What middle schooler doesn't get that warm and fuzzy feeling when we tell them we're going to play with fractions! Here's a small investigation to capture that mood of euphoria...

RULES OF THE GAME:

- No Calculators - No decimals!
- All fractions must be expressed in lowest terms

1. Write the fraction that divides the segment between A and B into two equal parts. How can we verify that this point satisfies the desired condition.

Complete: This fraction is the _______ of 1/4 and 1/3 and is ____-way between 1/4 and 1/3. The corresponding point is the _________ of segment AB.

2. Write the two fractions that divide the segment between A and B into three equal parts.

3. Do the same for four, five and six equal parts.

4. Reach/Extension: Describe a general procedure for dividing the segment AB into any desired number of equal parts (mathematically speaking, we would say n equal parts).

Comments:

(1) What prerequisite skills do students need to have in order to attempt this investigation? When planning a lesson like this, I found I had to consider this question first and review those needed skills. This avoided many issues that would otherwise slow down the lesson. I always tried to avoid the "You don't remember this?" comment. Sometimes this took superhuman effort on my part!

(2) Students should be organized into pairs or teams of 4. They can "divide" up the labor.

(3) There are several approaches to these problems. Many confident students with strong foundation skills (ok, this narrows it down to one student in the back of the room), recognize that a common denominator approach makes the most sense. You might see some very clever resourceful methods coming from your youngsters.

Note that 1/4 = 3/12, 1/3 = 4/12. In order to place a fraction in the 'middle', rewrite 1/4 = 6/24, 1/3 = 8/24. Don't be surprised to see some students invent similar methods for the other divisions.

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