## MAGIC RECTANGLE [DAY 2]

### OPTIMIZATION: AREA OF A RECTANGLE

Invistigate and determine the maximum area of a rectangle with constraints on the dinemsion.

#### Intentionality

#### Math Talk

#### Visual Math Talk Prompt #1

#### Visual Math Talk Prompt #2

#### Visual Math Talk Prompt #3

#### Visual Math Talk Prompt #4

#### Visual Math Talk Prompt #5

#### Visual Math Talk Prompt #6

#### Visual Math Talk Prompt #7

#### Visual Math Talk Prompt #8

#### Purposeful Practice

#### Resources & Downloads

#### Educator Discussion Area

## Intentionality & Unit Overview

### Length of Unit: 2 Days

### Access each lesson from this unit using the navigation links below

Students will continue building flexibility with constructing different algebraic forms of quadratic relations. More specifically students will use the method of completing the square to convert the standard form of a quadratic relation to the vertex form. .

## Intentionality…

The purpose of the Day 2 activities is to reinforce key concepts from Day 1. Students will engage in a math talk and will have an opportunity to complete independent purposeful practice. The math talk and purposeful practice serve to develop a deeper understanding of the following big ideas.

- A quadratic function can have multiple algebraic forms;
- Algebraic forms of a quadratic function can be manipulated using algebra;
- Area models can be helpful when manipulating algebraic forms of quadratic functions.

## Math Talk

Present the following series of **Math Talk Prompts** to students involving quadratic relations in standard form.

Students will be asked to manipulate algebra tiles to form squares and then write the vertex form a quadratic relation.

Some questions students will encounter and you should consider asking include:

- What defines a square?
- How can you ensure the dimensions of the rectangle are the same?
- What do you need to complete the square?

You may want to consider printing out all prompts so that groups of students can progress through each prompt at their own pace. Place an answer sheet somewhere in the room so they can check their answers along the way.

If you choose this set up then you’ll want to circulate around the room and monitor groups of students and be prepared to select and sequence student solutions so that you can point out “nice moves” to the whole group.

## Visual Math Talk Prompt #1

Show students the following math talk prompt.

You’ll notice that a series of four (4) buildings appear and students are asked:

**Create a square.****Write the algebraic expression that can be modelled using these tiles in standard form and vertex form.**

Students may first make rectangles instead of squares. Prompt these students with: What defines a square? How can we make the side lengths the same length?

Students will become comfortable with the idea of splitting the x-terms into two groups and placing each group along side the x squared tile. This partitioning will help students link the completing the square algorithm with this visual model.

Try to limit any mention of an algorithm at this point. Let students develop their own visual shortcuts and algebraic expression shortcuts.

If students struggle to write the vertex form encourage them to first list the dimensions of the square. For example, they could notice that the square has x + 2 as its length and its width. “How could we write an expression for its area?”

## Visual Math Talk Prompt #2

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

**Create a square.****Write the algebraic expression that can be modelled using these tiles in standard form and vertex form.**

Use the helpful tips and suggestions from Prompt #1 to help students with this prompt. This prompt is designed to help students practice the techniques and ideas formed in prompt #1.

## Visual Math Talk Prompt #3

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

**Create a square.****Write the algebraic expression that can be modelled using these tiles in standard form and vertex form.**

Use the helpful tips and suggestions from Prompt #1 to help students with this prompt. This prompt is designed to help students practice the techniques and ideas formed in prompt #1.

However, this prompt is slightly different as it will require students to use the zero principle to help complete the square.

Help students understand that we will need to bring in extra tiles to create the square but will need to bring in an equal number of opposite tiles so that this new expression is equivalent to the original expression.

## Visual Math Talk Prompt #4

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Then ask students:

**Create a square.**

This prompt introduces negative x-terms into the expression. Encourage students to form the square using the same techniques as in previous prompts, however ask them how the length of the square is affected? What are the dimensions of the square? Students should see that the length and width of the square should now be x – 4.

## Visual Math Talk Prompt #5

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## Visual Math Talk Prompt #6

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## Visual Math Talk Prompt #7

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## Visual Math Talk Prompt #8

## Purposeful Practice

## While Students Are Practicing…

## Questions: Multiplication

## Resources & Downloads

## Educator Discussion Area

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