Grade 7 Advanced Math Curriculum updated 9-30-05teachers.olatheschools.com/~kwilliamsirc/files/Curriculum/grades 6... · Grade 7 Advanced Math Curriculum ... the numbers does not

Grade 7 Advanced Math Curriculum

Indicator/Objective Code: ▲- KS Tested Indicator, - KS Constructed Response, N - no calculator © USD #233 BOE Approved November 2005 This material was developed for the exclusive use of USD #233 staff.

Further reproduction or distribution is prohibited without written permission from USD #233.

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Standard 1 Numbers and Computation The student uses numerical and computational concepts and procedures in a variety of situations.

Benchmark 1 Number Sense The student demonstrates number sense for rational numbers, the irrational number pi, and simple algebraic expressions in one variable in a variety of situations.

Indicator/Objective Critical Vocabulary Essential Concepts/Skills Implementation Assessment Examples Knowledge Indicators The student… 7A.M.NC.NS.1 knows, explains, and uses equivalent representations for rational numbers and simple algebraic expressions including integers, fractions, decimals, percents, and ratios; integer bases with whole number exponents; positive rational numbers written in scientific notation with positive integer exponents; time; and money. 7A.M.NC.NS.2 compares and orders rational numbers and the irrational numbers and algebraic expressions. 7A.M.NC.NS.3 explains the relative magnitude between rational numbers and between rational numbers and the irrational number pi and algebraic expressions. 7A.M.NC.NS.4 knows and explains what happens to the product or quotient when:

a. a positive number is multiplied or divided by a rational number greater than zero and less than one. b. a positive number is multiplied or divided by a rational number greater than one. c. a non zero real number is multiplied or divided by zero.

7A.M.NC.NS.5 explains and determines the absolute value of real numbers. Application Indicators 7A.M.NC.NS.6 ▲generates and/or solves real-world problems: equivalent

What Students Need to Know 1. How to compute mentally. 2. What relative magnitude means (the

size relationship one number has with another, is it larger, is it smaller, close, or about the same, how far apart numbers are.

3. How to determine whether or not solutions to real world problems that involve integers and fractions are reasonable.

4. How to look at a problem holistically before confronting the details of the problem.

What Students Need to Do/Apply 1. Generate and/or solve real-world

problems for fractions and decimal approximations of the irrational number pi.

2. Determine whether or not solutions to real-world problems using rational numbers, the irrational number pi, and simple algebraic expressions are reasonable.

Strategies

♦ Use district provided cyclical reviews.

♦ Compute daily with mental math.

Scope and Sequence 6th Graders explained and used equivalent representations for rational numbers expressed as fractions, terminating decimals, and percents; positive rational number bases with whole number exponents; time; and money. At this grade, student explained the relative magnitude between whole numbers, fractions greater than or equal to zero, and decimals greater than or equal to 0 and used equivalent representations for the same simple algebraic expression with understood coefficients of 1. 6th graders were tested upon comparing and ordering integers, fractions greater than or equal to zero, and decimals greater than or equal to 0 through thousandths place. 6th graders were also tested upon knowing and explaining numerical relationships between percents, decimals, and fractions between 0 and 1. Algebra 1 students will know, explain, and use equivalent representations for real numbers and algebraic expressions including integers, fractions, decimals, percents, ratios; rational number bases with integer exponents; rational numbers written in scientific notation; absolute value; time; and money. They will compare and order real numbers and/or algebraic expressions and explains the relative magnitude between them. At this level, students will know and explain what happens to the product or quotient when a real number is multiplied or divided by:

a. rational number greater than zero and less than one.

1. Which is not equivalent to 3/4? a. 75% b. .0075 c. .75 d. .075 2. Put the following rational numbers in order from least to greatest: 3.140 22/7 3 3.15 3 1/2 3. Between what 2 perfect squares is the number pi? Answer true or false. 4a. The product of 17 and a number between 0 and 1 is larger than 17. 4b. The number 17 gets smaller when you divide it by 2 1/2. 4c. What happens when you divide a number by 0? a. the quotient will be positive b. the quotient will be negative c. the quotient will be 0 d. the quotient will be undefined. 5. What is |20| ? 6. ▲Solve this problem. You are in the mountains. Wilson Mountain has an altitude of 5.28 x 103 feet. Rush Mountain is 4,300 feet tall. How much higher is Wilson Mountain than Rush Mountain?




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representations of rational numbers and simple algebraic expressions Vocabulary Students Know and Use equivalent representations integers exponent scientific notation rational numbers irrational number relative magnitude absolute value ratios pi

b. rational number greater than one. c. rational number less than zero.

Pacing Considerations Use chapters 1-2-3-4-5 In Harcourt’s Math Advantage textbook. The Long Range Pacer suggests 1st and 2nd quarters for this benchmark.




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Benchmark 2 Number Systems and Their Properties The student demonstrates an understanding of the rational number system and the irrational number pi: recognizes, uses, and describes their properties; and extends these properties to algebraic expressions in one variable.

Indicator/Objective Critical Vocabulary Essential Concepts/Skills Implementation Assessment Examples

Knowledge Indicators The student… 7A.M.NC.NSP.1 knows and explains the relationships between natural (counting) numbers, whole numbers, integers, rational, and irrational numbers using mathematical models. 7A.M.NC.NSP.2 identifies all the subsets of the real number system. 7A.M.NC.NSP.3 names, uses, and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects:

a. commutative properties of addition and multiplication (changing the order of the numbers does not change the solution). b. associative properties of addition and multiplication (changing the grouping of the numbers does not change the solution). c. distributive property [distributing multiplication or division over addition or subtraction. d. substitution property (one name of a number can be substituted for another name of the same number.

7A.M.NC.NSP.4 uses and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects:

a. identity properties for addition and

What Students Need to Know 1. Recognize rational numbers and pi. 2. Classify rational numbers. 3. Recognize and explain natural,

whole, integers, rational and real numbers.

4. Explain and use commutative, associative, distributive, identity, transitive, reflexive, and symmetric properties.

What Students Need to Do/Apply 1. Generate and/or solve real-world

problems with rational numbers and the irrational number pi using the concepts of these properties to explain reasoning: commutative and associative properties of addition and multiplication, distributive property, substitution property, symmetric property of equality, additive and multiplicative identities, 0 property of multiplication, addition and multiplication properties of equality, additive and multiplicative inverse properties.

2. Analysis and evaluates the advantages and disadvantages of using integers, whole numbers, fractions, decimals, or the irrational number pi and its rational approximations in solving a given real world problem.

Scope and Sequence 6th Graders classified subsets of the rational number system as counting and whole numbers, integers, fractions (including mixed numbers), or decimals. They identified prime and composite numbers and explain their meaning. At this grade, 6th graders used and described these properties with the rational number system and demonstrated their meaning including the use of concrete objects:

a. commutative and associative properties of addition and multiplication.

b. associative. c. identity properties for addition and

multiplication. d. symmetric property of equality. e. zero property of multiplication. f. distributive property. g. substitution property. h. addition property of equality. i. multiplication property of equality. j. additive inverse property.

They recognized and explained the need for integers and recognized that the irrational number pi can be represented by an approximate rational value. There were no tested indicators for this benchmark. Algebra 1 students explain and illustrate the relationship between the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, irrational numbers] using

1. Make a Venn diagram for counting numbers less than 10. Label one section even and one section prime. 2. Classify 0, -7, 3 1/2, -8.8 and 6 the into these categories: whole, counting, integer or rational 3a. At a delivery stop, Sylvia pulls out a flat of eggs. The flat has 5 columns and 6 rows of eggs. Express how to find the number of eggs in 2 ways. 3b. Which equation shows the associative property? a. a + (b + c) = (a + b) +c b. 3 x 8 = 8 x 3 c. 4(6-2) = 4 x 6 – 4 x 2 d. 7 + 3 + 5 = 9 + 0 + 6 3c. Trim is used around the outside edges of a bulletin board with dimensions 3 ft by 5 ft. Explain two different methods of solving this problem. 3d. V = IR [Ohm’s Law: voltage (V) = current (I) x resistance (R)] If the current is 5 amps (I = 5) and the resistance is 4 ohms (R = 4), what is the voltage?




