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NC Math 1 MVP 7.3: Factor Fixin'

Author:Douglas Bynum, Tim Brzezinski
Optima Prime has a quilt shop that makes and sells squares of fabric for quilting. At first, Optima’s Quilts only made square blocks for quilters and Optima spent her time making perfect squares. Customer service representatives were trained to ask for the length of the side of the block, , that was being ordered, and they would let the customer know the area of the block to be quilted using the formula . Optima found that many customers that came into the store were making designs that required a combination of squares and rectangles. So, Optima’s Quilts decided to produce several new lines of rectangular quilt blocks. Each new line is described in terms of how the rectangular block has been modified from the original square block. For example, one line of quilt blocks consists of starting with a square block and extending one side length by 5 inches and the other side length by 2 inches to form a new rectangular block. The design department knows that the area of this new block can be represented by the expression: . The diagram that represents the area and dimensions of the new block is drawn like this
Below is a version you can move to explore
Below is a version you can move to explore
They begin to use the diagram to find the area of the new block by labeling the area of the big square block . They calculate the area of each rectangular block as and the area of each of the small squares as .

Task 7.3 Question 1

Label each of the areas of the diagram in your printed copy of the MVP task and use them to find a different expression to represent the area of this new rectangular block two ways. The first way is: . What is a second expression for the area? Use the ^ key to enter an exponent when typing your expression and remember to write is as A(x)=____.

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In this portion of the task you will use the Geogebra Applet to draw each diagram.

Here are some additional new lines of blocks that Optima’s Quilts has introduced. Find two different algebraic expressions to represent each rectangle, and illustrate with a diagram why your representations are correct.

Task 7.3 Question 2

The original square block was extended 3 inches on one side and 4 inches on the other.

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 2a

The original square block was extended 3 inches on one side and 4 inches on the other. Find an expression to represent the area as a product of two factors. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 2b

The original square block was extended 3 inches on one side and 4 inches on the other. Find an expression to represent the area as a sum or difference of terms. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 3

The original square block was extended 4 inches on only one side.

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 3a

The original square block was extended 4 inches on only one side. Find an expression to represent the area as a product of two factors. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 3b

The original square block was extended 4 inches on only one side. Find an expression to represent the area as a sum or difference of terms. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 4

The original square block was extended 5 inches on each side.

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 4a

The original square block was extended 5 inches on each side. Find an expression to represent the area as a product of two factors. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 4b

The original square block was extended 5 inches on each side. Find an expression to represent the area as a sum or difference of terms. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 5

The original square block was extended 2 inches on one side and 6 inches on the other.

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 5a

The original square block was extended 2 inches on one side and 6 inches on the other. Find an expression to represent the area as a product of two factors. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 5b

The original square block was extended 2 inches on one side and 6 inches on the other. Find an expression to represent the area as a sum or difference of terms. Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.

Check my answer
Customers started ordering custom-made block designs by requesting how much additional area they want beyond the original area of . Once an order is taken for a certain type of block, customer service needs to have specific instructions on how to make the new design for the manufacturing team. The instructions need to explain how to extend the sides of a square block to create the new line of rectangular blocks. The customer service department has placed the following orders on your desk. For each, describe how to make the new blocks by extending the sides of a square block with an initial side length of . Your instructions should include diagrams, written descriptions and algebraic descriptions of the area of the rectangles in using expressions representing the lengths of the sides.

Task 7.3 Question 6

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 6a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 6b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 6c

Find an expression to represent the area, in simplest form. Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 7

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 7a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 7b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 7c

Find an expression to represent the area, in simplest form. Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 8

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 8a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 8b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 8c

Find an expression to represent the area, in simplest form. Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 9

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 9a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 9b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 9c

Find an expression to represent the area, in simplest form. Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.

Check my answer
Some of the orders are written in an even more simplified algebraic code. Figure out what these entries mean by finding the sides of the rectangles that have this area. Use the sides of the rectangle to write equivalent expressions for the area.

Task 7.3 Question 10

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 10a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 10b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 11

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 11a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 11b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 12

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 12a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 12b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 13

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 13a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 13b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 14

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 14a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 14b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 15

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

Task 7.3 Question 15a

Write a description of the following "order" For instance, "The original square block was extended __ inches on one side and __ inches on the other."

Check my answer

Task 7.3 Question 15b

Find an expression to represent the area, , as a product of two factors. When typing your expression and remember to write it as A(x)=_____.

Check my answer

Task 7.3 Question 16

Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)

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