Multiple Ways to Multiply With Multiple Digits

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Objective

Students will be able to use the "standard" algorithm with understanding of the properties of operations and the base-ten system.

Big Idea

There are multiple ways to check your answers when using the traditional multiplication algorithm.

Inclusion - Engaging Students in Their Learning

2 minutes

Inclusion Question:

Where have you used large numbers in multiplication before or, if you haven't, where do you think it is used in the "real world". 

Using an inclusion question to start each lesson gives student the "hook" to hang the new information on and brings up prior knowledge. 

I don't have the students talk at their tables, because exposure to multi-digit multiplication is significantly different in my multiage (4th and 5th) classroom. I want my fifth graders to share where they have used or seen multi-digit multiplication.

The first student I call shares, "You tell us your son has to do a lot of multiplication in middle school.  I will have to do multiplication in middle school."  I validate this student's comment with, "Yes, you will need to be able to do large multiplication as you go through school but has anyone seen it used outside of school."  I'm trying to get students to think of "real world" applications. 

I had to give more think time because I could see they were not linking this to real life applications.  After awhile a student raised their hand and the prior speaker tossed the koosh to them to let them know they could speak.  He said, "My uncle lays tile and he was showing me how to find the area of a floor.  He had to multiply 92 feet by 52 feet to find out the size of the room." 

Other students then came up with construction and space references.  Now that I had them thinking about where they could use or see multi-digit multiplication I move into teaching the different ways to check their work against the traditional algorithm. 

Introduction to Multiple Ways to do Multiplication

15 minutes

In 5th grade students need to be able to fluently multiply multi-digit whole numbers using the traditional algorithm. This is difficult for some students, especially if they are still counting on their fingers to do basic skills.  One way to reach the struggling students, as well as challenge others, is to have the students use other multiplication strategies along with the traditional algorithm to solve the problems. This is particularly appropriate for 4th graders, as they are expected to begin using the "standard" algorithm, but where it is still considered appropriate to use computational algorithms that utilize place value and operations understandings. For example, a 4th grader may be using an area model.

One of my students had mentioned the numbers 92 by 52 so I asked for student to tell me what would be a reasonable answer.  I had answers such as 4,500 and they explained because if you round each number and multiply 9 x 5 you get 45 and then there are two zeros so you add them on for 4,500.  Some students used front end rounding to have 100 x 50 for 5,000. 

After we had come up with a list of 4 reasonable answers I started with walking them through the traditional algorithm, reviewing the value of each column when they had to carry.  After completing this I asked if there were any other ways of doing the problem so we could check our answers. (We should always check our answers!) Sofie wanted to use Lattice and came up to solve the problem.  While Sofie is up writing, students sitting in the "audience" are making a copy of her work into their math journals for use as a reference.  

Another student came up to try to do the standard algorithm and made the common mistake of multiplying the 2 x 2 and then the 9 x 5 for an answer of 454.  He sat down without correcting his mistake. Later, when I was working with him, he said he had realized that his answer was not in the list of reasonable answers but he did not know how to fix it.  At a later point in the video another student came up and used the traditional algorithm correctly.  When I reviewed with my class and told them that AJ knew his answer was not reasonable, they went back into their math journals and crossed his work out.  Many already had, indicating it was not a reasonable answer. 

The student you see get up and walk in front of the board is Daichi, a new student from Japan.  He had gotten up to get his translation device and typed in the words reasonable and estimate since I had written it on the board, and kept pointing to the words. He had also heard the students using the words and I am sure was curious what it meant.  Daichi came up in the end of the video to show how he carried the numbers below the line - there is another example on the bottom of the board in the picture.  You can also see how the other students took care of Daichi by erasing the numbers, getting him a marker and then pointing at the board.  They are all doing a great job of having him feel included.  He did make a mistake on the math but I am sure that was because he was nervous being in front of the class.  When I turned the camera off he went back and changed the answer. 

To meet the needs of the diverse learners in my class (a multiage 4th and 5th grade class),  I allow all students to check their work with a calculator.  4th graders could use any format of multiplication to solve the problem but had to do 3 problems in the traditional algorithm.  5th graders had to use the traditional algorithm but if they still didn't understand it they could try with another format.  Not every student completed every problem because my focus was on the fluency of multiplying multi-digit whole numbers using the traditional algorithm - not the students speed at getting the problems completed. By having students connect place value and their prior work with operations to understand the algorithms they are using when multiplying multi-digit numbers, I am meeting Mathematical Practice 8 - Look for and express regularity in repeated reasoning.

I sat on the floor with a group of student who needed help working each problem in the traditional form.  My "expert" multiplication students helped others.  These students finished the entire page and completed some using other methods before they could go on to help other students. 

The other ways to do multiplication include decomposing the numbers.  If the problem is 68 x 15 the students

Practicing Multiple Ways of Multiplication

25 minutes

To meet the needs of the diverse learners in my class (a multiage 4th and 5th grade class),  I have different expectations based on the individual student, but generally there is a group of typical 4th and typical 5th grade skills.  

To align with the Common Core standards, 4th graders could use any format of multiplication to solve the problem but had to do 3 problems in the traditional algorithm (4.NBT.5).  5th graders had to use the traditional algorithm, but if they still didn't understand it they could try using another strategy such as lattice, decomposing, standard and the new way my new Japanese student showed us (5.NBT.5). 

Show Me: Multiplication

A 5th grader who loves to use lattice is showing the class how it is done. 

Not every student completed every problem, because my focus was on the fluency of multiplying multi-digit whole numbers using the standard algorithm - not the students speed at getting the problems completed. 

I sat on the floor with a group of student who needed help working each problem in the traditional form.  My "expert" multiplication students helped others.  These students finished the entire page and completed some using other methods before they could go on to help other students.

I use the site The Math Worksheets Site for skill practice pages, which is the one I used for this lesson. 

Closing - Student Reflection

2 minutes

I always have my students reflect on their learning because it increases their retention of the material.  Today I had my students reflect verbally on content and behavior.  I asked students to popcorn "What did you learn today about multi-digit multiplication?" Answers included comments such as:

I don't understand all the steps in the standard algorithm but I can use another way to find out the answers.  This will help me on tests (district benchmarks are a week away).

I used Lattice last year but you tell me in 6th grade we can't use it so I worked hard learning the traditional way. 

I don't understand it but I know how to take a guess on a test with more than one answer (multiple choice) by looking for a reasonable answer.

We will be continuing to practice multi-digit multiplication for the next few days.  I will also continue to fit it into morning Math Wake-Ups by having the students do one problem a day.