Learning Activity 1.1: Exponents

Introduction

Welcome to the world of mathematics. Mathematics is all around us and can be noticed in our lives every day. In this course, you will explore mathematical concepts and apply them to your learning. You will build your mathematical vocabulary, expand your mathematical toolset, and reflect on many topics from your prior education.

In order to be successful in mathematics, you will need:

numeracy skills
problem-solving skills
persistence and patience
a curious mind

This learning activity will give you a good introduction to the tools you will be using in the course, as well as the necessary skills you will need to implement in order to be successful. As you continue throughout the rest of the course, think of this learning activity as a guide and refer back to it when needed.

In the fascinating world we live in, populations vary from city to city, province to province, and country to country. Let’s begin by examining the following table of Ontario's population estimates from 2000 to 2020.

Ontario's population estimates from 2000 to 2020

Year	Population
2000	11,683,290
2002	12,094,174
2004	12,391,421
2006	12,661,878
2008	12,883,583
2010	13,135,778
2012	13,390,632
2014	13,617,553
2016	13,875,394
2018	14,308,697
2020	14,734,014

Have you ever wondered if there is a way to model and predict population growth?

The following is a graph of the population in Ontario estimates from 2000 to 2020.

Once the data is graphed it is easier to examine if there are trends. Examine the graph of Ontario population estimates from 2000-2020. Does it appear to be linear?

Notice that the scale for the population has been converted to scientific notation. Scientific notation is used in math and science to express very large or very small numbers in a more compact form. Scientific notation expresses the large or small number as a number between 1 and 10 and multiplied by a power of 10. For example 12,660,000 can be expressed as 1.266 × 10⁷.

In examining the scatterplot, you can also add a line of best fit and even find the equation. A line of best fit is exactly what you might expect, it is a line that best fits the data.

Let’s examine the following lines of best fit. Which one do you think is a better fit?

Line of best fit: $y = 132,198 x - 2.5257 \times 10^{8}$

Exponential curve of best fit: $y = 0.243753 {(1.0089)}^{x}$

When analyzing the preceding examples, you also should consider whether or not the population is growing at an exponential rate. Perhaps using an exponential equation would be a better fit than a line of best fit. In the preceding graph, the line of best fit appears to be best for the data as points are closer to the line than on the curve of best fit. It is interesting to note that if the population were graphed over a greater period of time, for example from 2000 to 2020, the graph would appear more curved and an exponential curve of best fit may model the data better than a line of best fit.

NOTE: Graphs were drawn using the regression formulas $y_{1} \sim m x_{1} + b$ and $y_{1} \sim a b^{x_{1}}$ .

In Unit 4, we will focus on different methods to help determine equations of lines and curves of best fit.

Acknowledgement (Opens in new window)

Navigating course components

Mathematics can be very fun and exciting. It requires practice and understanding. To succeed in this course, you will need to develop routines and a skill to record information. How you choose to record your information is ultimately up to you. You may choose to keep a paper or digital notebook. You may also choose to use organizers and other methods to help you better understand mathematical concepts.

Your work will be divided into components and each is important to your success in the course, and each one helps you learn how to learn, in addition to learning about mathematics. Press the following icons to learn more about each of these written components.

Organize new information and complete practice problems

Throughout this course you should keep a notebook (paper or digital) — this is where you will record and organize what you learn, include relevant reference material, attempt sample problems, and jot down important questions or ideas related to the course content. Although you should use your notebook consistently throughout the course to record information, the notebook icon will sometimes be used to remind you to make note of some important new information. When recording notes, you should keep the following in mind:

Organize material in a way that makes sense to you. Choose a method or organization which will help you locate your learning later on.
Use proper form and include mathematical equations when copying examples of solved problems. Complete, correct examples are great to have on hand when you are studying!
Whenever possible, use your own words. Using your own words is always more interesting than using someone else's.

Self-check quiz

Throughout the course you will be given the opportunity to practice what you have learned so far. This icon prompts you to complete a self-check quiz where you can apply and assess your learning. Note that these self-check quizzes are automatically scored, but they do not count toward your course grade. You may re-take these self-check quizzes as many times as you’d like, so feel free to use them for practice.

