The horizontal sequencing of the Mathematics Topics across the top of the matrix is not intended to imply an order in which the topics are to be taught. Rather than a disjointed series of compartmentalized skills, it represents a continuum of linked and related subjects.

The vertical Foundation Skills include the concepts of operations and computation along with the thinking, problem-solving, and communication skills referred to in the SCANS. Unlike the listing of mathematics topics, however, the order in which these foundation skills are listed is purposeful, since problem-solving and communication are the apex of these skills.

Concepts of
Operations
Add/Subtract
Multiply/Divide
Powers/Roots

Computation
Add/Subtract
Multiply/Divide
Powers/Roots
Manually and using
calculator

Reasoning
Estimation
Patterns
Relationships
Connections

Problem-solving
Personal
Work-related
Academic

x

Communication
Listening/speaking
Reading/writing

Adult Goal 1: Learners will develop and use mathematics to solve problems.

MST
1

Analysis, Inquiry, and Design

Adult Goal 2: Learners will develop and use estimation skills.

MST
1

Analysis, Inquiry, and Design

Adult Goal 3: Learners will develop and use whole number computation procedures for problem solving.

MST
1

Analysis, Inquiry, and Design

Adult Goal 4: Learners will develop facility in the use of a calculator.

MST
2

Information Systems

Adult Goal 5: Learners will recognize, understand, and use mathematics to communicate and reason.

MST
3

Mathematics

Adult Goal 6: Learners will develop a sense of number relationships.

MST
3

Mathematics

Adult Goal 7: Learners will develop concepts of mathematical operations.

MST
3

Mathematics

Adult Goal 8: Learners will develop and use skills in data analysis, statistics, and probability.

MST
3

Mathematics

Adult Goal 9: Learners will develop and use knowledge of fractions, decimals, and percents.

MST
3

Mathematics

Adult Goal 10: Learners will use algebra skills.

MST
3

Mathematics

Adult Goal 11: Learners will use geometry skills.

MST
3

Mathematics

Adult Goal 12: Learners will develop and use spatial sense and measurement.

MST
6

Interconnectedness: Common Themes

Adult Goal 13: Learners will develop and use patterns and relationships.

MST
6

Interconnectedness: Common Themes

Objective A: Learners will use a problem-solving process.

• Recognizing that a problem exists.
• Identify questions and understand what is being asked.
• Identify key information and organize data.
• Select an appropriate strategy.
• Perform appropriate computation.
• Evaluate correctness of answer.
• Evaluate effectiveness of thinking process (metacognition).

Objective B: Learners will develop and apply a variety of strategies and techniques to approach and solve problems.

• Model situations using oral, written, concrete, pictorial, graphical, and algebraic methods in order to restate a problem in simpler or more familiar terms.
• Identify facts needed.
• Use a calculator, computer, and/or other technologies to facilitate the process.

Objective C: Learners will apply critical thinking to the problem-solving process.

• Compare similar problems.
• Estimate reasonableness of answer.
• Identify problem-solving strategies used.
• Make generalizations which will apply to all similar data using inductive (from the specific to the general) and deductive (from the general to the specific) reasoning.

Objective A: Learners will develop and use estimation skills.

• Recognize when an estimate is appropriate.
• Develop and use estimation strategies, e.g., rounding, nearer to, larger than, smaller than, etc.
• Apply estimation in working with quantities, measurement, computation and problem solving.
• Estimate costs and income in distributing resources.
• Use estimation to determine if problem solution is reasonable.
• Establish upper and lower limits for an estimated answer, e.g., 4.5 x 6.13, 4 x 6 = 24, 5 x 6 = 30, therefore the answer is estimated as somewhere between 24 and 30 or 27, e.g., “ While at Penney's, Mary is considering buying two blouses on sale for $27.95 each, and a skirt priced at $34.50. She wants to pay cash for her purchases but has only $110 in her purse. The sales tax in her locality is 8.25%. Can she buy these items with the cash she has?” 27.95 = approximately 30, and 34.5 = approximately 35. If tax of 8.25% is rounded up to a convenient 10%, the total items would come to $30 +$30 +$35 or 95. Adding approximately $10 for tax comes to $105. Mary has enough cash.

Objective A: Learners will model, explain, and develop reasonable proficiency with basic number facts and procedures used to operate on them (algorithms).

• Addition
• Subtraction
• Multiplication
• Division
• Powers
• Roots

Objective B: Learners will use a variety of mental computation and estimation techniques.

• Combine numbers whose sums are ten and expand the concept to include multiples of ten.
• Add using distance from five or ten, e.g., 19 + 8 = 20 + 8 - 1.
• Multiply/divide by ten and powers of ten mentally.

