Maths Grade 7

PAN-ASIA MATHEMATICS: GRADE 7

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.
Compare, order, estimate, and translate among integer, fractions and mixed numbers (i.e., rational numbers), decimals and percents.
Continue to recognize, compare, order and graph integers and rational numbers on a number line; locate a number between two given numbers.
Differentiate between types of numbers (whole and real numbers, integers, rational and irrational number)
Define, compare, order, and apply frequently used irrational numbers, such as and .
Continue to reinforce computations with fractions(decimals, percents, ratio and proportion) and real numbers(integers, rationals, irrationals)
Use ratios and proportions in the solution of problems, in particular, problems involving unit rates, scale factors, and rate of change.
Understand and apply ratio and proportion to probability and geometry to solve numerical and algebraic problems.
Represent numbers in scientific notation, and use them in calculations and problem situations.
Recognize and write in exponential notation.
Demonstrate an understanding of absolute value, e.g., I –3I =I3I = 3.
Apply the rules of powers and roots to the solution of problems. Extend the Order of Operations to include positive integer exponents and square roots.
Identify and use place value in exponential, standard, and expanded form(including negative, positive, and zero powers) and apply then in meaningful problem solving situations.
Understand, use and perform four fundamental arithmetic operations on whole numbers, fractions, decimals, percents and integers: use order of operations.
Demonstrate an understanding of the properties of arithmetic operations on rational numbers. Use the associative, commutative, and distributive properties; properties of the identity and inverse elements (e.g., -7+7 = 0; ¾ x 4/3 = 1); and the notion of closure of a subset of the rational numbers under an operation (e.g., the set of odd integers is closed under multiplication but not under addition).
Use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems, e.g., multiplying by ½ or 0.5 is the same as dividing by 2.
Estimate and compute with fractions (including simplification of fractions), integers, decimals, and percents (including those greater than 100 and less than 1).
Estimate the outcomes of operations on whole numbers, fractions, decimals and percents and scientific notation.
Determine when an estimate rather than an exact answer is appropriate and apply in problems.
Use estimation to solve problems involving money, length area, perimeter, and volume.
Apply a variety of methods to check reasonableness of results.
Select and use appropriate operations – addition, subtraction, multiplication, division, and positive integer exponents – to solve problems with rational numbers (including negatives).

Geometry

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

Benchmarks:

Analyze characteristics and properties of two- and three-dimensional 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.
Analyze, apply, and explain the relationship between the number of sides and the sums of the interior and exterior angle measures of polygons.
Classify figures in terms of congruence and similarity, and apply these relationships to the solution of problems.
Demonstrate an understanding of the relationships of angles formed by intersecting lines, including parallel lines cut by a transversal.
Identify, draw, and describe line segments, rays, angles using letters and measuring angles with a protractor.
Identify and draw basic geometric figures(point, line, plane, intersect, line segment, endpoint, ray, and angles, vertex of an angle and side of an angle)
Use a straightedge, compass, or other tools to formulate and test conjectures, and to draw geometric figures.
Predict the results of transformations on unmarked or coordinate planes and draw the transformed figure, e.g., predict how tessellations transform under translations, reflections, and rotations.
Continue to explore, identify and use transformations such as flips, turns, rotations, translations and composite transformations.
Identify three-dimensional figures (e.g., prisms,) by their physical appearance, distinguishing attributes, and spatial relationships such as parallel faces.
Recognize and draw two-dimensional representations of three-dimensional objects, e.g., nets, projections, and perspective drawings.
Relate geometric shapes and ideas to the measurement, ratio and proportion.

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.
Select, convert (within the same system of measurement), and use appropriate units of measurement or scale.
Choose appropriate units of measurement and supply the concept of accuracy at predetermined levels.
Use customary and metric units for length, mass and capacity.
Describe and use estimates and actual measurements in real life situations.
Understand the process and relate measurement to number, data, and geometry.
Use powers of ten to metric measurement and scientific notation.
Given the formulas, convert from one system of measurement to another. Use technology as appropriate.
Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles.
Given the formulas, determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate.
Use ratio and proportion (including scale factors) in the solution of problems, including problems involving similar plane figures and indirect measurement.
Recognize and draw symmetric, similar, and congruent figures and solve problems using similarity of figures.
Develop and apply formulas for area, perimeter, and volume for standard figures, objects, and for figures both 2D and 3D.
Use models, graphs, and formulas to solve simple problems involving rates, e.g., velocity and density.

