# Standard 8: Look for & Express Regularity in Repeated Reasoning

## Classroom Observations

Integrating Standard Eight into classroom practice is not only a matter of planning for lessons that occasion students to look for general methods and shortcuts. It also requires teachers to attend to and listen closely to their students’ noticings and “a-ha moments,” and to follow those a-ha moments so that they generalize to the classroom as a whole. The video clips included here are intended to show as a composite how teachers create the conditions for students to look for and express regularity in repeated reasoning, and follow and elaborate students’ thinking when they begin to make these connections.

## The Standard

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. 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)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. 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.

## Connections to Classroom Practices

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.

In Katie Arrillaga’s 2nd grade Spanish bilingual classroom, she engages her students in a number talk, discussing whether or not their approaches to solving a two-digit addition problem (59 + 37) are reasonable. In this clip, her student Cesar begins with sharing one strategy for solving and addressing the challenge of how to use the 16 that results from adding 9 and 7. He begins, catches himself in the “intermediate results” stage, and changes his approach to use the “Australian Method” that the class had used elsewhere (and which is explained more fully in Arrillaga’s debrief.)

See this video in the context of an entire lesson.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

In this clip, Buljan works with her 3rd-grade students on a number talk, using a mental math approach to multiplying a two-digit number and a one-digit number. She encourages her students to identify regularity among the various approaches identified, and guides them toward identifying the commutative property.

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Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts... They continually evaluate the reasonableness of their intermediate results.

In Elysha Passeggi’s 4th grade classroom, students work on a number talk asking them to evaluate the reasonableness of a two-digit problem and solution—to “see if they might be true or false without thinking.” As her students discuss their rationales for determining why 62 + 78 is or is not equal to 238, various students offer responses that the numbers are “too small” or “would have to be bigger.” Early on, a student observes that “to get 200 you’d need at least 100 and 100.” Later in the discussion, a student advances this idea, stating that “two 2-digit numbers cannot…no matter what… if it’s addition…cannot be more than 198.” Other students then affirm that they agree with this student’s rule.

Starting at 29:23 ...

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts... They continually evaluate the reasonableness of their intermediate results.

Passeggi works with her 4th grade students in a daily number talk routine called “Can This Be True?” In this number talk, Passeggi challenges her students to look at the relationships between numbers to determine whether or not a given answer can be correct, without calculating the result. In this clip, after initial discussion by the class, Passeggi asks the students to generate ideas for testing the reasonableness of their solutions with their "shoulder partners." One student then proposes that "if there's one number on one side, and another number on the other side, and this number is bigger, then this [other] number has to be bigger." Passeggi then challenges the class to test out "Maddy's way" to see if the strategy is effective. In so doing, the students generate and develop their understanding of general methods for strategic solving.

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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.

Michelle Makinson challenges her students to individually select one grouping of representations from their chart, and move those representations over to a poster without the justifications. Each student then writes responses to several questions, each of which has a sentence stem prompt:

• Why did you select this grouping? / I selected this grouping because...

• How do you know that all the cards represent the same quantity? / I know that all the cards represent the same quantity because...

• When sharing your work with other pairs, what did you learn from the discussion? / When I shared my chart with another partnership I learned... from the discussion.

This clip also relates to standard 1 (make sense of problems & persevere in solving them).

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Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

Michelle Kious works with her 5th grade students on understanding multiple representations of mixed numbers. In this clip, her students engage in a final reflection on the lesson, and one observes that the day’s focus, using line segments to understand fractions and mixed numbers is similar to (but more challenging to this particular student) using an area model to represent fractions and mixed numbers.

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Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts….They continually evaluate the reasonableness of their intermediate results.

Erika Isomura engages her 4th and 5th graders in a mental math talk, inviting multiple iterations of dividing by 10, e.g., 8,000/10, 800/10, 80/10, 8/10, (8/10)/10. She asks students to justify and revise their answers if needed. She reviews norms for whole-class participation, and asks students to evaluate the reasonableness of their results. Erika then presents multiple iterations of dividing 8 by multiples of 10 (e.g., 8/10, 8/100, 8/1,000, 8/10,000), noting similarities and patterns. This clip also relates to standard 1 (make sense of problems and persevere in solving them).

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(Day 1 Math Talk)

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.

In the second day of the learning segment, Erika Isomura begins with a number talk with her 5th-grade students, focused on noticing how and why divisors get smaller.

Her students share their thoughts (e.g., “Divide by 10 each time”) and notice that the quotient is getting bigger by 10 each time.

In the number talk, Erika responds to a student’s wondering on the previous day about what happens if you work with a different number, for example, 2,300/100. Erika uses patterns suggested by the students themselves (e.g. “Yeini’s pattern”) to reduce by a power of 10 or increase by a power of 10 (e.g., 2,300 / .001). This clip also relates to standard 7 (look for and make use of structure).

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Day 2 Lesson Part A