Thought for the day: Some Creative Math Prompts are really problems without directions. Students’ first job is to figure out what the problem is!

Concepts: prime numbers; exponents and their properties; prime factorizations; analyzing and extending patterns

Beginning

I notice that each whole number from 1 through 15 has another number paired with it, but 0 does not. (We’ll call it a “code number.”)
I notice that all of the code numbers have only the digits 0 , 1, 2, and 3.
I wonder if codes for other numbers will ever contain digits greater than 3.
I wonder if the code is related to a different number base.
I notice that the powers of 2 (1, 2, 4, and 8) have "codes" of 0, 1, 2, and 3.
I wonder if there are patterns for powers of 3.
I notice a lot of patterns that look like they will work but then fall apart.

Exploring

I notice that the codes of prime numbers look like powers of ten.
I notice that when you double a number, you add 1 to its code.
I wonder what happens if I triple a code number.
I wonder why 0 doesn't have a code number.
I wonder if the code for 16 is 4 (extending the pattern for powers of 2 above).
I wonder how to find code numbers for composite numbers that are not powers of a number.

Creating

As they notice and wonder, students may create their own strategies for finding codes. Many of their strategies may flow out of things that they have wondered about. They may also

Create methods for making the code work for "tricky” numbers such as 1024.
Create systems for representing fractions in code.
Create systems for representing square roots in code.
Create new codes for other students to crack.

Reflecting and Extending

I notice that this code is all about prime factorizations.
I notice that this code is based on a new kind of place value.
I wonder why the code number is always the same, no matter how I factor the number.
I wonder what happens to code numbers when I divide the original numbers.
I wonder if writing prime factorizations in exponential form will give me new ways to think about the code.
I notice that the code is connected to properties of exponents.
I wonder if I could find code numbers for fractions.
I wonder if I could find code numbers for square roots.

Notes

You can find a more structured activity built around this image in in Exploration 10: Mathematical Mystery Code from my book, Advanced Common Core Math Explorations: Factors and Multiples.

This code in this image can be exasperating at first, because it is full of many "almost" patterns that begin and then fall apart! There are many ways to think about it. In the end, problem-solvers usually arrive the following:

• Prime numbers look like powers of 10 in the code. As the prime numbers increase, the powers of ten increase.

• To find the code number of a composite number, add the code numbers of its factors.

Part of the magic of the code is that you may factor the number in any way you like. For example, to find the code for 24:

24 = 2 • 12                    1 + 12 = 13
24 = 3 • 8                     10 + 3 = 13
24 = 4 • 6                     2 + 11 = 13
24 = 2 • 2 • 2 • 3         1 + 1 +1 + 10 = 13

The code number is 13 no matter how you factor it!

What's going on here? Actually, the code is a different way of showing a number's prime factorization (PF)The rightmost digit shows the number of 2s in the PF. The second-from-the-right digit shows the number of 3s in the PF. The next digit to the left shows the number of 5s in the PF, etc. It's a PF place value code! In fact, you can actually see the code in the exponents when you write a prime factorization in exponential form. You just need to begin with the largest prime factor on the left and show primes that are “missing” by including them with a 0 exponent. For example, the code for 84 is 1012 because

Algebra students can explore some of the ways in which the code illustrates properties of exponents. There are many additional questions to explore as well:

• Can you explain clearly why the rule "factor the number and add the codes" always works (and always gives the same answer no matter how you factor the number)?

• What happens to the code numbers when you divide? Why? (Does this suggest ideas for writing fractions in code?)

• Find the code number for 1024. What problem do you see? Can you think of a way to fix it? What is the next number that will have this problem?

• Why can't you use the code to represent the number 0?

• How might you use the code to represent fractions?

• How might you use the code to represent radicals such as the square root of 2 or the cube root of 6?

• Is it possible to represent all irrational numbers with the code?