Looking at Different Multiplication Methods

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SWBAT compare and analyze methods of double digit by double digit multiplication.

Big Idea

Students will participate in a group activity as they look for patterns and connections between multiplication methods.

Number Hook - Magic Trick

4 minutes

This is a dice trick that many students figure out after several repetitions. 

I call this trick Add 'em Up! 

First, I ask a student to roll two dice.  I then remind them that I have the ability to see through dice and can add up all the sides, (minus the top two since those can be seen ) without looking at the sides.  Once the dice are rolled, I subtract the top two numbers from 42.   Then I announce what all the faces of the dice add up to, minus the top two numbers after the students has had the chance to add up all the sides of the dice. 

The way this works:

This trick works by knowing that on each dice, there are 6 faces.  Opposite faces add up to 7, therefore there are three opposite sides totally the sum of 7 on one die and six for two dice.  6 and 1, 2, and 5, 3 and 4.  Six times 7 is 42.  Therefor, if I subtract the top numbers from 42,  and then tell the students what the unseen faces of dice total. 

Note: This trick's margin occurs by students adding the dice faces incorrectly.  Since I have done similar dice tricks, my students are figuring this trick out.  They are very close. They realize that opposite faces add up to seven and that this is the key to this trick.  They aren't quite putting all the pieces together, but are getting very close. 

Warm Up

5 minutes

Students solve 22 x 42 on an index card as a warm up and review for today's lesson.  The index cards will be used in the concept development of this lesson.  I also have students sit on the floor during the concept development as a change of pace and engagement strategy.  Students are accustomed to sitting in chairs and moving around the room, but we very rarely sit on the floor. This simple change for the day is one way I keep students engaged.  

Having them sit on the floor from the beginning of the lesson, I am able to rearrange my classroom tables to accommodate the activity during the Graffiti Wall activity in the concept development of this lesson. 

I have a student model how he/she arrived at their product.  In the video below, you can see a student talking through the model and how to use it.  I like this video because I think it clearly shows the amount of thinking that many fourth grade students do at this point with multiplication. He's working to solidify place value concepts about multiplication and make sense of the area model. 


Concept development

40 minutes

The shortcut method (standard algorithm) is the common method currently taught in many schools in the United States.  It is a complex method that is difficult for many fourth graders, especially when multiplying a two-digit number by a two-digit number. For many students, it is a process that is memorized rather than understood.  In this activity, I relate it to the Expanded Notation Method by first dropping steps and switching the order of multiplication from the highest place value (tens in this case) to the lowest place value (ones in this activity). The shortcut, or the standard algorithm is difficult for many fourth graders and is more of a focus in grade 5.

I begin this lesson by showing this Multiplication Chart with Double Digits which has several different multiplication methods and purposely does not have the standard algorithm. I ask students to observe and analyze the chart silently for 1 minutes.  Then, I ask students to talk with their learning partner and write on their whiteboards two things they notice and one question they have from looking at these methods.  

I incorporate Math Practice Standards 7 in this lesson by asking students to look for patterns and structures in the strategies and evaluate which strategy works for them.  My students are getting more used to this and are using phrases like, "this reminds me of" quite often. I then model each strategy for students, discussing and noting the connections between strategies as I model them. The methods I model are the area model for multiplication, and two versions of partial product methods in which students use expanded notation for numbers. One method has less writing and could be considered more of a shortcut while still relying on students place value and expanded notation skills. 

I give students 2 index cards. Each index card is different and each index card has a double digit by double digit multiplication number sentence on it. Hanging around the classroom, are long pieces of white paper I call Graffiti Walls. Students spend the rest of the class time moving to the various Graffiti Walls to find the product for the number sentence on their card. Each Graffiti Wall has a different multiplication method listed for students to show. By exposing students to multiple strategies of multiplication, they are able to deepen their conceptual understanding of multiplication and make connections and relationships between the strategies incorporating Math Practice Standards 7 and 8.

Students are assigned homework for this lesson.  On a piece of lined paper, I have students write down these 2 problems.  

58 x 39 = 

63 x 26 =

They may choose any method, except the standard algorithm, and multiple methods to find the products. My goal for this homework is for students to practice finding products using paper and pencil and begin to discover methods that are efficient for them.  I want students to start discovering which strategy resonates with them so they can refine this strategy and work towards fluency of this strategy.  At this point in the unit, I discourage the use of the shortcut method, or the standard algorithm.  I purposely call the standard algorithm "the shortcut" for my students so they can understand that the standard algorithm truly is a shortcut, but in order to understand the shortcut we must first be solid in place value understanding with multiplication and properties of operations. 

I tell students that in fourth grade, the focus in multiplication is about discovering patterns between methods and being able to use place value strategies and properties of operations in order to explain  how double digit by double digit multiplication works.

See the resource section for samples of what my students created on the graffiti walls. 

Student debrief - Wrap up

10 minutes

Today's exit ticket asks students to finish this sentence:

I like method ________ the best so far because _________________________________.


This type of exit tickets allows students to self reflect about what method of multiplication they prefer. I find it to be a nice change of pace from computation based exit tickets and allows me to stay connected to my students ideas and thinking.  In analyzing exit tickets, I can then group students together who prefer the area model and design lessons to extend their thinking and allow them to make connections with other methods.  Or, I could group students by mixed preference and allow each student to be the "expert" on that method and teach their partner.  For today's purpose, I want mostly to gauge what method most of my students prefer so I can plan future lessons to extend thinking. 

Click preferred method to see a student's exit ticket and my thoughts about this style of exit ticket.