Format of Guidelines

STRAND

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
	TOPIC	description of the extent to which a skill is developed, or of what behaviors the students will exhibit in completing the activities used to develop the skill.	vocabulary with which the students should be able to communicate (speak, write) mathematically.	computational or conceptual representation of skill indicating the extent to which the skill is to be presented at this level.	application of skill to problem-solving/real-life situation	inter-relatedness of skill to real-life or applications of skill to other areas of the curriculum or to other skills in mathematics	This space is intended to be utilized by the teacher to list the references and materials appropriate to the development of the skill: e.g. teacher guide, technology, supplementary materials, manipulatives, etc.

STRAND refers to the general area of mathematics into which each topic is categorized.

Strands include READINESS, SKILLS, NUMBER CONCEPTS, OPERATIONS, PROBLEM SOLVING, MENTAL

MATHEMATICS, ESTIMATION, GEOMETRY, MEASUREMENT, REASONING SKILLS, DATA AND

STATISTICS, TECHNOLOGY, RATIO-PROPORTION-PERCENT, AND PRE-ALGEBRA SKILLS.

A section of these guidelines is also designated as ENRICHMENT. This is intended for the horizontal expansion of topics

rather than the vertical progression of skills.

TOPIC – specific area within each strand. (For example, USING A PROBLEM-SOLVING MODEL is a topic in the strand of PROBLEM SOLVING).

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
I - D	INTEGERS	M7.1 The students will understand the concept of integers. M7.2 The students will compare and order integers.	integer positive negative whole number	+ 13 -7 + 16 > - 32 - 23 < + 4	Give the integer which represents the following: 22 degrees below zero a deposit of $5.00. At the end of a card game, Tom’s score was +8; Ellen’s score was –23. Who had the higher score?	Real life: Temperatures, game scores, feet above or below sea level, gains and losses in the stock market, etc.

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
M	PLACE VALUE	M7.3 The students will maintain understanding of place value millionths through billions	decimal millionths hundred-thousandths ten-thousandths thousandths hundredths tenths units tens hundreds thousands ten-thousands hundred-thousands millions ten-millions hundred-millions billions	Use place Value Chart.	Draw a place value chart for thirty eight and four hundred twelve hundred-thousandths In a reference book (e.g. an almanac) search for things that are described in terms of very large and very small numbers.	Study Skills – use of reference materials. Language Arts – Writing large numbers within a paragraph/sentence Social Studies, Science – larger and smaller numbers used in data. Physical Education – number used to record speed and distances Technology – limitations of calculator display.

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
M	READING AND WRITING NUMBERS (NUMERALS)	M7.4 The students will maintain skills in reading and writing numbers.	number numeral	Reading: Stress: decimal point is read as “and”. Stress: Correct reading of repeating decimals (e.g. _ 0.6 3 is read as “63 hundredths, bar 3.” Writing: Stress: correct formation of numerals and mathematical symbols; correct size of fractions and exponents.	Write in words: 4,003,016,024 (four billion, three million, sixteen thousand, twenty-four.) Write the inequality for the statement: Eight million is greater than eight millionths. (8,000,000 > 0.000008)	Language Arts – Handwriting; Reading; Vocabulary enrichment (number-related root words) Spelling Correct writing of numbers in sentences, outlines, tables, etc. Social Studies – reading and writing numerical data. Science – reading and writing numerical data. Technology – interpreting spreadsheet data.

