Experimenting with 3-act Activities

We’ve heard so much about 3-Act activities. Here’s this week’s experiment, with our verdict at the end.

Day 1…

Does \$1 million fit in a briefcase?

Show this GIF:

Ask… “Would \$1 million fit into a normal briefcase, and if so, could an average adult carry it? Assume the bills are all 100s.

When students asked for more info, we looked up the following  (on the internet):

• Weight of one \$100 bill:   one gram
• Height of a stack of 200 of such bills:  1 inch
• 1000 grams = 1 kilogram = about 2.2 pounds
• A fit adult can lift about 50 pounds

Then we turned them loose to work on this. Here are some of the strategies they used:

1. Draw a briefcase outline on your desk and see how many stacks you can fit in. They thought about 12 stacks would fit. 12 times 200 bills = 2400 bills for the first inch = \$240,000 per inch of depth. So 4 stacks deep (4 inches) should fit.

2. Do lots of multiplication (Not sure where this one is headed)

3. Draw a briefcase on the white board and lay bills side by side.

4. Each textbook weighs 1,600 grams (weight is printed on one of the first pages). So that’s 1,600 of the \$100 bills, or \$160,000. How many of those to make \$1 million? Can we lift it?

5.Use a binder that’s close to the right size

UNFORTUNATELY,  much of the work handed in looked like this:

We’ve spent so much time teaching careful word problem modeling and intricate fraction drawing, so this was disappointing. But good feedback.

So for Day 2, we decided to focus on problem solving AND good mathematical form.

One of us (Kathleen) felt students could focus on problem-solving AND neat showing of work.
One of us (Corrinne) felt that we should focus on one thing at a time, so she offered to record the thinking of students who were having trouble doing both (thinking and writing!) We’re still not sure who was right.

Day 2…

Which cup holds the most money?

We had to go to the bank for this one.

It didn’t seem right to just use photos, so we bought rolls and rolls of coins (and now we’ll have to take them back to a coin machine or something!)

We weighed each cup, wrote the weight, and let the students poke at the coins.

Then we gave teams each 4 baggies: Each bag held TEN coins, and was labeled with its weight.

• 10 Quarters: 60 g
• 10 Dimes:  23 g
• 10 Nickels: 50 g
• Ten pennies:  23 g

There was a lot of confusion at first. Most students would not have been able to solve this alone.  Kathleen interrupted the class after five or ten minutes and pointed out the reason they were getting lost:  They were not including UNITS at each step.

If you say “1 = 50 and 10 = 500, and 1000  = 10”, you’re soon totally lost. No one else knows what you mean, even you!

However, if you say “One nickel bag = 50 grams, so TEN bags = 500 grams,  then TWENTY bags = 1000 grams, so that means there are 20 bags of nickels in the cup. 20 bags have 200 nickels (ten per bag), so 200 x 5 cents = 1000 CENTS, or \$10.” Voila! It takes a little longer to write, but it makes sense.

We liked that this was a fairly easy-access activity.

Everyone could do some part of it –

even if it was only one denomination.

Here’s one student’s long road to

figuring out what 1000 pennies is:

And Kathleen wants to point out that offering more group points for showing work clearly led to much more decipherable submissions:

We chose these activities for a couple reasons:

1. We hoped they would reinforce multiplication by powers of 10. (There was a constant need to multiply or divide by 100, etc.)

2. We hoped it would spark interest in problem-solving. 3-Act activities are inherently motivational.

So how well did they meet our goals?

VERDICT:   We’ll give it a  B+

PLUSES:

Much of these two days was great.

1.Yes, there was engagement. The class was buzzing, everyone was trying something. Someone commented that “surely the hour can’t be up already”.

2. Yes, there was perseverance. Since the groups want points for their group, they kept at it – and some of the best ideas (testing whether we can lift that weight in textbooks, for example) came from students who don’t usually lead, don’t usually volunteer ideas because their classmates are ‘faster’.

3. There was use of mental multiplication by powers of ten, but less than we had hoped.  A lot of multiplication was done by hand, even when multiplying by 100. Good feedback, anyway.

4. We were pleased with the improved recording work done on Day 2, and the willing recognition of the students of the fact that “Maybe it helps to write out what you’re doing”.

CAVEATS:

1. There is definitely room for these explorations in our curriculum, but they’re not instructional, they’re good for review, for motivation, for perseverance. We’ll still use some direct instruction, especially visual models, followed by long recurrences of practice.  We’ll still use other motivational learning components like games and puzzles.

2. And, perhaps most importantly, we’ll still use investigations that have NO real-world application. Investigations that rely solely on the beauty of math itself. We strongly believe that some things about math are inherently interesting.  Why do consecutive odd numbers add to squares? Can you see it? Draw it?  Why do some patterns repeat and others don’t? Why does 0.99999…  equal 1? What does base 6 tell us about base 10? Why would aliens recognize prime numbers?

Math is not just a tool that is handy for science, commerce and medicine, although these roles give math relevance and respect.

No, it’s more than that. Math is humanity’s struggle with the unknown, our singular attempt to derive meaning from everything around us. It challenges our thirsty intellect, and leads us through hours – days – years – of conceptual wondering.

We’re big fans of that sense of wonder.

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