Math


The mathematics program at Brookwood provides a solid foundation in mathematical understanding from the very beginning concepts of number and counting in Pre-Kindergarten through Algebra 1 in Grade Eight. We aim to foster confidence in the face of unfamiliar problems, willingness to take risks, and appreciation for math as a tool with real-world applications. Graduates are well prepared for either a typical or advanced sequence of high school math courses.

We believe that children learn math best when they develop an understanding of the patterns and principles that underlie numerical operations. In order to do this, they need opportunities to explore math concepts in multiple ways, make discoveries, and share their findings. If you visit a typical Brookwood math lesson, you are likely to see students actively engaged with “hands on” materials, visual models, games, group problem-solving explorations, or computer technology. All of these tools bring greater depth and meaning to paper-and-pencil math as students increase their confidence and skill as mathematicians.

We use the Investigations in Number, Data, and Space curriculum for Grades K – Five. For more information about our use of this program, click here. Grades Six - Eighth use a combination of curriculum materials that leads students through a coherent sequence, starting with reinforcement of fraction and decimal computation skills in Grade Six, and ultimately completing an Algebra 1 course by the end of Grade Eight. Our aim is that all Brookwood graduates be thoroughly prepared for success in a standard or integrated sequence of high school mathematics courses.

Click here for online math websites for summer fun and practice!

Mathematics Curricular Goals
In general, the Math Department seeks to have Brookwood students:
 

  • Approach unfamiliar problems with confidence and an array of strategies
  • Reason independently
  • Demonstrate awareness of, and skill using, tools such as thought experiments, books, physical models, peer group discussions, diagrams, graphs, calculators, computer software, internet sources, etc., to appropriately support one’s individual learning style in any given mathematical situation
  • Explain how and why various calculation procedures work
  • Calculate fluently with positive and negative integers and non-integers
  • Interpret problems presented in a word-based/contextualized format; choose and apply appropriate strategies for solving
  • Apply algebraic thinking and procedural skills to a wide array of problem solving situations
  • Use estimation wisely and regularly to evaluate reasonableness of answers
  • Habitually double-check work
  • Communicate mathematical understanding accurately and clearly when writing, whether using symbols, words, or both
  • Use mathematical language accurately and effectively when speaking, whether explaining an idea or posing a question
  • Determine whether or not a mathematical argument or solution process is logically sound.  (It is expected that students will have preliminary skills in this area that will serve as a basis for learning formal proofs in later math courses.)


 

Grade Level Curricular Goals and Focal Points
At each respective grade level, our goal is to have students:

Pre-Kindergarten:

Number and Operation

  • Arrange a set of objects in one to one correspondence to 10
  • Count by ones to 20
  • Begin to realize how many are in a group without counting (up to 5)
  • Accurately use the language of more and less to describe quantities

Geometry and Measurement

  • Begin to recognize, describe, and name two- and three-dimensional shapes
  • Use words to describe spatial relationships (above, below, next to, etc.)
  • Combine shapes to make new shapes (e.g., two triangles make a “diamond” or rhombus, etc.)
  • Begin to make size comparisons
  • Measure lengths using objects such as blocks, pencils, paperclips, etc.

Data Analysis and Probability

  • Read a simple graph to identify how many are in a category, and whether a category has more or less than another
  • Sort objects by attributes
  • Identify and describe similarities and differences between objects

Patterns, Functions, and Algebra

  • Recognize and reproduce simple patterns in shape, color, numbers, etc.
  • Predict what comes next in a pattern
  • Identify and describe patterns in the environment


Kindergarten:

Number and Operation

  • Accurately count and keep track of a set of up to 20 objects
  • Connect number words, numerals, and quantities
  • Represent quantities with pictures and numerals
  • Write numbers up to at least 10
  • Develop and analyze visual images for quantities up to 10
  • Compare two quantities to decide which is greater/less; order quantities from least to most
  • Figure out what is one more and one fewer than a number
  • Model the action of combining (adding) and separating (subtracting) situations and find the result
  • Add small amounts accurately
  • Decompose numbers in different ways (i.e., five can be 2 and 3 or 1 and 4)

Geometry and Measurement

  • Describe the overall size, shape, and features of familiar 2- and 3-D shapes
  • Make 2-D and 3-D shapes by combining other shapes
  • Relate 3-D objects to 2-D pictures of those objects
  • Describe and compare lengths
  • Measure the length of an object by lining up multiple units
  • Record length measurements with pictures, numbers, and/or words


Data Analysis and Probability

  • Identify and describe attributes and use them to sort objects
  • Sort a set of data or objects in different ways
  • Collect and keep track of survey data
  • Use data to pose and answer questions
  • Use information about data to figure out what is missing


