Students notice repeated calculations, look for general expressions to annotate the calculation, and consider potential shortcuts. Students maintain oversight of a process as they work to solve problems, derive formulas, or make generalizations, while attending to details. They assess the reasonableness of their intermediate results.

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# Students notice repeated calculations, look for general expressions to annotate the calculation, and consider potential shortcuts

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### Related standards

- This standard is part of: 8
- This standard is derived from: Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts

#### Similar standards in other grades

Look for and express regularity in repeated reasoning.

Students routinely seek patterns or structure to model and solve problems. They recognize that patterns exist in ratio tables and make connections with the constant of proportionality in a table and the slope of a graph. Students recognize patterns and identify and develop strategies for creating equivalent expressions. Students identify complicated expressions or figures as compositions of simple parts.

Students notice repetitive actions in counting and computation, etc. When children have multiple opportunities to add and subtract ten, including multiples of ten, then they notice the pattern and gain a better understanding of place value. Students also notice that when adding two numbers, order of adding doesn’t affect the sum (commutative property). They also notice that three numbers create a family when adding or subtracting (2 + 3 = 5 and 5 - 2 = 3).

Students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical representation.

#### Similar standards elsewhere

Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.

Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.

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.

Notice repetitions in mathematics when solving multiple related problems. Use observations and reasoning to find shortcuts or generalizations. Evaluate the reasonableness of intermediate results.

They 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 students 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.