Free Area Model Division Teaching Resource

Are you preparing to teach division through the AREA MODEL, RECTANGLE/BOX METHOD, or PARTIAL QUOTIENTS METHOD? Well, I’ve got a free set of division task cards for you that will allow you to give you students opportunities to develop a deeper understanding of what division is, apply a variety of methods for dividing whole numbers, and enjoy problems that have real-world contexts.

If you have not read my last post about how I introduce the area model for division, be sure to check it out as I give you the step-by-step way that I create a need for students to think about division through a real-world example and engage them in applying the partial quotients/area model to track their thinking. If you have read that post to get ideas for INTRODUCING YOUR WHOLE NUMBER DIVISION UNIT, then you are ready to learn more about these free division task cards!

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After introducing area model division, I use these task cards to give students a real-world opportunity to practice dividing items. All of these problems are EQUAL SHARE division problems where we are dividing or sharing a certain number of items equally for a given number of people or groups.

Although I am calling these task cards AREA MODEL task cards, they can be used for the partial quotients division method too. So, let’s quickly define these two methods.

AREA MODEL DIVISION: In the area model for division, the components of division (divisor, dividend, and quotient) are displayed with a rectangle—the rectangle makes the connection between area (length x width) and division. This method is also known as the rectangular array model, the box method, and the array model. Students subtract multiples of the divisor in each step until they can no longer subtract the divisor. Each step of their subtraction is represented inside of the rectangles and partial quotients are placed above the rectangle.

PARTIAL QUOTIENTS DIVISION: Partial Quotients looks similar to the standard algorithm for division. Students subtract “parts” of the dividend in multiples of the divisor until they can no longer subtract the divisor from the dividend. They keep up with their “partial quotients” by writing them down the right side of the division problem.

While both methods for division allow students to rely on mental math strategies and partitioning the dividend until they can no longer “take away” a multiple of the divisor, the area model provides students with the rectangles as a way to organize their subtraction work. I like it the most because each new rectangle feels like a “fresh start.”

THE BENEFITS OF DIFFERENT MODELS FOR DIVISION?

  • With both of these models, students can start with any partial quotient that makes sense to them. They can think about the multiplication facts that they know and use multiples of 10 and 100 to take out larger parts of the quotient. (Some teachers call these “friendly numbers.”)

  • Students use number sense and their understanding of multiples, factors, and multiplication, rather than memorization of a set of steps that are DISCONNECTED FROM THE ACTUAL MEANING OF DIVISION.

  • When we teach students the standard algorithm for multiplication or division, we rely on the steps that SOMEONE ELSE CONCEPTUALIZED AND BOILED DOWN AS THE “STEPS TO FOLLOW.” These algorithms, when introduced (and taught for mastery and memorization) before students have been given opportunities to work with multiplication and division in concrete and representational ways, take away students’ opportunity to develop conceptual understanding of multiplication and division.

Obviously, I am a fan of using different methods of multiplication and division with 3rd, 4th, and 5th graders! If you check out that previous blog post, you’ll see one way I try to help students develop a conceptual understanding of division.

A CLOSER LOOK AT THE AREA MODEL/PARTIAL QUOTIENT TASK CARDS

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While I LOVE using word problem task cards, I designed these division task cards to help students get a better understanding of division through visualization and being able to imagine the situation. Each task card contains a cardboard box of something that students can relate to—books, pencils, donuts, cupcakes, erasers, folders—and these items need to be “shared among” a given number of students, teachers, friends, or stores.

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I’ve also included a second version of these task cards where students can “roll a divisor.” This creates an endless supply of practice problems that you can use in a station or small group after students have had experience with the original set of task cards.

Prepping the Area Model Division Task Cards

(1) Decide how you want students to engage with the task cards.

  1. Students can model the “equal sharing” and “partial quotients” with base-10 blocks. (Conceptual)

  2. Students can draw circles to represent the divisor and give out “equal parts” until they “run out” of the item they are sharing. (Representational)

  3. Students can be expected to structure their work using the area model or the partial quotients method to show their steps. (Abstract)

I RECOMMEND MOVING STUDENTS THROUGH EACH OF THESE PHASES. Using base-10 models is great for whole group instruction and small group instruction (for students who need more time to understand what division is). You can gradually release students to a more representational or abstract way of solving the problems. Allowing students to draw circles to represent the divisor (people or groups that the items are being shared among) allows them to grasp the “partial quotients” aspect of the division models that you will expect them to use to represent their work in later lessons.

(2) Choose the student recording sheet that you prefer. I’ve included three different recording sheets.

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  1. Students show their steps and work on notebook paper and record their final quotients on the recording sheet. This is a great option if you have students MODELING the situations with base-10 blocks or have them use circles for “divvying out” partial quotients.

  2. The second student answer sheet option contains space for students to show their steps using the AREA MODEL for all 30 problems. This is a good option if you plan to have students work on the task cards over multiple days and expect them to complete most of the task card problems.

  3. The third student answer sheet is the same as the second, except the “show your work” space is not numbered. Students simply write down the number of the task card they are showing their work for.

(3) Assign students to partners strategically. THINK:

  • Who needs the support of a student who can help guide them through the activity?

  • Who works well together?

  • Who might I pair together so that I can rotate around and support them often?

  • Who might I pair together and focus my attention on?

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