Lesson 15 of 19
Objective: Students will be able to use the distributive property to re-write numerical expressions as a different yet equivalent numerical expression.
- A gardener has 27 pansies and 36 daisies. He plants an equal number of each type of flower in each row. What is the greatest possible number pansies in each row?
- Fourteen boys and 21 girls will be equally divided into groups. Find the greatest number of groups that can be created if no one is left out.
- Macy is painting a design that contains two repeating patterns. One pattern repeats every 8 inches. The other pattern repeats every 12 inches. If both patterns begin at the same place, in how many inches will they begin together again?
These three problems reinforce the concepts of Greatest Common Factor and Least Common Multiple learned in yesterday's lesson.
Teaching Note: Greatest Common Factor is a prerequisite skill for today's lesson.
My classroom is normally set up in six groups of 4-5 students. In today's engagement activity, I will provide each of the six groups with 27 cubes presented as a set of 12 and a set of 15. I will then ask the students to create a rectangle with each set of cubes using the following criteria:
The rectangle created with each set of cubes needs to use all of the cubes. In addition, both rectangles need to be created with the same number of rows.
Once the student have figured out how to create the rectangles, I will ask the following questions:
- How did you create the first rectangle with the set of 12 cubes?
- How did you create the second rectangle with the set of 15 cubes?
- How many cubes do you have altogether?
- Look at the rectangles you have created. What are some ways that you can represent what you see using math?
- Can you represent what you see using a mathematical expression?
- How many different ways can you come up with to represent what you see?
As I listen to to my students answer these questions, I am looking for students to tell me that they can represent what they see in the following ways:
- 12 + 15 = 27
- (3 x 4) + (3 x 5) = 27
If I do not receive this response from my students, I will use more probing and strategic questions to help them arrive to this conclusion. For example, I might offer a leading questions like:
- Hey, do these rectangles remind you of an array?
- Can we use what we know about arrays to help us to come up with another way to represent what we see?
In today's instructional piece, I want to truly help my students understand the concept of distributive property. To do this, I will first present the students with the actual explanation/definition of the distributive property. I share some of the teaching moves that I will use with my students in my Distributive Property video.
The distributive property involves the operations of multiplication and addition or multiplication and subtraction. When we use the distributive property, we are multiplying each term inside the parentheses with the term outside of the parentheses. The distributive property, which is displayed below, holds true for all real numbers a, b, and c. Also notice that, if you view the formula in the opposite direction, we are just taking out the common factor of a.
Let's start with a simple application in arithmetic:
5(3 + 5)
Using the distributive property, we simplify as follows:
5(3) + 5(5) = 15 + 25 = 40
During the presentation of this explanation, I want to drive home the meaning of the word distribute and how it applies to this situation. I want to make sure that my students see that the factor on the outside of the parenthesis is being distributed evenly among the terms on the inside of the parenthesis.
After providing my students with this definition/explanation, I will ask them how does the activity that we completed today apply to the distributive property. It is my hope that they will see that the number of rows are representative of the a in the example above and that the number of columns are representative of the b and the c in the above example.
Next, I will complete one example to ensure their understanding of the presented material.
The example I will use is 16 + 24
Using the above expression, I will demonstrate to students how to factor using the distributive property. I want to make sure that the students understand the following:
- You must find the greatest common factor. Students need to understand that there is a possibility that there are more than one common factor between two quantities presented and that in order for the expression to be factored properly that the greatest amount should be used.
For example, 16 and 24 have a common factor of 2, 4, and 8... all of these factors can be used but, we should be using the greatest common factor at all times therefore,
- 2(8 + 12) is incorrect because more can be factored out.
- 4(4 + 6) is incorrect because more can be factored out.
- 8(2 + 3) is CORRECT because we have factored out the greatest amount possible.
Try It Out
To try out this concept of distributive property, I will have my students complete the following 4 problems from Try It Out:
Factor the following expressions using distributive property.
1.) 21 + 12
2.) 28 + 16
As my students work on problems 1 and 2, I am on the lookout for whether or not my students identify the greatest common factor before factoring.
Use the distributive property to simplify the following expressions.
3.) 5(4 + 7)
4.) 9(7 + 3)
I hope that my students are careful when following the algorithm for distributing. There may be some students who distribute the number that is on the outside of the parenthesis to the first term on the inside of the parenthesis only. I like to start with arithmetic expressions, even though we are breaking order of operations, because my students are able to check their own work.
While my students are completing these problems, I will be traveling the room checking their work for accuracy. Should I find that several students are making similar mistakes then, I will take the time to reteach and/or address any misconceptions.
To explore the Distributive Property a little more deeply further, my student will complete this Exploration Worksheet. I will ask them to complete this work in pairs.
The problems in this worksheet are similar to the work completed for the Try It Out section of the lesson. They will need to factor using the distributive property and they will also have to use the distributive property to simplify an expression. In this activity, I also ask my students to show what is happening in each problem using an area model.
To close out this lesson, I will have selected student pairs to come to place their work underneath the document camera. They will need to explain what they did to solve the problem that they are presenting. They will be expected to do the following:
- Present the problem.
- Explain what the problem was asking them to do.
- Describe step by step how they chose to solve the problem.
- Connect their visual representation to the mathematics.
The observing students will need to ask questions and critique the work of their peers. I will also be asking questions and critiquing the work. The type of questions that I will be asking will be strategic in nature, designed to provoke thought and and aide in students in being thorough and precise in their explanations.
Questions that I might ask to promote thoroughness and precision:
- Why did you do what you did at this point?
- So, what does that mean?
- So how does your area model represent the situation?
- Tell me what each part of your area model represents? How do you know?
- How did you know that the quantity you have there is the greatest common factor?