## 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 small decimals are used
- Provide examples of prime and composite numbers
- Explain the pattern of place value for any magnitude
- Identify multiples and factors of numbers
- Match equivalent fractions and decimals
- Sort sets of numbers as prime or composite
- 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
- Explain the properties of prime and composite numbers
- Explain the properties of zero and one as neither prime nor composite
- Apply arrays, repeated division, and factor trees to solve problems involving factors and multiples

**Skill Descriptor:** Describe percentage, ratio, and rate.

**Achievement Indicators:**

- Provide examples where ratios and rates are used
- Represent percent 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
- Explain when an approximate answer is appropriate
- Apply rounding to solutions
- 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:**

- Estimate products and quotients
- Apply front-end estimation to place the decimal point in sums, differences, products, and quotients
- Add, subtract, multiply, and divide decimals
- Solve problems including money and measurement

**Skill Descriptor:** Add, subtract, multiply, and divide fractions and mixed numbers to solve problems (limited to positive sums and differences).

**Global Competencies:** CTPS

**Achievement Indicators:**

- Express improper fractions as mixed numbers and vice-versa
- Estimate products and quotients
- Determine common denominators for sets of positive fractions and mixed numbers
- Multiply and divide positive fractions by whole numbers
- Add, subtract, multiply, and divide fractions and mixed numbers
- Simplify fractions and mixed numbers

**Skill Descriptor:** Apply the order of operations (excluding exponents).

**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 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

**Skill Descriptor:** Justify elements and compare graphs to solve problems.

**Achievement Indicators:**

- Describe patterns within columns of tables of values
- Describe relationships in tables of values using mathematical expressions
- Represent pattern rules using mathematical expressions
- Calculate values within table of values given other values and pattern rules
- Create tables of values from patterns
- Create tables of values from graphs
- Create and label graphs from tables of values or sets of data (limited to linear graphs with discrete elements)
- Predict and verify values of unknown terms
- Explain which sets of data can be represented by line graphs or series of points
- Justify appropriate types of graphs for displaying sets of collected data
- Interpret graphs to draw conclusions about data

**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 occurring in probability experiments 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 or without technology
- Compare experimental results to theoretical probability of probability experiments
- Demonstrate that as number of trials increases experimental probability approaches theoretical probability
- Explain the difference between theoretical probability 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:** Apply formulae to solve problems.

**Achievement Indicators:**

- Represent perimeter of polygons, volume of right rectangular prisms, and area of rectangles concretely and pictorially
- Generalize rules for determining perimeter of polygons, area of rectangles, area of parallelograms, area of triangles, volume of right rectangular prisms
- Explain strategies used to estimate area of parallelograms
- Calculate volumes of right prisms given the area of the base
- Explain the relationship between area of rectangles and area of triangles
- Explain the relationship between area of rectangles and area of parallelograms
- Explain the relationship between area of the base of right concrete 3D objects and the formula for the volume of the concrete 3D object
- Calculate perimeter of polygons, area of rectangles, area of triangles, area of parallelograms, and volume of right prisms to solve problems

**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 45°, 90°, and 180° angles without the use of a protractor
- Classify angles according to their measure
- Estimate measures of angles
- Measure angles in various orientations
- 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 concrete 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
- Compare concrete 3D objects to identify common attributes, different orientations, and different dimensions
- Sort concrete 3D objects by identifying edges, faces, vertices, length of the sides, shape of the base, etc.
- Relate concrete 3D objects to nets and skeletons
- Construct rectangular and triangular prisms using nets and skeletons
- Demonstrate that the orientation of concrete 3D objects does not affect volume

### 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
- Extend number lines to compare and order integers
- Compare integers using the symbols <, >, and =
- Explain the pattern on each side of the zero on a number line to compare and verify the position of integers

#### Big Idea: Operations

**Skill Descriptor:** Add and subtract positive and negative integers to solve problems.

**Achievement Indicators:**

- Add and subtract integers to solve problems
- Explain the relationship between adding integers and subtracting integers
- Demonstrate that the sum of opposite integers is
*zero*concretely, pictorially, or symbolically

**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
- Apply front-end estimation to place the decimal point in sums, differences, products, and quotients
- Multiply or divide 2-digit multipliers or 1-digit divisors (whole numbers or decimals) with and without the use of technology
- Add, subtract, multiply, and divide decimals (limited to thousandths)

**Skill Descriptor:** Add, subtract, multiply, and divide positive fractions and mixed numbers to solve problems.

**Global Competencies:** CTPS

**Achievement Indicators:**

- Express improper fractions as mixed numbers and vice-versa
- Estimate products and quotients
- Multiply and divide positive fractions by whole numbers
- Add, subtract, multiply, and divide fractions and mixed numbers
- Simplify fractions and mixed numbers
- 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:**