Objective A: Learners will recognize the importance and efficiency of calculators.

• Read and understand the directions.
• Identify and use memory and function keys.
• Identify any special features such as dual function keys, rounding capability, parentheses, order of operations, etc.
• Enter and manipulate negative values.

Objective B: Learners will investigate mathematical patterns and relationships such as fraction/decimal/percent conversions, multiplication tables, etc.

Objective C: Learners will solve problems with calculator.

• Choose correct operations and order in which numbers are entered.
• Use correct order of operations.
• Check for reasonableness of answer by using estimation, e.g., “While at Penney's, Mary is considering buying two blouses on sale for $27.95 each and a skirt priced at $34.50. She wants to pay cash for her purchases but has only $110 in her purse. The sales tax in her locality is 8.25%. Could she buy these items with the cash she has?” Solving this problem using a calculator may involve the use of percent key, parentheses, and knowledge of order of operations.

Objective D: Learners will simplify calculations by using short-cut problem-solving techniques.

• Constant key.
• Memory and recall keys.
• Parentheses key.
• Percentage key.

Objective E: Learners will solve problems involving scientific notations.

Objective A: Learners will develop common understandings of mathematical ideas.

• Recognize and use the vocabulary and symbols of mathematics
• Use the skills of reading (including use of inference and logical transition), listening, and viewing to interpret and evaluate mathematical ideas.
• Discuss mathematical situations and make conjectures and convincing arguments.

Objective B: Learners will draw logical conclusions about problem situations.

• Use physical materials, pictures, graphs, budgets, and diagrams to illustrate mathematical ideas.
• Use models, known facts, and relationships to explain thinking.
• Explain answers and solution processes.
• Follow a logical argument and judge its validity.
• Construct a logical argument and judge its validity.

Objective C: Learners will link conceptual and procedural knowledge.

• Apply relationships between operations such as addition/subtraction inverse, multiplication/division inverse, multiplication as repeated additions, division as repeated subtractions.
• Recognize different representations of the same concept as equivalent, e.g., horizontal and vertical addition, 20'/18" = 240"/18" = 20'/1.5'.

Objective D: Learners will recognize relationships among different topics.

• Whole numbers, decimals, fractions, and percents.
• Geometry and algebra, particularly in graphing.

Objective A: Learners will construct number meanings and sense.

• Interpret numbers in terms of whole/part relationships such as 7 = 3 + 4, 7 - 3 = 4; or 2 x 4 = 8, 8/2 = 4.
• Recognize the magnitudes of numbers, e.g., 3400 > 34.
• Use numbers in a variety of equivalent forms, e.g., 1/2 = 0.5 = 50%, 2000 = 2 x 103 .
• Understand a number line (positive and negative values).
• Predict the results of: Multiplying by a number less than one. Dividing by a number less than one. Adding/subtracting different combinati ons of positive or negative numbers (integers).

Objective B: Learners will demonstrate understanding of place value by explaining.

• Place values from units to billions.
• Place values from units to billionths.
• The concept of zero as a place holder.

Objective A: Learners will use pictures and/or objects (models) to develop meaning for operations and their functions.

• Addition
• Division
• Subtraction
• Powers
• Multiplication
• Roots

Objective B: Learners will apply the use of the four basic operations.

• Whole numbers.
• Fractions.
• Decimals/percents.
• Integers (positives, negatives, and zero).

Objective C: Learners will use the properties of numbers.

• Commutative property for addition/ multiplication (a +b = b + a, a x b = b x a).
• Associative property for addition/multiplication (a + b + c = b + c + a, a x b x c = b x c x a).
• Distributive property for multiplication over addition, a(b + c) = ab + ac
• Identity element for addition and subtraction (a + 0 = a, a - 0 = a).
• Identity element for multiplication and division (a x 1 = a, a /1 = a).

Objective D:* Learners will translate written or verbal expressions of mathematical problems into the language/symbols of mathematics and vice versa.

• “While at Penney's, Mary is considering buying two blouses on sale for $27.95 each, and a skirt priced at $34.50. She wants to pay cash for her purchases but has only $110 in her purse. The sales tax in her locality is 8.25%. Can she buy these items with the cash she has?” Translation: If 1.0825 x [34.5 + 2(27.95)] < 110, then Mary can buy the items.

Objective A: Learners will identify vocabulary and symbolism of statistics and probability.

• Median, average, mean, mode with real-life situations or problems.
• Use a calculator with Statistical Mode (elective for students with selected career goals).

