Reasoning and Sense Making Task Library

Introduction

Calculus In order for high school students to be engaged in reasoning and sense making in the classroom, the task—what students are asked to do—is critical. However, each item in this task library contains much more than a student work sheet. While linking the task directly with NCTM’s Focus in High School Mathematics: Reasoning and Sense Making, NCTM’s Principles and Standards for School Mathematics, and the Common Core State Standards, each item addresses:

Task Design: What the task is asking students to do (see Task Purpose, Task Overview, Focus on Reasoning and Sense Making, Focus on Mathematical Content, Materials and Technology, Assessment and the Student Activity sheet)
Teaching Design: How teachers might facilitate reasoning and sense making (see Use in the Classroom)
Student Engagement: What student might actually do in the classroom (see Focus on Student Thinking)

Task	Purpose
Horseshoes in Flight	Students analyze the structure of algebraic expressions and a graph to determine what information each expression readily contributes about the flight of a horseshoe. This task is particularly relevant to students who are studying (or have studied) various quadratic expressions (or functions). The task also illustrates a step in the mathematical modeling process that involves interpreting mathematical results in a real-world context.
Taking a Spin	Although students are often asked to find the angles of rotational symmetry for given regular polygons, in this task they are asked to find the regular polygons for a given angle of rotational symmetry, a reversal that yields some surprising results. This task would be most appropriate with students who have at least some experience in exploring rotational symmetry.
Tidal Waves	Students analyze a problem faced by the captain of a shipping vessel. Students may use a range of functions to model the situation and reflect on their usefulness. Because trigonometric functions can be useful, this task would be particularly appropriate for students who have had an introduction to graphing sine and cosine functions.
Eruptions: Old Faithful	Students analyze data and make predictions. They will create a variety of graphical displays to discover trends in the data, then use those graphs to support their predictions. This task is appropriate for students familiar with line graphs and other graphical displays of univariate data sets.
Fuel for Thought	Students use mathematical reasoning to determine appropriate numerical measures and patterns in fuel consumption in order to inform consumer choice for vehicle purchasing. The task promotes a sophisticated use of number sense including careful attention to units. It is accessible to beginning high school students.
Bank Shot	Students compare their own reasoning strategies and those of their classmates, focusing on the strategies’ usefulness in determining how to make certain bank shots in billiards. This task is intended to involve multiple geometric perspectives and would be appropriate for students with an understanding of similar triangles, rigid motions (especially reflections), and equations for lines and is designed to strengthen students’ understanding of these concepts.
As the Crow Flies	The distance formula is often presented as a “rule” for students to memorize. This task is designed to help students develop an understanding of the meaning of the formula. It would be appropriate as an introduction (or review) of the distance formula for students who are familiar with the Pythagorean theorem and coordinate systems.
Over the Hill	Students determine locations on a hillside for a cell phone tower erected to provide a signal to people on the other side of the hill. They identify necessary information, represent the problem with a scale model, and answer questions in context. This task is appropriate for students who have had experience in determining equations of linear functions through two points and in solving systems of linear equations.
Cash or Gas	State lotteries in Florida and other states give winners a choice between cash and another prize, such as free gas for life. In this task, students will evaluate two prize options by discussing and making reasonable assumptions to simplify a complex decision. This task is appropriate for students who can extrapolate quantities over time and are able to make conversions among different units of measure.

NCTM Position Statements

Procedural Fluency in Mathematics

Procedural fluency is a critical component mathematical proficiency and is more than memorizing facts and procedures.

Algebra as a Strand of School Mathematics for All Students

Algebra is not confined to a course or set of courses but is a strand that unfolds across a pre-K–12 curriculum.

Access and Equity in Mathematics Education

Practices that support access and equity require comprehensive understanding and require being responsive to students’ backgrounds, experiences, cultural perspectives, traditions, and knowledge.

Teaching Mathematics to English Language Learners

To ensure that all students can gain access to, interpret, and share information fluently, teachers must address multiple dimensions of instruction.

Mathematics in Early Childhood Learning

Young learners’ future understanding of mathematics requires an early foundation based on a high-quality, challenging, and accessible mathematics education.

Teacher Mentorship

Mentorship is important in shaping and developing the next generation of teachers, particularly as expectations for students become more rigorous.

Supporting the Common Core State Standards for Mathematics

The Common Core State Standards offer a foundation for the development of more rigorous, focused, and coherent mathematics curricula, instruction, and assessments that promote conceptual understanding and reasoning as well as skill fluency.

Computer Science and Mathematics Education

Computer science should be incorporated into the curriculum in a way that enhances, rather than limits, students’ college and career readiness in mathematics.

Preparing Pre-K-12 Teachers of Statistics

Professional development courses and workshops for future and current teachers need to model effective pedagogies for teaching statistics, in addition to focusing on developing understanding of statistical concepts, mastery of statistical content, and knowledge of the essential ideas of statistical thinking and problem solving. (A joint position statement of the American Statistical Association and the National Council of Teachers of Mathematics.)

Linking Mathematics Education Research and Practice

Collaboration between researchers and school personnel provides integrated perspectives for addressing critical issues in mathematics teaching and learning.

Calculus

The ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college. (A joint position statement of the Mathematical Association of America and the National Council of Teachers of Mathematics.)

Large-Scale Mathematics Assessments and High-Stakes Decisions

Large-scale mathematics assessments should not be used as the sole source of information to make high-stakes decisions about schools, teachers, and students.

Closing the Opportunity Gap in Mathematics Education

Much of the achievement gap in mathematics is a function of differential instructional opportunities. All students should have the opportunity to receive high-quality mathematics instruction, learn challenging grade-level content, and receive the support necessary to be successful.

Strategic Use of Technology in Teaching and Learning Mathematics

Strategic use of technology strengthens mathematics teaching and learning.

High Expectations

To teach mathematics with high expectations means that teachers recognize that each and every student, from prekindergarten through college, is able to solve challenging mathematical tasks.

Intervention

When implementing interventions, teachers must possess strong backgrounds in mathematical content knowledge for teaching, pedagogical content knowledge, and a wide range of instructional strategies.

Calculator Use in Elementary Grades

Calculators in the elementary grades serve as aids in advancing student understanding without replacing the need for other calculation methods.

The Metric System

Students need to develop an understanding of metric system units and relationships, as well as fluency in applying the metric system to real-world situations.

The Role of Elementary Mathematics Specialists in the Teaching and Learning of Mathematics

Every elementary school should have access to an elementary mathematics specialist to enhance the teaching, learning, and assessing of mathematics to improve student achievement. (A joint position of the Association of Mathematics Teacher Educators [AMTE], the Association of State Supervisors of Mathematics [ASSM], the National Council of Supervisors of Mathematics [NCSM], and the National Council of Teachers of Mathematics [NCTM] in response to the release of Elementary Mathematics Specialists: A Reference for Teacher Credentialing and Degree Programs [AMTE, 2010].)

Evaluation of Teachers of Mathematics

Professional growth and support should be the foremost goals of any teacher evaluation process, which should be led by those knowledgeable about effective mathematics instruction.

Providing Opportunities for Students with Exceptional Promise

Students with exceptional mathematical promise must be engaged in enriching learning opportunities to allow them to pursue their interests, develop their talent, and maintain their passion for mathematics.

Curricular Coherence and Open Educational Resources

A coherent, well-articulated curriculum is an essential tool for guiding teacher collaboration, goal-setting, analysis of student thinking, and implementation.

Building STEM Education on a Sound Mathematical Foundation

Students need a strong mathematics foundation to succeed in STEM fields and to make sense of STEM-related topics in their daily lives.

NCTM Position Statements