Joel Elementary School

Calendar Grid: Algebraic patterns and structures that introduce and reinforce a variety of key number and geometry skills 

Each month’s Calendar Grid workout features a unique set of markers (a card for each day of the month). Students explore patterns and develop algebraic thinking skills as they make predictions and generalizations. Each day, a new marker is added to the Calendar Grid pocket chart, gradually assembling a complete calendar for each month. The markers create multiple patterns related to a particular concept or skill, such as equivalent fractions; fraction and decimal equivalency; elapsed time; an ancient numeration system’ perimeter and area; line symmetry; algebraic functions; and a study of lines, angles, and polygons. As the month progresses, students begin to look for and make use of structure in repeating, growing, and number patterns, and they work with the teacher to record these observations on the Calendar Grid Observations Chart.

Calendar Collector Collections that promote deep understandings of estimation and counting, value, measurement, and data.

The Calendar Collector capitalizes on children’s natural interest in collecting, counting, and organizing objects and information. Each month, students focus on a different subject for real-world mathematical investigation—they might collect a designated amount each day (six inches, one decimeter, or one cup) or spin for a different amount each day (a different fraction or multi-digit number), or survey data from experiments such as measuring water evaporation on a line graph. During the first week of each month students are introduced to the collection procedure and begin collecting the month’s item each school day. They keep track of their collection using a variety of visual organizers such as graphs, frequency tables, and line plots, which they read, interpret, and discuss as a group. These graphic representations help them understand concepts more deeply as they ask and answer questions about the data using mathematical vocabulary. The Calendar Collector provides a great opportunity to explore multiplicative comparisons. As they accumulate quantities, students can work with comparative statements (e.g., We have three times as many inches as we had last week. How many inches do we have now?). Students work with multiplicative comparisons throughout the year; they become familiar with the language and work on matching multiplicative comparisons to equations and writing their own multiplicative comparisons. Toward the end of the month, students analyze the total collection. Most months, they also complete one or more related page in their Number Corner Student Book.

Computational Fluency: Activities, games, and practice pages designed to develop and maintain fluency.

The concept of fluency encompasses more than the speed at which students solve problems, and it extends beyond paper-and-pencil methods to mental math. Computational Fluency workouts are filled with activities, games, and practice pages to help students work efficiently, flexibly, and accurately with numbers. In fourth grade Computational Fluency, students typically participate in two activities a month. One activity focuses on the number line. Throughout the fall, students explore multiples, factors, prime and composite numbers, and more as they work with a 0–100 number line. For the rest of the year, they investigate fractions, decimals, and mixed numbers on the number line. The repeated structure of the number line work helps students see, reason, and make sense of our base ten system, whether thinking about fractions or whole numbers. The other activity is usually a game in which students do any number of things, from fact practice to working with a coordinate grid to thinking about the associative property of multiplication to exploring fractions. Playing games is an effective way for students to sharpen their skills; the games are fun, setting a light and comfortable tone optimal for learning. Discussion and reflection serve to deepen thinking and understanding.  

Problem String: Careful delivery of a series of related problems designed to bring out certain strategies and ideas through discussion and modeling.

 This whole-group activity engages students in a deeper look at operations, relationships between numbers, and much more. Problem strings are intentionally designed sequences of problems that help students construct understanding about numerical relationships. Ideally, each string brings out or emphasizes an efficient strategy or model, or both. Over time, students participating in strings amass a collection of strategies they can select and use as needed. Efficiency is emphasized, as is clarity and elegance. A string begins when the teacher presents the first problem, often embedded in a context that helps students solve the problem or visualize a model that will help solve the problem. The first problem is generally not very challenging, but it is important in that it lays the groundwork for future problems and creates an entry point for all students. Students solve the problem on paper and then share their answers and how they got them. The teacher continues the string with the next problem, allowing for “think time” as students solve the problem. As selected students share their work, the teacher represents their thinking for everyone to see. The sharing is not a free-for-all in which all strategies are shared. There is not enough time, and this is not the purpose of a string. Strings are designed to highlight a specific strategy or model, so the teacher looks over students’ work and invites sharing from students whose work reflects or leads into use of the strategy. The string continues in this way, with the delivery of one problem at a time, time to solve the problem, and then time to share. The discussion is key. Students are given ample opportunities to ask each other questions as well as being asked to paraphrase each others’ strategies. Strings often end with an “application problem,” a problem that is different from the previous problems, but one that students can solve with the strategy highlighted in the string. This helps students see how they can use the strategy without the support problems or how to create the support problems themselves in order to solve the problem. In fourth grade, strings focus on models and strategies for adding, subtraction, multiplication, and division; composing and decomposing fractions; adding and subtracting fractions; and multiplying fractions by whole numbers. The strategies students learn in problem strings are often echoed in other workouts within the same month.