Building place value and number sense with letter problems

I saw this wonderful problem from the 2009 AMC 8 last night:

Problem #24 from the 2009 AMC 8

Here’s the problem:

The letters A, B, C, and D represent digits. If AB + CA = DA and AB – CA = A, what is the value of D?

Many questions similar to this one appear on math contests for younger kids. At first these problems didn’t really stand out from all of the other problems, but lately I’ve come to see them as a neat way to get kids thinking about place value and number sense in general.

I also find the difference in approach between my older son (6th grade) and younger son (4th grade) to be fascinating.

Here are their approaches from last night.

My older son’s initial approach is to try to calculate the value of the individual digits:

 

After he finds the value of the various digits, I asked him to find a different approach that didn’t require so much calculation. Knowing the answer already helped, of course, but moving away from calculation also allowed him to see the place value ideas more clearly:

 

Now for my younger son – his initial approach involves much less calculation. He doesn’t have quite as much mathematical sophistication as his older brother (since he’s 2+ years younger) and he struggles a little to communicate the ideas that he’s seeing:

 

At the end of the last video he arrived at the idea to try out a few numbers. Once he starts down that path the place value ideas sort of emerge from the shadows and he finds his way to the end of the problem relatively quickly:

 

So, hopefully a nice example of how kids approach this type of arithmetic problem. Hopefully the example also shows how this type of problem can help kids think about place value and build number sense.