OUR MATH PROGRAMS

We provide individualized programs for children and adults to develop and improve math skills. Much of what we do is driven by the student's math curriculum. Our philosophy is to move from concrete to semi-concrete to abstract. We also help students create strategy notebooks for reference. These are like math recipe books and contain algorithms for the skills they have learned. For example, a student might have a page in the strategy notebook devoted to adding fractions with unlike denominators.

Elementary Math

We offer programs for students of any age who wish to develop their math skills. Programs may include instruction in whole numbers, fractions, decimals, time, money, measurement, elementary geometry, and word problems. The programs are designed to meet the specific needs of the learner. Instruction moves from the concrete to the abstract, when appropriate for the student. It often begins with hands-on materials and pictorial representation and then advances to symbolic computation and problem solving.

Advanced Math

Programs for Pre-Algebra, Algebra I and II, Geometry, Pre-Calculus, Calculus, Statistics, and Trigonometry are available. These instructional programs are suitable for secondary school students and for adults preparing for college entrance or high school equivalency examinations.

Here are some resources that we use with our students:

The Landmark Method for Teaching Arithmetic
Copyright © 1995 Landmark School, Inc. and Christopher Woodin

Children with learning disabilities are not necessarily deficient in mathematics due to an inability to grasp spatial tasks or estimate quantity. Their difficulties often lie in language dysfunction. These language-based problems preclude them from effectively developing mathematical abilities.

The Landmark Method for Teaching Arithmetic utilizes direct presentation of math grammar and encourages students to speak in complete sentences, to convey an entire thought, and to develop a consistent rehearsal pattern for math facts.

Students who benefit from direct presentation of math grammar are often those who have expressive language problems. These students typically provide one- or two- word answers to problems. This program helps students with language-based learning disabilities create math sentences and engage in class discussions.

Christopher Woodin © 1995 Landmark School, Inc. and Christopher Woodin

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On Cloud Nine® Math

To develop concept and numeral imagery, the On Cloud Nine® Math program integrates and applies imagery to the cognitive process of computing and conceptualizing math and mathematical principles.

The program moves through three basic steps to develop mathematical reasoning and computation using: manipulatives to experience math, imagery and language, and computation to apply math to problem solving. The use of manipulatives serves two purposes: to make numbers and mathematical concepts concrete, and to serve as a base for establishing numeral imagery. Students learn to image the concrete, attach language to their imagery and apply that imagery to the computation. For example, when asked to add the numbers 4 + 3, children who are drawing on their images may see 4 cars and 3 more trucks to show 7 vehicles. They know the answer because they can “see it.”

On Cloud Nine® Math is suitable for all grade levels and is for students who have difficulty learning math facts, grasping mathematical relationships, doing word problems, and higher math.

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Jamestown Math – Number Power
Copyright © Glencoe/McGraw-Hill

Jamestown Math targets a particular set of math skills with straightforward explanations, easy-to-follow, step-by-step instruction, real-life examples, and extensive reinforcement exercises. This is especially helpful for students who may have trouble with reading as well.

This program covers the full scope of mathematics from basic computation to the fundamentals of algebra and geometry. With Jamestown Math, we can review these math concepts at a lower readability level and help students review for standardized tests.

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Drops in the Bucket
Copyright © Frog Publications

Drops in the Bucket is a research-based, supplementary tool which provides systematic daily practice for skills and reinforcement to promote long-term memory. The systematic daily practice of Drops in the Bucket assures that the commonly taught and tested skills, vocabulary, and concepts will be maintained and strengthened, not introduced then forgotten.

Drops in the Bucket utilizes the learning principles of spaced practice, reinforcement, and repetition. Students develop both competence and confidence through this program.

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Exploratory Concepts and Skills by Grade: Grades PreK - 12

According to the Massachusetts Department of Education, these are the basic concepts and skills that students should establish in each grade range. We have highly trained tutors who can assist students at every level.