Patterns, Functions, and Algebra

  • Discriminate between a pattern and a random arrangement
  • Copy, construct, and extend simple repeating patterns, such as AB or ABC
  • Determine and describe what comes next in a repeating pattern


Grade One:

Number and Operation

  • Identify, count, read, write, and sequence numbers up to at least 105
  • Count by 1s forwards and backwards along any part of the number line between 0 and 100
  • Count by 2s 5s and 10s to at least 120
  • Find several combinations of two addends for any number up to 10
  • Demonstrate fluency with the two-addend combinations of 10
  • Interpret and solve addition and subtraction story problems
  • Add by counting on or by reasoning about known combinations
  • Use addition notation correctly
  • Subtract one small quantity from another

Geometry and Measurement

  • Use geometric language to describe and identify features of familiar 2-D shapes
  • Compose and decompose 2-D shapes in more than one way
  • Attend to and describe features of 3-D shapes such as overall size and shape, number and shape of faces, number of corners, etc.
  • Match a 2-D representation to a 3-D shape
  • Demonstrate accurate measuring techniques when measuring with non-standard units
  • Measure with inches to the nearest whole number; describe a measurement that falls between two whole numbers


Data Analysis and Probability

  • Devise categories and use them for sorting data, shapes, or objects
  • Represent a set of survey data with two categories
  • Interpret and draw accurate conclusions from various representations of survey data with two categories, including how many people responded to the survey

Patterns, Functions, and Algebra

  • Represent numbers using equivalent expressions
  • Begin to recognize the inverse relationship between addition and subtraction
  • Identify the unit of a repeating pattern
  • Determine what comes several steps beyond the visible part of an AB, ABC, AAB, or ABB pattern
  • Construct, extend, and describe a number pattern that has a constant increase through building, drawing, and/or counting


Grade Two:

Number and Operation

  • Count by 2’s, 5’s, and 10’s, up to at least 200
  • Know coin equivalencies and use them to solve problems
  • Recall addition and subtraction facts through 20
  • Interpret and solve problems about the number of tens and ones in a quantity
  • Interpret addition and subtraction story problems
  • Interpret and use the number line to model addition and subtraction
  • Model addition with a place value representation
  • Add two-digit numbers accurately and efficiently
  • Determine the difference between a number and any multiple of 10 up to 100
  • Have at least one strategy for subtracting two-digit numbers accurately
  • Double numbers and explain what doubling means
  • Find half of a set of objects
  • Interpret fractions as equal parts of a whole


Geometry and Measurement

  • Identify the number of sides of a polygon
  • Identify rectangles as four-sided closed shapes with four right angles
  • Create an array; identify the number of rows and the number of squares in each row
  • Accurately measure objects using inches and centimeters
  • Use a ruler to measure lengths longer than one foot
  • Recognize that, when measuring the same length, larger units yield smaller counts
  • Tell time to the five-minute interval
  • Create timelines and solve problems involving the beginning time, ending time, and duration of an event for events beginning and ending on the hour or half-hour

Data Analysis and Probability

  • Use a Venn diagram to sort data by two attributes
  • Create categories for a set of categorical data and organize the data into the chosen categories
  • Order and represent a set of numerical data
  • Describe a numerical data set, including the highest and lowest values and the mode
  • Read and interpret a variety of representations of numerical and categorical data
  • Compare two sets of numerical data

Patterns, Functions, and Algebra

  • Recognize that numbers can be added in any order without affecting the result (Commutative and Associative Properties); use this fact to add strings of five or more numbers quickly
  • Write an equation that represents the number relationships in a story problem
  • Make sense of and solve story problems involving an unknown increase or decrease
  • Extend a repeating numerical pattern and determine what element of the pattern will be in a particular position if the pattern keeps going
  • Complete and extend a table to match a situation involving a constant ratio
  • Name what each number in a table means
  • Use knowledge of doubling, halving, and/or number combinations to predict what numbers will be in a given row of a constant ratio table

 

Grade Three:

