The purpose of this comparison chart is to assist in understanding the revisions to the Grade 10 Applied course. The revised expectations (right hand column) are not presented in the order they will appear in the document, therefore it is critical to review the course as it is in the 2005 document
Grade 10 Applied
Foundations of Mathematics, Grade 10, Applied (MFM2P)1999DocumentProportional Reasoning
Most of the Proportional Reasoning strand was moved into MFM1P as a subgroup in the strand “Algebra and Number Sense”, and it was expanded upon and revised.
Overall ExpectationsBy the end of this course, students will:
solve problems derived from a variety of applications, using proportional reasoning; Moved to grade 9 applied / Foundations of Mathematics, Grade 10, Applied (MFM2P) Revised2005 Document
Measurement and Trigonometry
Overall Expectations
By the end of this course, students will:
solve problems involving similar triangles; revised / use their knowledge of ratio and proportion to investigate similar triangles and solveproblems related to similarity;
solve problems involving right triangles, using trigonometry. revised / solve problems involving right triangles, using the primary trigonometric ratios and thePythagorean theorem;
surface area moved here from grade 9 applied, imperial measure moved from grade 12 apprenticeship, and revised / solve problems involving the surface areas and volumes of three-dimensional figures, anduse the imperial and metric systems of measurement
Specific Expectations
Using Proportional Reasoning to Solve Problems from Applications
By the end of this course, students will:solve problems involving percent, ratio, rate, and proportion (e.g., in topics such as interest calculation, currency conversion, similar triangles, trigonometry, direct and partial variation related to linear functions) by a variety of methods and models (e.g., diagrams, concrete materials, fractions, tables, patterns, graphs, equations); Revised and moved to grade 9 / Specific Expectations
draw and interpret scale diagrams related to applications (e.g., technical drawings); Deleted, used as an e.g. in grade 9 applied
distinguish between consistent and inconsistent representations of proportionality in a variety of contexts (e.g., explain the distortion of figures resulting from irregular scales; identify misleading features in graphs; identify misleading conclusions based on invalid proportional reasoning). Deleted, could be incorporated into processes (representing, connecting)
Solving Problems Involving Similar Triangles
By the end of this course, students will:determine some properties of similar triangles (e.g., the correspondence and equality of angles, the ratio of corresponding sides) through investigation, using dynamic geometry software; revised / Solving Problems Involving Similar Triangles
By the end of this course, students will:
verify, through investigation (e.g., usingdynamic geometrysoftware, concretematerials), properties of similar triangles(e.g., given similar triangles, verify theequality of corresponding angles and theproportionality of corresponding sides);
solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying); revised / determine the lengths of sides of similartriangles, using proportional reasoning;
solve problems involving similar trianglesin realistic situations (e.g., shadows, reflections,scale models, surveying) (Sampleproblem: Use a metre stick to determinethe height of a tree, by means of the similartriangles formed by the tree, the metrestick, and their shadows.).
define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles
revised and / moved below
Solving Problems Involving the Trigonometry of Right Triangles
By the end of this course, students will:calculate the length of a side of a right triangle, using the Pythagorean theorem; revised and incorporated into other expectation / Solving Problems Involving
the Trigonometry
of Right Triangles
By the end of this course, students will:
Moved below
determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios; revised / determine, through investigation (e.g.,using dynamic geometry software, concretematerials), the relationship betweenthe ratio of two sides in a right triangleand the ratio of the two correspondingsides in a similar right triangle, and definethe sine, cosine, and tangent ratios (e.g.,sin A =);
determine the measures of the sides andangles in right triangles, using the primarytrigonometric ratios and the Pythagoreantheorem;
solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation); revised / solve problems involving the measures ofsides and angles in right triangles in reallifeapplications (e.g., in surveying, innavigation, in determining the height ofan inaccessible object around the school),using the primary trigonometric ratiosand the Pythagorean theorem (Sampleproblem: Build a kite, using imperialmeasurements, create a clinometer todetermine the angle of elevation when thekite is flown, and use the tangent ratio tocalculate the height attained.);
determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles; incorporated into the above / Incorporated into the above
describe applications of trigonometry in various occupations.revised / describe, through participation in anactivity, the application of trigonometryin an occupation (e.g., research andreport on how trigonometry is applied inastronomy; attend a career fair thatincludes a surveyor, and describe how asurveyor applies trigonometry to calculatedistances; job shadow a carpenter for a few hours, and describe how a carpenteruses trigonometry).
