Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 72720
The Distributive Property, Revealed! with a 100-dot
matrix
This lesson is designed as an introduction into the use of a 100-dot matrix to visualize the Distributive property of multiplication for one-digit by onedigit multiplication problems.
Subject(s): Mathematics
Grade Level(s): 4
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Internet Connection, Adobe
Acrobat Reader
Instructional Time: 1 Hour(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: multiplication, math, 100-dot, matrix, manipulatives, van de walle
Resource Collection: FCR-STEMLearn Mathematics General
ATTACHMENTS
100Dot Matrix.pdf
Multiplication Quiz.pdf
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to multiply one-digit by one-digit numbers through proper use of a 100-dot matrix.
Students will be able to visualize and describe multiplication models with respect to the Distributive Property.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should have prior knowledge of using addition to find the total number of objects arranged in rectangular arrays with up to ten rows and up to ten
columns.
Students should have prior knowledge of identifying arithmetic patterns and be able to explain them using properties of operations.
Guiding Questions: What are the guiding questions for this lesson?
"What does a x b mean? How else can you explain it?"
"How does the 100-dot matrix help us find solutions to multiplication problems?"
"How does modeling multiplication differ from modeling addition? Why?"
"How does the Distributive Property help us solve multiplication problems?"
Teaching Phase: How will the teacher present the concept or skill to students?
page 1 of 4 Hook: "Today we are going to learn a cool way to multiply. What better way to do that than with your buddy. Raise your hand if you want to sit next to your buddy to
learn multiplication. Alright! Stand quietly behind your chair and, when I tell you, go sit next to the friend you want to work with today. Remember that this is someone
that will help you do your best work."
Introduce the students to a 100-dot matrix (see link in Special Materials section) and the two half-sheets of construction paper they will use as covers.
Show the student that there are 10 dots going across the sheet vertically and 10 dots going horizontally.
Show the student that this particular 100-dot matrix is also divided into four quadrants, containing 25 dots organized in 5 X 5 arrays, alternating dark to light.
Once the student is familiar with the 100-dot matrix and its structure, decompose the problem given as follows: Example: 4 X 6 = 24
Explain that the first number, the 4 in this case, represents the number of groups (the number of dots that will go down along the left side). The second
number, the 6 in this case, represents the number of items inside each group (the number of dots that will be counted across the top). Finally, explain that the
number by itself on one side of the equal sign (i.e. the product) is the total.
Analyze the array that is created by identifying the arrays within the array. For example, they should see that:
4 X 6 = 4 X (5 + 1) = (4 X 5) + (4 X 1) = 20 + 4
Real-World Problems
Example 1: One-Digit by One-Digit Multiplication
Doug has 3 storage containers for his Beyblade toys. Each container holds 9 Beyblade toys at a time. If he filled every container with his
Beyblades, how many Beyblades did he store?
Allow students to explain the patterns and structure they see in the 100-dot matrix. For example:
3 X 9 = 3 X (5 + 4) = (3 X 5) + (3 X 4) = 15 + 12 = 27
Explain that 3 X (5 + 4) is equivalent to 3 X 9. Emphasize equivalence.
Example 2: Two-Digit by One-Digit Multiplication (Extension)
Sarah's mom wants to organize all of her family photos into an album. She went to the store and purchased an album containing 10 pages, holding
4 photos each. How many photos will Sarah's mom be able to insert in her new album?
Allow students to explain the patterns and structure they see in the 100-dot matrix. For example:
10 X 4 = (5 + 5) X (4) = (5 X 4) + (5 X 4) = 20 + 20 = 40
Explain to students that the 100-dot matrix and the equivalent expressions demonstrate the Distributive Property which indicates that multiplication distributes over
addition.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
In small groups, the students will continue solving more problems using the 100-dot matrix and answer the follow-up question in their math journal.
The teacher should be guiding students to see the structure of the 100-dot matrix with respect to the Distributive property. Teachers may consider asking students to
write equivalent expressions that demonstrate the Distributive Property.
For example, 5 X 8 = 5 X (5 + 3) = (5 X 5) + (5 X 3) = 25 + 15 = 40
For example, 9 X 9 = (5 + 4) X (5 + 4) = (5 X 5) + (4 X 5) + (5 X 4) + (4 X 4) = 25 + 20 + 20 + 16 = 81
Teachers could also create a sheet with multiple iterations of the 100-dot matrix (such as four matrices per page) and ask students to show the multiplication
problems on the 100-dot matrix using shading, noting the factors, and annotating the arrays within the arrays.
