Northwest Territories

M Measurement

• M.1 Develop spatial sense and proportional reasoning.

  • M.1.1 Solve problems that involve linear measurement, using:

    • M.1.1.a SI and imperial units of measure

    • M.1.1.b estimation strategies

    • M.1.1.c measurement strategies.

    • M.1.1.1 Provide referents for linear measurements, including millimetre, centimetre, metre, kilometre, inch, foot, yard and mile, and explain the choices.

    • M.1.1.2 Compare SI and imperial units, using referents.

    • M.1.1.3 Estimate a linear measure, using a referent, and explain the process used.

      • Estimate metric measurements (10-W.1)
      • Estimate imperial measurements of length (10-W.3)

    • M.1.1.4 Justify the choice of units used for determining a measurement in a problem-solving context.

    • M.1.1.5 Solve problems that involve linear measure, using instruments such as rulers, calipers or tape measures.

    • M.1.1.6 Describe and explain a personal strategy used to determine a linear measurement; e.g., circumference of a bottle, length of a curve, perimeter of the base of an irregular 3-D object.

  • M.1.2 Apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.

    • M.1.2.1 Explain how proportional reasoning can be used to convert a measurement within or between SI and imperial systems.

    • M.1.2.2 Solve a problem that involves the conversion of units within or between SI and imperial systems.

      • Convert rates and measurements: metric units (10-W.2)
      • Convert rates and measurements: imperial units of length (10-W.4)
      • Imperial mixed units of length (10-W.7)
      • Convert between metric and imperial units of length (10-W.8)

    • M.1.2.3 Verify, using unit analysis, a conversion within or between SI and imperial systems, and explain the conversion.

    • M.1.2.4 Justify, using mental mathematics, the reasonableness of a solution to a conversion problem.

  • M.1.3 Solve problems, using SI and imperial units, that involve the surface area and volume of 3-D objects, including:

    • M.1.3.a right cones

    • M.1.3.b right cylinders

    • M.1.3.c right prisms

    • M.1.3.d right pyramids

    • M.1.3.e spheres.

    • M.1.3.1 Sketch a diagram to represent a problem that involves surface area or volume.

    • M.1.3.2 Determine the surface area of a right cone, right cylinder, right prism, right pyramid or sphere, using an object or its labelled diagram.

      • Surface area of prisms and cylinders (10-Z.5)
      • Surface area of pyramids and cones (10-Z.6)
      • Surface area and volume of spheres (10-Z.9)

    • M.1.3.3 Determine the volume of a right cone, right cylinder, right prism, right pyramid or sphere, using an object or its labelled diagram.

      • Volume of prisms and cylinders (10-Z.7)
      • Volume of pyramids and cones (10-Z.8)
      • Surface area and volume of spheres (10-Z.9)

    • M.1.3.4 Determine an unknown dimension of a right cone, right cylinder, right prism, right pyramid or sphere, given the object's surface area or volume and the remaining dimensions.

      • Surface area of prisms and cylinders (10-Z.5)
      • Surface area of pyramids and cones (10-Z.6)
      • Volume of prisms and cylinders (10-Z.7)
      • Volume of pyramids and cones (10-Z.8)

    • M.1.3.5 Solve a problem that involves surface area or volume, given a diagram of a composite 3-D object.

    • M.1.3.6 Describe the relationship between the volumes of:

      • M.1.3.6.a right cones and right cylinders with the same base and height

      • M.1.3.6.b right pyramids and right prisms with the same base and height.

  • M.1.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

    • M.1.4.1 Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios.

    • M.1.4.2 Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a given acute angle in the triangle.

    • M.1.4.3 Solve right triangles.

      • Trigonometric ratios: sin, cos and tan (10-EE.1)
      • Solve a right triangle (10-EE.8)

    • M.1.4.4 Solve a problem that involves one or more right triangles by applying the primary trigonometric ratios or the Pythagorean theorem.

