San Diego City Schools
Mathematics
Content and Performance Standards
Board Review Copy, January 1998
Rigorous academic standards are essential for educational improvement. Coupled with appropriate instructional materials, instruction, and assessment, standards have the power to transform schooling and learning. The task of the Mathematics Grade-level Standards Committee was to draft rigorous, world-class mathematics content and performance standards for all San Diego City Schools students, grades K–12. Please keep the following in mind:
• The standards are an attempt to define excellence, not minimum competencies.
• The standards balance facts, skills, procedural knowledge, conceptual understanding, problem solving, application, reasoning, and mathematical communication.
• The standards define a core of essential knowledge and skills, but do not limit or dictate instructional strategies for ensuring students achieve them. The how-to is left to the professionals in the classrooms.
• The standards describe the content for each grade level or course. It is important to note that at this point, there has been no attempt to indicate which standards or content topics are most important for each grade level.
• The standards are designed to ensure that as students progress through the grades, they are effectively prepared for and successful in an increasingly rigorous mathematics curriculum.
In creating the standards, the committee carefully analyzed and drew from the best components of a variety of successful local, state, national, and international standards models. Model standards reviewed include the National Council of Teachers of Mathematics (NCTM) Standards, the California Academic Standards Commission Mathematics Content Standards, the Challenge Standards for Student Success, the Mathematically Correct Standards, the Education Round Table Standards for High School Graduates, New Standards, and the San Diego City Schools Consultation Draft standards. In addition, the committee reviewed state standards from Colorado, Maryland, Virginia, and Washington and district standards from Boston, Massachusetts; Charlotte-Mecklenburg, North Carolina; Chicago, Illinois; Jefferson County, Colorado; Los Angeles, California; Minneapolis, Minnesota; Milwaukee, Wisconsin; Philadelphia and Pittsburgh, Pennsylvania; and Rochester, New York. Local school standards from San Diego and international models including Japan also were carefully reviewed.
Adoption of increasingly rigorous math standards has important implications and inherent challenges:
• Current students, at the time the new standards are implemented, will not have had the benefit of beginning and progressing through a math program based upon these rigorous standards. As a result, it is anticipated that some will not be adequately prepared to meet the standards in the very first years of implementation and will need additional time and support to be successful.
• Instructional materials and evaluation tools must be changed or adopted to reflect and align with the new standards. Standards can only chart the course. Appropriate instructional materials supporting the achievement of the standards must be readily available and teacher- and student-friendly, if all students are to achieve these rigorous standards.
• The adoption of increasingly rigorous mathematics standards will result in need for intensified training in mathematics content for both new and experienced elementary teachers.
• Information and training for parents and community members will be essential.
Organization and Abbreviations
Content and performance standards for kindergarten through grade 7 have been organized into five strands: 1) number sense and operations; 2) function and algebra; 3) measurement and geometry; 4) data analysis, statistics, and probability; and 5) problem solving, mathematical reasoning, and communication. Above grade 7, standards are organized into four courses: algebra, geometry, intermediate algebra, and advanced algebra and trigonometry. The following abbreviations have been used in listing and enumerating the standards:
NO: Number Sense and Operations
FA: Functions and Algebra
MG: Measurement and Geometry
DA: Data Analysis, Statistics, and Probability
PS: Problem Solving, Mathematical Reasoning, and Communication
A: Algebra
G: Geometry
IA: Intermediate Algebra
AT: Advanced Algebra and Trigonometry
A list of key vocabulary is provided for each grade level, K–7. It is important for teachers and students to use appropriate mathematics terms. (Note: The words are not intended as a "spelling list" for students.)
Kindergarten
Number Sense and Operations
Content Standard K–NO1
The student understands and relates a sense of numbers and quantity in useful ways.
Performance Standards
The student will:
K–NO1.1 Count in various ways including counting objects up to 12, counting by ones up to 31 and backwards from 10, skip-counting by fives and tens to 50 and by twos up to 10 (2 to 10 and 1 to 9).
K–NO1.2 Identify written numbers from 0 to 31.
K–NO1.3 Select the correct numeral to indicate a quantity from 0 to 9, tracing over and writing the numeral.
K–NO1.4 Select a reasonable order of magnitude from two given quantities for a familiar situation (e.g., 5 or 50 jellybeans in a jar).
K–NO1.5 Identify ordinal positions from first to fifth, using concrete objects.
K–NO1.6 Compare two sets of 10 or fewer concrete items and identify one as containing more, less, or the same as the other set.
K–NO1.7 Divide a set of two, four, six, or eight concrete objects into equal halves.
K–NO1.8 Identify a penny, nickel, dime, quarter, and one-dollar bill.
K–NO1.9 Identify the cent sign and write amounts to nine cents using the cent sign.
Content Standard K–NO2
The student understands and describes simple addition and subtraction situations.
Performance Standards
The student will:
K–NO2.1 Identify one more and one less for numbers from one to nine.
K–NO2.2 Add and subtract whole numbers using up to 10 concrete items.
Functions and Algebra
Content Standard K–FA1
The student asks and answers questions involving quantities given a picture or situation.
Performance Standards
The student will:
K–FA1.1 When prompted by a picture or situation, identify whether things would be added or subtracted.
K–FA1.2 Identify and interpret + and – symbols.
Content Standard K–FA2
The student identifies, describes, and extends simple patterns.
Performance Standards
The student will:
K–FA2.1 Sort objects by attribute (e.g., color, shape, size) and identify the attribute.
K–FA2.2 Find the element of a set that does not belong and explain why it does not belong.
K–FA2.3 Identify and describe patterns of symbols, shapes, and objects (e.g., patterns on a calendar; counting by fives and tens; circle-circle-square, circle-circle-square, ….)
K–FA2.4 Extend and create simple patterns of symbols, shapes, and objects.
Measurement and Geometry
Content Standard K–MG1
The student selects appropriate tools and language to compare quantities and measure length, weight, capacity, time, and temperature.
Performance Standards
The student will:
K–MG1.1 Identify the instruments used to measure length (ruler), weight (scale), time (clock, calendar), and temperature (thermometer).
K–MG1.2 Make direct comparisons or use nonstandard units to measure length/height (shorter, longer, taller), weight (lighter, heavier), temperature (colder, hotter).
K–MG1.3 Compare the volumes of two given like-shaped containers by using concrete materials (e.g., jellybeans, sand, water, rice).
Content Standard K–MG2
The student demonstrates an understanding of time, using major units on clocks and calendars.
Performance Standards
The student will:
K–MG2.1 Tell time to the nearest hour, using analog and digital clocks, and demonstrate under-standing of morning, afternoon, and night.
K–MG2.2 Describe and tell time using calendars (e.g., days of the week, months of the year, seasons).
Content Standard K–MG3
The student identifies common geometric objects in the environment and describes their features.
Performance Standard
The student will:
K–MG3.1 Identify, describe, and draw or construct two-dimensional geometric objects (circle, triangle, square, rectangle) in the environment (e.g., clock faces, doorways) and describe their positions (e.g., next to, top, bottom).
Data Analysis, Statistics, and Probability
Content Standard K–DA1
The student collects and records information about the environment.
Performance Standards
The student will:
K–DA1.1 Notice and talk about quantities in the environment (e.g., How many students are wearing red? Brown? Blue?).
K–DA1.2 Collect information and record the results, using objects, pictures, or picture graphs.
Problem Solving, Mathematical Reasoning and Communication
Content Standard K–PS1
The student counts, adds, and subtracts as appropriate in problem solving.
Performance Standards
The student will:
K–PS1.1 Identify situations in which the answers to questions such as How many? How many more? How many altogether? can be obtained by counting, addition, or subtraction; identify the units needed for the answer (e.g., a number of apples); model the problem with concrete objects; and state the solution.
K–PS1.2 Restate in his or her own words problems involving counting, addition, or subtraction of small quantities.
K–PS1.3 Make up problems that can be solved by counting, addition, and subtraction.
Content Standard K–PS2
The student interprets and uses basic logical statements in informal language.
Performance Standard
The student will:
K–PS2.1 Identify true and false statements involving the use of either, or, both, and and, and the quantitative terms all, some, and none.
Content Standard K–PS3
The student communicates knowledge of basic skills, conceptual understanding, and problem solving, and demonstrates understanding of the mathematical communications of others.
Performance Standards
The student will:
K–PS3.1 Use appropriate mathematical vocabulary (see kindergarten Key Vocabulary).
K–PS3.2 Show ideas with a variety of concrete materials (connecting cubes, pattern blocks, buttons, beads, color tiles, etc.) and by pasting paper representations of materials.
K–PS3.3 Explain strategies used in solving problems and share ideas orally when probed by the teacher or when dictating to an adult for recording.
K–PS3.4 Understand and follow oral directions for appropriate mathematics activities.
Key Vocabulary
Number Sense and Operations
–, +, addition, answer, cent, dime, dollar, equal, estimation, fewer, halves, less, more, nickel, numeral, ordinal, penny, quarter, same, set, subtraction
Functions and Algebra
–, +, added, attribute, calendar, circle, color, element, objects, patterns, set, shape, size, square, subtracted, symbols
Measurement and Geometry
>, <, =, afternoon, analog, bottom, calendar, circle, clock, clock face, colder, compare, day, difference, digital, doorway, geometry, heavier, height, hotter, hour, larger, length, lighter, month, morning, next to, night, place value (ones, tens), rectangle, ruler, scale, seasons, shorter, smaller, square, sum, taller, temperature, thermometer, time, top, triangle, volume, week, weight, year
Data Analysis, Statistics, and Probability
environment, graph, quantity
Problem Solving, Mathematical Reasoning, and Communication
addition, all, and, both, counting, either, false, none, or, solution, some, subtraction, true
Grade 1
Number Sense and Operations
Content Standard 1–NO1
The student identifies and represents the number and order of objects.
Performance Standards
The student will:
1–NO1.1 Count forward by twos, fives, and tens to 100 and count backwards from 100 by ones.
1–NO1.2 Read and write numerals from 0 through 100.
1–NO1.3 Read and write numbers from 0 to 20 as words.
1–NO1.4 Identify the ordinal positions first through twenty-first using concrete objects and pictures.
1–NO1.5 Locate, sequence, and represent whole numbers on a number line.
1–NO1.6 Compare two sets of up to 12 objects and tell which has more and how many more.
1–NO1.7 Use the symbols >, <, and = to compare two sets or pictures of sets of up to 12 objects and two numbers from 0 to 100.
1–NO1.8 Identify the number of pennies equivalent to a nickel, a dime, and a quarter and compute sums of money up to $1.00.
1–NO1.9 Show different combinations of coins that equal the same amount of money up to $1.00 and use the dollar sign.
Content Standard 1–NO2
The student demonstrates an understanding of base-ten place value for two-digit numbers.
Performance Standards
The student will:
1–NO2.1 Identify the place value of each digit in numbers up to 100.
1–NO2.2 Use words, models (e.g., groups of objects), and expanded form (e.g., 24 means 20 + 4 or two bundles of 10 rods and 4 additional rods) to represent two-digit numbers.
1–NO2.3 Mentally identify one more than, one less than, ten more, ten less, with numbers and sums up to 100.
1–NO2.4 Select a reasonable order of magnitude from three given quantities for a familiar situation (e.g., 5, 50, and 500 jellybeans in a jar).
Content Standard 1–NO3
The student demonstrates the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, adding on) and uses those operations to solve problems.
Performance Standards
The student will:
1–NO3.1 Recall basic addition facts (sums 0–18) and the corresponding subtraction facts.
1–NO3.2 Add three single-digit numbers.
1–NO3.3 Add and subtract two-digit numbers both horizontally and vertically, without regrouping.
1–NO3.4 Solve story and picture problems involving one-step solutions, using basic addition and subtraction facts.
Content Standard 1–NO4
The student demonstrates a beginning understanding of the concept of division.
Performance Standards
The student will:
1–NO4.1 Identify one-half, one-third, and one-fourth, using concrete materials or pictures.
1–NO4.2 Divide groups of objects into equal sets.
Functions and Algebra
Content Standard 1–FA1
The student creates and solves problems using words, symbols, drawings, and objects.
Performance Standards
The student will:
1–FA1.1 Solve simple addition (sums 0–18) and corresponding subtraction equations with a blank, box, or other symbol in any position, such as 2 + 5 =___ , 7 – ___ = 5, ___ – 2 = 5.
1–FA1.2 Write and solve number sentences that express relationships involving addition and subtraction, including the correct interpretation and use of the +, –, and = symbols.
Content Standard 1–FA2
The student sorts objects and completes and describes patterns involving numbers, shape, size, rhythm, and color.
Performance Standards
The student will:
1–FA2.1 Sort concrete objects according to two attributes (such as color and shape).
1–FA2.2 Recognize, describe, and extend a wide variety of patterns, including size, color, shape, and quantity, and including increasing, decreasing, and repeating patterns, using concrete materials and patterns.
1–FA2.3 Identify the common property of the elements of a set, select matching additions to the set, and identify the item that does not belong in a set.
Measurement and Geometry
Content Standard 1–MG1
The student compares objects according to length, weight, and capacity using direct comparison, non-standard and standard units.
Performance Standards
The student will:
1–MG1.1 Estimate and then count the number of cubes in a rectangular box.
1–MG1.2 Compare weights of objects using a balance scale.