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multiplication (additive identity – zero added to any number is equal to that number; multiplicative identity – one multiplied by any number is equal to that number). b. symmetric property of equality. c. zero property of multiplication. d. addition and multiplication properties of equality. e. additive and multiplicative inverse properties.

7A.M.NC.NSP.5 recognizes that the irrational number pi can be represented by approximate rational values. Application Indicators There are no tested application indicators for this benchmark. Vocabulary Students Know and Use natural numbers whole numbers Venn diagrams properties commutative associative distributive identity properties of addition identify property of multiplication symmetric substitution additive inverse multiplicative inverse addition property of equality multiplicative property of equality rational value

Strategies ♦ Use technology web sites for

additional practice.

mathematical models. They identify all the subsets of the real number system [natural (counting) numbers, whole numbers, integers, rational numbers, and irrational numbers] to which a given number belongs. At this level, students are tested upon naming, using, and describing these properties with the real number system and demonstrates their meaning including the use of concrete objects:

a. commutative (a + b = b + a and ab = ba), associative [a + (b + c) = (a + b) + c and a(bc) = (ab)c], distributive [a (b + c) = ab + ac], and substitution properties.

b. identity properties for addition and multiplication and inverse properties of addition and multiplication (additive identity: a + 0 = a, multiplicative identity: a • 1 = a, additive inverse:, multiplicative inverse).

c. symmetric property of equality (if a = b, then b = a).

d. addition and multiplication properties of equality (if a = b, then a + c = b + c and if a = b, then ac = bc) and inequalities (if a > b, then a + c > b + c and if a > b, and c > 0 then ac > bc).

e. zero product property (if ab = 0, then a = 0 and/or b = 0). Algebra 1 students also use and describe the transitive and the reflexive properties: transitive property (if a = b and b = c, then a = c).

f. reflexive property (a = a). Pacing Considerations The 7th grade long range pacer suggests chapters 5 and 6 cover properties during 1st and 2nd quarters.

4a. Bob and Sue each read the same number of books. During the week, they each read 5 more books. Compare the number of books each read. Let b = the number of books Bob read and s = the number of books Sue read, then b + 5 = s + 5 4b. Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $15 = $10 + $5 is the same as $10 + $5 = $15. 4c. Jenny was thinking of two numbers. Jenny said that the product of the two numbers was 0. What could you deduct from this statement? Explain your reasoning. 4d. Solve. 4y – 7 = 5 4y – 7 + 7 = 5 + ____ 4e. What is 8 x 1/8? 5. What value would not give an approximation of pi?

a. 3.14

b. 7/22

c. 22/7




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Benchmark 3 Estimation The student uses computational estimation with rational numbers and the irrational number pi in a variety of situations.


Knowledge Indicators The student… 7A.M.NC.E.1 estimates quantities with combinations of rational numbers and/or the irrational number pi using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology. 7A.M.NC.E.2 N uses various estimation strategies and explains how they were used to estimate rational number quantities and the irrational number pi. 7A.M.NC.E.3 recognizes and explains the difference between an exact and approximate answer. 7A.M.NC.E.4 determines the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer and its potential impact on the result. 7A.M.NC.E.5 knows and explains why the fraction (22/7) or decimal (3.14) representation of the irrational number pi is an approximate value. Application Indicators No tested application indicators for this benchmark. Vocabulary Students Know and Use approximate underestimate overestimate

What Students Need to Know 1. Demonstrate a variety of

computational methods. 2. Reasonable results. 3. Recognize and explain the difference

between exact and approximate. 4. Discuss the appropriateness of

estimation. 5. Demonstrate a variety of

computational methods. 6. Explain and perform computations

with rational numbers. 7. Demonstrate arithmetic operations

and inverse relationships. What Students Need to Do/Apply 1. Adjust original rational number

estimate of a real-world problem based on additional information (a frame of reference).

2. Estimate to check whether or not the result of a real-world problem using rational numbers, the irrational number pi, and/or simple algebraic expressions is reasonable and makes predictions based on the information.

3. Determine a reasonable range for the estimation of a quantity given a real-world problem and explains the reasonableness of the range.

4. Determine if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate.

Scope and Sequence 6th Graders estimated quantities with combinations of rational numbers and/or the irrational number pi using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology. They used various estimation strategies and explain how they were used to estimate rational number quantities or the irrational number pi. 6th graders recognized and explained the difference between an exact and an approximate answer. They determined the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer and its potential impact on the result. Students were tested upon estimating to check whether or not the result of a real-world problem using rational numbers and/or the irrational number pi is reasonable and making predictions based on the information. Algebra 1 students will estimate real number quantities using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology. For this course, students will use various estimation strategies and explain how they were used to estimate real number quantities and algebraic expressions. They will know and explain why a decimal representation of an irrational number is an approximate value

1. Kathy has $120. Does she have enough to buy items priced $32.56, $12.83, and $6.99 and five items of $12.49? Explain your thinking. 2. Using compatible numbers, mentally add 19, 12, 7 and 11. 3. For the following situations, would you use exact number of approximate numbers? a. time for a race____________ b. calculating costs while waiting in line at the checkout ___________ 4. You are moving a sofa in an elevator. Would you want to overestimate or underestimate the weight of the sofa? Why? 5. Explain why 3.14 is only an approximation.




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Strategies ♦ Use district provided cyclical

reviews. ♦ Estimate daily as part of mental

math practice.

and know and explain between which two consecutive integers an irrational number lies. Pacing Considerations One week, then ongoing




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Benchmark 4 Computation The student models, performs, and explains computation with rational numbers, the irrational number pi, and first-degree algebraic expressions in one variable in a variety of situations.


Knowledge Indicators The student… 7A.M.NC.C.1 computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. 7A.M.NC.C.2 performs and explains these computational procedures:

a. ▲N adds and subtracts decimals from ten millions place through hundred thousandths place. b. ▲N multiplies and divides a four-digit number by a two-digit number using numbers from thousands place through thousandths place. c. ▲N multiplies and divides using numbers from thousands place through thousandths place by 10; 100; 1,000; .1; .01; .001; or single-digit multiples of each. d. ▲N adds, subtracts, multiplies, and divides fractions and expresses answers in simplest form. e. N adds, subtracts, multiplies, and divides integers. f. N uses order of operations (evaluates within grouping symbols, evaluates powers to the second or third power, multiplies or divides in order from left to right, then adds or subtracts in order from left to right) using whole numbers. g. simplifies positive rational numbers raised to positive whole

What Students Need to Know 1. How to compute mentally. 2. Order of operations. What Students Need to Do/Apply 1. Generate and/or solve real-world

problems using these computation procedures and math concepts: a. addition, subtraction,

multiplication and division of rational numbers with a special emphasis on integers.

b. 1st degree algebraic expressions in 1 variable.

c. percentages of rational numbers. d. approximation of the irrational

number pi. 2. Determine whether or not solutions to

real-world problems using rational numbers, the irrational number pi, and simple algebraic expressions are reasonable.

Strategies

♦ Practice daily: mental math and paper/pencil cyclical reviews provided by district.

♦ Use technology provided web sites.

Scope and Sequence 6th Graders computed with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. 6th graders were tested upon performing and explaining:

a. dividing whole numbers through a two-digit divisor and a four-digit dividend and expressing the remainder as a whole number, fraction, or decimal.

b. adding, subtracting, and multiplying fractions (including mixed numbers) expressing answers in simplest form.