      • Trigonometric ratios: find a side length (10-EE.6)
      • Trigonometric ratios: find an angle measure (10-EE.7)

    • M.1.4.5 Solve a problem that involves indirect and direct measurement, using the trigonometric ratios, the Pythagorean theorem and measurement instruments such as a clinometer or metre stick.

AN Algebra and Number

• AN.1 Develop algebraic reasoning and number sense.

  • AN.1.1 Demonstrate an understanding of factors of whole numbers by determining the:

    • AN.1.1.a prime factors

    • AN.1.1.b greatest common factor

    • AN.1.1.c least common multiple

    • AN.1.1.d square root

    • AN.1.1.e cube root.

    • AN.1.1.1 Determine the prime factors of a whole number.

      • Prime factorization (10-A.2)

    • AN.1.1.2 Explain why the numbers 0 and 1 have no prime factors.

    • AN.1.1.3 Determine, using a variety of strategies, the greatest common factor or least common multiple of a set of whole numbers, and explain the process.

      • Greatest common factor (10-A.3)
      • Least common multiple (10-A.4)

    • AN.1.1.4 Determine, concretely, whether a given whole number is a perfect square, a perfect cube or neither.

    • AN.1.1.5 Determine, using a variety of strategies, the square root of a perfect square, and explain the process.

      • Square roots (10-A.6)

    • AN.1.1.6 Determine, using a variety of strategies, the cube root of a perfect cube, and explain the process.

      • Cube roots (10-A.7)

    • AN.1.1.7 Solve problems that involve prime factors, greatest common factors, least common multiples, square roots or cube roots.

      • GCF and LCM: word problems (10-A.5)

  • AN.1.2 Demonstrate an understanding of irrational numbers by:

    • AN.1.2.a representing, identifying and simplifying irrational numbers

    • AN.1.2.b ordering irrational numbers

    • AN.1.2.1 Sort a set of numbers into rational and irrational numbers.

      • Classify rational and irrational numbers (10-A.8)

    • AN.1.2.2 Determine an approximate value of a given irrational number.

    • AN.1.2.3 Approximate the locations of irrational numbers on a number line, using a variety of strategies, and explain the reasoning.

    • AN.1.2.4 Order a set of irrational numbers on a number line.

    • AN.1.2.5 Express a radical as a mixed radical in simplest form (limited to numerical radicands).

      • Simplify radical expressions (10-D.1)

    • AN.1.2.6 Express a mixed radical as an entire radical (limited to numerical radicands).

    • AN.1.2.7 Explain, using examples, the meaning of the index of a radical.

    • AN.1.2.8 Represent, using a graphic organizer, the relationship among the subsets of the real numbers (natural, whole, integer, rational, irrational).

  • AN.1.3 Demonstrate an understanding of powers with integral and rational exponents.

    • AN.1.3.1 Explain, using patterns, why a?ⁿ = 1/aⁿ, a ≠ 0.

      • Negative exponents (10-B.3)

    • AN.1.3.2 Explain, using patterns, why a¹/ⁿ = ⁿ√a, n > 0.

    • AN.1.3.3 Apply the exponent laws:

      • AN.1.3.3.a (a to the m power) (a to the n power) = a to the power of m + n

        • Multiplication with exponents (10-B.4)
        • Multiplication with rational exponents (10-E.2)

      • AN.1.3.3.b (a to the m power) ÷ (a to the n power) = (a to the m – n power), a ≠ 0

        • Division with exponents (10-B.5)
        • Division with rational exponents (10-E.3)

      • AN.1.3.3.c (a to the m power) to the n power = a to the mn power

        • Power rule (10-B.7)
        • Power rule with rational exponents (10-E.4)

      • AN.1.3.3.d (ab) to the m power = (a to the m power)(b to the m power)

        • Power rule (10-B.7)
        • Power rule with rational exponents (10-E.4)

      • AN.1.3.3.e (a/b) to the n power = (a to the n power)/(b to the n power), b ≠ 0

    • AN.1.3.4 Express powers with rational exponents as radicals and vice versa.

