Nate started by having us imagine that we are young students who know numbers and adding, but not any formulas.
Particpants used three different-colored blocks to explore how many ways there are to choose any two of them (3 ways).
They then determined how many ways there are to choose any one of them (also 3). There was discussion about why those two answers are the same.
It was also stated that there is exactly 1 way to choose all 3 blocks and 1 way to choose zero blocks.
Nate wrote the data in a line: 1 3 3 1
The next quest was to answer how many ways there are to choose one block from a set of two of them (2 ways). Again, there is exactly 1 way to choose both blocks and 1 way to choose zero blocks.
Nate also remarked that ther was 1 way to choose 1 block from 1 block and 1 way to choose 0 blocks from 1 block and also 1 way to choose 0 blocks from 0 blocks.
Nate wrote the data above the previous data so that each row represented the number of possibilities of choices for the number of blocks being used like this:
0 blocks 1
1 block 1 1
2 blocks 1 2 1
3 blocks 1 3 3 1
Participants then worked on answering similare questions for 4 blocks, and 5 blocks. The midsection numbers were known by most, but we were encouraged to explain why the answers make sense in light of blocks and choices rather than using formulas (looking above to realize why we can add the two adjacent numbers to get a solution). We had a few volunteers explain the relationships of sums of numbers in Pascal's triangle to blocks and choices as we add one more color.
Nate introduced the notation for combinations "n choose m".
|
|
|
|
| Carrie Wright introduced Nate Carlson | A group compares their ideas. | Tierra and Cassie work together. |
|
 |
 |
| A large group attended the session. | Peggy showed the patterns of even numbers in Pascal's Triangle. | Groups had plenty of time to share their thinking. |
Common Core Standards for Mathematical Practice
8.MP.2. Reason abstractly and quantitatively.
8.MP.6. Attend to precision.
8.MP.7. Look for and make use of structure.
8.MP.8. Look for and express regularity in repeated reasoning.