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 the characteristics and limitations of a data sample. Identify different ways of selecting a sample, e.g., convenience sampling, responses to a survey, random sampling.
Select, create, interpret, and utilize various tabular and graphical representations of data, e.g., circle graphs, Venn diagrams, scatter plots, stem-and-leaf plots, box-and-whisker plots, histograms, tables, and charts.
Create simple algorithms to solve problems.
Use counting techniques (tree diagram, permutation and combination techniques) to determine the number of outcomes for situations.
Investigate the benefits of simple circuits and networks.
Use logic and inductive reasoning to make predictions related to a series of statements.
Find, describe, and interpret appropriate measures of central tendency (mean, median, and mode) and spread (range) that represent a set of data. Use these notions to compare different sets of data.
Analyze survey data to make predictions and to solve problems.
Read and interpret statistical data to make predictions, inferences, and decisions.
Use tree diagrams, tables, organized lists, basic combinations (“fundamental counting principle”), and area models to compute probabilities for simple compound events, e.g., multiple coin tosses or rolls of dice.
Carry out probability experiments, discuss the results.
Conduct experiments to determine experimental probabilities and construct a table to establish theoretical probabilities and compare two results.

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.
Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words, and when possible, symbolic expressions. Include arithmetic and geometric progressions.
Describe, and develop patterns using numbers, variables, and geometric figures.
Develop patterns into algebraic forms(functions, relations, general term)
Describe and represent patterns using models, tables, graphs, simple rules, and manipulatives
Develop and apply the idea of a function as a certain sort of relationship between quantities.
Use patterns and functions to solve problems.
Use patterns involving integers and positive rational numbers to solve problems.
Use a calculator to graph ordered pairs and convert numerical patterns from tables to graphs.
Evaluate simple algebraic expressions for given variable values, e.g., 3a2- b for a=3 and b=7.
Demonstrate an understanding of the identity (-x)(-y)=xy. Use this identity to simplify algebraic expressions, e.g., (-2)(-x+2 )= 2x – 4.
Continue to use concrete and abstract models to understand and describe the mathematical processes underlying the operations of addition, subtraction, multiplication, and division(and their relationship with one another) on fractions, decimals, and integers.
Develop and apply key concepts such as variable, equivalence, order, and inverse in the context of number, algebra, and geometry; use order of operations on algebraic expressions.
Create and use symbolic expressions; relate them to verbal, tabular, graphic representations.
Identify the slope of a line as a measure of its steepness and as a constant rate of change from its table of values, equation, or graph. Apply the concept of slope to the solution of problems.
Use calculators, computers, concrete manipulatives, and real life situations to explore and describe linear relationships and to solve simple linear equations.
Represent real-life situations to solve linear equations.
Identify the roles of variables within an equation, e.g., y=mx+b, expressing y as a function of x with parameters m and b.
Set up and solve linear equations and inequalities with one or two variables, using algebraic methods, models, and/or graphs.
Form and manipulate equations or inequalities to solve problems involving geometry, probability, and statistics.
Solve and graph solutions to inequalities.
Explain and analyze--both quantitatively and qualitatively, using pictures, graphs, charts, or equations--how a change in one variable results in a change in another variable in functional relationships, e.g., C= d, A= r2 (A as a function of r), A rectangle =lw (A rectangle as a function of l and w).
Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate.
Understand how algebra is used in the real world (as it relates to ratio and proportion).
Use tables and graphs to represent and compare linear growth patterns. In particular, compare rates of change and x- and y-intercepts of different linear patterns.
Use and find a function rule from a table of data, graphs and rules.

Discussion, Presentation, Composition

Standard Six: Express ideas in an organized way.

Benchmarks:

Use agreed upon rules to participate in discussions in large and small groups.
Express ideas in an organized way.
Explain their mathematical thinking in writing.
Maintain a system for collecting, referring to, and sharing their work.

Number Sense and Operations:

Students will…

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.

