Course Profile Foundations of Mathematics, Grade 10, Applied, Catholic

Unit 2: Modelling With Quadratics

Time: 35 hours

Unit Description

In this unit, quadratic relationships will be analysed using numerical, graphical, and algebraic methods. Students will build understanding of quadratic models through concrete experiences which incorporate a technical approach.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE1d, 2b, 2c, 4a, 4f, 5a, 5b, 5g, 7b.

Strand(s): Quadratic Functions

Overall Expectations: QFV.01P, QFV.02P, QFV.03P.

Specific Expectations: All those from the quadratic strand.

Activity Titles (Time + Sequence)

The activities in this unit are designed to allow students to build understanding of quadratic functions through concrete experiences. The effective use of technology, including graphing calculators, graphing software, and CBR devices, will facilitate the learning process. Students will begin by exploring realistic situations which are modelled by quadratic relationships. While the graphical model is the focus of the first three activities, the algebraic model is gradually developed in the subsequent activities. Various forms of the algebraic model are considered. Students are given the opportunity to make connections between the algebraic and graphical models of the quadratic by examining the effects of changes in a, h, and k on the graph of y = a(x - h)² + k. Finally students will solve problems involving given quadratic functions by interpreting the graphs.

Activity 2.1

The Twelve Days of Christmas
[a graphical exercise, where students construct tables of values and generate graphs of non-linear relationships]

150 minutes

Activity 2.2

Quadratic or Not
[collect data, graph with technology, determine characteristics of quadratic graphs and equations]

150 minutes

Activity 2.3

Follow-up Skills

Larger Than Life
[generate table of values from a realistic situation, graph the relation, model with the equation y = ax² + b, explore the effects of a and b on the equation]

Paper practice with the equation y = ax² + b

Inequalities

150 minutes

Activity 2.4

Follow-up Skills

Graphs on the Move
[investigate transformations (stretches, reflections, shifts) of the graph of y = x², to the form y = a(x - h)² + k]

Roles of a, h, k – paper practice

Assessment: Patterning with Parabolas

[create own patterns using graphs of quadratic relations with graphing calculators or graphing software]

225 minutes

75 minutes

Activity 2.5

Follow-up Skills

Experiments with the CBR

Part A: What Goes Up, Must Come Down

Part B: Ramp Cart
[collect data with the CBR, graph using technology, use y = ax² + bx + c and y = a(x - h)² + k forms of the quadratic, explore the effect a has on the equation]

Expand y = a(x - h)² + k into the form y = ax² + bx+ c

150 minutes

30 minutes

Activity 2.6

Follow-up Skills

If The Price is Right
[determine the zeros, intervals, maximum and minimum values of a quadratic function and interpret their meaning in the context of a revenue/price problem]

Maximum/minimum problems

Factor (common, factoring trinomials in form x² + bx + c, difference of squares)

Solve quadratic equations by factoring

150 minutes

345 minutes

Activity 2.7

Follow-up Skills

The Zero Factor
[graphical application of factoring]

Factor trinomials in form x² + bx + c, differences of squares factoring

Solve quadratic equations by factoring

Solve quadratic equations graphically using realistic situations

75 minutes

150 minutes

Activity 2.8

Summative Assessment Activity: Newton’s Apple

225 minutes

Prior Knowledge Required

Number Sense and Algebra

· manipulate 1st-degree polynomial expressions to solve 1st-degree equations (excluding equations with fractional coefficients)

· add and subtract polynomials; multiply a polynomial by a monomial; expand and simplify polynomial expressions involving one variable

· substitute into and evaluate algebraic expressions involving exponents

· solve problems, using the strategy of algebraic modelling

· demonstrate facility in operations with integers, as necessary to support other topics of the course

Relationships

· construct tables of values and graphs to represent non-linear relationships derived from realistic situations

· compare the graphs of linear and non-linear relations

· collect, organize and analyse data using appropriate equipment and/or technology

· describe trends and relationships in data

· construct tables of values, graphs, and formulas from descriptions of realistic situations and from data collected experimentally

· use interpolation and extrapolation to gather information from a graph

· distinguish between linear and non-linear relations by calculating finite differences

Analytic Geometry

· identify the properties of line segments (direction, positive/negative slope, parallelism, perpendicularity)

· calculate slope using various formulae

· identify slope as a constant rate of change

· graph lines by hand and using graphing technology

Measurement and Geometry

· solve problems involving area, in applications

Unit Planning Notes

· Practise using spreadsheets, graphing calculators, and graphing software in the context of the activities presented.

· Prepare to diagnose prior learning skills throughout the unit. Skill development activities should be developed to meet the needs that arise.

· Some students may require extended or enrichment activities to challenge their learning.

Teaching/Learning Strategies

· It is expected that direct, teacher-lead instruction will be integrated within the framework of the activities as required to facilitate student learning and success. After each investigation, teachers should ensure that the mathematics to be developed during that activity has been drawn out. Plan enough time at the end of each investigation to bring closure for the activity and perhaps write a concluding note or journal entry.

· Independent practice of new skills will be necessary throughout the course.

· It is recommended that students be assigned to groups with any special needs and strengths considered. Appropriate peer grouping to benefit those students requiring extra help is suggested.

· Students will be involved in a considerable amount of group work. It would be beneficial to take some time to review appropriate group work dynamics, sharing of work responsibilities, assigning group roles, etc.

· Students should be encouraged to take ownership and responsibility for their own learning.

· Appropriate opportunities for students to communicate solutions, ideas, concepts should be provided throughout the course.

Assessment and Evaluation

· It is recommended that students be involved in the development of some of the rubrics used in the assessment/evaluation process. Students should be encouraged to self-evaluate and to identify areas that need improvement.

· Sample generic rubrics for oral presentations, written reports, and various learning skills are provided in this profile. They may be adapted to suit the needs of your classroom.

· Rubrics should be used where appropriate to assess student work. They are particularly appropriate to assess the expectations under the thinking/inquiry/problem-solving, communication and application sections of the achievement chart.

· When rubrics are used in assessment, students should be provided with the specific rubric that will be used prior to completing the assigned task.

· Care should be taken in the design of traditional pencil and paper tests to ensure that a level 4 performance can be demonstrated.

· It is recommended that this unit be assessed using a summative assessment activity as well as a pencil and paper test, with emphasis on communication skills and understanding of concepts.

· Learning skills should be assessed in conjunction with the academic skills for each activity.

Resources

Graphing calculators, CBR devices, spreadsheet software

Zap-a-Graph (Ministry-licensed software)

Algebra in the Real World

Modelling Motion: High School Math Activities with the CBR. Texas Instruments Inc., 1997.

Real-World Math with the CBL System. Texas Instruments Inc., 1999.

Math and Science in Motion. Texas Instruments Inc., 1997.

Mathematics Teacher December 1995 Volume 88 Number 9

“A Graphical Approach to the Quadratic Formula”: Mathematics Teacher January 1996

“Algebra in a Technological World,” NCTM Addenda Series.

Green Globs: download to TI-83+ from TI web site

Explore Quadratic Functions with the TI-83+. Bob Alexander Publications.

Graphic Algebra. Key Curriculum Press, 1998.

