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BEDFORD COUNTY PUBLIC SCHOOLSAdvanced 6 th Grade Mathematics Curriculum Guide

SOLs taught (Click on SOL to find it in the pacing guide.)1st 9 weeks 6.1, 6.2, 6.3, 6.5,6.7,6.17,6.19,7.1,7.2,7.3,7.4,7.12,7.16,8.15c2nd 9 weeks 6.4, 6.6, 6.8, 6.14,6.18,7.11, 7.13, 7.14, 7.15, 8.15a3rd 9 weeks 6.9, 6.10, 6.12, 6.13, 6.15, 7.7, 8.104th 9 weeks 6.11, 6.16,6.20, 7.9, 7.10,8.16

1st nine weeks

Dates Standards Essential Knowledge/Skills/Understanding Resources2 days Standard 6.1 - The student

will describe and compare data, using ratios, and will use appropriate notations, such as , a to b, and a:b.

Prerequisite/Extension SOL

7.4 single and multistep practical problems with proportional reasoning

What is a ratio?

A ratio is a comparison of any two quantities. A ratio is used to represent relationships within a set and between two sets. A ratio can be written using fraction form( ), a colon (2:3), or the word to (2 to 3).

Describe a relationship within a set by comparing part of the set to the entire set.

Describe a relationship between two sets by comparing part of one set to a corresponding part of the other set.

Describe a relationship between two sets by comparing all of one set to all of the other set.

Describe a relationship within a set by comparing one part of the set to another part of the same set.

Represent a relationship in words that makes a comparison by

using the notations , a:b, and a to b.

Create a relationship in words for a given ratio expressed symbolically.

Key VocabularyRatioCorresponding partFractionCompare

DOE Lesson Plans

Textbook ResourcesPH 7-1, 7-1b

Resources and Links

Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources2 days Standard 7.4 The student will

solve single-step and multistep The student will use problem solving, mathematical communication, mathematical reasoning, connections, and

Key VocabularyRatio

Virginia Department of Education Mathematical Instructional Resources

http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-1.pdf


practical problems, using proportional reasoning.

Essential Questions

What makes two quantities proportional? Two quantities are proportional when one quantity is a constant multiple of the other.

Prerequisite/Extension SOLSOL 6.7SOL 8.3

representations to

Write proportions that represent equivalent relationships between two sets.

Solve a proportion to find a missing term.

Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may be used.

Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths. Calculators may be used.

Using 10% as a benchmark, mentally compute 5%, 10%, 15%, or 20% in a practical situation such as tips, tax and discounts.

Solve problems involving tips, tax, and discounts. Limit problems to only one percent computation per problem.

Corresponding partFractionCompare

DOE Lesson Plans

Textbook ResourcesPH2: 5-2, 5-3, 5-4, 5-5, 5-6

Resources and Links

Dates Standards Essential Knowledge/Skills/Understanding Resources8 days Standard 6.2 –

The student willinvestigate and describe fractions, decimals, and percents as ratios;b) identify a given

fraction, decimal, or percent from a representation;

c) demonstrate

What is the relationship among fractions, decimals and percents?

Fractions, decimals, and percents are three different ways to express the same number. A ratio can be written using fraction form ( ), a colon (2:3), or the word to (2 to 3). Any number that can be written as a fraction can be expressed as a terminating or repeating decimal or a percent.

Identify the decimal and percent equivalents for numbers written in fraction form including repeating decimals.

Represent fractions, decimals, and percents on a number line.

Describe orally and in writing the equivalent relationships among

Key VocabularyFractionDecimalPercentNumeratorDenominatorEquivalentPer cent (per 100) prefixTerminating decimalRepeating decimal

http://www.doe.virginia.gov/testing/solsearch/sol/math/7/mess_7-4_1.pdf


equivalent relationships among fractions, decimals, and percents; and

d) compare and order fractions, decimals, and percents.


5.2a) recognize/name fractions in their equivalent decimal form and vice versa; b) compare/order fractions and decimals.

7.1b) determine scientific notation for numbers greater than 0 c) compare/order fract/dec/percents, and scientific notation e) identify/describe absolute value for rational #’s

decimals, percents, and fractions that have denominators that are factors of 100.

Represent, by shading a grid, a fraction, decimal, and percent.

Represent in fraction, decimal, and percent form a given shaded region of a grid.

Compare two decimals through thousandths using manipulatives,

pictorial representations, number lines, and symbols (<,¿ ,≥,>, =).

Compare two fractions with denominators of 12 or less using manipulatives, pictorial representations, number lines, and

symbols (<,¿ ,≥,>, =).

Compare two percents using pictorial representations and symbols

(<,¿ ,≥,>, =).

Order no more than 3 fractions, decimals, and percents (decimals through thousandths, fractions with denominators of 12 or less), in ascending or descending order.

DOE Lesson Plans

Textbook ResourcesPH 1-5, 1-6, 4-5, 4-8, 7-1, 7-6

Resources and Links

Dates Standards Essential Knowledge/Skills/Understanding Resources2 days Standard 6.3 - The student will

a) identify and represent integers;

b) order and compare integers; and

c) identify and describe absolute value of integers.

What role do negative integers play in practical situations?

Some examples of the use of negative integers are found in temperature (below 0), finance (owing money), below sea level. There are many other examples.

How does the absolute value of an integer compare to the absolute value of its opposite?

They are the same because an integer and its opposite are the same

Key VocabularyIntegerAbsolute valuePositive (integer)Negative (integer)Opposite

DOE Lesson Plans

http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-3abc.pdf

http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-2bcd.pdf


Prerequisite/Extension SOL5.18 c) model one-step linear equations using add/sub

7.3 a) model operations (add, sub, mult, div) with integers

distance from zero on a number line.