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE		PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
I - D	NUMBER PATTERNS	M7.5 The students will discover number patterns; they will see the relationship between triangular and square numbers; they will discover patterns in Pascal’s triangle.	patterns triangular numbers figurate numbers square numbers Pascal’s Triangle	Explore dot patterns to find square numbers and their relation to triangular numbers. ▪ ▪▪ ▪▪▪ ▪▪▪▪ ▪▪ ▪▪▪ ▪▪▪▪ ▪▪▪ ▪▪▪▪ ▪▪▪▪			Problem Solving: Solving a Simpler Problem: How many squares are on a checkerboard? Start with a 2 x 2 square and develop the pattern to solve.
M	NUMBER SENSE PLACE VALUE	M7.6 The students will maintain understanding of place value millionths through trillions.	decimal millionths hundred-thousandths ten-thousandths thousandths hundredths tenth units tens hundreds thousands ten-thousands millions ten-millions hundred-millions billions trillions	Use place value chart			Draw a place value chart ffor thirty eight and four hundred twelve millionths. In a reference book (e.g. an almanac) search for things that are described in terms of very large and very small numbers.	Study Skills—use of reference materials Language Arts—Writing large numbers within a paragraph/sentence. Social Studies, Science— larger and smaller numbers used in data. Physical Education— number used to record speed and distances Technology—limitations of calculator display.
SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE			CONNECTIONS INTEGRATION	RESOURCES
M I - D M M M I-D-M	NUMBER PATTERNS Divisibility Perfect, Abundant, Deficient numbers Prime and Composite Numbers Sequences	M7.7 The students will determine if a number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10. M7.8 The students will determine if numbers are perfect, abundant or deficient. M7.9 The students will determine if numbers are prime of composite. M7.10 Using the Sieve of Erastothenes, the students will determine the prime numbers between 1 and 100. M7.11 The students will find the prime factorization of composite numbers. M7.12 The students will recognize and extend arithmetic and geometric sequences and will solve problems using these.	divisibility divisible factor perfect numbers abundant numbers deficient numbers prime number composite number Sieve of Erastothenes factor tree prime factorization Fundamental Theorem of Arithmetic relatively prime twin primes sequence terms arithmetic sequence geometric sequence common difference common ratio	48 is divisible by 2, 3, 4, 6, 8, 12, 16, 24 and 48 Perfect --a number is perfect if the sum of the factors less than the number is equal to the number ( The factors of 28 are 1, 2, 4, 7, and 14. The sum of these factors is 28). Abundant – a number is abundant if the sum of the factors less than the number is greater than the number.(12 is an abundant number). Deficient – sum of factors is less than the number (10 is a deficient number). Prime – a number having only 2 factors, itself and 1. Composite – a number with more than two factors. Sieve of Erasiothenes – Using a hundreds’ chart, cross out all multiples of 2, 3, 5, 7..Remaining numbers are prime numbers. factor tree – 240 / \ 60x 4 /\ /\ 15x 4x 2 x 2 /\ /\ | | /\ /\ | | 5x3x2x2x2x2 The prime factorization of 240 is 2⁴ ·3·5 arithmetic sequence- next term found by adding . (e.g. 3, 7, 11, 15, 19,..next terms found by adding 4 to last term.) Geometric Sequence – next term found through multiplication . (e.g. 4, 8, 16, 32, 64, 128…next term found by multiplying previous term by two.) Sequences found through using a combination of operations are neither arithmetic or geometric. (e.g. 4, 8, 9, 18, 19, 38, 39…the pattern is x2 + 1)	How many different ways can 4 dozen donuts be arranged on a tray? 2 rows of 24 3 rows of 16 4 rows of 12 6 rows of 8… Make a chart identifying the numbers from 25 – 40 as being deficient, abundant or perfect. Explain why 33 is considered a deficient number How many consecutive primes are there between 1 and 58 (Since 2 is the only even prime, there is only one set of consecutive primes). Bill’s father gave him a choice of payment plans for doing odd jobs at the family store during the month of July. Plan A: Starting with $4.81 on the first day of the month, he could double the amount each day. Plan B: $10. a day for each day he worked. If Bill worked 20 days out of the month which plan would Bill find most beneficial? (Under plan A, Bill would earn $10,485.75. Under plan B, he would earn only $200.00) Extend – How many days does Bill have to work in order for plan B to be more beneficial? (14) How much would Bill earn under each plan if he worked for 12 days? 15 Days? 25 days?		Science: patterns in nature Social Studies: election years Real Life: Patterns in architecture, industry, sports Math: Geometry- Using blocks or cubes, have the students discover that prime numbers can only make one rectangular arrangement. With composite numbers, multiple arrangements are possible. (e.g. with 24 cubes the arrangement could be 2 x 12, 3 x 8, 6 x 4, etc.) Math: Problem solving—Make a Table Strategy Science: Patterns Language Arts: Prepare an oral presentation explaining the different types of sequences. Prepare a job interview simulating the problem situation. Technology: Use the calculator/computer in working on sequences. Research: Leonardo Fibbonacci Science, Music: Applications of the Fibbonacci sequence. Art, Architecture: Applications of the Golden Rectangle

