7 Fractions. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability

Transcription

1 7 Fractions GRADE 7 FRACTIONS continue to develop proficiency by using fractions in mental strategies and in selecting and justifying use; develop proficiency in adding and subtracting simple fractions; use fractions to solve problems involving the area of a trapezoid; describe the reduction of two-dimensional shapes to create similar shapes and to describe reductions in dilatations; write fractions to determine the theoretical probability of an event or two independent events. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability generate multiples and factors of given numbers use fraction skills in solving problems involving measurement, e.g., the area of a trapezoid use fractions to describe reductions in dilatation and in reducing twodimensional shapes to create similar figures model everyday relationships involving rates use fractions to express the experimental and theoretical probability of an event solve problems involving whole number percents use fractions to describe related lines, e.g., perpendicular lines meet at 90 which is 1/2 of 180 translate phrases into algebraic expressions research and report on real- world applications of probabilities expressed in fraction, decimal, and percent form demonstrate an understanding of rate plot points on the Cartesian plane with simple fractional coordinates determine the theoretical probability of a specific outcome involving two independent events solve problems involving unit rates represent, compare, and order decimals to hundredths and fractions, using a variety of tools; select and justify the most appropriate representation of a quantity (i.e., fraction, decimal, percent) for a given context; divide whole numbers by simple fractions and by decimal numbers to hundredths, using concrete materials; use a variety of mental strategies to solve problems involving the addition and subtraction of fractions and decimals; add and subtract fractions with simple like and unlike denominators, using a variety of tools and algorithms; demonstrate, using concrete materials, the relationship between the repeated addition of fractions and the multiplication of that fraction by a whole number; determine, through investigation, the relationships among fractions, decimals, percents, and ratios; research and report on everyday applications of probabilities expressed in fraction, decimal, and percent form Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 1

2 8 Fractions GRADE 8 FRACTIONS develop proficiency in comparing, ordering, and representing fractions; develop proficiency in operations with fractions with and without concrete materials; solve problems involving addition, subtraction, multiplication, and division of simple fractions; substitute fractions for the variables in algebraic expressions up to three terms; continue to write fractions to determine the theoretical probability of an event, e.g., complementary event. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability determine common factors and multiples use fraction skills in solving problems involving measurement, e.g., the area of a circle graph the image of a point on the Cartesian coordinate plane with simple fractional coordinates evaluate algebraic expressions with up to three terms, by substituting fractions, decimals, and integers for the variables use fractions to express the experimental and theoretical probability of an event solve problems involving proportions determine relationships: area, perimeter, and side length of similar shapes, e.g., if 2 triangles are similar and the perimeter of one is 1/2 the perimeter of the other, compare their areas translate statements into algebraic expressions and equations identify the complementary event for a given event, and calculate the theoretical probability that a given event will not occur solve problems involving percent represent, compare, and order rational numbers; translate between equivalent forms of a number; use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution; represent the multiplication and division of fractions, using a variety of tools and strategies; solve problems involving addition, subtraction, multiplication, and division with simple fractions Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 2

3 9P Fractions GRADE 9 (APPLIED) FRACTIONS continue to solve problems involving percents, fractions, and decimals; use rationals as bases when evaluating expressions involving natural number exponents; use fractions in solving problems involving area of composite figures with triangles and/or trapezoids, and in solving problems involving volumes of pyramids or cones. Number Sense and Algebra (Applied) Number Sense and Algebra (Applied and Academic) Number Sense and Algebra (Academic) Measurement and Geometry Linear Relationships represent, using equivalent ratios and proportions, directly proportional relationships arising from realistic situations; substitute into and evaluate algebraic expressions involving exponents (i.e., evaluate expressions involving naturalnumber exponents with rational- number bases); solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate and proportion; solve problems involving area of composite figures, involving triangles and/or trapezoids interpret points on a scatterplot solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms. simplify numerical expressions involving integers and rational numbers, with and without the use of technology. solve first-degree equations involving one variable, including equations with fractional coefficients. develop, through investigation, the formulas for the volume of a pyramid or cone collect data; describe trends solve problems involving the volume of pyramids or cones construct tables of values and graphs for data determine and describe rates of change use initial value and rate of change to express a linear relation determine graphically a point of intersection Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 3

