BC Math 9 Chapter 5 by Edvantage Science

BC MATH 9

Author

Laura Epp

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BC Math 9 - Pilot Edition

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ISBN: 978-1-77430-521-8

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About

This Math 9 Chapter 5 is being piloted in classrooms during the 2025 school year. Edvantage Interactive is using this pilot to check clarity, pacing, examples, and assessment items so the final edition better supports student learning. We are grateful to the teachers who are trying new materials with their classes and taking time to share feedback. We also thank the students who are completing these lessons and pointing out where explanations can be clearer. Your participation helps us build a book that works in real classrooms.

For more information, please contact us.

Chapter 1 Rational Numbers

1.1 What is a Rational Number?

1.2 Equivalent forms of Rational Numbers

1.3 Comparing and Ordering Rational Numbers

1.4 Operations With Decimals

1.5 Adding and Subtracting Fractions

1.6 Multiplying and Dividing fractions

1.7 Order of Operations With Rational Numbers

1.8 Applications

1.9 Chapter Review

Chapter 2 Powers and Roots

2.1 What is a Power?

2.2 Exponent Laws Part 1

2.3 Exponent Laws Part 2

2.4 Combining Exponent Laws

2.5 Square Roots

2.6 Approximating Square Roots

2.7 Scientific Notation (optional)

2.8 Applications Part 1

2.9 Applications Part 2

2.10 Chapter Review

Chapter 3 Polynomials

3.1 What is a Polynomial?

3.2 Introduction to Algebra Tiles

3.3 Adding Polynomials

3.4 Subtracting Polynomials

3.5 Multiplying Polynomials

3.6 Dividing Polynomials

3.7 Combined Polynomial Operations

3.8 Applications of Polynomials

3.9 Chapter Review

Chapter 4 Linear Relations

4.1 What is a Linear Relation?

4.2 Graphing Linear Relations Part 1

4.3 Graphing Linear Relations Part 2

4.4 Interpreting Graphs of Linear Relations

4.5 Applications of Linear Relations

4.6 Chapter Review

Chapter 5 Linear Equations

5.1 What is a Linear Equation?

5.2 Solving 1 and 2 Step Equations With Integers

5.3 Solving 2 Step Equations With Rational Numbers

5.4 Solving Multi-step Equations With Brackets

5.5 Solving Multi-step Equations With Variables on Both Sides

5.6 How to Solve a Word Problem

5.7 Chapter Review

Chapter 6 Proportional Reasoning

6.1 What is Proportional Reasoning?

6.2 Converting Linear Metric Units

6.3 Scale Diagrams

6.4 Similar Triangles

6.5 Similar Polygons

6.6 Applications of Proportional Reasoning

6.7 Chapter Review

Chapter 7 Statistics (Data Analysis)

7.1 What is Statistics?

7.2 Factors Affecting Data Collection

7.3 Collecting and Organizing Data

7.4 Graphing Data

7.5 Interpreting Data (optional)

7.6 Chapter Review

Chapter 8 Financial Literacy

8.1 What is Financial Literacy?

8.2 Bank ing

8.3 Simple Interest

8.4 Making a Budget

8.5 Chapter Review

Cumulative Review of Math 9

Welcome to Edvantage Math: BC Math 9

Before diving into your course, we invite you to read this introduction. It outlines key ideas from modern cognitive science research; ideas that shaped how this textbook was designed to help you learn more effectively and remember for longer.

What Is Learning?

Let’s start with a research-based definition from Make It Stick: The Science of Successful Learning: “Learning is acquiring knowledge and skills and having them readily available from memory so you can make sense of future problems and opportunities.” Brown, Roediger, & McDaniel (2014), p. 2

This means that learning isn’t just about reading or listening. It’s about getting ideas and skills into your long-term memory so that you can apply them when you face new challenges. That’s what this course, and this textbook, are designed to help you do.

How Your Brain Learns: A Simplified Model

The diagram you’ll see below is a simplified version of the Cognitive Theory of Multimedia Learning (Mayer, 2005), which integrates decades of research on how people learn from words, images, and experiences.

The diagram highlights three key memory systems:

Reproduced with permission from Failsafe Strategies for Science and Literacy

• Sensory memory: Information enters through your senses (especially eyes and ears).

• Working memory: Where your brain holds and organizes new information.

• Long-term memory: Where learning is stored permanently as it’s integrated into your prior knowledge. Understanding how these systems interact is critical to you learning effectively.

Attention: The Gatekeeper of Learning

You can’t learn what you don’t notice. Your sensory memory only processes what you give attention to. If you're distracted by your phone, for example, you won’t retain what your teacher is saying, no matter how good the explanation.

That’s why attention is the first essential skill for learning. This textbook is structured to help you focus on what matters most in each lesson.

Cognitive Load: Why Less Is More

Your working memory can only hold about 4 to 7 pieces of new information at once and often fewer when the material is unfamiliar. If too much is thrown at you at once, your brain gets overwhelmed and can’t process it deeply enough to store in long-term memory.

That’s called cognitive overload, and it’s a major barrier to learning.

