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MVP NC Math 1 7.3 (Factor Fixin')
Interactive companion to the Mathematics Vision Project NC Edition Module 7 Lesson 3 task, Ready, Set, and Go. [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 undo, redo, or delete (Trash can) an object from the workspace. Red tiles represent negative values, while other colors represent positive values.[/b]
Table of Contents
- Ready
- MVP NC Math 1 Lesson 7.3 Ready
- Task 7.3: Factor Fixin'
- NC Math 1 MVP 7.3: Factor Fixin'
- Set
- MVP M1 7.3 Set: Representing binomial factors in an area diagram
- Go
- MVP NC Math 1 7.3 Go
MVP NC Math 1 Lesson 7.3 Ready
Creating binomial quadratic expressions
[size=200][size=150][b]Use the distributive property to multiply the expressions below.[br][/b][/size][/size][size=100][br][size=150][i]Write your answers in the form [/i][math]ax^2+bx+c[/math][i] and[u] use the "[b]^[/b]" key to input the exponent[/u]. (Your answers may look something like this: ax^2+bx+c). This is "standard" form for a quadratic.[/i][/size][/size]
Problem 1
[math]x\left(4x-7\right)[/math]
Problem 2
[math]5x\left(3x+8\right)[/math]
Problem 3
[math]3x\left(3x-2\right)[/math]
The answers to problems 1, 2, and 3 are quadratics that can be represented in the standard form [math]ax^2+bx+c[/math]. Which coefficient, [b]a[/b], [b]b[/b], or [b]c[/b] equals 0 for al of the exercises above?
Rewrite the following expressions as a product by dividing out the greatest common factor (gcf) in the two terms. (Your answers will look like the expressions in problems 1, 2, and 3 before you multiplied.)
Problem 5
[math]x^2+4x[/math]
Problem 6
[math]7x^2-21x[/math]
Problem 7
[math]12x^2+60x[/math]
Problem 8
[math]8x^2+20x[/math]
NC Math 1 MVP 7.3: Factor Fixin'
Optima Prime has a quilt shop that makes and sells squares of fabric for quilting. At first, [i]Optima’s Quilts[/i] 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, [math]x[/math], that was being ordered, and they would let the customer know the area of the block to be quilted using the formula [math]A\left(x\right)=x^2[/math].[br][br]Optima found that many customers that came into the store were making designs that required a combination of squares and rectangles. So, [i]Optima’s Quilts [/i]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: [math]A\left(x\right)=\left(x+5\right)\left(x+2\right)[/math].[br][br]The diagram that represents the area and dimensions of the new block is drawn like this
They begin to use the diagram to find the area of the new block by labeling the area of the big square block [math]x\cdot x=x^2[/math]. They calculate the area of each rectangular block as [math]1\cdot x=x[/math] and the area of each of the small squares as [math]1\cdot1=1[/math].
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: [math]A\left(x\right)=\left(x+5\right)\left(x+2\right)[/math]. 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)=____.
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 [i]Optima’s Quilts [/i]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
[b]The original square block was extended 3 inches on one side and 4 inches on the other.[/b]
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 3
[b]The original square block was extended 4 inches on only one side.[/b]
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 4
[b]The original square block was extended 5 inches on each side.[/b]
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 5
[b]The original square block was extended 2 inches on one side and 6 inches on the other.[/b]
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
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. [br][br]Use the ^ key to enter an exponent, if needed, when typing your expression and remember to write it as A(x)=_____.
Customers started ordering custom-made block designs by requesting how much additional area they want beyond the original area of [math]x^2[/math]. 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.[br][br]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 [math]x[/math]. Your instructions should include [b][i]diagrams[/i][/b], [b][i]written descriptions[/i][/b] and [b][i]algebraic descriptions[/i][/b] of the area of the rectangles in using expressions representing the lengths of the sides.