1–MG1.3 Measure and draw line segments in inches and centimeters.
1–MG1.4 Estimate and measure length in inches and weight in pounds.
1–MG1.5 Measure perimeter to the nearest inch and centimeter.
1–MG1.6 Order similar objects (e.g., smallest to largest, longest to shortest, lightest to heaviest).
1–MG1.7 Estimate and measure volume in cups and identify a cup, pint, quart, and gallon.
Content Standard 1–MG2
The student demonstrates an understanding of time and temperature and tells time to the half-hour.
Performance Standards
The student will:
1–MG2.1 Name the days of the week and the months of the year, both in order and out of sequence.
1–MG2.2 Tell time to the half-hour, using analog and digital clocks.
1–MG2.3 Orient events in time using today, yesterday, and tomorrow; morning and afternoon; this morning and yesterday morning, etc.
1–MG2.4 Associate temperature in degrees Fahrenheit with weather (e.g., 85° is really hot, 50° is colder).
Content Standard 1–MG3
The student identifies common geometric objects in the environment and describes their features.
Performance Standards
The student will:
1–MG3.1 Demonstrate orientation and relative position, such as closed/open, on/under/over, in front of/in back of, between, in the middle of, next to, beside, inside/outside, around, far from/near to, above/below, to the right of/to the left of, and here/there.
1–MG3.2 Identify and describe three-dimensional objects (cube, sphere, cone, cylinder), including identifying the faces as two-dimensional shapes (triangles, rectangles, squares, or circles).
1–MG3.3 Draw and describe triangles, squares, rectangles, and circles according to number of sides and corners.
1–MG3.4 Identify geometric shapes that have symmetry.
1–MG3.5 Identify congruent shapes and designs.
Data Analysis, Statistics, and Probability
Content Standard 1–DA1
The student collects, reads, interprets, and compares data from simple graphs.
Performance Standards
The student will:
1–DA1.1 Gather, record, read, interpret, and compare data in bar graphs, tally charts, and picture graphs.
1–DA1.2 Compare data using the concepts of largest, smallest, most often, and least often.
Problem Solving, Mathematical Reasoning, and Communication
Content Standard 1–PS1
The student shows critical evaluation of available information in problem situations involving counting, addition, and subtraction.
Performance Standards
The student will:
1–PS1.1 Identify quantities and units given as facts in a problem situation, determine which facts are relevant or irrelevant to the solution, and explain why some facts are irrelevant by showing that they are not required for the solution.
1–PS1.2 Identify whether or not a problem situation contains sufficient information to arrive at an answer and be able to explain what is lacking when there is insufficient information.
Content Standard 1–PS2
The student uses problem-solving strategies involving addition, subtraction, and basic measurements.
Performance Standards
The student will:
1–PS2.1 Formulate and carry out a plan to solve a problem by acting out the problem, using manipulatives or models, or drawing a picture.
1–PS2.2 Create a problem and state it orally when given a picture, a model, or a real-life situation and identify problem solutions as correct or incorrect.
Content Standard 1–PS3
The student communicates knowledge of basic skills, conceptual understanding, and problem solving and demonstrates understanding of mathematical communications of others.
Performance Standards
The student will:
1–PS3.1 Use appropriate mathematical vocabulary (see grade 1 Key Vocabulary).
1–PS3.2 Show ideas by drawing, using words and numbers, and by building with a variety of concrete materials, such as connecting cubes, pattern blocks, buttons, beads, and color tiles, and by pasting paper representations of materials.
1–PS3.3 Explain strategies used in solving problems and share ideas orally when probed by the teacher, and in writing and drawing.
1–PS3.4 Follow oral directions for appropriate mathematics activities.
Key Vocabulary
Number Sense and Operations
>, <, =, answer, backward, bundle, comparing, difference, digit, dime, dollar sign, equal, equivalent, expanded form, forward, fourth, half, horizontally, less than, locating, more than, nickel, ordinal, pennies, place value (ones, tens, hundreds), quarter, rod, sequencing, set, sum, symbol, third, vertically
Functions and Algebra
+, –, =, addition, attribute, color, decreasing pattern, element, equation, function, increasing pattern, number sentence, pattern, property, quantity, repeating pattern, set, shape, size, subtraction
Measurement and Geometry
above, afternoon, analog, around, balance, behind, below, beside, between, centimeter, circle, closed, comparing, cone, congruent, corner, cube, cup, customary, cylinder, day, design, digital, estimating, face, Fahrenheit, far from, gallon, half-hour, heaviest, here, inch, in back, in front, in the middle of, inside, largest, lightest, line segment, longest, metric unit, month, morning, near, next to, on, open, outside, over, perimeter, pint, position, pound, quart, rectangular, rectangle, scale, shape, shortest, side, smallest, sphere, square, symmetry, there, three-dimensional, to the left of, to the right of, tomorrow, triangle, two-dimensional, under, volume, week, weight, year, yesterday
Data Analysis, Statistics, and Probability
bar graph, data, gathering, largest, least often, most often, picture graph, smallest, tally chart
Problem Solving, Mathematical Reasoning, and Communication
insufficient, irrelevant, quantity, relevant, solution, sufficient, unit
Grade 2
Number Sense and Operations
Content Standard 2–NO1
The student extends an understanding of number and place value to exact and approximate whole numbers to 1,000.
Performance Standards
The student will:
2–NO1.1 Count forward and backwards by ones and count forward by twos, fives, and tens, starting at an arbitrary number.
2–NO1.2 Count by threes and fours to 48 and hundreds and fifties to 1,000.
2–NO1.3 Read and write numbers from 0 to 1,000.
2–NO1.4 Read and write numbers from 0 to 100 as words.
2–NO1.5 Write two- and three-digit numbers in expanded form (e.g., 567 = 500 + 60 + 7).
2–NO1.6 Use symbols and words (>, <, or =; greater than, less than, or equal to) to compare two whole numbers between 0 and 1,000.
2–NO1.7 Round numbers from 0 to 1,000 to the nearest 10 or 100.
2–NO1.8 Identify odd and even numbers from 0 to 100.
2–NO1.9 Identify the place value for each digit up to the hundreds.
2–NO1.10 Identify one more, one less, ten more, ten less, one hundred more, and one hundred less than a given number (solution in the range 0 to 1,000).
Content Standard 2–NO2
The student computes with whole numbers using addition, subtraction, multiplication, and division and uses those operations to solve problems.
Performance Standards
The student will:
2–NO2.1 Recall basic addition facts (sums 0–20) and the corresponding subtraction facts.
2–NO2.2 Add and subtract numbers on paper (sums to 1,000) with and without regrouping.
2–NO2.3 Add three two-digit numbers on paper with and without regrouping.
2–NO2.4 Use paper and pencil, mental computation, or concrete materials to estimate sums to 1,000 and the corresponding differences.
2–NO2.5 Solve word problems involving sums to 1,000 and corresponding differences.
2–NO2.6 Model products and quotients (e.g., with manipulatives) using repeated addition, arrays, counting by multiples, finding area, repeated subtraction, equal sharing, and forming equal groups.
2–NO2.7 Identify the part of a set and/or region that represents one-half, one-third, one-fourth, one-eighth, and one-tenth and write the corresponding fraction.
2–NO2.8 Recognize the multiplication sign, know what the terms factor and product mean in multiplication, and demonstrate that multiplication represents repeated addition.
2–NO2.9 Multiply single-digit numbers by 0, 1, 2, and 10.
Content Standard 2–NO3
The student counts and compares amounts of money.
Performance Standards
The student will:
2–NO3.1 Use a collection of coins and one-dollar bills to count, compare, and make change.
2–NO3.2 Read and write amounts of money using dollar and cent signs and the decimal point.
2–NO3.3 Show different combinations of coins and bills that equal the same amount of money (e.g., 15 quarters = $3.75).
Functions and Algebra
Content Standard 2–FA1
The student creates and solves problems involving addition and subtraction by modeling, representing, and interpreting number relationships.
Performance Standards
The student will:
2–FA1.1 Identify and use the inverse relationship between addition and subtraction to solve problems such as 4 + ___ = 7 and ___ + 3 = 7 and 7 – ___ = 3.
2–FA1.2 Use and describe the commutative (order) and associative (grouping) property of addition.
2–FA1.3 Use simple properties of addition and subtraction to devise algorithms or check results.
2–FA1.4 Identify and use the multiplication properties of zero and one.
2–FA1.5 Identify problem situations that match or do not match a given number sentence.
Content Standard 2–FA2
The student demonstrates an understanding of patterns and how they grow and describes them in general ways.
Performance Standards
The student will:
2–FA2.1 Recognize, describe, extend, and explain how to get the next term in patterns that grow in a linear way (e.g., 4, 8, 12, …; the number of ears on 1, 2, 3, 4, … horses).
2–FA2.2 Create and solve problems involving simple number patterns.
2–FA2.3 Identify and correct errors in patterns.
Content Standard 2–FA3
The student demonstrates an understanding of how two or more quantities are related to one another and how change in one causes change in another.
Performance Standards
The student will:
2–FA3.1 Demonstrate that a number is related to other numbers by being their sum, difference, or product (e.g., find all pairs of whole numbers whose sum is 14).
2–FA3.2 Explain how a change in input affects the output in a simple relationship (e.g., How far you can go depends on how much gasoline is in your car).
Measurement and Geometry
Content Standard 2–MG1
The student estimates, measures, and demonstrates understanding of the concepts of length, area, volume, and capacity.
Performance Standards
The student will:
2–MG1.1 Select appropriate formal or informal units to measure length, area, and volume and predict whether the measure will be greater or smaller when a different unit is used.
2–MG1.2 Determine the perimeter of a region by measuring or by adding given measures.
2–MG1.3 Estimate and determine the area and volume of figures by covering them with squares or counting the cubes.
2–MG1.4 Estimate and measure volumes in cups, pints, quarts, gallons, and liters and compare these volumes using the concepts of more, less, and equivalent.
2–MG1.5 Estimate and measure weight in pounds and kilograms.
Content Standard 2–MG2
The student demonstrates an understanding of the concept of time and temperature and the relationship among different units of time and tells time to the nearest quarter hour or five minutes.
Performance Standards
The student will:
2–MG2.1 Measure and tell time to the nearest quarter-hour or five minutes, using analog and digital clocks; compare time related to events (e.g., before/after, shorter/longer); and write time to the quarter-hour.
2–MG2.2 Solve simple problems of elapsed time to the hour.
2–MG2.3 Use a.m./p.m. and noon/midnight.
2–MG2.4 Order events by time sequence, identify equivalent periods of time (weeks in a year, days in a month, minutes in an hour), determine past and future days of the week, and identify specific dates on a calendar.
2–MG2.5 Write the date using words and numbers or only numbers.
2–MG2.6 Measure and record temperature in degrees Fahrenheit and Celsius to the nearest two degrees, using the degree sign.
Content Standard 2–MG3
The student identifies and describes the elements that compose common figures in the plane and common objects in space.
Performance Standards
The student will:
2–MG3.1 Compare and contrast plane and solid geometric shapes (circle/sphere, square/cube, triangle/pyramid, and rectangle/rectangular solid).
2–MG3.2 Identify, describe, and classify solid geometric shapes according to the number and shape of faces, edges, bases, and vertices.
2–MG3.3 Put shapes together to form larger or other shapes (e.g., two congruent right triangles form a rectangle) and find basic shapes within more complex figures (e.g., a hexagon is made up of six triangles).
2–MG3.4 Recognize and create symmetrical shapes and designs and identify and draw lines of reflection.
2–MG3.5 Identify lines as horizontal, vertical, perpendicular, and parallel.
2–MG3.6 Draw or construct congruent shapes and designs.
Data Analysis, Statistics, and Probability
Content Standard 2–DA1
The student collects, records, organizes, and displays data, noting patterns and making simple predictions.
Performance Standards
The student will:
2–DA1.1 Collect, record, and organize data to answer a question and display data from simple experiments using dot plots and bar graphs (including those with categorical or simple whole-number scales).
2–DA1.2 Make simple interpretations from data given in tables, picture graphs, or bar graphs.
2–DA1.3 List the possible outcomes of a simple event (e.g., tossing a coin, pulling colored marbles out of a bag) and predict whether outcomes are certain, likely, unlikely, or impossible.
Problem Solving, Mathematical Reasoning and Communication
Content Standard 2–PS1
The student uses appropriate problem-solving strategies.
Performance Standards
The student will:
2–PS1.1 Use addition to verify subtraction and vice versa.
2–PS1.2 Estimate to identify solutions that are reasonable or unreasonable in magnitude.
2–PS1.3 Solve addition and subtraction problems using data from simple charts, picture graphs, and number sentences (e.g., Find the answer for 4 + x = ___ when x = 2).
2–PS1.4 Recognize phrases in problem situations that suggest whether addition, subtraction, multiplication, or division is required and explain how the operations would be used to find the solutions.
Content Standard 2–PS2
The student identifies unions and intersections of sets.
Performance Standards
The student will:
2–PS2.1 Identify elements that belong in two sets (e.g., blue objects and triangles) and identify all elements that belong to the combination of two sets (e.g., the group of objects that is either red or square).
2–PS2.2 Use Venn diagrams to represent unions and intersections.
Content Standard 2–PS3
The student makes extensions of a problem and connections from one problem to another.
