Advanced Common Core Math Explorations: Factors and Multiples
Exploration 1: Building Blocks
This exploration is my personal favorite from the ACCME books. Over time, I have introduced the "building blocks" idea in three different ways.
- Use colored blocks to build each natural number in succession (beginning with 2), attaching blocks to represent multiplication. Introduce a new color when a number cannot be built by attaching blocks of colors already in use. See my article "Building Numbers from Primes" from the NCTM journal, Mathematics Teaching in the Middle School.
- Start with a 10 by 10 grid, and use a variation of the Sieve of Eratosthenes. Try to imagine how this might work! Repeated prime factors must be dealt with in order to create the prime factorizations: Every 4th square receives a second white block; every 8th square receives a third white block; every 9th square receives a second red block, etc.
- Give students a half-complete grid with no numbers on it and no explanations, and ask them to figure out what is going on. I and my students have enjoyed this approach the most. It is the one used in Problem #1.
The first approach may be best for typical students. I personally like the idea of the second approach, but I have never managed to make it work particularly well with students. (Please let me know if you find a good way to do this!) The third way works well with more advanced students, but with sufficient time and guidance may be appropriate for all leaners.
I often give Problem #1 at the beginning of the year to my advanced students as a way to establish a classroom culture of curiosity, collaboration, respect, effort, and patience. All of these attributes are needed for success, and this problem is a fun, non-threatening way to begin developing productive habits for the rest of the school year!
Most of my sixth grade students never reach Problem #2, because the Problem #1 takes us a few days! I could choose to rush them through it, but we have more fun and learn more if we take our time. Every year or two, I have a student who will "decode" the puzzle quickly - perhaps within 10 minutes or so. (Once, I had a sixth grader who did it in seconds!) Problem #2 comes in handy in this case, or later in the year after the kids have become very comfortable with the building blocks.
After we have finished, I make a copy of the completed, numbered grid for students to keep for the rest of the year. We use the grid to identify prime numbers, devise mental multiplication strategies, identify factors of numbers, find greatest common factors and least common multiples, simplify fractions, find common denominators, identify terminating and repeating decimals, explore properties of exponents, etc. Sometimes, we even use the blocks to investigate square roots (see Exploration 10: Factor Blocks and Radicals).