Objective B: Learners will investigate the use of statistics in real-world situations.

• Collect, organize, describe and interpret data.
• Construct, read, and interpret displays of data in tables, budgets, charts, and graphs.
• Evaluate statements based on data analysis.
• Formulate and solve problems involving collecting and analyzing data.
• Recognize data analysis as a tool for decision-making.

Objective C: Learners will explore the use of probability in real-world situations.

• Explore the concepts of chance.
• Make predictions that are based on experimental or theoretical probabilities, e.g., in genetics or tossing a coin.
• Model situations by devising simulations to determine probabilities in modifying existing system.
• Identify the pervasive use of probability in the real world.
• Formulate and solve problems that involve probability.

Objective D: Learners will understand and apply the concepts of range to establish acceptable upper and lower limits (tolerance).

• Read, interpret, and construct a chart, graph, or diagram related to concepts of tolerance and acceptable ranges.

Objective A: Learners will develop concepts of fractions, mixed numbers, decimals, and percents.

• Understand a fraction as a ratio, a relationship by division, a part or equal share of a whole, or a part or equal share of a group.
• Understand a proportion as two equal ratios, and that finding the least common denominator means you are solving for the missing element in a proportion.
• Understand a mixed number as the sum of a whole number and a fractional part that may also be expressed in fractional terms (4 1/2 = 4 + 1/2 = 8/2 + 1/2 = 9/2).
• Recognize a decimal as a fraction with a denominator which is a power of ten.
• Recognize a decimal as the fractional extension of the base ten system of numbers.
• Recognize percent as representing a ratio expressed in hundredths, e.g., 25% = 25:100.
• Recognize percent as a fraction with a denominator of 100 which can be expressed as a decimal, e.g., 12.5% = 12.5/100 = 125/1000 = 0.125.

Objective B: Learners will develop number sense for fractions, decimals, and percents.

• Recognize that all possible proper fractions (common, decimal, percent) fall between 0 and 1, i.e., 0 < x/y < 1.
• Recognize that as the denominator of any fraction gets larger, the size of the pieces the whole is being cut into gets smaller, e.g., 1/2 > 1/3 > 1/4 > 1/5.
• Identify how close to (or far away from) the fraction, decimal, and/or percent is to a half or a whole.
• Use models (Cuisenaire Rods, cutouts of circles, Deanes Blocks, paper, instruments, coins, dollars/cents, base ten grids, etc.) to relate conversions of common fractions to decimals and to percents.

Objective C: Learners will apply concepts of whole number operations: addition, subtraction, multiplication, division, powers and roots, to common fractions, decimals, and percents.

• Recognize that the concepts, definitions, and laws of addition, subtraction, multiplication, division apply also to common fractions, decimals, and/or percents.
• Use models (Cuisenaire Rods, cutouts of circles, Deanes Blocks, paper, instruments, coins, dollars/cents, base ten grids, etc.) to explore the result of number operations on common fractions, decimals, and/or percents.

Objective D: Learners will apply knowledge of ratios, proportions, common fractions, decimals, and percents to a variety of problem- solving situations.

• Learner will select and use the appropriate ratio, proportion, common fraction, decimal, and/or percent and an appropriate algorithm (procedure used to operate on them) to solve a variety of personal, work-related, or academic problems.

Objective A: Learners will translate verbal information into mathematical representations.

• Represent situations that involve variables with expressions.
• Recognize and represent situations that involve variables in equations.
• Translate written or verbal problems into mathematical symbols or equations.
• Translate open sentences and expressions using variables into written or verbal problems.

Objective B: Learners will use algebraic concepts to solve percent, ratio, and proportion problems.

• Recognize that converting fractions to find common denominators, converting fractions into decimals or into percents, and problems involving percent may all be expressed as proportions and converted into algebraic equations.

Objective C: Learners will use tables and graphs to solve problems.

• Recognize the connection between algebra and geometry by using algebraic equations to illustrate coordinate graphing and vice versa.
• Use computer programs to generate pie, bar, and line graphs.
• Provide statistical analyses for tables and graphs using the computer.

Objective D: Learners will apply algebraic strategies in solving a variety of real-world and mathematical problems.

• Substitute numbers for symbols and vice versa.
• Recognize order of operations.
• Perform operations using positive and negative numbers.
• Perform inverse operation to solve equations and inequalities.
• Solve equations and inequalities using identity factors.
• Investigate inequalities informally.
• Apply coordinate graphing.

Objective A: Learners will apply geometric properties and relationships.

