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### 6.1 Number & Operation

- 6.1.1 Read, write, represent and compare positive rational numbers expressed as fractions, decimals, percents and ratios; write positive integers as products of factors; use these representations in real-world and mathematical situations.
- 6.1.1.1 Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid.
- 6.1.1.2 Compare positive rational numbers represented in various forms. Use the symbols <, = and >.
- 6.1.1.3 Understand that percent represents parts out of 100 and ratios to 100.
- 6.1.1.4 Determine equivalences among fractions, decimals and percents; select among these representations to solve problems.
- 6.1.1.5 Factor whole numbers; express a whole number as a product of prime factors with exponents.
- 6.1.1.6 Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions.
- 6.1.1.7 Convert between equivalent representations of positive rational numbers.

- 6.1.2 Understand the concept of ratio and its relationship to fractions and to the multiplication and division of whole numbers. Use ratios to solve real-world and mathematical problems.
- 6.1.2.1 Identify and use ratios to compare quantities; understand that comparing quantities using ratios is not the same as comparing quantities using subtraction.
- 6.1.2.2 Apply the relationship between ratios, equivalent fractions and percents to solve problems in various contexts, including those involving mixtures and concentrations.
- 6.1.2.3 Determine the rate for ratios of quantities with different units.
- 6.1.2.4 Use reasoning about multiplication and division to solve ratio and rate problems.

- 6.1.3 Multiply and divide decimals, fractions and mixed numbers; solve real-world and mathematical problems using arithmetic with positive rational numbers.
- 6.1.3.1 Multiply and divide decimals and fractions, using efficient and generalizable procedures, including standard algorithms.
- 6.1.3.2 Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.
- 6.1.3.3 Calculate the percent of a number and determine what percent one number is of another number to solve problems in various contexts.
- 6.1.3.4 Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers.
- 6.1.3.5 Estimate solutions to problems with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of results in the context of the problem.

### 6.2 Algebra

- 6.2.1 Recognize and represent relationships between varying quantities; translate from one representation to another; use patterns, tables, graphs and rules to solve real-world and mathematical problems.
- 6.2.1.1 Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts.
- 6.2.1.2 Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations.

- 6.2.2 Use properties of arithmetic to generate equivalent numerical expressions and evaluate expressions involving positive rational numbers.
- 6.2.2.1 Apply the associative, commutative and distributive properties and order of operations to generate equivalent expressions and to solve problems involving positive rational numbers.

- 6.2.3 Understand and interpret equations and inequalities involving variables and positive rational numbers. Use equations and inequalities to represent real-world and mathematical problems; use the idea of maintaining equality to solve equations. Interpret solutions in the original context.
- 6.2.3.1 Represent real-world or mathematical situations using equations and inequalities involving variables and positive rational numbers.
- 6.2.3.2 Solve equations involving positive rational numbers using number sense, properties of arithmetic and the idea of maintaining equality on both sides of the equation. Interpret a solution in the original context and assess the reasonableness of results.

### 6.3 Geometry & Measurement

- 6.3.1 Calculate perimeter, area, surface area and volume of two- and three-dimensional figures to solve real-world and mathematical problems.
- 6.3.1.1 Calculate the surface area and volume of prisms and use appropriate units, such as cm² and cm³. Justify the formulas used. Justification may involve decomposition, nets or other models.
- 6.3.1.2 Calculate the area of quadrilaterals. Quadrilaterals include squares, rectangles, rhombuses, parallelograms, trapezoids and kites. When formulas are used, be able to explain why they are valid.
- 6.3.1.3 Estimate the perimeter and area of irregular figures on a grid when they cannot be decomposed into common figures and use correct units, such as cm and cm².

- 6.3.2 Understand and use relationships between angles in geometric figures.
- 6.3.2.1 Solve problems using the relationships between the angles formed by intersecting lines.
- 6.3.2.2 Determine missing angle measures in a triangle using the fact that the sum of the interior angles of a triangle is 180°. Use models of triangles to illustrate this fact.
- 6.3.2.3 Develop and use formulas for the sums of the interior angles of polygons by decomposing them into triangles.

- 6.3.3 Choose appropriate units of measurement and use ratios to convert within measurement systems to solve real-world and mathematical problems.
- 6.3.3.1 Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units.
- 6.3.3.2 Estimate weights, capacities and geometric measurements using benchmarks in measurement systems with appropriate units.

### 6.4 Data Analysis & Probability

- 6.4.1 Use probabilities to solve real-world and mathematical problems; represent probabilities using fractions, decimals and percents.
- 6.4.1.1 Determine the sample space (set of possible outcomes) for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables or pictorial representations.
- 6.4.1.2 Determine the probability of an event using the ratio between the size of the event and the size of the sample space; represent probabilities as percents, fractions and decimals between 0 and 1 inclusive. Understand that probabilities measure likelihood.
- 6.4.1.3 Perform experiments for situations in which the probabilities are known, compare the resulting relative frequencies with the known probabilities; know that there may be differences.
- 6.4.1.4 Calculate experimental probabilities from experiments; represent them as percents, fractions and decimals between 0 and 1 inclusive. Use experimental probabilities to make predictions when actual probabilities are unknown.