Two-Digit by Two-Digit Multiplication

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Objective

SWBAT represent two-digit by two-digit multiplication using area models.

Big Idea

Students will work towards mastery of multiplication by two-digit numbers.

Rationale

The inability to recall number facts and perform arithmetic fluently is often cited by teachers as a significant barrier to students’ mathematical performance. This lesson gives students some strategies to overcome these obstacles with respect to  multiplication. They learn to derive more difficult facts from easier facts, and they perform multiplication using an area model. The key feature of the area model is that it represents the product of two numbers as a rectangular region made up of unit squares. The area of a rectangle is the number of unit squares that make up the rectangle.

This lesson connects to algebra and algebraic thinking in various ways. In the lesson, I use multiple representations to model the process of multiplication. The approach exposes learners to both the Commutative and the Distributive Properties. These two properties are fundamental to the real number system and to success in manipulating algebraic symbols. In some respects, the area model for multiplication establishes the groundwork for conceptual understanding of the algebraic skills of polynomial multiplication and factoring.

Number Hook - Magic Trick

4 minutes

To see this part of the lesson unfold, watch: Classroom Video: Transitions

Here's a really cool Math Magic Trick that students can use to impress family and friends and build basic math skills too! .

Materials: 5 dice

Performing the Trick:

  1. Begin by telling the students that you can see through the dice all the way to the bottom numbers. Since you can see through the dice, you will be able to tell them the sum of the tops and bottoms of all five dice. Of course, exclude the top number since everyone can already see that one.
  2. Choose a student volunteer. Have the volunteer make a tower of all five dice. (I display this under the document camera for all to see.)
  3. Now, pretend that you are looking through the dice to see the bottom numbers. What you are actually doing is subtracting the top number from 35.  The opposite sides of a die always sum to 7. Since, 5 x 7 is 35 the number of dots on the tops and bottoms of five dice always add up to 35.
  4. Announce the sum of all the tops and bottoms of the dice. 
  5. Ask the volunteer to turn over the 5 dice and have the spectators sum the numbers. They will be amazed at how you did it!

 

 

 

Warm Up

4 minutes

To see this part of the lesson unfold, watch: Classroom Video: Warm Up

I start this lesson by playing this song as a review for my students.  

 

As third graders, students worked with to understand area as related to multiplication when mastering standards 3.MD.G.5, 3.MD.G.6, and  3.MD.G.7.  For most students this will be review, but I have found that most students benefit from the review.  Students always benefit from math vocabulary review and this song is a very engaging way to do that. 

I then quickly review previous lesson work by having students complete a problem on their whiteboards. Students solve 58 x 8 on their personal whiteboards and show the area model.  Students' personal white boards have a plain white side and a side that has dots on it.  If students need the dots to help with finding the product, they are allowed to uses this side. Most students are at the abstract stage of understanding the model, without the dots.  Most students will use the plain white side without dots. 

Then I explain that we will continue our use of the area model to help us find the products two-digit by two-digit numbers. 

Concept development

40 minutes

To see this part of the lesson unfold, watch: Classroom Video: Developing a Conceptual Understanding

Students begin this lesson with their personal whiteboards. My students have personal whiteboards that have dots on them in 10 x 10 squares.  I use this board with the dots today. I ask students to sketch arrays to visualize what is happening to the digits in a double digit by double digit multiplication problem.

Students start by drawing a double digit by double digit array like 24 x 37.  A typical student will count 24 rows of dots then count 37 columns.  When students have completed their rectangles, I ask how they can show the tens and the ones in 37 on their area model.  Some students will respond that they can draw a vertical line after measuring 30 to show 3 tens and 7 ones.  

Instructional Notes:

    1. In the past, students have struggled with this model when drawing the vertical line. This drawing may take more than one lesson for student to grasp.
    2. I am cognizant of how important it is to use math vocabulary precisely in this lesson. I will use words like decompose and Distributive Property with my students in this lesson.  

I then ask how we can model the tens and ones in 24.  Again, some student will respond that you can draw a horizontal line after 20 or 2 tens distinguishing 24 as 20 + 4.  You can see what that looks like in the area model photo in  the resources.

Once these lines are drawn decomposing the two multiplicands, we can interpret the area model as follows:

  • The top left section shows the tens times the tens or 20 x 30. 
  • The top right section shows the tens in 24 times the ones in 37 or 20 x 7. 
  • The bottom left shows the ones in 24 times the tens in 37 or 4 x 30. 
  • The bottom right shows the ones in 24 times the ones in 37 or 4 x 7.  

As I talk through this with my students, I write the equations inside the corresponding box on the area model.  I then refer back to the model and ask students to count the square units to see how well my numerical expressions describe the area of each piece.  Students draw lines on their boards to mark 600, 140, 120, and 28.

From prior learning, some students are able to make sense of the area model easily and quickly. Others may need repeated experiences.  In this lesson I model about three problems and then let learning partners work through problems together on their whiteboards. 

For Partner Practice, I list eight different problems on the board.  Learning partners must choose 4 to 6 of them to model on their whiteboards.  I do not assign specific problems for this assignment for several reasons.  First, I believe that giving my students choice in determining  the problems they work on builds their ownership in learning. Second, allowing students to choose their work is another strategy for differentiating instruction. Students will pick the problems that interests them the most and, at the same time, self-differentiate according to their capacity and needs.

While this assignment limits the numbers students can choose by the size of the dots on the whiteboards, students can still choose their learning path which enables them to feel challenged and be in charge of their learning.  This helps push my students in developing their growth mindset. By emphasize that there are strategies for learning material, (multiplication in this situation)—is one way to encourage a growth mindset. 

Student Debrief

10 minutes

To see this part of the lesson unfold, watch: Classroom Video: Student Self- Assessment

This is an exit strategy I call footprints. I ask students to write down some new knowledge they are walking out the door with today. Students write this on a small piece of paper, about sticky note size, and then tape it on a giant footprint outside our classroom door.  Later I look at these to determine if there are misconceptions that need cleared up.