    • AN.1.3.5 Solve a problem that involves exponent laws or radicals.

      • Multiplication and division with exponents (10-B.6)
      • Evaluate expressions using properties of exponents (10-B.8)
      • Identify equivalent expressions involving exponents (10-B.9)

    • AN.1.3.6 Identify and correct errors in a simplification of an expression that involves powers.

  • AN.1.4 Demonstrate an understanding of the multiplication of polynomial expressions (limited to monomials, binomials and trinomials), concretely, pictorially and symbolically.

    • AN.1.4.1 Model the multiplication of two given binomials, concretely or pictorially, and record the process symbolically.

      • Multiply two polynomials using algebra tiles (10-O.7)

    • AN.1.4.2 Relate the multiplication of two binomial expressions to an area model.

    • AN.1.4.3 Explain, using examples, the relationship between the multiplication of binomials and the multiplication of two-digit numbers.

    • AN.1.4.4 Verify a polynomial product by substituting numbers for the variables.

    • AN.1.4.5 Multiply two polynomials symbolically, and combine like terms in the product.

      • Multiply two binomials (10-O.8)
      • Multiply two binomials: special cases (10-O.9)
      • Multiply polynomials (10-O.10)

    • AN.1.4.6 Generalize and explain a strategy for multiplication of polynomials.

    • AN.1.4.7 Identify and explain errors in a solution for a polynomial multiplication.

  • AN.1.5 Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and symbolically.

    • AN.1.5.1 Determine the common factors in the terms of a polynomial, and express the polynomial in factored form.

      • GCF of monomials (10-P.1)

    • AN.1.5.2 Model the factoring of a trinomial, concretely or pictorially, and record the process symbolically.

      • Factor quadratics using algebra tiles (10-P.5)

    • AN.1.5.3 Factor a polynomial that is a difference of squares, and explain why it is a special case of trinomial factoring where b = 0.

    • AN.1.5.4 Identify and explain errors in a polynomial factorization.

    • AN.1.5.5 Factor a polynomial, and verify by multiplying the factors.

    • AN.1.5.6 Explain, using examples, the relationship between multiplication and factoring of polynomials.

    • AN.1.5.7 Generalize and explain strategies used to factor a trinomial.

    • AN.1.5.8 Express a polynomial as a product of its factors.

      • Factor out a monomial (10-P.2)
      • Factor quadratics with leading coefficient 1 (10-P.3)
      • Factor quadratics with other leading coefficients (10-P.4)
      • Factor quadratics: special cases (10-P.6)

RF Relations and Functions

• RF.1 Develop algebraic and graphical reasoning through the study of relations.

  • RF.1.1 Interpret and explain the relationships among data, graphs and situations.

    • RF.1.1.1 Graph, with or without technology, a set of data, and determine the restrictions on the domain and range.

    • RF.1.1.2 Explain why data points should or should not be connected on the graph for a situation.

    • RF.1.1.3 Describe a possible situation for a given graph.

    • RF.1.1.4 Sketch a possible graph for a given situation.

    • RF.1.1.5 Determine, and express in a variety of ways, the domain and range of a graph, a set of ordered pairs or a table of values.

      • Domain and range of relations (10-J.2)

  • RF.1.2 Demonstrate an understanding of relations and functions.

    • RF.1.2.1 Explain, using examples, why some relations are not functions but all functions are relations.

    • RF.1.2.2 Determine if a set of ordered pairs represents a function.

      • Identify functions (10-J.4)

    • RF.1.2.3 Sort a set of graphs as functions or non-functions.

      • Identify functions: vertical line test (10-J.5)

    • RF.1.2.4 Generalize and explain rules for determining whether graphs and sets of ordered pairs represent functions.

