Operations and Algebraic Thinking

Represent and solve problems involving multiplication and division.

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This 60-page pdf document demonstrates the connections between the CCSS content standards and the mathematical practice standards. It is a compilation of research, standards from several states, instructional strategies, common misconceptions, and examples for each standard at the grade 3 level. It is intended to help teachers understand what each standard means in terms of what students must know and be able to do. Additional flip books are cataloged separately for grades K-2 and 4-5. | Instructional Strategy, Reference Materials Grade Level: 3 | |

This web page provides links to resources aligned to the CCSS that guide and support third grade mathematics teaching and learning. Tasks developed by the Mathematics Assessment Resource Service (MARS) and Problems of the Month, (POM home page is cataloged separately) developed by the Noyce Foundation are included. The tasks were designed to measure students’ ability to solve non-routine problems, explain and justify their solutions, and promote high level thinking skills. They include the scoring rubric, student responses, and discussion of student understanding and misconceptions. Resources are listed for specific grade 3 standards and are also organized by progression for an alternate search route. | Instructional Strategy, Problem Set, Reference Materials Grade Level: 3 | |

This 1-hour webcast introduces some key notions of numeracy coaching from the experiences of Ontario classroom teachers, principals, consultants, superintendents, and math education researchers. These educators share examples of effective instructional strategies in problem solving. Their professional development model involves co-teaching and inquiry; teams of educators consult with a classroom teacher prior to a lesson, observe the lesson, review the results, and plan next steps. The video can be streamed or downloaded as a podcast. | Instructional Strategy, Video Grade Level: 1, 2, 3, 4, 5 | |

This 9-minute video teaches divisibility rules for the factors 2 through 10. The presenter explains each rule and provides examples. | Lecture/Presentation, Video Grade Level: 3, 4, 5 | |

This narrative document describes the progression of Counting and Cardinality and Operations and Algebraic Thinking across the K-5 grade band. It is informed both by research on children's cognitive development and by the logical structure of mathematics. The document discusses the most important goals for elementary students that of understanding and using numbers. The focus is on the basic operations—the kinds of quantitative relationships they model and consequently the kinds of problems they can be used to solve as well as their mathematical properties and relationships. | Reference Materials, Article Grade Level: K, 1, 2, 3, 4, 5 | |

In this math investigation students must determine how two people can share the food items for a picnic equally. The problem involves dividing whole numbers by two and using fractions to divide items equally. The problem is accompanied by a Teachers' Resource page that includes suggestions for approaching the problem, questions, extension ideas, and tips for support. | Activity Grade Level: 3, 4, 5 | |

In this lesson, students explore the patterns found in Pascal's Triangle by locating specific remainders when numbers in the triangle are divided by a chosen or random divisor. Students use modular arithmetic and the applet, "Clock Arithmetic"(cataloged separately) to identify remainders. Included are lists of prerequisites and suggested outlines. Links to individual activities and discussions are included. | Activity, Interactive Media, Lesson Plans Grade Level: 3, 4, 5, 6+ | |

This interactive Flash applet models the measurement interpretation of division. A child or teacher chooses a total number of objects and a divisor representing the size of equal groups. The applet allows the user to move the objects into equal groups and links the process to jumps on a number line. The applet can be used to introduce children to remainders and to reinforce the language and notation of division. It works well on an interactive white board or projector. A teacher's guide to this collection of applets is cataloged separately. | Activity, Interactive Media Grade Level: 2, 3, 4 | |

This brief article advocates the use of arrays to model the commutative and distributive properties as well as the inverse relationship between multiplication and division. The author explains how arrays also help children form mental pictures that support their memory and reasoning. | Instructional Strategy, Reference Materials, Article Grade Level: 2, 3, 4, 5 Resource is part of a PD collection | |

This problem allows students to explore introductory multiplication and division concepts in the context of sharing cookies ("biscuits" in the UK). Cookies are displayed in two different arrays, and students are challenged to divide them equally among two, three, five, and six people and to deal with remainders. The Teachers' Notes page offers rationale, suggestions for implementation, discussion questions, ideas for extension and support, and a printable sheet (pdf). | Activity Grade Level: 1, 2, 3 |