• Recognize and define terms which apply to point, line, line segment, vector, angle, tangent.
• Recognize and define terms which apply to figures, angles, polygons, circles.
• Classify figures in terms of congruence and similarity.
• Deduce properties of, and relationships between, figures from given assumptions.

Objective B: Learners will use coordinate representation to illustrate linear equations.

• Recognize and define terminology of coordinate geometry: slope, axis, coordinate pairs, intercept, figures, limit, quadrant, etc.
• Recognize the values of coordinates in the four quadrants.

Objective C: Learners will translate between the algebraic and coordinate representations.

• Plot solution set for given algebraic equations.
• Identify/create algebraic expressions given graphic representations.
• Solve problems using coordinate representation and vectors (elective for students with specialized career goals).
• Identify real-life situations in terms of graphic representation.

Objective D: Learners will use the special properties of triangles to solve real-life problems.

• Define terms related to triangles -- classification, names of angles, internal and external angles.
• Use special relation of sides of right triangles for problem solving: Pythagorean Theorem, Sine, Cosine, Tangent relations.
• Recognize similar triangles and their importance.

Objective A: Learners will describe, model, draw, and classify shapes and recognize them in the environment.

• Straight line
• Angles
• Triangles
• Quadrilaterals: including parallelogram, rhombus, rectangle, square
• Polygons, both regular and irregular
• Circles
• Solid figures including spheres and cones

Objective B: Learners will investigate and predict the results of combining, subdividing, and changing shapes.

• Combine two squares, rectangle and square, two rectangles.
• Subdivide square, rectangle, rhombus, parallelogram into two triangles.
• Subdivide irregular polygons into rectangles, and/or triangles.
• Regular polygons in circles.

Objective C: Learners will explore geometric relationships.

• Compare and contrast characteristics of geometric shapes.
• Explore relationships between geometric formula, e.g., areas of rectangle, parallelogram, triangle, etc.

Objective D: Learners will understand and identify the attributes of measurement.

• Length, width, depth, height (altitude), base
• Perimeter, area, volume
• Capacity, weight
• Time
• Temperature
• Angle

Objective E: Learners will develop the process of measuring and use the concepts related to units of measure, including conversions within systems.

• The English System.
• The Metric System.
• Select appropriate measurement tool to measure to the degree of accuracy desired in a particular situation.
• Identify personal measures approximating English and metric measures, i.e., first finger joint -- about 1 inch, distance across middle fingernail -- about 1 cm., distance from nose to fingers with arm extended -- about 1 yard.

Objective F: Learners will apply measurement skills to problem solving and everyday experience.

• Estimate and use measurements to describe and compare objects.
• Apply the concepts of perimeter, area, volume, angle measure, and weight.
• Discover formulas and optional procedures for measurement.
• Make and use measurements in problem and everyday situations.
• Read a variety of scales including both standard and metric, simple calipers, and dial gauges.

Objective A: Learners will identify, explore, and apply patterns and functions in mathematical situations.

• Investigate the effects of multiplying and dividing by multiples and powers of ten.
• Explore the patterns inherent in the addition/subtraction and multiplication/division tables.
• Analyze functional relationships to explain how a change in one quantity results in a change in another. e.g., In a formula such as A = lw, the inverse variation is: if “A” remains constant, then an increase in “l” will result in a decrease in “w.” The direct proportion is: if “l” remains the constant, (l = A/w), then an increase in “w” will result in an increase in “A,” or an increase in “A” will result in an increase in “w.”

Objective B: Learners will recognize, describe, and create a variety of patterns.

• Use a calculator to calculate decimal expansions for fractions to discover terminating, repeating, delayed repeating, and non-repeating decimal equivalents.
• Recognize patterns in geometric shapes.
• Recognize and use the patterns of the base ten system to explain the magnitude of numbers.
• Use the patterns of the basic addition and multiplication facts to aid in mental and paper and pencil calculations.

Objective C: Learners will represent and describe mathematical relationships.

• Interpret information contained in graphs, charts, tables.
• Construct graphs, charts, tables using given information.
• Use a Cartesian plane to illustrate mathematical relationships that result in straight and intersecting lines.

Objective D: Learners will apply mathematical thinking and modeling to solve problems that arise in other disciplines such as art, music, science, economics, business, and industry.

• Apply the concept of ratio and proportion to scale drawing.
• Recognize the meaning of the time signature in music and its relationship to the time value of the various types of note (a 1/2 note in 3/4 time is shorter than a 1/2 note in 4/4 time).
• Use knowledge of algebra to determine the number of atoms in multiple molecules of an element.
• Investigate the use of patterns in predicting social, economic, and business trends.
• Model systems to modify existing ones to improve products or services.