Challenge

Throughout this course you will be prompted to engage in a fun and exciting challenge, where you will have the opportunity to creatively apply your learning to the real world, your own interests, and personal goals. The challenges also help you build on your transferable skills.

Exponent laws

Exponent laws are algebraic ‘short cuts’ that are based on patterns that we can observe when working with powers. You may be familiar with exponent laws but we will review them and try some examples.

Take a moment to review the following exponent laws.

Review

Law	Description
Product	$a^{m} \times a^{n} = a^{m + n}$
Quotient	$a^{m} \div a^{n} = a^{m - n}$
Power	$(a^{m})^{n} = a^{m \times n}$
Power of a product	${(a b)}^{n} = a^{n} b^{n}$
Power of a quotient	${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$
Zero exponent	$a^{0} = 1; (a \neq 0)$
Negative exponent	$i) a^{- n} = \frac{1}{a^{n}}; (a \neq 0)$ $ii) {(\frac{a}{b})}^{- n} = {(\frac{b}{a})}^{n}; (a, b \neq 0)$

Notebook

Try your best to solve the following questions in your notebook and compare your answers to the suggested answers provided. Review the solutions to check whether you are following the correct format for solving questions.

Simplify each of the following.

1. $x^{6} x^{7}$

$x^{6} x^{7} = x^{6 + 7} = x^{13}$

2. $\frac{x^{13}}{x^{5}}$

$\frac{x^{13}}{x^{5}} = x^{13 - 5} = x^{8}$

3. ${(x^{6})}^{5}$

${(x^{6})}^{5} = x^{13 \times 5} = x^{30}$

4. $16^{0}$

$16^{0} = 1$

5. $2^{−3}$

$2^{−3} = \frac{1}{2^{3}} = \frac{1}{8}$

6. $\frac{x^{19} \times x^{26}}{x^{39} \times x^{5}}$

\frac{x^{19} \times x^{26}}{x^{39} \times x^{5}} = \frac{x^{19 + 26}}{x^{99 + 5}} = \frac{x^{45}}{x^{44}} = x^{45 - 44} = x^{1} = x

7. ${(\frac{3}{4})}^{−2}$

${(\frac{3}{4})}^{−2} = {(\frac{4}{3})}^{2}$
The negative exponent flips the fraction.
${(\frac{4}{3})}^{2} = \frac{4^{2}}{3^{2}} = \frac{16}{9}$

8. $(12 x^{2} y^{4}) (\frac{1}{2} x^{5} y)$

(12 x^{2} y^{4}) (\frac{1}{2} x^{5} y) = (12 \times \frac{1}{2}) (x^{2} x^{5}) (y^{4} y) = 6 x^{2 + 5} y^{4 + 1} = 6 x^{7} y^{5}

9. $(27 x^{4} y^{2} z^{6}) \div (- 3 x y z)$

(27 x^{4} y^{2} z^{6}) \div (- 3 x y z) = (27 \div (- 3)) (x^{4} \div x) (y^{2} \div y) (z^{6} \div z)

= - 9 x^{4 - 1} y^{2 - 1} z^{6 - 1} = - 9 x^{3} y z^{5}

10. ${(2 x^{2} y^{3})}^{3} {(3 x^{3} y)}^{- 2}$

{(2 x^{2} y^{3})}^{3} {(3 x^{3} y)}^{- 2} = (2^{3} {(x^{2})}^{3} {(y^{3})}^{3}) (3^{- 2} {(x^{3})}^{- 2} y^{- 2})

= (2^{3} x^{6} y^{9}) (3^{- 2} x^{- 6} y^{- 2}) = (2^{3} \times 3^{- 2}) (x^{6} x^{- 6}) (y^{9} y^{- 2})

= (8 \times \frac{1}{3^{2}}) (x^{6 - 6}) (y^{9 - 2}) = (8 \times \frac{1}{9}) x^{0} y^{7} = \frac{8}{9} y^{7}

Rational exponents

Rational exponents contain a fraction and can be rewritten using a root sign.

