In this module, students build their competencies in measurement as they relate multiplication to the conversion of measurement units. Throughout the module, students explore multiple strategies for solving measurement problems involving unit conversion.
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In Module 1, student's understandings of the patterns in the base ten system are extended from Grade 4's work with place value to include decimals to the thousandths place. In Grade 5, students deepen their knowledge through a more generalized understanding of the relationships between and among adjacent places on the place value chart, e.g., 1 tenth times any digit on the place value chart moves the digit one place value to the right. Toward the module's end, students apply these new understandings as they reason about and perform decimal operations through the hundredths place.
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In Module 2, students apply the patterns of the base ten system to mental strategies and multiplication and division algorithms.
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In Module 3, student's understanding of addition and subtraction of fractions extends from earlier work with fraction equivalence and decimals. This module marks a significant shift away from the elementary grades' centrality of base ten units to the study and use of the full set of fractional units from Grade 5 forward, especially as applied to algebra.
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In Module 4, students learn to multiply fractions and decimal fractions, and begin working with fraction division. Topic A opens the 38-day module with an exploration of fractional measurement. Students construct line plots by measuring the same objects using three different rulers accurate to 12 , 14 , and 18 of an inch.
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In this 25-day module, students work with two- and three-dimensional figures. Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms. The second half of the module turns to extending student's understanding of two-dimensional figures. Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths. They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes. This module fills a gap between Grade 4's work with two-dimensional figures and Grade 6's work with volume and area.
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In this 40-day module, students develop a coordinate system for the first quadrant of the coordinate plane and use it to solve problems. Students use the familiar number line as an introduction to the idea of a coordinate and construct two perpendicular number lines to create a coordinate system on the plane. They see that just as points on the line can be located by their distance from 0, the plane's coordinate system can be used to locate and plot points using two coordinates. They then use the coordinate system to explore relationships between points, ordered pairs, patterns, lines and, more abstractly, the rules that generate them. This study culminates in an exploration of the coordinate plane in real-world applications.
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In this module, students are introduced to the concepts of ratio and rate. Their previous experience solving problems involving multiplicative comparisons, such as 'Max has three times as many toy cars as Jack,' serves as the conceptual foundation for understanding ratios as a multiplicative comparison of two or more numbers used in quantities or measurements. Students develop fluidity in using multiple forms of ratio language and ratio notation. They construct viable arguments and communicate reasoning about ratio equivalence as they solve ratio problems in real-world contexts.
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In Module 2, students complete their understanding of the four operations as they study division of whole numbers, division by a fraction, and operations on multi-digit decimals. This expanded understanding serves to complete their study of the four operations with positive rational numbers, thereby preparing students for understanding, locating, and ordering negative rational numbers (Module 3) and algebraic expressions (Module 4).
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Students are familiar with the number line and determining the location of positive fractions, decimals, and whole numbers from previous grades. Students extend the number line (both horizontally and vertically) in Module 3 to include the opposites of whole numbers. The number line serves as a model to relate integers and other rational numbers to statements of order in real-world contexts. In this module's final topic, the number line model is extended to two-dimensions, as students use the coordinate plane to model and solve real-world problems involving rational numbers.
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In Module 4, students extend their arithmetic work to include using letters to represent numbers. Students understand that letters are simply "stand-ins" for numbers and that arithmetic is carried out exactly as it is with numbers. Students explore operations in terms of verbal expressions and determine that arithmetic properties hold true with expressions because nothing has changedšthey are still doing arithmetic with numbers. Students determine that letters are used to represent specific but unknown numbers and are used to make statements or identities that are true for all numbers or a range of numbers. Students understand the importance of specifying units when defining letters.
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Starting in Grade 1 , students compose and decompose plane and solid figures. They move to spatial structuring of rectangular arrays in Grade 2 and continually build upon their understanding of arrays to ultimately apply their knowledge to two- and three-dimensional figures in Grade 4 and Grade 5. Students move from building arrays to using arrays to find area and eventually move to decomposing three-dimensional shapes into layers that are arrays of cubes. In this module, students utilize their previous experiences in shape composition and decomposition in order to understand and develop formulas for area, volume, and surface area.