Identify an integer represented by a point on a number line.

Represent integers on a number line.

Order and compare integers using a number line.

Compare integers, using mathematical symbols (<, >, =).

Identify and describe the absolute value of an integer.

Textbook ResourcesPH 11-1, 11-2

Resources and Links

Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources5 days Standard 7.3

The student willa) model addition,

subtraction, multiplication and division of integers; and

b.) add, subtract, multiply, and divide integers.

Essential Questions

The sums, differences, products and quotients of integers are either positive, zero, or negative. How can this be demonstrated?This can be demonstrated through the use of patterns and models.


The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

Model addition, subtraction, multiplication and division of integers using pictorial representations of concrete manipulatives.

Add, subtract, multiply, and divide integers.

Simplify numerical expressions involving addition, subtraction, multiplication and division of integers using order of operations.

Solve practical problems involving addition, subtraction, multiplication, and division with integers.

Key VocabularyIntegerAbsolute valuePositive (integer)Negative (integer)Opposite

DOE Lesson Plans

Textbook ResourcesPH2 1-6, 1-7, 1-8

Resources and Links

http://www.doe.virginia.gov/testing/solsearch/sol/math/7/mess_7-3ab.pdf


Dates Standards Essential Knowledge/Skills/Understanding Resources1 day Standard 6.19 - The student will

investigate and recognizea) the identity properties for

addition and multiplication;

b) the multiplicative property of zero; and

c) the inverse property for multiplication.


5.19 – distributive property of multiplication over addition

7.16 a) apply properties with real numbers: commutative and associative properties for add/mult; b) distributive property; c) additive/multiplicative identity properties; d) additive/multiplicative inverse properties; e) multiplicative property of zero

How are the identity properties for multiplication and addition the same? Different? For each operation the identity elements are numbers that combine with other numbers without changing the value of the other numbers. The additive identity is zero (0). The multiplicative identity is one (1).

What is the result of multiplying any real number by zero?The product is always zero.

Do all real numbers have a multiplicative inverse? No. Zero has no multiplicative inverse because there is no real number that can be multiplied by zero resulting in a product of one.

Identify a real number equation that represents each property of operations with real numbers, when given several real number equations.

Test the validity of properties by using examples of the properties of operations on real numbers.

Identify the property of operations with real numbers that is illustrated by a real number equation.

NOTE: The commutative, associative and distributive properties are taught in previous grades.

Key VocabularyInverseIdentityProperty

DOE Lesson Plans

Textbook ResourcesPH 1-3

Resources and Links




apply the following properties of operations with real numbers:

a) the commutative and associative properties for addition and multiplication;b) the distributive property;c) the additive and multiplicative identity properties;d) the additive and multiplicative inverse properties; ande) the multiplicative property of zero.

Essential Questions

Why is it important to apply properties of operations when simplifying expressions? Using the properties of operations with real numbers helps with understanding mathematical relationships.

Prerequisite/Extension SOLSOL 6.19SOL 8.15c


Identify properties of operations used in simplifying expressions.

Apply the properties of operations to simplify expressions.


DOE Lesson Plans

Textbook ResourcesPH2: 1-9

Resources and Links

http://www.doe.virginia.gov/testing/solsearch/sol/math/7/mess_7-16abcde.pdf


Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources1 day Standard 8.15c The student will

c) identify properties of operations used to solve an equation.

Essential Questions


Identify properties of operations used to solve an equation from among:

- the commutative properties of addition and multiplication;- the associative properties of addition and multiplication;- the distributive property;- the identity properties of addition and multiplication;- the zero property of multiplication;- the additive inverse property; and- the multiplicative inverse property.


DOE Lesson Plans

Textbook Resources

Resources and Links

Dates Standards Essential Knowledge/Skills/Understanding Resources1 days Standard 6.7 - The student

will solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals.

Prerequisite/Extension SOL5.4 create/solve single/multi-step practical problems involving add/sub/mult/div of whole numbers

What is the role of estimation in solving problems? Estimation gives a reasonable solution to a problem when an exact answer is not required. If an exact answer is required, estimation allows you to know if the calculated answer is reasonable.

Solve single-step and multistep practical problems involving addition, subtraction, multiplication and division with decimals expressed to thousandths with no more than two operations.

Key VocabularyEstimateDecimal point

DOE Lesson Plans

Textbook ResourcesPH 1-7, 1-8, 1-9PH2 1-2, 1-3, 1-4

Resources and Links


http://www.doe.virginia.gov/testing/solsearch/sol/math/8/mess_8-15c.pdf


7.4 single and multistep practical problems with proportional reasoning

Dates Standards Essential Knowledge/Skills/Understanding Resources3 days and ongoing throughout the year

Standard 6.5 The student will investigate and describe concepts of positive exponents and perfect squares.

Prerequisite/Extension SOL7.1 a) investigate/describe negative exponents; d) determine square roots

What does exponential form represent?Exponential form is a short way to write repeated multiplication of a common factor such as

5 x 5 x 5 x 5 = 54

.

What is the relationship between perfect squares and a geometric square?

A perfect square is the area of a geometric square whose side length is a whole number.

Recognize and describe patterns with exponents that are natural numbers, by using a calculator.

Recognize and describe patterns of perfect squares not to exceed

202

, by using grid paper, square tiles, tables, and calculators.

Recognize powers of ten by examining patterns in a place value

chart: 104 = 10,000, 103 = 1000, 102 = 100, 101 = 10, 100

=1.