To address this, Edvantage Math: BC Math 9 is carefully designed to reduce unnecessary cognitive load so your brain can focus on understanding core ideas. Clear diagrams, structured problem-solving prompts, and scaffolded practice are just a few of the strategies we’ve built in to help.

The Importance of Forgetting (and Remembering)

Forgetting is normal. It happens to everyone. In fact, it begins as soon as we learn something. This is known as the Ebbinghaus Forgetting Curve and is illustrated in the graph below. But here’s the good news: forgetting can be slowed and even reversed by reviewing information at spaced intervals as shown in the graph.

This process is called retrieval practice, and it’s one of the most powerful learning strategies discovered in cognitive science. You’ll find many activities in this book designed to help you remember what you learn when you need it most.

This Book Supports How You Learn

Partially Worked Solutions - Try Together

Every sample problem is followed by a Try Together problem. This problem is a partially worked solution. This helps you focus on the key aspects of the problem without overloading your working memory with too many steps at once.

How is it Going?

At the end of each unit, you become the teacher. You'll mark a sample student’s quiz allowing you to retrieve, evaluate, and apply what you’ve learned. This kind of elaborative practice strengthens long-term memory and makes your learning more durable.

Remembering Ebbinghaus

Periodically, you’ll encounter questions that revisit earlier topics even if they’re not part of your current unit. This deliberate spacing and retrieval strengthens memory and prepares you to use math concepts flexibly, not just repeat them on a test.

Final Thoughts

If you’ve read this far, congratulations, you’ve already taken the first step in developing strategies that help you learn. This textbook is more than a source of information; it’s a tool to help you become a better learner.

Good luck on your journey through BC Math 9 and don’t forget to visit edvantagemath.com for additional resources and support.

References and Suggested Reading

Ashman, G. (2022). Little Guide for Teachers: Cognitive Load Theory.

Brown, P. C., Roediger, H. L., & McDaniel, M. A. (2014). Make It Stick: The Science of Successful Learning.

Christodoulou, D. (2014). Seven Myths About Education.

Kirschner, P., Hendrick, C. (2020). How Learning Happens: Seminal Works in Educational Psychology and What They Mean in Practice. Ebbinghaus, H. (1885/1913). Memory: A Contribution to Experimental Psychology.

Mayer, R. E. (2005). The Cambridge Handbook of Multimedia Learning.

Mirabelli, S., Sandner, L. (2023) Failsafe Strategies for Science and Literacy

5 Linear Equations

This chapter is your guide to understanding and solving linear equations. You will explore a range of strategies to find solutions, both symbolically and pictorially. Hands-on and visual activities are integrated throughout to provide a connection between theory and applications. While most equations will have one clear solution, some might surprise you with no solution at all, or many different answers. You will build foundational skills and confidence to help you solve problems in math and beyond.

Big Ideas of the Chapter:

• Solving equations algebraically and pictorially.

• Preservation of equality.

• Verifying the solution of an equation (checking your answer).

• Rewrite expressions in terms of the variables involved. For example: speed is distance divided by time can be written as distance is equal to speed multiplied by time.

• The distributive property is a(b + c) = ac + bc

• There are a variety of strategies that can be used to simplify solving (for example: multiplying by LCM to eliminate fractions and cross multiplying).

• Linear equations of the following forms have x as a variable, a, b, c, d as real numbers, where a is not zero:

• Linear equations can be used to solve problems in context.

The graph on the right shows that the amount of fuel in the gas tank depends on the distance driven by the vehicle. What do you notice, what do you wonder?

5.1 What is a Linear Equation?

Warm Up

Competency Focus - apply multiple strategies to solving problems, in both abstract and contextual form.

1. Solve the Math Mobile What number does the circle represent? The triangle? How do you know?

2. The Magic of Math

Ask your friend what their favorite number from 1 to 9 is. Multiply it by 3 and remember that number. Ask your friend to multiply 37 037 times the number from step 1 What happens? Try it with a different number. Explain the pattern?

Linear equations are mathematical equations that represent a straight line when graphed. The simplest example is y = x. When x = 0, y = 0. When x = 1, y = 1. y is the same value as x.

Why do we study linear equations? To help us understand the world around us, so we need to understand how things change. The simplest kind of change is a constant change. A linear equation helps us to describe and represent this kind of change, a change that happens at a constant rate. For example, the table on the next page is based on an hourly salary of 20 dollars an hour.

The linear equation for this table is S = 20t. We can use this linear equation to:

• Make predictions like: How much do you earn after an 8 hour shift?

• Solve problems like: How many hours do I need to work to buy headphones that cost $150?

Linear equations have a degree of 1, meaning that the largest exponent for the variables x and y is 1. This means you will see terms like y, 3y, x, 2x and , but you will NOT see terms with , x2 , x3 , y2 in linear equations.

Try This ...

Circle the linear equations: Hint: look at the exponents of x and y.

You may have heard a math teacher at school say this:

“Whatever you do to one side of an equation, you must do to the other side.”

This is the foundation of solving equations, and allows us to preserve or keep the equation balanced. The image below illustrates this statement using a balance instead of an equation.

