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. 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  + x 2 + 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.

Representative IMP Year 2 Lesson:

Don’t Fence Me In; More Fencing, Bigger Corrals; and Building the Best Fence (Do Bees Build It Best?), 323, 325, 328

Students build to the general formula to find the area of any regular polygon by investigating these three activities. Much like their experience in Year 1, they begin with a simple polygon (a quadrilateral) and work their way through polygons of increasing number of sides. By examining their work with these various polygons, students derive the area formula for a regular polygon.

Representative IMP Year 3 Lesson:

Squaring the Circle; Using the Squared Circle; Hexagoning the Circle; Octagoning the Circle; and Polygoning the Circle (Orchard Hideout), 120, 122, 123, 125

Students explore the case of the circumscribed polygon about a circle. Beginning with the square and working up to a polygon with n sides, students calculate the perimeter and area of the circumscribed polygon in terms of the radius of the circle. After examining their work, students will generalize a formula for the perimeter and area of any sided polygon.