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Ramapo Ridge Middle School

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Mathematics

Sixth Grade Math

In Grade 6 Math, instructional time will focus on connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; writing, interpreting, and using expressions and equations; and developing understanding of statistical thinking. Students will 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 will also build on their work with area in elementary school by reasoning about relationships among shapes to determine area, surface area, and volume. They will 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 will discuss, develop, and justify formulas for areas of triangles and parallelograms. Students will find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They will reason about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths.


Pre-Algebra 7

In Pre-Algebra 7, instructional time will focus on students extending their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students will use their understanding of ratios and proportionality to solve a wide variety of percent problems. Students will graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They will distinguish proportional relationships from other relationships. Students will develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. They will use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems. Students will continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity in Grade 8, they will reason about relationships among two-dimensional figures. They will solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. Students will build on their previous work with single data distributions to compare two data distributions and address questions about differences between populations. They will begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences.


Pre-Algebra 7/8

In Pre-Algebra 7/8, instructional time will focus on students developing a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students will recognize that every real number can be classified according to its characteristics. They will use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems. Students will work with numbers written in various forms, including powers and exponents. Students will extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students will graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They will distinguish proportional relationships from other relationships. Students will further their understanding of linear relationships to write and graph lines in the form y = mx + b and interpret the significance of the variables for real-world problems. Students will build on their previous work with single data distributions to compare two data distributions and address questions about differences between populations. They will begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences. Students will continue their work with solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. They
will solve real-world and mathematical problems involving area, surface area, and volume of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. Students will complete their work on volume by solving problems involving cones, cylinders, and spheres.


Pre-Algebra 8

In Pre-Algebra 8, instructional time will focus on students using linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students will also use a linear equation to describe the association between two quantities in bivariate data (ex: arm span vs. height for students in a classroom). Students will grasp the concept of a function and using functions to describe quantitative relationships. They will understand that functions describe situations where one quantity determines another, and they will describe how aspects of the function are reflected in the different representations. Students will use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students will understand the statement of the Pythagorean Theorem and its converse, and will explain why the Pythagorean Theorem holds. They will apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students will complete their work on volume by solving problems involving cones, cylinders, and spheres.


Algebra

In Algebra, many of the concepts presented in Algebra I are progressions of the concepts that were started in grades 6 through 8 in the Common Core; the content presented in this course is intended to extend and deepen the previous understandings. Algebra 1 is designed to help the student learn and use the language of mathematics to model real world patterns and phenomena and to solve problems. It includes a study of the interrelationship of algebraic, numerical and graphical representations of data. Algebra 1 makes the transition from the specifics of arithmetic to the generalizations of higher mathematics. Students will be using expressions and equations and be expected to understand quantities and the relationships between them. Students will be comparing and contrasting linear, absolute value, quadratic, and exponential functions in numerical, graphical and algebraic forms. Lastly, students work with descriptive statistics. Throughout the course, students will model mathematics with applicable data and situations. Additional topics for this course include: Solving and Graphing Linear Equations, Functions, Linear Inequalities, Absolute Value Equations and Inequalities, Solving Systems, Operations with Polynomials, Quadratics, Exponential Equations and Functions, Radical Expressions and Rational Exponents, Arithmetic and Geometric Models.


In addition to the appropriate content standards, all math courses embed the Standards for Mathematical Practice (SMP) into the daily curriculum. These process standards describe varieties of expertise that mathematics educators at all levels seek when developing mathematically thinking students. These practices rest on important processes and proficiencies with longstanding importance in mathematics education.

Make sense of problems and persevere in solving them The student makes meaning of a problem by analyzing givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution. There is a clear solution pathway demonstrated and exemplary students check their answers to problems using various methods. They can understand the approaches of others to solving complex problems and identify correspondences between approaches.

Reason abstractly and quantitatively The student demonstrates the ability to abstract and contextualize a given situation and represent it symbolically. Quantitative reasoning entails habits of creating a coherent representation of the problem, considering units involved; attending to the meaning of quantities. Formal mathematics language is used throughout the solution to share and clarify ideas.

Construct viable arguments and critique the reasoning of others The student determines appropriate domain, make conjectures, and builds a logical progression of statements to explore the validity of a conjecture. The student is able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed.The student can make sense of the arguments of others, decide whether they make sense, and ask useful questions to clarify the arguments.

Model with mathematics The student can apply the mathematics they know to solve complicated problems arising in everyday life, society, and the workplace. The student is able to identify quantities in a practical situation and map their relationships using tools such as diagrams, twoway tables, graph, flowcharts, and formulas. The student can analyze those relationships mathematically to draw conclusions.

Use appropriate tools strategically A correct strategy is chosen based on the mathematical situation. Evidence of solidifying prior knowledge and applying it to the problem-solving situation is present. The student is able to use a variety of tools, including technology, to explore and deepen their understanding of concepts.

Attend to precision The student must achieve a correct answer. The student is careful about specifying units of measure, labeling axes and calculating accurately and efficiently, expressing numerical answers with an appropriate degree of precision. Formal mathematical language is used throughout the solution to share and clarify ideas.

Look for and make use of structure The student identifies patterns and revises or edits them to solve problems.

Look for and express regularity in repeated reasoning Planning or monitoring of strategy is evident, noting patterns, structures, and regularities. Mathematical connections or observations are recognized.

Danielle Poleway

Supervisor of Mathematics

201-762-2322

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