Scale Model Solar System Complete!

This morning I guided the 3-4 graders through assembling our  scale model solar system.  I wanted them to really think about how to make a scale model, so I returned to each student the graph they had made last time  and I asked them to use the graph to figure out where they would put their planet, given that I had put Teacher Moné's beautiful Earth poster 2.5 meters from the Sun poster.  Of course, I found that I needed to break this task into smaller chunks for them to process.  We began by revisiting some of the steps we had done last week. Each child identified his/her planet on the graph, read its distance off the graph, and then we thought about what that distance means.  For example, Jupiter is at a distance of 5 on the graph.  Five what? The graph doesn't say.  But the graph itself is a scale model of the solar system.  We don't really care what  the actual distance is because we are simply stretching this scale model to become a larger scale model which will fill the school.  All we need to do is choose a reference point and stretch everything else accordingly.  The graph made this easy because it shows Earth as being at a distance of 1.  So if Jupiter is at 5, we simply need to put Jupiter 5 times farther from the Sun than Earth is from the Sun; in other words 5x2.5 meters or 12.5 meters.

To help the kids visualize this, I took a rubber band and marked three dots on it, representing Sun, Earth, and Jupiter.  This is a scale model much like the graph (if we ignore the vertical dimension of the graph).  If I stretch the rubber band, will Jupiter still be 5 times more distant from the Sun than the Earth is from the Sun?  Some kids said no and some said yes, so we took a vote.  Having to commit to a vote made the kids think harder and they voted overwhelmingly yes.  After the vote I did stretch the rubber band and I did get a bigger scale model.  In principle, if we got a really long rubber band, I could mark all the planets' distances at the scale of the graph and then stretch it out to get a giant scale model as big as the school, and that would tell us where to put each planet poster.  But since that's impractical, we do the math instead.

This seems to have been more or less the right level of conceptual challenge and the right level of math for the kids.  They found it a bit of a challenge, but a doable one that became satisfying rather than frustrating. After looking over each child's computation, we practiced some metacognition.  Alex was concerned that his number didn't make sense given what he knew about the relative positions of the Sun, Earth and Venus.  It turned out that he was misinterpreting his number as the Earth-Venus distance, but the point was a really important one: always check that your numerical results make sense! I have had so many students make a mistake punching numbers into a calculator, and get a number that obviously doesn't make sense given a moment's thought, but blithely write down the number as if any number displayed by a calculator must be correct.  In this case we wrote out the multiplication rather than use a calculator, but the same principle applies: check that the results actually make sense! This goes not only for numbers that you compute, but also for numbers that other people compute for you.

An especially effective way to double-check your number is to perform some completely different procedure; if you just perform the original procedure again, you may easily make the same mistake again.  So I thought of a way we could all check our numbers without recomputing anything.  I made a list of the students' results, starting with the closest planet and proceeding outward.  If the distance numbers didn't increase steadily, that would be a smoking gun indicating a mistake.  And we did find a mistake this way, so it was instructive.

Once we had our final numbers, we split into groups to measure off the distances and attach the posters to the walls.  We couldn't quite fit Neptune into the school grounds, and Orcus wasn't even close, but we put them up at the far end with a note saying where they should really be.  Even after choosing a scale so large that the orbit of Neptune was just outside the fence, the sizes of the planets are really small, smaller than a grain of sand for most planets.  Even Jupiter is only 2.4mm across.  Space is really big!

Looking at the finished product, I am really happy we did it and spent enough time on it to do it right. We certainly appreciate the solar system much better now, but we also learned new ways of thinking.