It is hard to think of a test question more widely mocked than the “Hannah and her sweets” question floating around twitter this week:

I don’t like this question at all as a standardized test question, but I thought that in the non pressurized setting of our living room that it would make for a fun project. This project turned out a little better than I expected because the boys actually stumbled on to the solution and were surprised to see that you could find the equation without knowing the value of n. Definitely a good math lesson to learn in a project.

Anyway, we started off by looking at the set up for the problem. Even though I’d shown them the question yesterday when it was all over twitter, I asked them to try to guess what the actual question was going to be. Turned out that the kids came up with some questions that I think were better than the one on the exam. They also had some neat ideas about how to think about the actual question:

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Next up we got some snap cubes and took a look at the problem more carefully. It was sort of funny when the boys asked me how many yellow snap cubes to bring over – ha! Since I wouldn’t tell them, they brought over a bunch đŸ™‚

Right off the bat we got to talk about some important ideas in probability. Some examples of the ideas that came up in the discussion: Does order of selection matter? When do you add and when do you multiply probabilities?

One little surprise here was that my older son started writing down probabilities for picking orange and yellow blocks in terms of n. My younger son then noticed that the probability we are trying to calculate with the n’s has to be equal to 1/3. So, totally by accident the boys found themselves on the path to solving the actual question asked in the problem.

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Now with some of the introductory probability ideas out of the way, we moved on to solving the exam question. The solution involves setting two fractions equal to each other. It was really cool to see the boys work together on this problem. Around 1:30 in the video below they’ve got the equation from the problem written down – fun little surprise! I didn’t expect the project to go in this direction at all.

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I didn’t want to end with just the equation, though, so I asked the boys to figure out what n was. This is a slightly unfair question since my younger son hasn’t studied quadratic equations, so in this part of the project my older son does most of the talking.

Beyond the ideas of basic probability and quadratic equations, this section of the project is interesting as a number sense activity. Is was great to see them work together to figure out what the square root of 361 was.

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For the last part of the project we double checked that we’d found the correct solution. If the bag has 6 orange sweets and 4 yellow ones, is the probability of picking two oranges in a row equal to 1/3?

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So, while I think the original question is a pretty poor question for a standardized test and definitely worthy of the twitter-mocking it received, it did work really well for us as a math project. There’s lots of good math to talk about with kids in this problem – basic probability, basic algebra, and even a little number sense. It was also fun to see the surprise that the boys had when they stumbled on the quadratic equation included in the original question.

I don’t quite get the twitter fuss. I guess probability questions will always create controversy.

What do your sons think if you change the probability from 1/3 to 3/4?

I think the revised question will not get them back inside from building tree forts with their cousins – ha ha.

I am planning on taking them through AoPS’s Introduction to Counting and Probability this summer, so we’ll have a chance to revisit questions like this in the coming months.

Yes, I’m sure tree forts > borrowed exam questions and their derivatives.

As I’ve discovered watching kids, equations like these are highly susceptible to guess and check strategies even without Alg. I. First you can bound the solution set by noting if half the candies were orange total probability would be ~ 1/4 which is less than 1/3. So orange is more than half of the set. That means you only have to plug in 1 – 5 for the # of yellows. Starting from 5 and going backwards means you have to check 2 times. Starting from one and going forwards means you only have to check 3 times.

That’s pretty much the approach I was expecting my kids to take. I was very surprised that they didn’t try to check any numbers at the beginning.