The lesson teaches students why the vertex form is more useful than the standard form. Students will:
- use Algebra Tiles™ to complete the square or to convert from standard form to vertex form.
- be able to convert between standard form and vertex form.
- get real-world problems solved by using quadratic math.

- How do we know if a quadratic, polynomial, or exponential function should be used to describe a real-world event? 
- How can graphs and tables that show quadratic equations help us understand what's going on in the world?

- Completing the square: A technique for finding the roots of quadratic equations that uses the terms \((x+{b \over 2})^2\) + c - \(({b \over 2})^2\) to substitute for x² + bx + c, resulting in a purely quadratic equation with no linear term (y = x² + bx + c).
- Vertex Form: A form for a quadratic function; y = a(x - h)² + k, where the coordinates of the vertex are (h, k).
- Minimum: The point(s) with the least y-value on the graph of a function; on a parabola that opens upward, the minimum occurs only at the vertex.
- Maximum: The point(s) with the greatest y-value on the graph of a function; on a parabola that opens downward, the maximum occurs only at the vertex.

- a classroom set of Algebra Tiles™ as well as a magnetic set for the teacher’s whiteboard or use tile template provided (M-A2-2-2_Algebra Tiles Template)
- graph paper
- Group Word Problems worksheet (M-A2-2-2_Group Word Problems and M-A2-2-2_Group Word Problems KEY)
- Lesson 2 Exit Ticket and KEY (M-A2-2-2_Lesson 2 Exit Ticket and M-A2-2-2_Lesson 2 Exit Ticket KEY)

- Teachers should make sure that all students participate in group activities and that each student's answer to each part of the activity is thought-provoking. If any students make mistakes in learning or doing the activity, the teacher should correct them right away. 
- The Exit Ticket exercise (M-A2-2-2_Lesson 2 Exit Ticket and KEY) shows that students understand and can represent vertex form and translate between standard form and vertex form. 
 

Active Engagement, Modeling, Explicit Instruction
W: Understanding and being able to show how completing the square can help with both solving quadratic problems and seeing how the shape of a quadratic function's graph is related to its behavior will be the goal of this lesson.
H: In this lesson, students are given a geometric and visual picture of the quadratic function y = x² + 6x + 8. They are asked to take it apart and put it back together again as squares and rectangles. Algebra Tiles™ is the model for exploring and visualizing.
E: In Activity 1 with the Algebra Tiles™, students can look at the sizes of different quadratic functions that have a range of b and c terms and terms that can have both positive and negative values.
R: The paired tasks in Activity 5 give students the chance to make their own examples of quadratic functions, change them from vertex form to standard form, and then share their work with a partner. They can change and improve their understanding of quadratic functions by giving themselves tests and asking a friend to grade them.
E: The Lesson 2 Exit Ticket tests how well students understand how the a, b, and c terms of the quadratic function relate to each other, as well as the shape of the parabola, the location of the tip, and the sizes of the Algebra Tiles™.
T: When students are having trouble picturing how the Algebra Tiles show the properties of quadratic functions, use the form y = ax² until they get used to how the x² part is shown. More advanced students should think about and draw what the squares would look like if the term was a rational number, like \(3 \over 4\).
O: This lesson starts with a problem that we introduced to the students in the first class. Solving problems is fun for them. Students have been studying intensively with the vertex form, and they now understand why they needed to learn about it. This lesson will help students understand why the vertex form is useful and how it can be used in real life.

By the end of this lesson, students will know how to use the "completing-the-square" method to convert quadratic functions from standard form to vertex form. That's because not all quadratic functions are put in vertex form, even though students learned how useful it can be in earlier lessons. Students will understand that quadratic relationships are all around us and that it's important to know the vertex, especially when talking about minimizing costs or maximizing money. Note that the vertex is where the maximum and minimum values occur. The y-coordinate represents either the maximum or minimum value.

"Who likes putting puzzles together? In the first lesson, I asked you to find the vertex of the equation: y = x² + 6x + 8 . Let's put together a small puzzle to find the vertex."

Everyone in the class should get a set of Algebra Tiles™. The tiles will be on each student's desk while they work. You can also use the template in (M-A2-2-2_Algebra Tiles) worksheet.

“Algebra Tiles™ will be used to make a square. We are going to say that x² is represented by a square with side length x. We'll now say that x is represented by a rectangle with length x and width 1. Then we have a single unit with side length 1.”

“The quadratic equation tells us how many tiles to use in each part. We require 1 x², 6 x rectangles, and 8 single units. Let's count them out.

"Using the tiles we have now, our goal is to make a square. Put the x² square in the upper left corner to start."

"How then would we evenly divide the 6 x rectangles?" They should say 3 and 3.

"Line up the rectangles: three rectangles along the right side of the x² square and three rectangles below the x² square."

"Let's start putting our single-unit pieces in the bottom right corner of the square."

The two graphics show the same equation. The only thing that makes them different is that the one on the right has the outside dimensions labeled .

"What are the dimensions of this square we made?" Students need to find x + 3 by x + 3. They might also say that they couldn't make a full square.

"What is an easy way to express x + 3 by x + 3?" They need to say (x + 3)².

"Did we make a full square now? We're missing a unit in the corner, so (x + 3)² can't be the answer. How should the equation be changed?" The students should say "minus" or "subtract 1."

"Does anyone know what (x + 3)² − 1 is? How does the equation look? Where is the vertex?" The equation now has a vertex, which is at (-3, -1).

