1: Mathematical Process

1.2: Reasoning and Proving

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;

Biconditional Statements

1.7: Communicating

1.7.1: communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.

Using Algebraic Expressions

2: Number Sense and Numeration

2.1: represent, compare, and order equivalent representations of numbers, including those involving positive exponents;

2.1.1: express repeated multiplication using exponential notation (e.g., 2 x 2 x 2 x 2 = 2⁴);

Exponents and Power Rules

2.1.3: represent, compare, and order rational numbers (i.e., positive and negative fractions and decimals to thousandths);

Comparing and Ordering Decimals
Percents, Fractions, and Decimals
Rational Numbers, Opposites, and Absolute Values

2.1.4: translate between equivalent forms of a number (i.e., decimals, fractions, percents) (e.g., 3/4 = 0.75);

Part-to-part and Part-to-whole Ratios
Percents, Fractions, and Decimals

2.1.5: determine common factors and common multiples using the prime factorization of numbers (e.g., the prime factorization of 12 is 2 x 2 x 3; the prime factorization of 18 is 2 x 3 x 3; the greatest common factor of 12 and 18 is 2 x 3 or 6; the least common multiple of 12 and 18 is 2 x 2 x 3 x 3 or 36).

Finding Factors with Area Models

2.2: solve problems involving whole numbers, decimal numbers, fractions, and integers, using a variety of computational strategies;

2.2.1: solve multi-step problems arising from real-life contexts and involving whole numbers and decimals, using a variety of tools (e.g., graphs, calculators) and strategies (e.g., estimation, algorithms);

Multiplying with Decimals
Percents, Fractions, and Decimals
Sums and Differences with Decimals

2.2.2: solve problems involving percents expressed to one decimal place (e.g., 12.5%) and whole-number percents greater than 100 (e.g., 115%) (Sample problem: The total cost of an item with tax included [115%] is $23.00. Use base ten materials to determine the price before tax.);

Part-to-part and Part-to-whole Ratios

2.2.3: use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution;

Estimating Sums and Differences

2.2.4: represent the multiplication and division of fractions, using a variety of tools and strategies (e.g., use an area model to represent 1/4 multiplied by 1/3);

Dividing Fractions
Dividing Mixed Numbers
Multiplying Fractions
Multiplying Mixed Numbers

2.2.6: represent the multiplication and division of integers, using a variety of tools [e.g., if black counters represent positive amounts and red counters represent negative amounts, you can model 3 x (–2) as three groups of two red counters];

Adding and Subtracting Integers

2.2.7: solve problems involving operations with integers, using a variety of tools (e.g., two-colour counters, virtual manipulatives, number lines);

Adding and Subtracting Integers
Adding on the Number Line
Addition of Polynomials

2.2.8: evaluate expressions that involve integers, including expressions that contain brackets and exponents, using order of operations;

Equivalent Algebraic Expressions I
Order of Operations

2.2.10: estimate, and verify using a calculator, the positive square roots of whole numbers, and distinguish between whole numbers that have whole-number square roots (i.e., perfect square numbers) and those that do not (Sample problem: Explain why a square with an area of 20 cm² does not have a whole-number side length.).

Operations with Radical Expressions
Simplifying Radical Expressions
Square Roots

2.3: solve problems by using proportional reasoning in a variety of meaningful contexts.

2.3.1: identify and describe real-life situations involving two quantities that are directly proportional (e.g., the number of servings and the quantities in a recipe, mass and volume of a substance, circumference and diameter of a circle);

Direct and Inverse Variation
Household Energy Usage

2.3.3: solve problems involving percent that arise from real-life contexts (e.g., discount, sales tax, simple interest) (Sample problem: In Ontario, people often pay a provincial sales tax [PST] of 8% and a federal sales tax [GST] of 7% when they make a purchase. Does it matter which tax is calculated first? Explain your reasoning.);

Compound Interest
Percent of Change
Percents and Proportions
Percents, Fractions, and Decimals
Real-Time Histogram

3: Measurement

3.2: determine the relationships among units and measurable attributes, including the area of a circle and the volume of a cylinder.

3.2.1: solve problems that require conversions involving metric units of area, volume, and capacity (i.e., square centimetres and square metres; cubic centimetres and cubic metres; millilitres and cubic centimetres) (Sample problem: What is the capacity of a cylindrical beaker with a radius of 5 cm and a height of 15 cm?);

Unit Conversions

3.2.2: measure the circumference, radius, and diameter of circular objects, using concrete materials (Sample Problem: Use string to measure the circumferences of different circular objects.);

Circumference and Area of Circles

3.2.3: determine, through investigation using a variety of tools (e.g., cans and string, dynamic geometry software) and strategies, the relationships for calculating the circumference and the area of a circle, and generalize to develop the formulas [i.e., Circumference of a circle = π x diameter; Area of a circle = π x (radius)²] (Sample problem: Use string to measure the circumferences and the diameters of a variety of cylindrical cans, and investigate the ratio of the circumference to the diameter.);

