Grade 9. Mathematical Process
9.MFM1P Foundations of Mathematics (Applied)
- MFM1P.1 Mathematical process expectations
- MFM1P.1.1 Problem Solving
- MFM1P.1.1.1 develop, select, apply, and compare a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;
- MFM1P.1.2 Reasoning and Proving
- MFM1P.1.2.1 develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;
- MFM1P.1.3 Reflecting
- MFM1P.1.3.1 demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);
- MFM1P.1.4 Selecting Tools and Computational Strategies
- MFM1P.1.4.1 select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
- Number lines (9-A.4)
- Model and solve equations using algebra tiles (9-N.1)
- Write and solve equations that represent diagrams (9-N.2)
- Solve equations: complete the solution (9-N.7)
- Graph inequalities (9-O.1)
- Write inequalities from graphs (9-O.2)
- Identify proportional relationships (9-R.1)
- Find the constant of variation (9-R.2)
- Interpret charts to find mean, median, mode and range (9-Y.2)
- MFM1P.1.5 Connecting
- MFM1P.1.5.1 make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
- Add and subtract rational numbers: word problems (9-B.5)
- Multiply and divide rational numbers: word problems (9-B.7)
- Write an equivalent ratio (9-C.2)
- Solve proportions: word problems (9-C.6)
- Scale drawings: word problems (9-C.7)
- Percent word problems (9-D.3)
- Percent of change: word problems (9-D.5)
- Pythagorean theorem: word problems (9-E.28)
- Circles: word problems (9-F.3)
- Word problems: mixed review (9-K.1)
- Word problems with money (9-K.2)
- Consecutive integer problems (9-K.3)
- Rate of travel: word problems (9-K.4)
- Weighted averages: word problems (9-K.5)
- Solve linear equations: word problems (9-N.10)
- Interpret the graph of a function: word problems (9-Q.10)
- Solve a system of equations by graphing: word problems (9-T.3)
- MFM1P.1.6 Representing
- MFM1P.1.6.1 create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
- Write variable expressions for arithmetic sequences (9-L.5)
- Write variable expressions for geometric sequences (9-L.6)
- Write variable expressions (9-M.1)
- Write variable equations (9-M.4)
- Write inequalities from graphs (9-O.2)
- Graph solutions to one-step linear inequalities (9-O.7)
- Graph solutions to two-step linear inequalities (9-O.9)
- Graph solutions to advanced linear inequalities (9-O.11)
- Complete a function table from an equation (9-Q.9)
- Graph a proportional relationship (9-R.3)
- Write direct variation equations (9-R.4)
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Complete a table and graph a linear function (9-S.13)
- Compare linear functions: graphs, tables and equations (9-S.14)
- Standard form: graph an equation (9-S.17)
- Graph a horizontal or vertical line (9-S.19)
- Point-slope form: graph an equation (9-S.20)
- Point-slope form: write an equation from a graph (9-S.22)
- Solve a system of equations by graphing (9-T.2)
- Solve a system of equations by graphing: word problems (9-T.3)
- Find the number of solutions to a system of equations by graphing (9-T.4)
- MFM1P.1.7 Communicating
- MFM1P.1.7.1 communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
- Percent of change (9-D.4)
- Symmetry (9-E.5)
- Transversal of parallel lines (9-E.9)
- Nets of three-dimensional figures (9-E.13)
- Perimeter, area and volume: changes in scale (9-E.22)
- Identify reflections, rotations and translations (9-E.23)
- Parts of a circle (9-F.1)
- Properties of addition and multiplication (9-H.1)
- Identify equivalent expressions involving exponents (9-I.14)
- Identify arithmetic and geometric sequences (9-L.1)
- Identify equivalent linear expressions (9-M.3)
- Solve equations: complete the solution (9-N.7)
- Find the number of solutions (9-N.8)
- Identify solutions to inequalities (9-O.3)
- Interpret a scatter plot (9-P.6)
- Identify independent and dependent variables (9-Q.2)
- Identify functions (9-Q.3)
- Identify functions: vertical line test (9-Q.4)
- Identify proportional relationships (9-R.1)
- Identify monomials (9-U.1)
- Polynomial vocabulary (9-V.1)
- Identify independent and dependent events (9-X.4)
- MFM1P.2 Number Sense and Algebra
- MFM1P.2.1 Overall Expectations
- MFM1P.2.1.1 solve problems involving proportional reasoning;
- MFM1P.2.1.2 simplify numerical and polynomial expressions in one variable, and solve simple first-degree equations.
- Solve equations using order of operations (9-M.6)
- Model and solve equations using algebra tiles (9-N.1)
- Write and solve equations that represent diagrams (9-N.2)
- Solve one-step linear equations (9-N.3)
- Solve two-step linear equations (9-N.4)
- Solve advanced linear equations (9-N.5)
- Solve equations with variables on both sides (9-N.6)
- Solve equations: complete the solution (9-N.7)
- Find the number of solutions (9-N.8)
- Solve linear equations: word problems (9-N.10)
- Solve linear equations: mixed review (9-N.11)
- Model polynomials with algebra tiles (9-V.2)
- Add and subtract polynomials using algebra tiles (9-V.3)
- Add and subtract polynomials (9-V.4)
- Multiply a polynomial by a monomial (9-V.6)
- Divide a polynomial by a monomial (9-V.8)
- MFM1P.2.2 Solving Problems Involving Proportional Reasoning
- MFM1P.2.2.1 illustrate equivalent ratios, using a variety of tools (e.g., concrete materials, diagrams, dynamic geometry software) (e.g., show that 4:6 represents the same ratio as 2:3 by showing that a ramp with a height of 4 m and a base of 6 m and a ramp with a height of 2 m and a base of 3 m are equally steep);
- MFM1P.2.2.2 represent, using equivalent ratios and proportions, directly proportional relationships arising from realistic situations (Sample problem: You are building a skateboard ramp whose ratio of height to base must be 2:3. Write a proportion that could be used to determine the base if the height is 4.5 m.);
- MFM1P.2.2.3 solve for the unknown value in a proportion, using a variety of methods (e.g., concrete materials, algebraic reasoning, equivalent ratios, constant of proportionality) (Sample problem: Solve x/4 = 15/20.);
- MFM1P.2.2.4 make comparisons using unit rates (e.g., if 500 mL of juice costs $2.29, the unit rate is 0.458¢/mL; this unit rate is less than for 750 mL of juice at $3.59, which has a unit rate of 0.479¢/mL);
- MFM1P.2.2.5 solve problems involving ratios, rates, and directly proportional relationships in various contexts (e.g., currency conversions, scale drawings, measurement), using a variety of methods (e.g., using algebraic reasoning, equivalent ratios, a constant of proportionality; using dynamic geometry software to construct and measure scale drawings) (Sample problem: Simple interest is directly proportional to the amount invested. If Luis invests $84 for one year and earns $1.26 in interest, how much would he earn in interest if he invested $235 for one year?);
- MFM1P.2.2.6 solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms (e.g., calculating simple interest and sales tax; analysing data) (Sample problem: Of the 29 students in a Grade 9 math class, 13 are taking science this semester. If this class is representative of all the Grade 9 students in the school, estimate and calculate the percent of the 236 Grade 9 students who are taking science this semester. Estimate and calculate the number of Grade 9 students this percent represents.).
