middle Block:
Mathematics
Learners work individually, with partners, or in teams within the context of critical thinking, reasoning and justification, and problem solving. They calculate percent, rate, and probability, and work with increasingly bigger and smaller numbers, interpret graphs, and critique data. Learners develop their understanding, communicate, and justify through concrete, pictorial, and/or symbolic representations of mathematics. Problem solving includes the use of mathematical tools (manipulatives), graphing, written work, and technology. Learners enact prior mathematics knowledge, create, and ask questions while applying understanding to concepts and events in their lives and other subject areas. Increasingly complex and formal mathematics allows learners to quantify and organize their world.
CONTEXTS AND CONCEPTS
APPLICATION
STRATEGIES
- Determining when to use algorithms, mental procedures, technology/tools, or other strategies
- Determining appropriate units of measure
- Discovering the most efficient strategies
- Determining the reasonableness of an answer and explaining thinking or rules
- Verifying solutions with substitution
PROCESSES
- Developing mathematical rules and algorithms
- Using manipulatives
- Modeling
- Determining unit rate
- Creating graphs
FLUENCY
- Conversion between formats, representations, and equivalents of numbers
- Ways to present information/data
- Automaticity (with perfect squares)
- Scale
COMMUNICATION
- Representing mathematics concretely, pictorially, and symbolically
- Algebraic expressions
- Using formal mathematical vocabulary
- Using formal mathematical symbols
- Decimal notation
- Bar notation
- Introducing terms repeating and period
- Equations
- Using variables
- Using coefficients
- Using the congruence symbol
Strand: Number
Big Idea: Number Sense
Skill Descriptor: Describe numbers, ways of representing numbers, relationships among numbers, and number systems.
Global Competencies: CM, CTPS, SASM
Achievement Indicators:
- Provide examples where large numbers and decimals are used
- Explain the properties of prime and composite numbers, including zero and one as neither prime nor composite
- Provide and sort sets of prime and composite numbers
- Explain the pattern of place value for any magnitude
- Identify multiples and factors of whole numbers
- Justify examples of fractions to their decimal equivalents
- Sort sets of fractions representing repeating and terminating decimals
- Predict decimal representations of fractions using patterns
- Explain strategies to find multiples and factors of numbers
- Apply arrays, repeated division, and factor trees to solve problems involving factors and multiples
Skill Descriptor: Describe percentage, ratio, and rate.
Achievement Indicators:
- Explain percentage indicators below within the 0%-100% range such as: 0%, 1%, 10%, 50%, and 100%
- Provide examples where ratios and rates are used
- Represent percent of numbers concretely or pictorially
- Represent part-to-part ratios as part-to-whole fractions
- Represent rates using words or symbolically
- Express ratios as percentages
- Express percentages in decimal and fractional forms and vice-versa
- Explain that percentage represents a ratio out of 100
- Justify when an approximate answer is appropriate
- Solve rate, ratio, and percent problems
Big Idea: Operations
Skill Descriptor: Add, subtract, multiply, and divide decimals to solve problems (1-digit whole number multipliers and 1-digit natural number divisors).
Global Competencies: CTPS, SASM
Achievement Indicators:
- Explain decimal indicators below using benchmark decimals such as: 0.1, 0.25, 0.5, 0.75, 0.01, 0.05
- Estimate sums, differences, products, and quotients
- Add, subtract, multiply, and divide decimals with and without models (limited to hundredths)
- Solve problems using decimal operations
- Justify the reasonableness of solutions, by applying the principles of place value (e.g., front-end, magnitude, and value of digit)
Skill Descriptor: Add, subtract, multiply, and divide benchmark fractions and mixed numbers to solve problems (limited to positive sums and differences).
Global Competencies: CTPS
Achievement Indicators:
- Explain fraction indicators below using friendly unit fractions such as: 1/2 , 1/4, 1/8 or 1/2, 1/5, 1/10 or 1/3, 1/6
- Simplify fractions and mixed numbers
- Estimate sums, differences, products, and quotients
- Determine common denominators for sets of positive fractions and mixed numbers
- Add and subtract fractions and mixed numbers with and without the use of models
- Multiply and divide positive fractions by whole numbers with and without the use of models
- Multiply and divide fractions and mixed numbers with and without the use of models
- Justify the reasonableness of solutions
Skill Descriptor: Apply the order of operations (excluding exponents, limited to whole numbers).