4 10P Fractions GRADE 10 (APPLIED) FRACTIONS solve first degree equations and rearrange formulas involving variable in the first degree, involving fractional coefficients; use ratios and proportions in problem solving when working with similar triangles and trigonometry; rearrange measurement formulas involving surface area and volume to solve problems; graph equations of lines, with fractional coefficients. Measurement and Trigonometry Modelling Linear Relationships Quadratic Relations of the Form y = ax^2 + bx + c use proportional reasoning (by setting two ratios written as fractions equal) to solve similar triangle problems graph lines (with fractional coefficients) by hand, using a variety of techniques collect data that can be represented as a quadratic relation determine, through investigation, the trigonometric ratios of sine, cosine, and tangent as ratios represented by fractions solve equations involving fractional coefficients solve problems involving quadratic relations solve problems involving the measures in right-angled triangles connect rate of change and slope rearrange measurement formulas when solving problems involving surface area and volume, including combination of these figures in the metric or imperial system determine the equations of a line perform conversions between imperial and metric systems determine graphically the point of intersection of 2 lines solve problems involving surface area and volume of pyramids and cylinders solve first-degree equations involving one variable, including equations with fractional coefficients; determine the value of a variable in the first degree, using a formula Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 4

5 7 Integers GRADE 7 INTEGERS identify, describe, and interpret everyday applications that involve integers, e.g., in primary and secondary data; order and compare integers using two-colour counters, manipulatives, and number lines; add or subtract two integers using concrete materials, e.g., coloured tiles, then (after doing the concrete manipulation) represent the operation using symbols; graph coordinate points in all four quadrants. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability identify and compare integers found in real-life contexts; represent and order integers, using a variety of tools; add and subtract integers, using a variety of tools. N/A plot points using all four quadrants of the Cartesian coordinate plane recognize patterns and use them to make predictions create and analyse designs involving translations, reflections, dilatations, and/or simple rotations of two-dimensional establish or verify strategies for operating shapes, using a variety of tools (rotation of with integers 90 can be associated with a rotation of positive/negative 90 ) primary and secondary data may involve the need to read, interpret, and draw conclusions from charts, tables, and graphs that involve integers determine, through investigation, the effect on a measure of central tendency of adding or removing a value or values identify and describe trends, based on the distribution of the data presented in table Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 5

6 8 Integers GRADE 8 INTEGERS continue to identify, describe, and interpret everyday applications that involve integers, e.g., in primary and secondary data; develop the rules for the multiplication and division of two integers; develop proficiency in multiplying and dividing integers, with and without manipulatives; develop proficiency in using order of operations with brackets and exponents that involve integers; become proficient in adding, subtracting, multiplying, and dividing integers without the use of manipulatives, i.e., using symbols; evaluate algebraic expressions and solve and verify simple linear equations. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability represent, compare, and order rational numbers use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution; N/A graph the image of a point, or set of points, on the Cartesian coordinate plane after applying a transformation to the original point(s) (i.e., translation; reflection in the x- axis, the y-axis, or the angle bisector of the axes that passes through the first and third quadrants; rotation of 90, 180, or 270 about the origin) evaluate algebraic expressions with up to three terms, by substituting fractions, decimals, and integers for the variables solve and verify linear equations involving a one- variable term and having solutions that are integers, by using inspection, guess and check, and a balance model primary and secondary data (categorical, discrete, or continuous) may involve the need to read, interpret, and draw conclusions from charts, tables, and graphs that involve integers identify and describe trends, based on the rate of change of data from tables and graphs, using informal language represent the multiplication and division of integers, using a variety of tools; solve problems involving operations with integers, using a variety of tools; evaluate expressions that involve integers, including expressions that contain brackets and exponents, using order of operations; evaluate algebraic expressions with up to three terms, by substituting fractions, decimals and integers for the variables; solve and verify linear equations involving a one variable term and having solutions that are integers, by inspection, guess and check, and a balance model Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 6

