Course Profile Functions and Relations, Grade 11, University Preparation, Catholic and Public

Unit 1: Exploring Functions: Connecting Algebra and Geometry

Time: 18 hours

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 | Activity 6 | Activity 7 | Activity 8

Unit Description

Students investigate quadratic functions and related concepts from algebraic and geometric perspectives, in order to deepen their understanding and prepare them for further explorations of functions and relations. A winter recreation theme is loosely woven throughout selected activities in the unit, providing a contextual framework for students to solve problems, both with and without the use of graphing technology. Students solve first-degree inequalities and graph their solutions on number lines. Skills involving operations with polynomials and rational expressions are consolidated, and then extended to the complex number system which is introduced in this unit. Students apply the method of completing the square in order to solve maximum/minimum problems involving quadratic functions. Algebraic and graphical methods are used to determine the roots of quadratic equations. The exponent laws are applied to expressions, which have powers containing integer and rational exponents. Students discover the nature of exponential functions and solve exponential equations.

Activity 1: A Range of Possibilities

Time: 75 minutes

Description

Students add, subtract, and multiply polynomials in inequalities. Students investigate solutions to inequalities using graphing calculator technology, and subsequently graph solutions on a number line.

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.01 - solve first-degree inequalities and represent the solutions on number lines;

OC1.02 - add, subtract, and multiply polynomials;

OC3.04 - demonstrate the correct use of mathematical language, symbols, visuals (e.g., diagrams, graphs), and conventions;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

· Solve first degree equations in one variable;

· Manipulate algebraic expressions;

· Perform basic graphing functions on a graphing calculator, including zooming and tracing.

Planning Notes

· Prepare worksheets.

· Students require graphing calculators.

· Teacher requires a graphing calculator with an overhead projection unit.

Teaching/Learning Strategies

Student Activity

Students:

· manipulate polynomials by adding, subtracting, and multiplying;

· use graphing calculators to visually investigate solutions of inequalities;

· solve inequalities algebraically and graph solutions on number lines.

Teacher Facilitation

· Begin with a discussion about the differences in solutions between equalities and inequalities.
Using a graphing calculator with an overhead projection unit attached, project the graph of:

a) y = 2x – 6. Direct the students to use the graph of this function to determine the root of the equation 2x – 6 = 0 (use the zoom and trace features to locate the x-intercept).

b) 2x -6 > 0. Discuss, and direct students to use the graph to solve the inequality. Use a test point to help determine the region defined by the inequality. Students should discover that x-values for which the graph is above the x-axis represent the solution set.

c) 2x – 6 # 0. Repeat the process from (b). Students should discover that x-values for which the graph is on, or below, the x-axis represent the solution set.

· As a class, represent the solutions to (a), (b) and (c) on the real number line. The concept of projecting the line onto the x-axis may help students to visualize the connection between the Cartesian illustration, in two dimensions, and the corresponding one-dimensional graph on the real number line. Discuss differences and similarities in the solutions of (a), (b), and (c).

· Organize the class into small groups. Have them repeat the above investigation using the following equations: (a) –4(x + 2) – 3(x + 4) = 3x, (b) –4(x + 2) – 3(x + 4) < 3x, and
(c) –4(x + 2) – 3(x + 4) $ 3x. There are different ways to solve these. Encourage students to experiment with the calculator and devise their own technique.

· One approach is to graph two functions: y₁ = –4(x + 2) – 3(x + 4) and y₂ = 3x. For clarity, use different line styles for each. The x-coordinate of the intersection point gives the solution to (a), while the set of x-values where y₁ appears below y₂ gives the solution to (b), and vice versa for (c), including the intersection point.

· Another approach is to move all non-zero terms to one side of the inequality, graph the corresponding linear relation, and focus on the x-intercept, as in the first example.

· Have students share their techniques with the class. After this, the teacher can model the algebraic techniques in order to verify the results, and provide an alternate solution approach.

· Distribute Sample Worksheet 1. Complete question 1 (parts i to iii) for the first equation on the worksheet, 3 – x < 6 – 2x.

· Using graphing calculator technology, have students work in pairs to complete the rest of the worksheet, and additional worksheets, or appropriate exercises from the textbook.

Sample Worksheet 1

1. (a) 3 – x < 6 – 2x

(b) 3(2x – 1) – 2(x + 1) # 3x + 8

(c) 3(4 – x) – 2 > 2(x – 3) + 6

i) Solve each inequality algebraically, ii) Represent the solution on a number line,
iii) Confirm your solution set by graphing the inequality using a graphing calculator.

2. Explain how the solution sets of the following inequalities differ: (a) x > 2 (b) x $ 2.

3. How does the nature of the solution set of the inequality x # -3 differ, when represented on:
(a) the real number line; (b) the Cartesian plane.

4. Consider the different techniques to solve inequalities. Discuss advantages and disadvantages of each.

Assessment & Evaluation of Student Achievement

Question 1, parts (i), (ii), and (iii) on Sample Worksheet 1 can be assessed for correct use of mathematical symbols, visuals, and conventions. The remaining questions can be assessed for Communication. Assessment should be of a formative nature in this activity.

Extension

Referring back to question 3, from Sample Worksheet 1, how would the nature of the solution sets in parts (a) and (b) differ from the solution set of x # -3, when represented on a 3-dimensional, x-y-z coordinate system, where z is an axis passing through the origin of the Cartesian plane, perpendicular to both the x and y axes?

Key Points to Look for in Answers to 3 and Extension: On the number line, x # -3 represents a ray of points to the left of x = -3. On the Cartesian Plane, x # -3 represents a 2-dimensional region of points or the area to the left of the line x = -3. In 3-space, x # -3 represents a 3-dimensional region of points on the space to the left of the plane x = 3 (which is parallel to the y-z plane).

Activity 2: Ski-Jumping to the Max!

Time: 150 minutes

Description

Students investigate the graphs of quadratic functions and examine their vertices and x-intercepts. Using graphing technology, students determine quadratic functions, which have desired maximum values. The method of completing the square is used to connect the algebraic and geometric significance of the vertex.

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.03 - determine the maximum or minimum value of a quadratic function whose equation is given in the form y = ax² + bx + c, using the algebraic method of completing the square;

OC1.05 - determine the real or complex roots of quadratic equations, using an appropriate method (e.g., factoring, the quadratic formula, completing the square), and relate the roots to the x-intercepts of the graph of the corresponding function;

OC3.01 - explain mathematical processes, methods of solution, and concepts clearly to others;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

· Complete the square in situations without fractions;

· Identify the vertex of a parabola expressed in the form y = a(x – h)² + k;

· Understand that the real roots of a quadratic equation are the x-intercepts of the graph of the corresponding quadratic function;

· Locate vertices and intercepts using graphing technology;

· Determine roots of quadratic equations, using the quadratic formula.

Planning Notes

· Prepare worksheets.

· Students require graphing calculators, or access to appropriate graphing software.

· Organize the class into pairs, or small, heterogeneous groups of about 3 or 4.

Teaching/Learning Strategies

Student Activity

Students:

· use graphing technology to graph a parabola;

· identify the vertex of a parabola from the its graph;

· complete the square;

· identify characteristics of a parabola expressed in the form y = a(x – h)² + k.

Teacher Facilitation

· Introduce the following scenario to the class: A new ski-jumping stunt is planned, as shown below An expert ski jumper will accelerate down the ramp, and leave it at the “exit point”, E, which is 8 m above the horizontal ground. She will leave the ramp at an angle of inclination of 30° to the ground, with an initial speed of v. Ignoring air resistance, the height, y, of the ski jumper can be described by the equation y = – 4.9t² + 0.5vt + 8, where t is the time, in seconds, starting from the time when the jumper leaves the ramp. The derivation of this equation is left as an exercise at the end of the activity. It should be discussed why “t” is a more appropriate variable than “x”, for this problem. The skier, an expert at such stunts, can control her exit speed, v, by adjusting her ski position and body tuck. She wants to determine the minimum speed she needs in order to clear the barrier with just a 1 metre margin of error, for dramatic effect.

· The barrier can be shifted closer to, or further from the ramp, however the barrier is not placed until after the optimum speed has been determined. Pose the question: “Why is this important?” Students should determine that as speed changes, the horizontal position of the maximum height will shift.

· Starting with a barrier height of 15 m, students are to use a graphing calculator to determine the optimum speed that will allow for a successful jump. Varying the parameter v through systematic trials, investigating the resulting parabolas and using the trace function to locate maxima is a typical approach to solving the problem.

· After students have discovered the optimum speed, reassemble the class. Have one or two groups present their method to the class. It should be determined that, for a barrier height of 15 m, a speed of approximately 25.1 m/s is required, which corresponds to the function y = -4.9t² + 12.55t + 8.

· At this point, review the algorithm of completing the square and apply it to one or two simple examples, which do not involve fractions or decimals.

· Now guide students through the process of completing the square on the function y = -4.9t² + 12.55t + 8, where a = -4.9, b = 12.55 and c = 8. Discuss the validity of rounding in this particular situation, with some attention to the carrying of an appropriate, or reasonable, number of significant digits.

y = -4.9(t² + 2.56t) + 8 factor out “a” from the first two terms

y = -4.9(t² + 2.56t + 1.28² – 1.28²) + 8 add and subtract inside the brackets

continue completing the square…

y = -4.9(t – 1.3)² + 16

· Prompt the students to look at the terms in this equation. Ask the following: “Can you see any possible relationships between these terms and the graph of the function? What is the physical significance of the values 1.3 and 16?” Have students repeat the exercise for a new barrier height. They are to determine the optimum speed, and then complete the square, in order to express the function in the form shown above. Assign a different barrier height (e.g., 10, 12, 18, 20, etc.) to each group, in order to generalize the results more readily. Once groups have had a chance to do this, have the class share with each other, using Sample Worksheet 1. Then have students complete the questions, alone or in small groups. The teacher may need to review, with one or two examples, application of the quadratic formula, in order to assist with question 3.

Sample Worksheet 1

Barrier height (m)	Optimum Initial Speed (m/s)	Quadratic Equation Standard Form	Quadratic Equation y = a(x – h)^2 + k
10
12
15	25.1	y = -4.9t^2 + 12.55t + 8	y = -4.9(t – 1.3)^2 + 16
18
20

1. What is the physical significance of h and k, in the equation y = -4.9(t – h)² + k?

2. For y = -4.9t² + 0.5vt + 8, what is the physical significance of the 8? Prove this is true.

3. Determine the t-intercepts of the quadratic equation -4.9 t² + 12.55 t + 8 = 0, by using the quadratic formula, and by using a graphing calculator.

4. The term “hang time” refers to the amount of time a ski jumper is in the air. Considering the diagram provided, how is the right t-intercept on the graph related to the hang time?

5. What is the physical significance of the y-intercept of the graph?

6. What is the geometric significance of the left t-intercept of the graph? Why does this point have no real physical meaning?

7. Why is it important that the barrier be moveable, closer to, or further from the ramp, when changing the barrier’s height?

Key Points to Look for in Answers to Questions

1. h is the time at which the jumper reaches the maximum height; k is the maximum height.

2. 8 is the initial height of the jumper, at time zero. One way to prove this is to set t = 0 in the equation and simplify for y.