They were also tested upon generating and/or solving one- and two-step real-world problems with rational numbers using these computational procedures: addition, subtraction, multiplication, and division of decimals through hundredths place. 6th graders learned to:

a. add and subtract decimals from millions place through thousandths place.

b. multiply and divide a four-digit number by a two-digit number using numbers from thousands place through hundredths place.

c. multiply and divide using numbers from thousands place through thousandths place by 10; 100; 1,000; .1; .01; .001; or single-digit multiples of each.

d. add integers.

1. Add 362 + 362 using mental math. 2a. ▲ What is the difference between 1.53615 and 1.49832? 2b. ▲ What is the quotient when 384.3 is divided by 1.8? 2c. ▲ What is the product of $1,000 x .05? 2d. ▲ Solve: 9/10 ÷ 1/8 = ___. 2e. What is the value of (-7 x -3) + -4? 2f. Evaluate using the order of operations. 6 + 16 ÷ 4 x 2= 2g. Evaluate 36 Show your work. 2h. Simplify 3(x + 2) + 8x. 3. Show another way to write 49 x 23. Ex: (40 x 23) + (9 x 23) or 49 x 23 = (49 x 20) + (49 x 3) or 49 x 23 = (50 x 23) – 23. 4. Show the prime factorization of 36. 5. ▲150% of 90 is what number?




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number powers. h. combines like terms of a first degree algebraic expression. i. approximates roots of numbers using calculators j. adds polynomials

7A.M.NC.C.3 recognizes, describes, and uses different ways to express computational procedures. 7A.M.NC.C.4 finds the prime factors, greatest common factor, multiplies, and least common multiple. 7A.M.NC.C.5 ▲finds percentages of rational numbers. Application Indicators No application tested multiple choice on the KS 7th grade test but constructed response. The student… 7A.M.NC.C.6 generates and/or solves real-world problems using these computation procedures and math concepts for addition, subtraction, multiplication and division of rational numbers with a special emphasis on fractions and expressions answers in simplest form. Vocabulary Students Know and Use order of operations like terms algebraic expression prime factors greatest common factor least common multiple

e. find the root of perfect whole number squares.

f. use basic order of operations (multiplication and division in order from left to right, then addition and subtraction in order from left to right) with whole numbers.

g. add, subtract, multiply, and divide rational numbers using concrete objects.

At this grade, students recognize, describe, and use different representations to express the same computational procedures. They identify, explain, and find the prime factorization of whole numbers. 6th graders find prime factors, greatest common factors, multiples, and the least common multiple. Finally, students found a whole number percent (between 0 and 100) of a whole number. Algebra 1 students will compute with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology. They will perform and explain these computational procedures:

a. addition, subtraction, multiplication, and division using the order of operations.

b. multiplication or division to find: i. a percent of a number.

ii. percent of increase and decrease. iii. percent one number is of another number. iv. a number when a percent of the number is given. c. manipulation of variable

quantities within an equation or




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inequality. d. simplification of radical

expressions (without rationalizing denominators) including square roots of perfect square monomials and cube roots of perfect cubic monomials.

e. simplification or evaluation of real numbers and algebraic monomial expressions raised to a whole number power and algebraic binomial expressions squared or cubed.

f. simplification of products and quotients of real number and algebraic monomial expressions using the properties of exponents.

g. matrix addition. h. scalar-matrix multiplication.

In this course, students will find prime factors, greatest common factor, multiples and the least common multiple of algebraic expressions. Pacing Considerations Long Range Pacer suggests 3 weeks, then ongoing




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Standard 2 Algebra The student uses algebraic concepts and procedures in a variety of situations.

Benchmark 1 Patterns The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations.


Knowledge Indicators The student… 7A.M.A.P.1 identifies, states, and continues a pattern presented in various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes:

a. ▲counting numbers including perfect squares, cubes, and factors and multiples. b. ▲positive rational numbers including arithmetic and geometric sequences. c. geometric figures. d. measurements. e. things related to daily life.

7A.M.A.P.2 generates a pattern (repeating, growing). 7A.M.A.P.3 extends a pattern when given a rule of one or two simultaneous changes (addition, subtraction, multiplication, division) between consecutive terms. 7A.M.A.P.4 ▲ ■ states the rule to find the nth term of a pattern with one operational change (addition or subtraction) between consecutive terms. Application Indicators No tested application indicators for this benchmark.

What Students Need to Know 1. Identify and continue patterns

presented in a variety of formats. 2. Generalize a pattern by using a

written description. What Students Need to Do/Apply 1. Generalize a pattern by giving the nth

term using symbolic notation. 2. Recognize the same general pattern

presented in different representations [numeric (list or table), visual (picture, table, or graph), and written.

Strategies

♦ Use district provided cyclical reviews.

♦ Use real life items and/or real world experiences, i.e. analyze daily activities. Look for a pattern in routines.

Scope and Sequence 6th Graders identified stated, and continued a pattern presented in various formats including numeric (list or table), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes include:

a. counting numbers including perfect squares, and factors and multiples (number theory).

b. positive rational numbers limited to two operations (addition, subtraction, multiplication, division) including arithmetic sequences (a sequence of numbers in which the difference of two consecutive numbers is the same).

c. geometric figures through two attribute changes.

d. measurements. e. things related to daily life.

Students generated a pattern (repeating, growing) and extended a pattern when given a rule of one or two simultaneous operational changes (addition, subtraction, multiplication, division) between consecutive terms. At this grade, students were tested upon stating the rule to find the next number of a pattern with one operational change (addition, subtraction, multiplication, division) to move between consecutive terms. Algebra 1 students will identify, state, and continue the following patterns using various formats including numeric (list or

1a. ▲Find the next term: 1, 4, 9, 16, … 1b. ▲What rule does this pattern follow?

1 1 1 1, , , ,...2 4 8 16

a. add 2 b. multiply by 2 c. subtract 2 d. divide by 2 1c. How many small squares will the 5th figure contain?

a. 25 b. 12 c. 16 d. 20 1d. What are the next two terms in the pattern? a. 24 in, 3 ft b. 30 in, 3 ft

c. 26 in, 122

ft

d. 30 in, 4 ft




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Vocabulary Students Know and Use counting numbers rational numbers perfect squares arithmetic sequences geometric sequences consecutive terms

table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written:

a. arithmetic and geometric sequences using real numbers and/or exponents.

b. patterns using geometric figures. c. algebraic patterns including

consecutive number patterns or equations of functions.

d. special patterns. For this course, students will generate and explain a pattern. They will classify sequences as arithmetic, geometric, or neither. Students will define:

a. a recursive or explicit formula for arithmetic sequences and find any particular term.

b. a recursive or explicit formula for geometric sequences and find any particular term.

Pacing Considerations One week, then on-going.

1e. If you mowed nine lawns, how much money would you make?

Lawns, l 1 2 3 4 5 Pay, p

a. $16.00 b. $80.00 c. $48.00 d. $72.00 2. Generate a pattern in which the consecutive terms are doubled. a. 5, 10, 15, 20 b. 2, 4, 16, 196 c. 3, 6, 12, 24 d. 1, 2, 3, 5 3. To find the next term in the pattern below, the rule multiply the previous number by 3, can be used. 8, 24, 72, 216, 648 What are the next two terms in the pattern? a. 651, 654 b. 1944, 5832 c. 648, 1944 d. 1296, 2592 4. ▲The first four numbers of a pattern are shown below.

89, 78, 67, 56 Which rule could be used to find the nth term of the pattern? a. 11n + 1 b. 56 + 11n c. 89 – 11n d. 100 – 11n




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Benchmark 2 Variables, Equations, and Inequalities The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations.


Knowledge Indicators The student… 7A.M.A.V.1 knows and explains that a variable can represent a single quantity that changes. 7A.M.A.V.2 knows, explains, and uses equivalent representations for the same simple algebraic expressions. 7A.M.A.V.3 shows and explains how changes in one variable affects other variables. 7A.M.A.V.4 explains the difference between an equation and an expression. 7A.M.A.V.5 solves:

a. one-step linear equations in one variable with positive rational coefficients and solutions. b. two-step linear equations in one variable with rational coefficients and constants and positive rational solutions. c. one-step linear inequalities with counting numbers and one variable explains and uses the equality and inequality symbols (=, ≠, <, ≤, >, ≥) and corresponding meanings (is equal to, is not equal to, is less than, is less than or equal to, is greater than, is greater than or equal to) to represent mathematical relationships with rational numbers.

