Glastonbury Public Schools Mathematics Curriculum

Grade 6

Grade 6 Big Idea: Variables represent unknowns in relationships.

Unknowns: How can symbolic representation of an unknown in a problem help us to solve problems?

Mathematics is learned through questions that arise while solving well-constructed problems. Our students begin with problems, they use strategies to solve the problems, and they learn the necessary mathematics along the way. Many classroom investigations are designed so that students will collaboratively or individually discover the mathematical properties. The properties are then discussed in class, summarized, and become part of the students’ mathematical knowledge to be applied to future problems.

The discovery of the mathematics is an essential part of the development of each student’s confidence as a mathematician. Knowledge that is gained through inquiry is more likely to be remembered for the long term. Teachers and parents work together to promote this discovery of math through investigation, problem solving, and reasoning. Our goal is for students to understand that mathematics makes sense.

Arithmetic in Base Ten

In this unit, students compute sums, differences, products, and quotients of multi-digit whole numbers and decimals, using efficient algorithms. They use calculations with whole numbers and decimals to solve problems set in real-world contexts.

Common Core State Standards: 6.NS.B.3, 6.EE.A.4, 6.NS.B.2, 6.NS.B.4,

Dividing Fractions

In this unit, students examine how the relative sizes of numerator and denominator affect the size of their quotient when numerator or denominator (or both) is a fraction. They acquire the understanding that dividing by ab has the same outcome as multiplying by b, then by 1a. They compute quotients of fractions. They solve problems involving lengths and areas of figures with fractional side lengths and extend the formula for the volume of a right rectangular prism to prisms with fractional edge lengths and use it to solve problems. They use tape diagrams, equations, and expressions to represent situations involving partitive or quotitive interpretations of division with fractions. Given a multiplication or division equation or expression with fractions, they describe a situation that it could represent. They use tape diagrams and equations in reasoning about situations that involve multiplication and division of fractions.

Common Core State Standards: 6.NS.A.1, 6.G.A.1, 6.G.A.2,

Area and Surface Area

Students will solve engaging geometric problems. The problems will allow for the brief review of addition, subtraction, and multiplication with decimals and fractions. In this unit, students learn to find areas of polygons by decomposing, rearranging, and composing shapes. They learn to understand and use the terms “base” and “height,” and find areas of parallelograms and triangles. Students approximate areas of non-polygonal regions by polygonal regions. They represent polyhedra with nets and find their surface areas. Throughout the unit, students will build the understanding that a generalized equation written with variables expresses a universal formula that can be used to solve problems. Finally, problems that involve finding the missing dimensions of polygons will create the need for a deeper understanding of variables and the understanding of solving equations by inspection.

Common Core State Standards: 6.G.A.1, 6.EE.A.2c, 6.EE.A.2a, 6.G.A.2, 6.G.A.4, 6.EE.A.1,

Introducing Ratios and Unit Rate

In this unit, students learn to understand and use the terms “ratio,” “rate,” “equivalent ratios,” “per,” “at this rate,” “constant speed,” and “constant rate,” and to recognize when two ratios are or are not equivalent. They represent ratios as expressions, and represent equivalent ratios with double number line diagrams, tape diagrams, and tables. They use these terms and representations in reasoning about situations involving color mixtures, recipes, unit pricing, and constant speed. 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. Students will uncover the understanding that the interpretation of the relationship of two quantities as a ratio facilitates problem solving.

Common Core State Standards: 6.RP.A.1, 6.RP.A.2, 6.RP.A.3(a-d)

Unit Rates and Percentages

In this unit, students learn to understand and use the terms “unit rate,” “speed,” “pace,” “percent,” and “percentage,” and recognize that equivalent ratios have equal unit rates. They represent percentages with tables, tape diagrams, and double number line diagrams, and as expressions. They use these terms and representations in reasoning about situations involving unit price, constant speed, and measurement conversion. Students further develop the understanding that the interpretation of two relationships as unit rates allows for comparison and problem solving.

