Maths Grade 6

PAN-ASIA MATHEMATICS – GRADE 6

Number Sense and Operations:

Standard One: Students will engage in problem solving, communicating, reasoning, connecting and representing.

Benchmarks:

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Understand meanings of operations and how they relate to one another.
Compute fluently and make reasonable estimates.
Demonstrate an understanding of positive integer exponents, in particular, when used in powers of ten, e.g. 102, 105.
Find and use common multiples, factors, prime and composite numbers.
Use divisibility tests (2,3,4,5,6,8,9,10,11, and their multiples.
Master the use of primes and properties of numbers such as, GCF and LCM to compute and/or approximate powers and roots.
Demonstrate an understanding of place value to billions and thousandths.
Identify and use place value in standard and expanded form.
Represent and compare very large (billions) and very small (thousandths) positive numbers in various forms such as expanded notation without exponents, e.g., 9724 = 9 x 1000+ 7 x 2 x 10 + 4.
Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection and as locations on the number line.
Identify and determine common equivalent fractions, mixed numbers, decimals, and percents.
Find and position integers, fractions, mixed numbers, and decimals (both positive and negative) on the number line.
Read, write, and understand conceptually integers (positive and negative), fractions, and decimals.
Compare and order integers (including negative integers), and positive fractions, mixed numbers, decimals, and percents.
Apply number theory concepts—including prime and composite numbers, prime factorization, greatest common factor, least common multiple, and divisibility rules for 2, 3, 4, 5, 6, 9, and 10—to the solution of problems.
Select and use appropriate operations to solve problems involving addition, subtraction, multiplication, division, and positive integer exponents with whole numbers, and with positive fractions, mixed numbers, decimals, and percents.
Use the number line to model addition and subtraction of integers, with the exception of subtracting negative integers.
Use a variety of approaches for computing and solving problems.
Demonstrate an understanding of the inverse relationship of addition and subtraction, and use that understanding to simplify computation and solve problems.
Use concrete and abstract models to understand and describe the mathematical processes underlying the operation of addition, subtraction, multiplication and division (and their relationship with one another) on fractions, decimals and integers
Accurately and efficiently add, subtract, multiply, and divide (with double-digit divisors) whole numbers and positive decimals.
Compute with whole numbers, fractions, decimals, and integers.
Use effective models and processes of computation with understanding (mental, oral, written, calculator, computer, etc.).
Accurately and efficiently add, subtract, multiply, and divide positive fractions and mixed numbers.
Simplify fractions.
Compute with whole numbers, fractions, decimals and integers
Add and subtract integers, with the exception of subtracting negative integers.
Use operations involving fractions, decimals, and integers
Estimate results of computations with whole numbers, and with positive fractions, mixed numbers, decimals, and percents. Describe reasonableness of estimates.
Round to the nearest ten, hundred, thousand, million, billion.
Round decimals and fractions to the nearest pre-assigned value.
Use estimation to solve problems involving, money, length, area, perimeter and volume.
Create and solve meaningful problems that require the use of numbers other than whole numbers.
Know and use order of operations including the use of parentheses.

Geometry

Standard Two: Students will engage in problem solving, communicating, reasoning, connecting and representing as they:

Benchmarks:

Analyze characteristics and properties of two geometric shapes and develop mathematical arguments about geometric relationships.
Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
Apply transformations and use symmetry to analyze mathematical situations.
Use visualization, spatial reasoning, and geometric modeling to solve problems.
Identify polygons based on their properties, including types of interior angles, perpendicular or parallel sides, and congruence of sides, e.g., squares, rectangles, rhombuses, parallelograms, trapezoids, and isosceles, equilateral, and right triangles.
Identify three-dimensional shapes (e.g., cubes, prisms, spheres, cones, and pyramids) based on their properties, such as edges and faces.
Identify relationships among points, lines, and planes, e.g., intersecting, parallel, perpendicular.
Graph points and identify coordinates of points on the Cartesian coordinate plane (all four quadrants).
Predict, describe, and perform transformations on two-dimensional shapes, e.g., translations, rotations, and reflections.
Identify and describe transformations, reflections, translations, slides, flips, and rotations.
Tessellate shapes across a plane
Identify types of symmetry, including line and rotational.
Identify point and line symmetry in given polygons.
Identify and describe transformations, reflections, translations, slides, flips, and rotations.
Determine if two shapes are congruent by measuring sides or a combination of sides and angles, as necessary; or by motions of series of motions, e.g., translations, rotations, and reflections.
Identify, describe and draw congruent and similar figures.