What Students Need to Know 1. Identify and explain variables. 2. Show and explain changes in

variables. 3. Explain the difference between an

equation and expression. 4. Explain and interpret symbols. 5. Solve two-step linear equations. 6. Solve two-step linear inequalities. 7. Evaluate formulas. 8. Demonstrate and explain the

relationship between ratios, proportions, and percents.

9. Represent solutions sets of linear equations.

What Students Need to Do/Apply 1. Solve real-world problems with one-

or two-step linear equations in one variable with whole number coefficients and constants and positive rational solutions intuitively and analytically.

2. Generate real-world problems that represent one- or two-step linear equations.

3. Explain the mathematical reasoning that was used to solve a real-world problem using a one- or two-step linear equation.

Strategies

♦ Use math models: algeblocks, algebra tiles, Hands on Equations.

Scope and Sequence 6th Graders explain and use variables and or symbols to represent unknown quantities and variable relationships. They solve 1-step linear equations with 1 variable and whole number solutions. They also explain and use equality and inequality symbols. Algebra 1 students will know and explain the use of variables as parameters for a specific variable situation. They will manipulate variable quantities within an equation or inequality. During this course, students will solve:

a. linear equations and inequalities both analytically and graphically without using a calculator.

b. quadratic equations with integer solutions (may be solved by trial and error, graphing, quadratic formula, or factoring).

c. radical equations with no more than one inverse operation around the radical expression.

d. equations where the solution to a rational equation can be simplified as a linear equation with a nonzero denominator.

e. equations and inequalities with absolute value quantities containing one variable with a special emphasis on using a number line and the concept of absolute value.

f. exponential equations with the

1. On Monday it was 12o warmer than Tuesday. If it was 87o on Tuesday, what was the temperature on Monday? a. 89o b. 75o c. 99o d. 100o 2. 3 2 9x y x y+ + − + + is the same as… a. 2 3 9x y+ + b. 4 3 9x y+ + c. 16xy d. 14xy x− 3. Using C dπ= , what is the difference to the nearest whole number in the circumferences if circle A has a diameter of 4m and diameter of circle B is 8m? a. ≈ 11m b. ≈ 13m c. ≈ 15m d. ≈ 17m 4. Write a numerical equation that uses multiplication that shows the number of days in 4 weeks. a. 4 + 7 = 11 b. 4 7 28= c. 4 7 d. 4 + 7




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7A.M.A.V.6 ▲knows the mathematical relationship between ratios, proportions, and percents and how to solve for a missing term in a proportion with positive rational number solutions and monomials. 7A.M.A.V.7 ▲evaluates simple algebraic expressions using positive rational numbers. 7A.M.A.V.8 identifies independent and dependent variables within a given situation. 7A.M.A.V.9 evaluate formulas using substitution. Application Indicators 7A.M.A.V.10 ▲represents real-world problems using variables and symbols to write linear expressions, one- or two-step equations. Vocabulary Students Know and Use variable linear equation expression equation

same base without the aid of a calculator or computer.

Algebra 1 students will be tested upon solving systems of linear equations with two unknowns using integer coefficients and constants. Pacing Considerations 2 weeks, then ongoing

5a. Solve the following equations. 8 40x =

a. x=2 b. x=3 c. x=4 d. x=5

X + 5 = 11 a. x=5 b. x=8 c. x=6 d. x=7

X – 3 = 9 a. x=10 b. x=11 c. x=12 d. x=13

153x=

a. x=5 b. x=-45 c. x=-5 d. x=45 5b. Solve for x. 2x + 3 = 7 a. x=2 b. x=3 c. x=5 d. x=4 5c. Solve for x. 3 12x > a. x=4 b. x=15 c. x=36 d. x=6




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6. ▲Solve the following proportion.

2 56x

=

a. 10 b. 2.4 c. 4.2 d. 5

7. ▲If 32

x = and y= 2, then what does

5xy + 2 equal? a. 15 b. 2 c. 17

d. 1152

8. John has three times as much money as his sister. If M is the amount of money his sister has, what is the equality that represents the amount of money that John has? 9. Today John is 3.25 inches more than half his sister’s height. If J = John’s height, and S = his sister’s height, then J = 0.5S + 3.25.




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Benchmark 3 Functions The student recognizes, describes, and analyzes constant and linear relationships in a variety of situations.


Knowledge Indicators The student… 7A.M.A.F.1 recognizes constant, linear, and non linear relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology. 7A.M.A.F.2 finds the values and determines the rule through two operations using a function table (input/output machine, T-table). 7A.M.A.F.3 demonstrates mathematical relationships using ordered pairs in all four quadrants of a coordinate plane. 7A.M.A.F.4 describes and/or gives examples of mathematical relationships that remain constant. 7A.M.A.F.5 explains the concepts of slope and x- and y-intercepts of a line. 7A.M.A.F.6 recognizes and identifies the graphs of constant and linear functions. 7A.M.A.F.7 identifies ordered pairs from a graph, and/or plots ordered pairs using a variety of scales for the x- and y-axis. Application Indicators No tested applications indicators for this benchmark. Vocabulary Students Know and Use

What Students Need to Know 1. Implement a variety of methods to

recognize and examine constant and linear relationships.

2. Explain and give examples of relationships which remain constant.

3. Construct ordered pairs to demonstrate relationships.

4. Convert between numerical, tabular, graphical, and verbal rules used to represent relationships.

5. Relate functions to equations and graphs.

What Students Need to Do/Apply 1. Interpret, describe, and analyze the

mathematical relationships of numerical, tabular, and graphical representations, including translations between the representations.

Strategies

♦ Use mental math. ♦ Use graphing utilities. ♦ Use concrete materials, i.e.,

coins.

Scope and Sequence 6th Graders recognize linear relationships using various methods including mental math, paper/pencil, and calculators. They find values and determine the rule with one operation using a function table, or a T-chart. They also generalize numerical patterns and state the rule. Using a function table they identify, plot and label ordered pairs using the 4 quadrants on a coordinate plane. Algebra 1 students evaluate and analyze functions using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology. They match equations and graphs of constant and linear functions and quadratic functions limited to y = ax2 + c. Students in Algebra 1 determine whether a graph, list of ordered pairs, table of values, or rule represents a function and determine x- and y-intercepts and maximum and minimum values of the portion of the graph that is shown on a coordinate plane. Students identify domain and range of:

a. relationships given the graph or table.

b. linear, constant, and quadratic functions given the equation(s). They use function notation and evaluate function(s) given a specific domain.

Students describe the difference between independent and dependent variables and

1. Knowing that one person has 2 hands, if there are ten people, how many hands are there?

a. 20 b. 10 c. 12 d. 0

2. Write the rule for the input/output chart using an algebraic expression.

Input Output 1 4 2 8 3 12 4 16

a. x + 3 b. x + 4 c. x + 3x d. x + 2x

3. What are the coordinates of point A and in what quadrant is point A located?

A




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identify independent and dependent variables. A tested indicator has students recognizing how changes in the constant and/or slope within a linear function changes the appearance of a graph. Pacing Considerations One week, then ongoing

4. Which of these is NOT an example of a constant relationship? a. You will get $10 to do a job no

matter how long it takes. b. You will buy 2 pairs of jeans in

August no matter how many times you go to the mall.

c. You will get 8 hours of sleep each night no matter what time you go to bed.

d. You will earn $6.00 an hour no matter how many hours you work.