- MFM1P.2.3 Simplifying Expressions and Solving Equations
- MFM1P.2.3.1 simplify numerical expressions involving integers and rational numbers, with and without the use of technology;
- Add, subtract, multiply and divide integers (9-B.1)
- Evaluate numerical expressions involving integers (9-B.2)
- Evaluate variable expressions involving integers (9-B.3)
- Add and subtract rational numbers (9-B.4)
- Add and subtract rational numbers: word problems (9-B.5)
- Multiply and divide rational numbers (9-B.6)
- Multiply and divide rational numbers: word problems (9-B.7)
- Evaluate numerical expressions involving rational numbers (9-B.8)
- Evaluate variable expressions involving rational numbers (9-B.9)
- MFM1P.2.3.2 relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations;
- MFM1P.2.3.3 describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three [i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or x²; volume, which is three dimensional, can be represented by (x)(x)(x), (x²)(x), or x³];
- MFM1P.2.3.4 substitute into and evaluate algebraic expressions involving exponents (i.e., evaluate expressions involving natural-number exponents with rational-number bases) [e.g., evaluate (3/2)³ by hand and 9.8³ by using a calculator]) (Sample problem: A movie theatre wants to compare the volumes of popcorn in two containers, a cube with edge length 8.1 cm and a cylinder with radius 4.5 cm and height 8.0 cm. Which container holds more popcorn?);
- MFM1P.2.3.5 add and subtract polynomials involving the same variable up to degree three [e.g., (2x + 1) + (x² – 3x + 4)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
- MFM1P.2.3.6 multiply a polynomial by a monomial involving the same variable to give results up to degree three [e.g., (2x)(3x), 2x(x + 3)], using a variety of tools (e.g., algebra tiles, drawings, computer algebra systems, paper and pencil);
- MFM1P.2.3.7 solve first-degree equations with nonfractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies) (Sample problem: Solve 2x + 7 = 6x – 1 using the balance analogy.);
- MFM1P.2.3.8 substitute into algebraic equations and solve for one variable in the first degree (e.g., in relationships, in measurement) (Sample problem: The perimeter of a rectangle can be represented as P = 2l + 2w. If the perimeter of a rectangle is 59 cm and the width is 12 cm, determine the length.).
- MFM1P.2.3.1 simplify numerical expressions involving integers and rational numbers, with and without the use of technology;
- MFM1P.3 Linear Relations
- MFM1P.3.1 Overall Expectations
- MFM1P.3.1.1 apply data-management techniques to investigate relationships between two variables;
- MFM1P.3.1.2 determine the characteristics of linear relations;
- MFM1P.3.1.3 demonstrate an understanding of constant rate of change and its connection to linear relations;
- MFM1P.3.1.4 connect various representations of a linear relation, and solve problems using the representations.
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Compare linear functions: graphs, tables and equations (9-S.14)
- Standard form: graph an equation (9-S.17)
- Graph a horizontal or vertical line (9-S.19)
- Point-slope form: graph an equation (9-S.20)
- Point-slope form: write an equation from a graph (9-S.22)
- MFM1P.3.2 Using Data Management to Investigate Relationships
- MFM1P.3.2.1 interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant [e.g., on a scatter plot of height versus age, interpret the point (13, 150) as representing a student who is 13 years old and 150 cm tall; identify points on the graph that represent students who are taller and younger than this student] (Sample problem: Given a graph that represents the relationship of the Celsius scale and the Fahrenheit scale, determine the Celsius equivalent of –5°F.);
- MFM1P.3.2.2 pose problems, identify variables, and formulate hypotheses associated with relationships between two variables (Sample problem: Does the rebound height of a ball depend on the height from which it was dropped?);
- MFM1P.3.2.3 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 (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques (e.g., making tables, drawing graphs) (Sample problem: Perform an experiment to measure and record the temperature of ice water in a plastic cup and ice water in a thermal mug over a 30 min period, for the purpose of comparison. What factors might affect the outcome of this experiment? How could you change the experiment to account for them?);
- MFM1P.3.2.4 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 (e.g., describe the trend observed in the data. Does a relationship seem to exist? Of what sort? Is the outcome consistent with your hypothesis? Identify and explain any outlying pieces of data. Suggest a formula that relates the variables. How might you vary this experiment to examine other relationships?) (Sample problem: Hypothesize the effect of the length of a pendulum on the time required for the pendulum to make five full swings. Use data to make an inference. Compare the inference with the hypothesis. Are there other relationships you might investigate involving pendulums?).
- MFM1P.3.3 Determining Characteristics of Linear Relations
- MFM1P.3.3.1 construct tables of values and graphs, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil), to represent linear relations derived from descriptions of realistic situations (Sample problem: Construct a table of values and a graph to represent a monthly cellphone plan that costs $25, plus $0.10 per minute of airtime.);
- MFM1P.3.3.2 construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools (e.g., spreadsheets, graphing software, graphing calculators, paper and pencil), for linearly related and non-linearly related data collected from a variety of sources (e.g., experiments, electronic secondary sources, patterning with concrete materials) (Sample problem: Collect data, using concrete materials or dynamic geometry software, and construct a table of values, a scatter plot, and a line or curve of best fit to represent the following relationships: the volume and the height for a square-based prism with a fixed base; the volume and the side length of the base for a square-based prism with a fixed height.);
- MFM1P.3.3.3 identify, through investigation, some properties of linear relations (i.e., numerically, the first difference is a constant, which represents a constant rate of change; graphically, a straight line represents the relation), and apply these properties to determine whether a relation is linear or non-linear.
- MFM1P.3.4 Investigating Constant Rate of Change
- MFM1P.3.4.1 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., rate of change = rise/run);
- MFM1P.3.4.2 determine, through investigation, connections among the representations of a constant rate of change of a linear relation (e.g., the cost of producing a book of photographs is $50, plus $5 per book, so an equation is C = 50 + 5p; a table of values provides the first difference of 5; the rate of change has a value of 5; and 5 is the coefficient of the independent variable, p, in this equation);
- MFM1P.3.4.3 compare the properties of direct variation and partial variation in applications, and identify the initial value (e.g., for a relation described in words, or represented as a graph or an equation) (Sample problem: Yoga costs $20 for registration, plus $8 per class. Tai chi costs $12 per class. Which situation represents a direct variation, and which represents a partial variation? For each relation, what is the initial value? Explain your answers.);
- MFM1P.3.4.4 express a linear relation as an equation in two variables, using the rate of change and the initial value (e.g., Mei is raising funds in a charity walkathon; the course measures 25 km, and Mei walks at a steady pace of 4 km/h; the distance she has left to walk can be expressed as d = 25 – 4t, where t is the number of hours since she started the walk);
- MFM1P.3.4.5 describe the meaning of the rate of change and the initial value for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the vertical intercept, 40, represents the $40 cost of renting the gym; the value of the rate of change, 2, represents the $2 cost per person), and describe a situation that could be modelled by a given linear equation (e.g., the linear equation M = 50 + 6d could model the mass of a shipping package, including 50 g for the packaging material, plus 6 g per flyer added to the package).
