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Thought for the day: You can often make new discoveries by making things simpler (for example, by focusing only on subtraction problems that have numerators of 1).

Concepts: subtracting and adding fractions; analyzing and extending patterns (possibly using algebra to describe patterns)

Beginning

I notice that the fractions in the subtraction expressions have numerators of 1.
I notice that the denominators in the left column differ by 1.
I notice that the denominators in the right column differ by 2.
I notice that the denominator of every answer is the product of the denominators in the subtraction expression.

Exploring

I wonder if all of these patterns will continue.
I wonder what causes all of the patterns.
I wonder what would happen if the numerators were still 1 but the denominators differed by 3 or some other number;
I wonder what would happen if the subtraction expressions had numerators of 2 instead of 1.
I wonder if I can use these patterns as shortcuts for doing certain subtraction calculations.

Creating

As they notice and wonder, students may create their own lists of subtraction equations like the ones in this prompt. Much of what they create may flow out of things that they have wondered about. For example, they may

Create longer lists from the equations in the prompt by extending the patterns.
Create new lists by using numbers other than 1 and/or other differences for the denominators. Search for new patterns.
Create algebraic equations that describe the patterns that they discover.
Use algebraic processes to create proofs of their discoveries.

Reflecting and Extending

I notice that the patterns I predicted seem to be true.
I notice a shortcut for subtracting fractions whose numerators are 1 and whose denominators differ by 1: the numerator of the difference is 1, and the denominator of the difference is the product of the denominators.
I notice that I can understand why this shortcut works by doing the calculations the usual way and noticing patterns in the process.
I notice that it is easier to describe some patterns algebraically than verbally. For example, the pattern for the left column is

and the pattern for the right column is

I wonder if I can prove these patterns by using the usual rules for subtraction on the expressions 1 / a – 1 / (a + 1) and 1 / a – 1 / (a + 2).
I wonder if these formulas are part of a larger subtraction pattern for fractions whose numerators are 1.
I wonder what happens if the I make the numerator of the second fraction smaller than the first one.
I wonder if the rules that I discover will still work when the variables are not whole numbers.

Notes

This Creative Math Prompt shows two beautiful subtraction patterns that most students (and teachers) are probably not aware of. Students may (1) describe the patterns (2) extend them (3) figure out what causes them, and (4) generalize them (create new patterns based on the ideas). Depending on their background, some students may even try to write algebraic expressions to describe and/or prove the patterns.

The larger pattern for fractions whose numerators are 1 is

Students may also explore what happens when both numerators are 2, 3, etc.

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