5. A fish tank is being filled with water with a 2-liter jug. There are already 5 liters of water in the fish tank. Therefore, you are showing how full the tank is as you empty 2-liter jugs of water into it. Y = 2x + 5 (symbolic) can be represented in a table (tabular) –

X Y 0 5 1 7

2 9 3 11

and as a graph (graphical) –




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Standard 3 Geometry The student uses geometric concepts and procedures in a variety of situations.

Benchmark 1 Geometric Figures and Their Properties The student recognizes geometric figures and compares their properties in a variety of situations.


Knowledge Indicators The student… 7A.M.G.GFP.1 recognizes and compares properties of two- and three-dimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology. 7A.M.G.GFP.2 classifies regular and irregular polygons having through ten sides as convex or concave. 7A.M.G.GFP.3 ▲identifies angle and side properties of triangles and quadrilaterals:

a. sum of the interior angles of any triangle is 180°. b. sum of the interior angles of any quadrilateral is 360°. c. parallelograms have opposite sides that are parallel and congruent. d. rectangles have angles of 90°, opposite sides are congruent. e. rhombi have all sides the same length, opposite angles are congruent. f. squares have angles of 90°, all sides congruent. g. trapezoids have one pair of opposite sides parallel and the other pair of opposite sides are not parallel.

7A.M.G.GFP.4 identifies and describes:

a. the altitude and base of a rectangular prism and triangular prism. b. the radius and diameter of a cylinder.

What Students Need to Know 1. Recognize, define, and use

vocabulary and properties of: circles, squares, rectangles, triangles, ellipses, trapezoids, and parallelograms.

2. Classify triangles and polygons. 3. Recognize or apply properties of

corresponding parts of similar and congruent triangles and quadrilaterals.

4. Demonstrate symbols for perpendicular, parallel, and right triangle.

What Students Need to Do/Apply 1. Solve real-world problems by

applying the properties of: a. plane figures (regular and

irregular polygons through 10 sides, circles, and semicircles) and the line(s) of symmetry.

b. solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases.

2. Decompose geometric figures made from regular and irregular polygons through 10 sides, circles, and semicircles: a. nets (two-dimensional shapes

that can be folded into three-dimensional figures).

b. prisms, pyramids, cylinders, cones, spheres, and hemispheres.

Scope and Sequence 6th Graders recognize and compare properties of plane figures and solids. They also name regular and irregular polygons, and recognize all existing lines of symmetry. 6th graders learn to use symbols for angles, line segments, rays, parallel and perpendicular. They can classify triangles. 6th grader learned to define circumference, radius, and diameter of circles and determine the radius and diameter of a circle, given one or the other. Algebra 1 students recognize and compare properties of two-and three-dimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology. They discuses properties of regular polygons related to:

a. angle measures. b. diagonals.

For this course, students will recognize and describe the symmetries (point, line, plane) that exist in three-dimensional figures. Students will recognize that similar figures have congruent angles, and their corresponding sides are proportional. They will use the Pythagorean theorem to:

a. determine if a triangle is a right triangle.

b. find a missing side of a right triangle.

Students will recognize and describes: a. congruence of triangles using:

1. The sides of a certain quadrilateral are unequal in length. Which statement must be true about the quadrilateral?

a. The sum of any 2 angles must be 180°.

b. The sum of all 4 angles must be 360°.

c. The sum of any 3 angles must be 180°.

d. One angle must be 90°.

2. Is this hexagon a regular or non-regular figure? Is this hexagon concave or convex?

3a. ▲ Triangle XYZ is shown below.

What is the measure of ? a. 39° b. 141° c. 165° d. 219°




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7A.M.G.GFP.5 identifies corresponding parts of similar and congruent triangles and quadrilaterals. 7A.M.G.GFP.6 uses symbols for right angle within a figure ( ), parallel (||), perpendicular (⊥), and triangle (Δ) to describe geometric figures. 7A.M.G.GFP.7 classifies triangles as:

a. scalene, isosceles, or equilateral. b. right, acute, obtuse, or equiangular.

7.M.G.GFP.8 determines if a triangle can be constructed given sides of three different lengths. 7A.M.G.GFP.9 generates a pattern for the sum of angles for 3-, 4-, 5-, … n-sides polygons.

7A.M.G.GFP.10 describes the relationship between the diameter and the circumference of a circle. 7A.M.G.GFP.11 uses the Pythagorean theorem to:

a. determine if a triangle is a right triangle. b. find a missing side of a right triangle where the lengths of all three sides are whole numbers.

Application Indicators No tested application indicators for this benchmark. Vocabulary Students Know and Use parallel perpendicular

3. Compose geometric figures made from: a. regular and irregular polygons

through 10 sides, circles, and semicircles.

b. nets (two-dimensional shapes that can be folded into three-dimensional figures).

c. prisms, pyramids, cylinders, cones, spheres, and hemispheres.

Strategies Geometry is the study of shapes, their properties, and their relationships to other shapes. Symbols and numbers are used to describe their properties and their relationships to other shapes. The fundamental concepts in geometry are point (no dimension), line (one-dimensional), plane (two-dimensional), and space (three-dimensional). Plane figures are referred to as two-dimensional. Solids are referred to as three-dimensional. The base, in terms of geometry, generally refers to the side on which a figure rests. Therefore, depending on the orientation of the solid, the base changes.

♦ Use technology (i.e., Geometer’s Sketchpad).

♦ Use manipulatives ( i.e., Tinker Toys, D-Sticks, Polygons, Pattern blocks).

♦ Create drawings and constructions.

Side-Side-Side (SSS), Angle-Side-Angle (ASA), Side-Angle-Side (SAS), and Angle-Angle-Side (AAS).

b. the ratios of the sides in special right triangles: 30°-60°-90° and 45°-45°-90°.

Algebra 1 students will also recognize, describe, and compare the relationships of the angles formed when parallel lines are cut by a transversal. Finally, they will recognize and identify parts of a circle: arcs, chords, sectors of circles, secant and tangent lines, central and inscribed angles. Pacing Considerations Angles, Triangles and Polygons – 1 week Perimeter, circumference, area – 3 weeks

3b. ▲The picture below shows an old-fashioned kite. The dotted lines represent the wooden braces behind the kite’s fabric.

What is the sum of the interior angles of the interior angles of the fabric area of the kite?

3c. ▲The picture below shows parallelogram RSTU and the length, in units, of two of its sides.

What is the length, in units, ofUT ?




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polygons scalene isosceles equilateral right triangle acute triangle obtuse triangle diameter circumference prism trapezoid convex concave

3d. ▲What is the measure of each interior angle of a rectangle?

3e. ▲ Which statement is not true for all rhombi? a. All sides are congruent. b. All angles are congruent. c. Opposite sides are parallel. d. Adjacent angles are supplementary. 3f. ▲Which statement about squares is true? a. squares have exactly 1 right angle. b. squares have exactly 2 right angles. c. squares have exactly 2 right angles. d. squares have exactly 4 right angles. 3g. ▲The shape shown below is not a trapezoid.

The reason it is not a trapezoid is that it has a. angles that sum to 360° b. 2 pairs of adjacent sides that are perpendicular c. 2 pairs of opposite sides that are parallel d. congruent adjacent angles

4a. Label the base and altitude of the rectangular prism.

4b. Find the diameter of this cylinder.

5. Determine whether the figures are similar. Write yes or no, and support your answer. 6. Describe the relationship between

.AB andCDsuur suur

7a. Classify the triangle according to the lengths of its sides. 35m, 25m, 15m 7b. Classify the triangle according to the measures of the angles, 90, 20, 70.

8. Can you create a triangle with the side lengths of 10 cm, 5cm, and 1cm? 9. Complete the table.

10. Mr. Pelak’s lawn sprinkler waters a circular region of lawn with a radius of 10 ft. What is the area watered? 11a. A triangle has sides with lengths of 10, 12, and 15. Is it a right triangle? Use the Pythagorean Theorem to justify your answer. 11b. Find the length of the missing side in this right triangle.