- MFM1P.3.5 Connecting Various Representations of Linear Relations and Solving Problems Using the Representations
- MFM1P.3.5.1 determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation (Sample problem: The equation H = 300 – 60t represents the height of a hot air balloon that is initially at 300 m and is descending at a constant rate of 60 m/min. Determine algebraically and graphically its height after 3.5 min.);
- MFM1P.3.5.2 describe a situation that would explain the events illustrated by a given graph of a relationship between two variables (Sample problem: The walk of an individual is illustrated in the given graph, produced by a motion detector and a graphing calculator. Describe the walk [e.g., the initial distance from the motion detector, the rate of walk].);
- MFM1P.3.5.3 determine other representations of a linear relation arising from a realistic situation, given one representation (e.g., given a numeric model, determine a graphical model and an algebraic model; given a graph, determine some points on the graph and determine an algebraic model);
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Standard form: graph an equation (9-S.17)
- Graph a horizontal or vertical line (9-S.19)
- Point-slope form: graph an equation (9-S.20)
- Point-slope form: write an equation (9-S.21)
- Point-slope form: write an equation from a graph (9-S.22)
- MFM1P.3.5.4 solve problems that can be modelled with first-degree equations, and compare the algebraic method to other solution methods (e.g., graphing) (Sample problem: Bill noticed it snowing and measured that 5 cm of snow had already fallen. During the next hour, an additional 1.5 cm of snow fell. If it continues to snow at this rate, how many more hours will it take until a total of 12.5 cm of snow has accumulated?);
- MFM1P.3.5.5 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 (e.g., given a partial variation graph and an equation representing the cost of producing a yearbook, describe how the graph changes if the cost per book is altered, describe how the graph changes if the fixed costs are altered, and make the corresponding changes to the equation);
- MFM1P.3.5.6 determine graphically the point of intersection of two linear relations, and interpret the intersection point in the context of an application (Sample problem: A video rental company has two monthly plans. Plan A charges a flat fee of $30 for unlimited rentals; Plan B charges $9, plus $3 per video. Use a graphical model to determine the conditions under which you should choose Plan A or Plan B.);
- MFM1P.3.5.7 select a topic involving a two-variable relationship (e.g., the amount of your pay cheque and the number of hours you work; trends in sports salaries over time; the time required to cool a cup of coffee), pose a question on the topic, collect data to answer the question, and present its solution using appropriate representations of the data (Sample problem: Individually or in a small group, collect data on the cost compared to the capacity of computer hard drives. Present the data numerically, graphically, and [if linear] algebraically. Describe the results and any trends orally or by making a poster display or by using presentation software.).
- MFM1P.4 Measurement and Geometry
- MFM1P.4.1 Overall Expectations
- MFM1P.4.1.1 determine, through investigation, the optimal values of various measurements of rectangles;
- MFM1P.4.1.2 solve problems involving the measurements of two-dimensional shapes and the volumes of three-dimensional figures;
- MFM1P.4.1.3 determine, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.
- Perimeter (9-E.1)
- Area (9-E.2)
- Area of compound figures (9-E.3)
- Count lines of symmetry (9-E.6)
- Draw lines of symmetry (9-E.7)
- Interior angles of polygons (9-E.8)
- Similar figures: side lengths and angle measures (9-E.17)
- Similar triangles and indirect measurement (9-E.18)
- Area and perimeter of similar figures (9-E.19)
- Circles: calculate area, circumference, radius and diameter (9-F.2)
- MFM1P.4.2 Investigating the Optimal Values of Measurements of Rectangles
- MFM1P.4.2.1 determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, toothpicks, a pre-made dynamic geometry sketch), and by examining various values of the area as the side lengths change and the perimeter remains constant;
- MFM1P.4.2.2 determine the minimum perimeter of a rectangle with a given area by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, a pre-made dynamic geometry sketch), and by examining various values of the side lengths and the perimeter as the area stays constant;
- MFM1P.4.2.3 solve problems that require maximizing the area of a rectangle for a fixed perimeter or minimizing the perimeter of a rectangle for a fixed area (Sample problem: You have 100 m of fence to enclose a rectangular area to be used for a snow sculpture competition. One side of the area is bounded by the school, so the fence is required for only three sides of the rectangle. Determine the dimensions of the maximum area that can be enclosed.).
- MFM1P.4.3 Solving Problems Involving Perimeter, Area, and Volume
- MFM1P.4.3.1 relate the geometric representation of the Pythagorean theorem to the algebraic representation a² + b² = c²;
- Pythagorean theorem: find the length of the hypotenuse (8-S.1)
- Pythagorean theorem: find the missing leg length (8-S.2)
- Pythagorean theorem: find the perimeter (8-S.3)
- Pythagorean theorem (9-E.27)
- Pythagorean theorem: word problems (9-E.28)
- Converse of the Pythagorean theorem: is it a right triangle? (9-E.29)
- MFM1P.4.3.2 solve problems using the Pythagorean theorem, as required in applications (e.g., calculate the height of a cone, given the radius and the slant height, in order to determine the volume of the cone);
- MFM1P.4.3.3 solve problems involving the areas and perimeters of composite two-dimensional shapes (i.e., combinations of rectangles, triangles, parallelograms, trapezoids, and circles) (Sample problem: A new park is in the shape of an isosceles trapezoid with a square attached to the shortest side. The side lengths of the trapezoidal section are 200 m, 500 m, 500 m, and 800 m, and the side length of the square section is 200 m. If the park is to be fully fenced and sodded, how much fencing and sod are required?);
- MFM1P.4.3.4 develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere (e.g., use three-dimensional figures to show that the volume of a pyramid [or cone] is 1/3 the volume of a prism [or cylinder] with the same base and height, and therefore that V(pyramid) = V(prism)/3 or V(pyramid) = (area of base)(height)/3;
- MFM1P.4.3.5 solve problems involving the volumes of prisms, pyramids, cylinders, cones, and spheres (Sample problem: Break-bit Cereal is sold in a single-serving size, in a box in the shape of a rectangular prism of dimensions 5 cm by 4 cm by 10 cm. The manufacturer also sells the cereal in a larger size, in a box with dimensions double those of the smaller box. Make a hypothesis about the effect on the volume of doubling the dimensions. Test your hypothesis using the volumes of the two boxes, and discuss the result.).
- MFM1P.4.4 Investigating and Applying Geometric Relationships
- MFM1P.4.4.1 determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials), and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons (Sample problem: With the assistance of dynamic geometry software, determine the relationship between the sum of the interior angles of a polygon and the number of sides. Use your conclusion to determine the sum of the interior angles of a 20-sided polygon.);
- MFM1P.4.4.2 determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials), and describe the properties and relationships of the angles formed by parallel lines cut by a transversal, and apply the results to problems involving parallel lines (e.g., given a diagram of a rectangular gate with a supporting diagonal beam, and given the measure of one angle in the diagram, use the angle properties of triangles and parallel lines to determine the measures of the other angles in the diagram);
- MFM1P.4.4.3 create an original dynamic sketch, paper-folding design, or other illustration that incorporates some of the geometric properties from this section, or find and report on some real-life application(s) (e.g., in carpentry, sports, architecture) of the geometric properties.
