1.1 Understand and apply concepts and procedures from number sense.

Grade 6

Grade 7

Grade 8

Number and Numeration

1.1.1

Understand the concept of integers as the set of natural numbers (1, 2, 3 …), their opposites (-1, -2, -3 …), and 0. W

·         Illustrate integer values using models and pictures (e.g., temperature, elevators, net worth/debt, riding a bus or subway). [CU]

·         Represent and identify integers on a model (e.g., number line, fraction line, or decimal grid). [SP, RL, CU]

·         Apply number theory concepts to rename a number quantity (e.g. Four, 4, 4.0, 8/2, 2x2, 6 – 2).

·         Apply rules of divisibility to show if a quotient is an integer. [SP, RL]

·         Explain the meaning of integers and give examples.

Understand the concept of rational numbers (integers, decimals, fractions). W

·          Demonstrate understanding of the concepts and symbolic representations of rational numbers including integers.

·          Create a model when given a symbolic representation of a rational number. [SP, RL, CU, MC]

·          Write the rational number when given a model (e.g., number line, area model, situation, diagram, picture). [SP, RL, CU, MC]

·          Identify and convert between equivalent forms of rational numbers (e.g., fractions to decimals, percents to fractions). [MC]

·          Identify prime, square, or composite numbers. [CU]

·          Explain the meaning of rational numbers and give examples.

Understand the concept of rational numbers, including whole number powers and square roots of perfect squares. W

·          Demonstrate understanding of the concepts and the symbolic representations of rational numbers including whole number powers and square roots of perfect squares.

·          Explain the meaning of a whole number exponent. [CU]

·          Read and use exponential notation to represent large numbers. [MC, SP, RL]

·          Identify a square number and find its root.

·          Identify different representations of rational numbers and select the best representation (e.g., percent for sales discount or sales tax, fraction for probability, and decimals for money, distance (4.35 kilometers), batting averages).

1.1.2

Understand the relative values of integers and non-negative rational numbers. W

·          Compare different representations of non-negative rational numbers by implementing strategies (e.g., like denominators, changing to the same form). [SP, RL, CU, MC]

·          Identify equivalence between non-negative integers, fractions, percents and decimals. [MC]

·          Compare and order integer values and explain which is greater and why (e.g., place the integers on a number line). [CU]

·          Locate integers on a number line.

Understand the relative values of rational numbers. W

·          Compare and order rational numbers using physical models or implementing strategies (e.g., like denominators, changing to the same form). [SP, RL, CU, MC]

·          Locate symbolic representations of rational numbers including fractions, decimals, and percents on a physical model (e.g., a number line, fraction line, decimal grid, and circle graph. [MC]

·          Explain the value of a given digit in a rational number (e.g., 2.3 is 2 ones and three tenths). [CU]

Understand the relative values of rational numbers, including whole number powers and square roots of perfect squares. W

·          Compare and order rational numbers using models or implementing strategies. [SP, RL]

·          Order different representations of rational numbers. [SP, RL]

·          Locate symbolic representations of rational numbers on a number line including whole number powers and square roots of square numbers. [SP, RL]

1.1 Understand and apply concepts and procedures from number sense.

Grade 6

Grade 7

Grade 8

Number and Numeration

1.1.3

Apply properties of addition and multiplication to non-negative rational numbers and understand the additive inverse property with integers. W

·          Illustrate the additive inverse property using physical models and pictures (e.g., number line). [CU]

·          Explain the additive inverse property and why it works. [CU]

·          Identify the opposite of a given integer.

·          Use the additive inverse property to solve problems. [SP, RL]

Apply properties of addition and multiplication, including inverse properties, to the rational number system. W

·          Use the inverse relationships of multiplication and division to simplify computations and solve problems. [SP, RL]

·          Identify errors and explain correct procedures in the application of order of operations. [SP, RL, CU]

·          Use the inverse properties of addition and multiplication to simplify computations with integers, fractions, and decimals. [SP, RL]

·          Identify the inverse elements when using the additive inverse and the multiplicative inverse properties (e.g., 8 + -8 = 0; 2 x ½ = 1.)

·          Explain the additive and multiplicative inverse properties.

Apply properties of addition, multiplication, and the distributive property to the rational number system. W

·          Illustrate and explain the distributive property of multiplication over addition (e.g., using an area model or picture). [CU, MC]

·          Use the distributive property to simplify expressions, including those using integers. [SP, RL]

·          Use the distributive property to factor expressions (e.g. 3▪9+3=3▪(9+1)). [SP, RL]

1.1.4

Understand the concepts of ratio and percent. W

·         Write ratios in part/part and part/whole relationships using objects, pictures, and symbols (e.g., using /, :, or to as representations for ratios). [CU]

·         Represent equivalent ratios or given percentages using objects, pictures, and symbols. [CU, MC]

·          Identify percent as 100 equal size parts of a set (e.g., 1% of 200 items is 2 items). [SP, RL]

·          Explain ratio and percents and give examples of each.