Number and Operation

  • Demonstrate fluency with subtraction combinations up to 20
  • Find combinations of 2-digit numbers that add to 100 quickly and accurately
  • Read, write, and sequence numbers up to 1,000
  • Identify the value of each digit in a 3-digit number
  • Break up 3-digit numbers into 100s, 10s, and 1s in different ways
  • Identify how many groups of 10 are in a 3-digit number
  • Add multiples of 10 and 100 to and subtract them from any 3-digit number
  • Solve addition problems with 3-digit numbers by breaking the numbers apart by place or by adding one number in parts
  • Solve 3-digit addition problems accurately and efficiently using at least one strategy: adding by place, adding in parts, or changing both numbers to make an equivalent problem
  • Solve subtraction story problems in contexts that include removing a part from a whole, comparing two quantities, or finding a missing part
  • Solve subtraction problems with 2- and 3-digit numbers using at least one of the following strategies: subtracting the second number in parts, adding up to reach the first number, or subtracting back to reach the second number
  • Solve multiplication and related division problems using skip counting or known multiplication combinations
  • Interpret and use multiplication and division notation
  • Demonstrate fluency with the multiplication combinations with products up to 50
  • Divide a single whole or a quantity into equal parts and name those parts as fractions or mixed numbers
  • Identify equivalent fractions involving halves, thirds, fourths, sixths, and eighths
  • Find combinations of fractions that are equal to one


Geometry and Measurement

  • Identify and accurately measure the perimeter of a shape using U.S. standard and metric units
  • Identify and find the area of given figures by counting whole and partial square units
  • Identify triangles as three-sided closed shapes with three sides and three angles
  • Identify right angles and recognize whether an angle is larger or smaller than a right angle
  • Identify and compare attributes of 3-D solids
  • Determine the number of cubes that will fit in the box made by a given pattern (volume)
  • Design patterns for boxes that will hold a given number of cubes

Data Analysis and Probability

  • Interpret a bar graph
  • Make a line plot for a set of numerical data
  • Summarize a set of data, describing concentrations of data and what those concentrations mean in terms of the situation the data represent, and determining the mode

Patterns, Functions, and Algebra

  • Use the inverse relationships of addition and subtraction, multiplication and division, to write equivalent problems and to double-check calculations
  • Recognize that numbers can be multiplied in any order (Commutative Property); use this knowledge to help solve multiplication problems and memorize multiplication combinations


Grade Four:

Number and Operation

  • Read, write, and sequence numbers up to 10,000
  • Add and subtract multiples of 10, 100, and 1,000 fluently
  • Solve multi-digit addition and subtraction problems accurately and efficiently, choosing from a number of mastered strategies
  • Demonstrate fluency with multiplication combinations through 12 x 12
  • Use arrays, pictures or models of groups, and story contexts to demonstrate understanding of multiplication situations and relationships
  • Find the factors of 2-digit numbers
  • Multiply by 10 and multiples of 10
  • Multiply 2-digit numbers efficiently
  • Use arrays, pictures, models, and story problems to demonstrate understanding of division situations and relationships
  • Solve division problems with 1-digit and some 2-digit divisors (for example, 11, 25, multiples of 10) using at least one strategy efficiently
  • Read, write, and interpret decimal fractions in tenths and hundredths, recognizing decimal notation as an extension of the place value system used for whole numbers
  • Order fractions with like and unlike denominators and locate them on a number line
  • Solve problems involving fraction relationships and fractional amounts of an area

Geometry and Measurement

  • Identify and classify quadrilaterals
  • Know that a right angle measures 90 degrees, and use this as a landmark to find angles of 30, 45, and 60 degrees
  • Find the area of polygons using a square unit of measure
  • Identify fractional parts of an area
  • Identify 2-D silhouettes of 3-D solids
  • Draw 2-D representations showing different perspectives of a 3-D object
  • Find the volume of a rectangular prism

Data Analysis and Probability

  • Organize and represent data about two groups to compare the groups
  • Determine the median of a numerical data set
  • Use evidence from a set of data to support an argument
  • Describe the probability of an event in terms of a scale from 0 to 1
  • Interpret probability as the likelihood of an event, where 0 is impossible and 1 is certain

Patterns, Functions, and Algebra

  • Connect tables and graphs to each other and to the situations they represent
  • Graph a table of values onto a coordinate grid
  • Describe how a graph shows change; relate steepness to the rate of change
  • Locate increasing, decreasing, and constant rate of change on a graph
  • In a constant change scenario, write rules to express the value of one quantity in terms of the other


Grade Five:

Number and Operation

  • Read, write, and sequence numbers up to 100,000
  • Find the factors of a number
  • Demonstrate fluency with division problems related to the multiplication combinations through 12 x 12
  • Solve multiplication and division problems efficiently
  • Read, write, interpret, and order decimals to thousandths
  • Round to a specified place value
  • Demonstrate an understanding of the relationships among fractions, decimals, and percents; convert with facility from one form to another
  • Add and subtract fractions through reasoning about fraction equivalents and relationships
  • Simplify (reduce) answers to fraction calculations as far as possible
  • Add and subtract decimals through reasoning about place value, equivalents, and representations
  • Use fraction-percent equivalents to solve problems about the percentage of a quantity

