Using an AB Calculus question as a estimation problem for kids

Last night I was curious what the AP calculus questions looked like and was flipping through the multiple choice questions from the 2012 AB calculus exam.

This one caught my eye because I felt that the wrong choices weren’t selected very well and the correct answer was obvious from the choices:

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Despite not liking the question so much for a calculus exam, I thought it would make a pretty neat estimation problem for kids. My older son was looking at exponentials and logarithms this week anyway, so using this question as an estimation problem sort of fit in naturally with what he was doing anyway this week.

I started off today’s project by introducing the problem and asking the kids how they would approach it.

My younger son had a bit of a difficult time understanding the problem – which wasn’t a big surprise. My older son wanted to start by estimating what the square root of e was. Not the starting idea I was expecting, but it made for a good estimating problem:

Now that we we’d guessed that e^{x/2} would look a lot like 1.65^x we decided to draw the region.

My younger son thought that the region would be a trapezoid – my older son thought it would be more of a curved shape.

Now that we had the shape drawn, we could estimate the area. We actually used the idea that it was nearly a quadrilateral to make that estimate.

Finally, we used our estimate of the area (3.71 square units) to see if we could identify the correct answer from the choices given in the original problem:

Two nice problems shared by David Coffey and Federico Chialvo

I saw two neat problems for kids on twitter this week and went through them with the boys this morning.

First up, the problem shared by David Coffey:

I couldn’t find any multicolored items to use, so we used snap cubes instead of cards. The first thing that we tried to do is see if we could solve the problem at all. The boys had a couple of ideas at the beginning, and it was interesting to hear the discussion. My younger son noticed that a procedure my older son was following was going in a circle, for example. My older son noticed that the last step would have to involve swapping out 3 orange cubes for blue cubes. That idea will help us find the minimum number of steps in the next video.

Eventually we found a way to swap the orange and blue cubes in 5 steps.

Next we tried to see if there was a way to swap the cubes in fewer than 5 steps. My younger son noticed that it would take at least 3 steps for a complete swap since you have to move 7 cubes, 3 at a time. He then noticed a way to make the complete swap in three steps.

At the end we compare both procedures that we found. This was a nice little activity.

The next problem we tackled this morning was from Federico Chialvo

The challenge on this problem is that my older son knows how to calculate each of these areas from studying geometry. I had him do that calculation at the end, but first I wanted to hear ideas that didn’t involve calculation. The “no calculations” requirement turned this into a fairly challenging task.

I split the 9 minute conversation into two pieces just to make it easier to watch. Their geometric instincts are to chop the hexagon into triangles and rectangles. Chopping up the hexagon in this way is interesting, but since you get 30-60-90 triangles my older son’s urge to calculate is hard to suppress 🙂

As we try to move away from calculating, they notice that the triangles they are looking at might actually form a square that is roughly equal in size to the original square. Eventually we see that the remaining rectangle has an area that is a little bit more than half the area of our original square. All of that information comes together to produce an estimate that the area of the hexagon is bit more than 2.5x the area of the square.

We finish up with a quick calculation of the area of the hexagon by my older son. The calculation shows that the hexagon is 6*\sqrt{3} times the area of the square. So, our original estimate of 2.5 times wasn’t that far off – yay!

A bit of a struggle with estimation

My wife was running a few errands with my older son this morning, so I decided to do today’s Family Math with just my younger son. The topic today was estimation and the specific problem we were thinking about was this:

If we drive from the Museum of Science in Boston to the Exploratorium in San Francisco, how many time will the wheels in our car turn around?

Fun question, and we began by talking about the different pieces of the puzzle we’d need to solve this problem (with a little help from the math cat):

After talking through the problem, we went outside to see if we could measure the circumference of our car wheel. We were unable to find a wheel of the same size (we tried a garbage can, a christmas tree stand, and a bike wheel) so we had to measure the diameter. I wanted to also make a chalk mark on the tire and measure the distance along the ground when it made one rotation, but I accidentally left my keys in my wife’s car. Oh well . . .

Next we came back inside to tackle the problem. In this segment my son was having trouble estimating some of the numbers we’d encounter in the problem, so instead of proceeding straight to the solution we went back outside.

We went back outside to take a closer look at some of the numbers in our problem. I thought it would be helpful for him to see what 88 inches looked like in relation to the length of our driveway, for example. That seemed to help him get a better estimate of how many times one of the car wheels would turn leaving our driveway.

The second trip outside seemed to help him get a better understanding of some of the distances involved in the problem, but he still had a tough time getting a good estimate of the number of times the wheel would turn going from Boston to California. He now understood that his estimate of 500 from before was too low, but he wasn’t sure how much to increase that estimate. We did a series of simple approximations and arrived at an estimate of 2 million.

Finally we get to the point where we plug in some numbers. We haven’t done much with unit conversions, so I had to help him through that a little bit. Eventually we arrived at a number of roughly 2.2 million for the number of times the wheel would turn.

So, a little bit more of a struggle with this one than I’d intended, but still a lot of fun. The lesson for me on this one is that I need to do a few more exercises that involve estimation and unit conversion. Hopefully I’ll remember to incorporate an exercise with those characteristics as least once per month.

Measuring Pi

After yesterday’s Family Math project I was thinking about a project with spheres so that we could talk about the area of the spherical caps from our printed shapes.  This morning i changed my mind and thought that a slightly more laid back project was in order, so we spent the morning trying to measure \pi.

The first thing that we talked about was basic definitions.  Just trying to set the stage for thinking about geometry, especially since I’ve not really spent much time talking about geometry with my younger son:

After getting through the definitions we started measuring.   We started with a can of chickpeas and then talked for a bit about why doing the same measurements with a cube would be different.

Now we moved on to some larger circles.  This required a larger area, so we moved to the garage.   Our first attempt here didn’t go so well as our estimate for \pi was about 3.5.  I wasn’t too disappointed, though, since learning that measurements don’t always produce what you expect is an important lesson.

Our last prop was a bicycle wheel.  This experiment required a little bit more room than our camera could handle, so we split it into three pieces.  We got an estimate for \pi that was a little low, but it the best estimate of the bunch.    After we finished the calculations on this one we talked through a few of the aspects of our measurements – is it easier to get an estimate for \pi with a large circle or a small circle, for example.

Definitely a fun little set of experiments.  Fun to see the boys rolling up their sleeves and taking a few measurements, too.