Circumference and Area of Circles

3.2.4: solve problems involving the estimation and calculation of the circumference and the area of a circle;

Circumference and Area of Circles

3.2.5: determine, through investigation using a variety of tools and strategies (e.g., generalizing from the volume relationship for right prisms, and verifying using the capacity of thin-walled cylindrical containers), the relationship between the area of the base and height and the volume of a cylinder, and generalize to develop the formula (i.e., Volume = area of base x height);

Prisms and Cylinders
Pyramids and Cones

3.2.7: solve problems involving the surface area and the volume of cylinders, using a variety of strategies (Sample problem: Compare the volumes of the two cylinders that can be created by taping the top and bottom, or the other two sides, of a standard sheet of paper.).

Surface and Lateral Areas of Prisms and Cylinders

4: Geometry and Spatial Sense

4.1: demonstrate an understanding of the geometric properties of quadrilaterals and circles and the applications of geometric properties in the real world;

4.1.1: sort and classify quadrilaterals by geometric properties, including those based on diagonals, through investigation using a variety of tools (e.g., concrete materials, dynamic geometry software) (Sample problem: Which quadrilaterals have diagonals that bisect each other perpendicularly?);

Classifying Quadrilaterals

4.1.2: construct a circle, given its centre and radius, or its centre and a point on the circle, or three points on the circle;

Inscribed Angles

4.2: develop geometric relationships involving lines, triangles, and polyhedra, and solve problems involving lines and triangles;

4.2.1: determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials, geoboard), relationships among area, perimeter, corresponding side lengths, and corresponding angles of similar shapes (Sample problem: Construct three similar rectangles, using grid paper or a geoboard, and compare the perimeters and areas of the rectangles.);

Similar Figures

4.2.2: determine, through investigation using a variety of tools (e.g., dynamic geometry software, concrete materials, protractor) and strategies (e.g., paper folding), the angle relationships for intersecting lines and for parallel lines and transversals, and the sum of the angles of a triangle;

Investigating Angle Theorems
Isosceles and Equilateral Triangles
Polygon Angle Sum
Triangle Angle Sum

4.2.3: solve angle-relationship problems involving triangles (e.g., finding interior angles or complementary angles), intersecting lines (e.g., finding supplementary angles or opposite angles), and parallel lines and transversals (e.g., finding alternate angles or corresponding angles);

Polygon Angle Sum
Triangle Angle Sum

4.2.5: solve problems involving right triangles geometrically, using the Pythagorean relationship;

Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

4.3: represent transformations using the Cartesian coordinate plane, and make connections between transformations and the real world.

4.3.1: graph the image of a point, or set of points, on the Cartesian coordinate plane after applying a transformation to the original point(s) (i.e., translation; reflection in the x-axis, the y-axis, or the angle bisector of the axes that passes through the first and third quadrants; rotation of 90°, 180°, or 270° about the origin);

Dilations
Rotations, Reflections, and Translations
Translations

4.3.2: identify, through investigation, real-world movements that are translations, reflections, and rotations.

Similar Figures

5: Patterning and Algebra

5.1: represent linear growing patterns (where the terms are whole numbers) using graphs, algebraic expressions, and equations;

5.1.3: determine a term, given its term number, in a linear pattern that is represented by a graph or an algebraic equation (Sample problem: Given the graph that represents the pattern 1, 3, 5, 7,…, find the 10th term. Given the algebraic equation that represents the pattern, t = 2n – 1, find the 100th term.).

Arithmetic and Geometric Sequences

5.2: model linear relationships graphically and algebraically, and solve and verify algebraic equations, using a variety of strategies, including inspection, guess and check, and using a “balance” model.

5.2.1: describe different ways in which algebra can be used in real-life situations (e.g., the value of $5 bills and toonies placed in a envelope for fund raising can be represented by the equation v = 5f + 2t);

Using Algebraic Expressions

5.2.2: model linear relationships using tables of values, graphs, and equations (e.g., the sequence 2, 3, 4, 5, 6,… can be represented by the equation t = n + 1, where n represents the term number and t represents the term), through investigation using a variety of tools (e.g., algebra tiles, pattern blocks, connecting cubes, base ten materials) (Sample problem: Leah put $350 in a bank certificate that pays 4% simple interest each year. Make a table of values to show how much the bank certificate is worth after five years, using base ten materials to help you. Represent the relationship using an equation.);

Arithmetic Sequences
Compound Interest
Function Machines 1 (Functions and Tables)
Function Machines 2 (Functions, Tables, and Graphs)
Function Machines 3 (Functions and Problem Solving)
Points, Lines, and Equations

5.2.3: translate statements describing mathematical relationships into algebraic expressions and equations (e.g., for a collection of triangles, the total number of sides is equal to three times the number of triangles or s = 3n);