Geometry and Measurement

  • Identify different quadrilaterals by their attributes, and know that some quadrilaterals can be classified in more than one way
  • Use known angle sizes to determine the sizes of other angles (30, 45, 60, 90, 120, and 150 degrees)
  • Calculate the perimeter and area of rectangles; state formulas for these calculations
  • Identify similar polygons
  • Calculate the volume of rectangular solids, state a formula for this calculation, and use standard units for volume
  • Identify how the dimensions of a rectangular prism change when the volume is changed
  • Explain the relationship between the volumes of prisms and pyramids with the same base and height

Data Analysis and Probability

  • Design and carry out an experiment in order to compare two groups
  • Give a quantitative description, involving median, mode, fractional parts of the data, etc., of a set of data presented in a line plot or bar graph.
  • Draw conclusions about how two groups compare based on summarizing the data for each group
  • Use decimals, fractions, and percents to describe and compare the theoretical probabilities of events with a certain number of equally likely outcomes

Patterns, Functions, and Algebra

  • Explain why doubling one factor in a multiplication expression and dividing the other by 2 results in an equivalent expression
  • Create tables and graphs to represent the relationship between two variables
  • Begin to use symbolic notation to represent the value of one variable in terms of another variable in situations with constant rates of change (linear functions)
  • Use tables and graphs to compare two linear functions with constant rates of change


Grade Six:

Number and Operation

  • Solve whole number division problems efficiently
  • Use and interpret exponential notation for whole number exponents
  • Use multiple representations, including number lines and arrays, to interpret and to communicate understanding of positive and negative integer, fraction, and decimal relationships and operations
  • Find the absolute value of a number
  • Add, subtract, multiply, and divide fractions and mixed numbers efficiently and accurately
  • Add, subtract, multiply, and divide decimals efficiently and accurately
  • Determine whether a decimal is terminating, repeating, or neither, and use bar notation correctly
  • Add, subtract, multiply, and divide negative integers and non-integers
  • Find any missing quantity in a percent problem, including percent of change
  • Use percents in applied contexts such as sales tax, discounts, etc.
  • Estimate the answer to any type of problem mentioned above to assess whether calculated answer makes sense

Geometry and Measurement

  • Memorize the distance-rate-time formula and recognize when to use it
  • Use a protractor to measure and draw angles
  • Use a compass to draw circles
  • Find the perimeter of any polygon
  • Find the circumference of a circle
  • Derive and use area formulas for triangles, parallelograms, trapezoids, circles and irregular figures based on these shapes
  • Find a missing length given the area, perimeter, or circumference

Data Analysis and Probability

  • Find the mean of a set of data
  • Create and interpret circle graphs


Patterns, Functions, and Algebra

  • Use the correct order of operations to simplify and evaluate expressions
  • Use a balance model to interpret equality; apply this model when solving equations
  • Write a complete set of four “fact family” operations for any given one-step calculation
  • Solve one-step algebraic equations mentally and by inverse operations or “fact family” relationships
  • Solve multi-step algebraic equations using manipulatives and/or symbol manipulation
  • Recognize commutativity and associativity of addition and multiplication
  • Use the distributive property to factor and expand expressions
  • Use the commutative, associative, and distributive properties when simplifying expressions and solving equations
  • Check solutions to equations


Grade Seven / Pre-Algebra:

Pre-Algebra Standard Sections:

Number and Operation

  • Add, subtract, multiply, and divide positive and negative numbers efficiently
  • Convert between scientific notation and standard form
  • Find square roots of perfect squares, and approximate square roots of other numbers
  • Use ratios and rates to compare two quantities
  • Calculate unit rates
  • Solve proportions efficiently
  • Choose and apply efficient strategies for solving percent problems, including percent of change

Geometry and Measurement

  • Solve for missing angle measures in polygons
  • Find unknown side lengths and angle measures in congruent and similar figures
  • Use the Pythagorean Theorem and its converse
  • Memorize the sine, cosine, and tangent ratios, and use them to solve for missing sides in right triangles.
  • Find surface area and volume of prisms, pyramids, cones, cylinders, and composite three-dimensional shapes
  • Identify vertical, supplementary, complementary, alternate interior and exterior, and corresponding angles, and use them to solve for missing angle measures

Data Analysis and Probability

  • Use the Counting Principle
  • Determine whether a situation involves permutations or combinations, and calculate how many are possible
  • Use permutations and combinations when calculating probabilities
  • Determine whether events are disjoint, overlapping, complementary, dependent, or independent, and find their probability

