##
* *Reflection: Checks for Understanding
Expand Algebraic Expressions Using the Distributive Property - Section 2: Guided Problem Solving

My students were able to do the first 4 problems with relative ease. Emphasis on relative; there were still some difficulties. While they were able to "do" the math, they still didn't understand that they were essentially finding the product of two numbers. Going back to simple numeric examples helped students see the connection to some degree. Even more beneficial was to substitute a value into the variable and evaluate. Then expand the expression and substitute the same value. Students then saw the equivalence and saw a bit more clearly that an expression in the form of a(bx + y) can be seen as the product of two numbers.

Using GP1 as an example:

15 (2b + 3) = 30b + 45

If we let b = 1:

15 (2*1 + 3) = 30 * 1 + 45

15 (5) = 30 + 45

75 = 75

Several students were still a bit stumped on GP7 - finding the area of a rectangle. They all know how to find an area of a rectangle; I'm sure a few think you can find the area of any shape by multiplying length times width! So just multiply 5 and (3x + 8) to find the area of the rectangle!

Students are exercising mathematical practices 2 and 7!

*The Product of Two Numbers*

*Checks for Understanding: The Product of Two Numbers*

# Expand Algebraic Expressions Using the Distributive Property

Lesson 2 of 20

## Objective: SWBAT expand linear expressions using the distributive property

## Big Idea: Expand expressions involving positive and negative rational numbers (mostly integers) using the distributive property.

*50 minutes*

#### Introduction

*10 min*

I will begin with the essential question: How can we use the distributive property to expand the expression? The distributive property is a 6th grade common core standard (6.EE.3 and 6.NS.4) however these focus on positive whole numbers. In 7th grade, students apply positive and negative rational numbers to the property.

It would be helpful to start with a simple problem involving constants. The area model could go along with this problem. So in the example below we might have a rectangle whose width is 5 and whose length is 8 + 2 units.

5 (8 + 2)

When we find the area we can multiply the 5 * 8 rectangle and the 5 * 2 rectangle and take the sum of the products. Students will also see this equals 5 * 10.

At our school our students spend about 90 minutes a week on a program called ST MATH. Many have already started solving distributive property problems in this program so I will bring up one of the puzzles in teacher mode from my introduction. Another similar representation can be found here. Either way, it is important to give a nice visual to help students understand how the property works.

I will then go through the examples. To tie the examples to the visual models, example 1 could be written as such: 4(2x + 7) means 4 groups of 2x + 7, so:

2x + 7

2x + 7

2x + 7

2x + 7

Which equals 8x + 28.

But we quickly want to get efficient **(MP8)** and just write:

4 * 2x + 4 * 7

8x + 28.

As needed I will go back to the models but they get a bit convoluted (in my opinion) when the number get large or negative.

One other note, based on my own understanding and strengths (weakness?), I generally find it easier to have differences re-written as sums before distributing. I also do this with like terms. I personally get confused when working with -7(5-4x) if I do not turn 5 - 4x to 5 + -4x. Since I am more comfortable with this method, I think I will be more effective teaching it. That being said, I need to make sure that my students can recognize all equivalent forms. So -35 + 28x is also 28x - 35, etc...

*expand content*

#### Guided Problem Solving

*15 min*

There are 7 problems here. The first 4 are similar to the first 4 of independent practice and the problems on the exit ticket. I will be very upfront and let my students know that these are the types of problems they should be able to solve in order to be successful with the lesson. The 5th and 6th problems involve a fraction and a decimal. Problem 7 involves the area of a rectangle.

I will set a short time limit for students to solving the first 4 problems. Depending how the intro goes, I may ask them to do more or less. If they are struggling, I may ask them to work on 1 at a time. Either way, each time they solve I want them to explicitly write out the products before simplifying. So for GP1, students should write 15 * 2b + 15 * 3 before simplifying to 30b + 45.

*expand content*

#### Independent Problem Solving

*20 min*

Students now work independently. Before I will be willing to answer questions, I will insist that students look at the guided problem solving section. Each problem (1-6) is very similar to the problems GP1-GP6 in guided problem solving. Of course, students may still make the common mistake of forgetting to distribute fully. For example, they may think 3(2x + 6) equals 6x + 6 when forgetting to multiply 3 * 6.

The last problem 8, is a chance for students to explain their thinking (**MP3**). This is an important problem because it makes sure students see all of the equivalent forms of an expression. This will be expected of them on the unit assessment and it is also important for their development as mathematicians.

*expand content*

#### Exit Ticket

*5 min*

The exit ticket has 4 problems. Again, these are similar to problems already solved at least twice throughout the lesson. The 4th problem brings distributive property together with combining like terms (the previous lesson).

Before we begin the exit ticket, we will briefly discuss how we used the distributive property to expand the expressions.

Students need to be able to solve at least 3 of the 4 correctly to show proficiency with the lesson.

*expand content*

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- LESSON 1: Simplifying Linear Expressions by Combining Like Terms
- LESSON 2: Expand Algebraic Expressions Using the Distributive Property
- LESSON 3: Factor Algebraic Expressions Using the Distributive Property
- LESSON 4: Linear Expressions and Word Problems
- LESSON 5: Solving Addition and Subtraction Equations Using Models
- LESSON 6: Solve Addition and Subtraction Equations using Inverse Operations
- LESSON 7: Solve One-Step Equations Using Inverse Operations
- LESSON 8: Diagrams of Two-Step Equations
- LESSON 9: Speeding Tickets
- LESSON 10: Translate Verbal Statements to Inequalities
- LESSON 11: Graph Inequalities on a Number Line
- LESSON 12: Solve Inequalities Using Addition and Subtraction
- LESSON 13: Solve Inequalities Using Multiplication and Division
- LESSON 14: Solve Two-Step Equations Using Inverse Operations
- LESSON 15: Equation and Inequality Reteach - Fraction Coefficients
- LESSON 16: Algebraic Expressions and Equations for Shape Patterns
- LESSON 17: Simplify Expressions Containing Fractions by Combining Like Terms
- LESSON 18: Simplify Rational Number Expressions Using the Distributive Property
- LESSON 19: Writing Algebraic Expressions to Solve Perimeter Problems
- LESSON 20: An Introduction To Programming in SCRATCH