Global Competencies: CTPS
Achievement Indicators:
- Apply the order of operations to solve multi-step problems with and without technology
Strand: Statistics and Probability
Big Idea: Data Analysis
Skill Descriptor: Collect and represent data.
Achievement Indicators:
- Create questions to answer, record results, and draw conclusions
- Design and administer questionnaires for collecting data and recording results
- Justify methods for collecting data to answer questions
- Justify appropriate types of graphs for displaying sets of collected data
- Interpret graphs to draw conclusions about data
Skill Descriptor: Analyze tables of values and graphs of linear relations of the forms: ๐ฆ=๐ฅ+๐, ๐ฆ=๐ฅโ๐, ๐ฆ=๐๐ฅ, y=(x/a), ๐โ 0.
Achievement Indicators:
- Describe patterns within columns of tables of values
- Calculate values within a table of values given other values
- Create tables of values given a linear relationships and graphs
- Create and label graphs from tables of values or sets of data (limited to linear graphs with discrete elements)
- Explain which sets of data can be represented by line graphs
Skill Descriptor: Identify points and transformations in the first quadrant of the Cartesian Plane using whole number ordered pairs.
Global Competencies: CTPS
Achievement Indicators:
- Label the axes of the first quadrant of the Cartesian Plane and identify the origin
- Plot points in the first quadrant of the Cartesian Plane given ordered pairs
- Match points in the first quadrant of a Cartesian Plane with their corresponding ordered pair
- Create designs using 2D shapes
- Describe direction of turns, fractions of turns, and points of rotation of 2D shapes
- Describe lines of reflection of 2D shapes
Big Idea: Chance and Uncertainty
Skill Descriptor: Determine the probability of outcomes to solve problems.
Global Competencies: CM, CTPS, ICE
Achievement Indicators:
- Determine theoretical probabilities (sample space) of outcomes occurring
- Predict experimental probabilities of outcomes using theoretical probability
- Conduct probability experiments with and without technology
- Represent probabilities as ratios, fractions, and percentages
- Compare experimental and theoretical probabilities.
- Demonstrate that as the number of trials increases experimental probability approaches theoretical probability
- Explain the difference between theoretical and experimental probability
Strand: Patterns and Relations
Big Idea: Algebra
Skill Descriptor: Develop equations using letter variables.
Global Competencies: CTPS
Achievement Indicators:
- Represent the preservation of equality for addition, subtraction, multiplication, and division concretely and pictorially
- Represent equivalent forms of equations
- Verify the preservation of equality of equations
- Develop equations using letter variables and the commutative property of addition and multiplication
Strand: Shape and Space
Big Idea: Measurement
Skill Descriptor: Investigate perimeter, area, volume, and their relationships.
Achievement Indicators:
- Represent perimeter of polygons concretely and pictorially
- Represent area of rectangles, triangles, and parallelograms concretely and pictorially
- Explain the relationship between area of rectangles and area of parallelograms
- Explain the relationship between area of rectangles and area of triangles
- Represent volume of right rectangular prisms concretely and pictorially
- Explain the relationship between area of the base of right concrete 3D objects and the volume of the concrete 3D objects
- Demonstrate that the orientation of concrete right rectangular prisms does not affect its volume
Skill Descriptor: Describe angles.
Global Competencies: CTPS
Achievement Indicators:
- Provide examples of angles found in the environment
- Describe measures of angles as measures of rotation of one of its sides
- Describe the relationship between and sketch 45ยฐ, 90ยฐ, and 180ยฐ angles without the use of a protractor
- Classify angles according to their measure
- Estimate measures of angles
- Measure, create, and label angles in various orientations using a protractor
- Demonstrate that the sum of the interior angles of a triangle is the same for all triangles and that the sum of the interior angles of quadrilaterals is the same for all quadrilaterals
Big Idea: 2-D Shapes and 3-D Objects
Skill Descriptor: Describe 3D objects.