7 9P Integers GRADE 9 (APPLIED) INTEGERS apply integers to simplify polynomial expressions and solve first-degree equations, in algebraic manipulation, e.g., measurement and formulas, and in solving problems; interpret the meaning of points in four quadrants in context; apply integers in calculating first differences to determine if relationship is linear, and in calculating the rate of change (rise). Number Sense and Algebra Measurement and Geometry Analytic Geometry Linear Relations determine, from the examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one variable N/A express the equation of a line in the form y = mx + b, given the form ax + by + c = 0 interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant add and subtract polynomials with up to two variables, using a variety of tools determine the slope of a line or line segment using the various formulas of slope determine if a relationship is linear by calculating first differences expand and simplify polynomial expressions involving one variable, using a variety of tools determine the equation of a line given slope and y -intercept, slope and a point, or two points on the line determine other representations of a linear relation, given one representation solve first-degree equations, including equations with fractional coefficients, using a variety of tools and strategies explain the geometric significance of m and b in the equation y = mx + b describe the effects on a linear graph and make the corresponding changes to the linear equation when the conditions of the situation they represent are varied algebraic manipulation (including formulas) simplify numerical expressions involving integers and rational numbers, with and without the use of technology. describe the meaning of the slope and y- intercept for a linear relation arising from a realistic situation determine graphically the point of intersection of two linear relations, and interpret the intersection point in the context of an application Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 7

8 10P Integers GRADE 10 (APPLIED) INTEGERS apply integers in manipulating and solving first-degree equations and in solving systems of two linear equations; apply integers in determining the equation of a line, in calculating slope of a line or line segment, in determining the equation of a line, in rewriting equations of the form y = mx + b and Ax + By + C = 0, and in describing the meaning of slope and y-intercept in the equation of a line; apply integers in manipulating quadratic expression, e.g., second degree polynomials, common factoring binomials and trinomials of one variable up to degree two, and factoring simple trinomials; apply integers in calculating the second differences to determine if a relationship is quadratic, and in identifying the key features of the graph of a parabola. Measurement and Trigonometry Modelling Linear Relationships Quadratic Relations of the Form y = ax^2 + bx + c N/A solve first-degree equations involving one variable, including equations with fractional coefficients determine the value of a variable in the first degree, using a formula expand and simplify second-degree polynomial expressions involving one variable that consist of the product of two binomials, using a variety of tools factor binomials (e.g., 4x2 + 8x) and trinomials (e.g., 3x2 + 9x 15) involving one variable up to degree two, by determining a common factor using a variety of tools express the equation of a line in the form y = mx + b, given the form ax + by + c = 0 factor simple trinomials of the form x2 + bx + c, using a variety of tools explain the geometric significance of m and b in the equation y = mx + b determine if a relationship is quadratic by calculating second differences determine properties of the slopes of lines and line segments (e.g., negative rate of change) identify the key features of a graph of a parabola determine the equation of a line, given its graph, the slope and y-intercept, the slope and a point on the line, or two points on the line solve systems of two linear equations involving two variables with integral coefficients, using the algebraic method of substitution or elimination Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 8

9 7 Patterning and Algebra GRADE 7 PATTERNING AND ALGEBRA continue to identify, create, describe, and extend linear growth patterns using a variety of strategies; use patterns to relate multiplication of fractions by a whole number to repeated addition of fractions; develop formulas for area of trapezoids, volume of right prisms, and surface area of right prisms; use patterns to sort and classify triangles and quadrilaterals, to describe congruent shapes; continue to use patterns to solve simple linear equations; describe trends and make inferences from data. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability relate multiplication of fractions by a whole number to repeated addition of fractions develop formulas for the area of trapezoids, volume of right prisms, and surface area of right prisms recognize patterns to help sort and classify triangles and quadrilaterals solve linear equations of the form c = ax, ax = c, ax + b = c, c = ax + b, using a variety of methods, e.g., inspection, guess and check use patterns to identify, describe, and interpret trends, and make inferences and conclusions from data presented in tables, graphs, and charts determine, through investigation, the relationships among fractions, decimals, percents, and ratios use patterns to describe congruent shapes represent linear growing patterns, using a variety of tools and strategies; make predictions about a population when given a probability determine, through investigation using a variety of tools, polygons or combinations of polygons that tile a plane, and describe the transformation(s) involved make predictions about linear growing patterns, through investigation with concrete materials; develop and represent the general term of a linear growing pattern, using algebraic expressions involving one operation; compare pattern rules that generate a pattern by adding or subtracting a constant, or multiplying or dividing by a constant, to get the next term with pattern rules that use the term number to describe the general term; model everyday life relationships involving constant rates, where the initial condition starts at 0, through investigation and using tables of values and graphs; model everyday life relationships involving constant rates, using algebraic equations with variables to represent the changing quantities in the relationship; translate phrases describing simple mathematical relationships into algebraic expressions, using concrete materials; evaluate algebraic expressions by substituting natural numbers for the variables; make connections between evaluating algebraic expressions and determining the term in a pattern using the general term; solve linear equations of the form ax = c or c = ax and ax + b = c, or variations such as b + ax = c and c = bx + a, by modelling with concrete materials, by inspection, or by guess and check, with and without the aid of a calculator Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 9