They will…

Compare, order, estimate, and translate among integer, fractions and mixed numbers (i.e., rational numbers), decimals and percents.
Continue to recognize, compare, order and graph integers and rational numbers on a number line; locate a number between two given numbers.
Differentiate between types of numbers (whole and real numbers, integers, rational and irrational number)
Define, compare, order, and apply frequently used irrational numbers, such as √2 and π
Continue to reinforce computations with fractions(decimals, percents, ratio and proportion) and real numbers(integers, rationals, irrationals)
Use ratios and proportions in the solution of problems, in particular, problems involving unit rates, scale factors, and rate of change.
Understand and apply ratio and proportion to probability and geometry to solve numerical and algebraic problems.
Represent numbers in scientific notation, and use them in calculations and problem situations.
Recognize and write in exponential notation.
Demonstrate an understanding of absolute value, e.g., I –3I =I3I = 3.
Apply the rules of powers and roots to the solution of problems. Extend the Order of Operations to include positive integer exponents and square roots.
Identify and use place value in exponential, standard, and expanded form(including negative, positive, and zero powers) and apply then in meaningful problem solving situations.
Understand, use and perform four fundamental arithmetic operations on whole numbers, fractions, decimals, percents and integers: use order of operations.
Demonstrate an understanding of the properties of arithmetic operations on rational numbers. Use the associative, commutative, and distributive properties; properties of the identity and inverse elements (e.g., -7+7 = 0; ¾ x 4/3 = 1); and the notion of closure of a subset of the rational numbers under an operation (e.g., the set of odd integers is closed under multiplication but not under addition).
Use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems, e.g., multiplying by ½ or 0.5 is the same as dividing by 2.
Estimate and compute with fractions (including simplification of fractions), integers, decimals, and percents (including those greater than 100 and less than 1).
Estimate the outcomes of operations on whole numbers, fractions, decimals and percents and scientific notation.
Determine when an estimate rather than an exact answer is appropriate and apply in problems.
Use estimation to solve problems involving money, length area, perimeter, and volume.
Apply a variety of methods to check reasonableness of results.
Select and use appropriate operations – addition, subtraction, multiplication, division, and positive integer exponents – to solve problems with rational numbers (including negatives).

Patterns, Relations and Algebra

Students will…

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.

They will…

Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words, and when possible, symbolic expressions. Include arithmetic and geometric progressions.
Describe, and develop patterns using numbers, variables, and geometric figures.
Develop patterns into algebraic forms(functions, relations, general term).
Describe and represent patterns using models, tables, graphs, simple rules, and manipulatives
Develop and apply the idea of a function as a certain sort of relationship between quantities.
Use patterns and functions to solve problems.
Use patterns involving integers and positive rational numbers to solve problems.
Use a calculator to graph ordered pairs and convert numerical patterns from tables to graphs.
Evaluate simple algebraic expressions for given variable values, e.g., 3a²- b for a=3 and b=7.
Demonstrate an understanding of the identity (-x)(-y)=xy. Use this identity to simplify algebraic expressions, e.g., (-2)(-x+2 )= 2x – 4.
Continue to use concrete and abstract models to understand and describe the mathematical processes underlying the operations of addition, subtraction, multiplication, and division(and their relationship with one another) on fractions, decimals, and integers.
Develop and apply key concepts such as variable, equivalence, order, and inverse in the context of number, algebra, and geometry; use order of operations on algebraic expressions.
Create and use symbolic expressions; relate them to verbal, tabular, graphic representations.
Identify the slope of a line as a measure of its steepness and as a constant rate of change from its table of values, equation, or graph. Apply the concept of slope to the solution of problems.
Use calculators, computers, concrete manipulatives, and real life situations to explore and describe linear relationships and to solve simple linear equations.
Represent real-life situations to solve linear equations.
Identify the roles of variables within an equation, e.g., y=mx+b, expressing y as a function of x with parameters m and b.
Set up and solve linear equations and inequalities with one or two variables, using algebraic methods, models, and/or graphs.
Form and manipulate equations or inequalities to solve problems involving geometry, probability, and statistics.
Solve and graph solutions to inequalities.
Explain and analyze __both quantitatively and qualitatively, using pictures, graphs, charts, or equations__how a change in one variable results in a change in another variable in functional relationships, e.g., C=π d, A=π r² (As a function of r), A_rectangle =lw (A_rectangle as a function of l and w).
Use linear equations to model and analyze problems involving proportional relationships. Use technology as appropriate.
Understand how algebra is used in the real world(as it relates to ratio and proportion).
Use tables and graphs to represent and compare linear growth patterns. In particular, compare rates of change and x- and y-intercepts of different linear patterns.
Use and find a function rule from a table of data, graphs and rules.