Task 7.3 Question 6
[b][math]x^2+5x+3x+15[/math][/b]
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" [math]x^2+5x+3x+15[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 6b
Find an expression to represent the area,[math]x^2+5x+3x+15[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 6c
Find an expression to represent the area,[math]x^2+5x+3x+15[/math] in simplest form. [br][br]Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 7
[b][math]x^2+4x+6x+24[/math][/b]
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" [math]x^2+4x+6x+24[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 7b
Find an expression to represent the area,[math]x^2+4x+6x+24[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 7c
Find an expression to represent the area,[math]x^2+4x+6x+24[/math] in simplest form. [br][br]Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 8
[b][math]x^2+9x+2x+18[/math][/b]
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" [math]x^2+9x+2x+18[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 8b
Find an expression to represent the area,[math]x^2+9x+2x+18[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 8c
Find an expression to represent the area,[math]x^2+9x+2x+18[/math] in simplest form. [br][br]Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 9
[b][math]x^2+5x+x+5[/math][/b]
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" [math]x^2+5x+x+5[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 9b
Find an expression to represent the area,[math]x^2+5x+x+5[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 9c
Find an expression to represent the area,[math]x^2+5x+x+5[/math] in simplest form. [br][br]Use the ^ key to enter an exponent when typing your expression and remember to write it as A(x)=_____.
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
[b][math]x^2+11x+10[/math][/b]
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" [math]x^2+11x+10[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 10b
Find an expression to represent the area,[math]x^2+11x+10[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 11
[b][math]x^2+7x+10[/math][/b]
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" [math]x^2+7x+10[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 11b
Find an expression to represent the area,[math]x^2+7x+10[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 12
[b][math]x^2+9x+8[/math][/b]
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" [math]x^2+9x+8[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 12b
Find an expression to represent the area,[math]x^2+9x+8[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 13
[b][math]x^2+6x+8[/math][/b]
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" [math]x^2+6x+8[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 13b
Find an expression to represent the area,[math]x^2+6x+8[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 14
[b][math]x^2+8x+12[/math][/b]
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" [math]x^2+8x+12[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 14b
Find an expression to represent the area,[math]x^2+8x+12[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 15
[b][math]x^2+7x+12[/math][/b]
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" [math]x^2+7x+12[/math] For instance, [i]"[/i][i]The original square block was extended __ inches on one side and __ inches on the other."[/i]
Task 7.3 Question 15b
Find an expression to represent the area,[math]x^2+7x+12[/math] , as a product of two factors. [br][br]When typing your expression and remember to write it as A(x)=_____.
Task 7.3 Question 16
[b][math]x^2+13x+12[/math][/b]
Draw your illustration: Click the buttons to add each element (blue, green, and yellow represent positive terms)
MVP M1 7.3 Set: Representing binomial factors in an area diagram
Directions:
Construct a diagram that represents the area described by the given expression. Use your diagram to find an expression in the form [math]ax^2+bx+c[/math] that also describes the area of your diagram.[br][br]NOTE: Make sure the arrow is selected in order to move algebra tiles in the workspace.
MVP 7.3 Set Problem 9
[math]A\left(x\right)=\left(x+6\right)\left(x+6\right)[/math]
Diagram
Write an expression to represent the area using standard form ([math]A\left(x\right)=ax^2+bx+c[/math]). Use the ^ key to type your expression below in the form A(x)=___.
MVP 7.3 Set Problem 10
[math]B\left(x\right)=\left(x+4\right)\left(x+9\right)[/math]
Diagram
Write an expression to represent the area using standard form ([math]B\left(x\right)=ax^2+bx+c[/math]). Use the ^ key to type your expression below in the form B(x)=___.
MVP 7.3 Set Problem 11
[math]C\left(x\right)=\left(x+2\right)\left(x+18\right)[/math]
Diagram
Write an expression to represent the area using standard form ([math]C\left(x\right)=ax^2+bx+c[/math]). Use the ^ key to type your expression below in the form C(x)=___.
