### 8 — Look For and Express Regularity in Repeated Reasoning

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Upper elementary students might notice when dividing $25$ by $11$ that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through $(1, 2)$ with slope $3$, middle school students might abstract the equation $(y-2)/(x-1) = 3$. Noticing the regularity in the way terms cancel when expanding $(x-1)(x+1)$, $(x-1)(x^2+x+1)$, and $(x-1)(x^3+x^2+x+1)$ might lead them to the general formula for the sum of a geometric series.

### New: Vignettes and videos

#### Kindergarten: Double Compare

Double Compare is a card game played with numeral cards, usually in pairs. In the classroom example below, the teacher notices that in some rounds of the game students add or count to find the sums on their numeral cards, while in other rounds some students do not add or count-they have other ways to reason about the two pairs of numbers. Read more (PDF)

As young students learn about addition and subtraction, they naturally begin to notice regularities across problems. Sometimes students point out what they are noticing-especially if being alert for patterns and regularities is expected in math class. Read more (PDF)

#### Grade 2: Related Story Problems

Although students may have methods to calculate and to solve different kinds of story problems, it is a different skill to look across related problems to notice generalizations about the behavior of the operations involved. Read more (PDF)

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#### Grade 2: How Do You Know That 23 + 2 = 2 + 23?

What does it mean to use “properties of operations” in the primary grades? The focus is not learning names for the properties, but on understanding how each operation behaves-for example, that the numbers in an addition expression can be rearranged without changing the sum. Read more (PDF)

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This extended example presents a sequence of eight lessons in which students 1) identify regularities they notice in pairs of related problems, 2) articulate a generalization about the behavior of an operation, 3) explore that generalization, and 4) develop arguments to prove that the generalization is true for all whole numbers. Read more (PDF)

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There are many variations of a task that some teachers call “Number of the Day” or “Today’s Number.” Typically, teachers provide a target number, and students are challenged to write expressions equivalent to that number. Read more (PDF)