Measurement

Standard Three: Students will engage in problem solving, communicating, reasoning, connecting and representing as they:

Benchmarks:

Understand measurable attributes of objects and the units, systems, and processes of measurement.
Apply appropriate techniques, tools, and formulas to determine measurements.
Apply the concepts of perimeter and area to the solution of problems. Apply formulas where appropriate.
Develop and apply formulas for area and perimeter for standard figures and objects, and for combined figures.
Identify, measure, describe, classify, and construct various angles, triangles and quadrilaterals.
Identify, name, and construct plane and simple solid figures.
Solve problems involving proportional relationships and units of measurements.
Apply the comparison of similar figures and the ratio of their corresponding lengths and areas to solve problems.
Find areas of triangles, parallelograms. Recognize that shapes with the same number of sides but with different appearances can have the same area. Develop strategies to find area of complex shapes.
Develop and apply formulas for area, perimeter, and volume for standard figures and objects, and for combined figures.
Identify, measure and describe circles and the relationships of the radius, diameter, circumference and area and use the concepts to solve problems.
Measure the circumference and diameter of at least five different circular objects and explain how the ratios approach pi.
Find surface areas of rectangular prisms.
Describe the meaning of and calculate:
1. area of rectangles and triangles
2. perimeter of simple polygons
Find the sum of the angles in simple polygons (up to eight sides) with and without measuring angles.
Explore identify, describe spatial relationships and properties of 2D shapes using manipulatives.

Data Analysis, Statistics and Probability

Standard Four: Students will engage in problem solving, communicating, reasoning, connecting and representing as they:

Benchmarks:

Formulate questions that can be answered with data and collect, organize and display relevant data to answer them.
Select and use appropriate statistical methods to analyze data.
Develop and evaluate inferences and predictions that are based on data.
Understand and apply basic concepts of probability.
Describe and compare data sets using the concepts of median, mean, mode, maximum and minimum and range.
Construct and interpret stem-and-leaf plots, line plots and circle graphs.
Construct and interpret line plots, stem and leaf plots and frequency distributions.
Use tree diagrams and other models to represent possible or actual outcomes of trials. Analyze the outcomes.
Utilize appropriate technology to simulate, display, graph and analyze data with multiple representations such as tables, charts, and graphs.
Predict the probability of outcomes of simple experiments and test the predictions. Use appropriate ratios between 0 and 1 to represent the probability of the outcome and associate the probability with the likelihood of the event.
Read and interpret statistical data to make predictions, inferences, and decisions.
Use the’ best-line through data’ concept to make predictions and to solve problems.
Cite at least five world situations and show how probability is used in each.

Patterns, Relations and Algebra

Standard Five: Students will engage in problem solving, communicating, reasoning, connecting and representing.

Benchmarks:

Understand patterns, relations and functions.
Represent and analyze mathematical situations and structures using algebraic symbols.
Use mathematical models to represent and understand quantitative relationships.
Analyze change in various contexts.
Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns and progressions, e.g., ABBCCC…; 1, 5, 9, 13, …; 3, 9, 27, ….
Describe and represent patterns using models, tables, graphs, and rules.
Locate ordered pairs to graph 2D figures/data.
Use patterns involving integers and positive rational numbers to solve problems.
Represent number patterns with verbal and written rules and standard algebraic notations: describe the four properties and relationships.
Replace variables with given values and evaluate/simplify, e.g., 2 (▲) + 3 when ▲ = 4.
Use expressions with variables (i.e., letters, shapes) to understand and describe algebraic relationships, mappings, formulas, expressions, equations and inequalities.
Use algebraic procedures and strategies to solve real-world and mathematical problems using equations and inequalities.
Use the properties of equality to solve problems, e.g., if  + 7 = 13, then  = 13 – 7, therefore  = 6; if 3 x  = 15, then ⅓ x 3 x  = ⅓ x 15, therefore  = 5.
Use multiple strategies and tools to solve equations.
Represent real situations and mathematical relationships with concrete models, tables, graphs, and rules in words and with symbols, e.g., input-output tables.
Solve linear equations using concrete models, tables, graphs, and paper-pencil methods.
Produce and interpret graphs that represent the relationship between two variables in everyday situations.
Identify and describe relationships between two variables with a constant rate of change. Contrast these with relationships where the rate of change is not constant.
Describe, represent and compare rates of change using graphs in meaningful context.

Discussion, Presentation, Composition

Standard Six: Express ideas in an organized way.

Benchmarks:

Explain their mathematical thinking in writing.
Maintain a system for collecting, referring to, and sharing their work.

Number Sense and Operations:

Standard One: Students will engage in problem solving, communicating, reasoning, connecting and representing.

Benchmarks:

1.1 Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
1.2 Understand meanings of operations and how they relate to one another.
1.3 Compute fluently and make reasonable estimates.

