NC Math 1 MVP 7.5: The Wow Factor
[i]Optima’s Quilts[/i] sometimes gets orders for blocks that are multiples of a given block. For instance, Optima got an order for a block that was exactly twice as big as the rectangular block that has a side that is 1” longer than the basic size, [math]x[/math],and one side that is 3” longer than the basic size.
Task 7.5 Question 1
[b]Draw a diagram of this block.[/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Task 7.5 Question 1a
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Task 7.5 Question 1b
Find an expression to represent the area as a sum or difference of terms. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Task 7.5 Question 2
Oh dear! This order was scrambled and the pieces are shown here. Put the pieces together to make a rectangular block and write two equivalent expressions for the area of the block.
Task 7.5 Question 2a
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Task 7.5 Question 2b
Find an expression to represent the area as a sum or difference of terms. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Task 7.5 Question 3
What do you notice when you compare the two equivalent expressions in problems #1 and #2?
Task 7.5 Question 4
Optima has a lot of new orders. Use diagrams to help you find equivalent expressions for each of the following:
Problem A
[math]5x^2+10x[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Problem B
[math]3x^2+21x+36[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Problem C
[math]2x^2+2x-4[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Problem D
[math]2x^2-10x+12[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Problem E
[math]3x^2-27[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Because she is a great business manager, Optima offers her customers lots of options. One option is to have rectangles that have side lengths that are more than one [math]x[/math]. For instance, Optima made this cool block:[br][br]
Task 7.5 Question 5
Write two equivalent expressions for this block. Use the distributive property to verify that your answer is correct.
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Find an expression to represent the area as a sum or difference of terms. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Here we have some partial orders. We have one of the expressions for the area of the block and we know the length of one of the sides. Use a diagram to find the length of the other side and write a second expression for the area of the block. Verify your two expressions for the area of the block are equivalent using algebra.
Task 7.5 Question 6 - Problem A
Area: [math]2x^2+7x+3[/math] Side: [math]\left(x+3\right)[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Task 7.5 Question 6 - Problem B
Area: [math]5x^2+8x+3[/math] Side: [math]\left(x+1\right)[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Task 7.5 Question 6 - Problem C
Area: [math]2x^2+7x+3[/math] Side: [math]\left(2x+1\right)[/math]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Task 7.5 Question 7
What are some patterns you see in the two equivalent expressions for area that might help you to factor?
Use this tool to help you answer the following questions. Click the Trash Can and then each object and anchor point to clear the workspace between problems.
Task 7.5 Question 8
Business is booming! More and more orders are coming in! Use diagrams or number patterns (or both) to write each of the following orders in factored form:
Task 7.5 Question 8 - Problem A
[math]3x^2+16x+5[/math]
Task 7.5 Question 8 - Problem B
[math]2x^2-13x+15[/math]
Task 7.5 Question 8 - Problem C
[math]3x^2+x-10[/math]
Task 7.5 Question 8 - Problem D
[math]2x^2+9x-5[/math]
Task 7.5 - Question 9
In [i]The x Factor, [/i]you wrote some rules for deciding about the signs inside the factors. Do those rules still work in factoring these types of expressions? Explain your answer.
Task 7.5 - Question 10
Explain how Optima can tell if the block is a multiple of another block or if one side has a multiple of [math]x[/math] in the side length.
There's one more twist on the kind of blocks that Optima makes. These are the trickiest of all because they have more than one [math]x[/math] in the length of both sides of the rectangle![br][br]Here's an example:[br][br]
Task 7.5 Question 11
Write two equivalent expressions for this block. Use the distributive property to verify that your answer is correct.
Find an expression to represent the area as a product of two factors. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.
Find an expression to represent the area as a sum or difference of terms. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression.