Alg1 Lesson 3.2: Area Models for Quadratic Functions

Lesson 3.2: Area Models for

Quadratic Functions

Obj: 6B, 6C, 7A, 8B: I can convert a given form of a
quadratic function into another form of the quadratic function.

EQ: How do I graph quadratic functions?

How can I formulate quadratic functions?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Poll

What do you plan to do most today?

• Write Notes in Journal

• Write notes in document then print it

• Ask and answer verbal questions

• Origami

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Warming Up with

Triangular Numbers

Triangular Numbers

How many dots in each stage?

How do you build the next triangle?

What kind of sequence is this?

Is 55 a triangular number?

Drag and Drop

How many dots in each stage? 1, 3, 6, 10, ​

Placeholder

, 21, …

How do you build the next triangle?

Add the # of the largest row plus 1.

What kind of sequence is this? ​

Placeholder

Is 55 a triangular number? ​

Placeholder

Drag these tiles and drop them in the correct blank above

Yes

No

Quadratic

Exponential

Linear

15

17

18

14

Multiple Choice

What is the explicit formula for this sequence?

• T(p) = 0.5p^2 + 0.5p

• T(p) = p^2 + p
• T(p) = 0.5p^2 - 0.5p

• T(p) = p^2 + 0.5p

• T(p) = p(p+1)(p+2)/6

Part 2: Exploring Area

How does the area of the square change
as the side length increases?

How does the perimeter of the square
change as the side length increases?

Multiple Choice

What is a function that can represent the perimeter of the square in terms of the
side length?

• P(n) = 6n^2

• P(n) = 4n
• P(n) = n^2
• P(n) = 8n
• P(n) = n^3

(n + 1)² = (n + 1)(n + 1) = n² + 2n + 1

An exponent of 2 means that the
factor has a multiplicity of 2, which
means, to multiply the term by
itself. Therefore, (n+1)² is the
same as (n+1)(n+1).

Completing the Square

4x(x + 3) = y Factored Form

4x² + 12x = y Standard Form

4x² + 12x + 9 = y + 9

4(x^2 + 3x + 9/4) = y + 9

4(2x + 3)(2x + 3) = y + 9

4(2x + 3)² = y + 9

4(2x + 3)² - 9 = y Vertex Form

Sequences

(1) The first differences are an arithmetic sequence, (2) the
second differences are constant, and (3) the formula for a
quadratic sequence will have a term with an exponent of 2.

Part 3: Maximizing Area

Garden Area Design Challenge

You want to fence in a rectangular region for your garden, and you have 16 sections of fencing, each 1 foot long. You want to make the area that you enclose as large as possible. What are the dimensions of the largest garden you can fence in, and what is the area? (Note: Assume that the dimensions must be whole numbers.)

All Combinations

What do you notice about these?

Observations

Drag and Drop

How can we calculate the length of the garden if we know the width of the garden? We subtract ​

Placeholder

the width from the ​

Placeholder

and then divide the result by ​

Placeholder

to get the length. The length has to be

L = ​

Placeholder

/2, or L = ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

twice

thrice

perimeter

area

2

4

6

(16-2w)

(8 - w)

(8 - 2w)

Multiple Choice

How do you figure out the area of a square garden using w and 8−w as the width and length? A(w) = w(8 - w)

• A(w) = w + (8 - w) is the area function.
• A(w) = w(8 - w) is the area function.
• A(w) = 8w is the area function.
• A(w) = w(w - 8) is the area function.

Graphing

Graph the area function,

A(w) = w(8 - w)

(where A(w) = y and w = x).

Part 4: Summary and

Practice

Key Terms (write in journal)

Parabola: the set of points, P, such that the distance from P to the focus is = to
the distance from P to the directrix.

Line of Symmetry: a line that divides a parabola into mirror images and passes
through the vertex (for quadratics, it will be in the x=h form).

Extrema: the maximum or minimum value of a quadratic.

Quadratic Function: A function f is quadratic if and only if: (1) a sequence of the
output values whose input values differ by a fixed amount, such as 1, have a
constant second difference, (2) the graph has the shape of a parabola, and (3) the algebraic expression for the function has the form f (x) = ax² + bx + c, where a, b, and c are constants, and a is not zero.

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.