Moved from grade 12 apprenticeship, and revised / Solving Problems Involving Surface Area and
Volume, Using the Imperial and Metric
Systems of Measurement
By the end of this course, students will:
use the imperial system when solvingmeasurement problems (e.g., problemsinvolving dimensions of lumber, areas ofcarpets, and volumes of soil or concrete);
Moved from grade 12 apprenticeship and revised / perform everyday conversions betweenthe imperial system and the metric system(e.g., millilitres to cups, centimetres toinches) and within these systems (e.g.,cubic metres to cubic centimetres, squarefeet to square yards), as necessary to solveproblems involving measurement (Sampleproblem: A vertical post is to be supportedby a wooden pole, secured on the groundat an angle of elevation of 60°, and reaching3 m up the post from its base. If woodis sold by the foot, how many feet ofwood are needed to make the pole?);
determine, through investigation, the relationshipfor calculating the surface area ofa pyramid (e.g., use the net of a squarebasedpyramid to determine that the surfacearea is the area of the square base plusthe areas of the four congruent triangles);
Moved from grade 9 applied and revised / solve problems involving the surface areasof prisms, pyramids, and cylinders, and thevolumes of prisms, pyramids, cylinders,cones, and spheres, including problemsinvolving combinations of these figures,using the metric system or the imperialsystem, as appropriate (Sample problem:How many cubic yards of concrete arerequired to pour a concrete pad measuring10 feet by 10 feet by 1 foot? If pouredconcrete costs $110 per cubic yard, howmuch does it cost to pour a concretedriveway requiring 6 pads?).
Linear Functions
Overall Expectations
By the end of this course, students will:apply the properties of piecewise linear functions as they occur in realistic situations; Deleted / Modelling Linear Relations
Overall Expectations
By the end of this course, students will:
Moved from grade 9 applied / graph a line and write the equation of a line from given information;
solve and interpret systems of two linear equations as they occur in applications; revised / solve systems of two linear equations, and solve related problems that arise from realisticsituations.
manipulate algebraic expressions as they relate to linear functions. revised / manipulate and solve algebraic equations, as needed to solve problems;
Specific Expectations
Applying Piecewise Linear Functions
By the end of this course, students will:explain the characteristics of situations involving piecewise linear functions (e.g., pay scale variations, gas consumption costs, water consumption costs, differentiated pricing, motion); Deleted
construct tables of values and sketch graphs to represent given descriptions of realistic situations involving piecewise linear functions, with and without the use of graphing calculators or graphing software; Deleted
answer questions about piecewise linear functions by interpolation and extrapolation, and by considering variations on given conditions. Deleted
Interpreting Systems of Linear Equations
By the end of this course, students will:determine the point of intersection of two linear relations arising from a realistic situation, using graphing calculators or graphing software; revised / Solving and Interpreting Systems of Linear
Equations
By the end of this course, students will:
determine graphically the point of intersectionof two linear relations (e.g., usinggraph paper, using technology) (Sample
problem: Determine the point of intersectionof y + 2x = –5 and y =x+ 3using an appropriate graphing technique,and verify.);
interpret the point of intersection of two linear relations within the context of a realistic situation; done in grade 9 / Done in grade 9
solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination; revised / solve systems of two linear equationsinvolving two variables with integralcoefficients, using the algebraic methodof substitution or elimination (Sampleproblem: Solve y = 2x + 1, 3x + 2y = 16for x and y algebraically, and verify algebraicallyand graphically.);
solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs. revised / solve problems that arise from realistic situationsdescribed in words or representedby given linear systems of two equationsinvolving two variables, by choosing anappropriate algebraic or graphical method(Sample problem: Maria has been hired byCompany A with an annual salary,S dollars, given by S = 32 500 + 500a,where a represents the number of yearsshe has been employed by this company.Ruth has been hired by Company B withan annual salary, S dollars, given byS = 28 000 + 1000a, where a representsthe number of years she has beenemployed by that company. Describe whatthe solution of this system would representin termsofMaria’s salary and Ruth’ssalary. After how many years will theirsalaries be the same? What will theirsalaries be at that time?).
