Students will learn to see mathematics as a language, a tool, and an art form with which they can communicate ideas, solve problems, and explore the world around them. By the end of eighth grade, students will be taught to see multiple ways of expressing mathematical ideas, to make multiple connections to real life situations, and to work with others in exploring possibilities. Students will have an understanding of how numbers are used and represented. They will be able to estimate and use basic operations to solve everyday problems and confront more involved calculations in algebraic, geometric, and statistical settings.
As a result of the Math Common Core implementation, our curriculum has shifted to become more rigorous and focused K-8. Rigor includes fluency, application and deep understanding of the mathematics. Focus allows teachers the opportunity to help students develop a deep understanding of the concepts. With the shifts it is most important that children master the grade level curriculum. It is equally important that children are exposed to another major shift: a shift in student mathematical practices. Kindergarten through eighth grade classrooms across District #37 are implementing mathematical practice standards through tasks and projects, which include the following:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
Mathematicians in kindergarten through fifth grade focus on operations and algebraic thinking, numbers and operations in base ten, geometry, and measurement and data, in addition to the mathematical practices. Materials used for math instruction for kindergarten through fifth grade are based upon the unit math targets. The following domains are the focus of the content.
Operations and Algebraic Thinking
The progression in Operations and Algebraic Thinking deals with the basic operations—the kinds of quantitative relationships they model and consequently the kinds of problems they can be used to solve as well as their mathematical properties and relationships. Primary students develop meanings for addition and subtraction as they encounter problem situations in kindergarten, and they extend these meanings as they encounter increasingly difficult problem situations in grade 1. They represent these problems in increasingly sophisticated ways. And they learn and use increasingly sophisticated computation methods to find answers. By grade 3 students focus on understanding the meaning and properties of multiplication and division and on finding products of single-digit multiplying and related quotients. Fourth graders extend problem solving to multi-step word problems using the four operations posed with whole numbers. As preparation for the Expressions and Equations Progression in the middle grades, students in grade 5 begin working more formally with expressions.
Number and Operations in Base Ten
Students’ work in the base-ten system is intertwined with their work on counting and cardinality, and with the meanings and properties of addition, subtraction, multiplication, and division. Work in the base-ten system relies on these meanings and properties, but also contributes to deepening students’ understanding of them. Work with computation begins with use of strategies and “efficient, accurate, and generalizable methods.” In Kindergarten, teachers help children lay the foundation for understanding the base-ten system by drawing special attention to 10. In first grade, students learn to view ten ones as a unit called a ten. At grade 2, students extend their base-ten understanding to hundreds. At grade 3, the major focus is multiplication, so students’ work with addition and subtraction is limited to maintenance of fluency within 1000 for some students and building fluency to within 1000 for others. At grade 4, students extend their work in the base-ten system; they use standard algorithms to fluently add and subtract. In grade 5, students extend their understanding of the base-ten system to decimals to the thousandths place, building on their grade 4 work with tenths and hundredths. They become fluent with the standard multiplication algorithm with multi-digit whole numbers. They reason about dividing whole numbers with two-digit divisors, and reason about adding, subtracting, multiplying, and dividing decimals to hundredths.
Fractions (grades 3-5)
In grades 1 and 2, students use fraction language to describe partitions of shapes into equal shares. In grade 3 they start to develop the idea of a fraction more formally, building on the idea of partitioning a whole into equal parts. The whole can be a shape such as a circle or rectangle, a line segment, or any one finite entity susceptible to subdivision and measurement. In grade 4, this is extended to include wholes that are collections of objects. grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction. This forms the basis for much of their other work in grade 4, including the comparison, addition, and subtraction of fractions and the introduction of finite decimals. In grade 4, students have some experience calculating sums of fractions with different denominators in their work with decimals, and grade 5 students extend this reasoning to situations where it is necessary to re-express both fractions in terms of a new denominator.