Grades PreK - K

Number Sense and Operations
Count by ones, beginning from any number in the counting sequence.
Represent quantities using concrete objects, and investigate the partitioning of sets. Identify equal parts of groups.
Create problems that can be solved using addition and subtraction.

Patterns, Relations, and Algebra
Explore skip counting by twos.

Geometry
Investigate symmetry of two- and three-dimensional shapes and constructions.

Measurement
Explore and use standard units to measure and compare temperature, length, & time.
Identify positions of events over time, e.g., earlier, later.

Data Analysis, Statistics, and Probability
Collect and organize data in lists, tables, and simple graphs.

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Grades 1 - 2

Number Sense and Operations
Use concrete materials to investigate situations that lead to multiplication and division.
Develop and use strategies for addition and subtraction of multi-digit whole numbers.
Investigate addition of common fractions, e.g., 1/2 + 1/2 = 1, 1/4 + 1/4 = 1/2.
Understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally.

Patterns, Relations, and Algebra
Investigate situations with variables as unknowns and as quantities that vary.

Geometry
Investigate symmetry in two-dimensional shapes with mirrors or by paper folding.
Explore intersecting, parallel, and perpendicular lines.
Create mental images of geometric shapes using spatial memory and spatial visualization.
Recognize and represent shapes from different perspectives.
Recognize geometric shapes and structures in the environment and specify their location.
Identify relative positions, e.g., closer, farther, higher, lower.
Find and name locations on maps and express simple relationships, e.g., near to, far away from, etc.

Measurement
Explore measurable attributes of objects, including length, perimeter, weight, area, volume, and temperature. Compare concrete objects using these measures.

Data Analysis, Statistics, and Probability
Investigate more likely, likely, and impossible outcomes by conducting experiments using spinners, counters, and other concrete objects.
List and count the number of possible pairings of objects from two sets.

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Grades 3 - 4

Number Sense and Operations
Extend multiplication and division to larger-digit numbers.
Use models to explore multiplication & division with fractions (to twelfths) & decimals.
Investigate number theory concepts, e.g., prime and composite numbers.
Investigate the concept of ratio.
Use concrete objects and visual models to add and subtract common decimals.
Explore numbers less than zero by extending the number line and by using familiar applications such as temperature.
Use concrete objects and visual models to add and subtract common decimals.
Investigate the distributive property of multiplication over addition for single-digit multipliers, e.g., 7 x 15 is equivalent to 7 x (10 + 5) is equivalent to 7 x 10 + 7 x 5.

Patterns, Relations, and Algebra
Use concrete materials to build an understanding of equality and inequality.
Explore properties of equality in number sentences: when equals are added to equals, then the sums are equal; when equals are multiplied by equals, then the products are equal, e.g., if ? = 5, then 3 x ? = 3 x 5.

Geometry
Predict and describe results of transformations (e.g., translations, rotations, and reflections) on two-dimensional shapes.
Investigate two-dimensional representations of three-dimensional objects.

Measurement
Develop the concepts of area and perimeter by investigating areas and perimeters of regular and irregular shapes created on dot paper, coordinate grids, or geoboards.
Use concrete objects to explore volumes and surface areas of rectangular prisms.
Investigate the use of protractors to measure angles.
Identify common measurements of turns, e.g., 360° in one full turn, 180° in a half turn, and 90° in a quarter turn.
Investigate areas of right triangles.
Understand that measurements are approximations and investigate how differences in units affect precision.

Data Analysis, Statistics, and Probability
Explore the concepts of median, mode, maximum and minimum, and range.
Discuss what data-collection methods are appropriate for various types of investigations.
Explore situations that involve probabilities of equally likely events.
Investigate the construction of simple circle graphs.

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Grades 5 - 6

Number Sense and Operations
Explore the addition and subtraction of positive and negative fractions.
Investigate the concepts of ratio and proportion.
Investigate the distributive property of multiplication over addition for double-digit multipliers, e.g., 12 x (10 + 3) is equivalent to 12 x 10 + 12 x 3.