Key VocabularyExponentPerfect squareBasePowerSquaredCubed

DOE Lesson Plans


Resources and Links

Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources2 days Standard 7.1The student will

a) investigate and describe the concept of negative exponents for powers of ten;


Key VocabularyFractionDecimalPercent



b) determine scientific notation for numbers greater than zero;c) compare and order fractions, decimals, percents and numbers written in scientific notation;d) determine square roots; ande) identify and describe absolute value for rational numbers.

Essential Questions When should scientific notation

be used?Scientific notation should be used whenever the situation calls for use of very large or very small numbers.

How are fractions, decimals and percents related?Any rational number can be represented in fraction, decimal and percent form.

What does a negative exponent mean when the base is 10?A base of 10 raised to a negative exponent represents a number between 0 and 1.

How is taking a square root different from squaring a number?Squaring a number and taking a square root are inverse

Recognize powers of 10 with negative exponents by examining patterns.

Write a power of 10 with a negative exponent in fraction and decimal form.

Write a number greater than 0 in scientific notation.

Recognize a number greater than 0 in scientific notation.

Compare and determine equivalent relationships between numbers larger than 0 written in scientific notation.

Represent a number in fraction, decimal, and percent forms.

Compare, order, and determine equivalent relationships among fractions, decimals, and percents. Decimals are limited to the thousandths place, and percents are limited to the tenths place. Ordering is limited to no more than 4 numbers.

Order no more than 3 numbers greater than 0 written in scientific notation

Determine the square root of a perfect square less than or equal to 400.

Demonstrate absolute value using a number line.

Determine the absolute value of a rational number.

Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle to solve practical problems.†

NumeratorDenominatorEquivalentPer cent (per 100) prefixTerminating decimalRepeating decimalExponentPerfect squareBasePowerSquaredCubed

DOE Lesson Plans

Textbook ResourcesPH2 2-8, 2-4, 2-6a, 8-6

Resources and Links

http://www.doe.virginia.gov/testing/solsearch/sol/math/7/mess_7-1d.pdf


operations.

Why is the absolute value of a number positive?The absolute value of a number represents distance from zero on a number line regardless of direction. Distance is positive.



identify and extend geometric and arithmetic sequences.


5.17 describe/express the relationship in number patterns

7.2 describe/represent arithmetic/geometric sequences using variable expressions

What is the difference between an arithmetic and a geometric sequence?

While both are numerical patterns, arithmetic sequences are additive and geometric sequences are multiplicative.

Investigate and apply strategies to recognize and describe the change between terms in arithmetic patterns.

Investigate and apply strategies to recognize and describe geometric patterns.

Describe verbally and in writing the relationships between consecutive terms in an arithmetic or geometric sequence.

Extend and apply arithmetic and geometric sequences to similar situations.

Extend arithmetic and geometric sequences in a table by using a given rule or mathematical relationship.

Compare and contrast arithmetic and geometric sequences.

Identify the common difference for a given arithmetic sequence.

Key VocabularyGeometric sequenceArithmetic sequenceCommon differenceCommon ratioTermConsecutive term

DOE Lesson Plans

Textbook ResourcesPH 3-1, 3-3PH2 9-2, 9-3, 9-4

Resources and Links



Identify the common ratio for a given geometric sequence.


The student will describe and represent arithmetic and geometric sequences using variable expressions.

Essential Questions

When are variable expressions used?Variable expressions can express the relationship between two consecutive terms in a sequence.

Prerequisite/Extension SOLSOL 6.17


Analyze arithmetic and geometric sequences to discover a variety of patterns.

Identify the common difference in an arithmetic sequence.

Identify the common ratio in a geometric sequence.

Given an arithmetic or geometric sequence, write a variable expression to describe the relationship between two consecutive terms in the sequence.

Key VocabularyGeometric sequenceArithmetic sequenceCommon differenceCommon ratioTermConsecutive term

DOE Lesson Plans

Textbook ResourcesPH2 4-6, 9-2, 9-3, 9-4

Resources and Links


The student will represent relationships with tables, graphs, rules, and words.


Describe and represent relations and functions, using tables, graphs, rules, and words. Given one representation, students will be able to

Key VocabularyOrdered pairFunctionTableGraphs



Essential QuestionsWhat are the different ways to represent the relationship between two sets of numbers?Rules that relate elements in two sets can be represented by word sentences, equations, tables of values, graphs or illustrated pictorially.


represent the relation in another form.DOE Lesson Plans

Textbook ResourcesPH2 9-4, 9-5

Resources and Links

Performance Task must be completed and graded on rubric by the end of Quarter 1Sample Performance Tasks:

Sample Performance Tasks:

SOL 6.1

Differentiated Best of Math I: Charmin (R) Choices

Students determine which toilet tissue is the better buy.

Differentiated Best of Math I: Dear Betty

Students compare 2 fruit snack products.

Differentiated Best of Math I: Dense Rocks



Students estimate the weight of a rock which cannot physically be weighed.

Differentiated Best of Math I: Dog Years Dilemma

Students determine the 'dog age' of a puppy based on his human age.

Differentiated Best of Math I: Frugal Freda's Fruit Foray

Students determine the fraction of edible flesh of Freda's favorite fruits, along with the cost per edible serving.

Differentiated Best of Math I: Gears

Given three gears, and a number of teeth in 2 of the 3 gears, students determine the number of teeth in the third gear.

Differentiated Best of Math I: Newspaper Advertisement

Students are given guidelines for generating a statistical exploration on a topic of their choice.

Differentiated Best of Math I: Raisins

Students determine how many raisins are in a box.

