## Common Core State Standards for Mathematics

Grade(s) 3 resources related to the following standard:
Operations and Algebraic Thinking
Understand properties of multiplication and the relationship between multiplication and division.
5. Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

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This interactive Java applet helps students explore the relationship between area and multiplication. Users are asked to draw all possible factorizations of a number as rectangular arrays by clicking and dragging across a grid. Options include the use of the commutative property (e.g., user must enter both 2x4 and 4x2 for factors of 8 represented by different orientations of the array), entering a number of the user's own choice, and an optional scoring feature allowing users to keep track of the number correct.
Interactive Media
Resource is part of a PD collection
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+
In this brief article NCTM President Linda Gojak encourages educators to reconsider the notion of fluency with regard to number facts, procedures, and other topics. She elaborates on the NCTM's position of broadening the definition to include conceptual understanding and flexible thinking.
Instructional Strategy, Reference Materials, Article
Grade Level: 1, 2, 3, 4, 5
Resource is part of a PD collection
This interactive applet helps students develop mental arithmetic skills and strategies for learning multiplication facts. Frog asks the user to type in a known multiplication fact and then asks the user to complete a related fact by using one of several strategies, including commutativity, doubling a factor, halving a factor, multiplying one factor by 10. The student's fact and Frog's related one are recorded in a window to help students discover relationships and internalize the strategies.
Activity, Interactive Media
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
KenKen puzzles develop fluency with basic whole number operations and logical reasoning. This website offers puzzles at six levels of complexity, a tutorial, solving tips, and links to strategy suggestions. Teachers may register to receive a selection of free KenKen puzzles for classroom use on a regular basis.
Activity, Game, Interactive Media
This is the iOS app for Ken Ken (two web-based versions are cataloged separately), a grid-based logic and math puzzle that develops fluency with basic whole number operations. There are unlimited puzzles available on this app. The first 31 are included with the download and more are available for purchase.
Activity, Game, Interactive Media, Mobile App