Methods of One-Digit by Two-Digit Multiplication

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SWBAT relate the area model of multiplication to numeric methods of multiplication.

Big Idea

Students look for patterns and structure in different one-digit by two digit multiplication strategies.

Warm Up

5 minutes

Each student is given a card when they walk in the room.  Each card has either a multiplication number sentence with a multiple of ten, or a product.  Students are asked to find their "match" as quickly as they can.  They may whisper to each other to get this done, but I ask them to use low voices.  I time the students and tell them how fast they were able to do this activity.  I end the day with this activity as well and see if students can beat their first time. 

This review activity serves two purposes. One, it is an engaging activity in which students can review, practice, and apply their skills from the previous lessons.  One of the shifts in the CCSS is rigor. Rigor requires conceptual understanding, procedural skill and fluency, and application with intensity.  This activity allows students to build fluency.  The CCSS defines procedural skill and fluency as students are expected to have speed and accuracy in calculation. Teachers structure class time and/or homework time for students to practice core functions such as multiplication facts so that students are able to understand and manipulate more complex concepts.  Multiplying by multiples of ten is a skill I want my students to be fluent in so they can be successful with double digit by double digit multiplication.  

The second reason I chose this activity as a fluency practice is it serves a quick formative assessment.  I can see and observe students that are struggling and make adjustments to instruction immediately.  If students struggle with this activity it gives me a clue that these same students will most likely struggle in the next lessons as I present further multiplication concepts. I can build in scaffolds during those lessons and continue to reteach this concept for students that have not mastered this skill.

Concept Development

45 minutes
This lesson's focus is CCSS 4.NBT.5 as well as Math Practice Standard 7 - Look For and Make Use of Structure

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

One way I develop students’ capacity to "look for and make use of structure" is to help them identify and evaluate efficient strategies for solutions.  This lesson gives students an opportunity to compare methods of multiplication and evaluate the methods for efficiency. Students usually are able to discern that drawing area models for every multiplication problem becomes labor intensive and takes a longer amount of time than the expanded notation method. 

I start this lesson by modeling the area model of multiplication and the expanded notation methods.  Check out the video below for information on these two methods. I don't show this video to students, but the way I model the two methods is very similar to this video.


 I model another number sentence on the smartboard, and show the area model and the expanded notation method for the same number sentence.  I put the two methods side by side in a T-chart. After letting students observe and analyze the two methods silently for about one minute, I then, I explicitly ask  questions like:

How are these methods similar?

How are these methods different?

What patterns do you notice or see in each method?

Which method is your favorite right now?

Which method feels easiest for you to do?

Which method will take a longer amount of time to solve?  

If they both seem efficient, and take the same amount of time, do you think when we learn double digit- by double digit multiplication, one might be more efficient?  Take less time? Note: this is when I really focus on Math Practice Standard 7 and breifly discuss when one of these methods might take longer than the other. At this point, using only double digits by one digit, both methods take about the same amount of time and are both efficient ways to multiply.  As students solve more complex problems, various methods become more and less efficient based on student knowledge and the size of numbers to multiply. 

 For the remainder of the lesson, students practice the area method and the expanded notation method on lined paper.  I let students choose which problems they want to work on for the assignment.  They must choose at least 5 one-digit by two-digit multiplication problems to model the area model for and use the expanded notation method to solve.  I ask them to choose a double digit larger than 21 simply because I want them getting practice with large numbers.  By limiting the smallest number to 21, students who are struggling with basic facts using 7, 8, and 9, can choose smaller double digits and still feel successful and challenged. Based on prior experience, students that do not have their multiplication facts mastered at this point struggle most with facts having 7, 8, or 9 as a factor. 

Giving students a choice for this assignment is beneficial for me and them. Students that have their basic math facts mastered often choose problems that use 7, 8, and 9 as digits in their numbers. Students who have not mastered their facts often choose lower digits like 2, 3, 4, and 5 when designing the problems they want to solve.  Since I want students to master the concept of double digit by one digit multiplication, letting them choose the numbers they want to use in their problems allows them to use facts they are comfortable with while still practicing an important fourth grade skill.

Students finish this assignment as homework if they don't finish in class.  About half of my students finished in class and half needed to take it home to finish.  

In the following video, you can listen to a student talking about what method she likes best.  I wanted to make sure I related my questioning to Math Practice Standard 7 and 8 by asking her about patterns and similarities between the two methods. As you can see in the video, she can identify similarities, but needs some more guidance in her math talk and using correct concept terms like ones and tens when noting the similarities between the place values and partial products in both methods. 

 Click here for a sample of student work in which I share my ideas about this student's progress. 

Click here for another sample of student work and what this student discovered by doing the two different methods side by side. 

Student Debrief

10 minutes

I end the lesson with the activity students started the lesson with. Each student is given a different card with either a multiplication number sentence with a multiple of ten, or a product.  Students then find their "match" as quickly as they can. Once they find their match, they stand with their partner on the outside edge of the classroom.   They may whisper to each other to get this done, but I ask them to use low voices.  I time the students again and I encourage them to beat their time from earlier. 

The students got much faster the second time through. This turned out to be a great formative assessment.  The last four to six students standing in the center of the room had difficulties multiplying by a multiple of ten. This was a great formative assessment because I really thought ALL my students could do this skill quite easily.  I was reminded that I still have a few students that need more support.