Differentiated Best of Math I: Sneakers

Given data on recent sneaker sales, students determine an appropriate number to order the following week.

SOL 6.2b

Differentiated Best of Math I: Candy Dilemma

A box of candies is eaten a fraction at a time. Students determine the number of candies that began in the box.

Differentiated Best of Math I: Distribution Dilemma

Students determine the number of movie tickets a seller began with, after giving away different fractional amounts.

Differentiated Best of Math I: Double Dilemma


Students determine the probability of using dice to throw doubles 1/3 of the time.

Differentiated Best of Math I: Easter Bunny Fractions

A garden is planted with unusual vegetables in fractional parts. Students determine the missing fractional parts.

Differentiated Best of Math I: M&M's (R)

Collect, organize and describe data, draw conclusions, compare theoretical to experimental probability

Differentiated Best of Math I: Newspaper Advertisement

Students are given guidelines for generating a statistical exploration on a topic of their choice.

Differentiated Best of Math I: Sneakers

Given data on recent sneaker sales, students determine an appropriate number to order the following week.

Differentiated Best of Math I: Tangram Fractions

Students find the fractional part of each Tangram piece as it relates to the whole.

SOL 6.2c



Differentiated Best of Math I: M&M's (R)

Collect, organize and describe data, draw conclusions, compare theoretical to experimental probability

SOL 6.2d



Students determine the fraction of edible flesh of Freda's favorite fruits, along with the cost per edible serving.

SOL 6.7



SOL 6.5

Differentiated Best of Math I: Big Numbers

Students show their significance of 30 billion dollars.

Differentiated Best of Math I: Checkerboard Dilemma

Students determine how many squares are on a checkerboard.

Differentiated Best of Math I: Day Lily Dilemma

Students determine the number of day lilies in all after following a pattern in their reproduction.

Differentiated Best of Math I: Division of Labor

Students determine the number of hours 2 people work.

Differentiated Best of Math I: Locker Dilemma

Students determine which lockers will be open, and which closed, after other students have open and closed them.

SOL 6.17

Differentiated Best of Math I: ABC Quilts


Given directions for making a quilt, students determine the amount of material needed.

Differentiated Best of Math I: Building Block Dilemma

Given the first three block towers, students determine the number of blocks needed for any number tower.



Differentiated Best of Math I: Tower of Hanoi

Students determine the number of moves it takes to solve a problem using spindles and disks.

SOL 7.1

Best of Secondary Math: And Now, Andy Rooney

Students determine if the national debt is big enough to allow for a stack of $1 bills would stretch to the moon and back.


Students determine the fraction of edible flesh of Freda's favorite fruits, along with the cost per edible serving

SOL 7.2

Differentiated Best of Math, 6-8: Building Block Dilemma

Given the first three block towers, students determine the number of blocks needed for any number of tower.

SOL 7.3Best of Secondary Math: Our Paper, Their Paper

Using patterns, students find the size of A4 paper.

Best of Secondary Math: Smooth Sailing


Using charts, students plot a safe sailing course through the harbor.

SOL 7.4



Differentiated Best of Math I: Dog House

Given a piece of plywood, students design the largest dog house possible.

Differentiated Best of Math I: Greener Grass

Students determine the better buy when re-turfing a football field.

Differentiated Best of Math I: Olympic Trophy Design

Students design a trophy topped with a scaled figure of the typical middle level student.

Differentiated Best of Math I: Turkey Day

Given a turkey and time to cook per pound, students determine the appropriate time to start cooking the turkey given additional parameters.

SOL 7.12Best of Secondary Math: Hacky-Sack Problem

Students determine the number of passes in a hacky-sack game given different sizes of teams.

Differentiated Best of Math I: Bridges

Students are given the first 3 bridge designs, and determine the 10th bridge design, noticing patterns and making generalizations.

Differentiated Best of Math I: Euler's Dilemma

Students investigate Euler's rule for classifying figures.


Differentiated Best of Math I: Runners' Dilemma

Two friends are running at different speeds. Students determine when one will catch up with the other.

Differentiated Best of Math I: Swinging Time

Students conduct an investigation to determine the slowest beat that could be produced by a metronome made with 60 centimeters of string.

http://www.doe.virginia.gov/instruction/mathematics/professional_development/institutes/2011/6-8/task_items_grade_6.pdf

2nd nine weeks


evaluate whole number numerical expressions, using the order of operations.


5.7 evaluate whole number numerical expressions using order of operations, limited to parentheses/add/sub/mult/div

7.13 a) write verbal expressions as algebraic expressions and sentences as equations and vice

What is the significance of the order of operations? The order of operations prescribes the order to use to simplify expressions containing more than one operation. It ensures that there is only one correct answer.

Simplify expressions by using the order of operations in a demonstrated step-by-step approach. The expressions should be limited to positive values and not include braces { } or absolute value | |.

Find the value of numerical expressions, using order of operations, mental mathematics, and appropriate tools. Exponents are limited to positive values.

Key VocabularyEvaluateExpressionParentheses

DOE Lesson Plans


Resources and Links




versa; b) evaluate algebraic expressions


solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.


5.18 a) investigate/describe concept of variable; b) write open sentence using variable; c) model one-step linear equations using add/sub; d) create problems based on open sentence

7.14 a) solve one and two-step linear equations; b) solve practical problems in one variable

When solving an equation, why is it necessary to perform the same operation on both sides of an equal sign?

To maintain equality, an operation performed on one side of an equation must be performed on the other side.

Represent and solve a one-step equation, using a variety of concrete materials such as colored chips, algebra tiles, or weights on a balance scale.

