Summary:

At the Helmwood Heights Elementary School in Elizabethtown, Kentucky, Kathy Doolittle's students use array structures to reason and conjecture about relationships between sums and products of numbers added and multiplied. Patterns are sought when numbers are added to themselves and when they are multiplied by each other. Groups are assigned one number and told to create a model of its sum and product. (If N is the assigned number, then an N x 2 array will represent its sum and an N x N array its product.) Students place arrays on grid paper, and label each one with the symbolic solutions to the addition and multiplication problems. Each group then posts its models and solutions around the classroom so students can detect patterns, seek relationships, and discuss findings. Anticipating that patterns and relationships are more easily detected if the array structures are arranged systematically, Mrs. Doolittle gives out charts with the numbers, sums, and products arranged in descending order. Groups then find and record numerous numerical relationships and geometric patterns and share observations.

Standard Connection:

(Practice Standard)—Common Core Practice Standard #8—Look for and express regularity in repeated reasoning—is evident in this fourth grade class. The students see connections among the array structures arranged in descending order in the "condensed" table created by the teacher. Using repeated reasoning, one group recognizes that the sum column produces array structures that are rectangles and the product column produces array structures that are squares. A second group, looking at the symbolic relationships, finds that perfect square numbers are produced when a number is multiplied by itself. A third group's observation centers on differences. Looking for and expressing regularities in repeated reasoning are skills students will need when they encounter relations and functions in algebra. Discovering regularities in computational problems can lead to "shortcut" taking without conceptual understandings. "Shortcuts" are fine as long as they are student discoveries, created from generalizing patterns and not from mimicking procedures.

(Content Standard)—The domain that best encompasses the content of this clip is Operations and Algebraic Thinking—4. OA. Mrs. Doolittle's fourth grades are getting experience in learning to generate and analyze patterns. As they get more and more practice in using their pattern detecting brain, they will see the patterns and order in mathematics.

Questions to Consider:

What value is there to the study of patterns in arithmetic, geometry, algebra, measurement, and data analysis and probability?

8. Look for and express regularity in repeated reasoning

4.OA Operations and Algebraic Thinking