10 8 Patterning and Algebra GRADE 8 PATTERNING AND ALGEBRA continue to identify, create, describe, and extend linear growth patterns; generalize patterns with equations; develop proficiency in multiplying and dividing fractions and integers; develop formulas for circumference and area of circles, volume of right prisms, and surface area of cylinders; sort and classify geometric figures and determine geometric relationships; connect patterns to equations; continue to describe trends and make inferences from data. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability represent the multiplication and division of fractions, using a variety of tools and strategies represent the multiplication and division of integers, using a variety of tools develop formulas for circumference of a circle, area of a circle, volume for right prisms, and surface area of a cylinder sort and classify quadrilaterals by geometric properties, including those based on diagonals, through investigation and using a variety of tools make connections between patterns and equations represent, through investigation with determine the Pythagorean relationship, concrete materials, the general term of a through investigation and using a variety of linear pattern, using one or more algebraic tools expressions; determine through investigation using concrete materials the relationship between the numbers of faces, edges, and vertices of a polyhedron represent linear patterns graphically, using a variety of tools; determine a term, given its term number, in a linear pattern that is represented by a graph or an algebraic equation; use patterns to identify, describe, interpret trends, and make inferences and conclusions from data (including rate of change of data) presented in tables, graphs, and charts describe different ways in which algebra can be used in everyday life situations; model linear relationships using tables of values, graphs, and equations, through investigation and using a variety of tools; translate statements describing mathematical relationships into algebraic expressions and equations; evaluate algebraic expressions with up to three terms, by substituting fractions, decimals, and integers for the variables; make connections between solving equations and determining the term number in a pattern, using the general term; solve and verify linear equations involving a one-variable term and having solutions that are integers, by using inspection, guess and check, and a balance model Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 10

11 9P Patterning and Algebra GRADE 9 (APPLIED) PATTERNING AND ALGEBRA connect the algebraic and geometric representations of a single-variable term up to degree three; determine relationships of optimal values of rectangles; connect the geometric and algebraic representations of the Pythagorean theorem; develop formulas for volume; continue to determine geometric relationships; continue to describe trends and make inferences from data. Number Sense and Algebra Measurement and Geometry Linear Relations describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three; determine the relationship of optimal values of rectangles, e.g., maximum area given perimeter; minimum perimeter given area describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses substitute into and evaluate algebraic expressions involving exponents; pose problems, identify variables, and formulate hypotheses associated with relationships between two variables; relate the geometric representation of the Pythagorean theorem to the algebraic representation a2 + b2 = c2 develop formulas for volumes of pyramids, cones, and spheres identify, through investigation, some properties of linear relations, and apply these properties to determine whether a relation is linear or non-linear describe the effects on a linear graph and make the corresponding changes to the linear equation when the conditions of the situation they represent are varied relate the geometric representation of the investigate geometric relationships, using a Pythagorean theorem to the algebraic variety of tools representation a2 +b2 =c2; determine other representations of a linear relation arising from a realistic situation, given one representation; solve problems that can be modelled with first-degree equations, and compare the algebraic method to other solution methods. solve for the unknown value in a proportion, using a variety of methods; multiply a polynomial by a monomial involving the same variable to give results up to degree three, using a variety of tools; carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology and techniques Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 11

