Mathematics

This unit is the Number strand of Grade 9 Mathematics (MTH1W), Ontario's de-streamed Grade 9 math course. It has three big parts.

1. Where numbers come from and how they fit together (B1). You will tell the story of place value and zero in Babylonian, Maya and Indian mathematics, and convert between base 10, base 20, base 60 and binary. Then you will define the natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers with proper set notation, compare them using closure, prove that every repeating decimal is a fraction, and explore the big ideas of density, infinity and limits (including why 0.999… = 1).

2. Powers (B2). You will use patterns to explain zero and negative exponents and how the sign of the base affects a power, write and calculate with scientific notation, and use all the exponent laws to simplify numeric and algebraic expressions like (−2x³y)⁴ and (2x²y)³ ÷ (4x⁴y²).

3. Rational numbers in action (B3). You will use integers to describe location, direction, amount and change; use unit fractions with tape measures, measuring cups and Egyptian fractions; explain how signs affect fractions, decimals, ratios and rates; add, subtract, multiply and divide positive and negative fractions and mixed numbers in formulas and linear relations; and solve problems with rates, proportions and percentages, from maple syrup and map scales to 13% HST and simple interest.

Remember from Grade 8: you met the real number system, scientific notation, square roots, integer operations, fraction operations and proportional reasoning. Grade 9 makes these ideas more precise, connects them with algebra, and prepares you for Grade 10 math and science. The math processes of Strand A (problem solving, reasoning and proving, reflecting, connecting, communicating, representing, selecting tools) and connections to culture, careers and daily life are woven into every lesson.

Each lesson has worked examples, a hands-on activity you can do at home, four sets of practice questions with full answers, and a short quiz. The last lesson is a unit review.

Curriculum expectations covered: B1.1, B1.2, B1.3, B2.1, B2.2, B3.1, B3.2, B3.3, B3.4, B3.5, with A1 and A2 woven in (Ontario Mathematics, Grade 9, MTH1W, 2021).

In this unit you will learn the core of Grade 9 algebra and how to make a computer do it for you. You will start with the story of algebra, from Babylonian clay tablets and The Nine Chapters on the Mathematical Art to Brahmagupta, al-Khwarizmi and the invention of the equals sign. Then you will write algebraic expressions from words, pictures and patterns; compare expressions with algebra tiles, tables of values, graphs and algebra to decide whether they are equivalent; and simplify expressions by collecting like terms, adding and subtracting polynomials, using the distributive property, factoring out common factors and multiplying binomials. Next you will solve and verify multi-step equations (including ones with brackets, fractions and decimals), rearrange formulas, and create equations to model real situations and make decisions. Finally, you will write, read and alter Python code that uses variables, expressions, functions, loops, equations and inequalities.

Remember from Grade 8: you added and subtracted monomials, added binomials with algebra tiles, evaluated expressions with rational numbers, solved equations with integers and decimals (including variables on both sides), solved inequalities, and wrote unplugged code with lists and loops. Grade 9 builds on all of it. You can run the Python examples in any Python editor, or trace them by hand; every program shows its exact output.

• Algebraic expressions and equations (C1): the development of algebra in different cultures; creating expressions that generalize relationships; comparing expressions to identify equivalent ones; simplifying expressions; creating and solving equations and verifying solutions.
• Coding (C2): using code to show variables, parameters, equations and inequalities; decomposing problems into computational steps; reading code to predict its outcome and altering it for new situations.

Throughout the unit you will use the mathematical processes (reasoning, representing, connecting, communicating) and connect algebra to history, careers and everyday decisions (Strand A).

Curriculum expectations covered: C1.1, C1.2, C1.3, C1.4, C1.5, C2.1, C2.2, C2.3, with A1 and A2 woven in (Ontario Mathematics, Grade 9, MTH1W, 2021). Linear and non-linear relations (C3 and C4) are in their own unit.

In this unit you will learn to read relations the way mathematicians, scientists and engineers do. You will tell linear relations (constant rate of change, straight-line graphs) from non-linear ones (curves) using graphs, tables of values, first differences and equations, and you will use a short Python program to calculate first differences. You will represent linear relations with models, tables, graphs and equations, connect the rate of change to the slope and the initial value to the y-intercept, and find the equation of a line from a graph, a table, two points or a hands-on model.

Then you will graph x = k, y = k, x + y = k, x − y = k, ax + by = k and xy = k, shade the regions for their inequalities, and translate, reflect and rotate the line y = ax. Finally, you will compare two linear relations and explain what their point of intersection means (Which ski pass? When does Maya catch up?), and build models from your own experiments to make and judge predictions.

Remember from Grade 8: you identified linear and non-linear patterns, described linear patterns by their constant rate and initial value, found when two linear patterns are equal, and solved equations and inequalities. Solving linear equations is taught in this course's unit on algebraic expressions and equations; this unit uses that skill. Every lesson has worked examples, an activity with things found at home, lots of practice with answers and a short quiz.

