Hook How do you expand linear expressions that involve multiplication, addition, and subtraction? For example, how do you expand 3(4 + 2x)? 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 the distributive property of multiplication over addition.

Objective In this lesson you will learn how to expand linear expressions with rational coefficients by using the properties of real numbers.

Vocabulary: Linear expression Rational coefficient Combine like terms Let’s Review Vocabulary: linear expression, rational coefficient, combine like terms

Properties of the Real Numbers: Commutative: 11 + 4 = 4 + 11 Associative: (4 + 3) + 9 = 4 + (3 + 9) Distributive: 5(6 + 2) = 5(6) + 5(2) Let’s Review Review the commutative, associative, and distributive properties of the real number system.

Failing to distribute negative numbers completely: -9(4 + 3) = -9(4) + 9(3) A Common Mistake A common mistake students make is to distribute a negative number to only the first term in parentheses.

Distributing multiplication over multiplication: 3(5 2) = 3(5) 3(2) A Common Mistake Students frequently try to distribute multiplication over multiplication. If we compute the value of each expression, we can see that the results are not equal. (Talk through the numbers.) Coach’s Commentary Students frequently try to do this with variables – for example, thinking that 3(xy) = (3x)(3y). When this happens, have them evaluate each expression for some values of x and y to convince them that the two results are not equal.

x 1 x 1 x 1 Core Lesson How do we expand 3(2x + 4)? One way to visualize this is by using an area model. Let a rectangle of no particular size represent the value x, and 4 small squares, each with an area of 1, represent the value 4. Placing them side by side results a rectangle with an area of 2x + 4. Since we are given 3 groups of 2x + 4, we simply use three of these rectangles. Now we can see that there are 3 groups of 2x and 3 groups of 4, or 2x + 2x + 2x plus 4 + 4 + 4, so the result is 6x + 12. Coach’s Commentary Using an area model gives students a concrete way to visualize the abstract concept of distribution.

Core Lesson Sometimes we have to use the distributive property on two quantities, then simplify by combining like terms. The commutative property can help us gather like terms. When subtraction is involved, you are simply distributing a negative number rather than a positive number. It is helpful to change subtraction to addition of a negative. For example, 11(3a – 2) – 6(8a – 9) = 11(3a + (–2)) + (–6)(8a + (–9)) = 33a + (–22) + (–48a) + 54 = –15a + 32. Coach’s Commentary Some students forget to distribute the negative to every term in parentheses.

Review In this lesson, you learned how to expand linear expressions with rational coefficients by using the properties of real numbers.

Guided Practice Simplify: 9(5k – 8) – 4(7 – 2k) Answer: 53k – 100

Extension Activities For a struggling student who needs more practice: Use a diagram to show why 4(3y + 2) = 12y + 8. 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.

Extension Activities For a student who gets it but needs more practice: Simplify 5(7y + 1) – 2(9y – 10) + 3(18 – 4y) Answer: 5y + 79

Extension Activities For a student who gets it and is ready to be challenged further: Write at least two different linear expressions that expand and simplify to a value of 40x + 27. Answers vary.

Quick Quiz 1. Simplify: 7(3x – 4) + 2(5 + 6x) 2. Simplify: 11(4 – 8w) – 6(–9w – 5) Answers: 33x – 18; 74 – 34w