MVP 7.3 Set Problem 12
[math]D\left(x\right)=\left(x+3\right)\left(x+12\right)[/math]
Diagram
Write an expression to represent the area using standard form ([math]D\left(x\right)=ax^2+bx+c[/math]). Use the ^ key to type your expression below in the form D(x)=___.
MVP 7.3 Set Problem 13
Look back at each expression that you wrote in the form [math]ax^2+bx+c[/math] that also describes the area of your diagram. The coefficients ([b](a) [/b]and [b](c)[/b] should be the same in each problem. Explain why you think the coefficient [b](b) [/b]of the middle term is different in each problem when the “outside” coefficients [b](a)[/b] and [b](c)[/b] are the same.
Draw a diagram that represents the area described by the given expression. Next, rewrite the area function in the form [math]\left(x+a\right)\left(x+b\right)[/math].
MVP 7.3 Set Problem 14
[math]f\left(x\right)=x^2+3x+5x+15[/math]
Diagram
Write an expression to represent the area using factored form ([math]f\left(x\right)=\left(x+a\right)\left(x+b\right)[/math]). Remember to include the f(x)=(__ + __)(__ + __).
MVP 7.3 Set Problem 15
[math]g\left(x\right)=x^2+7x+2x+14[/math]
Diagram
Write an expression to represent the area using factored form ([math]g\left(x\right)=\left(x+a\right)\left(x+b\right)[/math]). Remember to include the g(x)=(__ + __)(__ + __).
Figure out what these expressions mean by finding the sides of rectangles that have the given area. Use the sides of the rectangle to write an equivalent expression for the area (as the product of two factors).
MVP 7.3 Set Problem 16
[math]x^2+5x+6[/math]
Diagram
Write an expression to represent the area using factored form ([math]\left(x+a\right)\left(x+b\right)[/math]).
MVP 7.3 Set Problem 17
[math]x^2+7x+6[/math]
Diagram
Write an expression to represent the area using factored form ([math]\left(x+a\right)\left(x+b\right)[/math]).
MVP 7.3 Set Problem 18
[math]x^2+8x+12[/math]
Diagram
Write an expression to represent the area using factored form ([math]\left(x+a\right)\left(x+b\right)[/math]).
MVP 7.3 Set Problem 19
[math]x^2+7x+12[/math]
Diagram
Write an expression to represent the area using factored form ([math]\left(x+a\right)\left(x+b\right)[/math]).
MVP NC Math 1 7.3 Go
Connecting variables to coefficients in quadratic equations
The standard form of a quadratic equation is defined as [math]y=ax^2+bx+c,\left(x\ne0\right)[/math].[br][br]Identify a, b, and c in the following equations.[br] [br]Example: Given [math]y=4x^2+7x-6[/math], a=4, b=7, and c=-6
MVP 7.3 Go Problem 20
[math]y=5x^2+3x+6[/math][br][br]Be sure to write your answer in the form, "a=__, b=__, and c=__"
MVP 7.3 Go Problem 21
[math]y=x^2-7x+3[/math][br][br]Be sure to write your answer in the form, "a=__, b=__, and c=__"
MVP 7.3 Go Problem 22
[math]y=-2x^2+3x[/math][br][br]Be sure to write your answer in the form, "a=__, b=__, and c=__"
MVP 7.3 Go Problem 23
[math]y=6x^2-5[/math][br][br]Be sure to write your answer in the form, "a=__, b=__, and c=__"
MVP 7.3 Go Problem 24
[math]y=-3x^2+4x[/math][br][br]Be sure to write your answer in the form, "a=__, b=__, and c=__"
MVP 7.3 Go Problem 25
[math]y=8x^2-5x-2[/math][br][br]Be sure to write your answer in the form, "a=__, b=__, and c=__"
Note
It's important that you notice that the values for a, b, and c [b][i]NEVER [/i][/b]include the x.