- MFM1P.4.3.1 relate the geometric representation of the Pythagorean theorem to the algebraic representation a² + b² = c²;
- MFM1P.4.1 Overall Expectations
- MFM1P.3.1 Overall Expectations
- MFM1P.2.1 Overall Expectations
- MFM1P.1.7.1 communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
- MFM1P.1.6.1 create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
- MFM1P.1.5.1 make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
- MFM1P.1.4.1 select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
- MFM1P.1.1 Problem Solving
9.MPM1D Principles of Mathematics (Academic)
- MPM1D.1 Mathematical process expectations
- MPM1D.1.1 Problem Solving
- MPM1D.1.1.1 develop, select, apply, and compare a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;
- MPM1D.1.2 Reasoning and Proving
- MPM1D.1.2.1 develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;
- MPM1D.1.3 Reflecting
- MPM1D.1.3.1 demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);
- MPM1D.1.4 Selecting Tools and Computational Strategies
- MPM1D.1.4.1 select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
- MPM1D.1.5 Connecting
- MPM1D.1.5.1 make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
- Add and subtract rational numbers: word problems (9-B.5)
- Multiply and divide rational numbers: word problems (9-B.7)
- Write an equivalent ratio (9-C.2)
- Solve proportions: word problems (9-C.6)
- Scale drawings: word problems (9-C.7)
- Percent word problems (9-D.3)
- Percent of change: word problems (9-D.5)
- Pythagorean theorem: word problems (9-E.28)
- Circles: word problems (9-F.3)
- Word problems: mixed review (9-K.1)
- Word problems with money (9-K.2)
- Consecutive integer problems (9-K.3)
- Rate of travel: word problems (9-K.4)
- Weighted averages: word problems (9-K.5)
- Solve linear equations: word problems (9-N.10)
- Interpret the graph of a function: word problems (9-Q.10)
- Solve a system of equations by graphing: word problems (9-T.3)
- MPM1D.1.6 Representing
- MPM1D.1.6.1 create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
- Write variable expressions for arithmetic sequences (9-L.5)
- Write variable expressions for geometric sequences (9-L.6)
- Write variable expressions (9-M.1)
- Write variable equations (9-M.4)
- Graph inequalities (9-O.1)
- Write inequalities from graphs (9-O.2)
- Graph solutions to one-step linear inequalities (9-O.7)
- Graph solutions to two-step linear inequalities (9-O.9)
- Graph solutions to advanced linear inequalities (9-O.11)
- Complete a function table from a graph (9-Q.8)
- Complete a function table from an equation (9-Q.9)
- Graph a proportional relationship (9-R.3)
- Write and solve direct variation equations (9-R.5)
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Complete a table and graph a linear function (9-S.13)
- Compare linear functions: graphs, tables and equations (9-S.14)
- Standard form: graph an equation (9-S.17)
- Graph a horizontal or vertical line (9-S.19)
- Point-slope form: graph an equation (9-S.20)
- Point-slope form: write an equation from a graph (9-S.22)
- Write an equation for a parallel or perpendicular line (9-S.24)
- Solve a system of equations by graphing (9-T.2)
- Solve a system of equations by graphing: word problems (9-T.3)
- Find the number of solutions to a system of equations by graphing (9-T.4)
- MPM1D.1.7 Communicating
- MPM1D.1.7.1 communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
- Symmetry (9-E.5)
- Transversal of parallel lines (9-E.9)
- Nets of three-dimensional figures (9-E.13)
- Perimeter, area and volume: changes in scale (9-E.22)
- Identify reflections, rotations and translations (9-E.23)
- Parts of a circle (9-F.1)
- Properties of addition and multiplication (9-H.1)
- Properties of equality (9-H.4)
- Identify arithmetic and geometric sequences (9-L.1)
- Solve equations: complete the solution (9-N.7)
- Find the number of solutions (9-N.8)
- Identify solutions to inequalities (9-O.3)
- Interpret a scatter plot (9-P.6)
- Identify independent and dependent variables (9-Q.2)
- Identify functions (9-Q.3)
- Identify functions: vertical line test (9-Q.4)
- Identify proportional relationships (9-R.1)
- Identify monomials (9-U.1)
- Polynomial vocabulary (9-V.1)
- Identify independent and dependent events (9-X.4)
- MPM1D.2 Number Sense and Algebra
- MPM1D.2.1 Overall Expectations
- MPM1D.2.1.1 demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
- Exponents with integer bases (9-I.2)
- Exponents with decimal and fractional bases (9-I.3)
- Negative exponents (9-I.4)
- Multiplication with exponents – variable bases (9-I.9)
- Division with exponents – variable bases (9-I.10)
- Multiplication and division with exponents – variable bases (9-I.11)
- Evaluate expressions using properties of exponents (9-I.13)
- MPM1D.2.1.2 manipulate numerical and polynomial expressions, and solve first-degree equations.
- Identify equivalent linear expressions (9-M.3)
- Rearrange multi-variable equations (9-M.7)
- Model and solve equations using algebra tiles (9-N.1)
- Write and solve equations that represent diagrams (9-N.2)
- Solve one-step linear equations (9-N.3)
- Solve two-step linear equations (9-N.4)
- Solve advanced linear equations (9-N.5)
- Solve equations with variables on both sides (9-N.6)
- Solve equations: complete the solution (9-N.7)
- Find the number of solutions (9-N.8)
- Solve linear equations: word problems (9-N.10)
- Solve linear equations: mixed review (9-N.11)
- Model polynomials with algebra tiles (9-V.2)
- Multiply a polynomial by a monomial (9-V.6)
- Divide a polynomial by a monomial (9-V.8)
- MPM1D.2.2 Operating with Exponents
- MPM1D.2.2.1 substitute into and evaluate algebraic expressions involving exponents (i.e., evaluate expressions involving natural-number exponents with rational-number bases [e.g., evaluate (3/2)³ by hand and 9.8³ by using a calculator]) (Sample problem: A movie theatre wants to compare the volumes of popcorn in two containers, a cube with edge length 8.1 cm and a cylinder with radius 4.5 cm and height 8.0 cm. Which container holds more popcorn?);
- MPM1D.2.2.2 describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three [i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or x²; volume, which is three dimensional, can be represented by (x)(x)(x), (x²)(x), or x³];
- MPM1D.2.2.3 derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents;
- MPM1D.2.2.4 extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents.