  • RF.1.3 Demonstrate an understanding of slope with respect to:

    • RF.1.3.a rise and run

    • RF.1.3.b line segments and lines

    • RF.1.3.c rate of change

    • RF.1.3.d parallel lines

    • RF.1.3.e perpendicular lines.

    • RF.1.3.1 Determine the slope of a line segment by measuring or calculating the rise and run.

      • Find the slope of a graph (10-L.2)

    • RF.1.3.2 Classify lines in a given set as having positive or negative slopes.

    • RF.1.3.3 Explain the meaning of the slope of a horizontal or vertical line.

    • RF.1.3.4 Explain why the slope of a line can be determined by using any two points on that line.

    • RF.1.3.5 Explain, using examples, slope as a rate of change.

    • RF.1.3.6 Draw a line, given its slope and a point on the line.

    • RF.1.3.7 Determine another point on a line, given the slope and a point on the line.

      • Find a missing coordinate using slope (10-L.4)

    • RF.1.3.8 Generalize and apply a rule for determining whether two lines are parallel or perpendicular.

      • Slopes of parallel and perpendicular lines (10-L.22)

    • RF.1.3.9 Solve a contextual problem involving slope.

      • Find the constant of variation (10-K.2)
      • Find the slope from two points (10-L.3)

  • RF.1.4 Describe and represent linear relations, using:

    • RF.1.4.a words

    • RF.1.4.b ordered pairs

    • RF.1.4.c tables of values

    • RF.1.4.d graphs

    • RF.1.4.e equations.

    • RF.1.4.1 Identify independent and dependent variables in a given context.

      • Identify independent and dependent variables (10-J.3)

    • RF.1.4.2 Determine whether a situation represents a linear relation, and explain why or why not.

      • Identify linear functions (10-L.1)

    • RF.1.4.3 Determine whether a graph represents a linear relation, and explain why or why not.

      • Identify linear functions (10-L.1)

    • RF.1.4.4 Determine whether a table of values or a set of ordered pairs represents a linear relation, and explain why or why not.

      • Identify linear functions (10-L.1)

    • RF.1.4.5 Draw a graph from a set of ordered pairs within a given situation, and determine whether the relationship between the variables is linear.

    • RF.1.4.6 Determine whether an equation represents a linear relation, and explain why or why not.

      • Identify linear functions (10-L.1)

    • RF.1.4.7 Match corresponding representations of linear relations.

  • RF.1.5 Determine the characteristics of the graphs of linear relations, including the:

    • RF.1.5.a intercepts

    • RF.1.5.b slope

    • RF.1.5.c domain

    • RF.1.5.d range.

    • RF.1.5.1 Determine the intercepts of the graph of a linear relation, and state the intercepts as values or ordered pairs.

      • Standard form: find x- and y-intercepts (10-L.15)

    • RF.1.5.2 Determine the slope of the graph of a linear relation.

      • Find the slope of a graph (10-L.2)

    • RF.1.5.3 Determine the domain and range of the graph of a linear relation.

    • RF.1.5.4 Sketch a linear relation that has one intercept, two intercepts or an infinite number of intercepts.

    • RF.1.5.5 Identify the graph that corresponds to a given slope and y-intercept.

    • RF.1.5.6 Identify the slope and y-intercept that correspond to a given graph.

    • RF.1.5.7 Solve a contextual problem that involves intercepts, slope, domain or range of a linear relation.

      • Write linear functions to solve word problems (10-L.11)

  • RF.1.6 Relate linear relations expressed in:

    • RF.1.6.a slope–intercept form (y = mx + b)

    • RF.1.6.b general form (Ax + By + C = 0)

    • RF.1.6.c slope–point form (y – y? = m(x – x?))

    • RF.1.6.1 Express a linear relation in different forms, and compare the graphs.

    • RF.1.6.2 Rewrite a linear relation in either slope–intercept or general form.

    • RF.1.6.3 Generalize and explain strategies for graphing a linear relation in slope–intercept, general or slope–point form.