Standard Two: They will:

Benchmarks:

2.1 Demonstrate an understanding of positive integer exponents, in particular, when used in powers of ten, e.g. 102, 105.
2.2 Find and use common multiples, factors, prime and composite numbers.
2.3 Use divisibility tests (2,3,4,5,6,8,9,10,11, and their multiples.
2.4 Master the use of primes and properties of numbers such as, GCF and LCM to compute and/or approximate powers and roots.
2.5 Demonstrate an understanding of place value to billions and thousandths.
2.6 Identify and use place value in standard and expanded form.
2.7 Represent and compare very large (billions) and very small (thousandths) positive numbers in various forms such as expanded notation without exponents, e.g., 9724 = 9 x 1000+ 7 x 2 x 10 + 4.
2.8 Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection and as locations on the number line.
2.9 Identify and determine common equivalent fractions, mixed numbers, decimals, and percents.
2.10 Find and position integers, fractions, mixed numbers, and decimals (both positive and negative) on the number line.
2.11 Read, write, and understand conceptually integers (positive and negative), fractions, and decimals.
2.12 Compare and order integers (including negative integers), and positive fractions, mixed numbers, decimals, and percents.
2.13 Apply number theory concepts—including prime and composite numbers, prime factorization, greatest common factor, least common multiple, and divisibility rules for 2, 3, 4, 5, 6, 9, and 10—to the solution of problems.
2.14 Select and use appropriate operations to solve problems involving addition, subtraction, multiplication, division, and positive integer exponents with whole numbers, and with positive fractions, mixed numbers, decimals, and percents.
2.15 Use the number line to model addition and subtraction of integers, with the exception of subtracting negative integers.
2.16 Use a variety of approaches for computing and solving problems.
2.17 Demonstrate an understanding of the inverse relationship of addition and subtraction, and use that understanding to simplify computation and solve problems.
2.18 Use concrete and abstract models to understand and describe the mathematical processes underlying the operation of addition, subtraction, multiplication and division (and their relationship with one another) on fractions, decimals and integers
2.19 Accurately and efficiently add, subtract, multiply, and divide (with double-digit divisors) whole numbers and positive decimals.
2.20 Compute with whole numbers, fractions, decimals, and integers.
2.21 Use effective models and processes of computation with understanding (mental, oral, written, calculator, computer, etc.).
2.22 Accurately and efficiently add, subtract, multiply, and divide positive fractions and mixed numbers. Simplify fractions.
2.23 Compute with whole numbers, fractions, decimals and integers
2.24 Add and subtract integers, with the exception of subtracting negative integers.
2.25 Use operations involving fractions, decimals, and integers
2.26 Estimate results of computations with whole numbers, and with positive fractions, mixed numbers, decimals, and percents. Describe reasonableness of estimates.
2.27 Round to the nearest ten, hundred, thousand, million, billion.
2.28 Round decimals and fractions to the nearest pre-assigned value.
2.29 Use estimation to solve problems involving, money, length, area, perimeter and volume.
2.30 Create and solve meaningful problems that require the use of numbers other than whole numbers.
2.31 Know and use order of operations including the use of parentheses.

Patterns, Relations and Algebra

Standard Three: Students will engage in problem solving, communicating, reasoning, connecting and representing.

Benchmarks:

3.1Understand patterns, relations and functions.
3.2Represent and analyze mathematical situations and structures using algebraic symbols.
3.3Use mathematical models to represent and understand quantitative relationships.
3.4Analyze change in various contexts.

Standard Four: They will:

Benchmarks:

4.1 Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns and progressions, e.g., ABBCCC…; 1, 5, 9, 13, …; 3, 9, 27, ….
4.2 Describe and represent patterns using models, tables, graphs, and rules.
4.3 Locate ordered pairs to graph 2D figures/data.
4.4 Use patterns involving integers and positive rational numbers to solve problems.
4.5 Represent number patterns with verbal and written rules and standard algebraic notations: describe the four properties and relationships.
4.6 Replace variables with given values and evaluate/simplify, e.g., 2 (▲) + 3 when ▲ = 4.
4.7 Use expressions with variables (i.e., letters, shapes) to understand and describe algebraic relationships, mappings, formulas, expressions, equations and inequalities.
4.8 Use algebraic procedures and strategies to solve real-world and mathematical problems using equations and inequalities.
4.9 Use the properties of equality to solve problems, e.g., if X + 7 = 13, then X = 13 – 7, therefore X = 6; if 3 x X = 15, then ⅓ x 3 x X = ⅓ x 15, therefore X = 5.
4.10 Use multiple strategies and tools to solve equations.
4.11 Represent real situations and mathematical relationships with concrete models, tables, graphs, and rules in words and with symbols, e.g., input-output tables.
4.12 Solve linear equations using concrete models, tables, graphs, and paper-pencil methods.
4.13 Produce and interpret graphs that represent the relationship between two variables in everyday situations.
4.14 Identify and describe relationships between two variables with a constant rate of change. Contrast these with relationships where the rate of change is not constant.
4.15 Describe, represent and compare rates of change using graphs in meaningful context.