Curriculum expectations covered: C3.1, C3.2, C3.3, C4.1, C4.2, C4.3, C4.4 (Ontario Mathematics, Grade 9, De-streamed, MTH1W, 2021), with A1 (mathematical processes) and A2 (making connections) woven in. Algebraic expressions, equations and coding (C1 and C2) are in a separate unit.

In this unit you will learn to think like a data scientist and a mathematical modeller. You will start by looking at large data sets in the world around you (transit cards, apps, the census) and the benefits and risks of how data is collected, stored, represented and used, including privacy and Indigenous data sovereignty. Then you will analyse one-variable data in several ways: the mean, median, mode and range, quartiles, the interquartile range, box plots, histograms and stem-and-leaf plots, and the shape of a distribution. Next you will study two-variable data: scatter plots, the correlation coefficient r, lines of best fit, and testing linear, quadratic and exponential regression models with technology such as Desmos or a spreadsheet. Finally, you will carry out the mathematical modelling process from start to finish: pose a question of interest, plan and collect data, create a model, and report its answer, fit, limitations and predictions.

Remember from Grade 8: you sorted one-variable and two-variable data, made scatter plots, described relationships as positive, negative, strong or weak, spotted misleading graphs and built infographics. Grade 9 adds new statistics, new graphs and a full modelling project. Every lesson has worked examples, a hands-on activity, lots of practice and a short quiz.

Curriculum expectations covered: D1.1, D1.2, D1.3, D2.1, D2.2, D2.3, D2.4, D2.5, with Strand A (A1: mathematical processes; A2: connections to knowledge systems and real life) woven in (The Ontario Curriculum, Mathematics, Grade 9, MTH1W, 2021).

This is the Geometry and Measurement strand of Ontario's Grade 9 de-streamed mathematics course (MTH1W). You will become a measurement historian, a unit converter, a circle and triangle detective, a designer, an optimizer and a volume expert.

You will research how measurement systems and geometric ideas such as π developed in different times and communities, including First Nations, Métis and Inuit communities, and how they are used in careers. You will convert units within the metric and imperial systems (including area and volume units) and between systems, including historical Canadian units like the arpent and the surveyor's chain. You will learn the properties of chords, tangents, central and inscribed angles, and cyclic quadrilaterals, plus triangle properties and the circumcentre, incentre and centroid, then use them to create and analyse designs. You will show how changing dimensions affects perimeter, circumference, area, surface area and volume, and use tables, graphs, spreadsheets and a little Python to find optimal shapes. You will apply the Pythagorean relationship to real situations and composite shapes, and use the relationships between the volumes of prisms and pyramids and of cylinders and cones.

Remember from Grade 8: metric prefixes, the Pythagorean theorem, scale drawings, angle properties, and the area, surface area and volume of composite shapes, prisms and cylinders. Every lesson has worked examples, a hands-on activity, four sets of practice with full answers, and a quiz. Grade 10 math builds on these skills, especially optimization, right triangles and volume.

Curriculum expectations covered: E1.1, E1.2, E1.3, E1.4, E1.5, E1.6, with A1 (mathematical processes) and A2 (making connections) woven in (Ontario Mathematics, Grade 9, MTH1W, 2021).

In this unit you will learn to make informed financial decisions, the kind you will soon face with a first job, a phone, a car or a first apartment. You will start by learning from past and current financial situations, such as the high inflation of 2022 and the Bank of Canada's rate increases, and you will use both math and context to make choices. Next you will compare how interest rates, lengths of borrowing time, ways interest is calculated (compounding periods, add-on interest, credit cards, payday loans, "pay later" plans) and down payments change the overall cost of buying things, using formulas, spreadsheets and a little Python code. Then you will identify situations involving appreciation and depreciation and use straight-line and curved graphs to answer questions about them. Finally, you will read budgets displayed in different ways and modify them when circumstances change, giving a clear rationale. A final lesson puts everything together in one big decision.

Remember from Grade 8: you made financial plans with income, expenses and taxes, kept budgets balanced, tracked spending, calculated simple and compound interest (compounded yearly), used the Rule of 72, and compared credit cards and contracts. Grade 9 builds on all of it and connects to the linear and non-linear relations in the Algebra strand of this course.

Important: all prices, wages, interest rates, fees and rules in this unit are examples in Canadian dollars, chosen to be realistic. Real rates and rules change often, so always check current numbers before a real decision. Every lesson has worked examples, hands-on activities, lots of practice with full answers, and a short quiz.

Curriculum expectations covered: F1.1, F1.2, F1.3, F1.4, with A1 (mathematical processes) and A2 (making connections) woven in (Ontario Mathematics, Grade 9, MTH1W, 2021).