Building Conceptual Understanding

1. Building Conceptual Understanding Through The Effective Use of Technology

2. Your Learning Partners for Today … Look at the numbered card you received. How many of the following properties does it possess? • prime • even • square • cube • triangular • Fibonacci • deficient Form a group with the people whose # possesses the same number of properties.

3. Outline for today

4. Setting the Stage • The strand in the Grade 10, Academic Mathematics Course that deals with quadratic relations has the following overall expectations: • determine the basic properties of quadratic relations; • relate transformations of the graph of to the algebraic representation ; • solve quadratic equations and interpret the solutions with respect to the corresponding relations; • solve problems involving quadratic relations. Do these expectations build on concepts from previous grades or units in this course? How will these expectations impact my students in future mathematics courses?

5. Organizing a Unit • Some questions I need to ask myself as I plan: • If I want to know whether my students have achieved these overall expectations, what questions should I ask them? • Which of the specific expectations, that relate to each overall expectation, are more important in helping my students achieve that overall expectation? • What lessons do I need to create in order to teach these expectations? • How do I organize these lessons in a way that supports my students making connections about the math they’re learning?

6. Prior Learnings • My students have: • explored the graphs of quadratic relations in standard form using technology • are able to identify key aspects of a parabola, such as: • the axis of symmetry • the zeros of the parabola • the coordinates of the vertex • Investigated the effects of changing the values of parameters a, b, and c, in the standard form of a quadratic equation • factored quadratic expressions • sketched the graph of a quadratic relation in factored form

7. What’s My Lesson Goal? What do I want my students to have accomplished by the end of this lesson? • I want my students to: • understand the connection between the location of the vertex of a parabola and the equation of the quadratic relation in vertex form • recognize that mathematics plays a role in art & design (spatial intelligence)

8. Expectations and Lesson Goals What specific curriculum expectations do I want to address? • identify, through investigation using technology, the effect on the graph of of transformations … • explain the roles of a, h, and k in , using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of symmetry;

9. Will My Students get it? What question(s) should I ask at the end of the lesson that will inform me about whether my students understood the lesson goal? I might want to ask: For the parabola defined by , how does its shape compare to the parabola defined by ? What are the coordinates of the vertex? What is the axis of symmetry? Or Write the equation of the parabola pictured below, and describe how to draw it.

10. The Lesson Organizer

11. Organizing My Thoughts Lesson Title: Picturing Parabolas Course: 10, Academic

12. The Lesson Organizer

13. Starting with the end in mind …

14. The Lesson Organizer

15. Mind’s On Where’s the math?

16. Our Picture … Deconstructed

17. Action Investigate the roles of h and k in the graphs of and . In your groups, carry out the investigation, using the graphing calculator, by following the instructions provided. Calculator Rep. Timer Recorder Presenter

18. Consolidation In your math journal, write a reflection on today’s investigation, as you consider the following questions: Describe how changing the value of k, in the equation y = x2 + k, affects: i) the graph of y = x2 ii) the coordinates of each point on the parabola iii) the parabola’s vertex and axis of symmetry Describe how changing the value of h, in the equation y = (x – h)2 , affects: i) the graph of y = x2 ii) the coordinates of each point on the parabola iii) the parabola’s vertex and axis of symmetry For a parabola of the form y = (x – h)2 + k, describe the process you would use to sketch its graph, if you begin by drawing a graph of y = x2 .

19. Further Consolidation … Where’s the Homework? Create a graphic design, based on a set of parabolas that you enter into your graphing calculator. Transfer your picture from the graphing calculator to your PC, and using a “paint” program, give it some colour! Print a copy of your finished design, and submit it to your instructor along with the list of equations that you used to create your design.

20.  (h, k) x = h How Else Could You Teach These Concepts? • Teacher introduces the vertex form of a quadratic relation as the equation: • y = (x – h)2 + k, where the vertex is the point (h, k). y = x2 y = (x – h)2 + k • Teacher demonstrates several examples. • Teacher assigns practice questions. • Teacher takes up practice questions and summarizes.

21. Which Approach Would You Take?Why? • Who’s doing the thinking? • What mathematical processes are students using? • Which approach is more likely to engage more of your students?