- MPM1D.2.3 Manipulating Expressions and Solving Equations
- MPM1D.2.3.1 simplify numerical expressions involving integers and rational numbers, with and without the use of technology;
- Add, subtract, multiply and divide integers (9-B.1)
- Evaluate numerical expressions involving integers (9-B.2)
- Evaluate variable expressions involving integers (9-B.3)
- Add and subtract rational numbers (9-B.4)
- Add and subtract rational numbers: word problems (9-B.5)
- Multiply and divide rational numbers (9-B.6)
- Multiply and divide rational numbers: word problems (9-B.7)
- Evaluate numerical expressions involving rational numbers (9-B.8)
- Evaluate variable expressions involving rational numbers (9-B.9)
- MPM1D.2.3.2 solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion;
- MPM1D.2.3.3 relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations;
- MPM1D.2.3.4 add and subtract polynomials with up to two variables [e.g., (2x – 5) + (3x + 1), (3x²y + 2xy²) + (4x²y – 6xy²)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
- MPM1D.2.3.5 multiply a polynomial by a monomial involving the same variable [e.g., 2x(x + 4), 2x²(3x² – 2x + 1)], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil);
- MPM1D.2.3.6 expand and simplify polynomial expressions involving one variable [e.g., 2x(4x + 1) – 3x(x + 2)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
- MPM1D.2.3.7 solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);
- Solve one-step linear equations (9-N.3)
- Solve two-step linear equations (9-N.4)
- Solve advanced linear equations (9-N.5)
- Solve equations with variables on both sides (9-N.6)
- Solve equations: complete the solution (9-N.7)
- Solve linear equations: mixed review (9-N.11)
- Linear equations: solve for y (9-S.11)
- MPM1D.2.3.8 rearrange formulas involving variables in the first degree, with and without substitution (e.g., in analytic geometry, in measurement) (Sample problem: A circular garden has a circumference of 30 m. What is the length of a straight path that goes through the centre of this garden?);
- MPM1D.2.3.9 solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods (Sample problem: Solve the following problem in more than one way: Jonah is involved in a walkathon. His goal is to walk 25 km. He begins at 9:00 a.m. and walks at a steady rate of 4 km/h. How many kilometres does he still have left to walk at 1:15 p.m. if he is to achieve his goal?).
- MPM1D.2.3.1 simplify numerical expressions involving integers and rational numbers, with and without the use of technology;
- MPM1D.3 Linear Relations
- MPM1D.3.1 Overall Expectations
- MPM1D.3.1.1 apply data-management techniques to investigate relationships between two variables;
- MPM1D.3.1.2 demonstrate an understanding of the characteristics of a linear relation;
- MPM1D.3.1.3 connect various representations of a linear relation.
- Graph a proportional relationship (9-R.3)
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Complete a table and graph a linear function (9-S.13)
- Compare linear functions: graphs, tables and equations (9-S.14)
- Standard form: graph an equation (9-S.17)
- Graph a horizontal or vertical line (9-S.19)
- Point-slope form: graph an equation (9-S.20)
- Point-slope form: write an equation from a graph (9-S.22)
- MPM1D.3.2 Using Data Management to Investigate Relationships
- MPM1D.3.2.1 interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant [e.g., on a scatter plot of height versus age, interpret the point (13, 150) as representing a student who is 13 years old and 150 cm tall; identify points on the graph that represent students who are taller and younger than this student] (Sample problem: Given a graph that represents the relationship of the Celsius scale and the Fahrenheit scale, determine the Celsius equivalent of –5°F.);
- MPM1D.3.2.2 pose problems, identify variables, and formulate hypotheses associated with relationships between two variables (Sample problem: Does the rebound height of a ball depend on the height from which it was dropped?);
- MPM1D.3.2.3 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 (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques (e.g., making tables, drawing graphs) (Sample problem: Design and perform an experiment to measure and record the temperature of ice water in a plastic cup and ice water in a thermal mug over a 30 min period, for the purpose of comparison. What factors might affect the outcome of this experiment? How could you design the experiment to account for them?);
- MPM1D.3.2.4 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 (e.g., describe the trend observed in the data. Does a relationship seem to exist? Of what sort? Is the outcome consistent with your hypothesis? Identify and explain any outlying pieces of data. Suggest a formula that relates the variables. How might you vary this experiment to examine other relationships?) (Sample problem: Hypothesize the effect of the length of a pendulum on the time required for the pendulum to make five full swings. Use data to make an inference. Compare the inference with the hypothesis. Are there other relationships you might investigate involving pendulums?).
- MPM1D.3.3 Understanding Characteristics of Linear Relations
- MPM1D.3.3.1 construct tables of values, graphs, and equations, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil), to represent linear relations derived from descriptions of realistic situations (Sample problem: Construct a table of values, a graph, and an equation to represent a monthly cellphone plan that costs $25, plus $0.10 per minute of airtime.);
- MPM1D.3.3.2 construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools (e.g., spreadsheets, graphing software, graphing calculators, paper and pencil), for linearly related and non-linearly related data collected from a variety of sources (e.g., experiments, electronic secondary sources, patterning with concrete materials) (Sample problem: Collect data, using concrete materials or dynamic geometry software, and construct a table of values, a scatter plot, and a line or curve of best fit to represent the following relationships: the volume and the height for a square-based prism with a fixed base; the volume and the side length of the base for a square-based prism with a fixed height.);
- MPM1D.3.3.3 identify, through investigation, some properties of linear relations (i.e., numerically, the first difference is a constant, which represents a constant rate of change; graphically, a straight line represents the relation), and apply these properties to determine whether a relation is linear or non-linear;
- MPM1D.3.3.4 compare the properties of direct variation and partial variation in applications, and identify the initial value (e.g., for a relation described in words, or represented as a graph or an equation) (Sample problem: Yoga costs $20 for registration, plus $8 per class. Tai chi costs $12 per class. Which situation represents a direct variation, and which represents a partial variation? For each relation, what is the initial value? Explain your answers.);
- MPM1D.3.3.5 determine the equation of a line of best fit for a scatter plot, using an informal process (e.g., using a movable line in dynamic statistical software; using a process of trial and error on a graphing calculator; determining the equation of the line joining two carefully chosen points on the scatter plot).
- MPM1D.3.4 Connecting Various Representations of Linear Relations
- MPM1D.3.4.1 determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation (Sample problem: The equation H = 300 – 60t represents the height of a hot air balloon that is initially at 300 m and is descending at a constant rate of 60 m/min. Determine algebraically and graphically how long the balloon will take to reach a height of 160 m.);
- MPM1D.3.4.2 describe a situation that would explain the events illustrated by a given graph of a relationship between two variables (Sample problem: The walk of an individual is illustrated in the given graph, produced by a motion detector and a graphing calculator. Describe the walk [e.g., the initial distance from the motion detector, the rate of walk].);
- MPM1D.3.4.3 determine other representations of a linear relation, given one representation (e.g., given a numeric model, determine a graphical model and an algebraic model; given a graph, determine some points on the graph and determine an algebraic model);
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Standard form: graph an equation (9-S.17)
- Graph a horizontal or vertical line (9-S.19)
- Point-slope form: graph an equation (9-S.20)
- Point-slope form: write an equation (9-S.21)
- Point-slope form: write an equation from a graph (9-S.22)
- MPM1D.3.4.4 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 (e.g., given a partial variation graph and an equation representing the cost of producing a yearbook, describe how the graph changes if the cost per book is altered, describe how the graph changes if the fixed costs are altered, and make the corresponding changes to the equation).
- MPM1D.4 Analytic Geometry
- MPM1D.4.1 Overall Expectations
- MPM1D.4.1.1 determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;
- MPM1D.4.1.2 determine, through investigation, the properties of the slope and y-intercept of a linear relation;
- MPM1D.4.1.3 solve problems involving linear relations.