Internet Site
http://forum.swarthmore.edu/workshops/sum98/participants/sinclair/sample.htm
(investigating functions using spreadsheets)

Mathematics dictionary

Activity 2.1: The Twelve Days of Christmas

Time: 150 minutes

Description

Students will illustrate the data obtained from a familiar holiday carol graphically and will discover the shape of a quadratic relation. Students will determine which type of gift was the largest in number and which type of gift was the smallest in number.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads, understands and uses written materials effectively;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Quadratic Functions

Overall Expectations

QFV.02P - determine through investigation the relationships between the graphs and the equations of the quadratic functions;

QFV.03P - solve problems by interpreting graphs of quadratic functions.

Specific Expectations

QF2.01P - construct tables of values, sketch graphs to represent quadratic functions derived from descriptions of realistic situations;

QF3.02P - determine the maximum value of a quadratic function from its graph, using graphing calculators or graphing software.

Planning Notes

· Graphs in this activity will be completed by hand and using graphing software or graphing calculators

· Bring in Copies of “ The Twelve Days of Christmas” and a Mathematics dictionary

Prior Knowledge Required

Relationships

· construct tables of values and graphs to represent non-linear relationships derived from realistic situations

Analytic Geometry

· graph lines by hand and using graphing calculators

Teaching/Learning Strategies

Teacher Facilitation: Read “The Twelve Days of Christmas” together and present the problems to the class. (See Student Activity.) Students will work in pairs in order to help each other through the assignment. The teacher will circulate among the various groups to assist those needing some guidance. The extension activity involves the use of the regression function on the graphing calculator. This allows students to find the equation of the curve of best fit for their data and need not be further discussed at this time. Students should be encouraged to experiment with the regression functions available on the graphing calculators.

Student Activity

Students will complete the following questions.

The Twelve Days of Christmas

1. After twelve days of receiving gifts, the woman in the song decided to total up all of her gifts. She wanted to know the following:

a) How many of each type of gift did she receive?

i) How many partridges? (Remember she received a partridge every day for twelve days!)

ii) How many turtle doves?

iii) How many french hens?

Continue the total for all twelve days

Present the information in a table.

Gift	1	2	3	4	5	6	7	8	9	10	11	12
Total Number Received

b) Draw a graph to illustrate the data. When entering the data in the graphing calculator, let “X” be the gift number. For example Partridge is gift type “1",Turtle doves are gift type “2". Is this a linear or non-linear relationship?

c) Describe the shape of the curve.

d) Which gifts did she receive the most of and the least of? How is this illustrated on the graph?

e) How many gifts did she receive in total?

Extension:

2. a) To show her appreciation, the woman decided to return the same type of gift giving to her true love. On the first day she gave a gift “a”, on the second day she gave gift “a” and two gift “b’s”, on the third day she gave gift “a”, two gifts “b’s” and three gift “c’s” and so on. (There were 26 days of giving various gifts to her true love, who is a math teacher) Be creative in your choice of gifts. Choose math-related items for each day (e.g., “a” represents something such as an “abacus”, “b” could represent a “balance”, etc.). Present your data in a table and illustrate the data on a graph.

b) Include a list of your mathematical gifts and the dictionary meaning of each.

c) State which type of gift was the most numerous? Where is it located on the curve?

d) State which type of gift was the least in number. Where is it located on the curve?

e) Using the regression menu on the calculator, state the equation of best fit.

3. Homework assignment: The Parabola of Plenty!
Suppose that, as a class, you have decided to collect food as a group project for the whole month of October (31 days), following the model of “The Twelve Days of Christmas”. For example, on the first day you will collect 1 jar of jam; on the second day you will collect 1 jar of jam and 2 tins of tuna....etc. Which donation would be the least in number and which would be the most.
Try to determine which items would be most needed by your local food bank and ensure that these have the highest total numbers in your model. Include tables of values and graphs to support your answers.

Extension: Students could organize a school event to actually carry out the homework assignment and donate the items to the local food bank.

Assessment/Evaluation Techniques

· Assess this activity using the learning skills rubric (independent work, organizational skills, work habits, communication and initiative) and assess the homework assignment in the areas of knowledge/understanding and communication using the written report rubric found in the appendices.

Resources

Mathematics Teacher December 1995 Volume 88 Number 9

Appendices

Appendix A – Learning Skills Rubric

Appendix D – Written Report Rubric

Activity 2.2: Quadratic or Not

Time: 150 minutes

Description

This activity will allow students to investigate various relationships and distinguish which relationships are quadratic. The use of finite differences will help students make the decision about whether a relationship is quadratic or not. Using the tables of values generated by various situations, students will use graphing technology and the regression menu of a graphing calculator to establish the connection between the curve and the equation.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a self-directed, responsible, life long learner who demonstrates flexibility and adaptability.

Strand(s): Quadratic Functions

Overall Expectations

QFV.02P - determine through investigation, the relationships between the graphs and the equations of quadratic functions.

Specific Expectations

QFV2.01P - construct tables of values, sketch graphs derived from descriptions of realistic situations.

Planning Notes

· The use of graphing tools, either graphing calculators or graphing software, is recommended since these will allow students to quickly establish which relations are quadratic and which are not.

· Although only five different situations have been given, other types of relationships can be investigated in a similar fashion. With a larger number of relationships to investigate, students could be placed in groups and then present their charts, curves and equations to the whole class. Together, the class can draw conclusions on which curves produce quadratic equations.

Prior Knowledge Required

Number Sense and Algebra

· demonstrate facility in operations with integers

Relationships

· construct tables of values and graphs to represent non-linear and linear relations derived from descriptions of realistic situations

· demonstrate an understanding that straight lines represent linear relations and curves represent non-linear relations

· identify, by calculating finite differences in its table of values, whether a relation is linear or non-linear

Analytic Geometry

· graph lines by hand, using graphing calculators or graphing software

Teaching/Learning Strategies

Teacher Facilitation: Place students in pairs so that they can discuss the investigation and help each other through the various scenarios. Each student is to complete the handout individually in order to have a record of this activity. At the end of the activity students should be given a new situation to complete individually and hand in. Each student should be able to set up a table of values, state whether the relationship is quadratic or not by using first and second differences, and use technology to produce a graph of the situation and to generate the equation.

Student Activity

For each of the following examples, complete the table of values, calculate the finite differences, analyse the differences to hypothesize about the shape of the graph and graph the relations.

For each example answer the following questions:

a) Are the first differences the same or different?

b) Are the second differences the same or different?

c) Is the relation linear or non-linear?

d) Using the regression menu on the calculator, find the equation of the curve of best fit. State the equation.

(A) Pizza Prices

1. A local pizzeria charges a flat rate of $7.00 for a medium pizza with one topping and $0.95 for each additional topping. What is the cost of a pizza with six additional toppings? What is your hypothesis about the cost? Explain your answer.

2. Complete the chart.

Number of Toppings	Cost of Pizza	First Differences	Second Differences
0
1
2
3
4
5
6

What is the cost of a pizza with six additional toppings?