Geometry

Students will…

Analyze characteristics and properties of two- and three-dimensional 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.

They will…

Analyze, apply, and explain the relationship between the number of sides and the sums of the interior and exterior angle measures of polygons.
Classify figures in terms of congruence and similarity, and apply these relationships to the solution of problems.
Demonstrate an understanding of the relationships of angles formed by intersecting lines, including parallel lines cut by a transversal.
Identify, draw, and describe line segments, rays, angles using letters and measuring angles with a protractor.
Identify and draw basic geometric figures(point, line, plane, intersect, line segment, endpoint, ray, and angles, vertex of an angle and side of an angle)
Use a straightedge, compass, or other tools to formulate and test conjectures, and to draw geometric figures.
Predict the results of transformations on unmarked or coordinate planes and draw the transformed figure, e.g., predict how tessellations transform under translations, reflections, and rotations.
Continue to explore, identify and use transformations such as flips, turns, rotations, translations and composite transformations.
Identify three-dimensional figures (e.g., prisms,) by their physical appearance, distinguishing attributes, and spatial relationships such as parallel faces.
Recognize and draw two-dimensional representations of three-dimensional objects, e.g., nets, projections, and perspective drawings.
Relate geometric shapes and ideas to the measurement, ratio and proportion.

Measurement

Students will…

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

They will…

Select, convert (within the same system of measurement), and use appropriate units of measurement or scale.
Choose appropriate units of measurement and supply the concept of accuracy at predetermined levels.
Use customary and metric units for length, mass and capacity.
Describe and use estimates and actual measurements in real life situations.
Understand the process and relate measurement to number, data, and geometry.
Use powers of ten to metric measurement and scientific notation.
Given the formulas, convert from one system of measurement to another. Use technology as appropriate.
Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles. Given the formulas, determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate.
Use ratio and proportion (including scale factors) in the solution of problems, including problems involving similar plane figures and indirect measurement.
Recognize and draw symmetric, similar, and congruent figures and solve problems using similarity of figures.
Develop and apply formulas for area, perimeter, and volume for standard figures, objects, and for figures both 2D and 3D.
Use models, graphs, and formulas to solve simple problems involving rates, e.g., velocity and density.

Data Analysis, Statistics and Probability

Students will,,,

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.

They will…

Describe the characteristics and limitations of a data sample. Identify different ways of selecting a sample, e.g., convenience sampling, responses to a survey, random sampling.
Select, create, interpret, and utilize various tabular and graphical representations of data, e.g., circle graphs, Venn diagrams, scatterplots, stem-and-leaf plots, box-and-whisker plots, histograms, tables, and charts.
Create simple algorithms to solve problems.
Use counting techniques(tree diagram, permutation and combination techniques) to determine the number of outcomes for situations.
Investigate the benefits of simple circuits and networks.
Use logic and inductive reasoning to make predictions related to a series of statements.
Find, describe, and interpret appropriate measures of central tendency (mean, median, and mode) and spread (range) that represent a set of data. Use these notions to compare different sets of data.
Analyze survey data to make predictions and to solve problems.
Read and interpret statistical data to make predictions, inferences, and decisions.
Use tree diagrams, tables, organized lists, basic combinaations (“fundamental counting principle”), and area models to compute probabilities for simple compound events, e.g., multiple coin tosses or rolls of dice.
Carry out probability experiments, discuss the results.
Conduct experiments to determine experimental probabilities and construct a table to establish theoretical probabilities and compare two results.

Discussion, Presentation, Composition

Use agreed upon rules to participate in discussions in large and small groups.
Express ideas in an organized way.
Explain their mathematical thinking in writing.
Maintain a system for collecting, referring to, and sharing their work.

Students will be able to…