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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)

**Examples of noticing and wondering**

*I notice *that all of the fractions being subtracted have numerators of 1.*I notice *that the denominators in each equation in the left column differ by 1.*I notice *that the denominators in each equation in the right column differ by 2.*I notice *that the denominator of every answer is the product of the denominators of the fractions being subtracted.*I notice *that numerators of all of the answers are equal to the difference of the denominators of the fractions being subtracted.

*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 two numbers being subtracted had numerators of 2 instead of 1..*I wonder *if I can use these patterns as shortcuts for doing certain subtraction calculations.

**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.

Left column pattern:

When you subtract two fractions (larger – smaller) whose numerators are 1 and whose denominators differ by 1, the difference will be a fraction whose numerator is 1 and whose denominator is the product of the denominators of the numbers you are subtracting. (Students may be surprised to know that this is true even if the denominators are not whole numbers, but it is much harder to see.)

Algebraically: 1 / a – 1 / (a+1) = 1 / [a • (a + 1)]

Right column pattern:

When you subtract two fractions (larger – smaller) whose numerators are 1 and whose denominators differ by 2, the difference will be a fraction whose numerator is 2 and whose denominator is the product of the denominators of the numbers you are subtracting. (Again, this is true even if the denominators are not whole numbers, but it is harder to see.)

Algebraically: 1 / a – 1 / (a+2) = 2 / [a • (a + 1)]

Similar patterns hold for other differences between the denominators. Students may also explore what happens when both numerators are 2, 3, etc.

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