3. Was your hypothesis correct? If not, what error(s) did you make in your initial hypothesis?

(B) Floor Tiles

1. How many dark tiles are in a square floor of 144 tiles? What is your hypothesis?

2. Complete the chart

Width of Square Floor	Number of Dark Tiles	First Differences	Second Differences
3
4
5
. . .
12

What is the number of dark tiles in a square floor containing 144 tiles?

3. Was your hypothesis correct? If not, what error(s) did you make in your initial hypothesis?

1. A special millennium math bug doubles in population every day. The population of this new type of bug was 300 on February 20th. What is the population of this bug on the 15th and 25th of February? What is your hypothesis about the population on these dates?

2. Complete the chart.

Date	Number of Days	Population	First Differences	Second Differences
Feb. 15
. .
Feb. 20	0
Feb. 21	1
. .	3
Feb. 25	4

3. What is the population of the millennium math bug on Feb. 15th? Feb. 25th?
Was your hypothesis correct? If not, what error(s) did you make in your initial hypothesis?

(D) Sum of Natural Numbers

1. What is the sum of the first nine natural numbers? What is your hypothesis?

2. Complete the chart.

Number of Terms	Sum	First Differences	Second Differences
1	1
2	1 + 2 =
3	1 + 2 + 3 =
4	1 + 2 + 3 + 4 =
. .
9

What is the sum of the first nine natural numbers?

3. Was your hypothesis correct? If not, what error(s) did you make in your initial hypothesis?

(E) The Pizza Function

1. If you make six straight cuts in a circular pizza, what is the maximum number of pieces you can have? What is your hypothesis?

2. Complete the chart.

Number of Cuts	Number of Pieces	First Differences	Second Differences
0
1
2
.
.
6

What is the maximum number of pieces using six straight cuts?

3. Was your hypothesis correct? If not, what error(s) did you make in your initial hypothesis?

(F) Summary Report

i) Submit all charts, graphs and questions completed in this activity.

ii) State the equations of the three quadratic curves of best fit in this exercise.

iii) What are the similarities in these three curves? What are the similarities in the corresponding equations for these three curves?

iv) Invent an equation that would give you a quadratic curve. Test your equation using the graphing calculator. Sketch the curve displayed on your calculator.

Assessment/Evaluation Techniques

· A written report rubric assessing knowledge/understanding; thinking/inquiry/problem-solving and communication can be used on the work which was handed in. Assessment of independent work, teamwork, organizational skills, work habits, communication and initiative should also be made.

Resources

Algebra in the Real World.

Appendices:

Appendix A – Learning Skills Rubric

Appendix B – Observational Rubric

Appendix D – Written Report Rubric

Activity 2.3: Larger than Life

Time: 150 minutes

Description

In this activity, students will generate a table of values and obtain the graph for the relationship between the area of a projected image on a screen and the distance from the projector to the screen. They will then explore the shapes and positions of graphs with equations of the form y = ax² + b, in search of an algebraic representation of the same data.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Quadratic Functions

Overall Expectations

QFV.02P - determine, through investigation, the relationships between the graphs and the equations of quadratic functions.

Specific Expectations

QFV2.01P - construct tables of values, sketch graphs, and write equations of the form y = ax² + b to represent quadratic functions derived from descriptions of realistic situations.

Planning Notes

· Graphs should be completed using graphing calculators or graphing software

· An overhead projector or other projection device (minimally, a flashlight), on a moveable trolley, will be useful for demonstration of the way the size of an image changes as the projector is moved towards the screen. Use an extension cord to allow for maximum flexibility of position. If more than one projection device is available, this will save time.

· Students will need graph paper, scissors, and measuring tapes

· Masking tape to place markers on the floor (optional).

· Overhead transparencies of graph grids, or chart paper and markers, will be needed for groups to present their graphs to the class. Alternatively, the graphing calculator viewscreen could also be used for their presentations

Prior Knowledge Required

Relationships

· construct tables of values and graphs to represent non-linear relationships derived from realistic situations

Analytic Geometry

· graph lines by hand, and using graphing calculators or graphing software

Measurement and Geometry

· solve problems involving area, in applications.

Teaching/Learning Strategies

Teacher Facilitation: Set the scene for this investigation by demonstrating, with an overhead projector, the change in the size of a projected image as the overhead projector is moved towards or away from the screen. A graphing calculator overhead view screen provides a well defined image to use in this demonstration.

Tell students that they will be investigating the relationship between the area of an image and the distance from the projector to the screen.

Assign students to small groups of three or four.

Student Activity

Students will collect data as follows.

1. On graph paper, draw a regular shape such as a square, rectangle, right triangle, parallelogram or trapezoid. Make it large enough to occupy about one half of the graph paper. Cut out the shape leaving the rest of the graph paper in one piece. This will be your group’s object to be projected onto the screen.

2. When it is your group’s turn, place your “stencil” on the over head projector and position the projector so that it is exactly 1 metre from the screen.

3. Take enough measurements from the projected image to enable you to calculate its area.

4. Move the projector further away so that it is now 2 metres from the screen and repeat your measurements.

5. Repeat for distances from the screen of 3 m, 4 m, and 5 m.

6. Complete the table.

Distance from screen (metres)	Area of Image (metres²)
1
2
3
4
5

7. Calculate the first differences for your data. Will the graph be linear or non-linear?

8. Plot your data points on the overhead transparency grid (or chart paper, or viewscreen) you have been given. Make “distance from the screen”, the independent variable and “Area” the dependent variable.

9. Draw a smooth curve of best fit to represent the data.

10. Describe the graph in your own words.

Teacher Facilitation: When all groups have completed their graphs, have each group present their graph and a description of their object. Follow these presentations with a whole class discussion of the similarities and differences between the graphs presented. (Graphs will all be in the form y = ax², with different a’s due to the different shapes used by each group.) Explain to students that the next investigation will explore a basic algebraic model for the type of relationship depicted in their graphs.

Student Activity

In order to be able to do more work with this type of relationship we need to find an algebraic description for it. Since area is measured in square units, it is likely that the area of your shape is related to the square of the distance from the projector to the screen.

Exploration 1 – The graph of y = ax² – exploring the effects of changes in a

1. Using a graphing calculator (or graphing software), graph the following relationships. Sketch each graph on the same set of axes so that you can observe the effect of changing the value of a. Use a different symbol or colour for each graph.

y = x² y = 2x² y = 10x² y = 0.5x²y = 0.1x²

What effect does changing a have on the graph of y = x²?

2. Now graph these relationships, which have different negative values for a. Sketch each graph on the same set of axes.

y = -x² y = -2x² y = -10x² y = -0.5x²y = -0.1x²

What effect does changing a have on the graph of y = x²?

What happens to the graph when a is negative?

3. For graphs whose equations have the form y = ax², determine

· the coordinates of the vertex

· the type of symmetry (are there any lines of symmetry?)

· the x-intercepts

4. Predict what the graphs of the following equations would look like and explain the reasons the reasons for your prediction. Check your answers using a graphing calculator.

y = ¹/₃x² y = -5x² y = 0.001x²

Exploration II – The graph of y = x² + c and y = ax² + c exploring the effects of changes in a and c

1. Use a graphing calculator to complete the following graphs. Find the coordinates of the vertex for each equation listed and then complete the chart.