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Benchmark 2 Measurement and Estimation The student estimates, measures, and uses measurement formulas in a variety of situations.


Knowledge Indicators The student… 7A.M.G.ME.1 determines and uses rational number approximations (estimations) for length, width, weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure. 7A.M.G.ME.2 selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, area, and angle measurements. 7A.M.G.ME.3 converts within the customary system and within the metric system. 7A.M.G.ME.4 ▲ knows and uses perimeter and area formulas for circles, squares, rectangles, triangles, and parallelograms. 7A.M.G.ME.5 finds perimeter and area of two-dimensional composite figures of circles, squares, rectangles, and triangles. 7A.M.G.ME.6 ▲ uses given measurement formulas to find:

a. surface area of cubes. b. volume of rectangular prisms.

7A.M.G.ME.7 finds surface area of rectangular prisms using concrete objects.

What Students Need to Know 1. Demonstrate various estimation

techniques to determine rational number approximations of length, volume, surface area, area, perimeter, weight, capacity, temperature, or time.

2. Recognize, state, and use measurement formulas for perimeter, area, and volume.

3. Find area and perimeter of 2-D figures composed of squares, rectangles, triangles, and circles.

4. Apply various measurement techniques and appropriate tools to find accurate rational number representations for length, volume, surface area, perimeter, weight, temperature, and time.

What Students Need to Do/Apply 1. Solve real-world problems by:

a. converting within the customary and metric systems.

b. finding perimeter and area of circles, squares, rectangles, triangles, and parallelograms.

c. using appropriate units to describe rate as a unit of measure.

d. finding missing angle measurements in triangles and quadrilaterals.

Strategies

♦ Use appropriate tools: rulers (standard and metric),

Scope and Sequence 6th Grader were tested upon the metric system. They converted in the metric system using prefixes: kilo, hector, deka, deci, centi, and milli. They were also tested upon solving real-world problems by applying measurement formulas for perimeter of polygons, and the area of squares, rectangles, and triangles using the same unit of measure. 6th grader found the volume of rectangular prisms. Algebra 1 students will determine and use real number approximations (estimations) for length, width, weight, volume, temperature, time, distance, perimeter, area, surface area, and angle measurement using standard and non-standard units of measure. They will select and use measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate real number representations for length, weight, volume, temperature, time, distance, area, surface area, mass, midpoint, and angle measurements. For this course, students will approximate conversions between customary and metric systems given the conversion unit or formula. Students will also state, recognize, and apply formulas for perimeter and area of squares, rectangle, and triangles; circumference and area of circles; and volume of rectangular solids. Algebra 1 students will use given measurement formulas to find perimeter,

1. Using a rectangular prism estimate its length, width, perimeter, and volume using a centimeter scale. If the temperature drops 3.2 degrees per minute, estimate how long it will take to drop 9 degrees. 2. What would be the most accurate to measure the length of your math textbook meters, inches, or feet? Estimate the weight of your math textbook in grams. How would you estimate the volume of a glass of milk in cubic centimeters or square feet? 3. 2 yards is equivalent to how many feet? 2 meters is equivalent to how many centimeters? 4. ▲Lisa has a string and is wrapping it around a box, as shown below.




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7A.M.G.ME.8 uses appropriate units to describe rate as a unit of measure. 7A.M.G.ME.9 finds missing angle measurements in triangles and quadrilaterals. 7A.M.G.ME.10 uses given measurement formulas to find:

a. area of parallelograms and trapezoids. b. surface area of rectangular prisms, triangular prisms, and cylinders. c. volume of rectangular prisms, triangular prisms, and cylinders.

Application Indicators 7A.M.G.ME.11 ▲■ solves real-world problems by finding perimeter and area of two-dimensional composite figures of squares, rectangles, and triangles. Vocabulary Students Know and Use perimeter circumference area cube prism composite figure

protractors. ♦ Use manipulatives: pattern

blocks, cubes, tiles, thermometers, scales, and clocks.

area, volume, and surface area of two- and three-dimensional figures (regular and irregular). They will recognize and apply properties of corresponding parts of similar and congruent figures to find measurements of missing sides. Finally, students will know, explain, and use ratios and proportions to describe rates of change. Pacing Considerations Perimeter, circumference, area – 3 weeks

Lisa’s string wraps around the box exactly 3 times. How ling is her string?

5. A rectangular garden has a triangular patio next to it, as shown below.

What is the area, in square feet (ft.2of both the garden and the patio combines?




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6a. ▲ What is the total surface area in square meter ( m2 ) of a cube that measures 1.2 meters on each side? SA=6s2

6b. ▲A movie theater uses a rectangular bag to hold popcorn. The bag is 9inches (in.) high, 4 in. wide, and 6 in. long. What is the volume of the popcorn bag? (V = lwh).

7. Find the surface area of the cube.

Find the volume.

11 ft

8. A car can go 500 miles on a tank of gas. How far will it go if you have a 20 gallon gas tank?

11. ▲The front of a barn is rectangular in shape with a height of 10 feet and a width of 48 feet. Above the rectangle is a triangle that is 7 feet high with sides 25 feet long. What is the area of the front of the barn?




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Benchmark 3 Transformation Geometry The student recognizes and performs transformations on two- and three-dimensional geometric figures in a variety of situations.


Knowledge Indicators The student… 7A.M.G.TG.1 identifies, describes, and performs single and multiple transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on a two-dimensional figure. 7A.M.G.TG.2 identifies three-dimensional figures from various perspectives (top, bottom, sides, corners). 7A.M.G.TG.3 draws three-dimensional figures from various perspectives (top, bottom, sides, corners). 7A.M.G.TG.4 generates a tessellation. Application Indicators 7A.M.G.TG.5 ▲■ determines the actual dimensions and/or measurements of a two-dimensional figure represented in a scale drawing. Vocabulary Students Know and Use reflection rotation translation reduction magnification tessellation

What Students Need to Know 1. Recognize, describe, and perform

single and multiple transformations on 2-D figures.

2. Recognize and draw 3-D shapes as they would appear from a variety of visual perspectives.

3. Create a tessellation. What Students Need to Do/Apply 1. Describe the impact of

transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on the perimeter and area of squares and rectangles.

2. Investigate congruency and similarity of geometric figures using transformations.

Strategies Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure to a new location. To help students form images of shapes through different transformations, students can use concrete objects, figures drawn on graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do not change the figure itself. Other transformations like reduction (contraction/shrinking) or enlargement (magnification/growing)

Scope and Sequence 6th Graders were tested upon identifying and performing 1 or 2 transformations on 2-D figures. They also reduced and enlarged simple shapes with simple scale factors and also recognize 3-D figures from various perspectives. 6th graders also worked with tessellated figures. Algebra 1 students will describe and perform single and multiple transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on two- and three-dimensional figures. They will recognize a three-dimensional figure created by rotating a simple two-dimensional figure around a fixed line and generate a two-dimensional representation of a three-dimensional figure. For this course, students will determine where and how an object or a shape can be tessellated using single or multiple transformations and create a tessellation. Pacing Considerations One week

1. Reflect about the x-axis.

2. Name the figures you see from the top view and the side view of the cylinder. 3. Name the figure you would see from the following views.




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change the size of a figure, but not the shape (congruence vs. similarity).

♦ Use scale drawings to determine actual dimensions.

♦ Use technology (virtual manipulatives).

♦ Use manipulatives (i.e., Patterns Blocks, cubes, Point of View cards).

♦ Cooperative groups: Build It Activities, Spatial Visualization book, Middle School Math, Transformation Game)

4. Cut the basic unit out of graph paper. Use it to make a tessellation of at least two rows. 5. ▲




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Standard 3 Geometry The student uses geometric concepts and procedures in a variety of ways.

Benchmark 4 Geometry from and Algebraic Perspective The student related geometric concepts to a number line and a coordinate plane in a variety of situations.


