A major theme of students’ work in upper elementary school is understanding division and especially how division relates to place value and the concept of units. In third grade, students develop informal strategies for solving division problems in various contexts; in fourth and fifth grades, they develop fluency while extending their conceptual understanding to make sense of dividing larger numbers and dividing nonwhole numbers using decimal representation. The Illuminations activity "How Many Each? How Many Are Left?: Conceptualizing Division with Large Numbers" can be used early in the year in each grade level to assess students’ readiness to use efficient strategies to solve division problems. Teachers can differentiate by providing different numbers of counters; making the groups different sizes; and by providing different manipulatives to divide, such as using place-value blocks in fourth grade, or dollars, dimes, and pennies in fifth grade.

Additional links: Another tool for offering students more practice with the “partial products” method for fair-sharing division is to use “The Quotient Café” interactive from Illuminations. For more information about the division learning trajectory, elementary math coach Graham Fletcher, a member of NCTM’s Classroom Resources Committee, has prepared an eight-minute video illustrating some of the division concepts, methods, and procedures students learn from third through sixth grade.

Students come into high school statistics units with a wide range of understandings and skills. Assessing those skills through rich tasks early in the year can provide teachers with a snapshot of what students already understand and can do, and those rich tasks can serve as anchors that students and teachers return to over time to apply new understandings. The Mathematics Teacher article, “How Faithful Is Old Faithful? Statistical Thinking: A Story of Variation and Prediction,” describes an activity that opens a window into students’ skills and understanding of measures of center and graphical displays of data. The authors then show how the activity ties into deep conceptual understandings of big ideas in statistics, such as the need for data, the role of variation, and reasoning with statistical models. An analysis of sample student work helps the reader see how to assess these big concepts using the task. For access to the data used and a student recording sheet, also see the “Eruptions: Old Faithful” task in the Reasoning and Sense-Making Task Library.