Equation	*Vertex (h, k)*	Number of x-intercepts	x-intercepts
y = x²
y = x² + 4
y = x² -16
y = x² - 9
y = x² + 8
y = x² - 5

2. How is the number of x-intercepts related to k?

3. Find a possible relationship that can determine the x-intercepts – if they exist – when you are given k.

4. Complete the following chart.

Equation	*Vertex (h, k)*	Number of x-intercepts	x-intercepts
y = x² - 4
y = 2x² - 4
y = 4x² - 4
y = 0.5x² - 4
y = -x² - 4

5. Change the rule you found in Question 2 to take into account both k and a.

6. Alter the relationship you found in Question 3 to take into account both k and a.

7. Predict what the graphs of the following equations would look like and explain the reasons for your prediction. Check your answers using a graphing calculator.

y = 2x² - 9 y = -3x² + 9 y = - 0.5x² - 16

Teacher Facilitation: The teacher will need to circulate and provide help where needed, especially with Questions 2, 3, 5, and 6. On completion of the explorations, students should be asked to conjecture and then determine an equation for the graph they drew based on the area vs. distance data they collected earlier. Graphs are of the form y = ax², a > 0.

Extension: Students can create an animated movie which shows the effect of varying a in y = ax² by showing a sequence of graphs in quick succession so that they look as though they are moving.*

They could also create patterns by graphing a number of quadratic relations on one grid.*

Follow-up Skills: 150 minutes

· paper practice with the equation y = ax² + b

· QF1.03P - describe intervals on quadratic functions

Assessment/Evaluation Techniques

· Assessment in the learning skills area of teamwork is possible as groups work on the initial activity.

· Individual student communication skills and knowledge/understanding may be assessed during the presentation, using the presentation rubric.

· Assessment in the learning skills areas of independence and initiative, using the learning skills rubric is possible as students work on the explorations

· Assign a homework journal for students to explain the role of a and k in quadratic functions.

Resources

The explorations are adapted from “A Graphical Approach to the Quadratic Formula”, Mathematics Teacher, January 1996.

*Explore Quadratic functions with the TI-83. Bob Alexander Publications.

Appendices

Appendix A – Learning Skills Rubric

Appendix B – Observational Rubric

Appendix D – Presentation Rubric

Activity 2.4: Graphs on the Move

Time: 225 minutes

Description

Students will investigate the effects of simple transformations on the graph of y = x². Through these explorations, students will develop an understanding of the roles of a, h, and k in the graph of
y = a(x - h)² + k. While these investigations may be completed by hand, the process is facilitated by the use of graphing software and/or graphing calculators.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- a collaborative contributor who works effectively as an interdependent team member;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems.

Strand(s): Quadratic Functions

Overall Expectations

QFV.02P - determine through investigation, the relationships between the graphs and the equations of quadratic functions;

QFV.03P - solve problems by interpreting graphs of quadratic functions.

Specific Expectations

QF2.02P - identify the effect of simple transformations (e.g., translations, reflections, vertical stretch factors) on the graph and the equation of y = x², using graphing calculators or graphing software;

QF2.03P - explain the role of a, h, and k in the graph of y = a(x - h)² + k

QF3.01P - obtain the graphs of quadratic functions whose equations are given in the form y = a(x - h)² + k or the form y = ax² + bx + c, using graphing calculators or graphing software.

Planning Notes

· This activity extends the exploration of graphs of quadratic relations which was begun in Explorations I and II of Activity 2.3. It is essential that students have completed these previous explorations before they attempt this activity

· This activity is best carried out using Zap-a-Graph (or equivalent graphing software) and/or graphing calculators, but pencil and paper approaches may also be used

· Reserve computer lab time for two to three periods if graphing software is to be utilized

· If students have not previously used graphing software, an initial workshop on the features and operation of the program will be necessary

· Students may work in pairs to complete this activity. Pairing should be done with ability levels in mind

Prior Knowledge Required

Relationships

· graph lines by hand or using graphing calculators or graphing software.

Teaching/Learning Strategies

Student Activity

Working in pairs, students will investigate the effects of simple transformations on the graph of y = x². These explorations may be completed by hand; or using graphing software (Zap-a-Graph) or graphing calculators.

Teacher Facilitation: This activity builds on the previous Larger than Life activity (Activity 2.3); and extends the students’ experience with quadratics to the area of transformations and investigates the meaning of the y = a(x - h)² + k form of the quadratic.

If graphing technology is to be used, it may be necessary to conduct a preliminary workshop on the features and operation of the graphing software particularly if this is the first time students have used the program. Students may work in pairs to complete this investigation.

Some aspects of the worksheet may require guidance from the teacher. Include teacher-led segments where necessary for your class.

On the Move Worksheet

Part A: Comparing y = x² and y = ax² A review of results of Exploration I in Activity 2.3.

Look over the graphs you sketched in Exploration I. Fill in the blanks, using the key words compressed, down, up, stretched, to complete the following statements:

In comparing the graphs of y = x² and y = ax²:

(I) If a > 1: the graph of y = x² is: ___________ vertically by a factor of a.

(II) If 0 < a < 1: the graph of y = x² is: ___________ vertically by a factor of a.

(III) If a < 0: the graph of y = x² opens: ___________ .

(IV) If a > 0: the graph of y = x² opens: ___________ .

(IV) If a < -1: the graph of y = x² is: ___________ vertically. This graph would open: ___________ .

(V) If -1 < a < 0: the graph of y = x² is: ___________ vertically and would open: ___________ .

Part B: Comparing y = x² and y = x² + k A review of results of Exploration II in Activity 2.3.

Look over the graphs you sketched in Exploration II. Fill in the blanks, using the key terms: translated k units up; translated k units down; vertical, to complete the following statements:

(I) Compared with the graph of y = x², the graph of y = x² + k is: ___________.

(II) Compared with the graph of y = x². the graph of y = x² - k is: ___________.

(III) The effect of adding or subtracting k from the equation y = x² results in a vertical (up or down) shift in the graph of y = x². This effect is called a(n): ___________ translation.

Part C: Comparing the graphs of y = x² and y = (x ± h)²

Using your graphing calculators (or Zap-a-Graph) construct the following graphs and sketch the resulting graphs on the axes provided. Choose a different colour for each graph for easier analysis:

(i) y = x²

(ii) y = (x - 5)²

(iii) y = (x + 5)²

(iv) y = x² + ½

(v) y = x² - ½

(a) What is the effect of adding a constant (h) on the graph of y = x²?

(b) What is the effect of subtracting a constant (h) on the graph of y = x²?

(c) Without using graphing technology, sketch what you think the graph of: y = (x + 6)² and y = (x - 3)² look like. (Verify your hypothesis using graphing technology).

Part C Summary:

Fill in the blanks, using the key words: left, right, horizontal, to complete the following statements:

(I) For y = (x - h)², the graph of y = x² moves h units: ___________ .

(II) For y = (x + h)², the graph of y = x² moves h units: ___________ .

(III) The equation y = (x ± h)² results in a horizontal (left or right) shift in the graph of y = x². This effect is called a(n): ___________ translation.