1. 5X8
2. 9X9
3. 4X5
4. 8X4
5. 5X7
6. 6X3
7. 10X10
"Imagine that your best friend was absent from school because he had the flu. Your teacher asked you to help him out by teaching him how to multiply using the same
method she taught you with the 100-dot matrix. How would you explain it to him and make him a multiplying machine? Remember, that your friend does not know
how to multiply or how to use the 100-dot matrix."
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Using the 100-dot matrix, create:
10 one-digit by one-digit multiplication problems for a group member to answer. Include an "answer sheet" on a separate sheet of paper.
page 2 of 4 Create an array using the 100-dot matrix and have a group member try to determine the factors.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Have each group share their math journal entry to explain their answer to the group question, "Imagine that your best friend was absent from school because he had
the flu. Your teacher asked you to help him out by teaching him how to multiply using the same method she taught you with the 100-dot matrix. How would you
explain it to him and make him a multiplying machine? Remember, that your friend does not know how to multiply or how to use the 100-dot matrix."
Explain and answer one of the group's multiplication practice problems on the board.
Students may also explain the Distributive Property in their own words in their math journals.
Summative Assessment
Using a 100-dot matrix, students will complete a short 10-question "Multiplication Quiz" (see attached).
Formative Assessment
Teacher will circulate thoroughly throughout the room, listening to each student or group of students as they cooperatively decide how to calculate the problem, and
asking guiding questions such as:
"What do you have to find out?"
"What have you already tried?"
"Is this answer reasonable? Does it make sense? Why or why not?"
"Do you see a pattern as you work? What is it? Can it help you?"
"Show me what you mean so I can see it."
Feedback to Students
Feedback to students should take place in two parts, depending on what the teacher observes during individual or group practice.
1. If the student is able to properly use the 100-dot matrix (see Special Materials for a link to the 100-dot matrix) in order to solve one-digit by one-digit multiplication
problems, the teacher should encourage the student to continue with their good work, and perhaps, challenge them to create their own problems and solutions.
2. If the student(s) is not able to properly use the 100-dot matrix (see Special Materials) in order to solve one-digit by one-digit multiplication problems, the teacher
should ask guiding questions to determine where the student's weakness is. For example, the teacher can ask:
"What do you have to find out?"
"What have you already tried?"
"Is this answer reasonable? Does it make sense? Why or why not?"
"Do you see a pattern as you work? What is it? Can it help you?"
"Show me what you mean so I can see it."
"What does a x b mean? How else can you explain it?"
"How does modeling multiplication differ from modeling addition? Why?"
"Compare 4, 9 and 36. Use multiplication language." The teacher should listen out for the student to reply something along the lines of "Well, 36 is 4 times as
much as 9.
"How are the numbers that you are working with related?"
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
If a student is having difficulty using the 100-dot matrix to multiply after teacher assistance, pair them up with another student and have that student explain it again
using their own language or mode of explanation. Often, students respond more positively when one of their peers explains something they don't understand.
Extensions:
Teach students how to decompose two-digit by one-digit or two-digit by two-digit multiplication problems (e.g. 12 X 6, 15 X 22, etc.).
Provide students with a 100-dot matrix that does not include alternating colors and ask them to find as many ways as they can to express 6 X 4. In this, students can
find many ways to decompose the factors using the Distributive property.
Suggested Technology: Document Camera, Computer for Presenter, Internet Connection, Adobe Acrobat Reader
Special Materials Needed:
100-dot Matrix
2 half sheets of construction paper
Math Journal
Multiplication Quiz (see attached)
Optionally, you could create a sheet that includes multiple iterations of the 100-dot matrix, such as 4 per page. Students could then use
this to record the arrays and annotate the factors.
Further Recommendations:
Give students 2 different color pieces of construction paper to serve as their covers in order to more easily distinguish between the number of groups and the number
in each group.
Additional Information/Instructions
By Author/Submitter
page 3 of 4 This lesson is written with a large ESE population in mind. However, it can be modified and extended to fit the needs of any other population of students.
This lesson engages students in Mathematical Practice 7: Look for and make use of structure as students explore the distributive property with a 100-dot matrix that allows
students to visualize the distributive property within an array representation.
SOURCE AND ACCESS INFORMATION
Contributed by: Doug Mejia
Name of Author/Source: Doug Mejia
District/Organization of Contributor(s): Miami-Dade
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.3.OA.2.5:
Description
Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24
is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or
by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16,
one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
page 4 of 4