Classroom Resources Collaboration Center

  • In September, thoughts turn to getting to know our students as mathematical thinkers and learners. What do they understand about the major work of this grade or course? What key concepts or big ideas do they fully understand? What is the state of their  problem-solving skills?  Do they persevere and make sense of problems? Do they listen to and learn from one another? In this month’s featured resources, we focus on tasks that not only are great learning experiences for students but also  help teachers get to know their students. We also focus on tasks around major ideas in each grade-band or subject area, so that teachers can focus on big ideas that develop over the course of an entire school year (or even multiple years).

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    Helping students develop number sense and an understanding of whole numbers is perhaps the most important work of primary teachers. Beginning in kindergarten, students learn to count and record their counting using grouping strategies, which continues to build in first and second grade into making sense of place value. Students also reason about the relative size of whole numbers, how to compare numbers, and connect their counting and comparing strategies to addition and subtraction. The Counting Collections activity described in this Teaching Children Mathematics article is naturally differentiated and helps primary teachers learn about their students’ understanding of counting, grouping, place value, and large numbers.

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    Grades 3-5

    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.

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    Grades 6-8

    In middle school, a major focus of instruction is helping students represent linear and proportional relationships symbolically, as well as recognize other relationships that are not linear or proportional. Visual pattern tasks are low-threshold, high-ceiling investigations that can be used to assess students’ skills at identifying and describing patterns, representing patterns with tables and graphs, and writing and understanding algebraic expressions. One of NCTM’s new Activities with Rigor and Coherence (ARCs), “Growing Patterns”, presents a variety of pattern activities and examples that teachers can use to assess students’ comfort with functions, tables, patterns, and algebraic notation. For example, sixth-grade teachers might focus on Lesson 3, “Identify Relationships and Generate Rules: How Many Beans?” or Lesson 5, “Analyze Two Patterns and Find Rules: Rows of House Designs,” which involve proportional relationships, whereas seventh-grade and eighth-grade teachers might want to include other linear and nonlinear pattern examples from other lessons.

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    Algebra 1

    Videos like this YouTube video have been shared on Facebook millions of times. Students are interested in figuring out how the activity works, and how they can make their own “Math Magic” rules to fool and amaze their friends and family. The Mathematics Teacher article “Fostering Algebraic Understanding through Math Magic” supports teachers to use a math magic trick to assess students’ understanding of such key concepts as order of operations, the use of variables, and reversibility. Tasks such as this can invite open-ended exploration at the beginning of the year; they can then be revisited later in the year when students have more experience with functions and variables and can more fully resolve the questions brought up by the task.

    NCTM members also have access to another prompt about a different variety of math magic trick, with additional student work, from the Problems of the Week.

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    Algebra II/Pre-Calculus

    In the Mathematics Teacher article, “Conceptual Precalculus: Strengthening Students’ Quantitative and Covariational Reasoning,” the authors share three released items from the Mathematical Association of America’s calculus placement test. The items test students’ abilities to reason quantitatively and to reason about co-variation. Each item is multiple choice, which offers teachers opportunities to turn the three test items into rich formative-assessment tasks. For example, students could be asked to answer the question, form into groups based on the answer they chose, and defend their choice in a formal math debate. Alternatively, students could be given the correct answer and then asked to explain why each wrong answer is wrong. Because the article carefully lays out the kinds of conceptual understanding each problem is testing for, teachers can circulate and listen for quantitative and covariational reasoning in their students.

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    A core theme in students’ study of geometry in high school is connecting the understandings of similar triangles and proportional relationships that students bring from middle school to such high school topics as trigonometry and dilation. The tasks described in “Shadow Puppets: Exploring a Context for Similarity and Dilation” do not require students to use high school concepts like trigonometry, but they can be used to assess students’ readiness for such concepts, including how they use triangles to model situations involving light rays, a key context for many applications of trigonometry. High school geometry teachers who want to assess students’ understanding of similar figures, of modeling with geometry, and readiness for concepts such as dilation and trigonometry can use the Shadow Puppets task as an open-ended modeling prompt that can give them a wealth of information about student understanding and problem solving.

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    Probability and Statistics

    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.

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    Be sure to also check out these additional resources and tools for your classroom.

    Math Forum - Problems of the Week

    Get your weekly dose of math problems and puzzles from the Math Forum. You will also find more math resources and tools, as well as a Math Forum: Problems of the Week Blog, furthering discussion.

    Problems of the Week Samples

    Brain Teasers

    Want quick ideas for great back-to-school icebreaker classroom activities? We've got you covered. Challenge your new students and mathematics enthusiasts alike with these staff-picked puzzles. In need of more? Browse the entire Illuminations library and discover what's in store in this amazing resource.

    Sliding Triangle

    Golden Ratio

    Heart Shaped Words

    Browse Illuminations