Using an Area Model to Divide Tens and Hundreds
Lesson 4 of 16
Objective: SWBAT use an area model to divide tens and hundreds by a one digit divisor.
Math magic tricks can liven up any math class and create a sense of wonder and curiosity about math. Not only that, math magic creates a new context for algebraic reasoning as students go beyond "What's the answer?" to explore "What's the trick?"
Many math magic tricks call on students to compute with the four basic operations -- sometimes applied to very large numbers. In the context of math magic, computational practice is fun.
If a trick works for some students and not others, it's amazing how eager they can be to find and fix their mistakes so the trick will come out right.
Best of all, many students have an inner motivation to understand how math magic tricks work, and that curiosity can lead them to embrace and apply both new and familiar concepts and skills, including algebra.
This is the magic trick I started today's math class with. I call it 19.
Amazing Cool Number Crunching Card Trick where you come up with a number of cards in a pile using the following method.
Method: Shuffle a pack of cards well. Holding the pack face down, turn over the top card and place it face up on the table. Think of it as a pile. Starting with its face value, deal face up on top of it as many more cards as needed to reach 10.
For instance if it's a 3, deal seven cards on top of it; if it's a 5, deal five cards. Face cards count as 10, so no more cards are needed. An ace counts as 1 and needs nine more cards.
Continue making piles as above, keeping them separate, until you have used up all the cards in the deck. If there are not enough cards to complete a final pile, keep that incomplete column in your hand.
Now choose at random, any three piles that contain at least four cards each and turn these piles face down. Gather all the remaining cards in any order and add them to the cards (if any) in your hand.
Pick any two of the three face-down piles on the table, and turn up the top card on each of those two piles. Add their values together. Remove this number of cards from those in your hand, and then remove an additional 19 cards.
Count the remaining cards in your hand. Now turn up the top card of the third pile. Its value will equal the number of cards in your hand.
This lesson uses Math Practice Standard 4 - Model with Mathematics. Models can help make multiplication and division with integers more tangible. They provide a visual representation of the algorithms students use to perform these operations and help students see how the algorithms relate to what is actually happening with the manipulatives.
If you are unfamiliar with the area model for division, sometimes referred to as the rectangle or box method, you can watch a short explanatory video below. I did not create this video, however, it does a nice job at explaining the steps you will see in this lesson.
I start this lesson by showing this photo. I wait for about a minute before I ask students what they notice about the photo. Many students notice that it looks like an area model. As my students look at this photo many are able able find things they know like, 60 x 4 is 240, and 5 x 4 equals 20.
Then, I let students know that this is another division strategy that I call the rectangular array or the area model strategy. It uses a rectangle with known area and known side length to find the missing side length by decomposing the rectangle into sections. The sections of the rectangle shown are built up one piece at a time as I deal with each place value in the dividend.
I then engage students in a discussion that breaks apart this method's steps that goes something like this:
Step 1: I first drew a rectangle and wrote the dividend inside it. I wrote the divisor (or other factor that I knew) on the left side of the rectangle.
Step 2: Then I thought about what number times 4 would give me an answer close to 260 without going over. I thought, “60 x 4 = 240, so 60 gives me an answer close to 260 without going over.” I then wrote the 60 above the rectangle, multiplied the 60 x 4 and wrote the product, 240, under the 260 inside the rectangle. I subtracted 240 from 260 and wrote the difference, 20, below the rectangle.
Step 3: Since I still had 20 to deal with, I knew I needed to continue by expanding the rectangle to the right of the first section. I wrote 20 in that section.
Step 4: I then thought about what number times 4 would give me an answer close to 20 without going over. I thought, “5 x 4 = 20.” I wrote 5 above that section of the rectangle, multiplied the 5 x 4 and wrote the product, 20, under the 20 inside the rectangle. Then, I subtracted 20 from 20 and wrote the difference, 0, below that section of the rectangle.
Step 5: Since the difference in the last rectangle was 0, I knew I was finished with my work inside the rectangle. I added the quotients from each section to find the final quotient: 60 + 5 = 65.
After this initial problem, students then model a division problem with me in their math notebooks. We model together 56 divided by 4 using the rectangle method and checking our quotients by using multiplication. This allows students to make a connection between multiplication and division and use Math Practice Standard 7, look for and make use of structure. In this practice standard, students look closely to discern a pattern or structure and make connections between the relationship of multiplication and division.
We then model a few more problems as I gradually do less on the board and let students take over showing the model. Problems we did together were:
Then, I had students work on some problems with their learning partner. Near the end of the class period, I went over the correct answers with the class. Students solved:
In this video, you can hear a student working through a division problem using the area model and her productive struggle.