Performance Standards
The student will:
2–PS3.1 Identify relevant and irrelevant information in the statements of problem situations.
2–PS3.2 When directed to do so, show the solution with another material or in another way.
2–PS3.3 Compare the problem to other problems already solved and tell how it is related.
Content Standard 2–PS4
The student communicates knowledge of basic skills, conceptual understanding, and problem solving and demonstrates understanding of mathematical communications of others.
Performance Standards
The student will:
2–PS4.1 Use appropriate mathematical terms, vocabulary, and language, based on prior conceptual work (see grade 2 Key Vocabulary).
2–PS4.2 Show ideas in a variety of ways, including drawings, words, numbers, symbols, simple graphs and tables, and building with a variety of concrete materials.
2–PS4.3 Explain strategies used in solving problems and share ideas orally when probed by the teacher and in writing and drawing.
2–PS4.4 Present ideas appropriately for a particular audience or purpose.
2–PS4.5 Understand oral and simple written directions for appropriate mathematics activities.
Key Vocabulary
Number Sense and Operations
1,000, >, <, =, addition, area, array, cent sign, combination, computation, decimal, difference, division, dollar, equal to, even, expanded form, factor, hundred, less than, greater than, multiplication, odd, one-eighth, one-fourth, one-half, one-tenth, one-third, point, product, quotient, range, region, regrouping, repeated addition, rounding, subtraction, sum, symbol, ten, word
Functions and Algebra
algorithms, associative property, commutative property, difference, input, inverse relationship, linear, multiplication properties, number pattern, one, output, product, properties of addition, properties of subtraction, simple relationship, sum, zero
Measurement and Geometry
after, a.m., analog, area, bases, before, Celsius, circle, congruent, corner, cube, cup, date, day, degree sign, design, digital, edges, elapsed time, equivalent, faces, Fahrenheit, future, gallon, geometric, geometry, greater, horizontal, hour, kilogram, length, less, line of reflection, liter, longer, measurement, midnight, minute, month, more, noon, parallel, past, perimeter, perpendicular, pint, plane, p.m., pound, predicting, pyramid, quart, quarter hour, rectangle, rectangular solid, region, shape, shorter, smaller, solid, sphere, square, symmetry, temperature, time, triangle, week, vertical, vertices, volume, weight, year
Data Analysis, Statistics, and Probability
bar graphs, category, certain, data, dot plots, experiments, impossible, likely, picture graph, table, unlikely
Problem Solving, Mathematical Reasoning, and Communication
addition, division, elements, estimating, intersection, irrelevant, multiplication, number sentences, operations, relevant, sets, solutions, subtraction, union, Venn diagram
Grade 3
Number Sense and Operations
Content Standard 3–NO1
The student extends an understanding of number and place value to exact and approximate whole numbers to 999,999 and demonstrates an understanding of negative numbers, using a number line.
Performance Standards
The student will:
3–NO1.1 Read and write numbers from 0 to 999,999 with digits and words.
3–NO1.2 Identify numbers as even or odd.
3–NO1.3 Write numbers in expanded form to 999,999 and identify the place value for each digit up to the hundred-thousands.
3–NO1.4 Compare two whole numbers between 0 to 999,999 using symbols (>, <, or =) and words (greater than, less than, or equal to).
3–NO1.5 Round a whole number to the nearest 10, 100, or 1,000.
3–NO1.6 Use and interpret negative numbers using a number line.
3–NO1.7 Use a number line to solve simple subtraction problems with differences in the range of -10 to -1.
Content Standard 3–NO2
The student estimates, calculates, and demonstrates understanding of sums, differences, products, and quotients and the mutual relationships between them.
Performance Standards
The student will:
3–NO2.1 Complete addition problems (up to 100,000) of any two whole numbers and the corresponding subtraction problems.
3–NO2.2 Mentally estimate a sum to 999 and the corresponding difference.
3–NO2.3 Use mental computation strategies to simplify addition and subtraction problems (e.g., 102 + 84 = 100 + 86).
3–NO2.4 Recall multiplication up to 10 times 10 and corresponding division facts.
3–NO2.5 Multiply by 10, 100, and 1,000 mentally.
3–NO2.6 Estimate and find the product of two whole numbers, in which one factor is nine or less and the other is a multi-digit number up to four digits.
3–NO2.7 Identify perfect squares (and square roots) to 100 and recognize the exponent 2 as indicating a number squared.
3–NO2.8 Divide dividends up to four digits by one-digit divisors, with and without remainders.
3–NO2.9 Check division by multiplying and adding any remainder.
3–NO2.10 Solve two-step word problems.
Content Standard 3–NO3
The student demonstrates an understanding of decimals and simple fractions.
Performance Standards
The student will:
3–NO3.1 Identify fractions (to tenths) represented by drawings or concrete materials and represent a given fraction using both concrete materials and symbols.
3–NO3.2 Identify and write mixed numbers.
3–NO3.3 Demonstrate equivalent fractions (for example, 1/2 = 3/6), using concrete materials.
3–NO3.4 Use the signs >, <, and = to compare fractions with like denominators.
3–NO3.5 Compare the numerical values of two fractions having like denominators, using concrete materials.
3–NO3.6 Add and subtract with proper fractions having like denominators of 10 or less.
3–NO3.7 Read and write decimals to the hundredths.
3–NO3.8 Add and subtract decimals to hundredths.
3–NO3.9 Order money amounts written as decimals; count, compare, and make change for amounts up to $20.00, using a collection of coins and bills.
3–NO3.10 Model, estimate, and solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation (for multiplication and division, use whole number multipliers and divisors).
Functions and Algebra
Content Standard 3–FA1
The student selects appropriate symbols, operations, and properties to represent, describe, simplify, and solve simple number relationships.
Performance Standards
The student will:
3–FA1.1 Use variables as place holders for numbers in sentences involving addition, subtraction, multiplication, or division (m + 2 = 6; 2 K = 12; 7 + 8 = N).
3–FA1.2 Express simple unit conversions in symbolic form (e.g., # inches = # feet 12).
3–FA1.3 Identify and describe the commutative and associative properties of multiplication and the special properties of 0 and 1.
3–FA1.4 Write one or more number sentences to represent a word problem and solve and interpret the arithmetic answer(s) in the context of the problem.
3–FA1.5 Identify and use the inverse relationships between multiplication and division (e.g., 3 x 7 = 21, so 21 ÷ 3 = ____ ) to compute and check results.
Content Standard 3–FA2
The student represents and interprets a quantitative relationship in a formula, table, or graph.
Performance Standards
The student will:
3–FA2.1 Use a function rule to solve simple problems (e.g., the function rule is "adding 7" so 1 ->8, 3 ->10, 12 ->19) and graph the resulting ordered pairs of whole numbers on a grid.
3–FA2.2 Recognize, describe, and extend geometric, simple numeric, and linear patterns.
3–FA2.3 Use simple two-dimensional coordinate systems to find locations on a map (e.g., A4) and represent points and simple figures in a coordinate grid.
Measurement and Geometry
Content Standard 3–MG1
The student makes reasonable estimates and accurately measures distance, area, volume, and weight.
Performance Standards
The student will:
3–MG1.1 Know that 1 foot = 12 inches, 1 yard = 36 inches = 3 feet, 1 meter = 100 centimeters, one meter is a little more than one yard, etc.
3–MG1.2 Measure and draw line segments in inches (to 1/4 inch) and in centimeters.
3–MG1.3 Estimate and measure length in inches, feet, yards, centimeters, and meters and know the abbreviations.
3–MG1.4 Know that 1 quart = 2 pints, 1 gallon = 4 quarts, etc.
3–MG1.5 Estimate and measure liquid volume in cups, pints, quarts, gallons, and liters.
3–MG1.6 Estimate and measure weight in pounds/ounces and grams/kilograms and know the abbreviations.
Content Standard 3–MG2
The student tells time to the nearest minute, solves problems of elapsed time, and measures temperature at, above, or below zero.
Performance Standards
The student will:
3–MG2.1 Read digital and analog clock faces and tell time to the minute in both minutes before and minutes after the hour.
3–MG2.2 Know that 60 seconds equals one minute.
3–MG2.3 Solve problems of elapsed time (e.g., The game started at 1:15 and ended at 3:45. How long was the game? It is 1:15. What time will it be in 22 minutes?).
3–MG2.4 Measure and record temperature (Celsius and Fahrenheit) to the nearest degree at, above, or below zero.
Content Standard 3–MG3
The student analyzes plane and solid geometric objects and their attributes.
Performance Standards
The student will:
3–MG3.1 Use names for lines and line segments (e.g., line AB; segment CD).
3–MG3.2 Identify, name, describe, and classify polygons, including pentagons, hexagons, and octagons.
3–MG3.3 Name and describe (e.g., symmetry, number/shape of faces, straight/curved edges) common three-dimensional geometric objects (cube, rectangular solid, sphere, prism, pyramid, cone, cylinder).
3–MG3.4 Identify attributes of triangles and quadrilaterals; draw and describe the elements that distinguish those figures (e.g., equal sides, perpendicular sides, right angles, parallel sides, vertices).
3–MG3.5 Compute areas of rectangles in square inches and square centimeters.
3–MG3.6 Demonstrate understanding that a given area can be enclosed by several different perimeter lengths and that a given perimeter can encase figures of different areas (e.g., Given 12 square tiles, make all the rectangles you can that use all the tiles and find the perimeter of each rectangle).
3–MG3.7 Find the perimeter and the area of a polygon and give examples when one would want to find perimeter and area.
Data Analysis, Statistics and Probability
Content Standard 3–DA1
The student collects, records, organizes, graphs, and interprets the results of surveys and experiments and communicates the findings clearly.
Performance Standards
The student will:
3–DA1.1 Ask and answer questions relevant to the data presented in charts, tables, and graphs.
3–DA1.2 Predict probable outcomes from a collection of organized data.
Content Standard 3–DA2
The student determines the number of possible outcomes of an event and makes simple predictions.
Performance Standards
The student will:
3–DA2.1 Identify possible outcomes (e.g., with a spinner or die) and predict how likely they are to occur.
3–DA2.2 Find possible combinations and arrangements involving a limited number of items (e.g., How many ways can you line up a blue block, a red block, and a green block?).
Problem Solving, Mathematical Reasoning, and Communication
Content Standard 3–PS1
The student uses multiple solution strategies in simple multiplication and division problems.
Performance Standards
The student will:
3–PS1.1 Use concrete objects or drawings to model a multiplication problem and show that it can be solved by counting, skip counting, repeated addition, and multiplication (and similar demonstrations for simple division problems).
3–PS1.2 Select appropriate information and operations to express and solve word problems.
Content Standard 3–PS2
The student identifies and makes adjustments for units, as needed in solving problems.
Performance Standards
The student will:
3–PS2.1 Identify units for each fact in a problem situation, the units needed in the solution, and make simple unit conversions within a system of measurement (e.g., inches and feet, hours and minutes), as needed to correctly solve problems.
3–PS2.2 Identify unit conversion errors in incorrect problem solutions and correct them.
3–PS2.3 Determine a total cost or amount as a function of the number of units and the unit cost or value.
Content Standard 3–PS3
The student defends reasoning when finding solutions to problems generated from grade 3 key math content.
Performance Standards
The student will:
3–PS3.1 Use patterns to verify that all combinations have been found, when required to do so as part of the problem solution.
3–PS3.2 Make up and use a variety of strategies and approaches to solving problems and understand approaches that others use.
3–PS3.3 Use one strategy to verify a solution obtained by another strategy.
Content Standard 3–PS4
The student communicates knowledge of basic skills, conceptual understanding, and problem solving and demonstrates understanding of mathematical communications of others.
Performance Standards
The student will:
3–PS4.1 Use appropriate mathematical terms, vocabulary, and language, based on prior conceptual work (see grade 3 Key Vocabulary).
3–PS4.2 Show ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and building with a variety of concrete materials.
3–PS4.3 Explain strategies used in solving problems, show ideas, and support solutions with evidence, in both oral and written form.
3–PS4.4 Present ideas appropriately for a particular audience or purpose.
3–PS4.5 Understand oral and written directions for appropriate mathematics activities.
Key Vocabulary
Number Sense and Operations
decimal point, denominator, dividend, division, divisor, equivalent fraction, exponent (x 2), even, fraction, hundreds, hundred thousands, hundredths, making change, mental, mixed number, multiplication, negative, numerator, odd, ones, perfect square, place value, proper fraction, quotient, remainder, simplify, square exponents, square root, tenths, tens, ten thousands, thousands, whole number
Functions and Algebra
arithmetic, associative property, commutative property, division, function rule, geometric, grid, interpret, inverse relationships, linear pattern, multiplication, ordered pair, simple numeric, special properties of 0 and 1, subtraction, symbolic form, unit conversion, word problem, variables
Measurement and Geometry
above zero, analog, area, at zero, attribute, below zero, Celsius, centimeter (cm), cone, coordinate grid, coordinate system, cube, curved edge, cylinder, digital, elapsed time, element, equal side, Fahrenheit, feet (ft.), foot (ft.), gallon (gal.), gram (g), hexagon, inch (in.), kilogram (kg), line, line segment, liter (l), map, meter (m), minute, number, octagon, ounce (oz.), parallel side, pentagon, perpendicular side, pint (pt.), point, polygon, pound (lb.), prism, pyramid, quadrilateral, quart (qt.), rectangle, rectangular solid, right angle, second, shape of faces, sphere, square centimeter, square inch, straight edge, symmetry, three-dimensional, triangle, vertex, vertices, yard (yd.)