12 9D Patterning and Algebra GRADE 9 ACADEMIC PATTERNING AND ALGEBRA Number Sense and Algebra Measurement and Geometry Analytic Geometry Linear Relations describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three; substitute into and evaluate algebraic expressions involving exponents; pose problems, identify variables, and formulate hypotheses associated with relationships between two variables; determine the relationship of optimal values of rectangles, e.g., maximum area given perimeter; minimum perimeter given area determine the properties of linear and nonlinear relationships using graphs and equations determine the relationships of optimal determine the properties of slopes of lines values by varying the dimensions of squarebased prisms and and line segments cylinders relate the geometric representation of the Pythagorean theorem to the algebraic representation a2 +b2 =c2 describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses identify, through investigations, some properties of linear relations, and apply these properties to determine whether a relation is linear or non-linear connect various forms of slopes (rise/run, change in y/change in x, y/ x, (y2 determine the equation of a line of best fit y1)/(x2 x1) and lines y = mx + b, ax + by + using an informal process c = 0, x = a, y = b) relate the geometric representation of the Pythagorean theorem to the algebraic representation a2 +b2 = c2; develop formulas for volumes of pyramids, cones, and spheres, and surface area of pyramids through investigations, explain the geometric significance of m and b in y = mx + b determine other representations of a linear relation arising from a realistic situation, given one representation; solve problems that can be modelled with first-degree equations, and compare the algebraic method to other solution methods. determine, through investigation using a variety of tools, geometric relationships, e.g., angles of triangles, quadrilaterals, polygons, properties of polygons determine, through investigation, connections among the representations of a constant rate of change of a linear relation identify and explain any restrictions on the variables in a linear relation arising from a realistic situation use patterning to derive the multiplication, division, and power exponent laws; multiply a polynomial by a monomial involving the same variable; design and carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology and techniques; identify y = mx + b as a common form for the equation of a straight line; identify properties of the slopes of lines and line segments; determine the equation of a line, given its graph, the slope and y-intercept, the slope and a point on the line, or two points on the line; determine graphically the point of intersection of two linear relations Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 12

13 10P Patterning and Algebra GRADE 10 (APPLIED) PATTERNING AND ALGEBRA develop the trigonometric ratios of sine, cosine, and tangent; develop the formula for surface area of pyramids; continue to use patterns to distinguish between linear and non-linear relations; identify the properties of slope and properties of linear and quadratic relationships; use patterns to connect the various representations and connect the graphical representations y = x2 + bx + c and y = (x r)(x s) of quadratic relations. Measurement and Trigonometry Modelling Linear Relationships Quadratic Relations of the Form y = ax^2 + bx + c determine, through investigation, the trigonometric ratios of sine, cosine, and tangent use patterns to identify the properties of slope of lines and line segments, explain the geometric significance of m and b in the equation of a line, y = mx + b, and connect the special cases x = a and y = b to y = mx + b move between the numerical, graphical, and algebraic representations for quadratic relations develop the formula for surface area of pyramids connect the graphical representations, y= x2 + bx + c and y = (x r)(x s) of quadratic relations verify properties of similar triangles; determine relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle and define the sine, cosine, and tangent ratios; determine the relationship for calculating the surface area of a pyramid; determine the value of a variable in the first degree, using a formula; identify y = mx + b as a common form for the equation of a straight line; identify properties of the slopes of lines and line segments; determine the equation of a line, given its graph, the slope and y-intercept, the slope and a point on the line, or two points on the line; determine graphically the point of intersection of two linear relations; solve problems that arise from realistic situations described in words or represented by given linear systems of two equations involving two variables; expand and simplify second-degree polynomial expressions; factor binomials, simple trinomials, and the difference of squares; collect data that can be represented as a quadratic equation; determine that a quadratic relation of the form y = ax2 + bx + c can be graphically represented as a parabola, and determine that the table of values yields a constant second difference; compare the graphical representations of a quadratic relation in the form of y = x2 + bx + c and the same relation in the factored form y = (x r)(x s) and describe the connections between each algebraic representation and the graph; solve problems involving a quadratic relation by interpreting a given graph or a graph generated with technology from its equation Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 13