The fractions are used to express radical numbers. For instance, the exponent of $\frac{1}{2}$ implies that you want the square root of the base.

The rule for rational exponents is:

$\begin{array}{lll} {(a)}^{\frac{m}{n}} = \sqrt[n]{a^{m}} \end{array}$

(The n value is referred to as the index and when the index is 2 it is generally omitted.)

$\begin{array}{lll} = {(\sqrt[n]{a})}^{m} \end{array}$

Note that when dealing with square roots the index is omitted so $x^{\frac{1}{2}} = \sqrt[2]{x}$ which is just written as $\sqrt{x}$ . If you are considering a cube root though the index must be written so $x^{\frac{1}{3}} = \sqrt[3]{x}$ .

Example:

Evaluate ${(25)}^{\frac{1}{2}}$ .

${(25)}^{\frac{1}{2}}$ is asking you to find $\sqrt{25}$ which is $\pm 5$ .

This example is asking for the principal square root of 25 which is the positive root of 25.

Therefore ${(25)}^{\frac{1}{2}} = 5$ .

Rational exponents can also be used for cube roots as mentioned previously.

Let's review some additional rules for rational exponents.

${(\frac{x}{y})}^{- a} = {(\frac{y}{x})}^{a}$

$x^{\frac{1}{2}} = \sqrt{x}$

$x^{\frac{1}{a}} = \sqrt[a]{x}$

$x^{\frac{a}{b}} = (\sqrt[b]{x})^{a} = \sqrt[b]{(x^{a})}$

Examples:

Press the following tabs to explore the examples.

Evaluate the following.

$8 1^{\frac{3}{4}}$

$= {(\sqrt[4]{81})}^{3}$

$= {(3)}^{3}$

$= 27$

Usually finding the root first and then raising it to the exponent is the preferred choice because finding the root often results in a number that is smaller and more familiar.

Consider the following alternative method to solve this question.

$8 1^{\frac{3}{4}}$

$= \sqrt[4]{{(81)}^{3}}$

$= \sqrt[4]{531,441}$

$= 27$

You will find that the answers are exactly the same.

The first method will be less reliant on a calculator due to the smaller numbers.

${(- 81)}^{\frac{3}{4}}$

$= {(\sqrt[4]{(- 81)})}^{3}$

This example does not have a solution as it is not possible to take the fourth root of a negative number.

${- (81)}^{\frac{3}{4}}$

$= - \sqrt[4]{{(81)}^{3}}$

$= - {(3)}^{3}$

$= - 27$

Since the negative was outside the bracket it did not affect your ability to find the 4th root so the placement of brackets is critical in solving these questions.

$8 1^{- \frac{3}{4}}$

This time the negative sign is in the exponent which suggests the final answer may be a fraction.

$8 1^{- \frac{3}{4}}$

$= {(\frac{1}{81})}^{\frac{3}{4}}$ Flip the base and the exponent becomes negative.

$= {(\sqrt[4]{\frac{1}{81}})}^{3}$ Apply the rule for rational exponents.

$= {(\frac{\sqrt[4]{1}}{\sqrt[4]{81}})}^{3}$ Optional step to show how to work with the fourth root.

$= {(\frac{1}{3})}^{3}$

$= \frac{{(1)}^{3}}{{(3)}^{3}}$

$= \frac{1}{27}$

Challenge

Can you take the square root of a negative number? Try it on your calculator. Notice that you receive an error message. Try this again by taking an even root (perhaps a 4th root) of a negative number.

You cannot take the square root of a negative number or the 4th root of a negative number. In fact, you can never take an even-numbered root of a negative number. The question cannot be answered, as there is no real answer to this challenge.

Notebook

Now it’s your turn to test your math skills. Try the following questions in your notebook and compare your work with the answers to check your understanding. Review your solutions to ensure that your format is correct. You do not need to show all steps but be sure your solutions are organized and logical.