Knowledge Indicators The student… 7A.M.G.AP.1 finds the distance between the points on a number line by computing the absolute value of their difference. 7A.M.G.AP.2 uses all four quadrants of a coordinate plane to:

a. identify in which quadrant or on which axis a point lies when given the coordinates of a point. b. plot points. c. identify points. d. list through five ordered pairs of a given line.

7A.M.G.AP.3 uses a given linear equation with whole number coefficients and constants and a whole number solution to find the ordered pairs, organize the ordered pairs using a T-table, and plot the ordered pairs on the coordinate plane. 7A.M.G.AP.4 examines characteristics of two-dimensional figures on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology. 7A.M.G.AP.5 uses the coordinate plane to:

a. list several ordered pairs on the graph of a line and find the slope of the line. b. recognize that ordered pairs that lie on the graph of an equation are

What Students Need to Know 1. Demonstrate a variety of methods to

analyze 2-D geometry on the coordinate system.

2. Utilize the coordinate plane to: a. Identify in which quadrant or on

which axis a point lies when given an ordered pair.

b. Graph or identify points on the coordinate plane in all four quadrants.

c. Determine if a given point is on the line given the graph of the line.

d. List coordinate pairs on the graph of a line and describe existing patterns.

e. Find distance between points on a number line.

What Students Need to Do/Apply 1. Represent and/or generates real-

world problems using a coordinate plane to find: a. perimeter of squares and

rectangles. b. circumference (perimeter) of

circles. c. area of circles, parallelograms,

triangles, squares, and rectangles.

Strategies A number line (a mathematical model) is a diagram that represents numbers with equal distances marked off as points on a line, and is an example of one-to-one

Scope and Sequence 6th Graders organized integer data using a T-table and plot ordered pairs in all 4 quadrants. They were tested upon using all 4 quadrants of the coordinate plane to identify the ordered pairs of integer values on a given graph and plot the ordered pairs of integer values. 6th graders used a number line to order integers and positive rational numbers. Algebra 1 students will recognize and examine two- and three-dimensional figures and their attributes including the graphs of functions on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology. They will determine if a given point lies on the graph of a given line or parabola without graphing and justifies the answer. Students will calculate the slope of a line from a list of ordered pairs on the line and explain how the graph of the line is related to its slope. They will use the Pythagorean theorem to find distance (may use the distance formula) and recognize the equation y = ax2 + c as a parabola; represents and identifies characteristics of the parabola including opens upward or opens downward, steepness (wide/narrow), the vertex, maximum and minimum values, and line of symmetry; and sketches the graph of

1. What is the distance between point A and point B? -2 -1 0 1 2 3 4 2a-c. In which quadrant are points A, B, and C located? Write the ordered pairs for points A, B, and C. Plot the following points on the coordinate plane. (3, 2), (-2, 1), and (-4, -2)

2d. List five ordered pairs that create a line that passes through (3, 1).

A B




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solutions to that equation. c. recognize that points that do not lie on the graph of an equation are not solutions to that equation. d. determine the length of a side of a figure drawn on a coordinate plane with vertices having the same x- or y-coordinates.

Application Indicators No tested application indicators for this benchmark. Vocabulary Students Know and Use quadrant coordinate linear equation ordered pairs

correspondence (a relation). A number line can be used as a visual representation of numbers and operations. In addition, a number line used horizontally and vertically is a precursor to the coordinate plane; and the distance between two numbers on a number line is a precursor to absolute value.

A coordinate plane (coordinate grid) consists of a horizontal number line called the x-axis and a vertical number line called the y-axis. These two lines intersect at a point called the origin. The x-axis and the y-axis divide the plane into four sections called quadrants. Any point on the coordinate plane can be named with two numbers called coordinates. The first number is the x-coordinate. The second number is the y-coordinate. Since the pair is always named in order (first x, then y), it is called an ordered pair.

♦ Use number lines. ♦ Use calculators (when

appropriate). ♦ Use manipulatives (i.e.,

Geoboards, maps, Battleship game).

the parabola. This is also a tested indicator. Algebra 1 students will explain the relationship between the solution(s) to systems of equations and systems of inequalities in two unknowns and their corresponding graphs. Also tested is finding and explaining the relationship between the slopes of parallel and perpendicular lines and recognizing the equation of a line and transforms the equation into slope-intercept form in order to identify the slope and y-intercept and using this information to graph the line. Pacing Considerations Graphing – 1 week

3. Make a table of values for the equation and plot the points on a coordinate plane. y = 3x – 2

4. Using a coordinate plane Jack started at the school (1, 2), traveled to the post office (3 ½, 2), then went by the fire station (3 ½, 3), then visited the park (1, 3), and finally returned to the school. Determine the distance Jack traveled.




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5a. Graph the following equation by setting up a table of values. 2x + y = 3 Also, find the slope of the line.

5b. Graph the following ordered pairs, and draw in the line: (1, 2) (0, 1) (-1, 0). Is (5, 6) a solution to the equation of this line?

5c. Graph the following ordered pairs, and draw in the line: (-1, 0) (0, -3) (1, -6). Is (-3, 2) a solution to the equation of this line?

5d. Use the vertices given to find the dimensions of this rectangle. (0, 2) (5, 2) (0, -3) (5, -3)




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Standard 4 Data Analysis The student uses concepts and procedures of data analysis in a variety of way.

Benchmark 1 Probability The student applies the concepts of probability to draw conclusions, generate convincing arguments, and make predictions and decisions including the use of concrete objects in a variety of situations.


Knowledge Indicators The student… 7A.M.D.P.1 finds the probability of a compound event composed of two independent events in an experiment or simulation. 7A.M.D.P.2 explains and gives examples of simple or compound events in an experiment or simulation having probability of zero or one. 7A.M.D.P.3 uses a fraction, decimal, and percent to represent the probability of:

a. a simple event in an experiment or simulation. b. a compound event composed of two independent events in an experiment or simulation.

7A.M.D.P.4 finds the probability of a simple event in an experiment or simulation using geometric models. Application Indicators None tested application indicators for this benchmark. Vocabulary Students Know and Use compound event independent event simple event simulation

What Students Need to Know 1. Apply concepts of probability to

generate convincing arguments. 2. Draw conclusions and make

decisions in a variety of situations. What Students Need to Do/Apply 1. Conduct an experiment or simulation

with a compound event composed of two independent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions and make predictions about future events.

2. Analyze the results of an experiment or simulation of a compound event composed of two independent events to draw conclusions, generate convincing arguments, and make predictions and decisions in a variety of real-world situations.

3. Compare results of theoretical (expected) probability with empirical (experimental) probability in an experiment or situation with a compound event composed of two simple independent events and understands that the larger the sample size, the greater the likelihood that the experimental results will equal the theoretical probability.

4. Make predictions based on the theoretical probability of a simple event in an experiment or simulation.

Scope and Sequence 6th Graders were tested on listing all possible outcomes of an experiment or simulation with a compound event composed of 2 interdependent events in a clear and organized way. They were also tested on representing the probability of a simple event in an experiment or simulation using fractions and decimals. They learned that all possibilities range from 0 through 1 and can be written as a fraction, decimal, or percent. Algebra 1 students will find the probability of two independent events in an experiment, simulation, or situation and find the conditional probability of two dependent events in an experiment, simulation, or situation. Students will be tested upon explaining the relationship between probability and odds and computes one given the other. Pacing Considerations Probability – 1 week

1. What is the probability of getting two heads, if you toss a dime and a quarter? 2. What’s the probability of rolling a counting number less than seven on a six sided die? 3a. Using a 5 count spinner, what’s the probability of getting a 1 or 2? Write your answer as a fraction, decimal, and percent. 3b. Joe has 3 shirts, red, yellow, and blue and 4 ties, striped, solid, checked and geometric. What is the probability of choosing a red shirt and a striped tie? 4. What is the probability of landing on a 2 with a 4 count spinner? Write your answer as a decimal, fraction, and percent.




