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# Middle School Math

Students are required to take a full year of Mathematics in grades 5, 6, 7, and 8. Students are required to take a full year of Mathematics in grades 5, 6, 7, and 8. Middle school students have the opportunity to accelerate in mathematics beginning in 7th grade and work toward taking the Algebra 1 Regents Exam at the end of 8th grade.

Our curriculum is designed around the New York State Learning Standards for Mathematics and focuses on developing a deep understanding of math concepts, proficiency with key skills, and the ability to solve new and complex problems. The idea behind our instructional approach is not to just to do math, but also to understand how and why we do math. Students need to be engaged in dialogue and learning experiences that allow complex topics and ideas to be explored from many angles and perspectives. They also need to learn how to think and solve problems for which there is no one solution—and learn mathematical skills along the way. This content-rich approach serves as a means to advance children’s 21st-century mathematical skills —their abstract reasoning, their collaboration skills, their ability to learn from peers and through technology, and their flexibility as a learner in a dynamic learning environment.

In order to assist schools and districts with the implementation of the New York State Learning Standards for Math, the New York State Education Department has provided curricular materials called math modules, which are utlized as a resource in Williamsville and are available by clicking HERE. These math modules are marked by in-depth focus on fewer topics. They integrate the standards, rigorous classroom reasoning, extended classroom time devoted to practice and reflection through extensive problem sets, and high expectations for mastery.

In Grade 5, instructional time focuses on three areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimals into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.

1. Through their learning in the Number and Operations – Fractions and Operations and Algebraic Thinking domains, students:
• apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators;
• develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them; and
• use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)
2. Through their learning in the Operations and Algebraic Thinking and Number and Operations in Base Ten domains, students:
• develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations;
• apply understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths;
• develop fluency with decimal computations to hundredths, and make reasonable estimates of their results; and
• use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense.
3. Through their learning in the Measurement and Data and Geometry domains, students:
• recognize volume as an attribute of three-dimensional space;
• understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps;
• understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume;
• select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume;
• decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes; and
• measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

In Grade 6, instructional time focuses on five 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; (4) deepening understanding of area, surface area and volume; and (5) developing understanding of simple probabilities and statistical thinking.

1. Through their learning in the Ratios and Proportional Relationships domain, students:
• use reasoning about multiplication and division to solve ratio and rate problems about quantities;
• connect understanding of multiplication and division with ratios and rates 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; and
• expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions.
2. Through their learning in the Number System domain, students:
• use the meaning of fractions and relationships between multiplication and division to understand and explain why the procedures for dividing fractions make sense;
• extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, particularly negative integers; and
• reason about the order and absolute value of rational numbers and about the location of points on a coordinate plane.
3. Through their learning in the Expressions, Equations, and Inequalities domain, students:
• write expressions and equations that correspond to given situations, using variables to represent an unknown and describe relationships between quantities;
• understand that expressions in different forms can be equivalent, and use the properties of operations to rewrite and evaluate expressions in equivalent forms; and
• use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations.
4. Through their learning in the Geometry domain, students:
• find areas of polygons, surface areas of prisms, and use area models to understand perfect squares; and
• extend formulas for the volume of a right rectangular prism to fractional side lengths and use volume models to understand perfect cubes.
5. Through their learning in the Statistics and Probability domain, students:
• learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected; and
• understand the probability of a chance event and develop probability models for simple events.

In Grade 7, instructional time focuses on three areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; and (3) drawing inferences about populations based on samples.

1. Through their learning in the Ratios and Proportional Relationships domain, students:
• extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems;
• use their understanding of ratios and proportionality to solve a wide variety of percent problems;
• solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects;
• graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line; and
• distinguish proportional relationships from other relationships.
2. Through their learning in the Number System and the Expressions, Equations, and Inequalities domains, students:
• 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;
• extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division;
• explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers by applying properties of operations, and view negative numbers in terms of everyday contexts; and
• use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.
3. Through their learning in the Statistics and Probability domain, students:
• build on their previous work with single data distributions to compare two data distributions and address questions about differences between populations;
• begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences; and
• extend previous understandings of simple probabilities in grade 6 to calculate probabilities of compound events.

In Grade 8, instructional time focuses on three areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

1. Through their learning in the Number System, the Expressions, Equations, and Inequalities, and the Probability and Statistics domains, students:
• recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin;
• understand that the slope (m) of a line is a constant rate of change, as well as how the input and output change as a result of the constant rate of change;
• interpret a model in the context of the data by expressing a linear relationship between the two quantities in question and interpret components of the relationship (such as slope and y-intercept) in terms of the situation;
• solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line; and
• use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to represent, analyze, and solve a variety of problems.
2. Through their learning in the Functions and the Expressions, Equations, and Inequalities domains, students:
• grasp the concept of a function as a rule that assigns to each input exactly one output;
• understand that functions describe situations where one quantity determines another; and
• translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations of the function), and describe how aspects of the function are reflected in the different representations.
3. Through their learning in the Geometry domain, students:
• 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;
• show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines;
• understand the statement of the Pythagorean Theorem and its converse, and why the Pythagorean Theorem holds; and
• apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons.

University at Buffalo, Gifted Math Program

The University at Buffalo’s Gifted Math Program (GMP) offers advanced mathematics courses on UB’s campus twice a week – after the regular school day – throughout the academic year. This program is designed to challenge exceptional math students (i.e., upper 1% in their national peer group) and provide students with the opportunity to earn up to 22 college credit hours. Each year, the GMP admits a class of 60 students who are nominated for entry by parents or teachers from across Western New York. Applicants undergo interviews and a battery of tests, which includes the math section of the Preliminary Scholastic Aptitude Test. Those in grades 7 through 10 study an accelerated curriculum, while 11th and 12th graders take university-level courses in calculus and linear algebra.

While the GMP courses are designed to replace the mathematics courses in the students’ home school, students may elect to take a district math course, in addition to an “aligned” GMP course. The grades for both courses will be factored into a student’s grade point average (GPA), appear on one’s transcript, and each class (beginning in the 2020-21 school year) may earn a NYS seat credit. For example, the GMP program has designated that GMP III is aligned to Geometry. So, if a student takes GMP III and elects to take Geometry in the district, both the GMP course and district’s Geometry course may each receive one NYS seat credit, and both courses will be factored into one’s GPA and appear on one’s transcript.

F.A.Q:
1. What GMP courses are considered “aligned” to Williamsville’s math courses?
UB has identified the following Gifted Math courses as “aligned” to Williamsville’s leveled courses:
UB GMP 1 = (no high school association)
UB GMP 2 = Algebra 1
UB GMP 3 = Geometry
UB GMP 4 = Algebra 2
UB GMP 5 = Calc. 1 and Calc. 2
UB GMP 6 = Calc. 3 and Linear Algebra

2. If a student is enrolled in UB GMP and enrolled in an “aligned” math course, which course counts?
Districts who partner with UB GMP are required to “accept the University coursework in place of the home school mathematics curricula.” In other words, UB’s math courses are designed to supplant the district’s math courses; nevertheless, Williamsville does permit UB GMP students the opportunity to simultaneously enroll in a district math course that is considered to be aligned to UB’s courses. The district, however, does not permit any form of course auditing (see the “Grade Point Average” section in the course catalog); thus, the student will receive grades for both the aligned GMP course(s) and grades from any Williamsville course(s), and all course grades will be averaged into the student’s GPA and appear on one’s transcript. Students will also receive one NYS seat credit for each of the two aligned courses.