Verify Your Solution

When solving any equation, it is helpful to check your solution. Double checking is a useful practice in the real world too. You may have heard of “measure twice, cut once” in carpentry. Pilots also use a variety of double-check procedures to ensure safety and mitigate risks before, during, and after flights. Checking your solution will help you catch and fix your mistakes. It can also help avoid finding a solution that is impossible or unreasonable, a solution that seems right until you put it in context and find out you have divided by zero, gone back in time, or taken the square root of a negative number.

Sample Problem — Checking Your Solution

Is x = 2 a solution for 2x –1 = 3?

What to Think About

1. Check if x = 2 by substituting or replacing x with 2.

How to Do It

2. Simplify.

3. Is this true or false statement? If it is true, it is a solution. If it is false, it is not a solution.

Try Together— Checking Your Solution

Joe and Silas are solving the equation 2x + 3 = 5x – 6. Joe got a solution x = 4 and Silas got x = 3. Who is correct?

Joe’s Solution

1. Check if x = 4.

2. Replace x with 4.

2( ____ ) + 3 = 5( ___ ) – 6 ____ + 3= ____ – 6 ___ = ___

This is true / not true.

Is Joe correct?

Silas’ Solution

1. Check if x = 3 2. Replace x with 3 2(___) + 3 = 5(___) – 6 ___ + 3 = ___ – 6 ___ = ___

This is true / not true

Is Silas correct?

Which of the following equations have x = 1 as a solution

x + 2 = 3 x – 2 = 1 2x = -2 -x – 4 = -5 0.75x = 0.75

How Many Solutions?

A linear equation can have one solution, no solution, or infinitely many solutions. After you simplify both sides, compare the variables and constants:

• If you get x = a number there is one solution.

• If you get a false statement involving 2 numbers like 4 = 7, there is no solution.

• If you get a true statement involving 2 numbers like –3 = –3 then there are infinite solutions.

Sample Problem — How Many Solutions?

For the equation 2(x + 3) = 2x + 6, check to see if x = 0, 1, –10 and ½ are solutions. Don’t forget to use BEDMAS.

x = 0

2 (0 +3) = 2(0) + 6

2(3) = 6 6 = 6

∴ x is a solution

= 1

(1 +3) = 2(1) + 6

= 8

x is a solution

= –10 2 (–10 +3) = 2(–10) + 6 2(-7) = -20 + 6 -14 = -14

x is a solution

= ½

2 (½ +3) = 2(½) + 6

2(½ + ) = 1 + 6 7 = 7

∴ x is a solution

All four numbers are solutions. In fact, all real numbers will be solutions to this equation. There are an infinite number of solutions. In lesson 5.4, you will learn how to prove that these two expressions are equal.

Try Together— How Many Solutions?

When we solve linear equations we often get one solution. Is there always one solution? Consider x+1 = x+1 and x+1=x+2 when x = 0, x = 2 and x = 3.

x + 1 = x + 1

When x = 0

+ 1 = ___ + 1

= ___

This is true / false

When x = 2

+ 1 = ___ + 1

This is true / false

When x = 3

This is true / false

What do you notice? Try a few other numbers mentally. Why is this happening?

Your Turn — How Many Solutions?

Look at the following equations and determine whether they have no solution or infinite solutions. How do you know?

3x – 1 = 3x – 1 -x + 4 = -x + 7 x = x – 1.5

Important Note:

Sometimes when we solve problems there is a solution, but sometimes there are no solutions, or there are many, many possible solutions.

Extend and Explore Look at the graphs of the 2 straight lines y = x + 1 and y = x + 2. What do you notice about the two lines?

Do they ever cross each other?

How does that relate to solving x + 1 = x + 2?

Remembering Ebbinghaus – Simplifying Expressions

1. Which of these expressions is equal to 3x + 2? a. b. c.

2. Create 2 expressions that simplify to –x – 1. 3.

a. Multiplying by one third is the same as dividing by ____.

b. Dividing by one half is the same as multiplying by ___.

c. Multiplying by ¾ is the same as dividing by ___.

d. Dividing by 10/11 is the same as multiplying by ___.

4. Use the table of values for the linear equation y = x + 1 to plot 3 points on the grid provided. Draw a line through the points.

1. Identifying Linear Equations - Circle the linear equations:

2. Linear Equations - Fill in the blank

a. You can check your solution to a linear equation by _______________ your answer into the equation, to see if it makes a true or false statement.

b. Equations with a degree of one can have no solution, ______ solution, or infinite solutions.

c. The degree of a linear equation is always ______.

d. The graph of a linear equation looks like a ____________________.

e. As we solve an equation, we want to make sure we don’t change the equation. This is why we talk about the preservation of ________________.

3. Working with Linear Equations - Solve

4. Simplify the expressions

5. Preserve the Equality

Circle the equation on the right that is equivalent to the one on the left. Equation Which is Equivalent?

6. Verify Your Solution

Check to see if x=1 is a solution to the following equations. Write yes or no. Show your work.

7. How Many Solutions?

How many solutions are there for each equation? How do you know?

f. Create an equation that has many solutions.

g. Create an equation that has no solutions.