14 7 Perimeter Area Volume GRADE 7 PERIMETER AREA VOLUME calculate the area of composite two-dimensional shapes; determine the formulas for the area of a trapezoid, and the volume of right prisms; determine the surface area of right prisms with polygonal bases. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability understand fractions and decimals by connecting them to area determine, through investigation using a variety of tools and strategies, the relationship for calculating the area of a trapezoid, and generalize to develop the formula; sort and classify triangles and quadrilaterals by geometric properties use patterns systematically collect, organize, and analyse data relate multiples and factors to area models solve problems involving the estimation and calculation of the area of a trapezoid; identify right prisms translate phrases connect area to bar and circle graphs use order of operations in measurement formulas represent perfect squares and square roots using a variety of tools, (e.g., geoboards, connecting cubes, grid paper) estimate and calculate the area of composite two-dimensional shapes by decomposing into shapes with known area relationships; determine, through investigation using a variety of tools and strategies, the relationship between the height, the area of the base, and the volume of right prisms with simple polygonal bases, and generalize to develop the formula; understand similarity distinguish between and compare similar shapes and congruent shapes describing simple mathematical relationships into algebraic expressions evaluate algebraic expressions by substituting natural numbers for the variables explain the relationship between exponential notation and the measurement of area and volume determine, through investigation using a variety of tools, the surface area of right prisms; solve problems that involve the surface area and volume of right prisms and that require conversion between metric measures of capacity and volume. create and analyse designs involving translations, reflections, dilatations, and/or simple rotations of two-dimensional shapes understand polygons as they relate to tiling Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 14

15 8 Perimeter Area Volume GRADE 8 PERIMETER AREA VOLUME determine the formulas for the area and circumference of a circle, and the volume of a cylinder; determine the surface area of a cylinder. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability use order of operations in measurement formulas determine, through investigation using a variety of tools and strategies, the relationships for calculating the circumference and the area of a circle, and generalize to develop the formulas; sort and classify quadrilaterals by geometric properties use patterns in algebraic terms systematically collect, organize, and analyse primary and secondary data use exponents in measurement formulas solve problems involving the estimation and calculation of the circumference and the area of a circle; construct a circle, given its centre and radius, or its centre and a point on the circle, or three points on the circle solve and verify linear equations [some formulas] involving a one- variable term and having solutions that are integers connect area to bar and circle graphs develop understanding of squares and square roots through formula for area of circle determine, through investigation using a variety of tools and strategies, the relationship between the area of the base and height and the volume of a cylinder, and generalize to develop the formula; determine relationships among area, perimeter, corresponding side lengths, and corresponding angles of similar shapes evaluate algebraic expressions [formulas] with up to three terms, by substituting fractions, decimals, and integers for the variables compare two attributes or characteristics, using a scatter plot, and determine whether or not the scatter plot suggests a relationship develop estimation skills in working with π determine, through investigation using concrete materials, the surface area of a cylinder; determine the Pythagorean relationship using area models and diagrams (include the investigation of the areas of semicircles drawn on the sides of the right triangle) translate statements describing mathematical relationships into algebraic expressions and equations (e.g., the surface area is made up of 3 congruent rectangles plus 2 congruent equilateral triangles) solve multi-step problems arising from reallife contexts and involving whole numbers and decimals solve problems involving the surface area and the volume of cylinders, using a variety of strategies. solve problems involving right triangles geometrically, using the Pythagorean relationship use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions apply understanding of proportional relationships to geometric shapes and figures Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 15

