Thought for the day: Patterned lists help students discover meanings and make connections between math concepts.
Examples of noticing and wondering
I notice that I can make a lot of addition or subtraction equations from the numerators and denominators.
I notice that the smallest numerators and denominators are on the ends of the list.
I notice that the numerator (denominator) in the middle is the sum of the numerators (denominators) in the first and last fractions.
I notice that all of the fractions are in simplest form.
I notice that the fractions increase from left to right. (They are very close together!)
I wonder what will happen if I remove the second and fourth fractions from the list.
I wonder how the "adding numerators and denominators" idea connects to the fact that the fractions increase from left to right.
I wonder if it is possible to extend the list to the right and/or left.
I wonder what would happen if I started with new fractions on the right and left.
I wonder what would happen if I started with two equivalent fractions on the right and left.
Start with the left and right fractions and keep inserting fractions between each neighboring pair by adding their numerators and adding their denominators. If you do this to the the list above, you get:
You can make the list as long as you like. The fractions will get closer and closer together, and they will always increase from left to right.
Adding numerators and denominators separately is a common error that students make when adding two fractions, but the result is a real mathematical thing! It is called the mediant (not to be confused with the median). The mediant of two fractions is always between them! In the language of algebra: (1) The mediant of a/b and c/d is (a + c) / (b + d), and (2) (a + c) / (b + d) is between a/b and c/d (assuming that a, b, c, and d are positive numbers.)
Note: You may challenge students to explain why this is true. (Hint: Think of what happens if you pour together two lemonade recipes: (1) 3 cups concentrate with 5 cups water, and (2) 5 cups concentrate with 8 cups water.)
Students who are interested in learning more may experiment by starting with many different pairs of fractions. They may also research some of the following topics: Farey sequences, the Stern-Brocot tree, and Ford circles.