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
I – D –M	READING AND WRITING MATHEMATICAL SYMBOLS	M7.13 The students will read and write mathematical symbols	Symbol +, -, x, ÷, =, <, >, ≤, ≥, ≠, ≈, ~, —, …, º, %, @, ½-3½, p,		Provide practice using symbols in context.	Mary’s favorite foods included pizza, hamburgers, fries, and ice cream. {pizza, hamburgers, fries, ice cream}
M	COMPARING AND ORDERING DECIMALS	M7.14 Given real-life situations, the students will compare decimals	greater than less than equal equivalent	1. Compare on number line: number to the right is always the greater number. 2. Use place-value chart to compare numbers. 3. Using graph paper, create models of decimals to compare.	Items to compare: - sports’ statistics - gas mileage - barometric pressure - nutritional value of food - interest rates	Science: data for biology. Social Studies: distances on maps. Study Skills: Reading charts, tables, etc.
M	ORDERING NUMBERS FROM GREATEST TO LEAST AND VICE VERSA.	M7.15 Given a set of data, the students will order the data from least to greatest or from greatest to least.	greatest least	Order the following: from least to greatest: 5, 0.5, 0.55, 0.505, 50 (0.5, 0.505, 0.55, 5, 50) Order the following from greatest to least. 166, 16.6, 1.66 1.6666 (166. 16.6, 1.6666, 1.66)	Compare the percentages of a basketball league. Find statistics represented in the sports section of the newspaper— compare the percentages represented as decimals.	Language Arts: Write a step-by-step plan for comparing decimals. Technology: Utilize software programs which compare decimals. Mathematics, Social Studies, Science: Compare data using bar and line graphs.

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
	NUMBER LINE	The students will use the number line as an aid in developing number sense.				Social Studies: Time lines
M	ROUNDING DECIMALS AND WHOLE NUMBERS	M7.16 The students will round whole numbers and decimals to any place-value position.	rounding Teacher Alert: Be careful when working with estimation to distinguish between rounding the numbers to be estimated before completing the operation and merely rounding the answer to the operation.	Round 53,769,211 to the nearest thousand. (The digit nine is in thousands place. The number immediately to the right is 2 which is less than 5, so 53,769, 211 rounded to the nearest thousand is 53, 769,000. Round 4.0695 to the nearest hundredth: The digit 6 is in hundredths place. The digit to the right is a nine. (4.0695 rounded to the nearest hundredth is 4.07.)	Locate the heights of the world’s mountains an atlas or almanac. Round these numbers to the nearest thousand. Order these from greatest to least. Make a graph showing their relationships.	Mathematics: measurement, estimating: Real life skills: rounding money to the nearest dollar.

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
D – M	POWERS AND EXPONENTS	M7.17 The students will use powers and exponents in expressions	factors exponent base powers cubed squared	Exponent ⇩ 100,000 = 10⁵ ⇧ base Write 2³ as a product. (2³ = 2·2·2) Note use of raised dot as a symbol for multiplication	The height of a building can be expressed as 7³ Approximately how many floors can the building have if each floor occupies about 14 feet?	Testing Practice: Which of the following statements is correct? A. 3² = 2³ B. 4² = 2⁴ (B is correct. A. 9 > 8 B. 16 = 16)

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
D – M I – D	EXPONENTIAL NOTATION SCIENTIFIC NOTATION	M7.18 The students will express numbers in exponential form. M7.19 The students will use scientific notation to represent large numbers.	Exponential form scientific notation	28 x 10³ = 28 x 1000 = 28,000 A number written in scientific notation has 2 factors: -a number ≥1 but < 10 - a power of ten 2.8 x 10³ = 2.8 x 1000 = 2,800	The land area of South America is 6,875,000 square miles. Represent this number using exponential notation. (6.875 x 10³) The largest river in the world is the Amazon with a total basin area of 2,720,000 square miles. Represent this number in scientific notation. (2.72 x 10⁶)	Science, Social Studies: larger and smaller numbers.
ESTIMATING	The concept of estimation should be applied to all strands within the curriculum. For specific applications of estimation, please refer to the estimation strand within these guidelines.