Geometry

Standard Five: Students will engage in problem solving, communicating, reasoning, connecting and representing as they:

Benchmarks:

5.1 Analyze characteristics and properties of two geometric shapes and develop mathematical arguments about geometric relationships.
5.2 Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
5.3 Apply transformations and use symmetry to analyze mathematical situations.
5.4 Use visualization, spatial reasoning, and geometric modeling to solve problems./li>
5.5 Identify polygons based on their properties, including types of interior angles, perpendicular or parallel sides, and congruence of sides, e.g., squares, rectangles, rhombuses, parallelograms, trapezoids, and isosceles, equilateral, and right triangles.
5.6Identify three-dimensional shapes (e.g., cubes, prisms, spheres, cones, and pyramids) based on their properties, such as edges and faces.
5.7Identify relationships among points, lines, and planes, e.g., intersecting, parallel, perpendicular.
5.8Graph points and identify coordinates of points on the Cartesian coordinate plane (all four quadrants).
5.9Predict, describe, and perform transformations on two-dimensional shapes, e.g., translations, rotations, and reflections.
5.10Identify and describe transformations, reflections, translations, slides, flips, and rotations.
5.11Tessellate shapes across a plane
5.12Identify types of symmetry, including line and rotational.
5.13Identify point and line symmetry in given polygons.
5.14Identify and describe transformations, reflections, translations, slides, flips, and rotations.
5.15Determine if two shapes are congruent by measuring sides or a combination of sides and angles, as necessary; or by motions of series of motions, e.g., translations, rotations, and reflections.
5.16Identify, describe and draw congruent and similar figures.

Measurement

Standard Six: Students will engage in problem solving, communicating, reasoning, connecting and representing as they:

6.1Understand measurable attributes of objects and the units, systems, and processes of measurement.
6.2Apply appropriate techniques, tools, and formulas to determine measurements.

They will…

6.3Apply the concepts of perimeter and area to the solution of problems. Apply formulas where appropriate.
6.4Develop and apply formulas for area and perimeter for standard figures and objects, and for combined figures.
6.5Identify, measure, describe, classify, and construct various angles, triangles and quadrilaterals.
6.6Identify, name, and construct plane and simple solid figures.
6.7Solve problems involving proportional relationships and units of measurements.
6.8Apply the comparison of similar figures and the ratio of their corresponding lengths and areas to solve problems.
6.9Find areas of triangles, parallelograms. Recognize that shapes with the same number of sides but with different appearances can have the same area. Develop strategies to find area of complex shapes.
6.10Develop and apply formulas for area, perimeter, and volume for standard figures and objects, and for combined figures.
6.11Identify, measure and describe circles and the relationships of the radius, diameter, circumference and area and use the concepts to solve problems.
6.12Measure the circumference and diameter of at least five different circular objects and explain how the ratios approach pi.
6.13Find surface areas of rectangular prisms.
6.14Describe the meaning of and calculate:
6.15area of rectangles and triangles
6.16perimeter of simple polygons
6.17Find the sum of the angles in simple polygons (up to eight sides) with and without measuring angles.
6.18Explore identify, describe spatial relationships and properties of 2D shapes using manipulatives.

Data Analysis, Statistics and Probability

Standard Seven: Students will engage in problem solving, communicating, reasoning, connecting and representing as they:

Benchmarks:

7.1Formulate questions that can be answered with data and collect, organize and display relevant data to answer them.
7.2Select and use appropriate statistical methods to analyze data.
7.3Develop and evaluate inferences and predictions that are based on data.
7.4Understand and apply basic concepts of probability.<
7.5Describe and compare data sets using the concepts of median, mean, mode, maximum and minimum and range.
7.6Construct and interpret stem-and-leaf plots, line plots and circle graphs.
7.7Construct and interpret line plots, stem and leaf plots and frequency distributions.
7.8Use tree diagrams and other models to represent possible or actual outcomes of trials. Analyze the outcomes.
7.9Utilize appropriate technology to simulate, display, graph and analyze data with multiple representations such as tables, charts, and graphs.
7.10Predict the probability of outcomes of simple experiments and test the predictions. Use appropriate ratios between 0 and 1 to represent the probability of the outcome and associate the probability with the likelihood of the event.
7.11Read and interpret statistical data to make predictions, inferences, and decisions.
7.12Use the’ best-line through data’ concept to make predictions and to solve problems.
7.13Cite at least five world situations and show how probability is used in each.

Discussion, Presentation, Composition

Express ideas in an organized way.
Explain their mathematical thinking in writing.
Maintain a system for collecting, referring to, and sharing their work.