16 9P Perimeter Area Volume GRADE 9 (APPLIED) PERIMETER AREA VOLUME solve problems involving the perimeter and area of composite two-dimensional shapes; develop the formulas for the volume of a cone, a pyramid, and a sphere; use formulas developed in earlier grades as a good foundation for grade-level investigations. Number Sense and Algebra Measurement and Geometry Analytic Geometry (Applied) Analytic Geometry (Academic) Linear Relations simplify numerical expressions involving integers and rational numbers relate the geometric representation of the Pythagorean theorem to the algebraic representation a^2 + b^2 = c^2 determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools, and by examining various values of the area as the side lengths change and the perimeter remains constant; determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools, and by examining various values of the area as the side lengths change and the perimeter remains constant; describe the connections between various representations of relations describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three solve problems using the Pythagorean theorem, as required in applications determine the minimum perimeter of a rectangle with a given area by constructing a variety of rectangles, using a variety of tools, and by examining various values of the side lengths and the perimeter as the area stays constant; determine the minimum perimeter of a rectangle with a given area by constructing a variety of rectangles, using a variety of tools, and by examining various values of the side lengths and the perimeter as the area stays constant; determine the relationship between variables in measurement formulas, e.g., the radius and area of a circle form a relationship that is non- linear substitute into and evaluate algebraic expressions involving exponents investigate the optimal values of measurements of rectangles solve problems involving the areas and perimeters of composite two-dimensional shapes; solve problems involving the areas and perimeters of composite two-dimensional shapes; solve first-degree equations develop, through investigation, the formulas for the volume of a pyramid, a cone, and a sphere. develop, through investigation, the formulas for the volume of a pyramid, a cone, and a sphere. graph relationships that are determined by measurement formulas, e.g., investigate the relationship between the height and radius of a cone if the volume remains constant substitute into algebraic equations and solve for one variable in the first degree identify, through investigation with a solve problems that require maximizing the variety of tools, the effect of varying the area of a rectangle for a fixed perimeter or dimensions on the surface area [or minimizing the perimeter of a rectangle for volume] of square-based prisms and a fixed area; cylinders, given a fixed volume [or surface area]; create tables of values, plot points, and graph lines or curves of best fit illustrate equivalent ratios, using a variety of tools solve problems involving the volumes of prisms, pyramids, cylinders, cones, and spheres. pose and solve problems involving maximization and minimization of measurements of geometric shapes and figures; use algebraic modelling solve problems involving ratio, rates, and directly proportional relationships in various contexts determine, through investigation, the relationship for calculating the surface area of a pyramid; solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 16

17 10P Perimeter Area Volume GRADE 10 (APPLIED) PERIMETER AREA VOLUME determine the surface area of a pyramid; consolidate understanding of the surface area of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres; use imperial measurements, when appropriate. Measurement and Trigonometry Modelling Linear Relationships Analytic Geometry Quadratic Relations of the Form y = ax^2 + bx + c verify properties of similar triangles solve first-degree equations involving one variable, including equations with fractional coefficients determine, through investigation, the relationship for calculating the surface area of a pyramid; factor binomials and trinomials using a variety of tools and strategies (e.g., use an area model) solve problems involving the measures of sides and angles in right triangles in reallife applications determine the value of a variable in the first degree, using a formula solve problems involving the surface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, collect data that can be represented as a cones, and spheres, including problems quadratic relation (e.g., the length of the involving combinations of these figures, side of a square vs. the area of the square) using the metric system or the imperial system, as appropriate. use the imperial system when solving measurement problems Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 17

18 7 Proportional Reasoning GRADE 7 PROPORTIONAL REASONING continue to determine the relationship between fractions, decimals, and percent; develop an understanding of rates as a comparison of two measures with different units; solve problems involving whole number percents and unit rates using a variety of tools and strategies; determine conversions of metric units of measure; demonstrate an understanding of proportional relationships using percent, ratio, and rate. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability determine, through investigation, the relationships among fractions, decimals, percents, and ratios solve problems that require conversion between metric units of measure construct angle bisectors and perpendicular bisectors, using a variety of tools and strategies, and represent equal angles and equal lengths using mathematical notation represent linear growing patterns, using a variety of tools and strategies (for a proportional relationship) research and report on real- world applications of probabilities expressed in fraction, decimal, and percent form solve problems that involve determining whole number percents, using a variety of tools solve problems that require conversion between metric units of area demonstrate an understanding that enlarging or reducing two- dimensional shapes creates similar shapes make predictions about linear growing patterns, through investigation with concrete materials make predictions about a population when given a probability demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units identify, perform, and describe dilatations through investigation using a variety of tools model real-life relationships involving constant rates where the initial condition starts at 0, through investigation using tables of values and graphs determine the theoretical probability of a specific outcome involving two independent events solve problems involving the calculation of unit rates model real-life relationships involving constant rates, using algebraic equations with variables to represent the changing quantities in the relationship determine, through investigation, the relationships among fractions, decimals, percents, and ratios; solve linear equations of the type ax = c or c = ax solve problems that involve determining whole number percents, using a variety of tools; demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units; solve problems involving the calculation of unit rates Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 18