Solve a one-step equation by demonstrating the steps algebraically.

Identify and use the following algebraic terms appropriately: equation, variable, expression, term, and coefficient.

Key VocabularyEquationVariableCoefficientTermExpressionBalance scale

DOE Lesson Plans


Resources and Links

Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources4 days Standard 7.13 (with 6.18) The

student willa) write verbal

expressions as algebraic expressions


Write verbal expressions as algebraic expressions. Expressions will be

Key VocabularyEquationVariableCoefficientTerm

http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-18_2.pdf


and sentences as equations and vice versa; and

b) evaluate algebraic expressions for given replacement values of the variables.

Essential Questions

How can algebraic expressions and equations be written? Word phrases and sentences can be used to represent algebraic expressions and equations.


limited to no more than 2 operations.

Write verbal sentences as algebraic equations. Equations will contain no more than 1 variable term.

Translate algebraic expressions and equations to verbal expressions and sentences. Expressions will be limited to no more than 2 operations.

Identify examples of expressions and equations.Apply the order of operations to evaluate expressions for given replacement values of the variables. Limit the number of replacements to no more than 3 per expression.

ExpressionBalance scaleExponent

DOE Lesson Plans


Resources and Links

Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources4 days Standard 7.14 (with 6.18) The

student willa) solve one- and two-step linear equations in one variable; andb) solve practical problems requiring the solution of one- and two-step linear equations.

Essential Questions


Represent and demonstrate steps for solving one- and two-step equations in one variable using concrete materials, pictorial representations and algebraic sentences.

Solve one- and two-step linear equations in one variable.

Solve practical problems that require the solution of a one- or two-step linear equation.

Key VocabularyInverse Operations

DOE Lesson Plans


Resources/Lessons




When solving an equation, why is it important to perform identical operations on each side of the equal sign? An operation that is performed on one side of an equation must be performed on the other side to maintain equality.


Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources2 days Standard 8.15a

The student willa) solve multistep linear equations in one variable on one and two sides of the equation;



Solve two- to four-step linear equations in one variable using concrete materials, pictorial representations, and paper and pencil illustrating the steps performed.

Key VocabularyInverse OperationsEquationVariableCoefficientTermExpressionBalance scaleDOE Lesson PlansTextbook ResourcesPH2: 4-6

Resources/Lessons

Dates Standards Essential Knowledge/Skills/Understanding Resources

http://www.doe.virginia.gov/testing/solsearch/sol/math/8/mess_8-15a.pdf


3 days Standard 6.4 The student will demonstrate multiple representations of multiplication and division of fractions.

Prerequisite/Extension SOL5.3 a) identify/describe characteristics of prime/composite numbers; b) describe characteristics of even/odd numbers

When multiplying fractions, what is the meaning of the operation?

When multiplying a whole by a fraction such as 3 x , the meaning is the same as with multiplication of whole numbers: 3

groups the size of of the whole.

When multiplying a fraction by a fraction such as , we are asking for part of a part.

When multiplying a fraction by a whole number such as x 6, we are trying to find a part of the whole.

What does it mean to divide with fractions? For measurement division, the divisor is the number of groups and the quotient will be the number of groups in the dividend. Division of fractions can be explained as how many of a given divisor are

needed to equal the given dividend. In other words, for the

question is, “How many make ?”For partition division the divisor is the size of the group, so the quotient answers the question, “How much is the whole?” or “How much for one?”

Demonstrate multiplication and division of fractions using multiple representations.

Model algorithms for multiplying and dividing with fractions using appropriate representations.

Key VocabularyDivisorDividendQuotient

DOE Lesson Plans

Textbook ResourcesPH 6-1, 6-2, 6-3, 6-4, 6-5PH2 3-4, 3-5

Resources and Links


a) multiply and divide fractions and mixed

How are multiplication and division of fractions and multiplication and division of whole numbers alike? Fraction computation can be approached in the same way as whole number computation, applying those concepts to fractional parts.

Key VocabularyEstimateSimplest formInverse operation

http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-4_6-6ab_1.pdf


numbers; andb) estimate solutions

and then solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.


5.5 a) find sum/diff/product/quotient of two decimals through thousandths b) create/solve single/multistep problems involving decimals

What is the role of estimation in solving problems? Estimation helps determine the reasonableness of answers.

Multiply and divide with fractions and mixed numbers. Answers are expressed in simplest form.

Solve single-step and multistep practical problems that involve addition and subtraction with fractions and mixed numbers, with and without regrouping, that include like and unlike denominators of 12 or less. Answers are expressed in simplest form.

Solve single-step and multistep practical problems that involve multiplication and division with fractions and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.

“of” (multiply)

DOE Multiplication Lesson PlansDOE Division Lesson Plans

Textbook ResourcesPH 5-1, 5-2, 5-3, 5-4, 6-1, 6-2, 6-3, 6-4PH2 3-2, 3-3, 3-4, 3-5

Resources and Links

Dates Standards Essential Knowledge/Skills/Understanding Resources5 days Standard 6.14 - The student,

given a problem situation, willa) construct circle

What types of data are best presented in a circle graph?Circle graphs are best used for data showing a relationship of the parts to the whole.

Key VocabularyBar graphLine graphCircle graph




graphs;b) draw conclusions

and make predictions, using circle graphs; and

c) compare and contrast graphs that present information from the same data set.


5.15 collect/organize/interpret data, using stem-and-leaf plots/line graphs

7.11 a) construct/analyze histograms; b) compare/contrast histograms

Collect, organize and display data in circle graphs by depicting information as fractional.

