MVP NC Math 1 7.4 (The x Factor)
Interactive companion to the Mathematics Vision Project NC Edition Module 7 Lesson 4 task and Set. [b]Special Note: This activity uses an Algebra Tile GeoGebra applet to construct visual area models and may not be suitable for small screens. To use the applet, click on tool you would like to add to the workspace, then click inside the workspace to place the tile at that particular point. You must click the arrow in order to move (by dragging the tile) or rotate (by dragging one of the corners) the algebra tiles. Other tools available allow you to clear the workspace or delete (Trash can) an object from the workspace. Red tiles represent negative values, while other colors represent positive values.[/b]
Table of Contents
- Task 7.4: The x Factor
- NC Math 1 MVP 7.4: The x Factor
- Set
- MVP M1 7.4 Set: Factoring Quadratics
NC Math 1 MVP 7.4: The x Factor
Now that [i]Optima’s Quilts[/i] is accepting orders for rectangular blocks, their business is growing by leaps and bounds. Many customers want rectangular blocks that are bigger than the standard square block on one side. [br][br]
Task 7.4 Question 1
[b]Sometimes they want one side of the block to be the standard length, x, with the other side of the block 2 inches bigger. [/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 1a
Sometimes they want one side of the block to be the standard length, 𝑥, with the other side of the block 2 inches bigger. [br]Write two different expressions for the area of the block.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Task 7.4 Question 2
[b]Sometimes they want blocks with one side that is the standard length, x, and one side that is 2 inches less than the standard size.[br][/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 2a
Sometimes they want blocks with one side that is the standard length, 𝑥, and one side that is 2 inches less than the standard size[b].[/b] [br]Write two different expressions for the area of the block. Use your diagram and verify algebraically that the two expression are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
[b]There are many other size blocks requested, with the side lengths all based on the standard length, [math]x[/math]. Construct each of the following blocks, then use your diagrams to write two equivalent expressions for the area. Verify algebraically that the expressions are equal.[/b]
Task 7.4 Question 3
[b]One side is 1" less than the standard size and the other side is 2" more than the standard size.[/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 3a
Write two different expressions for the area of the block. Use your diagram and verify algebraically that the two expression are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Task 7.4 Question 4
[b]One side is 2" less than the standard size and the other side is 3" more than the standard size.[/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 4a
Write two different expressions for the area of the block. Use your diagram and verify algebraically that the two expression are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Task 7.4 Question 5
[b]One side is 2" more than the standard size and the other side is 3" less than the standard size.[/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 5a
Write two different expressions for the area of the block. Use your diagram and verify algebraically that the two expression are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Task 7.4 Question 6
[b]One side is 3" more than the standard size and the other side is 4" less than the standard size.[/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 6a
Write two different expressions for the area of the block. Use your diagram and verify algebraically that the two expression are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Task 7.4 Question 7
[b]One side is 4" more than the standard size and the other side is 3" less than the standard size.[/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 7a
Write two different expressions for the area of the block. Use your diagram and verify algebraically that the two expression are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
[b]An expression that has 3 terms in the form: [math]ax^2+bx+c[/math][/b] is called a trinomial. look back at the trinomials you wrote in questions 3 - 7.
Task 7.4 Question 8
How can you tell if the middle term ([math]bx[/math]) is going to be positive or negative?
Task 7.4 Question 9
One customer had an unusual request. She wanted a block that is extended 2 inches on one side and decreased by 2 inches on the other.
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.4 Question 9a
One of the employees thinks that this rectangle will have the same areas as the original square since one side was decreased by the same amount as the other side was increased. What do you think?
Task 7.4 Question 9b
Write two different expressions for the area of the block. Use your diagram and verify algebraically that the two expression are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Task 7.4 Question 10
The result of the unusual request made the employee curious. Is there a pattern or a way to predict the two expressions for area when one side is increased and the other side is decreased by the same number?
[b]Try modeling these two problems, look at your answer to #8, and see if you can find a pattern in the result.[/b]
Problem A
[math]\left(x+1\right)\left(x-1\right)[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Write a trinomial expression for the area of the block. Use your diagram and verify algebraically that the two representations are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Problem B
[math]\left(x+3\right)\left(x-3\right)[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Write a trinomial expression for the area of the block. Use your diagram and verify algebraically that the two representations are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
Task 7.4 Question 11
What pattern did you notice? What is the result of [math]\left(x+a\right)\left(x-a\right)[/math]?
Task 7.4 Question 12
Some customers want both sides of the block reduced. Draw the diagram for the following blocks and find a trinomial expression for the area of each block.
Problem A
[math]\left(x-2\right)\left(x-3\right)[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Write a trinomial expression for the area of the block. Use your diagram and verify algebraically that the two representations are equivalent.[br][br][i]Use the ^ key to enter an exponent, if needed, when typing your expression.[/i]
MVP M1 7.4 Set: Factoring Quadratics
Directions:
The area of a rectangle is given in the form of a trinomial expression. Find the equivalent expression that shows the lengths of the two sides of the rectangle.[br][br]One Algebra Tile GeoGebra applet is included should you choose to work with a visual representation.
Algebra Tile Applet
MVP 7.4 Set Problem 9
[math]x^2+9x+8[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 10
[math]x^2-6x+8[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 11
[math]x^2-2x-8[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 12
[math]x^2+7x-8[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 13
[math]x^2-11x+24[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 14
[math]x^2-14x+24[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 15
[math]x^2-25x+24[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 16
[math]x^2-10x+24[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 17
[math]x^2-2x-24[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 18
[math]x^2-5x-24[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 19
[math]x^2+5x-24[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 20
[math]x^2-10x+25[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 21
[math]x^2-25[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 22
[math]x^2+2x+1[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 23
[math]x^2-1[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.
MVP 7.4 Set Problem 24
[math]x^2-2x-1[/math][br][br]Find the equivalent expression that shows the lengths of the two sides of the rectangle.