Common Core State Standards: 6.RP.A.1, 6.RP.A.2, 6.RP.A.3, 6.G.A

Expressions and Equations

In this unit, students learn to understand and use the terms “variable,” “coefficient,” “solution,” “equivalent expressions,” “exponent,” “independent variable,” and “dependent variable.” They begin to write coefficients next to variables without a multiplication symbol, e.g., 10x rather than 10⋅x, and note that x is 1⋅x. They learn other situations in which the multiplication symbol can be omitted, e.g., 6⋅(3+2) can be written 6(3+2). They work with expressions that have positive whole-number exponents and whole-number, fraction, or variable bases, using properties of exponents strategically to evaluate these expressions, given a value for the variable. They find solutions for linear equations in one variable and simple equations that include exponents, e.g., 2x=32 and 100=x2. They use these terms and representations (including expressions with two variables) in reasoning about real-world and geometrical situations, understanding that some values of variables may not make sense in a given context. They represent collections of equivalent ratios as equations and use and make connections between tables, graphs, and linear equations that represent the same relationships.

Common Core State Standards: 6.NS.B.3, 6.EE.A.2, 6.RP.A.3b, 6.EE.B.5-7,6.EE.C.9,

Rational Numbers

In this unit, students interpret signed numbers in contexts (e.g., temperature above or below zero, elevation above or below sea level). They understand and use the terms “positive number,” “negative number,” “rational number,” “opposite,” “sign,” “absolute value,” “a solution to an inequality,” “less than,” “greater than,” and the corresponding symbols. They plot points with signed rational number coordinates on the number line, and recognize and use the connection between relative position of two points on the number line and inequalities involving the coordinates of the points. (These are limited to strict inequalities rather than inequalities such as 2≤x which occur in grade 7.) They understand and use absolute value notation, understanding that the absolute value of a number as its distance from zero on the number line. Students graph inequalities in one variable on number line diagrams, using a circle or disk to indicate when a given point is, respectively, excluded or included. They solve simple inequalities, understanding that there may be infinitely many solutions, and show solutions symbolically and on the number line. They interpret solutions of inequalities in contexts, understanding that some solutions do not make sense in some contexts. Students plot pairs of signed number coordinates in the plane, understanding the relationship between the signs of a pair of coordinates and the quadrant of the corresponding point, and use coordinates to calculate horizontal and vertical distances between two points. Students understand and use the terms “common factor,” “greatest common factor,” “common multiple,” and “least common multiple,” and solve problems set in real-world contexts in which common factors or multiples occur.

Common Core State Standards: 6.NS.B.4, 6.NS.C.5, 6.NS.C.6, 6.NS.C.7, 6.NS.C.8, 6.EE.B.2, 6.EE.B.5, 6.EE.B.6, 6.EE.B.7, 6.EE.B.8,

Data Sets and Distributions

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 will understand that the analysis of data answers research questions.

Common Core State Standards: 6.SP.A.1, 6.SP.A.2, 6.SP.A.3, 6.SP.B.4, 6.SP.B.5(a-d)

Putting it All Together

In this unit, students use concepts and skills from previous units. They may solve Fermi problems, use measurement conversions together with their knowledge of volumes or surface areas of right rectangular prisms or the relationship of distance, rate, and time. Or they may answer questions about ratios of two populations, work with percentages that include numbers expressed in the form a/b or as decimals. In answering questions about diagrams of rectangles with whole-number dimensions, they connect arithmetic features of the dimensions such as remainder or greatest common factor with geometric features of the diagrams. In answering questions about votes, voting methods, and equitable distribution, they use their knowledge of equivalent ratios, part–part ratios, percentages, and unit rates. During this last unit of sixth grade students revisit many of the topics from earlier in the year, especially expressions, equations, and ratios, making important mathematical connections and develop the understanding that variables represent unknowns in relationships.

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