Draw conclusions and make predictions about data presented in a circle graph.

Compare and contrast data presented in a circle graph with the same data represented in other graphical forms.

ConclusionConstructPredict

DOE Lesson Plans

Textbook ResourcesPH 7-8PH2 7-7

Resources and Links


The student, given data in a practical situation, willa) construct and analyze histograms; andb) compare and contrast histograms with other types


Collect, analyze, display, and interpret a data set using histograms. For collection and display of raw data, limit the data to 20 items.

Determine patterns and relationships within data sets (e.g., trends).

Make inferences, conjectures, and predictions based on analysis of a

Key VocabularyHistogramStem and leaf plotLine graphLineplotFrequency table

DOE Lesson Plans


http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-14abc.pdf


of graphs presenting information from the same data set.

Essential Questions

What types of data are most appropriate to display in a histogram? Numerical data that can be characterized using consecutive intervals are best displayed in a histogram.


set of data.

Compare and contrast histograms with line plots, circle graphs, and stem-and-leaf plots presenting information from the same data set.

Textbook ResourcesPH2: 11-1PH2: VA14

Resources/Lessons

Performance Task must be completed and graded on rubric by the end of Quarter 2


SOL 6.17



Differentiated Best of Math I: Building Block Dilemma

Given the first three block towers, students determine the number of blocks needed for any number tower.




Differentiated Best of Math I: Tower of Hanoi

Students determine the number of moves it takes to solve a problem using spindles and disks.

SOL 6.4

Differentiated Best of Math I: Stained Glass

Students determine the amount of glass and metal needed for a stained glass window.

SOL 6.6a

Differentiated Best of Math I: Stained Glass

Students determine the amount of glass and metal needed for a stained glass window.

SOL 6.6b

Differentiated Best of Math I: Candy Dilemma

A box of candies is eaten a fraction at a time. Students determine the number of candies that began in the box.

Differentiated Best of Math I: Distribution Dilemma

Students determine the number of movie tickets a seller began with, after giving away different fractional amounts.

Differentiated Best of Math I: Easter Bunny Fractions

A garden is planted with unusual vegetables in fractional parts. Students determine the missing fractional parts.


SOL 7.11

Best of Secondary Math: Barnie's Backpack Business

Students conduct a survey to identify the most desirable characteristics of a backpack.

SOL 7.13

Differentiated Best of Math, 6-8: Townhouse Tribulation

Students determine the number of square feet of brick required for a group of townhouses.

Differentiated Best of Math, 6-8: Variable Dilemma

Students solve a series of letter equations by substituting numbers.

Sample Performance Tasks:


3rd nine weeks

Dates Standards Essential Knowledge/Skills/Understanding Resources4 days Standard 6.15 - The student will What does the phrase “measure of center” mean? Key Vocabulary

Mean



a)describe mean as balance point; and

b) decide which measure of center is appropriate for a given purpose.


5.16 a) describe mean/median/mode; b) describe mean as fair share; c) find the mean/median/mode/range; d) describe range as a measure of variation

This is a collective term for the 3 types of averages for a set of data – mean, median, and mode.

What is meant by mean as balance point? Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of all the points above the mean is equal to the sum of the distances of all the data points below the mean. This is the concept of mean as the balance point.

Find the mean for a set of data.

Describe the three measures of center and a situation in which each would best represent a set of data.

Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data.

MedianModeNumerical averageBalance pointMeasure of center (central tendency)

DOE Lesson Plans


Resources/Lessons


determine congruence of segments, angles, and polygons.

Prerequisite/Extension SOL5.13 a) using plane figures will develop definitions of plane figures; b) investigate/describe results of combining/subdividing plane figures

Given two congruent figures, what inferences can be drawn about how the figures are related?The congruent figures will have exactly the same size and shape.

Given two congruent polygons, what inferences can be drawn about how the polygons are related?

Corresponding angles of congruent polygons will have the same measure. Corresponding sides of congruent polygons will have the same measure

Characterize polygons as congruent and noncongruent according to the measures of their sides and angles.

Determine the congruence of segments, angles, and polygons

Key VocabularySegmentAnglePolygonCongruentNoncongruentCorrespondingAttributes

DOE Lesson Plans





7.6 determine similarity of plane figures and write the proportions to express the relationships between similar quads and triangles

given their attributes.

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving practical and mathematical problems.†

Resources/Lessons


describe and identify properties of quadrilaterals.

Prerequisite/Extension SOL5.12 a) classify angles as right/acute/obtuse/straight; b) triangles right/acute/obtuse/equilateral/scalene/Isosceles

7.7 compare/contrast quadrilaterals based on properties

Can a figure belong to more than one subset of quadrilaterals?Any figure that has the attributes of more than one subset of quadrilaterals can belong to more than one subset. For example, rectangles have opposite sides of equal length. Squares have all 4 sides of equal length thereby meeting the attributes of both subsets.

Sort and classify polygons as quadrilaterals, parallelograms, rectangles, trapezoids, kites, rhombi, and squares based on their properties. Properties include number of parallel sides, angle measures and number of congruent sides.

Identify the sum of the measures of the angles of a quadrilateral as 360°.

Key VocabularyQuadrilateralParallelogramBisectCongruentAdjacentParallelPolygonRectangleTrapezoidKiteRhombus (rhombi)SquarePropertySum of the measures (interior angles)

DOE Lesson PlansTextbook ResourcesPH 8-5Resources/Lessons


The student will compare and contrast the following


Key VocabularyQuadrilateralParallelogram



quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.

