Thought for the day: Patterned lists help students discover meanings and make connections between math concepts.

Concepts: comparing and ordering fractions; identifying and extending complex patterns (possibly equivalent fractions and ratios)

Beginning

I notice that many of the numerators and denominators are sums of one another.
I notice that all of the fractions are in simplest form.
I notice that the smallest numerators and denominators are on the ends of the list.
I notice that quite a few numbers (3, 5, 8, and 13) appear twice.
I wonder if there is a pattern that I could use to extend the list.
I notice that the fractions increase from left to right. (They are very close together!)
I wonder if there is a more specific pattern in how the fractions change from left to right.

Exploring

I wonder why all of the fractions are in simplest form.
I notice
that the largest numerators and denominators are in the second and fourth positions of the list.
I wonder what will happen if I ignore the second and fourth fractions from the list.
I notice that the numerator (denominator) in the middle is the sum of the numerators (denominators) in the first and last fractions. (3 + 5 = 8 and 5 + 8 = 13.)
I notice that the same thing happens between the first and third fractions (3 + 8 = 11 and 5 + 13 = 18) and the third and fifth fractions (8 + 5 = 13 and 13 + 8 =21).
I notice that I can use this pattern to keep inserting new fractions between the fractions in the list.
I wonder if the fractions would always continue to increase from left to right if I did this.
I wonder why you always get an “in-between” fraction when you add the numerators and add the denominators of two fractions.

Creating

As they notice and wonder, students may create and explore their own patterns related to this image. Many of their creations may flow out of things that they have wondered about. For example, they may

Create longer lists of fractions by extending patterns in the prompt.
Create new lists by beginning with a variety of fractions at the beginning and end of the list.
Create new lists by creating new rules for inserting fractions between fractions.

In all of these cases, students may analyze their new lists, look for new patterns, see if old patterns continue to hold, and try to understand why these things happen.

Reflecting and Extending

I notice that (a + c) / (b + d) always seems to be between a / b and c / d.
I wonder if this is still true if fractions are not in simplest form.
I wonder if this is still true when some of the numerators and denominators are negative.
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.
I wonder if I could express any of the patterns algebraically.

Notes

You can learn more about using this image and related topics in Exploration 2: Ramps, Paints, and Hot-Air Balloons from my book, Advanced Common Core Math Explorations: Ratios, Proportions, and Similarity.

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.