3. The roots are –0.53 seconds and 3.1 seconds.

4. If the ground were horizontal between the ramp and the landing point then this would represent hang time. Typically the skier will land, for safety purposes, where the land is sloping downwards, so the hang time will be greater than this value.

5. This point represents the initial height of the jumper, and corresponds to t = 0.

6. This point represents the other point at which the jumper would be at ground level, if the parabolic function were extended to the left of the y-axis. Because this corresponds to t < 0, there is no physical meaning for this point. The skier’s motion is not accurately described by this function prior to t = 0.

7. The skier will reach maximum height at different horizontal distances from the ramp, as the initial speed changes.

Assessment & Evaluation of Student Achievement

Depending on the format of assessing the various components of the activity (e.g., brief oral presentations, written submissions, etc.), the teacher may use all, or parts of the following rubric. A category addressing Knowledge has been included in the rubric, however the teacher may wish to replace or augment the Knowledge assessment with a quiz, focussing on the process of completing the square. It is recommended that the assessment in this activity be used formatively, with the option of using a similar rubric for a summative performance task later. Refer also to the “Generic Rubrics” for Communication and Thinking/Inquiry/Problem Solving, from the OAME/OMCA “CARE Package”, which can be downloaded from http://www.oame.on.ca.

Criteria	Level 1 (50-59%)	Level 2 (60-69%)	Level 3 (70-79%)	Level 4 (80-100%)
Inquiry Determines optimum speeds and particular quadratic functions	- uses an approach which demonstrates a limited understanding of the problem	- uses an approach which demonstrates some understanding of the problem	- uses an approach which demonstrates a considerable understanding of the problem	- uses an approach which demonstrates a thorough understanding of the problem
Application Uses graphing technology effectively, (e.g., sets display window, zoom, trace, etc.)	- uses graphing technology with limited effectiveness	- uses graphing technology with some effectiveness	- uses graphing technology with considerable effectiveness	- uses graphing technology with a high degree of effectiveness
Knowledge Understands how to complete the square to optimize a quadratic function	- limited understanding of how to complete the square to optimize a quadratic function	- some understanding of how to complete the square to optimize a quadratic function	- considerable understanding of how to complete the square to optimize a quadratic function	- thorough understanding of how to complete the square to optimize a quadratic function
Communication Clarity of explanations	- demonstrates limited clarity in making explanations	- demonstrates some clarity in making explanations	- demonstrates considerable clarity in making explanations	- demonstrates thorough clarity in making explanations

Extensions

1. Students could investigate the effects of changing the ramp’s angle of inclination. The term 0.5v in the original equation comes from the vertical component of the velocity, v, when the jumper leaves the ramp: v = *v* sin 30°. By varying this angle, students could analyse the effect this has on maximum height and hang time.

2. Have students derive the formula y = -4.9t² + 0.5vt + 8 from the physics formula y = 0.5at² + v_yt + d, where a is the acceleration due to gravity, -9.8 m/s², v_y is the vertical component of the velocity (derived above) and d is the initial height of the skier.

Activity 3: Rooting Around the Parabola

Time: 150 minutes

Description

Students investigate quadratic functions and examine their x-intercepts. Using graphing technology, conceptual connections are drawn between the x-intercepts of a quadratic function, and the real or complex roots of its corresponding quadratic equation.

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.04 - identify the structure of the complex number system and express complex numbers in the form a + bi, where i² = -1 (e.g., 4i, 3 -2i);

OC1.05 - determine the real or complex roots of quadratic equations, using an appropriate method (e.g., factoring, the quadratic formula, completing the square), and relate the roots to the x-intercepts of the graph of the corresponding function;

OC3.02 - present problems and their solutions to a group, and answer questions about the problems and the solutions;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

· Complete the square on quadratic expressions with no fractions or decimals;

· Identify the vertex of a parabola expressed in the form y = a(x – h)² + k;

· Understand that the real roots of a quadratic equation are the x-intercepts of the graph of the corresponding quadratic function;

· Locate vertices and intercepts using graphing technology;

· Use the quadratic formula to solve for real roots of a quadratic equation.

Planning Notes

· Students require graphing calculators, or appropriate graphing software on a computer.

· Appropriate overhead projection technology will be required.

· Organize the class into pairs, or small, heterogeneous groups of about three or four.

Teaching/Learning Strategies

Student Activity

Students:

· use graphing technology to graph parabolas;

· determine the roots of a parabola by visually identifying the x-intercepts of its graph;

· algebraically determine the roots of a quadratic equation using an appropriate method;

· examine complex roots of a quadratic equation;

· conjecture and test an hypothesis;

· design and carry out an investigation using graphing technology.

Teacher Facilitation

· Review with the class the connection between real roots of quadratic equations and the x-intercepts of their corresponding quadratic functions, using simple examples.

· Demonstrate that the value of the discriminant, b² – 4ac, in the quadratic formula gives information regarding the nature of the roots of a quadratic equation, with a few simple examples. Draw connections between the information provided by the discriminant to the x-intercepts of the corresponding functions. Pose the question, “How do we interpret roots of quadratic equations whose corresponding functions do not intersect the x-axis?”

· To set the stage for the investigation, work with a simple quadratic function, such as y₁ = (x – 3)² + 4. Show that the graph clearly does not intersect the x-axis.

· Next, expand the corresponding quadratic equation to produce x² – 6x + 13 = 0 and apply the quadratic formula to introduce the concept of complex roots, in this case: 3 ± 2i. Take it to the point where you get , and then interrupt the solution with the following.