Students in kindergarten, grade 1, and grade 2 focus on three major aspects of geometry. Students build understandings of shapes and their properties, becoming able to do and discuss increasingly elaborate compositions, decompositions, and iterations of the two, as well as spatial structures and relations. In
grade 2, students begin the formal study of measure, learning to use units of length and use and understand rulers. Measurement of angles and parallelism are a focus in grades 3, 4, and 5. At grade 3, students begin to consider relationships of shape categories, considering two levels of subcategories (e.g., rectangles are parallelograms and squares are rectangles). They complete this categorization in grade 5 with all necessary levels of categories and with the understanding that any property of a category also applies to all shapes
in any of its subcategories. They understand that some categories overlap (e.g., not all parallelograms are rectangles) and some are disjoint (e.g., no square is a triangle), and they connect these with their understanding of categories and subcategories. Spatial structuring for two- and three-dimensional regions is used to understand what it means to measure area and volume of the simplest shapes in those dimensions: rectangles at grade 3 and right rectangular prisms at grade 5.
Measurement and Data
As students work with data in grades K-5, they strengthen and apply what they are learning in arithmetic. Kindergarten work with data involves counting and order relations. First- and second-graders solve addition and subtraction problems in a data context. In grades 3–5, work with data is closely related to the number line, fraction concepts, fraction arithmetic, and solving problems that involve the four operations. In geometric measurement, second graders learn to measure length with a variety of tools, such as rulers, meter sticks, and measuring tapes. Second graders also learn the concept of the inverse relationship between the size of the unit of length and the number of units required to cover a specific length or distance. Third graders focus on solving real-world and mathematical problems involving perimeters of polygons. Students in grade 3 learn to solve a variety of problems involving measurement and such attributes as length and area, liquid volume, mass, and time. In grade 4, students build on competencies in measurement and in understanding units that they have developed in number, geometry, and geometric measurement. Students also combine competencies from different domains as they solve measurement problems using all four arithmetic operations, addition, subtraction, multiplication, and division. In grade 5, students extend their abilities from grade 4 to express measurements in larger or smaller units within a measurement system. Grade 5 students also learn and use such conversions in solving multi-step, real world problems.
Mathematicians in fifth through eighth grade focus on units specific to course placement in addition to the mathematical practices. Materials used for math instruction for fifth through eighth grade are based upon the unit math targets.
Students working with the grades 6-8 learning targets in Grades 6-8 Math focus on the following general domains: Ratio and Proportional Relationships; the Number System; Expressions and Equations; Geometry; and, Statistics and Probability. Grades 7 and 8 Algebra content and units include the Number System; Expressions and Equations; Geometry; Statistics and Probability as well as an introduction to Functions. The general descriptions about these domains are described below. Grade 8 Geometry content and units include Circles; Congruence; Geometric Measurement and Dimension; Expressing Geometric Properties with Equations; Modeling with Geometry; and Similarity, Right Triangles, and Trigonometry.
Ratio and Proportional Relationships
The study of ratios and proportional relationships extends students’ work in measurement and in multiplication and division in the elementary grades. Students in grade 6 focus on understanding the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. They also build an understanding of the concept of a unit rate associated with a ratio and use rate language in the context of a ratio relationship.
In grade 7, students extend their reasoning about ratios and proportional relationships in several ways. Students use ratios in cases that involve pairs of rational number entries, and they compute associated unit rates. They identify these unit rates in representations of proportional relationships. They work with equations in two variables to represent and analyze proportional relationships. They also solve multi-step ratio and percent problems, such as problems involving percent increase and decrease.
The Number System
Within the grades 6-8 learning targets, students build on two important conceptions which have developed throughout K–5, in order to understand the rational numbers as a number system. The first is the representation of whole numbers and fractions as points on the number line, and the second is a firm understanding of the properties of operations on whole numbers and fractions. As grade 6 begins, students have a firm understanding of place value and the properties of operations. On this foundation they are ready to start using the properties of operations as tools of exploration, deploying them confidently to build new understandings of operations with fractions and negative numbers. They are also ready to complete their growing fluency with algorithms for the four operations. In grade 6 students learned to locate rational numbers on the number line; in grade 7 they extend their understanding of operations with fractions to operations with rational numbers. Students must rely increasingly on the properties of operations to build the necessary bridges from their previous understandings to situations where one or more of the numbers might be negative. In grade 7 students encountered infinitely repeating decimals, and in grade 8 they understand why this phenomenon occurs, a good exercise in expressing regularity in repeated reasoning. In addition, they glimpse the existence of irrational numbers.