1. $2 7^{\frac{1}{3}}$

$2 7^{\frac{1}{3}} = \sqrt[3]{27} = 3$

2. ${(- 27)}^{\frac{1}{3}}$

${(- 27)}^{\frac{1}{3}} = \sqrt[3]{(- 27)} = - 3$

3. $2 7^{- \frac{2}{3}}$

$2 7^{- \frac{2}{3}} = {(\sqrt[3]{27})}^{- 2} = 3^{- 2} = \frac{1}{3^{2}} = \frac{1}{9}$

Evaluating versus simplifying exponential expressions

Sometimes you are asked to evaluate an expression and sometimes you are asked to simplify.

Brainstorm: What is the difference between evaluating and simplifying exponential expressions? (Hint: A reference was made to this earlier in the learning activity.)

What’s the difference between evaluating an expression and simplifying an expression? Let’s find out together!

Similarities:

Both involve simplifying the expression using exponent rules.

Differences:

Final answers for an evaluate question will be a numerical answer while a simplifying final answer will still contain letters (variables).

Examples:

Press the following tabs to explore the examples.

A carpenter is creating rectangular tables. The design they are working with has the length of the table being twice the width.

The cost to build each table depends partly on the area of the top. The carpenter creates a sketch of the design, as shown in the diagram and determines a formula for the area of the table top.

They know that the area of a rectangle is calculated by multiplying length by width.

$A = w \times 2 w$

This can be simplified to the following:

$A = 2 w^{2}$

If a customer wants to have a table that has a width of 50 cm the formula for area can be used to determine the area of the table:

$A = 2 {(50)}^{2}$

$A = 5,000$

Therefore, the table has an area of $5,000 {cm}^{2}$ .

Evaluate $\frac{(2^{3}) (2^{5})}{(2^{4}) (2^{2})}$

$\frac{(2^{3}) (2^{5})}{(2^{4}) (2^{2})}$

$= \frac{2^{3 + 5}}{2^{4 + 2}}$

$= \frac{2^{8}}{2^{6}}$

$= 4$

Simplify $\frac{x^{3} x^{5}}{x^{4} x^{2}}$

$\frac{x^{3} x^{5}}{x^{4} x^{2}}$

$= \frac{x^{8}}{x^{6}}$

$= x^{2}$

Evaluate $\frac{x^{3} x^{5}}{x^{4} x^{2}}$ when $x = 3$ .

This example is similar to the previous but now we are asked to evaluate. It is usually easiest to simplify and then evaluate. The initial steps would be identical to example 2 and then a final step that involves the substitution is all that is required.

$\begin{aligned} \frac{x^{3} x^{5}}{x^{4} x^{2}} & = x^{2} & The simplified answer \\ = 3^{2} & Letting x = 3 \\ = 9 & Evaluating \end{aligned}$

Exponential equations are of the form $y = a^{x}$ , for example $y = 2^{x}$ . When solving these equations, the end goal is to determine the value of x which will make the equation true. These equations are solved in a different manner than linear or quadratic equations. There are various approaches to solve exponential equations but our approach will be to have like bases on both the left and right sides of the equation.

Challenge

How would you approach solving an exponential equation? Write an example of an exponential equation in your notebook.

Solve the following equations in your notebook. Compare your answer to the suggested answer.

Solve the following equations.

1. $4^{x} = 64$

Rewrite 64 with a base of 4.

$64 = 4^{3}$

Substitute into the equation.

$4^{x} = 4^{3}$

Since the bases are now equal, the exponents must also be equal.

$\begin{aligned} 4^{x} & = 4^{3} \\ x & = 3 \end{aligned}$

2. $6^{(2 - x)} = 6^{(- 1)}$

Rewrite both sides with a base 6. Recall using exponent laws that $\frac{1}{6} = 6^{- 1}$ .

$6^{2 - x} = \frac{1}{6}$

Since the bases are now equal, the exponents must also be equal and then you can solve the resulting equation.

$\begin{aligned} 2 - x & = - 1 \\ - x & = - 1 - 2 \\ - x & = - 1 - 2 \\ - x & = - 3 \\ \frac{- x}{- 1} & = \frac{- 3}{- 1} \end{aligned}$

Divide both sides by −1 to isolate the x.