Part D: Putting it all Together:

Consider the following:

(i) y₁ = 2x² + 3 (iii) y₁ = 3x² + 4 (v) y₁ = 5x² -1

y₂ = -2x² +3 y₂ = 3x² – 4 y₂ = 1/5x² +1

(ii) y₁ = 3(x + 2)² (iv) y₁ = 2(x + 3)² (vi) y₁ = 3(x + 1)²

y₂ = 3(x - 1)² y₂ = -2(x - 3)² y₂ = 1/3(x + 1)²

For each case above, describe how the graph of y₂ will differ from the graph of y₁. Use the key words and terms you used in Parts A, B, and C wherever possible. Give reasons for your descriptions. Verify your answer using graphing technology.

Consider the following: y = (x - 2)² - 4

(i) Graph this on your calculator and describe how this graph differs from the graph of y = x².

(ii) Examine the graph y = (x - 2)² - 4 on your calculator. Point (0, 0) on the graph y = x² moved to
(2, -4) on the graph y = (x - 2)² - 4. Where did the point (1, 1) move to? Explain your answer.

Part E: Identifying the Transformation:

1. For each of the following graphs, describe which factors (a, h, and k) were used to produce the transformation (from y = x²) shown:

2. Based on your answers to Question 1, write an equation for each graph in the form y = a(x - h)² + k.

Follow-up Skills: 75 minutes

· Further paper and pencil practice to identify the roles of a, h, and k in transformations; recognizing specific types of transformations.

Assessment/Evaluation Techniques

· Assessment in the Learning Skills areas of Independence and Initiative is possible as students work on the activity.

· Student worksheets may evaluated using a written work rubric, if desired.

· After paper and pencil practise, a quiz on key concepts of the section is recommended.

Other Assessment Methods:

1. Journal: Summarize the role of a, h, and k in the shape and position of a parabolic curve.

2. Complete the following assignment: Patterning with Parabolas: the process and product can be evaluated using a rubric.

Assignment: Patterning with Parabolas

Time: 75 minutes

Teacher Facilitation: Share the following design with the class and lead a discussion about how the graph may have been created (discuss how original curve may have been transformed to produce pattern shown):

Student Activity

Students will create their own parabolic designs by developing an equation in the form y = a(x - h)² + k; and applying various transformations to it, using graphing software and/or graphing calculators.

Student Worksheet

1. Develop an equation in the form: y = a(x - h)² + k

equation:

2. Make a design by repeatedly changing the values of a, h, or k (e.g., If you started with
y = 2(x - 3)² + 5, you could graph y = 2(x - 3)² + 4 or y = 2(x - 4)² + 5, or y = 3(x - 3)² + 5)

3. Graph the original equation and the transformations on the same set of axes using a graphing calculator or graphing software. Sketch your pattern on a sheet of graph paper.

4. Share your pattern with a classmate. Indicate to your classmate which graph was the original equation. Have your classmate try to determine the transformations that produced your pattern.

Teacher Facilitation: You may alter Question 2 to meet the needs of your class (e.g., Ask the class to follow a particular set of instructions: Construct a parabola. Make a design by repeatedly moving your parabola 2 units to the right and 1 unit up; or make a design by flipping your parabola upside down; move it 2 units to the right; repeat; etc.)

Assessment/Evaluation Techniques

This worksheet may be collected and evaluated using a written report rubric; attention should be given to creativity displayed in the pattern produced and to the understanding of the role of transformations in producing the pattern.

Resources

Internet site
http://forum.swarthmore.edu/workshops/sum98/participants/sinclair/sample.htm
(Investigating Functions Using Spreadsheets – allows students to make connections between equations, data ranges-differences, and graphs simultaneously)

Appendices

Appendix A – Learning Skills Rubric

Appendix D – Written Report Rubric

Activity 2.5: Experiments with the CBR

Time: 150 minutes

Description

This activity will provide students with the opportunity to use the CBR, collect experimental data, and illustrate the data graphically using a graphing calculator. The students will be introduced to another form of the quadratic equation y = a(x - h)² + k as the curve of best fit. They will observe the effect a has on the graph in an effort to find the equation which best represents their curve.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems.

Strand(s): Quadratic Functions

Overall Expectations:

QFV.02P - determine, through investigation, the relationships between the graphs and the equations of quadratic functions;

QFV.03P - solve problems by interpreting graphs of quadratic functions.

Specific Expectations

QF2.01P - sketch graphs derived from descriptions of realistic situations;

QF2.02P - identify the effect of simple transformations (reflection and stretch) using calculators or graphing software;

QF2.04P - expand and simplify an equation of the form y = a(x - h)² + k to obtain the form
y = ax² + bx + c;

QF3.02P - determine the zero and maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software.

Planning Notes

The teacher will need

· different types of bouncing balls

· CBR units, TI-83 Graphing calculators, calculator to CBR linking cable

· a board to be use as a ramp (about 2 metres long)

· different types of carts (large enough for the CBR to sense)

· books to place under one end of the ramp

Prior Knowledge Required

Relationships

· collect data, using appropriate equipment and/or technology

· organize and analyse data, using appropriate techniques and technology

Teaching/Learning Strategies

Teacher Facilitation: It is best to have students work in groups of three. Each group needs one person to give directions and record the data, another to work with the CBR and calculator and the third person to get the equipment needed and do the actual experiment. Have students read through the experiment first, discuss what role each will take, and make an equipment list, before allowing the activity to proceed. The teacher will check the equipment list before allowing the students to proceed with the experiment. Warn students to read carefully and follow the instructions in order!

Student Activity

Part A: What Goes Up Must Come Down

Follow and complete the handout

Instructions READ CAREFULLY and DO IN ORDER!

1. Connect the CBR to the calculator with the linking cable.

2. Turn on the calculator, press apps, select ranger program, press enter.

3. You will see prgmRANGER on the screen, press enter. Follow the instructions on the calculator until you see the MAIN MENU.

4. In the MAIN MENU, select 3: APPLICATIONS, press enter.

5. In the UNITS menu, select 1: METERS, press enter.

6. In the APPLICATIONS menu, select 3: BALL BOUNCE, press enter and follow the instructions on the calculator.

7. Drop the ball at the same time you press the trigger button on the CBR. When the CBR stops recording, wait for the calculator to transfer the data to the graph.

8. Your graph should show at least four good bounces. If you are not pleased with your graph repeat the experiment by pressing enter and selecting 5: REPEAT SAMPLE. Try again!

9. When you are pleased with your ball bounces, sketch a distance-time relation. Include units.

Collecting Data

1. You are going to select one parabola and eventually transfer this curve onto another set of axes for closer examination. Select the TRACE button on the calculator. Use the right cursor key to move the tracer to the bottom left point of the parabola. Record the co-ordinates of this point in the chart below. Continue using the right cursor key to find the maximum point (called the VERTEX). Record the co-ordinates of this point in the chart. Again using the cursor key, locate the bottom right point and record the co-ordinates of this point.

	Time (X)	Distance (Y)
Bottom Left Point	(Xmin)
Maximum Point (Vertex)		(Ymax)
Bottom Right Point	(Xmax)

Note: Ymin must be chosen as the lowest value in the Y column

2. Press enter. On the PLOT menu, select 7:QUIT to exit out of the ranger program.

3. Press WINDOW key on the calculator. Input your data from the chart in #1. Use the cursor down key (not the enter key) to input the information!