Linear Functions
Solving Equations on the Number Line
Using Algebraic Equations
Using Algebraic Expressions

6: Data Management and Probability

6.1: collect and organize categorical, discrete, or continuous primary data and secondary data and display the data using charts and graphs, including frequency tables with intervals, histograms, and scatter plots;

6.1.1: collect data by conducting a survey or an experiment to do with themselves, their environment, issues in their school or community, or content from another subject, and record observations or measurements;

Estimating Population Size

6.1.2: organize into intervals a set of data that is spread over a broad range (e.g., the age of respondents to a survey may range over 80 years and may be organized into ten-year intervals);

Box-and-Whisker Plots
Describing Data Using Statistics
Reaction Time 1 (Graphs and Statistics)
Stem-and-Leaf Plots

6.1.3: collect and organize categorical, discrete, or continuous primary data and secondary data (e.g., electronic data from websites such as E-Stat or Census At Schools), and display the data in charts, tables, and graphs (including histograms and scatter plots) that have appropriate titles, labels (e.g., appropriate units marked on the axes), and scales (e.g., with appropriate increments) that suit the range and distribution of the data, using a variety of tools (e.g., graph paper, spreadsheets, dynamic statistical software);

Graphing Skills
Histograms

6.1.4: select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph (i.e., from types of graphs already studied, including histograms and scatter plots);

Histograms
Stem-and-Leaf Plots

6.1.5: explain the relationship between a census, a representative sample, sample size, and a population (e.g., “I think that in most cases a larger sample size will be more representative of the entire population.”).

Polling: City

6.2: apply a variety of data management tools and strategies to make convincing arguments about data;

6.2.1: read, interpret, and draw conclusions from primary data (e.g., survey results, measurements, observations) and from secondary data (e.g., election data or temperature data from the newspaper, data from the Internet about lifestyles), presented in charts, tables, and graphs (including frequency tables with intervals, histograms, and scatter plots);

Correlation
Describing Data Using Statistics
Histograms
Real-Time Histogram
Trends in Scatter Plots

6.2.2: determine, through investigation, the appropriate measure of central tendency (i.e., mean, median, or mode) needed to compare sets of data (e.g., in hockey, compare heights or masses of players on defence with that of forwards);

Box-and-Whisker Plots
Describing Data Using Statistics
Mean, Median, and Mode
Movie Reviewer (Mean and Median)
Populations and Samples
Reaction Time 1 (Graphs and Statistics)
Reaction Time 2 (Graphs and Statistics)

6.2.3: demonstrate an understanding of the appropriate uses of bar graphs and histograms by comparing their characteristics (Sample problem: How is a histogram similar to and different from a bar graph? Use examples to support your answer.);

Histograms

6.2.4: compare two attributes or characteristics (e.g., height versus arm span), using a scatter plot, and determine whether or not the scatter plot suggests a relationship (Sample problem: Create a scatter plot to compare the lengths of the bases of several similar triangles with their areas.);

Correlation
Trends in Scatter Plots

6.2.5: identify and describe trends, based on the rate of change of data from tables and graphs, using informal language (e.g., “The steep line going upward on this graph represents rapid growth. The steep line going downward on this other graph represents rapid decline.”);

Solving Using Trend Lines
Trends in Scatter Plots

6.2.6: make inferences and convincing arguments that are based on the analysis of charts, tables, and graphs (Sample problem: Use data to make a convincing argument that the environment is becoming increasingly polluted.);

Earthquakes 1 - Recording Station

6.3: use probability models to make predictions about real-life events.

6.3.1: compare, through investigation, the theoretical probability of an event (i.e., the ratio of the number of ways a favourable outcome can occur compared to the total number of possible outcomes) with experimental probability, and explain why they might differ (Sample problem: Toss a fair coin 10 times, record the results, and explain why you might not get the predicted result of 5 heads and 5 tails.);

Geometric Probability
Independent and Dependent Events
Probability Simulations
Spin the Big Wheel! (Probability)
Theoretical and Experimental Probability

6.3.2: determine, through investigation, the tendency of experimental probability to approach theoretical probability as the number of trials in an experiment increases, using class-generated data and technology-based simulation models (Sample problem: Compare the theoretical probability of getting a 6 when tossing a number cube with the experimental probabilities obtained after tossing a number cube once, 10 times, 100 times, and 1000 times.);

Geometric Probability
Independent and Dependent Events
Probability Simulations
Spin the Big Wheel! (Probability)
Theoretical and Experimental Probability

6.3.3: identify the complementary event for a given event, and calculate the theoretical probability that a given event will not occur (Sample problem: Bingo uses the numbers from 1 to 75. If the numbers are pulled at random, what is the probability that the first number is a multiple of 5? is not a multiple of 5?).

Geometric Probability
Independent and Dependent Events
Probability Simulations
Theoretical and Experimental Probability