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Georgia Standards of Excellence states the following:

GSE Mathematics Grade 6 Critical Areas The middle school standards specify the mathematics that all students should study in order to be high school ready. The middle school standards are listed in conceptual categories including Number Sense, Algebra, Expressions and Equations, Geometry, and Statistics and Probability.

In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

Descriptions of the four critical areas follow: (1) Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates. (2) Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems. Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane. (3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3𝑥 = 𝑦) to describe relationships between quantities.

(4) Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. Students recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected.

Students in Grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They prepare for work on scale drawings and constructions in Grade 7 by drawing polygons in the coordinate plane.

GSE Mathematics Grade 6 Unit Descriptions The sixth grade standards are arranged into units that will extend their knowledge and understanding of elementary topics into increasingly formalized and applicable skills as they transition into upper grades. The Standards for Mathematical Practice are a key component as they are applied in each course to equip students in making sense of problems and building a set of tools they can use in real-world situations. Rather than racing to cover many topics in a “mile-wide, inch-deep curriculum”, the standards ask mathematics teachers to significantly narrow and deepen the way time and energy are spent in the classroom. The elementary grades focused on concepts, skills, and problem solving related to addition and subtraction, multiplication and division of whole numbers, and fractions. In grade 6, the focus is on ratios and proportional relationships, and early algebraic expressions and equations.

Unit 1: By the end of fifth grade, students have had a variety of experiences working with whole numbers and fractions. In the first unit, work with whole numbers continues into dividing multi-digit numbers using the standard algorithm. All four operations with decimals, as well as dividing fractions by fractions, are emphasized from a hands-on approach in order to build understanding, not rely on memorization of rules and procedures. Students also find common factors and multiples as they progress in their understanding of composition and decomposition of numbers and become fluent in number sense.

Unit 2: Ratios and rational relationships form an important undergirding of the entire sixth grade mathematics curriculum. Understanding ratio and “rational thinking” is critical to all future mathematics courses, and from the second unit throughout the year students revisit and continue to use the skills developed in this unit as they explore other topics. Their work with ratios includes unit rate and using rate to solve real-world problems.

Unit 3: The formal study of algebra begins in earnest in sixth grade, as students move from arithmetic understandings to algebraic expressions. Students learn to translate verbal phrases into algebraic expressions and utilize exponential notation in appropriate situations.

Unit 4: Extending the work begun in Unit 3, Unit 4 has students reason about and solve one-variable equations and inequalities. Mathematics is all about answering questions, finding the solutions to unknowns, and making sense of real-life situations. Students also learn that often two things are not balanced or equal, but are unequal, and they explore inequalities using tools such as number lines to become fluent in grasping the magnitude of numbers.

Unit 5: The study of geometry is interesting and fun for many students, as it is often more concrete and visual than some other domains of mathematics. Sixth grade students extend their understanding of the meaning of area and volume from elementary grades, now often having fractional edge lengths to work with instead of only whole number lengths. This represents the types of measurements they very often encounter in real-life, and helps students understand magnitude and applications of operations on fractional numbers. Additionally, the fifth unit has students find area by composing and decomposing figures into familiar shapes, triangles and rectangles. They also use nets of three-dimensional figures to find surface area.

Unit 6: Sixth grade provides the first formal introduction to the study of statistics. Students begin by learning what questions are statistical in nature. That is, they are questions which will generate a range of responses. Unit 6 introduces the idea that data can be collected to answer a statistical question, then described by its center, spread, and overall shape. Statistical measures allow the description of a set of data and the spread of the data in single number summary, and tasks in Unit 6 acquaint students with this new domain.

Unit 7: Up to this point, students have only encountered numbers with values greater than or equal to zero (Natural Numbers, Counting Numbers, Whole Numbers). Unit 7 introduces students conceptually to circumstances best described with negative numbers, numbers with a value less than zero- the set of Integers. Operations with Integers are deliberately postponed to seventh grade, but by introducing students to Integers in sixth grade, they have the opportunity to explore situations appropriately represented by negative numbers, and graph points in all four quadrants of the coordinate plane. Using a number line, students learn about numbers and their “opposites”, and absolute value (distance from zero). This unit is intentionally at the end of sixth grade, as students are NOT expected to do any operations with Integers. Instead, this unit is to be an introduction. It leads directly into the first seventh grade unit, Operations with Rational Numbers.

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