Data Analysis, Statistics, and Probability
arrangements, possible combinations, predicting, probable outcome, variable
Problem Solving, Mathematical Reasoning and Communication
chart, combination, counting, diagram, division, error, feet, function, graph, hour, inch, measurement, minute, multiplication, number, picture, repeated addition, skip counting, solution, strategy, symbol, table, unit conversion, value, verify, word, word problem
Grade 4
Number Sense and Operations
Content Standard 4–NO1
The student extends understanding of number and place value to include decimals to thousandths.
Performance Standards
The student will:
4–NO1.1 Read, write, order, compare (in symbols and words), and round numbers from .001 to over 1,000,000.
4–NO1.2 Write numbers in expanded form from .001 to more than 1,000,000 and identify the place value for each digit (e.g., 21.49 = 20 + 1 + .4 + .09).
Content Standard 4–NO2
The student demonstrates an understanding of addition, subtraction, multiplication, and division and uses these operations to estimate or find solutions to problems.
Performance Standards
The student will:
4–NO2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to thousandths.
4–NO2.2 Recall multiplication up to 12 12.
4–NO2.3 Estimate and multiply by two-digit and three-digit numbers.
4–NO2.4 Create and solve problems that involve multiplication of two whole numbers, one factor 99 or less and the second factor 4 digits or less (two-step including multiplication and division).
4–NO2.5 Multiply mentally by 10, 100, 1,000, and 10,000.
4–NO2.6 Identify multiples of a given number and common multiples of two given numbers.
4–NO2.7 Identify factors of a given number and common factors of two given numbers.
4–NO2.8 Write a cube number as repeated multiplication.
4–NO2.9 Estimate and divide dividends up to four digits by one- digit divisors.
Content Standard 4–NO3
The student identifies, represents, interprets, and uses fractions and decimals.
Performance Standards
The student will:
4–NO3.1 Identify and write equivalent fractions.
4–NO3.2 Put fractions in lowest terms.
4–NO3.3 Change improper fractions to mixed numbers and vice versa.
4–NO3.4 Compare the numerical values of mixed numbers and fractions having like and unlike denominators of 12 or less and find their approximate location on a number line.
4–NO3.5 Add and subtract with fractions having like and unlike denominators of 12 or less.
4–NO3.6 Multiply and divide a fraction or a decimal by a whole number.
4–NO3.7 Read and write decimals as fractions (for example, 0.39 = 39/100) and vice versa.
4–NO3.8 Relate fractions to decimals, using concrete objects.
4–NO3.9 Determine and express simple ratios.
Content Standard 4–NO4
The student demonstrates an understanding of negative integers and their relationship to positive integers.
Performance Standards
The student will:
4–NO4.1 Interpret negative integers (to 20) as temperatures below zero, distances below sea level, distances to the left of zero on an integer number line, or solutions to problems in which a larger number is subtracted from a smaller number.
4–NO4.2 Model the concept that a whole number and its opposite add to be zero.
Functions and Algebra
Content Standard 4–FA1
The student uses and interprets variables, mathematical symbols, and properties to write and simplify expressions and sentences.
Performance Standards
The student will:
4–FA1.1 Model, describe, and use the distributive property.
4–FA1.2 Write and simplify numerical expressions involving one or more of the following operations: +, –, , and ÷.
4–FA1.3 Develop, interpret, and use formulas (e.g., A = bh) to answer questions about quantities and their relationships.
4–FA1.4 Recognize the addition of a negative number as the subtraction of a positive number.
4–FA1.5 Use mental computation strategies for multiplication, for example, breaking a problem into partial products: 3 x 27 = (3 x 20) + (3 x 7) = 60 + 21 = 81.
Content Standard 4–FA2
The student uses information given verbally, in tables, and on graphs to create and interpret other representations of the same information.
Performance Standards
The student will:
4–FA2.1 Describe and explain the effect that the variation of one quantity has on the variation of a second quantity.
4–FA2.2 Graph and name ordered pairs on a four-quadrant coordinate grid.
Measurement and Geometry
Content Standard 4–MG1
The student measures, estimates, and demonstrates an understanding of the relationships between distance, capacity, and weight measures.
Performance Standards
The student will:
4–MG1.1 Estimate and measure length in parts of an inch (1/2, 1/4, and 1/8), inches, feet, yards, millimeters, centimeters, and meters.
4–MG1.2 Estimate and measure areas, recognizing and using appropriate units in both the metric and customary system, e.g., square centimeters (cm2), square meters (m2), square kilometers (km2), square inches (in2), square yards (yd2), square miles (mi2).
4–MG1.3 Estimate and measure liquid capacity in teaspoons, tablespoons, cups, pints, quarts, gallons, milliliters, and liters and know the abbreviations.
4–MG1.4 Estimate and measure weights in pounds/ounces and in grams/kilograms.
4–MG1.5 Know the following equivalences among U. S. customary units of measurements and solve problems involving changing units of measurements: 1 ft. = 12 in., 1 yd. = 3 ft. = 36 in., 1 mi. = 5,280 ft, = 1,760 yd., 1 lb. = 16 oz., 1 ton = 2,000 lb., 1 cup = 8 fl. oz., 1 pt. = 2 c., 1 qt. = 2 pt., 1 gal. = 4 qt.
4–MG1.6 Know the following equivalences among metric units of measurement and solve problems involving changing units of measurement: 1 cm = 10 mm, 1 m = 1,000 mm = 100 cm, 1 km = 1,000 m, 1 cg = 10 mg, 1 g = 1,000 mg = 100 cg, 1 kg = 1,000 g, 1 cl =10 ml, 1 liter = 1,000 ml = 100 cl.
4–MG1.7 Estimate the conversion between ounces and grams, pounds and kilograms, inches and centimeters, yards and meters, miles and kilometers, and quarts and liters.
Content Standard 4–MG2
The student measures and draws line segments and measures, draws, identifies, and classifies angles.
Performance Standards
The student will:
4–MG2.1 Estimate, draw, measure, and classify right, acute, or obtuse angles and associate an angle with an amount of turning (1/4 turn with 90 degrees, 1/2 turn with 180 degrees).
4–MG2.2 Identify and draw points, line segments, rays, and lines that appear to be parallel, perpendicular, or intersecting (using appropriate words and symbols).
Content Standard 4–MG3
The student draws, analyzes, and classifies plane and solid geometric objects and determines and differentiates between their attributes.
Performance Standards
The student will:
4–MG3.1 Classify and draw quadrilaterals as squares, rectangles, rhombi, parallelograms, trapezoids, or triangles (scalene, isosceles, equilateral and right, acute, obtuse).
4–MG3.2 Determine and use formulas to find the perimeter of polygons and the areas of squares, rectangles, and triangles.
4–MG3.3 Demonstrate by example the radius, diameter, circumference, and area of a circle.
4–MG3.4 Determine whether the sides of a plane figure and the edges or faces of a solid object are congruent, parallel, or perpendicular. (e.g., the ceiling is parallel to the floor; the wall is perpendicular to the floor.)
Data Analysis, Statistics and Probability
Content Standard 4–DA1
The student collects, records, graphs, and interprets data and communicates the findings clearly.
Performance Standards
The student will:
4–DA1.1 Pose questions that can only be answered by collecting data and select ways of systematically collecting data to answer them.
4–DA1.2 Read, interpret, and display data in line plots, graphs, tables, or charts .
4–DA1.3 Identify verbal, numerical, and graphical representations that convey the same information.
4–DA1.4 Read and interpret scales on axes, maps, and drawings.
Content Standard 4–DA2
The student determines the possible outcomes of an event and the probability of an event occurring.
Performance Standards
The student will:
4–DA2.1 Determine (using concrete materials or lists) all possible outcomes of an event involving four or fewer items (e.g., the number of ways Jose, Mary, and Flo can finish first, second, and third in a race) and explain a process for ascertaining that there have been no omissions or duplications.
4–DA2.2 Determine and explain how likely an event is (e.g., If I have a bag with four red marbles and five blue marbles, I have five chances out of nine of choosing a blue marble.) and interpret this as the probability that the event will occur.
Problem Solving, Mathematical Reasoning, and Communication
Content Standard 4–PS1
The student carries out, explains, and justifies his or her work in an organized approach to problem solving using grade 4 mathematics skills.
Performance Standards
The student will:
4–PS1.1 Formulate or reformulate a statement of a problem, identify facts and collect information, consider various solution strategies, attempt problem solutions, and verify the reasonableness of solutions and/or verify the accuracy of the solution using another approach, explaining each step of the process.
4–PS1.2 Draw diagrams to represent problem situations and label the facts and the unknowns on the diagrams.
4–PS1.3 Identify the missing information that is needed to find a solution to a given problem.
Content Standard 4–PS2
The student defends reasoning when finding solutions to problems generated from grade 4 key math content.
Performance Standards
The student will:
4–PS2.1 Use patterns to verify that all combinations have been found, when required to do so as part of the problem solution.
4–PS2.2 Articulate reasoning for each step when a problem requires multiple steps to solve, including showing multiple steps in symbolic form and explaining each symbol or expression.
4–PS2.3 Explain a pattern that can be used in similar situations.
4–PS2.4 Explain how a problem is similar to other problems already solved.
4–PS2.5 Explain how the mathematics used in a problem is like other concepts in mathematics.
4–PS2.6 Explain how a problem solution can be applied to other school subjects in real-world situations.
4–PS2.7 Make a solution into a general rule that applies to other circumstances.
Content Standard 4–PS3
The student communicates knowledge of basic skills, conceptual understanding, and problem solving and demonstrates understanding of mathematical communications of others.
Performance Standards
The student will:
4–PS3.1 Use appropriate mathematical terms, vocabulary, and language, based on prior conceptual work (see grade 4 Key Vocabulary).
4–PS3.2 Show ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.
4–PS3.3 Explain clearly and logically solutions to problems and support solutions with both oral and written evidence.
4–PS3.4 State his or her purpose and address the audience when communicating.
4–PS3.5 Comprehend mathematics from reading assignments and other sources.
Key Vocabulary
Number Sense and Operations
1,000,000, approximate, common denominator, cube number, cube (x3) = x • x • x decimal, denominator, dividend, divisor, equivalent fractions, expanded form, exponent (x3), factors, improper , raction, like, lowest term, mixed number, multiple, negative integer, positive integer, ratio, repeated multiplication, rounding, simple ratio, thousandths, unlike
Functions and Algebra
+, –, x, ÷, additive, distributive property, formula, four-quadrant coordinate grid, graph, indicators of division (÷, /, long-division box), indicators of multiplication (x, dot, parenthesis, juxtaposition, *), inverse, mental computation, negative number, numerical expressions, ordered pair, partial product, positive number, quantity, relationship, variation, x-axis, y-axis, x-coordinate, y-coordinate
Measurement and Geometry
three-dimensional, acute angle, area, centigram (cg), centimeter (cm), circumference, congruent, conversion, cup (c.), customary, diameter, equilateral, equivalence, formula, fluid ounce (fl. oz.), feet (ft.), foot (ft.), gallon (gal.), gram (g), inch (in.), interpreting, intersecting, isosceles, kilogram (kg), kilometer (km), liquid capacity, liter (l), line (and symbolic notation), line segment (and symbolic notation), meter (m), metric, mile (mi.), milligram (mg), milliliter (ml), millimeter (mm), obtuse angle, orientation, ounce (oz.), parallel (p), parallelogram, perimeter, perpendicular (P), pint (pt.), plane, polygon, pound (lb.), quadrilateral, quart (qt.), radii, radius, ray (and symbolic notation), rectangle, reflection, rhombi, rhombus, right angle, rotation, scale, scalene, similar, slides, solid, square, square centimeter, square inch, square kilometer, square meter, square mile, square yard, tablespoon (tbsp.), teaspoon (tsp.), ton (T.), trapezoid, triangle, yard (yd.)
Data Analysis, Statistics, and Probability
chart, collect, data, display, duplication, experiment, formulate, graph, graphing, interpret, likelihood of an event, line plot, mean, median, mode, omission, possible outcome, probability, range, recording, survey, systematically, table
Problem Solving, Mathematical Reasoning, and Communication
chart, combination, diagram, formulate, graph, model, number, picture, solution, symbol, table, word
Grade 5
Number Sense and Operations
Content Standard 5–NO1
The student extends understanding of number and place value including very large and very small numbers.
Performance Standards
The student will:
5–NO1.1 Estimate, determine, and interpret the meaning of very large numbers (e.g., How many dollars is a million pennies?).
5–NO1.2 Read, write, and interpret whole number powers of 10 (e.g. 104 = 10,000).
5–NO1.3 Round decimals (and decimal quotients) to the nearest tenth; to the nearest hundredth; to the nearest thousandth.
5–NO1.4 Move the decimal point when dividing by 10, 100, or 1,000.
Content Standard 5–NO2
The student extends understanding of addition, subtraction, multiplication, and division with whole numbers and uses these operations to solve problems.
Performance Standards
The student will:
5–NO2.1 Multiply and divide whole numbers.