Manipulating Algebraic Expressions
By the end of this course, students will:write linear equations by generalizing from tables of values and by translating written descriptions; Done in grade 9 appliedwithin applications only / Specific Expectations
Manipulating and Solving Algebraic Equations
By the end of this course, students will:
Done in grade 9 applied
rearrange equations from the form y=mx + b to the form Ax + By + C=0, and vice versa; moved from grade 9 applied, revised to one direction only / express the equation of a line in the formy = mx + b, given the formAx + By + C = 0.
solve first-degree equations in one variable, including those with fractional coefficients, using an algebraic method; revised / solve first-degree equations involving onevariable, including equations with fractionalcoefficients (e.g. using the balanceanalogy, computer algebra systems, paperand pencil) (Sample problem: Solve+ 4 = 3x – 1 and verify.);
isolate a variable in formulas involving first-degree terms. revised / determine the value of a variable in thefirst degree, using a formula (i.e., byisolating the variable and then substitutingknown values; by substituting knownvalues and then solving for the variable)(e.g., in analytic geometry, in measurement)(Sample problem: A cone has a volume of100 cm3. The radius of the base is 3 cm.What is the height of the cone?);
Moved from grade 9 applied, and revised
/ Graphing and Writing Equations of LinesBy the end of this course, students will:
connect the rate of change of a linear relationto the slope of the line, and definethe slope as the ratio m =
Moved from grade 9 applied, and revised
/ identify, through investigation, y = mx + bas a common form for the equation of astraight line, and identify the special casesx = a, y = b;Moved from grade 9 applied, and revised
/ identify, through investigation with technology,the geometric significance of mand b in the equation y = mx + b;Moved from grade 9 applied, and revised
/ identify, through investigation, propertiesof the slopes of lines and line segments(e.g., direction, positive or negative rate ofchange, steepness, parallelism), usinggraphing technology to facilitate investigations,where appropriate;Moved from grade 9 applied, and revised
/ graph lines by hand, using a variety oftechniques (e.g., graph y =x – 4using the y-intercept and slope; graph2x + 3y = 6 using the x- andy-intercepts);Moved from grade 9 applied, and revised
/ determine the equation of a line, given itsgraph, the slope and y-intercept, the slopeand a point on the line, or two points onthe line.Quadratic Functions
Overall Expectations
By the end of this course, students will:manipulate algebraic expressions as they relate to quadratic functions; revised / Quadratic Relations of the form y = ax2 + bx + c
Overall Expectations
By the end of this course, students will:
manipulate algebraic expressions, as needed to understand quadratic relations;
determine, through investigation, the relationships between the graphs and the equations of quadratic functions; revised / identify characteristics of quadratic relations;
solve problems by interpreting graphs of quadratic functions.revised / solve problems by interpreting graphs of quadratic relations
Specific Expectations
Manipulating Algebraic Expressions
By the end of this course, students will:multiply two binomials and square a binomial; revised / Specific Expectations
Manipulating Quadratic Expressions
By the end of this course, students will:
expand and simplify second-degree polynomialexpressions involving one variable thatconsist of the product of two binomials[e.g., (2x + 3)(x + 4)] or the square of abinomial [e.g., (x + 3)2], using a variety oftools (e.g., algebra tiles, diagrams, computeralgebra systems, paper and pencil) andstrategies (e.g. patterning);
expand and simplify polynomial expressions involving the multiplying and squaring of binomials; Deleted
describe intervals on quadratic functions, using appropriate vocabulary (e.g., greater than, less than, between, from . . . to, less than 3 or greater than 7); Deleted
factor polynomials by determining a common factor; revised / factor binomials (e.g., 4x2 + 8x) and trinomials(e.g., 3x2 + 9x – 15) involving onevariable up to degree two, by determining acommon factor using a variety of tools(e.g., algebra tiles, computer algebra systems,paper and pencil) and strategies (e.g.,patterning);
factor trinomials of the form x2 + bx + c; revised / factor simple trinomials of the formx2 + bx + c (e.g., x2 + 7x + 10,x2 + 2x – 8), using a variety of tools (e.g.,algebra tiles, computer algebra systems,paper and pencil) and strategies (e.g.,patterning);
factor the difference of squares; revised / factor the difference of squares of the formx2– a2 (e.g., x2– 16).