Global Competencies: CTPS
Achievement Indicators:
- Describe faces and edges of 3D objects using language such as parallel, intersecting, perpendicular, vertical, or horizontal
- Provide examples from the environment with edges, faces, and sides that are parallel, intersecting, perpendicular, vertical, and horizontal
- Sort 3D objects by identifying edges, faces, vertices, length of the sides, shape of the base, etc.
- Construct rectangular and triangular prisms using nets and skeletons
- Compare 3D objects to identify common attributes, different orientations, and different dimensions
Strand: Number
Big Idea: Number Sense
Skill Descriptor: Describe numbers, ways of representing numbers, relationships among numbers, and number systems.
Achievement Indicators:
- Provide examples of integers and where they are used
- Explain the pattern on each side of the zero on a number line to compare, order, and verify the position of integers
- Compare integers using the symbols <, >, and =
Big Idea: Operations
Skill Descriptor: Add and subtract positive and negative integers to solve problems.
Achievement Indicators:
- Explain the relationship between adding integers and subtracting integers
- Demonstrate that the sum of opposite integers is zero concretely, pictorially, or symbolically
- Add and subtract integers to solve problems
Skill Descriptor: Add, subtract, multiply, and divide decimals to solve problems (1-digit divisors and 2-digit multipliers).
Global Competencies: CTPS
Achievement Indicators:
- Estimate sums, differences, products, and quotients
- Add, subtract, multiply, and divide decimals with and without models (limited to thousandths)
- Solve problems using decimal operations, considering order of operations
- Justify the reasonableness of solutions, by applying the principles of place value. (e.g., front-end, magnitude, and value of digit)
- Multiply or divide 2-digit multipliers or 1-digit divisors (whole numbers or decimals) with and without the use of technology
Skill Descriptor: Add, subtract, multiply, and divide positive fractions and mixed numbers to solve problems.
Global Competencies: CTPS
Achievement Indicators:
- Simplify fractions and mixed numbers
- Estimate sums, differences, products, and quotients
- Determine common denominators for sets of positive fractions and mixed numbers
- Add and subtract fractions and mixed numbers with and without the use of models
- Multiply and divide positive fractions by whole numbers with and without the use of models
- Multiply and divide fractions and mixed numbers with and without the use of models
- Justify the reasonableness of solutions
Strand: Statistics and Probability
Big Idea: Data Analysis
Skill Descriptor: Calculate measures of central tendency to solve problems.
Achievement Indicators:
- Determine the range of sets of data
- Identify any outliers in sets of data
- Collect data and calculate the mean, the median, and the mode for sets of data
- Provide contexts where mean, median, or mode is the most appropriate measure of central tendency
- Explain why mean, median, and mode for sets of data may be the same or different
- Explain the effect of outliers on measures of central tendency
- Collect data and apply measures of central tendency to identify and solve problems
Skill Descriptor: Analyze linear relations of the forms shown below and compare graphs to solve problems. (e.g., ๐ฆ=๐ฅ+๐, ๐ฆ=๐ฅโ๐, ๐ฆ=๐๐ฅ, y=(x/a), ๐โ 0).
Global Competencies: CTPS
Achievement Indicators:
- Create tables of values using substitution for variables
- Create tables of values of linear relations involving a single transformation
- Create and label graphs of linear relations from tables of values (limited to discrete elements)
- Describe linear relationships displayed in tables of values and graphs using a mathematical expression
- Predict and verify values of unknown terms using linear mathematical expressions and graphs
Skill Descriptor: Identify points and transformations in the four quadrants of the Cartesian Plane using integral ordered pairs.
Global Competencies: CTPS, ICE
Achievement Indicators:
- Label the axes of the four quadrants of the Cartesian Plane and identify the origin
- Plot points in the four quadrants of the Cartesian Plane given integral ordered pairs
- Match points in the four quadrants of the Cartesian Plane with their corresponding ordered pair
- Determine the distance between points along horizontal and vertical lines
- Describe position, direction, translation, rotation, reflection, and other 2D shape transformations
- Represent 2D shape transformations concretely or pictorially
Big Idea: Chance and Uncertainty
Skill Descriptor: Determine the probability of outcomes to solve problems.