$\begin{aligned} x & = 3 \end{aligned}$

3. $9^{2 x + 1} = 27$

Rewrite both sides with a common base. In this case, the common base will be 3 as $3^{2} = 9$ and $3^{3} = 27$ .

$\begin{aligned} {(3^{2})}^{2 x + 1} & = 3^{3} \\ 4 x + 2 & = 3 \\ 4 x & = 3 - 2 \\ 4 x & = 1 \\ x & = \frac{1}{4} \end{aligned}$

4. $3 (4^{x}) = 192$

The first step in solving this equation, is to isolate the $4^{x}$ by dividing both sides of the equation by 3.

$\begin{aligned} \frac{3 (4^{x})}{3} & = \frac{192}{3} \\ 4^{x} & = 64 \end{aligned}$

Now both sides can be written with a common base. In this case, the common base will be 4 as $4^{3} = 64$ .

$\begin{aligned} 4^{x} & = 4^{3} \\ x & = 3 \end{aligned}$

Applications

We were introduced to population growth at the beginning of this learning activity but now let’s dive deeper into understanding population growth using exponential equations.

Whether it’s the growth of a rabbit population in a forest, or bacteria in a petri dish, these populations will typically grow at an exponential rate. Our examples will assume there are no limiting factors, such as food limitations or predators.

Example:

A population of cells is being studied to better understand a disease. This particular type of cell doubles every day. If a sample of these cells is estimated to consist of 1,000 cells initially, the size of the sample after n days can be modelled with the equation $S = 1,000 (2)^{n}$ , where S represents the sample size of cells, and n represents the number of days that have passed. How long would it take to reach 512,000 cells?

Press the following tabs to explore the steps to solve the question.

Set the initial formula equal to 512,000 as we know $S = 512,000$ .

$512,000 = 1,000 (2)^{n}$

Switch the left and right sides so that the variable is on the left side.

$1,000 (2)^{n} = 512,000$

Divide both sides by 1,000 to isolate $2^{n}$ .

$\begin{aligned} \frac{1,000 (2)^{n}}{1,000} & = \frac{512,000}{1,000} \\ 2^{n} & = 512 \end{aligned}$

Write both sides with a common base. In this case 2 works as $2^{9} = 512$ .

$2^{n} = 2^{9}$

Since the bases are common, we can set the exponents equal to each other.

$n = 9$

Therefore, the cell population will reach 512,000 in 9 days.

Notebook

In your notebook, solve the following questions using full solutions. When you are ready, use the answer key to check your understanding.

1. Evaluate $6 4^{\frac{2}{3}}$

$64^{\frac{2}{3}}$

$= (\sqrt[3]{64})^{2}$

$= 4^{2}$

$= 16$

2. Evaluate $64^{\frac{- 2}{3}}$

$64^{\frac{- 2}{3}}$

$= {(\frac{1}{64})}^{\frac{2}{3}}$

$= {(\sqrt[3]{\frac{1}{64}})}^{2}$

$= {(\frac{\sqrt[3]{1}}{\sqrt[3]{64}})}^{2}$

$= {(\frac{1}{4})}^{2}$

$= \frac{1^{2}}{4^{2}}$

$= \frac{1}{16}$

3. Evaluate ${(- 64)}^{\frac{2}{3}}$

${(- 64)}^{\frac{2}{3}}$

$= (\sqrt[3]{- (64)})^{2}$

$= {(- 4)}^{2}$

$= 16$

4. Evaluate $\frac{{(2^{3})}^{4} (2^{34})}{{(2^{5})}^{8}}$

$\frac{{(2^{3})}^{4} (2^{34})}{{(2^{5})}^{8}}$

$= \frac{2^{12} 2^{34}}{2^{40}}$

$= \frac{2^{46}}{2^{40}}$

$= 2^{6}$

$= 64$

5. Simplify. Write answer with positive exponents only. $\frac{{(x^{4} y^{- 2})}^{2} {(x^{2} y^{4})}^{\frac{1}{2}}}{{(x y^{3})}^{5}}$

$\frac{{(x^{4} y^{- 2})}^{2} {(x^{2} y^{4})}^{\frac{1}{2}}}{{(x y^{3})}^{5}}$

$= \frac{(x^{8} y^{- 2}) (x y^{2})}{(x^{5} y^{15})}$

$= \frac{x^{9} y^{- 2}}{x^{5} y^{15}}$

$= x^{4} y^{- 17}$

$= \frac{x^{4}}{y^{17}}$

6. Solve for $x$ . $3^{2 x - 1} = 27$

Both sides of the equation can be written with 3 since $3^{2} = 27$

$3^{2 x - 1} = 27$

Now that bases are the same, set the exponents equal to each other and solve for $x$ .