solve quadratic equations by factoring. Deleted
Investigating the Connection Between the Graphs and the Equations of Quadratic Functions
By the end of this course, students will:construct tables of values, sketch graphs, and write equations of the form y=ax2 + b to represent quadratic functions derived from descriptions of realistic situations (e.g., vary the side length of a cube and observe the effect on the surface area of the cube); revised / Identifying Characteristics of Quadratic
Relations
By the end of this course, students will:
collect data that can be represented as aquadratic relation, from experiments usingappropriate equipment and technology(e.g., concrete materials, scientific probes,graphing calculators), or from secondarysources (e.g., the Internet, Statistics Canada);graph the data and draw a curve of best fit,if appropriate,with or without the use oftechnology (Sample problem: Make a 1 mramp that makes a 15° angle with the floor.Place a can 30 cm up the ramp. Record thetime it takes for the can to roll to the bottom.Repeat by placing the can 40 cm,50 cm, and 60 cm up the ramp, and so on.Graph the data and draw the curve of bestfit.);
determine, through investigation usingtechnology, that a quadratic relation of theform y = ax2 + bx + c (a ≠ 0) can be graphicallyrepresented as a parabola, and determinethat the table of values yields a constantsecond difference (Sample problem:Graph the quadratic relation y = x2– 4,using technology. Observe the shape of thegraph. Consider the corresponding table ofvalues, and calculate the first and seconddifferences. Repeat for a different quadraticrelation. Describe your observations andmake conclusions.);
identify the key features of a graph of aparabola (i.e., the equation of the axis ofsymmetry, the coordinates of the vertex, they-intercept, the zeros, and the maximum orminimum value), using a given graph or agraph generated with technology from itsequation, and use the appropriate terminologyto describe the features;
compare, through investigation using technology,the graphical representations of aquadratic relation in the formy = x2 + bx + c and the same relation inthe factored form y = (x – r)(x – s) (i.e., thegraphs are the same), and describe the connectionsbetween each algebraic representationand the graph [e.g., the y-intercept isc in the form y = x2 + bx + c; thex-intercepts are r and s in the formy = (x – r)(x – s)] (Sample problem: Use agraphing calculator to compare the graphsof y = x2 + 2x – 8 and y = (x + 4)(x – 2).In what way(s) are the equations related?What information about the graph can youidentify by looking at each equation? Makesome conclusions from your observations,and check your conclusions with a differentquadratic equation.).
identify the effect of simple transformations (i.e., translations, reflections, vertical stretch factors) on the graph and the equation of y=x2, using graphing calculators or graphing software; Deleted
explain the role of a, h, and k in the graph of y=a(x – h)2 + k; Deleted
expand and simplify an equation of the form y=a(x – h)2 + k to obtain the form y=ax2 + bx + c. Deleted
Solving Problems Involving Quadratic Functions
By the end of this course, students will:obtain the graphs of quadratic functions whose equations are given in the form y=a(x – h)2 + k or the form y=ax2 + bx + c, using graphing calculators or graphing software; incorporated into processes (selecting tools) / Solving Problems by Interpreting Graphs of
Quadratic Relations
By the end of this course, students will:
determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software; revised / solve problems involving a quadratic relationby interpreting a given graph or agraph generated with technology from itsequation (e.g., given an equation representingthe height of a ball over elapsed time,use a graphing calculator or graphing softwareto graph the relation, and answerquestions such as the following: What is themaximum height of the ball? After whatlength of time will the ball hit the ground?Over what time interval is the height ofthe ball greater than 3 m?);
solve problems involving a given quadratic function by interpreting its graph (e.g., given a formula representing the height of a ball over elapsed time, graph the function, using a graphing calculator or graphing software, and answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball touch the ground? Over what interval is the height of the ball greater than 3 m?). revised / solve problems by interpreting the significance of the key features of graphsobtained by collecting experimental datainvolving quadratic relations (Sampleproblem: Roll a can up a ramp. Using amotion detector and a graphing calculator,record the motion of the can until itreturns to its starting position, graph thedistance from the starting position versustime, and draw the curve of best fit.Interpret the meanings of the vertex andthe intercepts in terms of the experiment.Predict how the graph would change ifyou gave the can a harder push.Test yourprediction.).
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