Understand the concept of of ratio, percent, and direct proportion. W

·          Express proportional relationships using objects, pictures, and symbols. [CU, MC]

·          Explain the meaning of a proportion. [CU]

·          Represent a new relationship from a given ratio (e.g., part/part to part/whole; given a ratio of girls to boys, find the ratio of girls to class). [MC]

·          Represent percentages less than 1% or greater than 100% using objects, pictures, and symbols. [CU, MC]

·          Complete or write a proportion for a given situation. [CU, MC]

Apply ratio, percent, and direct proportion in situations. W

·          Solve problems involving ratio and proportion (e.g., similar figures, scale drawings, rates, find unit pricing, increase or decrease a recipe, find the portions for a group converting between different units of measure, or finding medicinal dosages). [SP, RL, CU, MC]

·          Solve problems involving percentages (e.g., percent increase/decrease, tax, commission, discount). [SP, RL, CU, MC]

·          Explain advantages and disadvantages of different representations in a given situation (e.g., using 1/3 versus 33 1/3 %). [CU]

1.1 Understand and apply concepts and procedures from number sense.

Grade 6

Grade 7

Grade 8

Computation

1.1.5

Understand the meaning of addition and subtraction on integers and the multiplication and division on non-negative rational numbers. W

·          Explain the meaning of addition and subtraction of integers using real world models (e.g., reducing debt, temperature increase or decrease, yards gained and lost, movement of a hot-air balloon). [CU]

·          Explain the meaning of multiplying and dividing non-negative fractions and decimals using visual and physical models (e.g., sharing a restaurant bill, cutting a board into equal-sized pieces, drawing a picture of an equation or situation). [CU]

Understand the meaning of multiplication and division on integers. W

·          Explain the meaning of multiplication and division of integers using visual and physical models. [CU]

·          Create a problem situation involving multiplication or division of integers. [SP, RL, CU, MC]

·          Demonstrate understanding of solutions received when non-negative rational numbers are divided by fractions. [SP, RL]

Understand the meaning of operations on rational numbers (including square roots of perfect squares and whole number powers). W

·          Compare and contrast operations on rational numbers using pictures and symbols. [CU]

·          Create a problem situation to match a given rational number equation. [SP, RL, CU, MC]

·          Identify a rational number equation to match a given situation. [CU, MC]

·          Explain the meaning of negative and zero exponents. [CU]

1.1.6

Apply computational procedures with fluency for addition and subtraction on non-negative rational numbers. W

·       Find the sums or differences of non-negative fractions or decimals.

·       Write and solve real-world problem situations to find sums or differences of decimals or fractions. [SP, RL, CU, MC]

·       Use the least common multiple and the greatest common factor of whole numbers to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the simplified form for a fraction).

Apply computational procedures with fluency for addition and subtraction on integers, multiplication and division on non-negative rational numbers. W

·          Find the sum, difference, product, or quotient using non-negative decimals and fractions with unlike denominators.

·          Find the sums and differences using integers.

·          Apply percentages in a variety of situations (e.g. taxes, discounts, interest). [SP, RL, MC]

·          Use addition, subtraction, multiplication, and division to solve real-world problems involving non-negative rational numbers and integers.  [SP, RL, CU, MC]

Apply computational procedures on rational numbers (including whole number powers and square roots of perfect squares). W

·          Compute with rational numbers using order of operations.

·          Compute fluently with rational numbers in all forms except exponential.

·          Write and solve problems that involve computation with rational numbers. [SP, RL, CU, MC]

1.1 Understand and apply concepts and procedures from number sense.

Grade 6

Grade 7

Grade 8

Computation

1.1.7

Understand and apply strategies and tools as appropriate to tasks involving addition and subtraction on non-negative rational numbers.

·          Select and justify appropriate strategies and tools from among mental computation, estimation, calculators, and paper and pencil to compute in a problem situation. [SP, RL]

·          Describe strategies for mentally solving problems involving fractions and decimals. [CU]

Understand and apply strategies and tools as appropriate to tasks involving the four basic operations on integers and non-negative rational numbers.