5–NO2.2 Determine the greatest common factor and the least common multiple of two or three whole numbers.
5–NO2.3 Identify prime and composite numbers up to 100.
5–NO2.4 Solve division problems with remainders by rounding a decimal quotient.
Content Standard 5–NO3
The student demonstrates an understanding of fractions and decimals and computes with them.
Performance Standards
The student will:
5–NO3.1 Determine the least common denominator (LCD) of fractions with unlike denominators.
5–NO3.2 Compare fractions with like and unlike denominators, using the signs >, <, and =, and greater than, less than, and equal to.
5–NO3.3 Add and subtract with fractions and mixed numerals (with like and unlike denominators), with and without regrouping, and express answers in simplest form.
5–NO3.4 Multiply mixed numbers and fractions.
5–NO3.5 Write fractions as decimals (e.g., 1/4 = 0.25; 17/25 = 0.68; 1/3 = 0.3333...or 0.33, rounded to the nearest hundredth).
5–NO3.6 Write terminating decimals as fractions.
5–NO3.7 Add and subtract decimals through hundred-thousandths.
5–NO3.8 Estimate and find the product of two numbers expressed as decimals through thousandths.
5–NO3.9 Given a dividend expressed as a decimal, estimate and find the quotient through ten-thousandths and a whole number.
5–NO3.10 Arrange fractions, mixed numbers, and decimals in order, represent them on a number line, and explain the process used.
5–NO3.11 Find the given percent of a number.
5–NO3.12 Express equivalences between fractions, decimals, and percentages, and know the percentage equivalent for halves, quarters, fifths, and tenths.
Content Standard 5–NO4
The student models, explains, and computes with positive and negative integers.
Performance Standards
The student will:
5–NO4.1 Illustrate and determine the value of a sum of a positive and a negative integer or two negative integers.
5–NO4.2 Identify or represent integers on a number line.
5–NO4.3 Read, write, and order positive and negative decimals to the nearest hundred thousandth.
5–NO4.4 Compare the value of two negative or positive decimals through hundred-thousandths using the symbols >, <, or =, and greater than, less than, and equal to.
Functions and Algebra
Content Standard 5–FA1
The student describes and uses properties and variables to express a verbal, numeric, geometric, or graphical relationship; models; and solves algebraic equations in one variable.
Performance Standards
The student will:
5–FA1.1 Identify the reciprocal of a given fraction; show that the product of a given number and its reciprocal = 1.
5–FA1.2 Use a letter to represent an unknown number and write and simplify a simple algebraic expression in one variable for a given situation.
5–FA1.3 Evaluate simple algebraic expressions by substitution.
5–FA1.4 Solve one-step linear equations in one variable involving whole number coefficients and rational solutions.
5–FA1.5 Solve simple problems involving rate of speed, unit cost, or unit weight.
5–FA1.6 Know the names and use of the commutative and associative properties for addition, and the commutative, associative, and distributive properties for multiplication, and illustrate understanding by usage and by identifying examples and counterexamples
Content Standard 5–FA2
The student investigates, describes, and extends numerical and geometric patterns.
Performance Standards
The student will:
5–FA2.1 Identify and describe triangular, square, and cubic numbers, and investigate and extend patterns involving them, using drawings and concrete materials as needed.
5–FA2.2 Create, describe, and extend patterns formed by powers and arithmetic sequences.
Measurement and Geometry
Content Standard 5–MG1
The student measures, compares, and interprets lengths, areas, volumes, weights, and time using appropriate units and tools.
Performance Standards
The student will:
5–MG1.1 Convert to common units of measurement in problems involving addition and subtraction of different units.
5–MG1.2 Differentiate between perimeter, area, and volume and determine which of these is appropriate to use in a given problem.
5–MG1.3 Determine the duration of a time interval (e.g., 11:00 a.m. to 3:30 p.m. the next day).
Content Standard 5–MG2
The student uses cubic units to measure volume and identify and differentiate between one-, two-, and three-dimensional parts of a plane or solid object.
Performance Standards
The student will:
5–MG2.1 Identify the diameter, radius, chord, central angle, arc, circumference, and area of a circle.
5–MG2.2 Model and compute the volume of geometric solids (rectangular solids and prisms), select and use appropriate units in both metric and customary systems (cubic centimeters [cm3], cubic meters [m3], cubic inches [in3], cubic yards [yd3].
5–MG2.3 Identify and determine a method to calculate the surface of a cube and a rectangular solid.
5–MG2.4 Recognize and describe bilateral and rotational symmetry in two- and three-dimensional figures.
Content Standard 5–MG3
The student identifies and draws one-, two-, and three-dimensional geometric objects and uses their properties.
Performance Standards
The student will:
5–MG3.1 Know that regular polygons have sides of equal lengths and angles of equal measure.
5–MG3.2 Identify and draw diagonals of polygons.
5–MG3.3 Estimate, measure, identify, and construct perpendicular and parallel lines, angles, midpoints, and bisectors (perpendicular and angle), using appropriate tools (e.g., straight edge, ruler, compass, protractor, and drawing software).
5–MG3.4 Associate 3/4 turn with 270˚ and a full turn with 360˚.
Data Analysis, Statistics and Probability
Content Standard 5–DA1
The student displays, analyzes, develops questions, interprets, and makes predictions based on sets of data.
Performance Standards
The student will:
5–DA1.1 Formulate questions and answer them by carrying out a survey or experiment, accurately recording data, and clearly communicating the results.
5–DA1.2 Create and interpret circle graphs and display information in alternative graphical formats (e.g., show information in a bar graph as a circle graph).
5–DA1.3 Draw conclusions, note trends, and make a recommendation based on data analysis.
5–DA1.4 Organize and display single-variable data in appropriately designed and labeled tables, charts, and graphs (e.g., use a bar graph, histogram or frequency table for grouped data).
5–DA1.5 Explain what measure of central tendency (mean, median, mode) is most representative for a given set of data.
Content Standard 5–DA2
The student selects and uses a method for determining the likelihood of a simple event.
Performance Standard
The student will:
5–DA2.1 Use experiments with concrete materials (e.g., dice, spinners, cards) to determine a relative frequency for an event and compare that with the expected probability derived from a tree diagram, sample space, or area comparison.
Problem Solving, Mathematical Reasoning, and Communication
Content Standard 5–PS1
The student translates among verbal, graphic, and symbolic representations of problems involving grade 5 mathematics and their solutions.
Performance Standard
The student will:
5–PS1.1 Translate among problem statements, diagrams, and symbolic expressions for situations involving perimeter, area, and volume and those that involve one- and two-step arithmetic solutions, and explain or justify these translations.
Content Standard 5–PS2
The student recognizes and explains simple logical errors.
Performance Standards
The student will:
5–PS2.1 Recognize that general rules may not apply in all cases (e.g., any number divided by itself equals 1, which does not apply for 0), and that special cases do not necessarily lead to a general rule, (e.g., 32 degrees F = 0 degrees C, does not mean that C = F – 32) and search for counterexamples in these situations.
5–PS2.2 Recognize that stating a premise as false does not guarantee that the conclusion also is false (e. g., All squares have four corners. This object is not a square. Therefore, this object does not have four corners.) and explain the logical error.
5–PS2.3 Recognize the misapplication of a formula or rule in solving a problem (e.g., 50 miles at 10 miles per gallon = 500 miles travel), and explain the error.
Content Standard 5–PS3
The student defends reasoning when finding solutions to problems generated from grade 5 key math content.
Performance Standards
The student will:
5–PS3.1 Defend the choice of using addition, subtraction, multiplication, and/or division of whole numbers and/or fractions and decimals when solving a non-routine problem.
5–PS3.2 Use and invent a variety of approaches and understand and evaluate those of others.
5–PS3.3 Involve problem-solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table.
5–PS3.4 Explain, where helpful, how to break a problem into simpler parts.
5–PS3.5 Solve for unknown or undecided quantities using algebraic thinking, graphing, sound reasoning, and other strategies.
5–PS3.6 Make sensible, reasonable estimates.
5–PS3.7 Make justified, logical statements.
5–PS3.8 Verify and interpret results with respect to the original problem situation.
5–PS3.9 Generalize solutions and strategies to new problem situations.
Content Standard 5–PS4
The student communicates knowledge of basic skills, conceptual understanding, and problem solving and demonstrates understanding of mathematical communications of others.
Performance Standards
The student will:
5–PS4.1 Use mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams (see grade 5 Key Vocabulary).
5–PS4.2 Organize work, explain facets of solution orally and in writing, label drawings, and use other techniques to make meaning clear to the audience.
5–PS4.3 Use mathematical language to make complex situations easier to understand.
5–PS4.4 Exhibit developing reasoning abilities by justifying statements and defending work.
5–PS4.5 Show understanding of concepts by explaining ideas not only to teachers but to fellow students or younger children.
5–PS4.6 Comprehend mathematics from reading assignments and from other sources.
Key Vocabulary
Number Sense and Operations
>, <, =, composite number, decimal, decimal point, decimal quotient, dividend, equal to, factor, greater than, greatest common factor, hundredth, least common denominator (LCD), least common multiple, less than, negative, negative integer, non-terminating decimal, number, percent (%), place value, positive, powers of 10, prime, product, ratio, reciprocal, rounding, tenth, ten thousandths, terminating decimal, thousandth, unlike denominator
Functions and Algebra
algebraic expression, arithmetic sequence, coefficients, cubic number, fraction, linear equation, multiplicative inverse, power, product, rate, rational solution, reciprocal, square, substitution
Measurement and Geometry
3/4 turn = 270˚, full turn = 360˚, angle, arc, area of a circle, bilateral, central angle, chord, circumference of a circle, compass, cube, cubic centimeter (cm3), cubic inches (in3), cubic meter (m3) , cubic yard (yd3), customary system, diameter, drawing software, duration, geometric solid, metric, midpoints, parallel line, perpendicular, pi, prism, protractor, radius, rectangular solid, rotational symmetry, ruler, straight edge, surface, time interval
Data Analysis, Statistics, and Probability
area comparison, bar graph, drawing conclusion, expected probability, frequency table, histogram, mean, median, measure of central tendency, mode, note trend, relative frequency, range, sample space, tree diagram
Problem Solving, Mathematical Reasoning, and Communication
area, diagram, false, justified, logical, perimeter, premise, problem statement, reasoning, symbolic, volume, translate
Grade 6
Number Sense and Operations
Content Standard 6–NO1
The student reads, writes, computes, and estimates whole numbers, fractions, decimals and percentages, and ratios and proportions.
Performance Standards
The student will:
6–NO1.1 Read, write, and compute with whole numbers in scientific notation (e.g., 325,600 = 3.256 105).
6–NO1.2 Find common multiples and factors, including the least common multiple and the greatest common factor, using prime factorization.
6–NO1.3 Determine whether a number is a prime number or a composite number and explain the concepts of prime and composite numbers.
6–NO1.4 Write fractions as equivalent terminating or repeating decimals and explain the process used.
6–NO1.5 Interpret and model percentage terms of parts of 100, determine the percentage equivalent to decimals (expressed as tenths or hundredths) and simple fractions (e.g., 1/2 is 50%, 3/5 is 60%, 1/20 = 5%).
6–NO1.6 Identify the reciprocal of a given fraction and know that the product of a given number and its reciprocal is 1 (multiplicative inverse).
6–NO1.7 Illustrate and solve division of fractions (e.g., 5/8 ÷ 3/4 = 5/8 4/3 or 20/24 or 5/6).
6–NO1.8 Describe and compare two quantities using ratios and appropriate notations (a/b, a to b, a:b) and give ratios in lowest terms.
6–NO1.9 Illustrate and solve proportions for a missing value (e.g., determine the value of N if 4/7 = N/21).
Content Standard 6–NO2
The student identifies, represents, adds, subtracts, multiplies, and divides positive and negative rational numbers.
Performance Standards
The student will:
6–NO2.1 Identify, compare, and order rational numbers and represent them on a number line.
6–NO2.2 Illustrate and solve addition, subtraction, multiplication, and division problems using positive and negative numbers, including fractions and decimals.
Functions and Algebra
Content Standard 6–FA1
The student writes verbal expressions/sentences as algebraic expressions/ equations, graphs them, and interprets the results in all three representations.
Performance Standards
The student will:
6–FA1.1 Write and solve one- and two-step linear equations and inequalities in one variable, using strategies involving inverse operations, integers, fractions, and decimals.
6–FA1.2 Apply order of operations and the commutative, associative, and distributive properties to solve problems involving combinations of the four operations.
Content Standard 6–FA2
The student analyzes tables, graphs, and rules to determine functional relationships.
Performance Standards
The student will:
6–FA2.1 Translate between verbal, numeric, graphical, and symbolic representations of linear functional relationships.
6–FA2.2 Demonstrate understanding that rate is a measure of one quantity per unit value of another quantity.
6–FA2.3 Solve multi-step problems involving rates, average speed, distance, and time.
Content Standard 6–FA3
The student investigates and describes geometric and exponential patterns.
Performance Standards
The student will:
6–FA3.1 Apply a functional perspective to questions in geometry (e.g., the number of sides of a regular polygon and the sum of its interior angles).
6–FA3.2 Identify, represent, extend, and create number patterns involving multiples, squares, and cubes.
