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Wheeler Lower School Mathematics Program Grades 4-5 Goals: 1.For all students to become mathematically proficient 2.To prepare students for success in middle and upper school

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What do children need to become mathematically proficient? Strong Understanding of Mathematics Understanding what concepts mean and how they interconnect with one another

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Problem Solving Skills Ability to use knowledge to solve problems in a variety of contexts and real world situations Computational Fluency Ability to add, subtract, multiply and divide efficiently and accurately

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Mathematical Reasoning Ability to use logic and deductive thinking Meta-Cognition Ability to reflect on one’s own thinking and process

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Communication Skills Ability to use mathematics language to share thinking and justify solutions A Set of Beliefs, Dispositions & Attitudes A willingness to engage with mathematics, seeing it as sensible, useful, and doable

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Mathematics in Fourth Grade – Content & Skills Number and Operations: Whole Numbers Work focuses on extending knowledge of the base-ten number system to 10,000. Multiplication and division are the major focus of students ’ work in number and operations in Grade 4. Students use models, representations, and story contexts to help them understand and solve multiplication and division problems. In addition and subtraction, students refine and compare strategies for solving problems with 3-4 digits. By the end of the year, students are expected to solve addition and subtraction problems efficiently; know their multiplication combinations to 12 x 12 and use the related division facts, and to solve 2 x 2 digit multiplication problems and division problems with 1-2 digit divisors. Number and Operations: Fractions and Decimals The major focus of work is on building students ’ understanding of the meaning, order, and equivalencies of fractions and decimals. They work with fractions in the context of area, as a group, and on a number line. Students are introduced to decimal fractions as an extension of the place value system. They reason about fraction comparisons, order fractions on a number line, and use representations and reasoning to add fractions and decimals.

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Geometry and Measurement Students expand their understanding of how the attributes of two-dimensional (2-D) and three- dimensional (3-D) shapes determine their classification. Students consider attributes of 2-D shapes, such as number of sides, the length of sides, parallel sides, and the size of angles. Students also describe attributes and properties of geometric solids (3-D shapes). Measurement work includes linear measurement (with both U.S standard and metric units), area, angle measurement, and volume. Students work on understanding volume by structuring and determining the volume of a rectangular prism. Patterns and Functions Students create tables and graphs for situations with a constant rate of change and use them to compare related situations. By analyzing tables and graphs, students consider how the starting amount and the rate of change define the relationship between the two quantities and develop rules that govern that relationship. Data Analysis and Probability Students collect, represent, describe, and interpret numerical data. Their work focuses on describing and summarizing data for comparing two groups. They develop conclusions and make arguments, based on the evidence they collect. In their study of probability, students work on describing and predicting what events are impossible, unlikely, likely, or certain. Students reason about how the theoretical chance (or theoretical probability) of, for example, rolling 1 on a number cube compares to what actually happens when a number cube is rolled repeatedly.

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Mathematics in Fifth Grade – Content & Skills Number and Operations: Whole Numbers Students practice and refine the strategies they know for addition, subtraction, multiplication, and division of whole numbers as they improve computational fluency and apply these strategies to solving problems with larger numbers. They expand their knowledge of the structure of place value and the base-ten number system as they work with numbers in the hundred thousands and beyond. By the end of the year, students are expected to know their division facts and to efficiently solve computation problems involving whole numbers for all operations. Number and Operations: Fractions, Decimals, and Percents The major focus of the work with rational numbers is on understanding relationships among fractions, decimals, and percents. Students make comparisons and identify equivalent fractions, decimals and percents. They order fractions and decimals, and develop strategies for adding fractions and decimals to the thousandths.

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Geometry and Measurement Students develop their understanding of the attributes of 2D shapes, examine the characteristics of polygons, including a variety of triangles, quadrilaterals, and regular polygons. They also find the measure of angles of polygons. In measurement, students use standard units of measure to study area and perimeter and to determine the volume of prisms and other polyhedra. Patterns and Functions Students examine, represent, and describe situations in which the rate of change is constant. They create tables and graphs to represent the relationship between two variables in a variety of contexts and articulate general rules using symbolic notation for each situation. Students create graphs for situations in which the rate of change is not constant and consider why the shape of the graph is not a straight line. Data Analysis and Probability. Work focuses on comparing two sets of data collected from experiments developed by the students. They represent, describe, and interpret this data. In their work with probability, students describe and predict the likelihood of events and compare theoretical probabilities with actual outcomes of many trials. They use fractions to express the probabilities of the possible outcomes.

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Multiplication Strategies Used by Wheeler Students Arrays model multiplication in terms of the area of a rectangle. 4 rows of 8 squares is 32 squares or the product of the dimensions of the rectangle. Example: 4 x 8 = 32 Building Arrays This strategy provides a concrete visual model of ‘how many rows of how many squares.’

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Example 12 x 8 = + (2 x 8) / Children begin to develop a sense of double digit by single digit multiplication by seeing larger arrays as the sum of smaller arrays they already know. 80 + 16 = 96 (10 x 8)

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Example 12 x 8 = 4 x 8 + 4 x 8 + 4 x 8 = 96 Children develop deeper understanding of multiplication when they find more than one way to solve a problem.

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Box Method Multiplication This box is an abstraction of the gridded array. The smaller boxes show the partial products of the multiplication problem. Example: 12 X 6 6 6012 10 + 2 60 + 12 = 72

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Double Digit by Double Digit Example: 12 x 13 10 + 3 10 + 2 100 30 206 100 + 30 + 20 + 6 = 156

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Example: 24 x 5 20 + 4 X 5 20 100 120 Expanded Notation Multiplication Example: 12 x 23 10 + 2 x 20 + 3 6 30 40 200 276

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27 323 x 6 x 7 42 21 120 140 162 2100 2261 Partial Products Multiplication No need for x-ing out decimal places. No need for funny little numbers scribbled on the top of the problem. Students make less place value errors when they write down the entire partial product.

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Division Strategies Used At Wheeler 168 ÷ 14 = 12 645 ÷ 6 = 107 3/6 14 168 - 140 28 - 28 0 12 10 2 645 - 600 45 - 36 9 - 6 3 6 100 6 1 107 R 3 Subtracting Out Convenient Groups

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Expanded Notation 945 ÷ 9 = 105 900 + 459 100 + 5 777 ÷ 24 = 105 480 + 240 + 48 + 924 20 + 10 + 2 R 9/24

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How Can Parents Help Children Become Mathematically Proficient? Help them develop automaticity of basic facts with addition, subtraction, multiplication and division. Encourage children to independently work through nightly homework before you step in and help. Instead of showing them the way you learned in school, ask about the different strategies they are learning.

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Keep pointing out the different ways that you use mathematics in your life so your children understand how meaningful it is. Make sure they understand that being good at mathematics doesn’t come from a special gene, but is something that is learned and developed over time. Ask them to share their thinking with you and talk about the steps they took to solve a problem and how they know their solutions are correct.

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