·          Select and justify appropriate strategies and tools from among mental computation, estimation, calculators, and paper and pencil to compute in a problem situation. [SP, RL]

·          Convert between fractions, decimals, whole numbers, and percents mentally, on paper, or with a calculator. [MC]

Understand and apply strategies and tools as appropriate to tasks involving computation on rational numbers.

·          Select and justify appropriate strategies and tools from among mental computation, estimation, calculators, and paper and pencil to compute in a problem situation. [SP, RL]

·          Describe strategies for mentally solving problems involving integers and exponents. [CU]

Estimation

1.1.8

Apply estimation strategies to determine the reasonableness of answers in situations involving addition and subtraction on non-negative rational numbers. W

·          Identify when an approximation is appropriate

·          Use estimation to determine the reasonableness of answers

·          Apply estimation strategies prior to computation of whole numbers, decimals, and fractions to determine reasonableness of answers. [SP, RL]

·          Use estimation to predict or to verify the reasonableness of calculated results.

·          Identify appropriate estimated answers for a given situation.

·          Articulate various strategies used during estimation involving fractions and decimals. [CU]

Apply estimation strategies to determine the reasonableness of answers in situations involving the four basic operations on integers and non-negative rational numbers. W

·          Identify when an approximation is appropriate in situations

·          Use estimation to determine the reasonableness of answers.

·          Apply estimation strategies prior to computing addition and subtraction of integers and operations on non-negative rational numbers to determine reasonableness of answers. [SP, RL]

·          Justify why estimation would be used rather than an exact computation. [CU]

·          Describe a situation where estimation is sufficient in real life contexts. [CU, MC]

·          Use estimation to predict or to verify the reasonableness of calculated results.

Apply estimation strategies to determine the reasonableness of answers in situations involving computation on rational numbers, including whole number powers and square roots of perfect squares. W

·          Identify when an approximation is appropriate

·          Use estimation to determine the reasonableness of answers in situations.

·          Explain situations involving real numbers where estimates are sufficient and others for which exact value is required. [CU]

·          Justify why an estimate would be used rather than an exact answer in a given situation. [CU]

·          Articulate various strategies used during estimation involving integers. [CU]

·          Use estimation to predict or to verify the reasonableness of calculated results

1.2 Understand and apply concepts and procedures from measurement.

Grade 6

Grade 7

Grade 8

Attributes, Units and Systems

1.2.1

Understand the concepts of volume and extend the concept of area to surface area of rectangular prisms. W

·          Compare the relative capacity of two containers (e.g., paper cylinders formed horizontally and vertically and filled with popcorn). [SP, RL, CU, MC]

·          Represent the volume for given rectangular prisms using pictures or models. [CU]

·          Compare the surface area of two different rectangular prisms.

·          Describe and provide examples for surface area measurement (e.g. gift wrapping, painting a room, amount of material needed to build a box).

Understand how a change in a linear dimension affects other linear measurements, (perimeter, circumference) and area measurements. W

·          Figures used are rectangles, triangles, and circles.

·          Describe the relationships among linear dimensions (e.g. radius of a circle, length of a side or base) and area of the figure (e.g., change the radius or length of a side, and check the change in area – describe that change). [CU]

·          Explain and give examples of changing one, two, or 3 dimensions in a rectangular prism and how it affects the surface area and volume.

Understand how a change in a linear dimension affects volume and surface area of rectangular prisms and right cylinders. W

·          Figures used are rectangular prisms and right cylinders.

·          Compare the impact that a change in one dimension has on volume and surface area in right cylinders and rectangular prisms. [SP, RL]

·          Describe the relationships among linear dimensions, volume, and surface area (e.g., changing the length of a side affects the surface area and volume). [CU]

1.2.2

Understand the differences between square and cubic units. W

·          Identify cubic units to measure volume (e.g., linking cubes, cubic centimeter).

·          Identify and read incremental units for capacity (e.g., milliliters, cups, ounces).

·          Use the appropriate units when describing a situation (e.g., 5 square meters of carpet, 5 cubic meters of water).

·          Describe and compare the use of area and volume (e.g. covering and filling). [CU]

·          Explain why volume measurement is labeled as cubed. [CU], [MC]

·          Explain and give examples of how the area and surface area are related (e.g., surface area is the sums of the areas of all the sides of a rectangular prism).

Understand derived units of measurement. W

·          Explain the concept of a rate. [CU]

·          Explain how division of measurements produces a derived unit of measurement (e.g., miles traveled divided by hours traveled yields the derived unit [miles per hour]). [CU]

·          Find a rate of change in a real world situation. [SP, RL, MC]

·          Use dimensional analysis to find equivalent rates (e.g., mph to ft/sec).