Essential Questions

Why can some quadrilaterals be classified in more than one category?Every quadrilateral in a subset has all of the defining attributes of the subset. For example, if a quadrilateral is a rhombus, it has all the attributes of a rhombus. However, if that rhombus also has the additional property of 4 right angles, then that rhombus is also a square.


Compare and contrast attributes of the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid.

Identify the classification(s) to which a quadrilateral belongs, using deductive reasoning and inference.

BisectCongruentAdjacentParallelPolygonRectangleTrapezoidKiteRhombus (rhombi)SquarePropertySum of the measures (interior angles)

DOE Lesson Plans


Resources and Links


a) define pi (π) as the What is the relationship between the circumference and

diameter of a circle?

Key VocabularyPi



ratio of the circumference of a circle to its diameter;

b) solve practical problems involving circumference and area of a circle, given the diameter or radius;

c) solve practical problems involving area and perimeter; and

d) describe and determine the volume and surface area of a rectangular prism.

Prerequisite/Extension SOL5.9 Identify/describe diameter/radius/chord/circumference of a circle.

The circumference of a circle is about 3 times the measure of the diameter.

What is the difference between area and perimeter? Perimeter is the distance around the outside of a figure while area is the measure of the amount of space enclosed by the perimeter.

What is the relationship between area and surface area? Surface area is calculated for a three-dimensional figure. It is the sum of the areas of the two-dimensional surfaces that make up the three-dimensional figure.

Derive an approximation for pi (3.14 or ) by gathering data and comparing the circumference to the diameter of various circles, using concrete materials or computer models.

Find the circumference of a circle by substituting a value for the diameter or the radius into the formula C = d or C = 2r.

Find the area of a circle by using the formula A = r2.

Apply formulas to solve practical problems involving area and perimeter of triangles and rectangles.

Create and solve problems that involve finding the circumference and area of a circle when given the diameter or radius.

Solve problems that require finding the surface area of a rectangular prism, given a diagram of the prism with the necessary dimensions labeled.

Solve problems that require finding the volume of a rectangular prism given a diagram of the prism with the necessary dimensions labeled.

RatioCircumferenceDiameterRadiusAreaPerimeterSurface areaVolumeFormula

DOE #1 Lesson PlansDOE #2 Lesson Plans

Textbook ResourcesPH 9-3, 9-4, 9-5, 9-6, 9-8, 9-9

Resources and Links


The student willa) verify the Pythagorean Theorem; and


Identify the parts of a right triangle (the hypotenuse and the legs).

Key VocabularyPythagorean TheormRight TriangleHypotenuse

http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-10cd.pdf



b) apply the Pythagorean Theorem.

Essential QuestionsHow can the area of squares generated by the legs and the hypotenuse of a right triangle be used to verify the Pythagorean Theorem? For a right triangle, the area of a square with one side equal to the measure of the hypotenuse equals the sum of the areas of the squares with one side each equal to the measures of the legs of the triangle.


Verify a triangle is a right triangle given the measures of its three sides.

Verify the Pythagorean Theorem, using diagrams, concrete materials, and measurement.

Find the measure of a side of a right triangle, given the measures of the other two sides.

Solve practical problems involving right triangles by using the Pythagorean Theorem.

Legs

DOE Lesson Plans


Resources and Links

Dates Standards Essential Knowledge/Skills/Understanding Resources3 days Standard 6.9 - The student will make

ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system.

What is the difference between weight and mass? Weight and mass are different. Mass is the amount of matter in an object. Weight is the pull of gravity on the mass of an object. The mass of an object remains the same regardless of its location. The weight of an object changes dependent on the gravitational pull at its location.

How do you determine which units to use at different times? Units of measure are determined by the attributes of the

Key VocabularyBallpark comparisonWeight/massU.S. Customary System (standard)Metric systemConversionLengthTemperature




5.8 a) find perimeter/area/volume; b) differentiate among area/perimeter/volume; c) Identify equivalent measurements within the metric system; d) estimate/measure U.S. Cust/metric units; e) choose appropriate unit of measure with U.S. Cust/metric

7.5 a) describe volume /surface area of cylinders; b) solve practical problems involving volume/surface area of rectangular prisms and cylinders; c) describe how changes in measured attribute affects volume/surface area

object being measured. Measures of length are expressed in linear units, measures of area are expressed in square units, and measures of volume are expressed in cubic units.

Why are there two different measurement systems?Measurement systems are conventions invented by different cultures to meet their needs. The U.S. Customary System is the preferred method in the United States. The metric system is the preferred system worldwide.

Estimate the conversion of units of length, weight/mass, volume, and temperature between the U.S. Customary system and the metric system by using ballpark comparisons.

Ex: 1 L 1qt. Ex: 4L 4 qts.

Estimate measurements by comparing the object to be measured against a benchmark.

Volume (cubic units)Area (square units)

DOE Lesson Plans

Textbook Resources

Resources and Links


Sample Performance Tasks:http://www.doe.virginia.gov/instruction/mathematics/professional_development/institutes/2011/6-8/task_items_grade_6.pdf

4th nine weeks

Dates Standards Essential Knowledge/Skills/Understanding Resources




7 days Standard 6.16 - The student willa) compare and contrast

dependent and independent events; and

b) determine probabilities for dependent and independent events.


5.14 make predictions/determine probability by constructing a sample space

7.9 investigate/describe the difference between experimental/theoretical probability7.10 determine the probability of compound events; Basic Counting Principal

How can you determine if a situation involves dependent or independent events? Events are independent when the outcome of one has no effect on the outcome of the other. Events are dependent when the outcome of one event is influenced by the outcome of the other.