· Engage in an introductory discussion of the complex number system. Explain that any complex number can be expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is defined implicitly by i² = -1. Provide a mathematical context for the discussion by identifying complex numbers as a set of numbers of which the real, rational, irrational, integer, whole, and natural number sets are subsets. A broader context can be drawn by identifying specific areas of study in engineering, physical sciences, and mathematics, in which complex numbers are used to model algebraic and geometric concepts. Some examples include: alternating current and voltage in electrical power systems, quantum mechanics (study of the nature of matter at the sub-atomic level), electro-optics, and fractal models such as the Mandelbrot Set. The teacher may wish to consolidate the basic concepts using one or two additional examples at this point.

· Return attention to the function y₁ = (x – 3)² + 4, and finish deriving the complex roots of the corresponding quadratic equation: 3 ± 2i.

· Draw a comparison to the graph of y₂ = -(x – 3)² + 4, which is a reflection of the graph of the function y₁, in the line y = 4.

· Engage the students in a class discussion including the following: “How are these two functions related to each other, algebraically and geometrically? What are the x-intercepts of y₂?” As a class, algebraically determine the roots of the corresponding quadratic equation (x – 3)² + 4 = 0.

· The following part of the activity should be performed within small groups. Have the students do the following: Given two functions of the form y₁ = a(x – h)² + k, and y₂ = -a(x – h)² + k, conjecture a relationship between the roots of their corresponding quadratic equations. Design and carry out an investigation, which will serve to test your hypothesis, using appropriate graphing technology. Analyse a number of different quadratic functions in all four quadrants. Note: Sample hypotheses/conclusions are provided, at each achievement level, at the end of the Assessment & Evaluation of Student Achievement section below.

· Once groups have had time to perform the investigation, have some groups present their findings to the class. Their presentation should include their stated hypothesis, the approach to the investigation, with a demonstration using projected graphing technology, and their findings. Classmates should be encouraged to ask questions during the presentation and presenters should defend their arguments and answer questions from the class.

· Have students answer the questions below, and assign appropriate skill building exercises.

Follow Up Questions

1. Consider the following statement: “If the discriminant of a quadratic equation is less than zero, then the quadratic equation has no roots.” Discuss the accuracy of this statement, based on your knowledge of various number sets.

2. Given a function in the form y = a(x – h)² + k, how can you easily determine the nature of the roots of the corresponding quadratic equation, by mentally analysing the terms?

Key Points to Look for in Answers to Questions

1. True statement if you confine the analysis to real numbers. There are complex roots.

2. If ak > 0 (a and k have the same sign), there are complex roots. If ak < 0, there are real roots.

Assessment & Evaluation of Student Achievement

The teacher may use all or parts of the following rubric for assessment which should generally be formative. Refer also to the “Generic Rubrics” for Communication and Thinking/Inquiry/Problem Solving, from the OAME/OMCA “CARE Package,” which can be downloaded from http://www.oame.on.ca.

Criteria	Level 1 (50-59%)	Level 2 (60-69%)	Level 3 (70-79%)	Level 4 (80-100%)
Inquiry* Conjectures an hypothesis related to the investigation	- statement reveals limited understanding of the purpose of the investigation	- statement reveals some understanding of the purpose of the investigation	- statement reveals a considerable understanding of the purpose of the investigation	- statement reveals thorough understanding of the purpose of the investigation
Inquiry Performs an investigation	- investigates with limited effectiveness	- investigates with some effectiveness	- investigates with considerable effectiveness	- investigates with a high degree of effectiveness and poses extending questions
Inquiry* Arrives at a conclusion	- demonstrates limited ability to arrive at a conclusion	- demonstrates some ability to arrive at a conclusion	- demonstrates considerable ability to arrive at a conclusion	- demonstrates a high degree of ability to arrive at a conclusion
Application Uses graphing technology effectively, (e.g., sets display window, zoom, trace, etc.)	- uses graphing technology with limited effectiveness	- uses graphing technology with some effectiveness	- uses graphing technology with considerable effectiveness	- uses graphing technology with a high degree of effectiveness
Communication Clearly explains answers to questions (oral and written) related to the activity	- explains with limited clarity	- explains with some clarity	- explains with considerable clarity	- explains with a high degree of clarity

*Sample Hypotheses/Conclusions

Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity.

Level 1: The quadratic equation corresponding to one of the two functions will have real roots. The other one will have complex roots.

Level 2: The quadratic equation corresponding to the function whose graph intersects the x-axis will have real roots. The other will have complex roots.

Level 3: The quadratic equation corresponding to the function whose graph intersects the x-axis twice will have real roots. The other will have complex roots. In the case where k = 0, both equations will have one real root, and this root is the same for both equations.

Level 4: In the case where k = 0, both equations will have one real root, and this root is the same for both equations. If one quadratic equation has complex roots a ± bi, then the other will have real roots r₁ and r₂, where |a – r₁| = |a – r₂| = b. Students who can otherwise, in less sophisticated terms, correctly quantify the relationship between the real and complex roots of the two quadratic equations, deserve Level 4 credit.

Extensions

1. A similar investigation can be carried out for other types of functions, (e.g., absolute value, square root).

2. (a) Show that for the function y = a(x – h)² + k, that the roots are given by the equation:

(optional hint: expand the right hand side and substitute into the Quadratic Formula)

2. (b) Show that the result in 2(a) yields real roots when ak < 0 and complex roots when ak > 0 (optional hint: focus on the radicand and consider cases)

3. Introduce the complex plane as a coordinate system for graphing complex numbers.

Activity 4: Complex Basics are Basically not Complex!

Time: 150 minutes

Description

Students investigate complex numbers graphically on the complex plane using graphing calculators. Students apply the properties of complex numbers by adding, subtracting, multiplying and dividing complex numbers in the form a + bi, and express results in simplest form.

