Some daily highlights from our Camp
In measuring the body proportions of some cartoon characters using rubber bands, we had to fold some paper. (It’s a long story, but your girls can explain.) In folding our papers, Laila had an interesting question. “The top paper is folded in fourths, and the bottom paper is folded in fourths, but since the top is way smaller than the bottom, maybe I should call it eights, or something else?” They discussed this for several minutes, and I then summarized the discussion by saying, “Maybe there is no such thing as ‘one fourth’, but maybe it always has to be one fourth of something.”
Using Cuisenaire Rods, we made models of various halves, thirds, and fourths, and I asked the girls to arrange their answers in a way that was both beautiful and showed the logic of the math. This team was obsessed with the idea of the fractions portraying a fireworks, and I went along with that. We discussed how that idea might dictate the placement of the various fractions, and they came up with an imaginative answer.
Enough photos of papers and gizmos. This is a candid of Meliya in pure thought. I showed the girls a way of visualizing multiplication, then we practiced on some problems such as 13 x 14 = ? Although we started solving these using wooden gizmos, the room became eerily silent when they solved using nothing but their imaginations.
The point of this was not to learn yet another way of multiplying, but to demonstrate that multiplication can be pictured. That is a valuable lesson that they will use this year, although these particular problems are harder than what they will see in school. That picture of multiplication is directly related to the “lattice method” that they’ve already learned in school, although the students don’t realize that yet.
DAY TWO: Math Field Trip
I can’t believe that I hadn’t noticed this before, despite several previous visits to the exhibit. It looks at first glance as if each of the smaller rectangles in this work is 1/20 of the whole, but I was startled to notice that the rectangles are subtly different. Why? What’s going on? Wooden rulers appeared, and 6/9 of the students rushed to measure the rectangles.
Figuring out ratio of length:width by subdividing. Sophie was the first to start doing this intuitively, but when I started referring to this as “Sophie’s trick”, there were several protestations of “I did it too!”
2 cups make a pint
2 pints make a quart
2 quarts make a half-gallon
2 half-gallons make a gallon
Although this list is conventional for us, if you think about it, there is something odd with it, something that breaks the pattern.
Our language has lost the older term for “half-gallon”, which used to be called “pottle”. Our early folk measurement system was elegantly based on the powers of 2, and the units continued far less than a cup and far greater than a gallon:
…. 2 jills make a cup, …. 2 gallons make a peck, 2 pecks make a pail, ….
I used this today as an introduction to thinking multiplicatively instead of additively. In other words, what is the next number — 2, 4,… ? Fourth graders will say that the pattern continues “2, 4, 6, 8…” and while that is true enough, fifth graders should start to see that there is another possibility, “2, 4, 8, 16,…” In the adult word, the second answer is far more important.
In the photo above, students are solving, “How many gallons in a hogshead?” Ten minutes later, the answer is 256.
Along the way, we again bumped into reciprocals. “If 16 cups make one gallon, how many gallons make one cup?” We had had a variation of this problem on Monday, when Stella complained, “This is making my head explode.” Although I haven’t formally taught reciprocals since then, her head is doing a lot better now. She raised her hand to answer the question, and then matter-of-factly said the correct answer. By Friday, I predict she would answer the question with some adolescent impatience, as in how could I be so dumb as to even ask her the question?
Amanda and Leila solved the problem in a surprising way, although the fact that I call it “surprising” may just mean that I don’t understand how they think. It also meant that they were very happy.
When Hannah and Nico (not pictured, sorry) worked on the problem, they realized the answer would be 16 x 16, and Hannah asked for paper and pencil to figure that out. But I reminded them, “Wait a second — we just did something like that the other day. Can’t you just solve this by imagining the picture instead of always running to paper and pencil?” And so they did.
We ended the morning by developing a visual understanding of the four basic operations with whole numbers, and then extending that visual understanding to fractions.