Global Competencies: CTPS
Achievement Indicators:
- Provide examples of two independent events
- Determine theoretical probabilities (sample space) of outcomes occurring involving two independent events
- Conduct probability experiments with and without technology
- Represent probabilities as ratios, fractions, and percentages
- Compare experimental and theoretical probabilities for outcomes involving two independent events
Strand: Patterns and Relations
Big Idea: Algebra
Skill Descriptor: Solve linear equations.
Achievement Indicators:
- Provide examples of expressions and equations
- Provide examples of contexts where linear relations represent patterns
- Identify constant terms, numerical coefficients, and variables in expressions and equations
- Represent linear equations concretely, pictorially, and symbolically
- Represent relationship and patterns using linear relations
- Evaluate expressions by substituting values for the unknowns
- Apply and explain the process of the preservation of equality for addition, subtraction, multiplication, and division
- Verify the preservation of equality of equations
- Represent problems with linear equations
- Justify the process to solve linear equations
Strand: Shape and Space
Big Idea: Measurement
Skill Descriptor: Apply the formulae for perimeter of polygons, area of rectangles, area of triangles, area of parallelograms, and volume of right prisms to solve problems.
Achievement Indicators:
- Generalize rules for determining perimeter of polygons, area of rectangles, area of parallelograms, area of triangles, and volume of right rectangular prisms
- Explain strategies used to estimate area of rectangles, triangles, and parallelograms
- Calculate volumes of right prisms given the area of the base
- Calculate the perimeter of polygons, area of rectangles, area of triangles, area of parallelograms, and volume of right prisms using the appropriate formula to solve problems
Skill Descriptor: Perform geometric constructions.
Achievement Indicators:
- Describe parallel line segments, perpendicular line segments, bisectors of angles, and perpendicular bisectors
- Create line segments parallel and perpendicular to other line segments
- Create bisectors of angles using multiple methods
- Create perpendicular bisectors of line segments using multiple methods
- Verify the equality or congruency of bisectors and perpendicular bisectors
Big Idea: 2-D Shapes and 3-D Objects
Skill Descriptor: Draw and verify nets.
Global Competencies: CTPS
Achievement Indicators:
- Relate 3D objects to nets and skeletons and vice versa
- Construct 3D objects, including rectangular and triangular prisms, using nets and skeletons
- Match nets to 3D objects
- Create and verify nets for right circular cylinders, right rectangular prisms, and right triangular prisms
Strand: Number
Big Idea: Number Sense
Skill Descriptor: Describe numbers, ways of representing numbers, relationships among numbers, and number systems.
Achievement Indicators:
- Provide examples where percentage may be between 0% and 1%, and examples where percentage may be more than 100%.
- Represent percentages greater than 100 using models (e.g., number lines, number frames, )
- Represent proper and improper fractional percentages using models (e.g., number lines, number frames, etc.)
- Record percentages in decimal and fractional forms and vice-versa
- Compare percentage benchmarks to positive fractions, mixed numbers, and decimals
- Solve percentage, combined problems, and percent-of-a-percent problems
- Represent given perfect squares as square regions concretely or pictorially
- Determine the squares of given numbers
- Identify numbers with square roots within a given range
- Determine the square roots of given perfect squares
- Determine all factors of given perfect squares
- Estimate the square root of numbers that are not perfect squares using benchmarks
- Demonstrate if given numbers are perfect squares concretely, pictorially, or symbolically (e.g., prime factorization)
Big Idea: Operations
Skill Descriptor: Multiply and divide integers to solve problems (2-digit by 1-digit).
Global Competencies: CTPS
Achievement Indicators:
- Provide examples of contexts that require multiplying and dividing integers
- Represent the multiplication and division of integers concretely, pictorially, or symbolically
- Determine the rule for a sign of a product or quotient of integers (e.g., through patterning)
- Multiply and divide integers to solve problems
Skill Descriptor: Apply the order of operations including whole numbers, positive fractions, and decimals.
