Multiplication: Arrays and an Expanded Algorithm

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SWBAT use an array and expanded algorithm to help solve multiplication problems.

Big Idea

Multiplication problems can be solved using different strategies.


5 minutes

Rationale for teaching with a task:

After I have worked directly with the students on a skill, I like to use a task.  A task gives the students more practice on the skill while working in groups.  Allowing the students to work in groups gives the students different perspectives from their classmates.  Students can learn from each other.  As the students work on a task, I am the facilitator, walking around monitoring and questioning the students to lead them to the solution.


I let the students know that today we will do a task.  I remind the students of the structure and routine of a task.  First, the students have private work time to think about and plan how to solve the task.  Next, the students work in groups to explore the concept of the lesson.  Finally, the students share/analyze/and discuss the task as a whole class.  Each student has a copy of the task at their desk and grid paper for the area model.  

In today's lesson, the students use their understanding of multiplying using the expanded algorithm to solve a real-world task without direct instruction.  They will be guided to the answer through questioning by me as they work in their groups.  They have to multiply a whole number of up to four digits by a one-digit whole number using strategies based on place value (4.NBT.5). Before they begin their private work time, I ask the students "When might you need to multiply a two-digit number by a two-digit number."  One student responds, "If a group is going to the movies and it costs $15 each to get in."  By asking these questions and guiding their thinking, I am letting the students know that this is an important skill in their everyday lives.  They will use this skill as adults.


10 minutes

I give the students about 5 minutes of independent time to read and plan to solve the Football Team Task.   The students have Centimeter Grid Paper on their desks which they can use at this time to plan how to solve the task.   The grid paper will help the students understand how to multiply using place value.  After the 5 minutes of independent planning, the lesson goes to the next phase of group exploration.


The high school football team has to play 16 games this season.  They will need 21 bottles of water and 37 bottles of Gatorade at each game.  How many bottles of water and Gatorade will they need in all?


1.  Use the expanded algorithm to break apart into 4 simpler problems.  Find the product.

2.  Use the grid paper to illustrate an area model. (MP5)

3.  Write to explain how the area model can help you solve the problem.


 Challenge:  (For the groups who finish early)

What if 2 games get canceled?  How many bottles of water and Gatorade will be subtracted?  How many bottles will they need at the games now?

What strategy did you use to find the answer?



Group Exploration

20 minutes

During the group exploration/discovery phase, the students work in pairs.  Each group has a copy of the task.  The students must work together to complete all requirements of the task. The students are required to use the expanded algorithm to break apart the problem into 4 simpler problem, then illustrate with an area model (4.NBT.5).  The students reason abstractly and quantitatively by decontextualizing the information from the task and representing it symbolically (MP2).  During this phase, the students do not receive direct instruction.  In this lesson, they apply skills previously learned.   The students are guided to the conceptual understanding through questioning by their classmates, as well as by me.  The students communicate with each other and agree upon the 4 simpler problems and the area model.  This takes discussion, critiquing, and justifying of answers by both students (MP3).  As the groups discuss this task, they must be precise in their communication within their groups using the appropriate math terminology for this skill. Each pair has grid paper to use for their models, thus giving them a visual of the product for the multiplication problem. As I walk around, I am listening for the students to use "talk" that will lead to the answer.  I am holding the students accountable for their own learning. 

 During the phase, I monitor and assess the students' progression of understanding through questioning.  Possible questions to help lead to the solution are as follows:

1.  What is the task asking you to find?  

2. After you find the 4 partial products, what do you do next?

3.  How can the grid paper help you solve this problem?



15 minutes

During this phase of the lesson, student solution paths are shared.  While the students were working in groups and I was walking around questioning, I identified solution paths to be shared as a whole class for this phase.  

I call groups to the front to share their solutions.  This is a teaching opportunity for the few students who may still not know how to add according to place value.  I lead this part of the lesson by asking assessing questions.  The students may also have questions that they would like to ask. 

During this phase, I like to organize the sharing of the solution paths in a strategic manner.   I begin with a group that did an excellent job at breaking the problem apart into 4 simpler problems.   Next, I have a group share who solved their problem differently from the other pairs.  There was only one group who added the bottles of water and Gatorade.  You can see by the sample Student Work.

I feel that this is a well rounded lesson on using area model/arrays and expanded algorithms to multiply a 2-digit number by a 2-digit number because the students are held accountable for their own learning.  They have been given the tools and resources necessary to accomplish solving the task.


10 minutes

After the share/discuss/analyze phase of the lesson, close the lesson out by having the students do an Exit Ticket Arrays and Expanded Algorithms.  This will enable me to see how well the students understood how to use a place value chart to add multi-digit numbers.  

The students receive an exit ticket to complete their answers.  I collect these exit tickets to evaluate the students' understanding.  Those students who need remediation will work with me in small group the next day.

After analyzing the students' answers to the exit ticket, I found that 11 students were able to complete the exit tickets with the correct answer.  A sample of Student Work is provided.  This student used place value to find the product of a 2-digit by 2-digit number.  There were 7 students who made some type of error on the exit ticket.  I found mistakes like:  not paying attention to place value and adding incorrectly.  In the Video of Sample Student Work - Expanded Algorithm, you will see examples of the student's errors.  I will work with these 7 students to help them master this skill.