Strand(s) & Learning Expectations

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.06 - add, subtract, multiply, and divide complex numbers in rectangular form;

OC3.04 - demonstrate the correct use of mathematical language, symbols, visuals (e.g., diagrams, graphs), and conventions;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

· Understand basic graphing calculator functions;

· Simplify expressions with exponents.

Planning Notes

· Students may require graphing calculators.

· A graphing calculator, a graphing calculator projection unit, and an overhead projector may be used for the lesson demonstration involving the Complex plane.

· Prepare worksheets.

Teaching/Learning Strategies

Student Activity

Students:

· graph complex numbers on the Complex plane;

· add, subtract, multiply, and divide complex numbers in the form a + bi, and use i² and = -1 and
i = to represent the results in simplest terms.

Teacher Facilitation

· The graphing parts of this activity can be done with or without graphing calculators. Instructions are provided to illustrate how they can be used to graph complex numbers.

· Begin with a general discussion about the applications of complex numbers. For example, complex numbers are used by mathematicians, physicists, and engineers, and have applications in current circuits, in quantum mechanics, and in aerodynamic design. Refer to the Teacher Facilitation section of Activity 3, for additional, specific examples of where complex numbers are used.

· Introduce the properties i² = -1, and i = , and the complex number form a + bi.

· Explain how a complex number, such as 2 + 3i, can be represented by a point on the complex plane. Using a graphing calculator and a graphing calculator projection unit, with an overhead projector, provide visual representation of complex numbers on the complex plane. Do this as follows: To plot the complex number a + bi, use a to replace x, b to replace y, and use the normal point plotting function. Now the rectangular grid can be thought of as the complex plane, where the x- and y-axes are replaced by the real and imaginary axes, respectively. Then discuss the similarities and differences between the Cartesian and Complex planes.

· Explain the concept of conjugates. Graphically represent a complex number and its conjugate on the complex plane and discuss the visual representation. Briefly explore the properties of conjugates (i.e., symmetry in the Real axis).

· As a class, complete an example adding two complex numbers . Link student’s previous knowledge of collecting like terms to simplifying complex numbers a + bi, by collecting real and imaginary terms. With the teacher at the overhead projection unit and the students on their own graphing calculator, graph the two original numbers and the result on the graphing calculator. Discuss the visual representation. Repeat this process with examples of subtracting two complex numbers , multiplying two complex numbers , and dividing two complex numbers .

· Have students work in pairs to complete Sample Worksheet 1.

· Distribute additional worksheets, or assign appropriate work from the textbook.

Sample Worksheet 1

1. a) Are any numbers that are in the complex number system also in the real number system?

b) Are any numbers that are in the real number system also in the complex number system?

c) Justify your answers with examples.

2. a) Investigate the pattern which emerges by looking at successive powers of iⁿ, for natural n. Discuss both the algebraic results and their graphic representations.

b) Use the results to simplify each of the following: i²⁶, i¹², i²⁷, and i¹³.

c) Summarize your results in (a) and (b) by explaining the pattern that emerges with odd and even exponents.

d) Predict the value of i⁸⁷ , then confirm your result through calculation.

3. The conjugate of (2 + 5i) is (2 – 5i) .

a) Calculate the product of these conjugates.

b) What type of number is your result? (i.e., a real number? a complex number?)

c) Explain why this type of result will be the same when any pair of complex number conjugates are multiplied together.

4. Plot the following numbers on the Complex plane:

a) 5i b) 6 c) 4 + 3i d) -2 – 7i

5. Perform the following complex number operations. Plot the original numbers and the results on the complex plane: a) (2 – 9i) +(-3 + 6i) b) (4 – 2i)(-3 + 8i) c)

6. When two complex numbers in the form a + bi are multiplied together, is the result always a complex number? Explain your answer using examples (i.e., Can you create examples that show the result is: (i) a complex number? (ii) a real number?).

7. A quadratic equation has roots 1 ± 2i. Determine the equation of a corresponding quadratic function.

8. Find two numbers whose sum is 8 and product is 25.

Teacher Facilitation

Once students have had an opportunity to complete the worksheet, introduce the “i” key on the graphing calculator, and have students use this feature to check their answers. This is a good opportunity to introduce the “decimal to fraction” and “fraction to decimal” functions, as well.

Assessment & Evaluation of Student Achievement

Students hand in completed Sample Worksheet 1 to be assessed. Communication can be assessed in questions 1, and 3. Inquiry and Communication can be assessed in questions 2, 6, 7, and 8. Parts of the rubric from Activity 3 can be adapted for this activity. Knowledge/Understanding can be assessed throughout the worksheet, using an objective marking scheme. The assessment in this activity should be largely formative.

Extension

Students can perform an investigation comparing the following: illustration of operations (addition, subtraction, multiplication, and division) of real numbers on the real number line, and illustration of the same operations of complex numbers on the complex plane.

Activity 5: Can We Please be Rational?!

Time: 150 minutes

Description

A physical geometric model is used to introduce the concept of a rational expression. Students use scientific and graphing calculators, in order to investigate properties of rational functions. Students simplify, add, subtract, multiply, and divide rational expressions, and state restrictions on variables.

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.07 - add, subtract, multiply, and divide rational expressions, and state the restrictions on the variable values;

OC3.01 - explain mathematical processes, methods of solution, and concepts clearly to others;

OC3.03 - communicate solutions to problems and to findings of investigations clearly and concisely, orally and in writing, using an effective integration of essay and mathematical forms;

OC3.04 - demonstrate the correct use of mathematical language, symbols, visuals, (e.g., diagrams, graphs, and conventions);

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

· Perform operations (i.e., adding, subtracting, multiplying, and dividing) with polynomials;

· Apply factoring techniques (including common factoring, difference of squares, trinomials in the form ax² + bx + c where a = 1, and where a ¹ 1).