Patterns, Relations, and Algebra
Use physical models to investigate and describe how a change in one variable affects a second variable.
Use models to develop understanding of slope as constant rate of change.
Model situations with proportional relationships and solve problems.

Geometry
Use manipulatives and technology to model geometric shapes.
Investigate tessellations (tilings).
Explore the angles formed by intersecting lines.
Identify and draw shapes and figures from different views/perspectives.
Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Measurement
Explore various models for finding the area of a triangle, parallelogram, and trapezoid, and develop strategies for more complex shapes.
Investigate volumes and surface areas of a variety of three-dimensional objects.
Explore volume and surface areas of rectangular prisms, cylinders, and spheres.

Data Analysis, Statistics, and Probability
Set up and analyze capture-recapture experiments.
Generate and group data, record the data using frequency tables and interpret the tables.
Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatter plots.
Compare different representations of the same data and evaluate how well each representation shows important aspects of the data.

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Grade 7 - 8

Number Sense and Operations
Investigate the meaning of significant digits.
Investigate negative integral exponents and their use in scientific and calculator notation.

Patterns, Relations, and Algebra
Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive (e.g., Pascal’s triangle), and linear functional relationships.
Use tables, graphs, and appropriate technology to explore quadratic and exponential growth patterns.
Investigate the use of systems of equations, tables, and graphs to represent mathematical relationships.
Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

Geometry
Formulate and test conjectures about shapes that tessellate.
Investigate trigonometric ratios in right triangles.
Investigate right triangle relationships, such as those in 45°–45°–90° and 30°–60°–90° triangles.
Explore proofs of the Pythagorean Theorem.

Measurement
Given the formula, find surface area and volume of pyramids and cones.
Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.
Investigate formulas to determine the circumference and area of circles, and the perimeter and area of polygons.

Data Analysis, Statistics, and Probability
Investigate the notion of fairness in games.
Make predictions, conduct experiments, and discuss discrepancies to develop understanding of actual versus predicted outcomes.
Conduct repetitive experiments (e.g., repeated throwing of three or more dice) and compare the outcomes to predicted probabilities.

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Grade 9 - 10

Number Sense and Operations
Analyze relationships among the various subsets of the real numbers (whole numbers, integers, rationals, and irrationals).
Explore higher powers and roots using technology.
Explore the system of complex numbers and find complex roots of quadratic equations.

Patterns, Relations, and Algebra
Explore matrices and their operations. Use matrices to solve systems of linear equations.
Investigate recursive function notation.

Geometry
Apply properties of chords, tangents, and secants to solve problems.
Use deduction to establish the validity of geometric conjectures and to prove theorems in Euclidean geometry.

Measurement
Explore the scientific use of different systems of measurement, e.g., centimeter-gram-second (CGS), Scientific International (SI).

Data Analysis, Statistics, and Probability
Explore designs of surveys, polls, and experiments to assess the validity of their results and to identify potential sources of bias; identify the types of conclusions that can be drawn.
Describe the differences between the theoretical probability of simple events and the experimental outcome from simulations.

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Grade 11 - 12

Number Sense and Operations
Investigate special topics in number theory, e.g., the use of prime numbers in cryptography.
Use polar-coordinate representations of complex numbers and DeMoivre’s theorem to multiply, take roots, and raise numbers to a power.
Plot complex numbers using both rectangular and polar coordinate systems.

Patterns, Relations, and Algebra
Prove theorems using mathematical induction.
Investigate parametrically defined curves and recursively defined functions, including applications to dynamic systems.

Geometry
Investigate and compare the axiomatic structures of Euclidean and non-Euclidean geometries.
Explore the use of conic sections in engineering, design, and other applications.
Investigate the notion of a fractal.
Use graphs (networks) to investigate probabilistic processes and optimization problems.

Data Analysis, Statistics, and Probability
Use technology to perform linear, quadratic, and exponential regression on a set of data.
Design surveys and apply random sampling techniques to avoid bias in the data collection.

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