$2 x - 1 = 3$

$2 x = 3 + 1$

$2 x = 4$

$x = 2$

7. Solve for $x$ . $2^{2 x - 1} = 2^{x + 9}$

Since bases are already the same, you just have to set the exponents equal to each other and then solve for $x$ .

$2 x - 1 = x - 9$

$2 x - x = 9 + 1$

$x = 10$

8. Solve for $x$ . $5 (4^{x}) = 10$

Divide both sides of the equation by 5 to isolate the $4^{x}$ .

$\frac{5 (4^{x})}{5} = \frac{10}{5}$

$4^{x} = 2$

Bases are not the same but the 4 on the left side of the equation can be expressed as a base 2 to match the right side since $4 = 2^{2}$ .

${(2^{2})}^{x} = 2$

${2^{2}}^{x} = 2$

Now that bases are the same, set the exponents equal to each other and solve for $x$ . The exponent on the right side will be 1.

$2 x = 1$

Divide both sides of the equation by 2 to isolate $x$ .

$\frac{2 x}{2} = \frac{1}{2}$

$x = \frac{1}{2}$

Conclusion

Congratulations! You have now completed Learning Activity 1.1. By working through all of the examples, you have practiced the following concepts:

evaluating powers with rational exponents
simplifying algebraic expressions involving exponents
solving problems involving exponential equations using common bases.

Self-check quiz

Check your understanding!

Complete the following self-check quiz to determine where you are in your learning and what areas you need to focus on.

This quiz is for feedback only, not part of your grade. You have unlimited attempts on this quiz. Take your time, do your best work, and reflect on any feedback provided.

Press Quiz to access this tool.

Next Steps

In Learning Activity 1.2, we will learn to interpret graphs and use graphical models.

Stay tuned to understand how graphs can help us in the real world.

Connecting to transferable skills

Ontario worked with other provinces in Canada to outline a set of competencies that are requirements to thrive. Ontario then developed its transferable skills framework as a set of skills for students to develop over time. These competencies are ones that are important to have in order to be successful in today’s world.

Read the following document entitled Transferable Skills Outline (Opens in new window) to explore the framework and the descriptors for each skill. Download, print, or copy the information in the document into your notes - you'll refer to it in each unit.

Press the following tabs to explore the skills.

Critical thinking and problem solving

Explore this!

Explore the following video to learn more about critical thinking and problem solving.

Definition

Critical thinking and problem solving involve examining complex issues and problems from a variety of different points of view in order to make informed judgments and decisions. Learning is deeper when the experiences are meaningful, real world, and authentic.

Students consistently:

solve meaningful, real-life problems
take steps to organize, design, and manage projects using inquiry processes
analyze information to make informed decisions
see patterns, make connections, and transfer learning from one situation to another
see the connections between social, economic, and ecological systems

Innovation, creativity, and entrepreneurship

Explore this!

Check out the following video to learn more about innovation, creativity, and entrepreneurship.

Definition

Innovation, creativity, and entrepreneurship involve the ability to turn ideas into action to meet the needs of a community. The ability to contribute new-to-the-world thinking and solutions to solve complex problems involves leadership, risk taking, and independent/unconventional thinking. Experimenting with new strategies, techniques, and perspectives through research is part of this skill set.

Students consistently:

formulate insightful questions to generate opinions
take risks in thinking; experiment to find new ways of doing things
demonstrate leadership in a range of creative projects
motivate others in an ethical and entrepreneurial spirit

Self-directed learning

Explore this!

Access the following video to learn more about self-directed learning.

