Standards for Mathematical Practice

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This article talks about the origins of our number system and the important roles zero plays in it. The author informs the reader that the early number systems of the Egyptians, the Babylonians, the Romans and Greeks were developed for counting but not for calculations. Not until we had a base ten number system with place value and a symbol for zero did our number system become useful. | Reference Materials, Article Grade Level: 3, 4, 5, 6+ | |

This 440-page report explores how students in pre-K through grade 8 learn mathematics, with a focus on number and operations, and recommends how teaching, curricula, and teacher education should change to improve mathematics learning. (A 39-page summary, Helping Children Learn Mathematics, is catalogued separately.) The authors identify five interdependent components of mathematical proficiency and describe how students develop this proficiency. The report presents a portrait of mathematics learning: research findings on what children know about numbers before they arrive in pre-K and the implications for instruction; details on the processes by which students acquire mathematical proficiency; what is known from research about teaching for mathematics proficiency and developing proficiency in teaching. Visitors to this website may read the full text online, download a copy (pdf), or purchase a hard copy. | Reference Materials Grade Level: Pre-K, K, 1, 2, 3, 4, 5, 6+ | |

This virtual manipulative allows a teacher or student to represent a linear equation and solve it through the use of a balance beam. Unit blocks (representing 1s) and x-boxes (representing the unknown), are placed on the pans of a balance beam. Once the beam balances to represent the given linear equation, the learner can choose to perform any arithmetic operation to both sides of the equation keeping the beam balanced. The goal, of course, is to get a single X-box on one side, with however many unit blocks needed for balance thus giving the value of x. There is also the option of creating your own equations to solve. | Activity, Interactive Media Grade Level: 5, 6+ | |

This interactive Java learning tool, which was constructed using the cross-platform open educational resource GeoGebra, allows a teacher to demonstrate and explore evaluating algebraic expressions with students. Choose from four algebraic expressions and use sliders to select the value of the variable (-10 to 10) and any parameters. Show or hide the algebraic expression, the expression with substitution, the notation shown for multiplication or division, and the result. Values range from -10 to 10 and can be integers or intervals of one tenth. | Demonstration, Interactive Media Grade Level: 4, 5, 6+ | |

This interactive Java learning tool allows a teacher to demonstrate and explore with students number line models for two simple variable expressions for multiplication. The first expression is a variable with an understood coefficient of one and the second is the multiplication of a variable by a coefficient. Sliders are used to select the value of the variable (-10 to 10) and the coefficient (0 to 10). Check boxes show or hide the algebraic expression, the expression with substitution, the notation shown for multiplication, the result, and a model for the multiplication, shown as jumps along a number line. | Demonstration, Interactive Media Grade Level: 5, 6+ | |

This unit consists of four lessons in which students explore several meanings and representations of multiplication, including number lines, sets, arrays, and balance beams. They also learn about the commutative property of multiplication, the results of multiplying by 1 and by 0, and the inverse property of multiplication. Students write story problems and create pictographs. The unit includes activity sheets, assessment ideas, links to related applets, reflective questions for students and teachers, extensions and a bibliography of children's literature with a multiplication focus. | Activity, Instructional Strategy, Lesson Plans, Unit of Instruction Grade Level: 3, 4, 5 | |

In this 14-min video British teacher Rosalind Caren demonstrates group activities designed to develop number sense, fluency with addition and subtraction fact families, and reasoning skills. Caren exhibits effective questioning techniques and routines. Headteacher/math coordinator Kate Frood describes the guiding principles and expectations of teaching at the school. Following a class observation Frood provides constructive feedback to Caren and her teaching assistants. | Instructional Strategy, Lesson Plans, Video Grade Level: 1, 2 | |

This professional development video ( video 1) from Minnesota Math and Science Teacher Academy Center in Region 11 presents an algebra training session on justification. Viewers watch Dr. Terry Wyberg, a methods professor at the University of Minnesota, discuss five statements after teachers have had an opportunity to decide if each statement is always, sometimes or never true and a reason why. Dr. Wyberg shares that in math a response of always or never mean you need justification. Justification is central to mathematics and even young children cannot learn mathematics without engaging in justification. He also presents information about the use of math conventions with the use of algebraic representation for his audience. | Video Grade Level: 3, 4, 5 Resource is part of a PD collection | |

In this problem learners practice addition, subtraction and multiplication while developing logical reasoning. Solvers apply the given clues along with their understanding of the domino numbering system to determine which dominoes are missing from a set. The Teachers' Notes page includes suggestions for implementation, discussion questions, ideas for extension and support, printable sheets of dominoes (pdf), and a link to an interactive Dominoes Environment (cataloged separately). | Activity Grade Level: 3, 4, 5 | |

This set of eight interactive activities lets the user explore angles from many different perspectives. Activities include (1) visualizing the size of an angle; (2) examining objects that will stand or fall with right and non-right angles; (3) identifying obtuse, right, acute and straight angles; (4) guessing angle measures with different levels of precision; (5) exploring regular shapes and their angle measures; (6) studying angles in a fractal tree that is drawn with user inputs of the same angle measure between the branches at each stage; (7) exploring angle measures through firing a cannon (8) drawing with a Logo activity. | Activity, Game, Interactive Media Grade Level: 3, 4, 5 |