Patterns, Functions, and Algebra

  • Translate words into variable expressions, equations, or inequalities
  • Identify like terms and use them to simplify expressions and solve equations
  • Use formal paper-and-pencil notation to solve multi-step equations and inequalities involving positive and negative integers, fractions, and decimals
  • Use substitution and the order of operations to check solutions
  • Graph inequalities on a number line
  • Define variables for a word problem, then use to write and solve an equation or inequality
  • Simplify and perform the four operations on fractions that involve variables
  • Determine whether an ordered pair is a solution of an equation in two variables
  • Graph linear equations by creating a table of values and by using the slope and y-intercept
  • Find slope by counting rise and run on a graph and by using ordered pairs without graphing
  • Assess whether a graphed line has positive, negative, zero, or no slope
  • Recognize the slope-intercept form of a linear equation
  • Move flexibly among the graph, slope and intercept information, ordered pairs or tables, and equations of a line


Pre-Algebra Advanced Sections:

All benchmarks for Pre-Algebra Standard Sections, plus the following:

Number and Operation

  • Calculate with numbers in scientific notation by using exponent properties

Geometry and Measurement

  • Analyze, describe, and create drawings made by geometric transformations: translation, reflection, rotation, and dilation
  • Use a compass and straightedge to execute various geometric constructions: parallel and perpendicular lines, angle bisection, etc. (time permitting)

Patterns, Functions, and Algebra

  • Recognize polynomials and write in standard form
  • Add, subtract, and multiply polynomials
  • Use exponent properties to rewrite expressions

Grade Eight / Algebra One:

Algebra 1 Standard Sections:


Number and Operation

  • Use the exponent properties with numbers in scientific notation to simplify calculations
  • Simplify square roots and write in simplest radical form

Geometry and Measurement

  • Apply algebraic equations and solution processes to geometric figures

Patterns, Functions, and Algebra

  • Recognize use of the associative, commutative, and distributive properties
  • Use the distributive property with subtraction of a quantity and with division
  • Determine whether an equation has one solution, has no solution, or is an identity
  • Use reciprocals to solve equations with fractional coefficients
  • Write a formula in terms of any of its variables
  • Solve for a variable that is in the denominator of a fraction
  • Identify the domain and range of a relation
  • Determine whether a relation is a function
  • Use function notation
  • Evaluate a function for a specific domain value or set of domain values
  • Graph absolute value equations, linear inequalities, and quadratic functions
  • Classify a set of data points, graph, or equation as linear, quadratic, or other
  • Move flexibly among tables, graphs, and equations for linear and quadratic relationships
  • Use slope-intercept form to write equations, graph lines, and identify slope and y-intercept
  • Find both intercepts given a linear equation
  • Analyze the relationship between the slopes of parallel lines and between the slopes of perpendicular lines
  • Determine an equation of a line given two points or given the slope and one point
  • Recognize, use, and move flexibly between slope-intercept form, standard form, and point-slope form of a line
  • Use the exponent properties to simplify and evaluate variable expressions with exponents 
  • Simplify, add, subtract, multiply, and factor polynomials 
  • Simplify, multiply, and divide rational expressions 
  • Solve quadratic equations by factoring and by the quadratic formula; report solutions in decimal form and in simplest radical form
  • Relate the solution(s) of a quadratic equation, including equations with no solution, to a graph of the related function
  • Graph linear systems; use the graph to determine the solution
  • Solve linear systems, including those with zero or infinitely many solutions, by substitution and by elimination; use the solution(s) to describe the graph

Algebra 1 Advanced Sections:

All benchmarks for Algebra 1 Standard Sections, plus the following:
 

Number and Operation 

  • Translate expressions with negative exponents into fraction form and vice-versa

 

Patterns, Functions, and Algebra

  • Solve equations and inequalities involving absolute values
  • Solve compound inequalities 
  • Write a function rule for a table of values that have a linear relationship 
  • Classify a set of data points, graph, or equation as linear, quadratic, rational, exponential, or other 
  • Move flexibly among tables, graphs, and equations for linear, quadratic, rational, and exponential relationships
  • Recognize the difference of two squares in factored and expanded form
  • Add and subtract rational expressions
  • Use the Completing the Square algorithm for solving quadratic equations and to derive the Quadratic Formula
  • Simplify, add, subtract, multiply, and divide radical expressions; rationalize denominators
  • Solve radical equations and check for extraneous solutions (time permitting)
  • Solve rational equations and check for solutions that result in a zero denomintaor (time permitting)
  • Graph rational and exponential functions (time permitting)
  • Write equations for situations involving exponential growth or decay (time permitting)