Sometimes starting with the given, creating a diagram, and working backward to fill it in is a useful technique.

"My favorite aunt gave me some money for my birthday. I spent one-third of it on a new CD. I spent half the remainder to take my friend to the movies. Then I bought a magazine with half of what was left. When I went home, I still had $6. How much did my aunt give me for my birthday?"

One way to solve this problem combines drawing a diagram and working backward.

I drew this rectangle to represent the money my aunt gave me. One-third went for the CD, so I shaded that much in using red. Half of the remaining amount I spent at the movies. (This was one-third of the whole). I shaded that in blue. I then spent half of what was left, so I shaded that in yellow. Then last part, in white, was what I had left, 6, and that's one-sixth of my whole rectangle. So the gift was $36.

Diagrams, beyond simple models of the problem, can also be introduced. In topics such as statistics, a tree diagram can provide a means to organize data and prompt insights into solutions. Here is an example:

There are four green chips and four blue chips in a bag. Without looking, remove one chip with your right hand. Do not look at the chip.

a. What is the probability that this is a blue chip?

Still without looking, remove a second chip with your left hand.

b. What is the probability that this second chip is blue? Remember that you do not know the color of the chip in your right hand.

A group of students working on this problem with guidance from their teacher arrived at this diagram:

Making a tree diagram organizes the data.

From the diagram, we see the following:

a) The probability that the first chip is blue is found on the bottom half of the first branch of the tree. The probability is 4/8, or 1/2.

b) The probability that the second chip is blue is found on the GB or BB branches of the tree. The sum of these two probabilities is: 4/14 + 3/14 = 7/14, or 1/2.

As you work through this technique, you should be aware of the steps you take. Do you understand why you add or multiply at different stages of solving a probability problem? Self-monitoring -- that is, checking your work and your understanding as you go -- is key. This is something both you and your students should employ.

Here is how one student explained his solution to the question, "What was the probability of getting two green chips?"

"The branches of the tree diagram show all the possibilities for removing two chips from the bag. I started by tracing individual paths in the tree -- first the top branch moving from left to right. Right away, I saw that the probability of removing a green chip first is 4/8, because four of the eight chips in the bag are green. Then I went to the next branch -- green -- and I could see that three times out of seven the second chip is going to be green. That made sense, because now there are seven chips left and of those four are blue, so the other three have to be green. But then I had to figure out the odds that both the first and the second chips were green. First I starting adding branches for each chip along the tree so I could see how many different ways you could get green-green. But then I noticed that you could just use the original tree and multiple the odds of the first green by the second green: 4/8 • 3/7 = 3/14."

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