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6.1
Students will expand number sense to include operations with rational numbers.
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6.1.1
Represent rational numbers in a variety of ways.
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6.1.1.a
Recognize a rational number as a ratio of two integers, a to b, where b is not equal to zero.
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6.1.1.b
Change whole numbers with exponents to standard form (e.g., 2 to the 4th power = 16) and recognize that any non-zero whole number to the zero power equals 1 (e.g., 9 to the 0 power = 1).
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6.1.1.c
Write a whole number in expanded form using exponents (e.g., 876,539 = 8 x 10 to the 5th power + 7 x 10 to the 4th power + 6 x 10³ + 5 x 10² + 3 x 10¹ + 9 x 10 to the 0 power).
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6.1.1.d
Express numbers in scientific notation using positive powers of ten.
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6.1.2
Explain relationships and equivalencies among rational numbers.
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6.1.2.a
Place rational numbers on the number line.
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6.1.2.b
Compare and order rational numbers, including positive and negative mixed fractions and decimals, using a variety of methods and symbols, including the number line and finding common denominators.
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6.1.2.c
Find equivalent forms for common fractions, decimals, percents, and ratios, including repeating or terminating decimals.
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6.1.2.d
Relate percents less than 1% or greater than 100% to equivalent fractions, decimals, whole numbers, and mixed numbers.
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6.1.2.e
Recognize that the sum of an integer and its additive inverse is zero.
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6.1.3
Use number theory concepts to find prime factorizations, least common multiples, and greatest common factors.
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6.1.3.a
Determine whether whole numbers to 100 are prime, composite, or neither.
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6.1.3.b
Find the prime factorization of composite numbers to 100.
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6.1.3.c
Find the greatest common factor and least common multiple for two numbers using a variety of methods (e.g., list of multiples, prime factorization).
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6.1.4
Model and illustrate meanings of operations and describe how they relate.
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6.1.4.a
Relate fractions to multiplication and division and use this relationship to explain procedures for multiplying and dividing fractions.
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6.1.4.b
Recognize that ratios derive from pairs of rows in the multiplication table and connect with equivalent fractions.
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6.1.4.c
Give mixed number and decimal solutions to division problems with whole numbers.
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6.1.5
Solve problems involving multiple steps.
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6.1.5.a
Select appropriate methods to solve a multi-step problem involving multiplication and division of fractions and decimals.
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6.1.5.b
Use estimation to determine whether results obtained using a calculator are reasonable.
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6.1.5.c
Use estimation or calculation to compute results, depending on the context and numbers involved in the problem.
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6.1.5.d
Solve problems involving ratios and proportions.
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6.1.6
Demonstrate proficiency with the four operations, with positive rational numbers, and with addition and subtraction of integers.
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6.1.6.a
Multiply and divide a multi-digit number by a two-digit number, including decimals.
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6.1.6.b
Add, subtract, multiply, and divide fractions and mixed numbers.
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6.1.6.c
Add and subtract integers.
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6.2
Students will use patterns, relations, and algebraic expressions to represent and analyze mathematical problems and number relationships.
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6.2.1
Analyze algebraic expressions, tables, and graphs to determine patterns, relations, and rules.
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6.2.1.a
Describe simple relationships by creating and analyzing tables, equations, and expressions.
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6.2.1.b
Draw a graph and write an equation from a table of values.
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6.2.1.c
Draw a graph and create a table of values from an equation.
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6.2.2
Write, interpret, and use mathematical expressions, equations, and formulas to represent and solve problems that correspond to given situations.
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6.2.2.a
Solve single variable linear equations using a variety of strategies.
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6.2.2.b
Recognize that expressions in different forms can be equivalent and rewrite an expression to represent a quantity in a different way.
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6.2.2.c
Evaluate and simplify expressions and formulas, substituting given values for the variables (e.g., 2x + 4; x = 2; therefore, 2 (2) + 4 = 8).
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6.3
Students will use spatial and logical reasoning to recognize, describe, and analyze geometric shapes and principles.
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6.3.1
Identify and analyze attributes and properties of geometric shapes to solve problems.
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6.3.1.a
Identify the midpoint of a line segment and the center and circumference of a circle.
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6.3.1.b
Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
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6.3.1.c
Develop and use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle in a triangle or quadrilateral.
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6.3.2
Visualize and identify geometric shapes after applying transformations on a coordinate plane.
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6.3.2.a
Rotate a polygon about the origin by a multiple of 90° and identify the location of the new vertices.
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6.3.2.b
Translate a polygon either horizontally or vertically on a coordinate grid and identify the location of the new vertices.
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6.3.2.c
Reflect a polygon across either the x- or y-axis and identify the location of the new vertices.
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6.4
Students will understand and apply measurement tools and techniques and find the circumference and area of a circle.
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6.4.1
Describe and find the circumference and area of a circle.
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6.4.1.a
Explore the relationship between the radius and diameter of a circle to the circle's circumference to develop the formula for circumference.
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6.4.1.b
Find the circumference of a circle using a formula.
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6.4.1.c
Describe pi as the ratio of the circumference to the diameter of a circle.
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6.4.1.d
Decompose a circle into a number of wedges and rearrange the wedges into a shape that approximates a parallelogram to develop the formula for the area of a circle.
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6.4.1.e
Find the area of a circle using a formula.
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6.4.2
Identify and describe measurable attributes of objects and units of measurement, and solve problems involving measurement.
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6.4.2.a
Recognize that measurements are approximations and describe how the size of the unit used in measuring affects the precision.
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6.4.2.b
Convert units of measurement within the metric system and convert units of measurement within the customary system.
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6.4.2.c
Compare a meter to a yard, a liter to a quart, and a kilometer to a mile.
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6.4.2.d
Determine when it is appropriate to estimate or use precise measurement when solving problems.
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6.4.2.e
Derive and use the formula to determine the surface area and volume of a cylinder.
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6.5
Students will analyze, draw conclusions, and make predictions based upon data and apply basic concepts of probability.
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6.5.1
Design investigations to reach conclusions using statistical methods to make inferences based on data.
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6.5.1.a
Design investigations to answer questions.
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6.5.1.b
Extend data display and comparisons to include scatter plots and circle graphs.
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6.5.1.c
Compare two similar sets of data on the same graph and compare two graphs representing the same set of data.
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6.5.1.d
Recognize that changing the scale influences the appearance of a display of data.
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6.5.1.e
Propose and justify inferences and predictions based on data.
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6.5.2
Apply basic concepts of probability and justify outcomes.
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6.5.2.a
Write the results of a probability experiment as a fraction between zero and one, or an equivalent percent.
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6.5.2.b
Compare experimental results with theoretical results (e.g., experimental: 7 out of 10 tails; whereas, theoretical 5 out of 10 tails).
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6.5.2.c
Compare individual, small group, and large group results of a probability experiment in order to more accurately estimate the actual probabilities.
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