Quadratics Part 4- Finding the Area

Now that you are an expert at making rectangles and sharing the dimensions, you can do even more interesting stuff! If you have a length and a width, how do you find the area? By multiplying the length and the width.

This is easy when we have a length of 8 and a width of 4. We then multiply 8 x 4 and know that the area is 32!

But, what if we have a length of (x+2) and a width of (x+3) like the image below? How do we multiply that?!

What we need to figure out, then, is how to multiply (x+2) by (x+3).

HOW TO MULTIPLY BINOMIALS

Before multiplying binomials, let's review the distributive property.

We learned in class that it's important to polite. That means that if you knock on someone's door, you have say hello to everyone inside!

Example 1: 5 (4x + 2) means that we must share the 5 with 4x AND with 2.

5 times 4x = 20x

5 times 2 = 10

So, we get 20x and 10 = 20x + 10.

Example 2: x (2x + 4) means that we must share the x on the outside with x AND 4.

x times x = 2x^2

x times 4 = 4x

So, we get 2x^2 and 4x = 2x^2 + 4x.

Try out these:

1. x (x + 6)

2. x (2x + 1)

3. 2x (x + 3)

4. 3x (2x + 1)

Now that we have that ironed that out, we're ready for the big leagues. Looking back at Example 2, we see x (2x + 4).

But, what if it was (x + 1) (2x + 4)? I use the distributive property just like before!

First, I have to share the x with (2x + 4). I am going to pretend that the 1 doesn't exist. When I do that, I get 2x^2 + 4x, just like we did above.

Now, the plot thickens. Instead of ignoring the 1 like I did above, I'm now going to ignore the x. I'm going to share 1 with (2x + 4). When I do that, I get 2x (1 times 2x = 2x) and 4 (1 times 4 = 4). So, it is 2x + 4.

All I need to do is combine all the multiplication I just did. I have 2x^2 + 4x and I have 2x + 4.

When I put it all together, I get: 2x^2 + 4x + 2x + 4.

Again, all I did was apply the distributive property to figure out the answer. First, I shared x with everything in the second parentheses. Then, I shared 1 with everything in the second parentheses. Then, I combined!

Example 2 of multiplying binomials:

(2x + 1) (x + 5)

First, I am going to share the 2x with everything inside (x + 5). When I do that, I get 2x times x = 2x^2 and 2x times 5 = 10x. I put that all together and get 2x^2 + 10x.

Second, I am going to share the 1 with everything inside (x + 5). When I do that, I get 1 times x = 1x and 1 times 5 = 5. I put that all together and get 1x + 5.

Now, I am going to combine everything together and write: 2x^2 + 10x + 1x + 5.

There I have it!

Now, practice doing the same thing with the problems below:

1. (x + 2) (x + 4)

2. (x + 4) (2x + 3)

3. (2x + 2) (x + 3)

4. (3x + 1) (x + 2)

5. (2x + 4) (x + 5)

6. (6x + 2) (4x + 3)

7. (5x + 1) (2x + 4)

Quadratics Part 3- Making Rectangles with Dimensions

Now that you understand the dimensions, we are going to step up our game to the next level. You are going to combine your knowledge of creating rectangles and side lengths together!

Let's use the following example to learn this new part: x^2 + 6x + 5.

First, we need to make a tile image of the shape below. Go ahead and do that! Now that you have that, we want to figure out how to write out the length and width.

We know that the dimensions of the big square (x^2) is x by x.

We know that the dimension of the rod (x) is x by 1.

We know that the dimension of the little square (1) is 1 by 1.

Let's look at the length of the image below (the left side of the shape). We are only looking at the side lengths of the shape, not the whole area. Looking at the left side, I see 2 green rods and 1 blue square. The dimension of one rod is x by 1. The left side of the shape has the smaller part of the rod, 1. There are two of them, so we have 1 and 1 to represent the green parts of the length. The blue parts of the length is part of the square. The dimension of one large square is x by x. The one blue part that is part of the length is one dimension: x.

So, in total for the length, we have 1 x and 2 1s. I can write that more easily as x and 2 = x + 2.

We can apply the same thinking to find the width. We see one long blue side length (x) and 3 small green parts (1s). In total for the width, we have 1 x and 3 1s. I can write that more easily as x and 3 = x + 3.

So, my length is (x + 2) and my width is (x + 3).

To practice with the dimensions, draw rectangles with each shape below AND indicate the length and width.

1. x^2 + 7x + 6
2. x^2 + 4x + 4
3. 2x^2 + 8x + 6
4. 2x^2 + 7x + 3
5. 2x^2 + 8x + 8
6. 3x^2 + 10x + 3
7. 3x^2 + 8x + 5

Quadratics Part 2- Dimensions!

Now that you are experts at creating the rectangles, let's talk about what they mean!

The image below shows the same key that we used to figure out what tiles we need to make the rectangle shape.

Key:

We all agree that x^2 tells me the number of big squares, x tells me the number of rods and 1 tells me the number of small squares. But, now we need to take it to the next level.

Let's talk about the dimensions of each shape.

The large square (x^2) has a length of x and a width of x, as shown below. To find the area, we multiply length (x) times width (x). Here, the area is x times x = x^2. The side lengths are x and x.

The rod (x) has a length of x and a width of 1. We know the length is the same as the big squares because it's the same length. To find the area, we would multiply length (x) by width (1). Here, the area is x times 1 = x. The side lengths are x and 1.

The small square (1) has a length of 1 and a width of 1. We know the length and width are the same as the rod's width, because it's the same length. To find the area, we would multiply length (1) by width (1). Here, the area is 1 times 1 = 1. The side lengths are 1 and 1.

Quadratics Part 1- What do all the tiles mean?

We use tiles to help us understand the Quadratics Unit. There are small square tiles, rods and large square tiles.

Key:

The large squares represent the value of x^2.

The rods represent the value of x.

The small square represents the value of 1.

Example 1:

x^2 + 5x + 6

For the example above, we would use 1 large square, 5 rods and 6 small squares. Your goal is to create a rectangle with the shapes you have. Draw a picture of each rectangle you create.

After you try this out, scroll down to see if you got the right image!

If you want extra practice, create rectangles with the tiles below. Draw out your pictures on paper and bring them to class!

1. x^2 + 7x + 6
2. x^2 + 4x + 4
3. 2x^2 + 8x + 6
4. 2x^2 + 7x + 3
5. 2x^2 + 8x + 8
6. 3x^2 + 10x + 3
7. 3x^2 + 8x + 5

Welcome!

Hi everyone,

I am excited to start this blog to make math class even more convenient for you! Look here for lesson reviews, test reviews and interesting articles/facts about math/high school/college!

The first few blogs are going to be focused on our Quadratics Unit. If you are confused about the work we are doing in class, read the blog posts in order to answer any questions you have! And, if you are still confused, email me at msdandiya@gmail.com.

Thanks for checking the website out-- I'm excited about this!

- Ms. D