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Strategies Ideas from probability reinforce concepts in the other Standards, especially Number and Computation and Geometry. Students need to develop an intuitive concept of chance – whether or not something is unlikely or likely to happen. Probability experiences should be addressed through the use of concrete objects (process models); spinners, number cubes, or dartboards (geometric models); and coins (money models). Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments. Some examples of uses of probability in every day life include: There is a 50% chance of rain today. What is the probability that the team will win every game?

♦ Use virtual manipulatives. ♦ Use spinners and other math

models.




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Standard 4 Data Analysis The student uses concepts and procedures of data analysis in a variety of situations.

Benchmark 2 Statistics The student collects, organizes, displays, and explains numerical (rational numbers) and non-numerical data sets in a variety of situations with a special emphasis on measures of central tendency.


Knowledge Indicators The student… 7A.M.D.S.1 ▲ organizes, displays, and reads quantitative (numerical) and qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays:

a. frequency tables. b. bar, line, and circle graphs. c. Venn diagrams or other pictorial displays. d. charts and tables. e. stem-and-leaf plots (single). f. scatter plots. g. box-and-whiskers plots.

7A.M.D.S.2 selects and justifies the choice of data collection techniques (observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, or purposeful sampling) in a given situation. 7A.M.D.S.3 conducts experiments with sampling and describes the results. 7A.M.D.S.4 determines the measures of central tendency (mode, median, mean) for a rational number data set. 7A.M.D.S.5 identifies and determines the range and the quartiles of a rational number data set.

What Students Need to Know 1. Generate, organize, and interpret

rational numbers and other data in a variety of situations.

2. Apply measures of central tendency when drawing conclusions for the data.

What Students Need to Do/Apply 1. Use data analysis (mean, median,

mode, range) of a rational number data set to make reasonable inferences and predictions, to analyze decisions, and to develop convincing arguments.

2. Explain advantages and disadvantages of various data displays for a given data set.

3. Determine and explain the advantages and disadvantages of using each measure of central tendency and the range to describe a data set.

Strategies

♦ Use Excel to create graphs, etc.

Scope and Sequence 6th Graders were taught to use scatter plots. They used statistical measures to determine mean, median, mode, and range with whole numbers and decimals. 6th graders also selected and justified the choice of data collection techniques. At this grade, sampling was used to collect data and describe the results. Algebra 1 students will organize, display, and read quantitative (numerical) and qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays:

a. frequency tables. b. bar, line, and circle graphs. c. Venn diagrams or other pictorial

displays. d. charts and tables. e. stem-and-leaf plots (single and

double). f. scatter plots. g. box-and-whiskers plots. h. histograms.

They will explain how the reader’s bias, measurement errors, and display distortions can affect the interpretation of data. For this course, students will calculate and explain the meaning of range, quartiles and interquartile range for a real number data set. Students will compare and contrast the dispersion of two given sets of data in terms of range

1a. ▲




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7A.M.D.S.6 identifies potential outliers within a set of data by inspection rather than formal calculation. Application Indicators 7A.M.D.S.7 ▲recognizes and explains:

a. ■ misleading representations of data. b. the effects of scale or interval changes on graphs of data sets.

Vocabulary Students Know and Use

and the shape of the distribution including:

a. symmetrical (including normal). b. skew (left or right). c. bimodal. d. uniform (rectangular).

Tested indicators include explaining the effects of outliers on the measures of central tendency (mean, median, mode) and range and interquartile range of a real number data set and approximating a line of best fit given a scatter plot and makes predictions using the equation of that line. Pacing Considerations Two weeks

1b. ▲Which bar graph shows that the daily high temperature was greater on Wednesday than it was on Monday or Friday?




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1d. ▲Which team sport was the most popular? How many students liked it?

1e. ▲Name the intervals in the stem-and- leaf plot.

Use the stem-and-leaf plot shown below:

a. What is the lowest test score? The highest test score? The range of scores? b. In what interval, shown as a stem, do most of the scores occur? c. Were there more test scores in the sixties or in the eighties? d. Were more than half the scores less than 78? Explain.

1f. ▲The scatter plot below shows the relationship between two variables.

The y-axis scale is changed to be 0 to 50. Which statement describes the effect this scale change has on the appearance of the scatter plot? a. There is no effect on the appearance of the scatter plot. b. The spread of the data points will widen to cover the entire scatter plot. c. The spread of the data points will become more concentrated in the top half of the scatter plot. d. The spread of the data points will become more concentrated in the bottom half of the scatter plot.

1g. ▲Use the box-and-whisker graph below, which shows the distribution of money spent at a grocery store.

What is the median amount spent? What are the greatest and least amounts spent? What is the range of money spent? Do most of the shoppers spend more than $30? Explain. 2. Make a bar graph with the given data. Choose between a line graph and a bar graph.

3. Last month, 2 out of every 12 customers of the Sweat Shoppe bought a treadmill. The owner estimates that about 8 of his first 41 customers this month will buy a tread-mill. Design a simulation to check his estimate. 4. The weekly mileage totals for runners are listed below. Determine the mode, median, and mean for the data set.

10, 60, 55, 15, 20, 25, 50, 30, 35, 40, 45 5. Using the data set below determine the range and the quartiles. 9, 10, 12, 13, 8, 9, 31, 9

1c. ▲A poll of 100 people was taken to find which political candidate was preferred, L or M. The poll showed that 8 people did not prefer either candidate, 11 people preferred both candidates equally, 41 people preferred only candidate L, and 40 people preferred only candidate M. Which Venn diagram correctly shows the results of the poll? A.

B.

C.

D.




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Standard Problem Solving Apply previously acquired knowledge, skills, and understandings in new and unfamiliar situations.

Benchmark Applications for Numbers and Computation, Benchmarks 1-2-3-4, applications for Algebra, Benchmarks 1-2-3-4, applications for Geometry, Benchmarks 1-2-3-4, Benchmarks for Data analysis, Benchmarks 1-2


Knowledge & Application Indicators The student . . . 7A.M.PS.A.1 solves problems using a variety of strategies and communicates the problem solving process and results. Vocabulary Students Know and Use problem solving solution process applied solve strategy Polya plan

What Students Need to Know 1. Choose appropriate strategies. 2. Apply strategies to find an answer that

can be defended. 3. Practice the level of difficulty that is

appropriate for outcomes studied in this course.

What Students Need to Do/Apply 1. Demonstrate 10 strategies of

problem solving: guess and check, draw a picture, look for a pattern, make a model, act it out, use easier numbers, make an organized list, make a table or chart, use logic, work backwards.

2. Apply George Polya plan of problem solving: read problem carefully, devise a plan, carry out the plan, and check the work.

Strategies

♦ Use cooperative learning. ♦ Use technology. ♦ Problem solve daily.

Scope and Sequence 6th Graders are in the process of mastering 10 problem solving strategies. They use the 4-step Polya Plan and problem solve daily. The problems they solve on an on-going basis include word, process, puzzle, and applied. Problem-solving occurs at all grade levels. Pacing Considerations Daily problem solving should occur on an on-going basis and continue through the school year.

♦ Jenny was thinking of two numbers. Jenny said that the product of the two numbers was 0. What could you deduct from this statement? Explain your reasoning.

♦ The total price (P) of a car, including

tax (T), is $14, 685. 33. If the tax is $785.42, what is the sale price of the car (S)?

♦ If 5 candy bars cost $1.00, what does

one candy bar cost? Explain your reasoning.

♦ In the store everything is 25% off.

When calculating the discount, which representation of 25% would you use and why?

♦ A goat is staked out in a pasture with

a rope that is 7 feet long. The goat needs 200 square feet of grass to graze. Does the goat have enough pasture? If not, how long should the rope be?

Documents

Grade 7 Advanced Math Curriculum updated 9-30-05teachers.olatheschools.com/~kwilliamsirc/files/Curriculum/grades 6... · Grade 7 Advanced Math Curriculum ... the numbers does not