Expressions and Equations
In grades 6-8, students start to use properties of operations to manipulate algebraic expressions and produce different but equivalent expressions for different purposes. This work builds on their extensive experience in K–5 working with the properties of operations in the context of operations with whole numbers, decimals and fractions. In grade 6 they begin to work systematically with algebraic expressions, and they start to incorporate whole number exponents into numerical expressions. In grade 7 students start to simplify general linear expressions with rational coefficients. Building on work in grade 6, where students used conventions about the order of operations to parse, and properties of operations to transform, simple expressions, students now encounter linear expressions with more operations and whose transformation may require an understanding of the rules for multiplying negative numbers. In grade 8 students add the properties of integer exponents to their repertoire of rules for transforming expressions. They prepare in grade 8 by starting to work systematically with the square root and cube root symbols. They begin to understand the idea of a function. Students in grade 8 also start to solve problems that lead to simultaneous equations.
Students working through the grade 6 learning targets work with problems involving areas and volumes to extend previous work and to provide a context for developing and using equations. Students’ competencies in shape composition and decomposition, especially with spatial structuring of rectangular arrays should be highly developed. These competencies form a foundation for understanding multiplication, formulas for area and volume, and the coordinate plane. Using the shape composition and decomposition skills acquired in earlier grades, students learn to develop area formulas for parallelograms, then triangles. They learn how to address three different cases for triangles: a height that is a side of a right angle, a height that “lies over the base” and a height that is outside the triangle.
Composition and decomposition of shapes are used throughout geometry from grade 6 to high school and beyond. Compositions and decompositions of regions continues to be important for solving a wide variety of area problems, including justifications of formulas and solving real world problems that involve complex shapes. Decompositions are often indicated in geometric diagrams by an auxiliary line, and using the strategy of drawing an auxiliary line to solve a problem are part of looking for and making use of structure. Recognizing the significance of an existing line in a figure is also part of looking for and making use of structure. This may involve identifying the length of an associated line segment, which in turn may rely on students’ abilities to identify relationships of line segments and angles in the figure. These abilities become more sophisticated as students gain more experience in geometry. In grade 7, this experience includes making scale drawings of geometric figures and solving problems involving angle measure, surface area, and volume.
Statistics and Probability
Through the grade 6 learning targets, students build on the knowledge and experiences in data analysis developed in earlier grades. Students working through the grade 7 standards move from concentrating on analysis of data to production of data, understanding that good answers to statistical questions depend upon a good plan for collecting data relevant to the questions of interest. The grade 8 mathematics standards have students apply their experience with the coordinate plane and linear functions in the study of association between two variables related to a question of interest.
Before they learn the term “function,” students begin to gain experience with functions in elementary grades. In kindergarten, they use patterns with numbers to learn particular additions and subtractions. A trickle of pattern standards in grades 4 and 5 continues the preparation for functions where a rule is explicitly given. The grades 4–5 pattern standards expand to the domain of Ratios and Proportional Relationships in grades 6–7. In grade 6, as they work with collections of equivalent ratios, students gain experience with tables and graphs, and correspondences between them. In grade 7, students recognize and represent an important type of regularity in these numerical tables—the multiplicative relationship between each pair of values—by equations of the form y = cx , identifying c as the constant of proportionality in equations and other representations. The notion of a function is formally introduced in Grade 8 Math. Linear functions are a major focus, but students are also expected to give examples of functions that are not linear.
Grades 7 and 8 Algebra is a combination of domains of the Grade 8 Algebra content and High School domains, and these include the following: Geometry; Statistics & Probability; Seeing Structure in Expressions; Creating Equations; Reasoning with Equations & Inequalities; Building Functions; Interpreting Functions; Quantities; Interpreting Categorical & Quantitative Data; Linear, Quadratic, & Exponential Models; Arithmetic with Polynomials & Rational Expressions; and, the Real Number System.
Grade 8 Geometry is a combination of the following High School domains: Circles; Congruence; Geometric Measurement and Dimension; Expressing Geometric Properties with Equations; Modeling with Geometry; and Similarity, Right Triangles, and Trigonometry. The Geometry sequence is currently being developed, and the resources used for the program are aligned with the Grant High School Geometry course.
The Illinois Learning Standards are correlated with the Common Core State Standards: Common Core State Standards. District 37's learning targets are aligned with the Illinois Learning Standards.
The Math Proficiency maps below show the math standards and learning target sequences for each course. Assessments are aligned to these proficiency maps.