- Linear equations: solve for y (9-S.11)
- Write linear functions to solve word problems (9-S.12)
- Is (x, y) a solution to the system of equations? (9-T.1)
- Solve a system of equations by graphing (9-T.2)
- Solve a system of equations by graphing: word problems (9-T.3)
- Find the number of solutions to a system of equations by graphing (9-T.4)
- MPM1D.4.2 Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph
- MPM1D.4.2.1 determine, through investigation, the characteristics that distinguish the equation of a straight line from the equations of nonlinear relations (e.g., use a graphing calculator or graphing software to graph a variety of linear and non-linear relations from their equations; classify the relations according to the shapes of their graphs; connect an equation of degree one to a linear relation);
- MPM1D.4.2.2 identify, through investigation, the equation of a line in any of the forms y = mx + b, Ax + By + C = 0, x = a, y = b;
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Write equations in standard form (9-S.15)
- Standard form: graph an equation (9-S.17)
- Equations of horizontal and vertical lines (9-S.18)
- Point-slope form: write an equation (9-S.21)
- Point-slope form: write an equation from a graph (9-S.22)
- MPM1D.4.2.3 express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.
- MPM1D.4.3 Investigating the Properties of Slope
- MPM1D.4.3.1 determine, through investigation, various formulas for the slope of a line segment or a line (e.g., m = rise/run, m = the change in y/the change in x or m = delta y/delta x, m = (y2 – y1)/(x2 – x1)), and use the formulas to determine the slope of a line segment or a line;
- MPM1D.4.3.2 identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;
- Slope-intercept form: find the slope and y-intercept (9-S.5)
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Standard form: graph an equation (9-S.17)
- MPM1D.4.3.3 determine, through investigation, connections among the representations of a constant rate of change of a linear relation (e.g., the cost of producing a book of photographs is $50, plus $5 per book, so an equation is C = 50 + 5p; a table of values provides the first difference of 5; the rate of change has a value of 5, which is also the slope of the corresponding line; and 5 is the coefficient of the independent variable, p, in this equation);
- MPM1D.4.3.4 identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using graphing technology to facilitate investigations, where appropriate.
- MPM1D.4.4 Using the Properties of Linear Relations to Solve Problems
- MPM1D.4.4.1 graph lines by hand, using a variety of techniques (e.g., graph y = 2/3x – 4 using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);
- MPM1D.4.4.2 determine the equation of a line from information about the line (e.g., the slope and y-intercept; the slope and a point; two points) (Sample problem: Compare the equations of the lines parallel to and perpendicular to y = 2x – 4, and with the same x-intercept as 3x – 4y = 12. Verify using dynamic geometry software.);
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Point-slope form: write an equation (9-S.21)
- Point-slope form: write an equation from a graph (9-S.22)
- Slopes of parallel and perpendicular lines (9-S.23)
- Write an equation for a parallel or perpendicular line (9-S.24)
- MPM1D.4.4.3 describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the vertical intercept, 40, represents the $40 cost of renting the gym; the value of the rate of change, 2, represents the $2 cost per person), and describe a situation that could be modelled by a given linear equation (e.g., the linear equation M = 50 + 6d could model the mass of a shipping package, including 50 g for the packaging material, plus 6 g per flyer added to the package);
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Write linear functions to solve word problems (9-S.12)
- Is (x, y) a solution to the system of equations? (9-T.1)
- Solve a system of equations by graphing (9-T.2)
- Solve a system of equations by graphing: word problems (9-T.3)
- Find the number of solutions to a system of equations by graphing (9-T.4)
- MPM1D.4.4.4 identify and explain any restrictions on the variables in a linear relation arising from a realistic situation (e.g., in the relation C = 50 + 25n, C is the cost of holding a party in a hall and n is the number of guests; n is restricted to whole numbers of 100 or less, because of the size of the hall, and C is consequently restricted to $50 to $2550);
- MPM1D.4.4.5 determine graphically the point of intersection of two linear relations, and interpret the intersection point in the context of an application (Sample problem: A video rental company has two monthly plans. Plan A charges a flat fee of $30 for unlimited rentals; Plan B charges $9, plus $3 per video. Use a graphical model to determine the conditions under which you should choose Plan A or Plan B.).
- MPM1D.5 Measurement and Geometry
- MPM1D.5.1 Overall Expectations
- MPM1D.5.1.1 determine, through investigation, the optimal values of various measurements;
- MPM1D.5.1.2 solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures;
- Perimeter (9-E.1)
- Area (9-E.2)
- Area of compound figures (9-E.3)
- Volume of prisms and pyramids (9-E.10)
- Volume of cones and cylinders (9-E.11)
- Volume of spheres (9-E.12)
- Surface area of prisms and pyramids (9-E.14)
- Surface area of cones and cylinders (9-E.15)
- Surface area of spheres (9-E.16)
- Similar figures: side lengths and angle measures (9-E.17)
- Similar triangles and indirect measurement (9-E.18)
- Area and perimeter of similar figures (9-E.19)
- Similar solids (9-E.20)
- Volume and surface area of similar solids (9-E.21)
- Perimeter, area and volume: changes in scale (9-E.22)
- Circles: calculate area, circumference, radius and diameter (9-F.2)
- Circles: word problems (9-F.3)
- MPM1D.5.1.3 verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.
- MPM1D.5.2 Investigating the Optimal Values of Measurements
- MPM1D.5.2.1 determine the maximum area of a rectangle with a given perimeter by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, toothpicks, a pre-made dynamic geometry sketch), and by examining various values of the area as the side lengths change and the perimeter remains constant;
- MPM1D.5.2.2 determine the minimum perimeter of a rectangle with a given area by constructing a variety of rectangles, using a variety of tools (e.g., geoboards, graph paper, a premade dynamic geometry sketch), and by examining various values of the side lengths and the perimeter as the area stays constant;
- MPM1D.5.2.3 identify, through investigation with a variety of tools (e.g. concrete materials, computer software), the effect of varying the dimensions on the surface area [or volume] of square-based prisms and cylinders, given a fixed volume [or surface area];
- MPM1D.5.2.4 explain the significance of optimal area, surface area, or volume in various applications (e.g., the minimum amount of packaging material; the relationship between surface area and heat loss);
- MPM1D.5.2.5 pose and solve problems involving maximization and minimization of measurements of geometric shapes and figures (e.g., determine the dimensions of the rectangular field with the maximum area that can be enclosed by a fixed amount of fencing, if the fencing is required on only three sides) (Sample problem: Determine the dimensions of a square-based, open-topped prism with a volume of 24 cm³ and with the minimum surface area.).