Xmin = __________

Xmax = __________

Xscl = 1

Ymin = ___________

Ymax = ___________

Yscl = 1

Xres = 1

When you are finished, press GRAPH key.

Sketch a distance-time relation. Don’t forget the units on your grid.

Questions

1. Another form of the quadratic equation is y = a(x - h)² + k. This form is called the vertex form of a quadratic. h is the x-coordinate of the vertex and k is the y-coordinate of the vertex. Using your vertex and a = -1, enter the equation into Y=. Press Graph. Does this curve fit your parabola? If not, change the value for a until you have a good fit.

2. What is the quadratic equation which best describes your parabola?

3. You will find the equation for the second parabola by returning to the ranger program. Will this equation be the same as the first parabola? Explain why or why not.

i) Press PRGM, select RANGER, press enter twice, follow the instructions on the calculator until you get to the MAIN MENU, select 4:PLOT menu, select 1:DIST-TIME.

ii) Repeat the instructions in Collecting Data.

	Time (X)	Distance (Y)
Bottom right point	(Xmin)
Maximum Point (Vertex)		(Ymax)
Bottom left point	(Xmax)

Note: Ymin must be chosen as the lowest value in the Y column.

Input the following data in WINDOW

Xmin = __________

Xmax = __________

Xscl = 1

Ymin = __________

Ymax = __________

Yscl = 1

Xres = 1

The equation of the second parabola [in the form y = a(x - h)² + k]:______________________

4. Repeat for the third parabola.

	Time (X)	Distance (Y)
Bottom left point	(Xmin)
Maximum Point (Vertex)		(Ymax)
Bottom right point	(Xmax)

Note: Ymin must be chosen as the lowest value in the Y column.

The equation of the third parabola [in the form y = a(x - h)² + k]:

5. How long did it take to complete the first bounce? Second bounce? Third bounce?

6. How high was the first bounce? Second bounce? Third bounce?

PART B: RAMP CAR EXPERIMENT

Instructions

1. Place about three textbooks under the one end of the ramp.

2. Place the cart at the bottom of the ramp and do a few practice runs with the cart. Give the cart just enough push so it can reach the top of the ramp and come back down.

3. Place the CBR at the bottom of the ramp. Connect the CBR to the calculator using a linking cable. Run the Ranger Program on the calculator until you get to the MAIN MENU, select 1:SETUP/SAMPLE, press enter.

4. Use the cursor keys and the enter key to change the information to match the illustration.

MAIN MENU

çSTART NOW

REALTIME:

TIME (S):

DISPLAY:

BEGIN ON:

SMOOTHING:

UNITS:

DIST

[ENTER]

LIGHT

METERS

When you have finished changing the screen, move the cursor to START NOW. Press enter. Read the instructions on the screen.

5. Give the cart a push at the same time as you press enter. Be careful. Do not allow the cart to hit the CBR. Wait for the information to show up on the screen.

6. If you are not pleased with the graph, repeat the process by pressing enter, select 5: REPEAT SAMPLE from the PLOT MENU.

7. You want a well-shaped parabola! If there is extra data on the curve you will remove it by pressing enter. On the PLOT MENU, select 4: PLOT TOOLS and then select 1:SELECT DOMAIN.

8. The calculator asks for “LEFT BOUND?”. Move the cursor to the lowest left hand point on the parabola, press enter. A vertical line will appear on the screen. Now the calculator asks for “RIGHT BOUND?”. Move the cursor to the lowest right hand point on the parabola, press enter. The calculator will now re-adjust the screen to accommodate the parabola you have chosen. Repeat Steps 7 and 8 if you are not pleased with the curve.

9. Sketch this on a Distance-Time grid.

Collecting Data

1. Use the cursor key to find the vertex of the parabola and the minimum points on your parabola. Record this information in the chart below.

	Time (X)	Distance (Y)
Bottom left point	(Xmin)
Maximum Point (Vertex)		(Ymax)
Bottom right point	(Xmax)

Note: Ymin must be taken as the lowest value in the Y column.

2. Press enter. On the PLOT MENU, select 7:QUIT to exit out of the ranger program.

3. Press WINDOW key on the calculator. Input your data from the chart above. Use the cursor down key (not the enter key) to input the information!

Xmin=__________

Xmax=__________

Xscl= 1

Ymin=__________

Ymax=__________

Yscl= 1

Xres= 1

When you are finished, press the GRAPH key.

Sketch the Distance-Time relation.

Questions

1. Using the vertex form of a quadratic [y = a(x - h)² + k], input this equation into the calculator.

a is -1

h is ________________________________ (x-coordinate of the vertex)

k is________________________________ (y-coordinate of the vertex)

Press Graph. Does this curve fit your parabola? If not, change the value for a until you have a good fit.

2. What is the quadratic equation which best describes your parabola?

3. a) How long did it take the cart to go up the ramp and finish at the same place it started? How would you find this on the graph?

b) What is the furthest distance of the cart from the CBR? How would you find this on the graph?

4. a) If you placed the CBR at the top of the ramp instead of the bottom and repeated the experiment, would the graph look the same or different? Explain your answer.

b) Sketch the graph.

5. Repeat the experiment by placing the CBR at the top of the ramp. Record the information in the chart

The equation in the form y = a(x - h)² + k is: _______________________________________

Sketch the relation.

Was your predicted curve correct? If not, what error(s) did you make in your initial hypothesis?

6. Explain the role of a in the equation y = a(x - h)² + k;

if a is a negative number:

if a is a positive number:

7. Given y = 2x² + 5x + 7:

(a) Graph the equation.

(b) Find the vertex from the graph.

(d) Try different a values to make the new form of the equation fit the graph in part (a).

Follow-up Skills: 30 minutes

· QF2.04P – expand and simplify an equation of the form y = a(x - h)² + k to obtain the form
y = ax² + bx + c

· Use the equations found in Activity 2.5: graph them, expand them, graph again to check.

Assessment/Evaluation Techniques

Through observation, make anecdotal comments on teamwork, independent work, organization skills, work habits, communication and initiative (see Rubric provided in Appendix). Students will submit their lab report for evaluation of knowledge/understanding; thinking/inquiry/problem-solving and communication using the written report rubric (see Appendix D).

Resources

Modeling Motion: High School Math Activities With The CBR. Texas Instruments Incorporated, 1997.

Real-World Math with the CBL System. Texas Instruments Incorporated, 1999.

Math and Science in Motion. Texas Instruments Incorporated, 1997.

Appendices

Appendix A – Learning Skills Rubric

Appendix B – Observational Rubric

Appendix D – Written Report Rubric

Activity 2.6: If the Price is Right

Time: 150 minutes

Description

In this activity, students will develop an algebraic model for the relationship between revenue and ticket price for a school fundraiser. Fixed costs will then be added into the model and the price which maximizes profit will be found graphically. The break-even points will also be found. Finally, the effect of adding a variable cost will be explored as an extension.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Quadratic Functions

Overall Expectations

QFV.01P - manipulate algebraic expressions as they relate to quadratic functions;

QFV.03P - solve problems by interpreting graphs of quadratic functions.

Specific Expectations

QF1.03P - describe intervals on quadratic functions.

