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1 48 and Curriculum Focal Points NCTM Curriculum Focal NCTM Curriculum Focal Points and, Grade 4 Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication. Students use understandings of multiplication to develop quick recall of the basic multiplication facts and related division facts , , , , 192, , , , Number and Numeration Goal 3 Find multiples of whole numbers less than 10; find whole-number factors of numbers. Operations and Computation Goal 3 Demonstrate automaticity with multiplication facts through 10 º 10 and proficiency with related division facts; use basic facts to compute fact extensions such as 30 º 60. Students apply their understanding of models for multiplication (i.e., equal-sized groups, arrays, area models, equal intervals on the number line), place value, and properties of operations (in particular, the distributive property) as they develop, discuss, and use efficient, accurate, and generalizable methods to multiply multidigit whole numbers , , , , , , , Operations and Computation Goal 7 Use repeated addition, skip counting, arrays, area, and scaling to model multiplication and division. Patterns, Functions, and Algebra Goal 4 Apply the Distributive Property of Multiplication over Addition to the partialproducts multiplication algorithm. Reprinted with permission from Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, copyright 2006 by the National Council of Teachers of Mathematics. All rights reserved. Please visit for more information.

2 NCTM Curriculum Focal Students select appropriate methods and apply them accurately to estimate products or calculate them mentally, depending on the context and numbers involved. Students develop fluency with efficient procedures, including the standard algorithm, for multiplying whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems , , , , , , , , , , , , , See also for Algorithms Handbook pp ,, and Algorithms in which contain online animations and associated print resources. Operations and Computation Goal 6 Make reasonable estimates for whole number and decimal addition and subtraction problems and whole number multiplication and division problems; explain how the estimates were obtained. Patterns, Functions, and Algebra Goal 4 Apply the Distributive Property of Multiplication over Addition to the partialproducts multiplication algorithm. Number and Operations: Developing an understanding of decimals, including the connections between fractions and decimals. Students understand decimal notation as an extension of the baseten system of writing whole numbers that is useful for representing more numbers, including numbers between 0 and 1, between 1 and 2, and so on , , , , , Grade 4 Correlation 49 Students relate their understanding of fractions to reading and writing decimals that are greater than or less than 1, identifying equivalent decimals, comparing and ordering decimals, and estimating decimal or fractional amounts in problem solving , , , , , , , Number and Numeration Goal 6 Compare and order whole numbers up to 1,000,000,000 and decimals through thousandths; compare and order integers between -100 and 0; use area models, benchmark fractions, and analyses of numerators and denominators to compare and order fractions.

3 50 and Curriculum Focal Points NCTM Curriculum Focal Students connect equivalent fractions and decimals by comparing models to symbols and locating equivalent symbols on the number line , , , , , , Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes. Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. Number and Numeration Goal 2 Read, write, and model fractions; solve problems involving fractional parts of a region or a collection; describe and explain strategies used; given a fractional part of a region or a collection, identify the unit whole , , Measurement and Reference Frames Goal 2 Students select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle. Algebra NCTM Connections to the Focal Students continue identifying, describing, and extending numeric patterns involving all operations and nonnumeric growing or repeating patterns , , , , , , , , , , , Measurement and Reference Frames Goal 2 Measurement and Reference Frames Goal 2 Patterns, Functions, and Algebra Goal 1 Extend, describe, and create numeric patterns; describe rules for patterns and use them to solve problems; use words and symbols to describe and write rules for functions that involve the four basic arithmetic operations and use those rules to solve problems.

4 NCTM Connections to the Focal Through these experiences, students develop an understanding of the use of a rule to describe a sequence of numbers or objects , , Patterns, Functions, and Algebra Goal 1 Extend, describe, and create numeric patterns; describe rules for patterns and use them to solve problems; use words and symbols to describe and write rules for functions that involve the four basic arithmetic operations and use those rules to solve problems. Geometry Students extend their understanding of properties of two-dimensional shapes as they find the areas of polygons. They build on their earlier work with symmetry and congruence in grade 3 to encompass transformations, including those that produce line and rotational symmetry. By using transformations to design and analyze simple tilings and tessellations, students deepen their understanding of twodimensional space , , , , , Geometry Goal 3 Identify, describe, and sketch examples of reflections; identify and describe examples of translations and rotations. Measurement As part of understanding two-dimensional shapes, students measure and classify angles , , , , Geometry Goal 1 Identify, draw, and describe points, intersecting and parallel line segments and lines, rays, and right, acute, and obtuse angles. Data Analysis Students continue to use tools from grade 3, solving problems by making frequency tables, bar graphs, picture graphs, and line plots. They apply their understanding of place value to develop and use stem-and-leaf plots , , , , , Data and Chance Goal 1 Collect and organize data or use given data to create charts, tables, bar graphs, line plots, and line graphs. Grade 4 Correlation 51 Number and Operations Building on their work in grade 3, students extend their understanding of place value and ways of representing numbers to 100,000 in various contexts , , , , , , , , ,

5 52 and Curriculum Focal Points NCTM Connections to the Focal Students use estimation in determining the relative sizes of amounts or distances. Students develop understandings of strategies for multidigit division by using models that represent division as the inverse of multiplication, as partitioning, or as successive subtraction. By working with decimals, students extend their ability to recognize equivalent fractions. Students earlier work in grade 3 with models of fractions and multiplication and division facts supports their understanding of techniques for generating equivalent fractions and simplifying fractions , , , , , , , , , , , , , , , , Operations and Computation Goal 6 Make reasonable estimates for whole number and decimal addition and subtraction problems, and whole number multiplication and division problems; explain how the estimates were obtained. Measurement and Reference Frames Goal 1 Estimate length with and without tools; measure length to the nearest 1_ 4 inch and 1_ centimeter; estimate the size of angles without tools. 2 Operations and Computation Goal 7 Use repeated addition, skip counting, arrays, area, and scaling to model multiplication and division , , , , , , ,