- MPM1D.5.3 Solving Problems Involving Perimeter, Area, Surface Area, and Volume
- MPM1D.5.3.1 relate the geometric representation of the Pythagorean theorem and the algebraic representation a² + b² = c²;
- Pythagorean theorem: find the length of the hypotenuse (8-S.1)
- Pythagorean theorem: find the missing leg length (8-S.2)
- Pythagorean theorem: find the perimeter (8-S.3)
- Pythagorean theorem (9-E.27)
- Pythagorean theorem: word problems (9-E.28)
- Converse of the Pythagorean theorem: is it a right triangle? (9-E.29)
- MPM1D.5.3.2 solve problems using the Pythagorean theorem, as required in applications (e.g., calculate the height of a cone, given the radius and the slant height, in order to determine the volume of the cone);
- MPM1D.5.3.3 solve problems involving the areas and perimeters of composite two-dimensional shapes (i.e., combinations of rectangles, triangles, parallelograms, trapezoids, and circles) (Sample problem: A new park is in the shape of an isosceles trapezoid with a square attached to the shortest side. The side lengths of the trapezoidal section are 200 m, 500 m, 500 m, and 800 m, and the side length of the square section is 200 m. If the park is to be fully fenced and sodded, how much fencing and sod are required?);
- MPM1D.5.3.4 develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere (e.g., use three-dimensional figures to show that the volume of a pyramid [or cone] is the volume of a prism [or cylinder] with the same base and height, and therefore that V(pyramid) = V(prism)/3 or V(pyramid) = (area of base)(height)/3;
- MPM1D.5.3.5 determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a square-based pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);
- MPM1D.5.3.6 solve problems involving the surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres, including composite figures (Sample problem: Break-bit Cereal is sold in a single-serving size, in a box in the shape of a rectangular prism of dimensions 5 cm by 4 cm by 10 cm. The manufacturer also sells the cereal in a larger size, in a box with dimensions double those of the smaller box. Compare the surface areas and the volumes of the two boxes, and explain the implications of your answers.).
- MPM1D.5.4 Investigating and Applying Geometric Relationships
- MPM1D.5.4.1 determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials), and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons (Sample problem: With the assistance of dynamic geometry software, determine the relationship between the sum of the interior angles of a polygon and the number of sides. Use your conclusion to determine the sum of the interior angles of a 20-sided polygon.);
- MPM1D.5.4.2 determine, through investigation using a variety of tools (e.g., dynamic geometry software, paper folding), and describe some properties of polygons (e.g., the figure that results from joining the midpoints of the sides of a quadrilateral is a parallelogram; the diagonals of a rectangle bisect each other; the line segment joining the midpoints of two sides of a triangle is half the length of the third side), and apply the results in problem solving (e.g., given the width of the base of an A-frame tree house, determine the length of a horizontal support beam that is attached half way up the sloping sides);
- MPM1D.5.4.3 pose questions about geometric relationships, investigate them, and present their findings, using a variety of mathematical forms (e.g., written explanations, diagrams, dynamic sketches, formulas, tables) (Sample problem: How many diagonals can be drawn from one vertex of a 20-sided polygon? How can I find out without counting them?);
- MPM1D.5.4.4 illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example, with or without the use of dynamic geometry software (Sample problem: Confirm or deny the following statement: If a quadrilateral has perpendicular diagonals, then it is a square.).
- MPM1D.5.3.1 relate the geometric representation of the Pythagorean theorem and the algebraic representation a² + b² = c²;
- MPM1D.5.1 Overall Expectations
- MPM1D.4.1 Overall Expectations
- MPM1D.3.1 Overall Expectations
- MPM1D.2.1.1 demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
- MPM1D.2.1 Overall Expectations
- MPM1D.1.7.1 communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
- MPM1D.1.6.1 create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
- MPM1D.1.5.1 make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
- MPM1D.1.1 Problem Solving
9.MPM1H Mathematics Transfer Course, Applied to Academic
- MPM1H.1 Number Sense and Algebra
- MPM1H.1.1 Overall Expectations
- MPM1H.1.1.1 demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
- MPM1H.1.1.2 manipulate numerical and polynomial expressions, and solve first-degree equations.
- Distributive property (9-H.2)
- Simplify variable expressions using properties (9-H.3)
- Simplify variable expressions involving like terms and the distributive property (9-M.2)
- Identify equivalent linear expressions (9-M.3)
- Rearrange multi-variable equations (9-M.7)
- Solve one-step linear equations (9-N.3)
- Solve two-step linear equations (9-N.4)
- Solve advanced linear equations (9-N.5)
- Solve equations with variables on both sides (9-N.6)
- Solve equations: complete the solution (9-N.7)
- Solve linear equations: word problems (9-N.10)
- Solve linear equations: mixed review (9-N.11)
- MPM1H.1.2 Operating with Exponents
- MPM1H.1.2.1 derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents;
- MPM1H.1.2.2 extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents.
- MPM1H.1.3 Manipulating Expressions and Solving Equations
- MPM1H.1.3.1 add and subtract polynomials with up to two variables [e.g., (3x²y + 2xy²) + (4x²y – 6xy²)], using a variety of tools (e.g., computer algebra systems, paper and pencil);
- MPM1H.1.3.2 multiply a polynomial by a monomial involving the same variable [e.g., 2x²(3x² – 2x + 1)], using a variety of tools (e.g., computer algebra systems, paper and pencil);
- MPM1H.1.3.3 expand and simplify polynomial expressions involving one variable [e.g., 2x(4x + 1) – 3x(x + 2)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
- MPM1H.1.3.4 solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);
- Solve proportions (9-C.5)
- Model and solve equations using algebra tiles (9-N.1)
- Write and solve equations that represent diagrams (9-N.2)
- Solve one-step linear equations (9-N.3)
- Solve two-step linear equations (9-N.4)
- Solve advanced linear equations (9-N.5)
- Solve equations with variables on both sides (9-N.6)
- Solve equations: complete the solution (9-N.7)
- Find the number of solutions (9-N.8)
- Solve linear equations: word problems (9-N.10)
- Solve linear equations: mixed review (9-N.11)
- Linear equations: solve for y (9-S.11)
- MPM1H.1.3.5 rearrange formulas involving variables in the first degree, with and without substitution (e.g., in analytic geometry, in measurement) (Sample problem: A circular garden has a circumference of 30m. Write an expression for the length of a straight path that goes through the centre of this garden.);
- MPM1H.1.3.6 solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods (Sample problem: Solve the following problem in more than one way: Jonah is involved in a walkathon. His goal is to walk 25 km. He begins at 9:00 a.m. and walks at a steady rate of 4 km/h. How many kilometres does he still have left to walk at 1:15 p.m. if he is to achieve his goal?).
- MPM1H.2 Linear Relations
- MPM1H.2.1 Most of the expectations in the Linear Relations strand of the Grade 9 applied and academic courses are common to the two courses. Some of these expectations may require review since the concepts and skills in this strand are critical for student success in the Analytic Geometry strand of this transfer course.
- MPM1H.3 Analytic Geometry
- MPM1H.3.1 Overall Expectations
- MPM1H.3.1.1 demonstrate an understanding of the characteristics of a linear relation;
- MPM1H.3.1.2 determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;
- MPM1H.3.1.3 determine, through investigation, the properties of the slope and y-intercept of a linear relation;
- Find the constant of variation (9-R.2)
- Find the slope of a graph (9-S.2)
- Find the slope from two points (9-S.3)
- Slope-intercept form: find the slope and y-intercept (9-S.5)
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Standard form: find x- and y-intercepts (9-S.16)
- Slopes of parallel and perpendicular lines (9-S.23)
- MPM1H.3.1.4 solve problems involving linear relations.