QF1.04P - factor polynomials by determining the common factor

QF1.05P - factor trinomials of the form x² + bx + c, differences of squares

QF3.02P - determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software.

Planning Notes

· Graphs are to be completed using graphing calculators.

Prior Knowledge Required

· Analytic Geometry – graph lines using graphing calculators or graphing software

Teaching/Learning Strategies

Teacher Facilitation: Allow students time to consider and discuss the question and hypothesize possible answers before they get into any algebraic manipulation.

If computer labs are available, this activity could be completed using spreadsheets.

When using the graphing calculators, ensure that appropriate window settings are established.

Student Activity

The student council at your high school has decided to hold a fundraising semi-formal dance. The proceeds from the dance will be donated to the local food bank. The council has decided to consider three factors in an attempt to maximize the funds raised at this event:

(1) the number of tickets sold

(2) the costs of holding the dance

(3) the price of the tickets

The costs associated with holding this dance are:

(i) student D.J.: $215.00

(ii) promotion: $32.00

(iii) decorations: $83.00

(iv) security: $70.00

The council decided to conduct a student survey to help them determine the price of the tickets. The survey asked students to indicate the ticket prices that they would be willing to pay.

Ticket Price (dollars) X	Number of Students Willing to Pay Y	Revenue from Ticket Sales (X)(Y)
15	255
20	202
25	147
30	98

Part 1

(A) What is your hypothesis regarding the best ticket price? Explain.

(B) Calculate the revenue from ticket sales for each ticket price. Which appears to be the best ticket price?

(C) Graph the number of students willing to pay vs. ticket price from the above survey on a piece of graph paper. Construct a line of best fit.

(D) Determine the equation of your line of best fit in the form y = mx + b (determine the slope and y-intercept and write the equation).

(E) What relationship is described by the equation in D?

(F) Use your graphing calculator to obtain a scatter plot of the data from the table. Use the Y= key to input your equation from D. How well does your equation fit the data?

Part 2

(A) How will the student council calculate the profit from this dance?

(B) Complete the chart to find the expression for revenue (Hint: use the expression found in Part 1).

Ticket Price (dollars)

Number of Students Willing to Pay

Revenue from Ticket Sales (X)(Y)

Revenue = Ticket price ´ Number of Students

= X ´ Y

= x ´ ( )

= x( )

(D) What are the total costs for holding this fundraiser?

(E) Complete the following to obtain an algebraic expression for the profit in terms of revenue and costs. Use the expression for Revenue from (C) and your answer from (D) for the costs.

Profit = Revenue - Costs

= x( ) -

(F) Using your expression from (E) to calculate profit, complete the following table:

Price ($)	Profit ($)
5
10
15
20
25
30
35
40
45
50

(G) Use your graphing calculator (or graph paper and pencil) to graph the relationship from the table.

(H) Describe the trends in profits as the ticket price increases. Over what ticket price interval will the student council make a profit?

(I) Where does the curve cross the x-axis? What does this mean in the context of this problem?

(J) What ticket price will maximize profits?

Part 3

Prepare a report for the student council with your recommendation for the optimal ticket price. Include the charts and graphs in your report.

Extension: Investigate the effect, if any, of the addition of catering (at $15 per person) on the optimal ticket price.

Follow-up Skills: 345 minutes

· more practice with max/min problems including area problems and packaging (box) problems and problems concerning the height of a ball (or projectile) over time

· QF1.04P - factor polynomials by determining the common factor

· QF1.05P - factor trinomials of the form x² + bx + c, differences of squares

Assessment/Evaluation Techniques

· Knowledge/understanding, application and communication skills will be assessed in the report.

· Learning skills (independence and initiative) may be assessed as students work on the activity.

Resources

The problem is an adaptation of an example in “Algebra in a Technological World” NCTM Addenda Series.

Appendices

Appendix A – Learning Skills Rubric

Appendix D – Written Report Rubric

Activity 2.7: The Zero Factor

Time: 75 minutes

Description

Students will graph quadratic equations by factoring to determine zeros. They will then locate the x-coordinate of the vertex midway between the zeros, and calculate the y coordinate of the vertex by substituting the x-coordinate into the equation of the quadratic and solving. This activity allows students to combine their factoring skills with their graphing skills.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is:

- a reflective and creative thinker who evaluates situations and solves problems;

- a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills.

Overall Expectations

QFV.01P - manipulate algebraic expressions as they relate to quadratic functions

QFV.03P - solve problems by interpreting graphs of quadratic functions

Specific Expectations

QF1.04P - factor polynomials by determining a common factor

QF1.05P - factor trinomials of the form x² + bx + c;

QF1.06P - factor the difference of squares

QF1.07P - solve quadratic equations by factoring

QF3.03P - solving problems involving a given quadratic function by interpreting its graph

Planning Notes

· Graphing calculators will be needed. The use of the calculator overhead display unit will enhance the visual part of the lesson and aid in the discussion about determining coordinates of the vertex.

Prior Knowledge Required

Number Sense and Algebra

· substitute into and evaluate algebraic expressions involving exponents

· solve first-degree equations

Teaching/Learning Strategies

Teacher Facilitation: The teacher will lead the class through an analysis of quadratics graphed using the graphing calculator. A full class discussion of each graph should lead to finding an algebraic method of graphing the parabola.

Student Activity

Using the graphing calculator, graph y = 3x² - 6x using a graphing window with the dimensions -2 £ x £ 3 and -3 £ y £ 5. Examine the graph to determine the x-intercepts, and the coordinates of the vertex.

Teacher Facilitation: Lead the students to determine the x-intercepts by common factoring the binomial and solving 0 = 3x(x - 2). Examine the line segment joining the two x-intercepts (2, 0) and (0, 0). The students should see that the vertex occurs midway between these two points (the x-coordinate of the vertex is x = 1). Substituting this x value into the equation y = 3x² - 6x, the y-coordinate is calculated as -3, giving the vertex as (1, -3). Some further discussion: this graph passes through the origin. Will all quadratics of the form y = ax² + bx pass through the origin? (Yes) Consider other equations of this form before students can answer this question.

Student Activity

Graph y = x² - 25. Set window -6 £ x £ 6, -25 £ y £ 10, scale 5. Examine the graph to determine the x-intercepts and the coordinates of the vertex.

Teacher Facilitation: Lead the students to determine the x-intercepts by factoring the difference of squares and solving 0 = (x - 5)(x + 5). Examining the line joining the x-intercepts (5, 0) and (-5, 0) the students should see that the vertex occurs midway between the x-intercepts (i.e., where x = 0). Thus, the x coordinate of the vertex will be 0. Substituting x = 0 into the equation y = x² - 25 the y coordinate is calculated as -5, giving the vertex point as (0, -5). Some further discussion: This graph has its line of symmetry on the y axis. Will all quadratics of the form y = ax² + c have this property? (Yes) Discuss why.

Student Activity

Graph y = x² + 2x - 8. Examine the graph to determine the x-intercepts and the coordinates of the vertex by examining the graph. Examine the equation to determine an algebraic method to find each of the values.

