Here are some other strategies that are helpful to students.

This strategy is best demonstrated using the following problem:

Problem: The Birthday Gift

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.

Solution: The Birthday Gift

The money my aunt gave me is represented by this rectangle. I spent one-third for the CD, so I shade in one-third: I spent half the rest on the movies so I shade half of the remainder (which I notice is one-third of the whole). I then spent half of what was left, so I shade this portion as well: I had $6 left over, which represents the white part. In other words, the white part is 1/6 of the whole. That means that the gift was 6 6, or $36.

Questions to Consider: The Birthday Gift

• In what problem-solving situations might this (drawing a diagram and working backward) be a useful strategy?
• What might this technique offer the students?
• How can you support students in learning to apply this strategy?

To understand this strategy for organizing data, take a look at the problem and solution below:

Problem: My Favorite Number

My favorite number is a two-digit number, and it equals twice the sum of its digits. What is my favorite number?

Solution: My Favorite Number

Let's make x be the tens digit of the number and y be the ones digit. The value of the number is then 10x + y. The sum of the digits is x + y, and twice the sum is 2(x + y). These are equal, so we write 10x + y = 2(x + y), which becomes 10x + y = 2x + 2y. Then we collect the variables: 8x = y. Now we remember that both x and y are one-digit numbers. That means that x must be 1 and y is 8. The number is 18.

Watch a brief video segment (duration 0:22) to hear a reflection from Nan Sepeda, a middle school mathematics teacher, about using a simple problem to introduce a more difficult one.

In addition, here are some phrases that are commonly used to name techniques for solving problems:

• Look for a pattern
• Identify a sub-goal
• Check for hidden assumptions
• Guess and check
• Find a different way to solve
• Look for other solutions
• Change your point of view
• Act it out

  Look at low threshold, high ceiling problems