the first value in a sequence is the `1text(st)` term, the second value is the `2text(nd)` term, and so on; a term is also any of the monomials that make up a polynomial.. In this sequence, there are `4` numbers, which means there are `4` terms. Three is the `1text(st)` term, `5` is the `2text(nd)` term, `7` is the `3text(rd)` term, and `9` is the `4text(th)` term.

Although Mario only laid out enough strips to make `4` triangles, he knows that he could make more just by continuing to add `2` more strips for each additional triangle. Without having to actually make more triangles, he can predict that the next term in the mathematical sequence will be `2` more than the last one. Since `4` triangles took `9` strips of leather, `5` triangles will take `11` strips.

Mario’s mathematical sequence: `3,5,7,9,11`

Mario is using what is called inductive reasoningA form of logical thinking that makes general conclusions based on specific situations, inductive reasoning takes the path of observation to generalization to conjecture.—he assumes the pattern he discovered by counting will continue to grow in the same way. He reasons that after `11`, the next term will be `13`, followed by `15`, and so on. Inductive reasoning allows us to predict the behavior of patterns beyond what we see. In this sequence, the values of the first `4` terms were determined by observation, and the next `3` by inductive reasoning.

Mario’s mathematical sequence: `3, 5, 7, 9, 11, 13, 15`,...

In mathematics, we use three dots at the end of a sequence to indicate that the pattern continues forever.

Mario has learned that there are two ways for him to figure out how many leather strips he needs to decorate belts of different lengths. He can lay out the pattern physically with strips of leather, and then count the number of strips that make it up. For a `10`-triangle pattern, that adds up to `21`.

`3 + 2 + 2 + 2 + 2 + 2+2+2+2+2 = 21`

He can also write down the mathematical sequence that describes the strip pattern, and count out the number of terms to find how many strips are needed. The `10text(th)` term in the sequence, which represents a `10`-triangle pattern, is again `21`.

`3, 5, 7, 9, 11, 13, 15, 17, 19, 21`,...         

There’s also a third way, a shortcut to the counting strategies, one that uses multiplication. Mario’s belt pattern starts with `3` and then grows by repeatedly adding `2` strips. A repeated addition sequence like this can be described by multiplication. He can write a simple equation that describes the pattern: `3 + 2(text(the number of additional triangles))`. To make a `10`-triangle pattern, Mario starts with `3` strips and then adds `2` more strips `9` times. `3 + 2(9)` is `21`, the same answer he got from the slower counting out methods.

`3 + 2(9) = 21`

Some patterns are simple, showing a constant change from one term to the next. For example, Mario’s belt decorations display a simple pattern: they add `2` more strips for every new triangle. Many other patterns change in more complicated ways. Let’s look at an example.

Sophie has `48` pages to read for her English homework. She begins reading at `6` p.m. The first hour she reads half of the assignment. Then she starts texting her friends, and she begins to read less and less as a result. Each hour from then on she reads half of the remaining pages.

Let’s write the mathematical sequence that describes Sophie’s reading pattern. Each term in the sequence will be the number of pages left to read.

She starts with `48` pages, then reads half of those in the first hour. So at `7` p.m.,  Sophie has read `24` pages, and has `24` left to go.

`48, 24` 

By `8` p.m., Sophie has read half of `24` pages, which means she has `12` pages left to read.

`48, 24, 12`

By `9` p.m., half of those `12` pages are read, and the other half are not—that’s `6`.

`48, 24, 12, 6` 

By `10` p.m., she’s read half of the `6` pages, leaving her `3` more to go.

`48, 24, 12, 6, 3`

By `11` p.m., Sophie has read another half of the `3` pages she hadn’t finished. She decides to go to bed and finish the last `1 1/2` pages over breakfast. Good night, Sophie!

`48, 24, 12, 6, 3, 1.5`

Sophie’s reading formed a sequence in which each term was half of the term before it. These types of patterns may be a little harder to recognize, but they can still be described mathematically and predicted using inductive reasoning.

As we’ve begun to learn, mathematics provides a useful way of exploring patterns, whether they are as small and familiar as the face of a flower or as exotic as the shape of a galaxy. Using mathematics and inductive reasoning, both simple and complex sequences can be described precisely and then predicted beyond what is seen and counted.