I ran into my first real problem with Khan Academy the other night.

Both kids struggled a lot with the

*Rate Problems 2*section in Khan. These are problems that require multiple steps to solve, and a more complicated mental picture than the earlier word.
I really like these problems. They exercise our understanding of the relationships between rates and times and other factors. They compel us to think conceptually about the big picture instead of just crunching the numbers.

The kids might solve this by working out how long it takes 1 person to paint 3 walls, then how long it takes 1 person to paint 1 wall, then how long it takes 6 people to paint 1 wall, and finally how long it takes 6 people to paint 7 walls.

I urge the kids to make a chart, fill in what they know, and then figure out how they can fill in additional boxes on the chart until they have what they need.

Every now and then, however, Khan throws up a variant of the problem like this:

After Jessica carefully worked it out, I showed her how this one was actually easier. If the number of people and walls are the same in both cases, then the answer is also the same. 7 people take as long to paint 1 wall each as 1 person takes to paint 1 wall or as 11 people take to paint 1 wall each.

“It’s like a freebie,” I said. “You don’t even have to do the arithmetic.”

This Jessica remembered. As she continued to struggle through these exercises, every now and then she would proclaim, “A freebie!” and I would immediately hear the happy sound that Khan plays when you get one right. But she was otherwise struggling, and she didn't succeed in getting five in a row before we called it a night.
The next day she took a Mastery challenge. She aced it easily, and when the report came up at the end, it showed she had “Mastered”

*Rate Problems 2*.
But she hadn’t come anywhere close to mastering it. She had just got lucky. The Mastery Challenge had served up a freebie.

Since I was watching, I promptly added more *Rate Problems 2*exercises to her list and continued to work through these with her. Actually I first took her all the way back to the earlier

*Rate Problems*topics. And eventually she really could do them all, and she knew it.