· Use the zoom and trace features of a graphing calculator to determine points on a graph.

Planning Notes

· Students require graphing calculators.

· Prepare worksheets.

Teaching/Learning Strategies

Student Activity

Students:

· simplify, add, subtract, multiply and divide rational expressions;

· determine the restrictions on the variables.

Teacher Facilitation

· Have the students think about the land on which some parking lots are located. A survey of their thoughts may result in the land identified as being in either a square or rectangular shape.

· Have them draw a square and then label a side length with x.
Develop an expression for the area in terms of x.

A = x(x)

A = x²

· Pose the question: “Suppose zoning laws made us reduce the area of the lot by 9, while one side was increased by 3. What would be the new width, in terms of the original length, x?”

· Show how this can be expressed algebraically:

· Discuss with the class the nature and properties of a rational expression.

· Have students perform the following numerical analysis:

a) Use a scientific or graphing calculator to divide any given number by successively smaller numbers (getting closer and closer to zero), and describe what happens to the result. Explain why the calculator gives an error message when you try to divide by zero.

b) Apply this concept to explain why the function is said to be undefined for x = -3.

c) Give a geometric explanation of the restriction on x.

· Refocus attention to the context of a physical lot once again.

· Pose the question: “Would an initial length of -2 be an acceptable value for this situation? Explain. Consider the mathematical model and the physical situation which it describes.”

· Turn attention away from the physical model and consider another rational function:

· Have students perform the following graphical analysis:

a) At standard zoom setting, graph the function.

b) Noting an unusual behaviour around x = -3, zoom into this region, using the following window settings. In each case, trace the curve from both the left and right sides of x = 3. Note the y-coordinate as x gets very close to 3. Note also what happens to the y-coordinate if you are able to trace so that x = 3 exactly. Have students organize their findings in a chart:

Zoom Trial	Xmin	Xmax	Ymin	Ymax	Observations
1	2.5	3.5	-50	50
2	2.9	3.1	-100	100
3	2.99	3.01	-1,000	1,000
4	2.999	3.001	-10,000	10,000

· Discuss with students why the function is said to be undefined at x = 3. By introducing the terminology and concepts associated with asymptotes, you can lay a foundation for later studies in the Investigation of Loci and Conics unit. As a nice lead-in to future studies in Calculus, you can touch on the concept of the right and left-hand limits as x approaches 3.

· Return attention to the function and have students graph this function, using the standard zoom setting. Remind students to toggle the other function off.

· Pose the following questions:

a) “What shape is this graph?”

b) “Why is it a line? Does the equation look linear?”

c) “Does this graph make sense for x = -3, based on the earlier investigation?”

· At this point introduce the method of simplifying this rational expression, by factoring the numerator, and then dividing out the common factor, x + 3.

· Pose the question: “Does the equation look linear, now?”

· Discuss the need for recognizing a “hole” at the point (-3, -6), in order to make the graph accurate. This is a good opportunity to identify the limitations of graphing technology and the need for critical thinking on the part of the user.

· Lead students through the (a) parts of questions 2 to 5 of Sample Worksheet 1. The teacher may need to review some factoring techniques with the class at this time.

· Assign the rest of the worksheet to be completed independently.

· Assign appropriate exercises in order to consolidate skills.

Sample Worksheet 1

1. Explain in your notes or journal the difference between mathematical restrictions on variables, and restrictions resulting from the physical nature of the area problem posed in class.

2. Consider the following expressions: a) b)

State the restrictions on each expression. If the denominators were representing a length of an object, would your answers be mathematical restrictions, physical ones, or both?

3. Another type of algebraic situation involves two or more variables in the denominator. State the restrictions on the following: a) b)
(hint: What makes the denominator equal zero?)

4. For each of the following, factor the numerator and denominator, state the restrictions, and then simplify (that is divide out common factors from the numerator and denominator).

a) b) c)

6. Consider a rectangle with dimensions given by: length = and width =

Show that a simplified expression for the area of the rectangle is A = 1/x.

7. Use the given expressions for length and area, from question 6, to derive the expression for width, by applying division. Explain your solution, using mathematics combined with a written explanation.

Teacher Facilitation

· Extend the study of rational expressions to include addition and subtraction, using the concept of adding or subtracting areas in order to provide a rationale.

· Review the process of determining a common denominator CD for rational numbers, and then extend this method to rational expressions, using examples such as:
(a) 2, 3 CD: 6 (b) 9, 3 CD: 9 (c) m, n CD: mn (d) x, x + 1 CD: x(x+1)

· Use traditional worked examples, followed by appropriate exercises, to consolidate skills in adding and subtracting rational expressions.

Assessment & Evaluation of Student Achievement

Questions 1, 6, and 7 from Sample Worksheet 1 can be used to assess Communication and Application. Parts of the rubrics used earlier in the unit can be adapted for this purpose. Refer also, to the “Generic Rubrics” for Communication and Thinking/Inquiry/Problem Solving, from the “CARE Package”, which can be downloaded from http://www.oame.on.ca. A quiz can be used to assess Knowledge, after students have had opportunity to consolidate skills.

Activity 6: Power Play

Time: 75 minutes

Description

Students discover the nature of powers containing rational exponents. Students extend exponent laws to expressions involving powers containing integer and rational exponents.

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.08 - simplify and evaluate expressions containing integer and rational exponents, using the laws of exponents;

OC3.01 - explain mathematical processes, methods of solution, and concepts clearly to others;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

· Apply exponent laws to simplify expressions involving powers which contain natural exponents;

· Use the zoom and trace features of a graphing calculator to determine points on a graph.