19 8 Proportional Reasoning GRADE 8 PROPORTIONAL REASONING continue to solve problems that involve percent and rates; identify and solve problems involving proportions, using concrete materials; solve problems using proportional reasoning in meaningful contexts; identify and describe real-life situations involving quantities that are directly proportional. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability identify and describe real-life situations involving two quantities that are directly proportional solve problems that require conversions involving metric units of area, volume, and capacity determine, through investigation using a variety of tools, relationships among area, perimeter, corresponding side lengths, and corresponding angles of similar shapes represent patterns using an algebraic expression determine the theoretical and experimental probability of an event occurring or not occurring solve problems involving proportions, using concrete materials, drawings, and variables solve problems involving percent that arise from real-life contexts determine, through investigation using a variety of tools and strategies, the relationships for calculating the circumference and the area of a circle, and generalize to develop the formulas model a proportional relationship using tables of values, graphs, and equations compare two attributes or characteristics using a scatter plot and determine whether or not the scatter plot suggests a relationship compare two attributes or characteristics, using a variety of data management tools and strategies solve problems involving rates translate between equivalent forms of a number (i.e., decimals, fractions, percents) identify and describe real-life situations involving two quantities that are directly proportional; solve problems involving proportions, using concrete materials, drawings, and variables; solve problems involving percent that arise from real- life contexts; solve problems involving rates Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 19

20 9P Proportional Reasoning GRADE 9 (APPLIED) PROPORTIONAL REASONING solve for an unknown in a proportional relationship by using a variety of methods including algebraic methods; solve problems that require a fraction, decimal, or percent to be rewritten in an equivalent form; make comparisons using unit rates; represent and solve problems involving ratios, rates, and directly proportional relationships in context. Number Sense and Algebra Measurement and Geometry Linear Relations illustrate equivalent ratios, using a variety of tools determine, through investigation, that the rate of change of a linear relation can be found by choosing any two points on the line that represents the relation, finding the vertical change between the points (i.e., the rise) and the horizontal change between the points (i.e., the run), and writing the ratio rise/run (i.e., the rate of change = rise/run) develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere represent, using equivalent ratios and proportions, directly proportional relationships arising from realistic situations solve for the unknown value in a proportion, using a variety of methods compare the properties of direct variation and partial variation in applications, and identify the initial value describe the meaning of the rate of change and the initial value for a linear relation arising from a realistic situation solve problems involving the volumes of prisms, pyramids, cylinders cones, and spheres make comparisons using unit rates solve problems involving ratios, rates, and directly proportional relationships in various contexts, using a variety of methods solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms illustrate equivalent ratios, using a variety of tools; represent, using equivalent ratios and proportions, directly proportional relationships arising from realistic situations; solve for the unknown value in a proportion, using a variety of methods; make comparisons using unit rates; solve problems involving ratios, rates, and directly proportional relationships in various contexts, using a variety of methods; solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 20

21 10P Proportional Reasoning GRADE 10 (APPLIED) PROPORTIONAL REASONING use their knowledge of ratios and proportions to investigate and use the property of similar triangles that the ratios of corresponding sides are equal; solve problems involving similar triangles in realistic situations. Measurement and Trigonometry Modelling Linear Relationships Analytic Geometry Quadratic Relations of the Form y = ax^2 + bx + c verify, through investigation, properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides) connect the rate of change of a linear relation to the slope of the line, and define the slope as the ratio m = rise/run verify, through investigation, properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides); students may realize through investigation that a quadratic relation of the form y = ax2 is a dilatation of the graph of y = x2 determine the lengths of sides of similar triangles, using proportional reasoning identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b determine the lengths of sides of similar triangles, using proportional reasoning; solve problems involving similar triangles in realistic situations identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism), using graphing technology to facilitate investigations, where appropriate solve problems involving similar triangles in realistic situations. determine, through investigation, the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios perform everyday conversions between the imperial system and the metric system and within these systems, as necessary to solve problems involving measurement Ontario Curriculum Continuum Grade 7-10 (Barr).xlsx Page 21