Teacher Facilitation: Lead the students to determine the x-intercepts by factoring the trinomial and solving 0 = (x + 4)(x - 2). Examining the line joining the x-intercepts (-4, 0) and (2, 0) the students should see that the vertex occurs midway between the x-intercepts (where x = -1). Thus, the x-coordinate of the vertex will be -1. Substituting x = -1 into the equation y = x² + 2x - 8 the y-coordinate is calculated as -9, giving the vertex point as (0, -9). Some further discussion: This graph has two x-intercepts. Will all quadratics of the form y = x² + bx + c have two x-intercepts? (No) Discuss. Will all quadratics of the form y = x² + bx + c factor? (No)

Student Activity

Practice graphing quadratics that can be factored as in the examples studied today.

Follow-up Skills: 150 minutes

· QF1.05P – factoring trinomials in form x² + bx + c

· QF1.06P – factoring a difference of squares

· QF1.07P – solving quadratic equations by factoring

· QF3.03P - solving problems involving a given quadratic function by interpreting its graph

Assessment/Evaluation Techniques

Assess factoring skills informally as students work on the activity. Assess knowledge and understanding using paper and pencil tasks involving graphing without the use of a graphing calculator. Assess communication skills as students write about graphing without a graphing tool to aid them.

Appendices

Appendix B – Observational Rubric

Activity 2.8: Summative Assessment Activity: Newton’s Apple

Time: 225 minutes

Description

This activity engages students in an exploration of an object’s motion under the force of gravity. Students begin by finding a quadratic rule that fits given height and time data for an apple falling from rest, using graphing calculators or software. They will then obtain height and time values by extrapolation/interpolation using this rule. Students next consider the motion of an apple thrown upwards with given initial velocity and again use graphs to obtain answers to questions about heights, times and intervals of time. Finally the problem is extended to a consideration of the simultaneous motion under gravity of two different “objects”. Students are then asked to create a scenario of their own for presentation to the class.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads, understands, and uses written materials effectively a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a self-directed, responsible, life long learner who demonstrates flexibility and adaptability.

Strand(s): Quadratic Functions

Overall Expectations

QFV.01P - manipulate algebraic expressions as they relate to quadratic functions;

QFV.02P - determine through investigation the relationships between the graphs and the equations of the quadratic functions;

QFV.03P - solve problems by interpreting graphs of quadratic functions.

Specific Expectations

QF1.03P - describe intervals on quadratic functions;

QF2.01P - construct tables of values, sketch graphs to represent quadratic functions derived from descriptions of realistic situations;

QF3.01P - obtain the graphs of quadratic functions whose equations are given in the form y = a(x - h)² + k or the form y = ax² + bx + c, using graphing calculator of graphing software;

QF3.02P - determine the maximum value of a quadratic function from its graph, using graphing calculators or graphing software;

QF3.03P - solving problems involving a given quadratic function by interpreting its graph.

Planning Notes

· Students work in pairs on this activity

· Written submissions should be completed individually

· Students will need to use a viewscreen or overhead projection device for their presentations.

· Encourage students to attempt an animated presentation of their extension problem.

Prior Learning Required

Students will have completed Unit 2. It is essential that students have had the opportunity to work with equations representing the height of a ball over elapsed time, as suggested in the follow-up to Activity 2.6

Student Activity: Newton’s Apple

Part 1

A student standing on a bridge accidentally drops his apple off the bridge into a river below. The data below describes how far the apple falls over a 2 second time interval:

Time (s)	0	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	2	2.2
Height above water (m)	44	44	43	42	41	39	37	35	32	28	25	20

1. Use your graphing calculator to plot this data.

2. Using the Y= key and the rule y = ax² + b, enter different values for a and b to find the parabola which fits the data well. Write down the equation that fits best:

3. Use your graph to predict how far the apple will have fallen after 2.5 seconds:

4. How high was the bridge?

5. At what time will you hear the splash?

6. In the context of this question, what do the negative height values mean? Do you think the same relationship will hold in the negative range of values? Explain.

7. Suppose the bridge was 20 m high. What effect would this have on your graph? How long would it be before you heard the splash in this example?

8. How long would it take for an apple dropped from the CN Tower to reach the ground? (The CN Tower, in Toronto, is approximately 553 m high (~5.5 football fields.)) Do you think you would be allowed to drop an apple from the CN Tower? Explain your answer.

Part 2

Suppose now that as the student dropped the apple from the bridge, a row boat was passing under the bridge, and someone in the boat caught the apple and attempted to throw it back to the student.
If the apple is thrown upwards at 19 m/s from a point 1.5 m above the water the quadratic equation
h = -4.9t² + 19t + 1.5, (where h is the height, in metres, above the water, and t is the time, in seconds, from the time the apple is thrown), describes the position of the apple.

1. Graph this equation using a graphing calculator.

Use the graph to determine answers to the following questions

2. How long does it take for the apple to reach the height of the student’s outstretched arm at the level of the top of the bridge.

3. If the student fails to catch the apple, how long is it before he gets a second chance to catch the apple?

4. What is the maximum height reached by the apple? How long does it take for the apple to reach this height?

5. For what interval of time is the apple above the level of the bridge?

6. If the student misses the apple again, when does he hear the splash? How long is the apple in the air (after it is thrown upward)?

7. How would the equation of the apple’s motion change if it was thrown upwards at 25 m/s? 32 m/s? How would this affect the graph of the equation? Graph the new equations to check your answer.

8. Suppose the student had not dropped the apple from the bridge but had thrown it upwards from the edge of the bridge at 19 m/s. What equation would describe this motion? Use the graph of this equation to determine how high the apple would go? How long would it take for the apple to reach the water?

9. How would the equation of position change if the student threw the apple downwards at 19 m/s? What difference would this make to the time before the splash?

10. If you threw an apple downwards at 25 m/s from the CN Tower how much sooner would it reach the ground than if you had dropped it?

Part 3

A crow, flying at a height of 33 m above the river, sees the apple being thrown upward from the rowing boat, tucks in its wings and dives down to intercept the apple. Use the vertical motion model h = -4.9t² + vt + s, (where h is the height in metres, v is initial velocity in metres per second, and s is the initial height in metres), to investigate possible scenarios for the outcome (Assume that once in motion, both the crow and the apple are moving only under the influence of gravity). For example, consider cases where the crow intercepts the apple (a) on its way up, (b) at the maximum height (c) on the way down, as well as cases where the crow is not successful in the attempt.

Write a report which details the results of your investigations. Include copies of the graphs you use in these investigations.

Be prepared to present your investigations and the graphs showing the different scenarios to the class.

Create a problem involving vertical motion under gravity for the class to work on.

Extension

Move the problem to another planet. Have students investigate the effect, on the graphical models for the apple scenarios, of a change in the value of the acceleration due to gravity.

Assessment/Evaluation Techniques

Use appropriate rubrics in assessing the thinking/inquiry; application and communication skills of the written reports and oral presentations. Include peer and self-evaluation for presentations and to assess teamwork in working in pairs.

Resources

Explore Quadratic Functions with the TI-83. Bob Alexander Publications.

Graphic Algebra. Key Curriculum Press.

Algebra I: An Integrated Approach. Larson, Kanold, Stiff.

Appendices

Appendix C – Presentation Rubric

Appendix D – Written Report Rubric