Planning Notes

· Students will require graphing calculators.

· Prepare worksheets.

Teaching/Learning Strategies

Student Activity

Students:

· investigate the properties of powers containing rational exponents;

· apply exponent laws to simplify expressions involving powers which contain integer and rational exponents.

Teacher Facilitation

· Review with the class, the meaning of powers with natural bases, as a repeated multiplication process, using examples such as 4³ = 4 ´ 4 ´ 4.

· Next, review the meaning of powers containing a zero, or negative exponent, using examples such as 4⁰ = 1 and =. Discuss the fact that these definitions of powers are not intuitively consistent with the concept of repeated multiplication.

· Pose the question: “How should we interpret powers having rational exponents, e.g., 4^1/2? Considering that 4⁰ = 1 and 4¹ = 4, should the value of 4^1/2 = 2.5 (arrived at through linear interpolation)?” Have students determine the value of 4^1/2 using their scientific or graphing calculator.

· Have students consider that 4² = 16, and hypothesize the value of 4^3/2, or 4^1.5, and of 4^-1/2, then test their hypotheses using their calculators.

· At this point, have students graph the function y = 4^x, using a graphing calculator. Have them use the zoom and trace functions to verify that the values above lie on this graph. Discuss the curvature of the graph, noting the asymptotic behaviour as x approaches negative infinity.

· Challenge students to discover another way of expressing any power to the exponent , i.e., x^1/2 = ? This will be revisited as students work through Sample Worksheet 1.

· Distribute Sample Worksheet 1 and have students work through the questions independently, or in small groups.

Sample Worksheet 1

1. (a) Investigate powers having different bases raised to the exponent ½.

(b) Square the result in each case. Explain what you notice.

2. Consider that 3² ´ 3⁴ in expanded form is (3)(3) ´ (3)(3)(3)(3) = 3²⁺⁴ = 3⁶

The algebraic way of generalizing this result is: m^x ´ m^y = m^x+y (Product Rule)

Apply the product rule to: 4^1/2 ´ 4^1/2

3. Apply the product rule to: 27^1/3 ´ 27^1/3 ´ 27^1/3

4. Determine a value of 27^1/3, using a scientific, or graphing calculator.

5. Use the results above to give another mathematical meaning of 4^1/2 and 27^1/3. Explain, using words and mathematical symbols.

6. (a) Generalize the result of question 5 to explain the meaning of x^1/n. Use words and symbols.

(b) Use numerical examples to illustrate your explanation.

7. Consider that can be expanded into = 5^{6 – 2} = 5⁴

The algebraic way of generalizing this result is: m^x ¸ m^y = m^{x – y} (Quotient Rule)

(a) Apply the quotient rule to 6³ ¸ 6⁵, and express as a power with a negative exponent.

(b) Show that 6³ ¸ 6⁵ = by writing the expression as an expanded rational expression.

(c) What must be true about the expressions from (a) and (b)?

8. Use three methods to show that to show that 3^{– 3} ¸ 3^{– 7} = 81

9. Use three methods to determine the value of 27^{– 1/3}

10. Explain why the graph of y = 4^x, asymptotically approaches the x-axis, as x approaches negative infinity.

Teacher Facilitation

· At this point, the class should reassemble to discuss their findings, and share their methods.

· The teacher should, through worked examples, consolidate concepts, and demonstrate that the “power of a power” exponent law also applies to integer and rational exponents.

· Use examples such as: a) b) If w = -1, s = 2, evaluate

Assessment & Evaluation of Student Achievement

Questions 1, 5, and 6 can be assessed for Communication. Questions 3, 4, and 7 can be assessed for Application. Questions 9 and 10, can be assessed for Thinking/Inquiry/Problem Solving. Parts of rubrics used earlier in this unit may be adapted for this purpose. A quiz with an objective marking scheme can be used to measure Understanding, after students have had a chance to consolidate skills.

Extension

Students can explore properties of powers containing complex bases, and investigate whether or not exponent laws hold.

Activity 7: It’s Snowing Cats and Dogs!

Time: 75 minutes

Description

Students extend their modelling and critical thinking skills to analyse an exponential growth pattern. Students discover that a quadratic model, which appears to effectively describe a natural phenomenon, breaks down as more data is collected, leading to the discovery of the exponential function. Students apply the properties of exponential functions to answer questions, and draw connections to, the algebraic and graphical forms of exponential relationships.

Strand(s) & Learning Expectations

Strand(s): Tools for Operating and Communicating with Functions

Overall Expectations

OCV.01 - demonstrate facility in manipulating polynomials, rational expressions, and exponential expressions;

OCV.03 - communicate mathematical reasoning with precision and clarity throughout the course.

Specific Expectations

OC1.09 - solve exponential equations (e.g., 4^x = 8^{x + 3}, 2^2x -2^x = 12);

OC3.03 - communicate solutions to problems and to findings of investigations clearly and concisely, orally and in writing, using an effective integration of essay and mathematical forms;

OC3.05 - use graphing technology effectively (e.g., use appropriate menus and algorithms; set the graph window to display the appropriate section of a curve).

Prior Knowledge & Skills

· Model linear and quadratic relations using graphing technology;

· Perform finite difference analysis on data;

· Apply exponent laws.

Planning Notes

· Students require graphing calculators or access to appropriate graphing software.

· Prepare worksheets.

Teaching/Learning Strategies

Student Activity

Students:

· plot and analyse data from a simulated situation;

· discover the nature of exponential functions;

· answer questions and solve problems with involving exponential equations.

Teacher Facilitation

· Organize the class into pairs or small groups.

· Some of the techniques required for the graphing technology component of this activity may require introduction or review (e.g., generating and storing regression equations).

