Patterns in Context Part A: What Do You See? (40 minutes)

Problem A4

Here is a recipe to perform on a number:

What does this recipe do to the numbers from 1 to 10? Record your answers in a table. Can you explain how this is related to the patterns you saw in the table from Problem A1 (below)?

Problem A5

Comment on each of the following descriptions of the table given in Problem A1. Do all of these descriptions produce the same list of outputs? Are all of these descriptions valid for the table?
Note 4

1. As the input increases by 1, the output increases by 4.
2. If you add 2 to 1 and double it, you get 6. If you add 3 to 2 and double it, you get 10. If you add 4 to 3 and double it, you get 14. Or, if you add the input to the next input, double that, you get the output.
3. The units digits are in the sequence 6, 0, 4, 8, 2, so the next number would be 26, then 30, 34, 38, and 42, and then 46, 50, 54, 58, 62, etc.
4. To get the output, multiply the input by 4 and add 2.
5. To get the output, triple the input, then add 2 more than the input.
6. After 6 as an input, the output numbers repeat over again: 6, 10, 14, 18, 22, 26, etc.
7. After 6, the output numbers remain constant: 26, 26, 26, etc.

Problem A6

Which number comes next in this sequence: 1, 2, 3, … ? Find as many different answers and explanations as you can.

Problem A7

Let’s take a closer look at the four skills for thinking about patterns.

• Finding patterns involves looking for regular features of a situation that repeats.
• Describing patterns involves communicating this regularity in words or in a mathematically concise way that other people can understand.
• Explaining patterns involves thinking about why the pattern continues forever, even down the line in cases you haven’t looked at.
• Predicting with patterns involves using your description to predict pieces of the situation that aren’t given.

Look back at Problems A1-A6. Describe how you use these skills in each problem. In other words:

• Finding is observing the pattern you see
• Describing is putting what you see into words or symbols
• Explaining is figuring out why the pattern continues
• Predicting is using your description or rule for a new value

Note 2

The content of this session centers on patterns. Go over the processes of finding, describing, explaining, and predicting with patterns. These skills help us to understand and make sense of situations, something to remember while working on problems throughout the session.

Note 3

Groups: Work in pairs on Problems A1-A4. When you’re finished, share your answers to Problem A1, coming up with as many different ways of describing the patterns as possible. At this point, we’re not necessarily interested in descriptions given in symbolic form. Descriptions such as “each output is 4 more than the previous” are valid because there are many ways to describe patterns, each with its own merit. At some point we will want to make a case for using a variable as a way of being concise, but that will come later.

Note 4

Read Problem A5.

Groups: Talk about each part of the question as a full group. Some people may be troubled by parts (c), (f), and (g), rejecting these scenarios as being far-fetched. An important point we are making in this session, though, is that many different rules can describe a table of data. When the data is attached to a context or situation, we can be more certain of an associated unique rule. Therefore it is important to see parts (c), (f), and (g) as possible descriptions.

Problem A1

Some possible answers: each output number is 4 more than the last; the output numbers that appear are all the even numbers that aren’t multiples of 4 (starting with 6); the output number is 2 more than 4 times the input number. Also, adding one input to the following input yields half the first output.

Problem A2

You can’t be sure, because the pattern is not completely specified, but it would be likely that the 100th number is 402. This follows the third rule listed above — that the output number is 2 more than 4 times the input number.

Problem A3

Again, you can’t be completely sure, but it would be likely that the 25th number is 102, because 2 more than 4 times 25 is 102. Following the same pattern, 1004 would not appear in the output column, since 1004 is not 2 more than 4 times any whole number.

Problem A4

Example: Pick 7, and then follow the algorithm. 7 >> 21 >> 19 >> 38 >> 44 >> 30 is the output. The numbers at the end are the same as the pattern described in the table. Here’s why: Pick n instead, which stands for a variable number. Follow the algorithm. n >> 3n >> 3n – 2 >> 6n – 4 >> 6n + 2 >> 4n + 2 is the output, which is the rule described in Problem A1.

Problem A5

Some comments:

1. This describes the way the table is built, but doesn’t say what values the table begins with.
2. If the first input is n, the sum of the first input and the next is n + (n + 1) = 2n + 1. Doubling this gives 4n + 2, which is the formula for the table.
3. This is a good digit-based description of the table.
4. It’s 4n + 2 again. An efficient, closed-form description.
5. Triple the input is 3n, then 2 more than the input is n + 2, so the sum is 4n + 2.
6. There’s no way of being sure that 4n + 2 is the correct pattern. It is definitely not the only pattern that starts 6, 10, 14, 18, 22, 26, …
7. This is another example of a different continuation to the pattern. Because the given table ends with 26, this is a valid continuation.

Problem A6

The next number could be any number with a justified pattern. 4 is the clear choice, but so is 5, the next Fibonacci number, or 10, the next number you would get when counting in base 4, or 1, the next beat in a waltz. A formula could be found for any 4th number in that sequence.