Measurement and Geometry
Content Standard 6–MG1
The student uses ratios and proportional reasoning to convert within measurement systems.
Performance Standards
The student will:
6–MG1.1 Determine perimeter, area, volume, and circumference of a given geometric figure.
6–MG1.2 Create and solve proportions to convert between units of time (e.g., 60 min./1 hr. = 700 min./x hr.).
6–MG1.3 Know estimates of pi and use them to determine the circumference and the area of a circle.
6–MG1.4 Model and compute the volume of a cylinder, selecting and modeling a volume of approximately unit 1 cc/cm, 1 in, 1 ft, 1 m, 1 yd, and select an appropriate unit with which to represent a given volume.
6–MG1.5 Determine whether geometric figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding parts of similar figures.
Content Standard 6–MG2
The student identifies and describes the properties of and relationships among two- and three-dimensional figures.
Performance Standards
The student will:
6–MG2.1 Identify, describe, and classify angles as vertical, adjacent, complementary, and/or supplementary.
6–MG2.2 Define, describe, and draw polygons, given information about them.
6–MG2.3 Identify and describe properties of simple geometric figures and parallel and perpendicular lines.
6–MG2.4 Identify congruent angles and sides and axes of symmetry and show how congruent plane figures can be made to correspond through reflection, rotation, and translation.
6–MG2.5 Visualize, represent, and interpret two-dimensional views of three-dimensional objects made from rectangular solids.
Data Analysis, Statistics and Probability
Content Standard 6–DA1
The student reads and analyzes data taken from tables or graphs and uses the concept of average to answer relevant questions.
Performance Standards
The student will:
6–DA1.1 Identify ordered pairs of data from a graph (bar, circle, line, or broken-line) and interpret the meaning of the data in terms of the situation depicted by the graph.
6–DA1.2 Describe the possible effects of missing or incorrect information.
6–DA1.3 Find and model the mean, median, mode, and range of a set of data.
6–DA1.4 Determine a missing quantity if the mean of a set of quantities is known.
Content Standard 6–DA2
The student determines theoretical and experimental probabilities and uses these to make predictions about events.
Performance Standards
The student will:
6–DA2.1 Carry out and interpret a simulation (by hand or using computer software) to predict the likelihood of an event.
6–DA2.2 Compute the probability of an event using counting principles, sample spaces, and geometric arguments and compare it with results from a simulation.
6–DA2.3 Demonstrate understanding that the sum of the probabilities of all possible events is 1.0, that the probability of any event lies between 0.0 and 1.0 inclusively, and that the probability of an event not happening is 1 minus the probability of the event’s happening.
6–DA2.4 Organize and display data in appropriately designed and labeled tables, charts, and graphs.
Problem Solving, Mathematical Reasoning, and Communication
Content Standard 6–PS1
The student incorporates the grade 6 mathematics of proportions, rates, inequalities, and exponents into problem-solving strategies.
Performance Standards
The student will:
6–PS1.1 Identify rates and proportions in problem situations, carry units through all steps of the solution, and describe each step in terms of the units involved, using appropriate mathematical language.
6–PS1.2 Identify and correct errors when rates or proportions are inverted in incorrect problem solutions.
6–PS1.3 Explain and justify each step of a solution to an inequality problem.
6–PS1.4 Describe situations involving patterns of increasing exponents, explain how they relate to repeated multiplication, and illustrate the mathematics of the example symbolically.
6–PS1.5 Extract pertinent information from situations and figures and identify what additional information is needed.
6–PS1.6 Invoke problem-solving strategies, such as illustrating with sketches to help make sense of complex situations or organize information in a table.
6–PS1.7 Solve problems for unknown or undecided quantities, using algebra, graphing, sound reasoning, and other strategies.
Content Standard 6–PS2
The student recognizes and explains simple logic.
Performance Standards
The student will:
6–PS2.1 Use estimation and reasonableness of solutions as a guide and identify and correct errors in the application of scientific notation to problem solutions.
6–PS2.2 Formulate conjectures and argue, short of formal proof, why they must be or seem to be true.
6–PS2.3 Explain, where helpful, how to break a problem into simpler parts.
6–PS2.4 Verify and interpret results of a problem.
6–PS2.5 Generalize solutions and strategies to new problems.
Content Standard 6–PS3
The student communicates knowledge of basic skills, conceptual understanding, and problem solving, and demonstrates understanding of mathematical communications of others.
Performance Standards
The student will:
6–PS3.1 Use mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations, formulas, charts, graphs, and diagrams (see grade 6 Key Vocabulary).
6–PS3.2 Organize work, explain facets of a solution orally and in writing, label drawings, and use other techniques to make meaning clear to the audience.
6–PS3.3 Use mathematical language to make complex situations easier to understand.
6–PS3.4 Exhibit developing reasoning abilities by justifying statements and defending work.
6–PS3.5 Demonstrate understanding of concepts by explaining ideas not only to teachers but to fellow students or younger children.
6–PS3.6 Demonstrate comprehension of mathematics from reading assignments and other sources.
Key Vocabulary
Number Sense and Operations
appropriate notation, common multiples, factor, greatest common factor, least common multiple, negative rational number, number line, positive rational number, prime factorization, proportion, ratio, rational number, reciprocal, repeating decimal, scientific notation, terminating decimal
Functions and Algebra
algebraic expression, associative property, average speed, commutative property, decimal, distance, distributive property, equations, exponent, exponential pattern, fraction, functional relationship, generalize, geometric growth, graphical, inequality, integer, interior angle, inverse operation, linear equation, numeric, quantity, rate, regular polygon, relationship, rule, scientific notation, square root, symbolic, time, variable, verbal
Measurement and Geometry
adjacent, area, axes of symmetry, circumference, classifying angles, cylinder, complementary, congruent, convert, parallel, perimeter, perpendicular, plane figure, polygon, property, proportion, quadrilateral, reflection, regular solid, rotation, similar figures, supplementary, three-dimensional, translation, triangle, two-dimensional, vertical, volume
Data Analysis, Statistics, and Probability
bar, broken line, chart, circle, counting principles, graph, line, mean, median, mode, modeling, ordered pair, predict, sample space, set of data, simulation, theoretical probability
Problem Solving, Mathematical Reasoning, and Communication
accuracy, chart, complex situation, conjecture, diagram, equation, estimation, formula, formulate, generalize, graph, increasing exponents, inequality, interpret, inverted, numerical table, justify, proportion, rate, reasonableness, scientific notation, symbolically, verify
Grade 7
Number Sense and Operations
Content Standard 7–NO1
The student applies, explains attributes, and computes with real numbers expressed in a variety of forms.
Performance Standards
The student will:
7–NO1.1 Read, write, and compute with rational numbers in scientific notation (positive and negative powers of 10), approximate numbers using scientific notation, and explain the process.
7–NO1.2 Use the laws of exponents to solve problems.
7–NO1.3 Model and express rational numbers as fractions, terminating or repeating decimals, or percentages and describe the equivalence relationship among these representations.
7–NO1.4 Demonstrate understanding of the square root symbol, determine the two integers between which a particular square root lies, and explain how he or she knows.
7–NO1.5 Interpret the absolute value of a real number as its distance from zero on a number line and determine the absolute value of real numbers.
Content Standard 7–NO2
The student uses correct order of operations and number properties to add, subtract, multiply, and divide positive and negative rational numbers, extract roots, and determine whole number powers of positive rational numbers.
Performance Standards
The student will:
7–NO2.1 Add, subtract, multiply, and divide rational numbers, integers, fractions, and decimals, and raise rational numbers to whole number powers.
7–NO2.2 Explain and use the inverse relationship between exponentiation/root extraction.
7–NO2.3 Explain and apply properties of real numbers (associative, commutative, distributive, identity, inverse) to simplify numerical expressions.
Content Standard 7–NO3
The student reasons proportionately and uses ratios and rates to solve problems.
Performance Standards
The student will:
7–NO3.1 Use proportions to solve problems involving a change of scale (drawing, models, maps) or a comparison of two quantities.
7–NO3.2 Identify and interpret situations involving direct variation and represent these situations on a coordinate graph.
7–NO3.3 Solve consumer application problems involving discount, markup, commission, profit, and simple compound interest.
Functions and Algebra
Content Standard 7–FA1
The student identifies, describes, represents, extends, and creates linear and non-linear number patterns.
Performance Standards
The student will:
7–FA1.1 Identify Pythagorean triples and describe patterns found in Pascal’s triangle.
7–FA1.2 Identify, describe, and generalize patterns involving geometric growth, square roots, or exponents.
Content Standard 7–FA2
The student relates the equation, coordinate graph, and set of ordered pairs of a linear function.
Performance Standards
The student will:
7–FA2.1 Graph a linear function in two variables on the coordinate plane, given a set of ordered pairs.
7–FA2.2 Graph a linear function in two variables on the coordinate plane given its equation, slope, and y-intercept, or given both its x- and its y-intercepts.
Content Standard 7–FA3
The student expresses quantitative relationships using algebraic terminology, expressions, equations, and inequalities.
Performance Standards
The student will:
7–FA3.1 Use variable and appropriate operations to write an expression, equation, inequality, or system of equations or inequalities representing a verbal description.
7–FA3.2 Apply the property of real number operations to evaluate algebraic expressions for given replacement values of variables.
7–FA3.3 Discuss the different uses of variables in expressions (e.g., 2w + 2L), equations (e.g., y = x – 4), formulas (e.g., c = 9d), and properties (e.g., a + b = b + a).
Content Standard 7–FA4
The student interprets and evaluates expressions involving integer powers and roots of monomials.
Performance Standards
The student will:
7–FA4.1 Interpret whole number powers as repeated multiplication, negative integer powers as reciprocals, and evaluate monomials that have them.
7–FA4.2 Simplify square roots of perfect square monomials.
Content Standard 7–FA5
The student writes verbal expressions/sentences as algebraic expressions/equations, graphs them and interprets the results in all three representations.
Performance Standard
The student will:
7–FA5.1 Write and solve two-step linear equations and inequalities in one variable, using strategies involving inverse operations, integers, fractions, and decimals.
Measurement and Geometry
Content Standard 7–MG1
The student chooses appropriate units of measure and uses proportional reasoning to convert within and between measurement and monetary systems.
Performance Standards
The student will:
7–MG1.1 Select, use, and explain a method for comparing weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, 4.5 meters is about 1190 inches).
7–MG1.2 Convert between monetary systems (e.g., US $ and francs).
7–MG1.3 Use rates (e.g., speed, density) and other derived units to solve problems.
Content Standard 7–MG2
The student knows and uses formulas for perimeter, circumference, area, and volume of common geometric objects and uses these to derive methods for finding or approximating measures of less common objects.
Performance Standards
The student will:
7–MG2.1 Estimate and find the area of polygons by subdividing them into rectangles and triangles.
7–MG2.2 Reason proportionately to find measures and the ratios in situations involving similar figures.
Content Standard 7–MG3
The student describes and explains relationships among one-, two-, and three-dimensional objects.
Performance Standards
The student will:
7–MG3.1 Identify and construct line segments, altitudes, medians, angle bisectors, and perpendicular bisectors.
7–MG3.2 Classify quadrilaterals using their properties (deductive reasoning).
7–MG3.3 Use the Pythagorean Theorem to find or approximate the length of the missing side of a right triangle or the diagonal of a square or rectangle.
7–MG3.4 Determine the number of diagonals and the measures of central, interior, and exterior angles of regular polygons.
7–MG3.5 Identify and sketch central and inscribed angles, arc, radii, diameters, and chords of circles.
Data Analysis, Statistics and Probability
Content Standard 7–DA1
The student demonstrates an understanding of displaying, analyzing, and interpreting data he or she has generated or taken from resources.
Performance Standards
The student will:
7–DA1.1 Construct and interpret frequency distributions, line plots, stem-and-leaf plots, box-and-whisker plots, and scattergrams.
7–DA1.2 Determine measures and appropriate uses of central tendencies (mean, median, and mode), frequency, and distribution (range, inter-quartile range) for a set of data.
7–DA1.3 Define, describe, and use algebraic terminology correctly (equation, inequality, variable, expression, term, constant, coefficient).
Content Standard 7–DA2
The student demonstrates an understanding of using probability to answer questions about the likelihood of an event.
Performance Standards
The student will:
7–DA2.1 Construct a sample space to determine theoretical and experimental probabilities and represent it in the form of a list, chart, picture, or tree diagram.
7–DA2.2 Determine the probability of a given simple event and express that probability as a ratio decimal or percentage.
7–DA2.3 Identify and describe the number of possible arrangements of several objects, using a tree diagram or the Fundamental Counting Principle.
Problem Solving, Mathematical Reasoning, and Communication
Content Standard 7–PS1
The student uses mathematical techniques and language accumulated by grade 7 in explaining and justifying each step in problem solutions.
Performance Standards
The student will:
7–PS1.1 Appropriately frame the symbolic representation of a problem (e.g., with introductory statements such as "Let x equal the quantity of fuel in gallons.").
7–PS1.2 Refer to properties by name or symbolic form (e.g., ab = ba) in justifying each step in manipulating an equation or expression.
7–PS1.3 Explain the sequence of steps in solving a problem (e.g., first convert all time values to minutes so that they will be in the same scale; then multiply durations times repetitions to get totals for each event; then add these products to get a grand total; and finally convert to hours and minutes so the solution is easier to comprehend) and state the reasons for selecting a particular sequence.
