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 = -4.9t2 + 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 + vyt + d, where a is the acceleration due to gravity, -9.8 m/s2, vy 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 =