Global Competencies: CTPS
Achievement Indicators:
- Apply the order of operations to solve multi-step problems with and without technology
- Apply combinations of operations and number forms to solve problems
Strand: Statistics and Probability
Big Idea: Data Analysis
Skill Descriptor: Compare and critique graphs and the presentation of data.
Global Competencies: CM, CTPS, SGC
Achievement Indicators:
- Examine and compare representations of data using a variety of circle graphs
- Convert percentages displayed in circle graphs into quantities
- Create and label circle graphs, including title and legend, with and without technology
- Interpret circle graphs to answer questions and solve problems
- Compare strengths and limitations of different types of graphs
- Recognize bias and discrimination, or stereotypes and misconceptions against Wabanaki peoples in graphs and representations of data
- Explain how the format of graphs may misrepresent or lead to misinterpretation of data
- Justify the choice of graphical representations for contexts and their data sets
Skill Descriptor: Analyze linear relations of the forms shown below and compare graphs to solve problems. (e.g., ๐ฆ=๐๐ฅ+๐, ๐ฆ=๐๐ฅโ๐, y=(x/a)+b, ๐โ 0, y=(x/a)-b, ๐โ 0).
Global Competencies: CTPS
Achievement Indicators:
- Create tables of values using substitution for variables
- Create tables of values of linear relations involving two transformations
- Determine if the plotted points of graphs should be displayed with solid lines
- Justify the types of number sets used to create linear graphs (i.e., whole numbers, real numbers, integers, etc.)
- Create and label graphs of linear relations from tables of values
- Describe linear relationships displayed in tables of values and graphs using a mathematical expression
- Predict and verify values of unknown terms using linear mathematical expressions and graphs
- Determine a missing coordinate in an ordered pair of a linear relation
Big Idea: Chance and Uncertainty
Skill Descriptor: Determine the probability of outcomes to solve problems.
Global Competencies: CM, CTPS, ICE
Achievement Indicators:
- Determine theoretical probabilities (sample space) of outcomes occurring involving two independent events
- Distinguish between dependent and independent events
- Conduct probability experiments, with and without technology, for outcomes involving two independent events
- Represent probabilities as ratios, fractions, and percentages
- Compare experimental and theoretical probabilities for outcomes involving two independent events
- Solve probability problems involving two independent events
Strand: Patterns and Relations
Big Idea: Algebra
Skill Descriptor: Solve linear equations.
Global Competencies: CTPS
Achievement Indicators:
- Represent linear equations pictorially and symbolically
- Apply the distributive property to solve linear equations
- Verify solutions to linear equations using different methods
- Solve linear equation problems
Strand: Shape and Space
Big Idea: Measurement
Skill Descriptor: Apply the formulae for volume of right cylinders to solve problems.
Achievement Indicators:
- Calculate the volume of right cylinders using the appropriate formula to solve problems
- Explain the relationship between area of the base of a right cylinder and the volume of a right cylinder
- Generalize rules for determining volume of right cylinders
- Apply rules for determining volume of right cylinders
- Demonstrate that the orientation of concrete right cylinders does not affect its volume
Skill Descriptor: Apply properties of circles to solve problems.
Global Competencies: CTPS
Achievement Indicators:
- Demonstrate diameter is twice the radius in circles
- Demonstrate the sum of central angles of circles is 360ยฐ
- Demonstrate circumference is approximately three times the diameter in circles
- Demonstrate pi is the ratio of the circumference to the diameter
- Demonstrate the value of pi is approximately 3.14
- Generalize rules to determine area of circles
- Apply formulae to determine the area of circles
Skill Descriptor: Apply the Pythagorean Theorem to solve problems.
Global Competencies: CTPS
Achievement Indicators:
- Represent the Pythagorean Theorem concretely, pictorially, symbolically, or using technology
- Demonstrate that the Pythagorean Theorem only applies to right triangles
- Explain the Pythagorean Theorem
- Apply the Pythagorean Theorem to verify if triangles are right triangles
- Determine the measure of the third side of a right triangle to solve problems, given the measures of the other two sides
- Recognize Pythagorean Triples