    • RF.1.6.4 Graph, with and without technology, a linear relation given in slope–intercept, general or slope–point form, and explain the strategy used to create the graph.

      • Slope-intercept form: graph an equation (10-L.6)
      • Graph a horizontal or vertical line (10-L.18)
      • Point-slope form: graph an equation (10-L.19)

    • RF.1.6.5 Identify equivalent linear relations from a set of linear relations.

    • RF.1.6.6 Match a set of linear relations to their graphs.

  • RF.1.7 Determine the equation of a linear relation, given:

    • RF.1.7.a a graph

    • RF.1.7.b a point and the slope

    • RF.1.7.c two points

    • RF.1.7.d a point and the equation of a parallel or perpendicular line

    • RF.1.7.1 Determine the slope and y-intercept of a given linear relation from its graph, and write the equation in the form y = mx + b.

      • Slope-intercept form: write an equation from a graph (10-L.7)

    • RF.1.7.2 Write the equation of a linear relation, given its slope and the coordinates of a point on the line, and explain the reasoning.

      • Slope-intercept form: write an equation (10-L.8)
      • Point-slope form: write an equation (10-L.20)

    • RF.1.7.3 Write the equation of a linear relation, given the coordinates of two points on the line, and explain the reasoning.

      • Slope-intercept form: write an equation (10-L.8)

    • RF.1.7.4 Write the equation of a linear relation, given the coordinates of a point on the line and the equation of a parallel or perpendicular line, and explain the reasoning.

      • Write an equation for a parallel or perpendicular line (10-L.23)

    • RF.1.7.5 Graph linear data generated from a context, and write the equation of the resulting line.

    • RF.1.7.6 Solve a problem, using the equation of a linear relation.

      • Write linear functions to solve word problems (10-L.11)

  • RF.1.8 Represent a linear function, using function notation.

    • RF.1.8.1 Express the equation of a linear function in two variables, using function notation.

    • RF.1.8.2 Express an equation given in function notation as a linear function in two variables.

    • RF.1.8.3 Determine the related range value, given a domain value for a linear function; e.g., if f(x) = 3x – 2, determine f(–1).

    • RF.1.8.4 Determine the related domain value, given a range value for a linear function; e.g., if g(t) = 7 + t, determine t so that g(t) = 15.

    • RF.1.8.5 Sketch the graph of a linear function expressed in function notation.

      • Complete a table and graph a linear function (10-L.12)

  • RF.1.9 Solve problems that involve systems of linear equations in two variables, graphically and algebraically.

    • RF.1.9.1 Model a situation, using a system of linear equations.

    • RF.1.9.2 Relate a system of linear equations to the context of a problem.

    • RF.1.9.3 Determine and verify the solution of a system of linear equations graphically, with and without technology.

      • Solve a system of equations by graphing (10-M.2)
      • Solve a system of equations by graphing: word problems (10-M.3)

    • RF.1.9.4 Explain the meaning of the point of intersection of a system of linear equations.

    • RF.1.9.5 Determine and verify the solution of a system of linear equations algebraically.

      • Is (x, y) a solution to the system of equations? (10-M.1)
      • Solve a system of equations using substitution (10-M.8)
      • Solve a system of equations using elimination (10-M.10)
      • Solve a system of equations using any method (10-M.12)

    • RF.1.9.6 Explain, using examples, why a system of equations may have no solution, one solution or an infinite number of solutions.

      • Find the number of solutions to a system of equations by graphing (10-M.4)
      • Find the number of solutions to a system of equations (10-M.5)

    • RF.1.9.7 Explain a strategy to solve a system of linear equations.

    • RF.1.9.8 Solve a problem that involves a system of linear equations.

      • Solve a system of equations using substitution: word problems (10-M.9)
      • Solve a system of equations using elimination: word problems (10-M.11)
      • Solve a system of equations using any method: word problems (10-M.13)