Determine whether two events are dependent or independent.

Compare and contrast dependent and independent events.

Determine the probability of two dependent events.

Determine the probability of two independent events.

Key VocabularyDependent eventsIndependent eventsProbabilityOutcomesSimultaneously

DOE Lesson Plans


Resources and Links


investigate and describe the difference between the experimental probability and theoretical probability of an event.

Essential Questions

What is the difference between the theoretical and experimental probability


Determine the theoretical probability of an event.

Determine the experimental probability of an event.

Describe changes in the experimental probability as the number of trials increases.

Investigate and describe the difference between the probability of an event found through experiment or simulation versus the

Key VocabularyOutcomeEventTheoretical ProbabilityComplementExperimental Probability

DOE Lesson Plans




of an event? Theoretical probability of an event is the expected probability and can be found with a formula. The experimental probability of an event is determined by carrying out a simulation or an experiment. In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability.


theoretical probability of that same event.Textbook ResourcesPH2: 12-1, 12-2

Resources/Lessons


The student will determine the probability of compound events, using the Fundamental (Basic) Counting Principle.

Essential Questions

What is the Fundamental (Basic) Counting Principle? The Fundamental (Basic) Counting Principle is a computational procedure used to determine the number of possible outcomes of several events.

What is the role of the Fundamental (Basic) Counting Principle in


Compute the number of possible outcomes by using the Fundamental (Basic) Counting Principle.

Determine the probability of a compound event containing no more than 2 events.

Key VocabularyCompound EventsIndependent EventsDependent Events

DOE Lesson Plans


Resources/Lessons



determining the probability of compound events?

The Fundamental (Basic) Counting Principle is used to determine the number of outcomes of several events. It is the product of the number of outcomes for each event that can be chosen individually.



a) identify the coordinates of a point in a coordinate plane; andb) graph ordered pairs

in a coordinate plane.


5.11 measure right/acute/obtuse/straight angles

7.8 represent transformations of polygons in the coordinate plane by graphing

Can any given point be represented by more than one ordered pair?The coordinates of a point define its unique location in a coordinate plane. Any given point is defined by only one ordered pair.

In naming a point in the plane, does the order of the two coordinates matter? Yes. The first coordinate tells the location of the point to the left or right of the y-axis and the second point tells the location of the point above or below the x-axis. Point (0, 0) is at the origin.

Identify and label the axes of a coordinate plane.

Identify and label the quadrants of a coordinate plane.

Identify the quadrant or the axis on which a point is positioned by examining the coordinates (ordered pair) of the point.

Graph ordered pairs in the four quadrants and on the axes of a coordinate plane.

Identify ordered pairs represented by points in the four quadrants and on the axes of the coordinate plane.

Key VocabularyCoordinatesPointCoordinate planeOrdered pairsAxesQuadrantOriginx-axisy-axis

DOE Lesson Plans

Textbook ResourcesPH 11-8PH2 10-1

Resources/Lessons



Relate the coordinate of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line.†

1 day Standard 6.20 - The student will graph inequalities on a number line.


7.15 a) solve one-step inequalities; b) graph solutions on number line

7.12 represent relationships with tables, graphs, rules, and words

In an inequality, does the order of the elements matter?Yes, the order does matter. For example, x > 5 is not the same relationship as 5 > x. However, x > 5 is the same relationship as 5 < x.

Given a simple inequality with integers, graph the relationship on a number line.

Given the graph of a simple inequality with integers, represent the inequality two different ways using symbols (<, >, <, >).

Key VocabularyInequalitiesOpen circle (> or < )Closed circle ( ≤ or ≥ )

DOE Lesson Plans


Resources/Lessons

Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources2 days Standard 7.15 The student will use problem solving, mathematical Key Vocabulary



The student will a) solve one-step inequalities in one variable; and b) graph solutions to inequalities on the number line.

Essential Questions

How are the procedures for solving equations and inequalities the same? The procedures are the same except for the case when an inequality is multiplied or divided on both sides by a negative number. Then the inequality sign is changed from less than to greater than, or greater than to less than.

How is the solution to an inequality different from that of a linear equation?In an inequality, there can be more than one value for the variable that makes the inequality true.

Prerequisite/Extension SOL 8.15aSOL 6.20SOL 8.16

communication, mathematical reasoning, connections, and representations to

Represent and demonstrate steps in solving inequalities in one variable, using concrete materials, pictorial representations, and algebraic sentences.

Graph solutions to inequalities on the number line.

Identify a numerical value that satisfies the inequality.

InequalitySolution of an Inequality

DOE Lesson Plans

Textbook ResourcesPH2 4-7, 4-8, 4-9

Resources/Lessons

Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources



3 days Standard 8.16 The student will graph a linear equation in two variables.

Essential Questions

What types of real life situations can be represented with linear equations?Any situation with a constant rate can be represented by a linear equation.



Construct a table of ordered pairs by substituting values for x in a linear equation to find values for y.

Plot in the coordinate plane ordered pairs (x, y) from a table.

Connect the ordered pairs to form a straight line (a continuous function).

Interpret the unit rate of the proportional relationship graphed as the slope of the graph, and compare two different proportional relationships represented in different ways.†

Key VocabularyGraph of an EquationLinear Equation

DOE Lesson Plans


Resources/Lessons


SOL 8.16Differentiated Best of Math, 6-8: Runners' DilemmaTwo friends are running at different speeds. Students determine when one will catch up with the other.

Sample Performance Tasks:http://www.doe.virginia.gov/instruction/mathematics/professional_development/institutes/2011/6-8/task_items_grade_6.pdf



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