- Provide contexts where mean, median, or mode is the most appropriate measure of central tendency
- Determine the range of sets of data
- Identify any outliers in sets of data
- Calculate mean, median, and mode for sets of data
- 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
- Apply measures of central tendency to solve problems

**Skill Descriptor:** Use tables of values and compare graphs to solve problems.

**Global Competencies:** CTPS

**Achievement Indicators:**

- Describe patterns and relationships found in graphs
- Match sets of linear relations and sets of graphs
- Create tables of values using substitution for variables
- Create tables of values using linear relations
- Create graphs of linear relations from tables of values (limited to discrete elements)

**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, magnitude, 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 occurring in probability experiments 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
- Compare experimental results to theoretical probability for outcomes involving two independent events
- Express probability as ratio, fraction, and percentage

### 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, or 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
- Record the process to solve linear equations

### Strand: Shape and Space

#### Big Idea: Measurement

**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 concrete 3D objects to nets and skeletons
- Construct concrete 3D objects, including rectangular and triangular prisms, using nets and skeletons
- Match nets to concrete 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%
- Identify numbers with square roots that are between two numbers
- Represent given perfect squares as square regions concretely or pictorially
- Represent fractional percentages
- Represent percentages greater than 100
- Estimate the square root of numbers that are not perfect squares using benchmarks
- Determine all factors of given perfect squares
- Determine the square roots of given perfect squares
- Determine the squares of given numbers
- Demonstrate if given numbers are perfect squares concretely, pictorially, or using prime factorization
- Record percentages in decimal and fractional forms and vice-versa
- Solve percentage, combined problems, and percent-of-a-percent problems

#### 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
- Apply a rule for determining the sign of the product and quotient of integers
- Multiply and divide integers
- Solve division of integers problems (2-digit by 2-digit) with the use of technology

**Skill Descriptor:** Apply the order of operations (including whole numbers, integers, positive fractions, and decimals).

**Global Competencies:** CTPS

**Achievement Indicators:**

- Compare percentage benchmarks to positive fractions, mixed numbers, and decimals
- Estimate sums differences, products, and quotients, using benchmarks
- Add, subtract, multiply, and divide decimals
- Add, subtract, multiply, and divide fractions and mixed numbers
- Apply the order of operations to solve multi-step problems with and without technology
- Apply the order of operations with flexibility, efficiency, and accuracy (fluency)
- 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:**

- Provide examples of circle graphs
- Compare strengths and limitations of different types of graphs
- Compare circle graphs and representations of data
- Recognize bias and discrimination, or stereotypes and misconceptions against Wabanaki peoples in graphs and representations of data
- Convert percentages displayed in circle graphs into quantities
- Create and label circle graphs, including title and legend, with and without technology
- Justify the choice of graphical representations for contexts and their data sets
- Explain how the format of graphs may misrepresent or lead to misinterpretation of data
- Interpret circle graphs to answer questions and solve problems

**Skill Descriptor:** Analyze two-variable linear relations and compare graphs to solve problems.

**Global Competencies:** CTPS

**Achievement Indicators:**

- Describe relationships between the variables of graphs
- Determine missing values in the ordered pairs of equations
- Determine if plotted points of graphs are shown with solid lines
- Create tables of values by substituting for variables in linear relations
- Graph equations of linear relations (limited to discrete data)

#### Big Idea: Chance and Uncertainty

**Skill Descriptor:** Determine the probability of outcomes occurring in probability experiments 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
- Compare experimental results to theoretical probability 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 or 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 formulae to solve problems.

**Achievement Indicators:**

- Apply the formulae for perimeter of polygons, area of rectangles, area of triangles, area of parallelograms, and volume of right prisms to solve problems
- Generalize rules for determining volume of right cylinders
- Apply rules for determining volume of right cylinders
- Apply formulae for the volume of right cylinders

**Skill Descriptor:** Apply properties of circles to solve problems.

**Global Competencies:** CTPS

**Achievement Indicators:**

- Demonstrate diameter is twice the radius in circles
- 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
- Demonstrate the sum of central angles of circles is 360°
- 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
- Apply the Pythagorean Theorem to verify if triangles are right triangles
- Explain the Pythagorean Theorem
- Demonstrate that the Pythagorean Theorem only applies to right triangles
- Solve Pythagorean Triple problems

#### Big Idea: 2-D Shapes and 3-D Objects

**Skill Descriptor:** *No skill descriptors for this year*