7–PS1.4 Check solutions (e.g., by working backwards or substituting solutions back into original formulations) and explain how the process verifies the solution.
7–PS1.5 Extract pertinent information from situations and figures and identify what additional information is needed.
7–PS1.6 Invoke problem-solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table.
7–PS1.7 Solve for unknown or undecided quantities, using algebra, graphing, sound reasoning, and other strategies.
Content Standard 7–PS2
The student recognizes and explains simple logical errors.
Performance Standards
The student will:
7–PS2.1 Integrate concepts and techniques from different areas of mathematics.
7–PS2.2 Make sensible and reasonable estimates.
7–PS2.3 Make justified and logical statements.
7–PS2.4 Generalize solutions and strategies to new problems.
7–PS2.5 Formulate conjectures and argue, short of formal proof, why they must be or seem to be true.
7–PS2.6 Use and invent a variety of approaches and understand and evaluate those of others.
Content Standard 7–PS3
The student communicates knowledge of basic skills, conceptual understanding, and problem solving and demonstrates understanding of the mathematical communications of others.
Performance Standards
The student will:
7–PS3.1 Use mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations, formulas, charts, graphs, and diagrams (see grade 7 Key Vocabulary).
7–PS3.2 Organize work, explain facets of a solution orally and in writing, label drawings, and use other techniques to make meaning clear to the audience.
7–PS3.3 Use mathematical language to make complex situations easier to understand.
7–PS3.4 Exhibit reasoning abilities by justifying statements and defending work.
7–PS3.5 Demonstrate understanding of concepts by explaining ideas not only to teachers but to fellow students or younger children.
7–PS3.6 Demonstrate comprehension of mathematics from reading assignments and from other sources.
Key Vocabulary
Number Sense and Operations
absolute value, associative property, attribute, commission, commutative property, compound interest, consumer application problem, coordinate graph, decimal, direct variation, discount, distributive property, equivalence relationship, exponentiation, exponent, extract roots, identity property, integer, inverse, markup, negative power, number property, numerical expression, order of operation, percent, power, profit, proportionately, ratio, rational number, real number, repeating decimal, scale, scientific notation, square root, symbol, terminating decimal
Functions and Algebra
algebraic expression, algebraic terminology, coefficient, constant, coordinate graph, coordinate plane, cube, decimal, equation, expression, formula, fraction, inequality, integer, inverse operation, linear equation, linear function, monomial, multiple, negative integer power, non-linear, number pattern, ordered pair, Pascal’s triangle, perfect square monomials, property, property of real number operations, Pythagorean triples, quantitative relationships, reciprocals, set of ordered pairs, square, square root, term, system of equations, variable, verbal description, whole number, whole number powers
Measurement and Geometry
altitude, angle bisector, arc, area, central angle, chords of circles, circumference, classify quadrilaterals, diagonal, diameter, exterior angle, inscribed angle, interior angle, line segment, median, one-dimensional object, perimeter, perpendicular bisector, polygon, radii, three-dimensional object, two-dimensional object, volume
Data Analysis, Statistics, and Probability
box-and-whisker plot, central tendencies, distribution, experimental probability, frequency, frequency distribution, Fundamental Counting Principle, inter-quartile range, line plot, mean, median, mode, range, scattergram, stem-and-leaf plot, theoretical
Problem Solving, Mathematical Reasoning, and Communication
chart, complex situation, diagram, equation, expression, formula, generalize, graph, integrate, justify, logical, manipulate, manipulating, numerical table, sequence, solution, symbolic representation, verify
Algebra
According to the notion fostered by more than a century of teaching, algebra is largely a matter of simplifying expressions made up of numbers and letters, solving equations, and learning the rules for manipulating symbols. Although important, these skills by no means encompass the whole of algebra. In fact, courses that emphasize the isolated practice of these skills, outside a context that provides meaning, can deprive students of an opportunity to master some of the most powerful intellectual tools our civilization has developed— algebraic reasoning in its many forms and the use of algebraic representations such as functions, graphs and spreadsheets. Without an understanding of algebra’s concepts and symbols, neither higher mathematics nor quantitative science is possible.
To assure that students have an opportunity to master algebra, the following content and performance standards have been developed:
Content Standard A.1: Problem Solving
The student demonstrates problem-solving ability by using algebraic concepts and skills to solve problems for which specific and detailed steps toward solution are not provided. In the process, the student demonstrates mastery of all three elements of finding a solution: formulation, implementation, and conclusion.
Performance Standards
A.1.1 The student represents and solves a variety of textbook and real-life problems involving linear, quadratic, exponential and absolute value expressions algebraically.
A.1.2 The student translates among verbal, graphical, tabular, and symbolic representations of a problem.
Content Standard A.2: Symbol and Symbol Manipulation
The student simplifies expressions and solves problems in symbolic form.
Performance Standards
A.2.1 The student solves linear equations and inequalities in one unknown and applies these skills to solve practical problems.
A.2.2 The student demonstrates an understanding of the different conceptual uses of symbols (e.g., literal numbers, variables) and communicates their meaning.
A.2.3 The student uses algebraic techniques, including linear combinations and substitution, to solve quantitative problems involving systems of two linear equations in two variables and understands the relationship between the graphic and symbolic forms.
A.2.4 The student applies the laws of exponents to perform operations on expressions with integral exponents and uses scientific notation when appropriate.
A.2.5 The student adds, subtracts, and multiplies polynomials and divides polynomials with monomial and binomial divisors. The student also simplifies algebraic expressions by combining like terms and by addition, subtraction, multiplication, and division of polynomial components of these expressions.
A.2.6 The student factors various factorable polynomials in one or two variables, including trinomials, quadratics, differences of two squares, expressions requiring regrouping or repeated factorizations, and expressions with lead coefficients greater than one and extracts monomial and binomial factors from expressions of the third or fourth degree.
A.2.7 The student uses graphing, factoring, or the quadratic formula to solve problems involving quadratic equations.
A.2.8 The student simplifies square root radical expressions, including those with constants and variables, and expresses given real numbers using radical sign.
Content Standard A.3: Communication and Reasoning
The student demonstrates the ability to reason mathematically and uses appropriate language and symbols to communicate this conceptual understanding of algebraic ideas.
Performance Standard
A.3.1 Using oral and written language, the student justifies steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects, pictorial representation, and the properties of the real number system.
Content Standard A.4: Functions and Relations
The student demonstrates an understanding of the concept of function. The student demonstrates this understanding by using it to solve problems and by representing it in multiple ways (graphically, symbolically, tabularly, and verbally as well as with numbers and diagrams). The student also demonstrates facility in using skills that are associated with the concept.
Performance Standards
A.4.1 The students analyzes a given set of data for the existence of a pattern, represents the pattern algebraically and graphically, if possible, and determines whether it is a function.
A.4.2 The student determines the domain and range of a relation, given a set of ordered pairs, and identifies the relations that are functions.
A.4.3 The student recognizes relations that are linear; finds slopes and understands their significance; and determines the three representations of a linear function (graphic, symbolic, and tabular), given specific information about the function.
A.4.4 The student selects, justifies, and applies an appropriate technique to graph a linear function. Techniques include slope intercept, x and y intercepts, and two points.
A.4.5 The student determines the slope of a line when given the symbolic form, the graph, or two points. The student understands the slope as a rate of change between two quantities and recognizes when two lines are perpendicular or parallel.
A.4.6 The student writes the equation of a line when given the graph, two points, or the slope and a point. The student determines the length of a segment with given end points.
A.4.7 The student, given the symbolic or graphic representation, finds the value of x given y, or y given x and finds the zeros algebraically.
A.4.8 The student demonstrates an understanding of families of functions, their characteristics, and the relationships between the various representations.
A.4.9 The student recognize quadratic relationships.
A.4.10 The student analyzes functional relations between quantities and determines how a change in one affects the other.
A.4.11 The student describes algebraically the relationship between quantities and answers questions about them, using various methods and tools.
A.4.12 The student analyzes the effects of the changes of coefficients and constants on the graphs of functions.
A.4.13 The student analyzes a relation to determine whether a direct or inverse variation exists and represents it algebraically and graphically, if possible.
Content Standard A.5: Technology
The student demonstrates the ability to use the tools of new technologies to solve algebraic problems.
Performance Standards
A.5.1 The student demonstrates the ability to use a calculator to solve problems and answer questions about real-world problems.
A.5.2 The student demonstrates the ability to use computer software to explore algebraic concepts.
Content Standard A.6: Statistics, Data Analysis, and Probability
The student demonstrates knowledge of basic skills, conceptual understanding, and problem-solving in statistics.
Performance Standard
A.6.1 The student demonstrates the ability, given a set of data points, to draw a line approximating a line of best fit, to find the equation of the line, and to use the graph and/or equation to make predictions.
Geometry
Mathematics is a language we use every day, often without knowing it. It builds and draws on conceptual understanding and skills and helps us make decisions and solve problems. This draft document tries to connect the notion of problem solving to conceptual understanding and skill development by embedding problem solving within the content strands at every grade. Accordingly, students are asked, for example, to:
• Ask relevant questions about problem situations.
• Decide between relevant and extraneous information.
• Choose appropriate operations tools and approaches to problem situations.
• Decide whether an exact or approximate answer is called for.
• Apply specific techniques in new situations.
• Explain, check, justify, prove, and judge the reasonableness of results.
• Create new approaches and connect knowledge and understanding in new ways.
Content Standard G.1: Problem Solving
The student demonstrates problem-solving ability by using geometric concepts and skills to solve problems for which specific and detailed steps toward solution are not provided. In the process, the student demonstrates mastery of all three elements of finding a solution: formulation, implementation, and conclusion.
Performance Standards
G.1.1 The student represents and solves a variety of textbook and real-life problems involving perimeter, circumference, area, volume, lateral area, and surface area, and non-routine geometric figures.
G.1.2 The student represents and solves a variety of problems involving classic construction, transformations, and coordinate geometry.
Content Standard G.2: Geometric Concepts
The student selects and uses appropriate units, tools, and degrees of accuracy to solve problems involving geometric figures and measures.
Performance Standards
G.2.1 The student knows, uses, derives formulas for, and solves problems involving perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures
G.2.2 The student describes how changes in the dimensions of an object affect the object’s peri-meter, area, and volume (e.g., tripling the radius of a sphere multiplies its volume by 27).
G.2.3 The student finds and uses measures of sides and interior and exterior angles of triangles and polygons to classify figures (e.g., as isosceles, obtuse, convex, regular) and to solve problems (e.g., to determine the number of degrees in a central angle of a regular polygon).
G.2.4 The student describes the relationships between vertical angles, angles that are supplementary and complementary, and angles formed when parallel lines are cut by a transversal and expresses and uses these to find missing angle measures in such systems of anges.
G.2.5 The student uses the Pythagorean Theorem, its converse, properties of special right triangles (e.g., sides in the ratio 3-4-5, angles of 30-60-90 degrees), and right triangle trigonometry to find missing information about triangles.
G.2.6 The student develops and implements a plan for obtaining indirect measures using similarity, proportional reasoning and trigonometric ratios.
G.2.7 The student compares, contrasts, classifies, and solves problems involving quadrilaterals (square, rhombus, rectangle, parallelogram, trapezoid, kite, cyclic on the basis of their definitions and properties (e.g., opposite sides, consecutive angles, diagonals).
G.2.8 The student applies the properties of angles, arcs, chords, radii, tangents, and secants to solve problems involving circles.
G.2.9 The student applies the triangle inequality properties (given information concerning the lengths of sides and/or measures of angles) to determine whether a triangle exists and the relative position of the sides and angles.
G.2.10 The student compares, contrasts, classifies and solves problems about solid geometric figures, including objects appearing in the real world.
Content Standard G.3: Communication and Reasoning
The student demonstrates the ability to reason geometrically and uses appropriate language and symbols to communicate this conceptual understanding of geometric ideas.
Performance Standards
G.3.1 The student constructs and judges the validity of a logical argument consisting of a set of premises and a conclusion. The student will:
• Identify the converse, inverse, and contrapositive of a conditional statement.
• Translate a short verbal argument into symbolic form.
• Diagram arguments involving quantifiers (all, no, none, some) using Venn diagrams.
• Use valid forms of deductive reasoning, including the law of syllogism.
• Use counterexamples to disprove a statement.
G.3.2 The student demonstrates an understanding of Euclidean geometry as an axiomatic system and of the nature of proof, by identifying and giving examples of undefined terms, axioms, theorems, inductive and deductive reasoning.
G.3.3 Using coordinate geometry and deductive, inductive, algebraic and/or transformational arguments, the student proves the Pythagorean theorem and proves theorems for the following:
• Parallel lines
• Congruent polygons
• Similar polygons
• Properties of quadrilaterals
• Properties of circles
G.3.4 Using a compass and straightedge, the student constructs the following basic construc-tions and uses combinations of them to create more complex constructions (center of a circle, tangents to circles, etc.).
• A line segment congruent to a given line segment
• The bisector of a line segment
• A perpendicular to a given line from a point not on the line
• A perpendicular to a given line at a point on the line
• The bisector of a given angle
• An angle congruent to a given angle
Content Standard G.4: Spatial Manipulation and Visualization
The student demonstrates an ability to visualize, manipulate, and describe objects, paths, and regions in two-dimensional and three-dimensional space.