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In this module, students move from simply representing data into analyzing data. In Topic A, students begin to think and reason statistically by first recognizing a statistical question as one that can be answered by collecting data. Students learn that the data collected to answer a statistical question have a distribution that is often summarized in terms of center, variability, and shape. Beginning in Topic A, and throughout the module, students see and represent data distributions using dot plots and histograms.
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In Module 1, students build upon their Grade 6 reasoning about ratios, rates, and unit rates to formally define proportional relationships and the constant of proportionality. In Topic A, students examine situations carefully to determine if they are describing a proportional relationship. Their analysis is applied to relationships given in tables, graphs, and verbal descriptions.
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In Grade 6, students formed a conceptual understanding of integers through the use of the number line, absolute value, and opposites and extended their understanding to include the ordering and comparing of rational numbers. This module uses the Integer Game, a card game that creates a conceptual understanding of integer operations and serves as a powerful mental model students can rely on during the module. Students build on their understanding of rational numbers to add, subtract, multiply, and divide signed numbers. Previous work in computing the sums, differences, products, and quotients of fractions and decimals serves as a significant foundation as well.
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In Grade 6, students interpreted expressions and equations as they reasoned about one-variable equations. This module consolidates and expands upon student's understanding of equivalent expressions as they apply the properties of operations (associative, commutative, and distributive) to write expressions in both standard form (by expanding products into sums) and in factored form (by expanding sums into products). They use linear equations to solve unknown angle problems and other problems presented within context to understand that solving algebraic equations is all about the numbers. It is assumed that a number already exists to satisfy the equation and context; we just need to discover it. A number sentence is an equation that is said to be true if both numerical expressions evaluate to the same number; it is said to be false otherwise. Students use the number line to understand the properties of inequality and recognize when to preserve the inequality and when to reverse the inequality when solving problems leading to inequalities. They interpret solutions within the context of problems. Students extend their sixth-grade study of geometric figures and the relationships between them as they apply their work with expressions and equations to solve problems involving area of a circle and composite area in the plane, as well as volume and surface area of right prisms.
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In Module 4, students deepen their understanding of ratios and proportional relationships from Module 1 by solving a variety of percent problems. They convert between fractions, decimals, and percents to further develop a conceptual understanding of percent (introduced in Grade 6 Module 1) and use algebraic expressions and equations to represent and solve multi-step percent scenarios. An initial focus on relating 100% to the whole serves as a foundation for students. Students begin the module by solving problems without the use of a calculator to develop a greater fluency and deeper reasoning behind calculations with percent. Material in early lessons is designed to reinforce student's understanding by having them use mental math and basic computational skills. To develop a conceptual understanding, students use visual models and equations, building on earlier work with these strategies. As the lessons and topics progress and more complex calculations are required to solve multi-step percent problems, teachers may let students use calculators so that their computational fluency does not interfere with the primary concept(s) being addressed.
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In this module, students begin their study of probability, learning how to interpret probabilities and how to compute probabilities in simple settings. They also learn how to estimate probabilities empirically. The concept of probability provides a foundation for the thinking required to make inferential reasoning that is developed in the second half of this module. Additionally, students build on their knowledge of data distributions that they studied in Grade 6, compare data distributions of two or more populations, and are introduced to the idea of drawing informal inferences based on data collected from random samples.
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In Module 6, students delve further into several geometry topics they have been developing over the years. Grade 7 presents some of these topics (e.g., angles, area, surface area, and volume) in the most challenging form students have experienced yet. Module 6 assumes students understand the basics; the goal is to build fluency in these difficult problems. The remaining topics (i.e., working on constructing triangles and taking slices, or cross sections, of three-dimensional figures) are new to students.
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In Module 1, student's knowledge of operations on numbers is expanded to include operations on numbers in integer exponents. Module 1 also builds on student's understanding from previous grades with regard to transforming expressions. Students were introduced to exponential notation in Grade 5 as they used whole number exponents to denote powers of ten. In Grade 6, students expanded the use of exponents to include bases other than ten as they wrote and evaluated exponential expressions limited to whole-number exponents. Students made use of exponents again in Grade 7 as they learned formulas for the area of a circle and volume.
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