In the Make a Table strategy activities, we suggest making a gallery of all different types of tables you could make for one math problem. That’s because different tables can show different patterns and help with different kinds of noticing. Here are some ways tables can be different:

- Some tables show calculations, others just show values/results
- Some tables are labeled, some are just numbers
- Some tables are horizontal, some are vertical
- Some tables have lots of columns, some have just a few
- Sometimes the columns in a table are in one order, sometimes they are in another

This week’s Pre-Algebra Problem of the Week, *Falling Leaves*, inspired many students to make tables. Lots of different tables were made. In a table gallery walk, we encourage you to share one thing you noticed about each table that you really like, and things that you wonder about the table. Here are four different tables from this week’s PoW, and a noticing and wondering about each. In the comments I’m excited to hear what you noticed and wondered, and how you might compare/contrast the different tables.

**Table 1, Avery and Nolan from W.P. Sandin:**

I noticed: the “Drop per minute” number is always one more than the “Total leaves on the ground” number in the previous row. The 3 columns made it easy for me to see that pattern.

I wondered: what formulas were used to calculate each value. For example, if they used a computer to make a spreadsheet, what calculations would they tell it to do.

**Table 2, Griffin from Patriot Elementary:**

Mins.

1: 1 leaf

2: 1(leaves on ground)+1=2

3: 3(leaves on ground, 1+2)+1=4

4: 7(leaves on ground, 3+4)+1=8

No need to continue, the pattern is doubling

5: 15(laves on ground, 7+8)

I notice: I could see the calculations being done to get each new value, and it really showed the pattern that each row you were adding a power of 2 + one less than a power of 2 + 1, and so it makes sense you would get a power of 2 as the final sum.

I wonder: if adding a column for the leaves on the ground quantity would make the table a little easier to read? Right now I am a bit confused which calculations are referring to which value.

**Table 3, Henri-Michael, Goshen Elementary: **

1 min = 1 leaf = 20

2 min = 1+1=2 leaves = 21

3 min = 1+2+1= 4 leaves =22

4 min = 1+2+4+1=8 leaves =23

I notice: writing the power of 2 that each row sums to helps me see that the exponent is always one less than the number of minutes, which isn’t always easy to see. I also noticed that the sums, written this way, are all sums of powers of 2 as well!

I wonder: if labeling more of the quantities might make it clear what the middle calculation represents? For example labeling it “leaves already on the ground” and the calculations labeled leaves as “leaves falling this minute”

**Table 4, Inteshar, Ravita, Ishnad, and Vasiliki from IS 141Q The Steinway School:**

Minute On the Ground

- 1
- 2
- 4
- 8
- 16
- 32

I notice: The doubling pattern is super clear in this table since all we see is the column with the powers of two.

I wonder: This table is great for seeing the way the minutes relate to the end results of the calculations, but it’s not as easy to show how the end results were achieved. What other quantities were used to calculate the number of leaves falling? How could we organize those quantities in other columns or by showing them in a formula or calculation?

I noticed that many of my kids had some version of the charts listed above.

I wonder why so few made the connection to the power of 2. I wonder what was the pivot that wasn’t being made to see that connection.

One thing I wonder is, if a student sees “hey it’s doubling every time!” if it isn’t easier to just push *2 twelve or twenty-four time? What if we had asked about the 100th or 1000th minute?

There’s also a connection to the mathematical practice “Make Use of Structure” — how do we help students know that “doubling each time” means powers of two and that it is really useful to represent that structure?

One thing that might make this problem hard is it sounds like it’s going to be a sequence that grows more slowly — the description talks about adding 1 to the number on the ground. It sounds like Fibonacci or an arithmetic sequence. The fact that it’s geometric might be harder to swallow.