Look for and Express Regularity in Repeated Reasoning

Part 6 of 8, Strategies for Integrating the Mathematical Practices into Instruction

By Dr. Michele Douglass

Of all the mathematical practices, I find this one hardest to implement. It is probably from my years of being taught how to just manipulate numbers versus how to use patterns to generalize.

This practice reverses the thinking of the previous practice (MP#7). For this practice, we want students to use patterns that we might give them to generalize a situation. For example, instead of teaching rules for adding integers, how could students look at patterns to generalize or come up with the rule? Think about a lesson that has students examining graphs and matching equations in order to generalize the slope-intercept equation.

“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. … 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.” —Common Core State Standards (CCSS), MP8: Look for and Express Regularity in Repeated Reasoning

Students can begin using this practice with the tried and true guess and check method. If they are using repeated reasoning to make the next guess, that can lead to a method for solving a problem. CCSS Mathematical Practice #8 is also useful in making sense of the formulas we use by finding a pattern or relationship in numbers generated during an exploration of the topic. For example, students can explore the area of a triangle by building rectangles, drawing the diagonal, and realizing that the area of a triangle is half of the area of a rectangle—thus the formula is developed.

As I work with teachers on planning lessons or tasks that use patterns, we are working to make connections for students. There must be purposeful connections between procedures and concepts or between the manipulation of tools and a procedure. It is important to design lessons where patterns are revealed and lessons that encourage students to make generalizations. Some of these tasks involve students working with prior knowledge in a nonroutine way. Let me describe a few lessons and the connections that students can make by making generalizations.

Example 1

Part A: Using a calculator, find the sum for each of the following problems.

Part B: Sort the problems above into two groups. Define how you sorted each group by naming or titling each group.

Part C: Sort each of the above groups into two groups, now forming four groups of problems. Define how you sorted each group by naming each one and create two additional problems that would be in each group.

The learning that takes place through this lesson is incredible. You can extend the learning by using flip chips as a model for students to justify each answer as well. Instead of teaching students the “rules for addition of integers,” they generalize the sets of problems. Part A isn’t that interesting, but you do need to be sure students get the correct responses. Part B begins a great dialogue. Most of the time students group the problems in one of two ways: by the sign of the sum or by the signs of the addends. In other words, problems are grouped as either being positive or negative sums, or problems are grouped by adding like signs together or by adding unlike addends together.

Part C is when students have to look at greater details in the problems to be able to sort the problems into new groups. They typically end up with the four groups that match the rules we teach: Positive + Positive = Positive; Negative + Negative = Negative; Positive + Negative = Sign of greater absolute value. However, students have generalized from a pattern to determine the rules on their own. You could extend to a Part D with new problems for them to test their generalizations, providing them practice in using the rules or generalizations.

Example 2

How would you describe the pattern you see in the products of the following pairs of numbers? Create another problem that will follow this same pattern? Why does it exist?  

I have used this pattern in the lower grades as they are learning about patterns. It can also be revisited in algebra as we teach this pattern in factoring. It is x² – 1 = (x + 1)(x – 1). Let’s take the first problem of 3 x 3 = 9 and 2 x 4 = 8. If x is 3, then x² – 1 = 8 and (x + 1)(x – 1) = 8. It helps students to begin with number patterns before moving into using variables to describe the pattern.  

Example 3

Have students use the distributive property to multiply the first set of problems. Ask them if they see a pattern and use the pattern to predict the product of the second set of problems.

Set A: (x + 2)(x + 2)              (x + 5)(x + 5)          (x + 7)(x + 7)          (x + 8)(x + 8)

Set B: (x + 1)(x + 1)              (x + 3)(x + 3)          (x + 9)(x + 9)          (x + 11)(x + 11)

When students have multiplied several of these problems, a few of them inevitably start saying, “I see a pattern here, so do I have to do all this work?” This is where you move the learning into generalizing using a pattern. You can’t stop here. Students need to test their pattern as well as have opportunities to recognize it from either side of the equal sign.

I have shared just a few examples. Here are some websites that include classroom vignettes of lessons that embed this specific practice:

• Illustrative Mathematics: https://www.illustrativemathematics.org/MP8
• The Teaching Channel: https://www.teachingchannel.org/videos/common-core-teaching-division
• Implementing the Mathematical Practice Standards: http://mathpractices.edc.org/content/dividing-fractions—servings-yogurt

Remember that using the best-designed lessons isn’t the only necessary component in developing this practice in students. Be sure you facilitate conversations in your classroom that include asking questions such as the following:

• What generalizations can you make?
• What mathematical consistencies do you notice?
• How would you prove that …?
• What would happen if …?
• What predictions or generalizations can this pattern support?
• How would this strategy work with another number? Does it work all the time? How do you know?
• Can you find a shortcut to solve the problem? How would your shortcut make the problem easier?

References and Additional Resources:

Common Core State Standards Initiative Website: http://www.corestandards.org/Math/Practice/

Implementing the Mathematical Practice Standards Website: http://mathpractices.edc.org/

Inside Mathematics Website: http://www.insidemathematics.org/index.php/mathematical-practice-standards

Mathematics Assessment Resource Service Website: http://map.mathshell.org/materials/stds.php

Michele Douglass, Ph.D., is the president of MD School Solutions, Inc., a company that contracts with school districts on content and pedagogy with teachers and leaders. Her experience ranges from math instructor to director of curriculum and instruction at Educational Testing Services. She has authored several math curricula, as well as technology programs and professional development, including NUMBERS® with John Woodward, Ph.D., and Mary Stroh.