Definition

Self-directed learning means: becoming aware and demonstrating ownership in your learning. Belief in your ability to learn (growth mindset), combined with strategies for planning, monitoring, and reflecting on your past, present, and future goals promote lifelong learning, well-being, and adaptability in an ever-changing world.

Students consistently:

are aware of how they learn best
ask for support when needed
set goals and make a plan to achieve their goals
practice new skills they want to improve
reflect on their own learning to determine strengths
learn to adapt to change and become resilient in the face of adversity
become managers of different aspects of their lives to enhance their health and overall well being

Collaboration

Explore this!

Explore the following video to learn more about collaboration.

Definition

Collaboration involves participating ethically and effectively in teams. Being versatile across different situations, roles, groups, and perspectives allows you to co-construct knowledge, meaning, content, and learn from, and with others in physical and online spaces.

Students consistently:

participate in teams in respectful and positive ways
learn from others; contribute to the learning of others
assume various roles on a team as needed being respectful of a diversity of perspectives including Indigenous ways of knowing
address disagreements and manage conflict in sensitive and constructive ways
network with a variety of people and groups on an ongoing basis

Communication

Explore this!

Check out the following video to learn more about communication.

Definition

Communication involves receiving and expressing meaning (e.g., reading and writing, viewing and creating, listening and speaking) in different contexts and with different audiences and purposes. Effective communication increasingly involves understanding both local and global perspectives, including using a variety of media appropriately, responsibly, and safely with regard to your digital footprint.

Students consistently:

communicate effectively in a variety of media
use digital tools appropriately to create a positive digital footprint
listen to understand
ask effective questions
understand the cultural importance of language

Global citizenship and sustainability

Explore this!

Access the following video to learn more about global citizenship and sustainability.

Definition

Global citizenship and sustainability involve understanding diverse worldviews and perspectives in order to address political, ecological, social, and economic issues that are crucial to living in a in a sustainable world. Being aware of what it means to be an engaged citizen and how the appreciation for the diversity of people and perspectives contributes to a sustainable world are part of this skill set.

Students consistently:

take actions and make responsible decisions to support the quality of life for all
understand the histories, knowledge, contributions, and inherent rights of Indigenous people
recognize discrimination and work to promote the principles of equity
contribute to their local and global community
participate in an inclusive, accountable, sustainable, and ethical manner, both in groups and in online networks

Digital literacy

Explore this!

Explore the following video to learn more about digital literacy.

Definition

Digital literacy involves the ability to solve problems using technology in a safe, legal, and ethically responsible manner. Digitally literate students recognize the rights and responsibilities, as well as the opportunities, that come with living, learning, and working in an interconnected digital world.

Students consistently:

select and use appropriate digital tools to collaborate, communicate, create, innovate, and solve problems
use technology in a way that is consistent with supporting their mental health and well-being
use digital tools effectively to solve problems and inform decisions
demonstrate a willingness and confidence to explore new or unfamiliar digital tools and emerging technologies
manage their digital footprint by engaging in social media and online communities respectfully, inclusively, safely, legally, and ethically

The transferable skills described in these videos have been adapted from the Ministry‘s definitions and descriptions that are available for viewing on the Ministry of Education‘s Curriculum and Resources site: Transferable Skills(Opens in a new window)

Note the indicators that you think you will develop in this course. Throughout this course, you should revisit these skills to reflect on which ones you develop and if your original predictions were correct.

As you continue through this unit and the rest of the course, keep your notebook updated and be mindful of opportunities to apply and develop transferable skills.

Learning goals

Success criteria

Introduction

Navigating course components

Exponent laws

Review

Notebook

Rational exponents

Examples:

Challenge

Notebook

Evaluating versus simplifying exponential expressions

Similarities:

Differences:

Examples:

Challenge

Applications

Example:

Notebook

Conclusion

Self-check quiz

Check your understanding!

Next Steps

Connecting to transferable skills

Critical thinking and problem solving

Innovation, creativity, and entrepreneurship

Self-directed learning

Collaboration

Communication

Global citizenship and sustainability

Digital literacy