Performance Standards
G.4.1 The student uses geometric language and diagrams to describe the intersection of two planes or a plane and a solid object (the cross section it cuts).
G.4.2 The student builds a three-dimensional model from a two-dimensional drawing and draws a two-dimensional representation of a three-dimensional object (e.g., nets, prisms, cones, pyramids).
Content Standard G.5: Technology
The student demonstrates the ability to use the tools of new technologies to solve geometric problems.
Performance Standards
G.5.1 The student demonstrates the ability to use a scientific calculator to solve geometric problems and answer questions about real-world problems.
G.5.2 The student demonstrates the ability to use computer software to explore geometric concepts and make conjectures.
Intermediate Algebra
Mathematics is a language we use every day, often without knowing it. It builds and draws on conceptual understanding and skills and helps us make decisions and solve problems. This draft document tries to connect the notion of problem solving to conceptual understanding and skill development by embedding problem solving within the content strands at every grade. Accordingly, students are asked, for example, to:
• Ask relevant questions about problem situations.
• Decide between relevant and extraneous information.
• Choose appropriate operations tools and approaches to problem situations.
• Decide whether an exact or approximate answer is called for.
• Apply specific techniques in new situations.
• Explain, check, justify, prove, and judge the reasonableness of results.
• Create new approaches and connect knowledge and understanding in new ways.
Content Standard IA.1: Problem Solving
The student demonstrates problem-solving ability by using algebraic concepts and skills to solve problems for which specific and detailed steps toward solution are not provided. In the process, the student demonstrates mastery of all three elements of finding a solution: formulation, implementation, and conclusion.
Performance Standard
IA.1.1 The student represents and solves problems involving systems of equations, systems of inequalities, matrices, functions and relations, conic sections, sequences, series, probability, statistics, and complex numbers.
Content Standard IA.2: Symbol and Symbol Manipulation
The student simplifies expressions and solves problems in symbolic form.
Performance Standards
IA.2.1 The student adds, subtracts, multiplies, divides, reduces, and evaluates rational expressions.
IA.2.2 The student adds, subtracts, multiplies, divides, and simplifies expressions containing radicals and fractional exponents
IA.2.3 The student solves quadratic, rational, radical, absolute value, and factorable polynomial equations in one variable and non-linear systems of equations in two variables.
IA.2.4 The student selects, justifies, and applies an appropriate algebraic technique (factoring, completing the square, quadratic formula) to solve a quadratic equation over the set of complex numbers and interprets the results graphically.
IA.2.5 The student formulates and solves problems involving joint and combined variations.
IA.2.6 The student use factoring techniques (e.g., common factor, difference of cubes, grouping) and synthetic division to solve factorable polynomial equations of degree four or less.
IA.2.7 The student understands and is able to explain the relationship among the coefficients, factors, roots, and x-intercepts of polynomial functions.
IA.2.8 The student compares, contrasts, and extends arithmetic and geometric patterns of growth and uses them to make predictions.
IA.2.9 The student applies the properties of arithmetic and geometric sequences and series to solve problems, including writing the first n terms, finding the nth term, and evaluating summation formulas.
IA.2.10 The student finds the sum of finite arithmetic and geometric sequences and infinite geometric sequences with |r| < 1, and uses and interprets notation.
IA.2.11 The student uses the Binomial Theorem to determine the kth term in (a + b)^n and recognizes the relationship between the coefficients and Pascal’s triangle.
IA.2.12 The student adds, subtracts, and multiplies (scalar product) matrices and finds the determinate and inverse of a matrix, when possible.
IA.2.13 The student understand the use of matrices to organize information and performs simple operations by representing and solving systems of two linear equations in two variables.
IA.2.14 The student solves linear equations and inequalities involving absolute value (e.g.,
|3x – 4| ≤ 5) and quadratic inequalities (e.g., x2 – 7x –1 ≥ -7) and graphs their solutions on a number line.
IA.2.15 The student solves linear-quadratic and quadratic-quadratic systems of equations and inequalities algebraically and graphically.
IA.2.16 The student converts between logarithmic and exponential forms and uses the Laws of Logarithms to manipulate expressions.
Content Standard IA.3: Communication and Reasoning
The student demonstrates the ability to reason algebraically and uses appropriate language and symbols to communicate this conceptual understanding of algebraic ideas.
Performance Standards
IA.3.1 Using oral and written language, the student justifies steps used in simplifying expressions and solving equations and inequalities over the real and complex numbers. Justifications will include the use of concrete objects, graphs, tables, formulae, equations, and the properties of complex numbers.
IA.3.2 The student explains and critiques the process of a survey or experiment; describes how that process might have contributed to or influenced the results (e.g., the student might discuss reliability of sampling procedures, bias, missing or incorrect information); and describes misuses of statistical or numerical data.
Content Standard IA.4: Functions and Relations
The student extends his or her understanding of standard functions and relations to include conic sections, as well as quadratic, exponential, logarithmic, rational, radical, absolute value functions and factorable polynomials.
Performance Standards
IA.4.1 Given information about the figures a, the student identifies and writes equations for circles, parabolas, ellipses, and hyperbolas whose axes are parallel to the x and y axes.
IA.4.2 Given an equation in standard form, the student determines the center and radius of a circle; the center, vertices, foci, and axes of an ellipse or hyperbola; and the vertex, axis of symmetry, focus, and directrix of a parabola and uses this information to graph the figure.
IA.4.3 The student identifies a general quadratic equation in two variables (no xy term) as a circle, a parabola, an ellipse, or a hyperbola and writes its equation in standard form.
IA.4.4 The student determines the domain, range, and zeros of standard functions and relations and the inverse of a standard function.
IA.4.5 The student represents a standard function as a table of values, an equation, or a graph and translates among these representations.
IA.4.6 The student demonstrates and explains the effect that changing a parameter has on the graph of a function (e.g., the student might describe the effect on the graph of y = x2 + 6x + C as
C grows from –5 to 5).
IA.4.7 The student determines the composition of two standard functions, including the necessary restrictions on the domain.
IA.4.8 The student relates arithmetic and geometric sequences to linear and exponential functions and expresses them in explicit and recursive form.
IA.4.9 The student determines which type of function (including step and piecewise) best models a situation, writes an equation noting the constraints on the domain of the function imposed by the situation, and uses this equation to answer questions about the situation.
IA.4.10 The student determines and interprets the meaning of the maximum or minimum values of a quadratic function in terms of applied, real-world situations.
Content Standard IA.5: Technology
The student demonstrates the ability to use the tools of new technologies to solve algebraic problems.
Performance Standards
IA.5.1 The student demonstrates the ability to use a graphing calculator to solve algebraic problems and answer questions about real-world problems.
IA.5.2 The student demonstrates the ability to use computer software to explore algebraic concepts.
Content Standard IA.6: Statistics, Data Analysis, and Probability
The student demonstrates knowledge of basic skills, conceptual understanding and problem solving ability in the areas of application, measurement, collection of data, analysis, making conclusions, and the defense of conclusions, in statistics and probability.
Performance Standards
IA.6.1 The student analyzes and interprets real-world data using graphs (frequency distributions, histograms, scatter plots) and analysis tools (estimated and calculator- or computer-generated regression lines, correlation coefficients, and residuals).
IA.6.2 The student applies curve-fitting techniques to data assesses the goodness-of-fit of a regression line or curve and its usefulness as a model for the data, and makes predictions by interpolating or extrapolating from the data or its graph.
IA.6.3 The student chooses an appropriate probability model (replacement, non-replacement, combination, permutation) and uses it to arrive at a theoretical probability for a simple or compound chance event.
Advanced Algebra and Trigonometry
Mathematics is a language we use every day, often without knowing it. It builds and draws on conceptual understanding and skills and helps us make decisions and solve problems. This draft document tries to connect the notion of problem solving to conceptual understanding and skill development by embedding problem solving within the content strands at every grade. Accordingly, students are asked, for example, to:
• Ask relevant questions about problem situations.
• Decide between relevant and extraneous information.
• Choose appropriate operations tools and approaches to problem situations.
• Decide whether an exact or approximate answer is called for.
• Apply specific techniques in new situations.
• Explain, check, justify, prove, and judge the reasonableness of results.
• Create new approaches and connect knowledge and understanding in new ways.
Content Standard AT.1: Problem Solving
The student demonstrates problem-solving ability by using algebraic concepts and skills to solve problems for which specific and detailed steps toward solution are not provided. In the process, the student demonstrates mastery of all three elements of finding a solution: formulation, implementation, and conclusion.
Performance Standards
AT.1.1 The student represents and solves problems involving trigonometry, vectors, matrices, logarithms, and exponential functions.
AT.1.2 The student analyzes and solves problems involving periodic phenomena (e.g., biological rhythms, sound waves, tidal variations).
Content Standard AT.2: Symbol and Symbol Manipulation
The student simplifies expressions and solves problems in symbolic form.
Performance Standards
AT. 2.1 The student solves quadratic, rational, logarithmic, exponential, radical, absolute value, and factorable polynomial equations in one variable and non-linear systems of equations in two variables.
AT.2.2 The student uses parametric equations to model and solve application problems and to write equivalent equations by eliminating the parameter, paying special attention to any restrictions on the variables.
AT.2.3 The student adds, subtracts, finds scalar products, dot products, and norms of vectors, noting the field properties which apply.
AT.2.4 The student determines, interprets, and uses a unit directional vector, perpendicular components, and norms to express vectors in the coordinate plane.
AT.2.5 The student graphs and interprets complex numbers as vectors and in polar form.
AT.2.6 The student finds the exact values of the trigonometric functions of multiples of 30∞(/6) and 45∞ (/4), and, given standard values of trigonometric functions, finds the angle measure.
AT.2.7 The student proves basic trigonometric identities and makes substitutions using the basic identities.
AT.2.8 The student solves trigonometric equations that include both infinite solutions and restricted domain solutions and solves basic trigonometric inequalities.
AT.2.9 The student demonstrates understanding of the addition, half angle, double angle, sum to product and product to sum formulae and can use them.
Content Standard AT.3: Communication and Reasoning
The student demonstrates the ability to reason algebraically and uses appropriate language and symbols to communicate this conceptual understanding of algebraic ideas.
Performance Standards
AT.3.1 The student finds a formula for the sum of number patterns (e.g., first n consecutive even integers, k consecutive square numbers beginning with 16) and, using mathematical induction, proves the formula.
AT.3.2 The student writes and uses recursive formulas to express iterative patterns of change, including those of exponential growth and decay.
Content Standard AT.4: Functions and Relations
The student extends his or her understanding of standard functions and relations to include conic sections, as well as analytical geometry, exponential and logarithmic functions, and trigonometric functions.
Performance Standards
AT.4.1 The student determines the zeros, y-intercepts, end behavior, and symmetry of polynomial functions and graphs the functions.
AT.4.2 The student determines the zeros, asymptotes, y-intercepts, end behavior, and symmetry of rational functions and graphs the functions.
AT.4.3 Given the graph of a function, the student graphs its inverse.
AT.4.4 The student determines and graphs the sum, difference, and composition of two or more functions and the composition of a function with itself and determines the domain and range of the resultant function.
AT.4.5 The student determines the inverse of an algebraic or trigonometric function given as an equation, a graph, or a set of ordered pairs; determines its domain and range; and discusses the relationships among these representations.
AT.4.6 The student knows the definitions of standards functions and their reciprocals and can graph them.
AT.4.7 The student applies transformations to the graph of a basic function (e.g., trigonometric functions) and predicts and analyzes the results on the graph of the function.
AT.4.8 The student graphs polar equations (e.g., roses, limniscates), analyzes the results of parameter changes on the graphs, and classifies the equations according to their graphs.
AT.4.9 The student understands and explains the relationship between triangle trigonometry and the unit circle/wrapping function approach to trigonometry.
AT.4.10 Given the value of one trigonometric function, the student finds the values of other trigonometric functions.
AT.4.11 The student identifies key characteristics (e.g., domain, range, amplitude, period, phase shift, and vertical shift) of trigonometric functions and graphs the functions and their inverses.
AT.4.12 The student graphs and analyzes step and piecewise defined functions.
AT.4.13 The student defines and applies the properties of limits of functions, including infinite sequences, series, and the slope of the tangent to a curve of a point.
Content Standard AT.5: Technology
The student demonstrates the ability to use the tools of new technologies to solve algebraic and trigonometric problems.
Performance Standards
AT.5.1 The student demonstrates the ability to use a graphing calculator to solve advanced algebraic and trigonometric problems and answer questions about real-world problems.
AT.5.2 The student demonstrates the ability to use computer software to explore advanced algebraic and trigonometric concepts.
Content Standard AT.6: Measurement and Geometry
The student understands and uses periodic function, trigonometric relationships, and vectors to represent and answer questions about quantities.
Performance Standards
AT.6.1 The student uses similarity, right triangles, the Law of Sines, and the Law of Cosines to determine measurements of objects that are difficult to measure directly.
AT.6.2 The student draws a system of vectors and finds the resultant graphically.
AT.6.3 The student determines the area of a triangle when the value of trigonometric functions are needed to determine the altitude.
AT.6.4 The student identifies, creates, and solves practical problems, using a system of vectors and their horizontal and vertical components.