· Students may be familiar with the correlation coefficient, r, which measures how well a line fits a set of data points. When applying regression analysis to non-linear data, a more appropriate measure of best fit is the coefficient of determination, r². The closer r² is to 1, the better the curve fits the data. Since a line is simply a type of curve, r² can also be applied to linear data. The teacher should explain this to the class, using a couple of simple examples.

· Introduce the scenario described on Sample Worksheet 1.

Sample Worksheet 1

Suppose you are the managers of a small ski lodge, with no artificial snow production equipment. You are receiving calls early in the ski season requesting information about the skiing conditions, particularly concerning the amount of snow expected for the upcoming weekend. The weather report has alerted that a heavy snowfall is due, and is expected to follow a consistent pattern for several days. Following the first signs of snow, you measure the amount of accumulated snow, every four hours, and record the following:

Time (hours)	4	8	12	16	20	24
Amount of Snow (cm)	1.7	3.0	5.3	9.2	16	28

In order to answer customers’ questions with some degree of accuracy, you decide to mathematically model this snowfall pattern, using graphing technology.

1. Plot the data from the table. Compare the pattern to functions with which you are familiar
(e.g., linear, quadratic). Use the regression feature of your graphing calculator to generate:

(a) a line of best fit,

(b) a quadratic curve of best fit.

For both of these, record the equation, the coefficient of determination, r², and a sketch of the graph. How well do these regression models fit the data?

2. Look at the first and second differences for your data. Do these satisfy the criteria for a linear or quadratic relationship? Explain your thinking.

Suppose you record the next two measurements as follows:

Time (hours)	28	32
Amount of Snow (cm)	49	84

3. Add this data to your table and graph. How well do the best fit line and best fit quadratic relation fit the data now?

4. Experiment with different types of regression. Find an equation that fits the data with a coefficient of determination at least r² > 0.999. Record the equation and store it as y₁.

5. Consider the following two equations, which probably look different from the one you just discovered:(i) y₂ = 4 ^0.1x (ii) y₃ = 2 ^0.2x

(a) Graph each of these functions over your original data points. How well do these fit your data? Explain how this is possible.

(b) Prove that y₂ and y₃ are identical functions.

(c) Prove that y₁ is not identical to y₂ and y₃.

6. In order for successful cross-country skiing, about 30 cm of snow is required, while successful downhill conditions require about 60 cm. Suppose the snowfall started at 3:00 p.m. on Thursday. When could you recommend to your customers that skiing conditions will be satisfactory for:

(a) cross-country? (b) downhill?

Teacher Facilitation

· Once students have had time to perform the activity above, discuss the answers to the questions. Sample solution to Question 5(b): 4 ^0.1x = (2 ²) ^0.1x = 2 ^0.2x

Sample approach to Question 5(c): show that a point, which satisfies y₁ does not satisfy y₂, by substitution.

· At this time, remove the contextual framework, and focus on skill development. Introduce methods for solving exponential equations. Use examples such as:

a) 4^x = 64 b) 5^2x = 5⁴ c) 4^-2x = d) 2^2x – 2^x = 12

Assign appropriate work from a worksheet, or the textbook, to consolidate algebraic skills.

Assessment & Evaluation of Student Achievement

Criteria	Level 1 (50-59%)	Level 2 (60-69%)	Level 3 (70-79%)	Level 4 (80-100%)
Inquiry Performs an investigation	- investigates with limited effectiveness	- investigates with moderate effectiveness	- investigates effectively	- investigates efficiently and effectively
Application Uses graphing technology effectively, (e.g., sets display window, zoom, trace, etc.)	- uses graphing technology with limited effectiveness	- uses graphing technology with some effectiveness	- uses graphing technology with considerable effectiveness	- uses graphing technology with a high degree of effectiveness
Communication Clearly explains answers to questions (oral and written) related to the activity	- explains with limited clarity	- explains with some clarity	- explains with considerable clarity	- explains with a high degree of clarity

Accommodations

Student skill in using the graphing technology may vary. It may be appropriate to pair, or group, students accordingly, as the focus of the investigation should be on the discovery of the mathematical principles.

Activity 8: Summative Assessment

Time: 75 minutes

Overall Expectations: OCV.01, OCV.03.

Specific Expectations: All expectations within unit

Description

A comprehensive, balanced summative assessment addressing all four Achievement Chart categories should be administered at the end of this unit. Three sample questions are provided which model how the teacher may assess for Communication, Application, and Thinking/Inquiry/Problem Solving. Traditional questions can be used to assess Knowledge and Understanding. These are samples only; the teacher should develop a complete summative assessment which addresses all expectations within this unit.

Planning Notes

· Graphing calculators will be required for some parts of the assessment.

· Prepare a complete, comprehensive summative assessment.

Sample Questions

1. Sample Communication Assessment

Consider a quadratic function whose corresponding quadratic equation has no real roots. Explain what this information reveals about the:

(a) x-intercepts of the graph of the quadratic function;

(b) value of the discriminant of the quadratic equation;

(c) direction of opening and relative position of the vertex, with respect to the x-axis, of the parabola.

Use complete sentences, mathematical symbols and/or diagrams to explain your answers.

2. Sample Application Assessment

A quadratic equation has roots 5 + 3i and 5 – 3i. Determine the vertex of the graph of its corresponding quadratic function. A graphing calculator is not permitted.

3. Sample Thinking/Inquiry/Problem Solving Assessment

Consider the following growth patterns of two bacteria cultures. Culture A, with an initial population of 100, doubles every hour. Culture B, with an initial population of 30, triples every hour. After what elapsed time, to the nearest minute, will the two cultures have the same population? A graphing calculator is permitted.

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