- MPM1H.3.2 Understanding Characteristics of Linear Relations
- MPM1H.3.2.1 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 (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques (e.g., making tables, drawing graphs) (Sample problem: Design and perform an experiment to measure and record the temperature of ice water in a plastic cup and ice water in a thermal mug over a 30 min period, for the purpose of comparison. What factors might affect the outcome of this experiment? How could you design the experiment to account for them?);
- MPM1H.3.2.2 construct equations to represent linear relations derived from descriptions of realistic situations, and connect the equations to tables of values and graphs, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil) (Sample problem: Construct a table of values, a graph, and an equation to represent a monthly cellphone plan that costs $25, plus $0.10 per minute of airtime.);
- Rate of travel: word problems (9-K.4)
- Solve linear equations: word problems (9-N.10)
- Complete a function table from an equation (9-Q.9)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Write linear functions to solve word problems (9-S.12)
- Complete a table and graph a linear function (9-S.13)
- MPM1H.3.2.3 determine the equation of a line of best fit for a scatter plot, using an informal process (e.g., using a movable line in dynamic statistical software; using a process of trial and error on a graphing calculator; determining the equation of the line joining two carefully chosen points on the scatter plot).
- MPM1H.3.3 Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph
- MPM1H.3.3.1 determine, through investigation, the characteristics that distinguish the equation of a straight line from the equations of nonlinear relations (e.g., use a graphing calculator or graphing software to graph a variety of linear and non-linear relations from their equations; classify the relations according to the shapes of their graphs; connect an equation of degree one to a linear relation);
- MPM1H.3.3.2 identify, through investigation, the equation of a line in any of the forms y = mx + b, Ax + By + C = 0, x = a, y = b;
- Write direct variation equations (9-R.4)
- Slope-intercept form: graph an equation (9-S.6)
- Slope-intercept form: write an equation from a graph (9-S.7)
- Slope-intercept form: write an equation (9-S.8)
- Slope-intercept form: write an equation from a table (9-S.9)
- Slope-intercept form: write an equation from a word problem (9-S.10)
- Write equations in standard form (9-S.15)
- Equations of horizontal and vertical lines (9-S.18)
- Point-slope form: graph an equation (9-S.20)
- Point-slope form: write an equation (9-S.21)
- Point-slope form: write an equation from a graph (9-S.22)
- MPM1H.3.3.3 express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.
- MPM1H.3.4 Investigating the Properties of Slope
- MPM1H.3.4.1 determine, through investigation, various formulas for the slope of a line segment or a line (e.g., m = rise/run, m = the change in y/the change in x or m = delta y/delta x, m = (y2 – y1)/(x2 – x1), and use the formulas to determine the slope of a line segment or a line;
- MPM1H.3.4.2 identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;
- MPM1H.3.4.3 determine, through investigation, connections between slope and other representations of a constant rate of change of linear relation (e.g., if the cost of producing a book of photographs is $50, plus $5 per book, then the slope of the line that represents the cost versus the number of books produced has a value of 5, which is also the rate of change; the value of the slope is the value of the coefficient of the independent variable in the equation of the line, C = 50 + 5p, and the value of the first difference in a table of values);
- MPM1H.3.4.4 identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using graphing technology to facilitate investigations, where appropriate.
- MPM1H.3.5 Using the Properties of Linear Relations to Solve Problems
- MPM1H.3.5.1 graph lines by hand, using a variety of techniques (e.g., graph y = 2/3x – 4 using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);
- MPM1H.3.5.2 determine the equation of a line from information about the line (e.g., the slope and y-intercept; the slope and a point; two points) (Sample problem: Compare the equations of the lines parallel to and perpendicular to y = 2x – 4, and with the same x-intercept as 3x – 4y = 12. Verify using dynamic geometry software.);
- MPM1H.3.5.3 describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the y-intercept, 40, represents the $40 cost of renting the gym; the value of the slope, 2, represents the $2 cost per person);
- MPM1H.3.5.4 identify and explain any restrictions on the variables in a linear relation arising from a realistic situation (e.g., in the relation C = 50 + 25n, C is the cost of holding a party in a hall and n is the number of guests; n is restricted to whole numbers of 100 or less, because of the size of the hall, and C is consequently restricted to $50 to $2550).
- MPM1H.4 Measurement and Geometry
- MPM1H.4.1 Overall Expectations
- MPM1H.4.1.1 solve problems involving the surface areas and volumes of three-dimensional figures;
- Volume of prisms and pyramids (9-E.10)
- Volume of cones and cylinders (9-E.11)
- Volume of spheres (9-E.12)
- Nets of three-dimensional figures (9-E.13)
- Surface area of prisms and pyramids (9-E.14)
- Surface area of cones and cylinders (9-E.15)
- Surface area of spheres (9-E.16)
- Similar solids (9-E.20)
- Volume and surface area of similar solids (9-E.21)
- Perimeter, area and volume: changes in scale (9-E.22)
- MPM1H.4.1.2 verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.
- MPM1H.4.2 Solving Problems Involving Surface Area and Volume
- MPM1H.4.2.1 solve problems involving the volumes of composite figures composed of prisms, pyramids, cylinders, cones, and spheres;
- MPM1H.4.2.2 determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a square-based pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);
- MPM1H.4.2.3 solve problems involving the surface areas of prisms, pyramids, cylinders, cones, and spheres, including composite figures (Sample problem: Break-bit Cereal is sold in a single-serving size, in a box in the shape of a rectangular prism of dimensions 5 cm by 4 cm by 10 cm. The manufacturer also sells the cereal in a larger size, in a box with dimensions double those of the smaller box. Compare the surface areas and the volumes of the two boxes, and explain the implications of your answers.);
- MPM1H.4.2.4 identify, through investigation with a variety of tools (e.g. concrete materials, computer software), the effect of varying the dimensions on the surface area [or volume] of square-based prisms and cylinders, given a fixed volume [or surface area];
- MPM1H.4.2.5 explain the significance of optimal surface area or volume in various applications (e.g., the minimum amount of packaging material; the relationship between surface area and heat loss);
- MPM1H.4.2.6 pose and solve problems involving maximization and minimization of measurements of geometric figures (i.e., square-based prisms or cylinders) (Sample problem: Determine the dimensions of a square-based, open-topped prism with a volume of 24 cm³ and with the minimum surface area.).
- MPM1H.4.3 Investigating and Applying Geometric Relationships
- MPM1H.4.3.1 determine, through investigation using a variety of tools (e.g., dynamic geometry software, paper folding), and describe some properties of polygons (e.g., the figure that results from joining the midpoints of the sides of a quadrilateral is a parallelogram; the diagonals of a rectangle bisect each other; the line segment joining the midpoints of two sides of a triangle is half the length of the third side), and apply the results in problem solving (e.g., given the width of the base of an A-frame tree house, determine the length of a horizontal support beam that is attached half way up the sloping sides);
- MPM1H.4.3.2 pose questions about geometric relationships, investigate them, and present their findings, using a variety of mathematical forms (e.g., written explanations, diagrams, dynamic sketches, formulas, tables) (Sample problem: How many diagonals can be drawn from one vertex of a 20- sided polygon? How can I find out without counting them?);
- MPM1H.4.3.3 illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example, with or without the use of dynamic geometry software (Sample problem: Confirm or deny the following statement: If a quadrilateral has perpendicular diagonals, then it is a square.).
- MPM1H.4.1.1 solve problems involving the surface areas and volumes of three-dimensional figures;
- MPM1H.4.1 Overall Expectations
- MPM1H.3.1 Overall Expectations
- MPM1H.1.1 Overall Expectations
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