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
D	RATIOS	M7.20 The students will identify ratios and will write equivalent ratios.	ratio rate unit rate (second term of ratio is one)	The names of twelve of the fifty states begin with a vowel. Express this as a ratio in three different ways: 12 to 50 12: 50 12 or 6 50 25	8 of the boys in the class wore green for St. Patrick’s Day. Five of the girls wore green. Write the ratio for a class of twenty five students.	Mathematics: Fractions, Problem Solving Geometry
D	PROPORTION	M7.21 The students will identify proportions.	proportion extremes/means cross products equivalent ratios	Model using fraction bars, fraction circles, or other manipulatives. ░ ▒ ▒ ▒ ░ ▒ ▒ ▒ ░ ▒ ▒ ▒ █ ░ ░ ░ █ ░ ░ ░ 3 = 2 9 6		Mathematics: equivalent fractions

NUMBER CONCEPTS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
	PERCENTAGE	Review all concepts of percentage as introduced and developed at level 6. Further development and application is indicated in the Ratio, Proportion and Percentage section of these guidelines. Students should commit to memory the most common decimal-fraction-rate of percentage equivalents.
I	BASES OTHER THAN TEN	M7.22 The students will become aware of number bases other than base ten.	base base ten decimal system base two binary system	Review Base 10: 136 means: 1 x 10² + 3 x 10¹ + 6 x 10º - 100 + 30 + 6 Note: the only digits in a number base are the digits less than the base itself. 1 0 1two = 1 x 2² + 0 x 2¹ + 1 x 2º = 4 + 0 + 1 = 5	Express 21 as a binary number: (1 x 2⁴) + (0 x 2³) + (1 x 2²) + (0 x 2¹) = (1 x 2º) = 16+0+4+0+1 Therefore: 10101two = 21ten	Technology: The use of the binary number system in computer programming. Enrichment: Challenge students to compare numbers in different number bases. 36tens---101two Challenge students to use bases other than base two and base ten.	Enrichment for above average students.
ROMAN NUMERALS	Review the concept and application of Roman Numerals as presented at earlier levels. Reminder: Do not draw lines connecting Numerals at top and bottom.

OTHER SETS OF NUMBERS

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
I - D	NATURAL NUMBERS	M7.23 The students will identify the natural numbers as counting numbers included in the set of numbers beginning with one and extending into infinity.	natural number counting number set of numbers infinity whole number	N = {1,2,3…} 3 is a natural number 4.5 is not a natural number 0 is not a natural number	Make a table identifying the following numbers as natural, whole, integers or rational numbers: 36 -3 2.5 3 ½	Mathematics: Problem Solving, Reasoning Skil
I – D	INTEGERS	M7.24 The students will understand integers as the extension of natural numbers to include zero and negative numbers.	integer positive negative	J=(…-3, -2, -1, 0, 1, 2, 3…) -43 is an integer -43.5 is not an integer the symbol J to represent the set of integers (I is reserved for Irrational numbers—not presented at this level.)		Q THE USE OF A GRAPHIC J TO REPRESENT THE SETS W OF NUMBERS WILL HELP THE STUDENTS TO SEE N THE RELATIONSHIPS/ EXTENSIONS
I – D	RATIONAL NUMBERS	M7.25 The students will understand the set of rational numbers as an extension of the set of integers	rational number (Q) fraction decimal	Q = {any number that can be expressed as the ratio of a to b where a and b are integers and b is not equal to zero.} This includes fractions and terminating/repeating decimals.			Give adequate practice in identifying numbers in their sets.

OPERATIONS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
	USE CONCRETE OBJECTS TO MODEL OPERATIONS	Manipulative materials—connecting cubes, blocks centimeter cubes, beans, etc. should be used frequently when modeling concepts. Understanding is built when skills are presented following the concept continuum.
	OPERATIONS INTRODUCED AND DEVELOPED IN PROBLEM-SOLVING CONTEXT	When operations are presented, it is important to give purpose to the learning by presenting the material in a problem-solving/real life context. Equations should never be presented in isolation until mastery has occurred.
	USE SYMBOLS FOR OPERATIONS	Refer to “symbols” in Readiness and Number Concepts section of these guidelines.
	ADD SINGLE-DIGIT AND MULTI-DIGIT NUMBERS	Review and maintain material presented at earlier levels. Extend through millions. Students should make frequent use of the calculator to check material which they have already mastered.

