Building Conceptual UnderstandingThrough The Effective Use of Technology

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.

Outline for today

1. Starting to Plan: Expectations 2. Prior Learning3. What’s My Goal?4. How to Plan … the PPQT4. Minds’ On5. Action: An investigation using

technology6. Consolidation: Reflecting and Practising7. Other approaches to this lesson8. Final thoughts

Setting the StageThe 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?

Organizing a UnitSome 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?

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

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)

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;

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?

OrWrite the equation of the parabola pictured below, and describe how to draw it.

The Lesson Organizer

Organizing My Thoughts

Goal(s) for a Specific LessonStudents will explore transformations of quadratic relations, using technology, and use what they’ve learned to create a graphic design. Curriculum Expectations• 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;

Big Idea(s) Addressed by the Expectations

Lesson Title: Picturing Parabolas Course: 10, Academic

The Lesson Organizer

Starting with the end in mind …Consolidate/DebriefFor 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?

Write the equation of the parabola pictured below, and describe how to draw it.

The Lesson Organizer

Mind’s On

Where’s the math?

Our Picture … Deconstructed

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

RecorderPresenter

ConsolidationIn 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 .

Further Consolidation …

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.

Where’s the Homework?

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).

(h, k)

x = h

y = (x – h)2 + ky = x2

• Teacher demonstrates several examples.• Teacher assigns practice questions.• Teacher takes up practice questions and summarizes.

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?