"So whenever we're missing a piece, we have to take it away. If we have y = x² + 6x + 11, what would the equation look like in vertex form?" Students should add 3 more units to their square. "As before, the equation in vertex form will be y = (x + 3)² + 2. The only difference is that three was added to the k value, making it −1 + 3 = 2."

The two graphics show the same equation; the only thing that makes them different is that the one on the right shows the outside dimensions labeled.

"It means we need to add that much more to the equation if we have too many pieces. Using "completing-the-square" is how we change an equation from standard form to vertex form."

Activity 1: Using Algebra Tiles™

Students can use the Algebra Tiles™ to work out the following solutions that you write on the board. They need to use their tiles to figure out the vertex and the vertex form of the parabola. Tell them to work on each one separately and not move on to the next one until you're satisfied with their work.


1. y = x² + 8x + 2 [y = (x + 4)² − 14; vertex (−4, −14)] 

2. y = x² + 2x + 3 [y = (x + 1)² + 2; vertex (−1, 2)]

3. y = x² + 10x + 12 [y = (x + 5)² − 13; vertex (−5, −13)]

4. y = x² + 4x + 11 [y = (x + 2)² + 7; vertex (−2, 7)]

5. y = x² − 6x + 4 [y = (x - 3)² − 5; vertex (3, −5)]


"Has anyone seen a pattern? What would happen if I had an odd number of x tiles? We might not always have the tiles on hand, so we might need to draw a picture. We'll be practicing this situation in the next activity."

Activity 2: Think-Pair-Share

"Use the formula to change the following standard form equations to vertex form with a partner." When you're done, check your work with another pair."

Basic form: y = ax² + bx + c

To change to vertex form, use this formula: y = \((x+{b \over 2})^2\) + c - \(b^2 \over 4\)


1. y = x² + 5x + 6 [y = (x + 2.5)² − .25; vertex (−2.5, −.25)]

Illustrate item 1 by showing steps to \((x+{5 \over 2})^2\) + 6 - \(5^2 \over 4\)

2. y = x² + 7x − 4 [y = (x + 3.5)² − 16.25; vertex (−3.5, −16.25)]

3. y = x² − 9x + 25 [y = (x - 4.5)² + 4.75; vertex (4.5, 4.75)]


Display the formula for students who don't like using the tiles. Write this method down in your students' notebooks. For example, look at other references in different books that talk about completing the square.

Activity 3: Group Work

Distribute the Group Word Problems worksheet (M-A2-2-2_Group Word Problems and KEY) to each group of four students. Note that the vertex is the location of the maximum and minimum. The y-coordinate represents the highest and lowest values.

Activity 4: Think-Pair-Share

"Can you tell me what we need to do to change an equation from vertex form to standard form? Come up with ideas with a partner for how to write the quadratic equation y = (x + 4)² – 5 in standard form." Teach them that it doesn't involve a square or a formula. They already have what they need to change this. First, have them think about it alone or with a classmate. Then, have them share their thoughts with the whole class.

“Use the order of operations. What does it mean to make something square? What does it mean to combine similar terms and simplify?"

Show students how simple it is to convert from vertex form to standard form.

(x + 4)² − 5 = (x + 4)(x + 4) − 5 = x² + 4x + 4x + 16 − 5 = x² + 8x + 11

Students are to solve the following equations in standard form:

1. y = (x - 3)² − 17 [y = x² − 6x − 8]

2. y = (x + 7)² − 40 [y = x² + 14x + 9]

3. y = (x + 8)² − 65 [y = x² + 16x − 1]

4. y = (x - 6)² − 1 [y = x² − 12x + 35]

Activity 5: Pair Work

"You will write two equations in standard form and two equations in vertex form on your own. Change your equations to the other form and save your results. Next, give your partner the four equations you wrote down. After that, you and your partner will change the numbers to the other form. Check your work against each other when you're both done."

Give the Lesson 2 Exit Ticket (M-A2-2-2_Lesson 2 Exit Ticket and KEY) to students to see how well they understand.

The following strategies will help you change the lesson to fit the needs of your students all year.

Routine: Groups and teamwork are used all the time so that students can help each other. Students should focus on explaining mathematical ideas using the right words for those ideas. To get the most out of the lesson and make useful notes, you need to be able to take accurate notes. 

"How do we finish the square if there is a number in front of x²? We can use the tiles or factoring."

Example: y = 2x² + 4x + 6

“Since 2 is the number in front of x², we will attempt to create 2 squares.”

"Because we have two squares with dimensions of x + 1 by x + 1, we may write it as 2(x + 1)², resulting in four single units. Therefore, y = 2(x + 1)² + 4 is the equation. The vertex is located at (−1, 4) and it has a stretch factor of 2."

"We also may have factored the equation to convert it from standard form to vertex form. Divide the integer in front of x² by its factor."

For instance, y = 2x² + 4x + 6.

y = 2(x² + 2x + 3)

"Use only the values in parenthesis (x² + 2x + 3) to complete the square."

(x + 1)² + 2

y = 2[(x + 1)² +2]

"Now distribute the 2 that we factored out."

2(x + 1)² + 4

Extension:

Students can utilize factoring or tiles to solve the following problems if they require more learning opportunities:

1. y = 3x² + 12x − 18 [y = 3(x + 2)² – 30]

2. y = −2x² − 10x + 8 [y = -2(x + 2.5)² + 20.5]

3. y = −x² + 6x − 3 [y = -(x - 3)² + 6]

4. y = 4x² − 16x − 12 [y = 4(x - 2)² – 28)]