1.2 Understand and apply concepts and procedures from measurement.

Grade 6

Grade 7

Grade 8

Attributes, Units and Systems

1.2.3

· 

Understand how the unit of measure selection affects the precision of measurement.  W

·          Select the appropriate measurement tool to match the precision needed (e.g., if needing a very precise measurement, a tool is needed that uses units that will give that precision).

Understand why different situations require different levels of precision. W

·          Explain the relationships among units within both the customary and metric system (kilograms to grams, feet to inches)

·          Justify the use of a unit of measure (e.g., meters or kilometers, inches or feet). [CU]

Procedures and Estimation

1.2.4

Understand and apply systematic procedures to measure volume and capacity for solid shapes. W

·          Compare the appropriateness of standard to nonstandard units in measuring volume or capacity. [CU]

·          Choose the appropriate standard unit for measuring volume or capacity (e.g., cubic inches vs. cubic feet, cups vs. gallons). [SP, RL]

·          Use a variety of methods to explain procedures for finding volume. [SP, RL, CU]

·          Use volume and capacity to describe and compare figures (e.g., fill containers with cubes to find which has a greater volume) [SP, RL, CU]

·          Measure volume of rectangular prisms and label appropriately. [SP, RL, CU]

·          Measure the capacity of containers using appropriate tools and label (e.g., graduated cylinders, measuring cups, tablespoons). [SP, RL, CU]

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1.2 Understand and apply concepts and procedures from measurement.

Grade 6

Grade 7

Grade 8

Procedures and Estimation

1.2.5

Apply formulas to find measurements of circles, triangles and rectangular prisms.  W

·          Apply formulas to determine missing measurements for circles, rectangular prisms and triangles.

·          Explain how to use a formula for finding the area and circumference of a circle. [CU]

·          Find and compare rectangular prisms that have a given volume (e.g., if two rectangular prisms have the same volume, and one has twice the height of the other, determine how the areas of their bases compare). [SP, RL]

·          Justify the standard formula for finding the area of a right triangle (e.g., 1/2 of a rectangle). [CU], [MC]

·          Explain why linear units are used to find the circumference of a circle. [CU]

·          Use given dimensions to determine surface area and volume.

Understand and apply formulas, including the Pythagorean Theorem, to right prisms, right cylinders, and triangles. W

·          Apply formulas, including the Pythagorean Theorem, to determine missing measurements for right prisms and right cylinders and triangles.

·          Explain how to use a formula for finding the surface area and volume of a solid. [CU]

·          Find missing sides or area of right triangles (e.g., use the Pythagorean Theorem to find any of the missing values]

·          Calculate measures of objects for which no direct information is given (e.g., similar figures, ratio, proportion, scale). [SP, RL, MC]

·          Compare material costs of various right cylinder and right prism containers with a given volume. [SP, RL, MC]

1.2.6

Understand and apply strategies to obtain reasonable estimates of volume and capacity. W

·          Identify situations in which estimated measures are sufficient; estimate volume or capacity

·          Identify situations when approximate measurements are sufficient.

·          Use estimation to justify reasonableness of a volume of a rectangular prism. [CU]

·          Estimate a measurement of volume or capacity using standard or nonstandard units (e.g., estimate the capacity of a bowl in cups and handfuls). [SP, RL]

·          Apply a process that can be used to find a reasonable estimate of volume and capacity (e.g., fill a container with rice or popcorn). [SP, RL, CU]

Understand and apply strategies to obtain reasonable estimates of: circle measurements; triangles; and surface area for a rectangular solid or area of a parallelogram. W

·          Identify situations in which estimated measures are sufficient; estimate circle and triangle measurements.

·          Justify the reasonableness of an estimate. [SP, RL]

·          Apply common approximations of pi (3.14; 22/7) to calculate the approximate circumference and the area of circles. [SP, RL, CU]

·          Apply a process that can be used to find a reasonable estimate of circle measurements (e.g., wrap a string around it). [SP, RL, CU]

Apply strategies to obtain reasonable estimates of: volume and surface area measurements for right cylinders; right prisms; and of the lengths of sides of right triangles W

·          Identify situations in which estimated measures are sufficient; estimate volume and surface area for right cylinders, right prisms, and the lengths of sides of right triangles.

·          Approximate distance or height in a problem situation using similar triangles or Pythagorean relationships (e.g., height of a flagpole using proportional reasoning, distance across a lake using Pythagorean relationship). [MC,CU,SP, RL]

1.3 Understand and apply concepts and procedures from geometric sense.

Grade 6

Grade 7