OPERATIONS – GRADE 7

PROPERTIES OF OPERATIONS	Review the following properties of operations: Properties + X Commutative a + b = b + a ab=ba Associative (a+b) +c = a + (b+c) (ab)c = a(bc) Identity a + 0 = a 1 . a = a Distributive ab + ac = a(b + c) Apply these properties to each set of numbers: Natural, Whole, Integers	Integrate into the context of other lessons.
ESTIMATION SUMS DIFFERENCES PRODUCTS QUOTIENTS	Estimation in all operations and with all sets of numbers should be reviewed and maintained. Estimation is a key skill and should be interwoven into every lesson involving operations.	Integrate into the context of other lessons

OPERATIONS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
I	OPERATIONS WITH POWERS AND EXPONENTS	M7.26 The students will learn that when multiplying the same base number, the operation can be simplified by adding the exponents; when dividing by the same base number, the operation can be simplified by subtracting the exponents.	power exponent base negative exponents	6²●6⁵=6⁷ 4⁹÷4⁶=4³	When the exponents of ten decrease by one, what happens to the products?	Technology— Computer programs dealing with numbers and bases. Use a calculator to evaluate equations with exponents.
M	INTERPRET REMAINDERS	M7.27 The students will interpret the remainder in a division problem and will decide whether the remainder should be written as a fraction or a decimal or whether the answer should be rounded up or rounded down.	remainder	Each small box of candy takes up three square feet of space on a layer of a box. How many small boxes can be placed on a layer of containing 40 square feet? (using a decimal or fraction does not make sense.) $23.00 was divided among four students. How much will each receive? Rounding the decimal to hundredths place makes the most sense.		Language Arts – Have the students create their own original word problems requiring reasonable answers. Have them conduct a newspaper scavenger hunt for situations which require exact answers.

OPERATIONS – GRADE 7

SCOPE and SEQUENCE	SKILL	OBJECTIVE	VOCABULARY	EXAMPLE	PROBLEM SOLVING EXAMPLE	CONNECTIONS INTEGRATION	RESOURCES
I - D	SQUARE ROOT	M7.28 After comprehending the concept of square root, the students will find the square root of a number in one of two ways: 1) Estimation 2) Calculator 1)	square root radical sign	Estimation: To find the square root of 225 Note that for every two digits in the square there will be one digit in the square root. Therefore, there will be 2 digits in the square root of 225. This narrows the answer to a number between 10 and 99. Use the guess and test method to narrow down even further. 12 x 12 = 144 too low 19 x 19 = 361 too high 15 x 15 = 225	The seventh grade packed 144 cans of food in cartons. The number of cans in each carton equaled the number of cartons they packed. How many was this?	Math: Estimation, problem solving. Study Skills: Students should memorize the table of squares and square roots for whole numbers through 12. Language Arts: Idioms—What is a “square” root?

OPERATIONS – GRADE 7

VOCABULARY OF OPERATIONS

Students should read, write and speak using the correct mathematical vocabulary. Provide ample opportunity during each lesson for the students to become aware of the vocabulary and to use it. Avoid expressions such as “timesing numbers” instead of multiplying, etc.

APPLICATION OF OPERATIONS TO OTHER SETS OF NUMBERS
D - M	INTEGERS	The students will apply previously learned skills to integers and will extend these skills to larger and smaller numbers within the set of integers: Operations Properties 2)	Addition, subtraction, multiplication and division applied to integers. Apply properties to all sets of numbers.	Problem-solving with all operations and in all sets of numbers.	Multiple applications in all content areas and in many aspects of real life.

PROBLEM SOLVING – GRADE 7

SCOPE and SEQUENCE	TOPIC -SKILL	OBJECTIVE	VOCABULARY	EXPLANATION - EXAMPLE	CONNECTIONS	RESOURCES
M	USE A PLAN FOR PROBLEM SOLVING	M7.29 The students will use a 4/5 –step plan for problem-solving 3)	Understand Plan Do Check Answer Statement Alternate Plan 1. Explore 2. Plan 3. Solve 4. Examine strategy ⋆Alert!! Addition, Subtraction, Multiplication and division are operations not strategies. An operation is not a plan in itself, but a means to carry out a plan.	Understand: What is known? What must be found out? Are there any hidden questions? Do I need an estimate or an exact answer? Plan: Choose a strategy. (a procedure to follow in solving the problem.) Do: Carry out the plan. Check: Is the answer reasonable? Does it answer the given question? If an operation was used to carry out a plan, use its complement to check the computation. Check using an alternate strategy. Answer Statement: Re-phrase the question in terms of an answer statement.	Language Arts— Reading Comprehension Reading for main ideas and details Declarative and Interrogative Sentences Study Skills- -mapping, outlining Following directions Logical reasoning Decision making Critical thinking skills Science: Scientific Method Research: George Polya, Mathematician	A Problem Solving Chart, listing the steps in the problem-solving method would benefit students.

PROBLEM SOLVING – GRADE 7

SCOPE

and

SEQUENCE