1. For example, how do you factor
Hook
How do you factor a linear expression?
For example, how do you factor 4x + 8?
Coach’s Commentary
I chose this example because it is easy to demonstrate with an area model, which gives students a concrete way to visualize factoring by using the distributive property.

2. Objective
In this lesson you will learn how to factor linear expressions with rational coefficients by using the distributive property.

3. Vocabulary: Linear expression Rational coefficient Constant
Let’s Review
Vocabulary: linear expression, rational coefficient, constant

4. Let’s Review
Even though 24 and 32 have 1, 2, 4, and 8 in common, the greatest of these is 8.

5. A Common Mistake
A common mistake students make is to use a common factor that is not the greatest common factor.

6. x 1 x 1
Core Lesson
How do we factor (4x + 8)? One way to visualize this is by using an area model.
Let 4 rectangles of no particular size represent the value 4x, and 8 small squares, each with an area of 1, represent the value 8.
Placing them side by side results a rectangle with an area of 4x + 8.
Now rearrange the 4 x’s and 8 1’s into another rectangle with the same area, but different dimensions – a height of 4 and a length of x + 2.
Let’s use the distributive property to calculate the area of the new rectangle.
The area of the new rectangle is 4 times x + 2, which equals 4(x) + 4(2), or 4x + 8.
Because the two rectangles have the same area, this tells us that 4x + 8 = 4(x + 2).
Coach’s Commentary
Using an area model gives students a concrete way to visualize the abstract concept of factoring a linear expression.

7. Core Lesson
To factor by using the distributive property, we must first identify the greatest common factor of the coefficient and the constant term. In this case, the coefficient is 4 and the constant term is 8.
Examine factors of 4 and 8, conclude that 4 is the GCF.
Then express each term as the product of 4 and another factor.
Now we can write 4x + 8 as 4(x) + 4(2).
By the distributive property, this is equal to 4(x + 2).
Coach’s Commentary
The most common errors in factoring are: 1) using a common factor less than the GCF, and 2) subtracting the common factor rather than dividing it. Remind the students that factoring is a division process, not a subtraction process.

8. Core Lesson
When subtraction is involved, we factor in the same way.
First, identify the greatest common factor of the coefficient and the constant term. In this case, the coefficient is 24 and the constant term is –32.
Examine factors of 24 and 32, conclude that 8 is the GCF.
Then express each term as the product of 8 and another factor.
Now we can write 24x – 32 as 8(3x) – 8(4).
By the distributive property, this is equal to 8(3x – 4).
Coach’s Commentary
Notice that the sign of the constant term is not important when determining the greatest common factor, but remind students to use correct signs so that the factored expression is equivalent to the original expression.

9. Review
In this lesson, you have learned how to factor linear expressions with rational coefficients by using the distributive property.

10. Guided Practice
Factor: 12x - 18
Answer: 6(2x - 3)

11. Extension Activities
For a student who gets it and is ready to be challenged further: Factor 2m m - 7
Answer: 4(2m + 3)

12. Use a diagram to show why 6y + 15 = 3(2y + 5).
Extension Activities
For a struggling student who needs more practice: Use a diagram to show why 6y + 15 = 3(2y + 5).
Coach’s Commentary
Asking a student to explain the reason why a mathematical procedure works will help the student to solidify his or her reasoning.

13. Extension Activities
For a student who gets it and is ready to be challenged further: Factor: – 3x – 18
Answer: – 3(x + 6)

14. Quick Quiz
1. Factor: 9y + 15
2. Factor: 24x - 60
Answers:
3(3y + 5)
12(2x – 5)