Long Division: Strategies and Extensions for Upper Elementary Math
By Mary Montero
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I’m so excited to chat all about teaching long division! Ah, division. Does the mere mention of it make the hair on your arms stand up? And the timing! Why do we always seem to be teaching long division during the holiday season, when energy is high and focus is low? No matter the time of year, having a clear, well-designed succession of division lessons can eliminate a lot of the usual angst that comes with teaching long division in 4th grade…and beyond!
This 4th grade standard calls for students to use place value strategies to find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. That’s a mouthful, but let’s tackle it!
When we’re thinking about division and planning our division unit, the first thing I think about is the specific strategy progression that I’m going to use to teach the concept. Students simply must have a conceptual understanding of HOW division procedures work. Even though it’s not a standard in 4th grade, ultimately, our goal is proficiency with the standard algorithm for long division so that when they begin to move through the many steps of the procedure, they can conceptualize each step and understand what they’re doing rather than just memorizing DOES MCDONALDS SELL CHEESEBURGERS. I have absolutely nothing against the acronym, and I have definitely used it on an anchor chart, but understanding the actual process will help them significantly more than memorizing the steps that are easy to get out of order and mix up. You can see my division anchor charts and other long division strategies on this post.
Multiplication and Division Review
The first thing I need to do at this point is be sure I’ve already reviewed multiplication and division with my students extensively. We talk about how they are inverse operations, we talk about what division is, model basic division, and make sure students know how to use multiplication to check their division answers (and vice versa). Almost ALL of the division strategies we are going to chat about build on similar multiplication strategies, which always tend to seem less daunting for students, so I encourage you to tackle those multiplication strategies first. Need a multiplication strategy refresher? Read this blog post all about multiplication strategies.
Long Division Strategies
Let’s talk division strategies! When I’m talking about strategies, there are several that I always introduce before moving on to the standard algorithm. These are critical for students to understand WHY the standard algorithm works. WHY do I present all of these strategies to students? Because for most children, ONE of these will cause their lightbulb moment when they say, “AH! I understand WHY I am doing what I’m doing with this particular procedure right now, and I know WHY I do this step next.” THAT is my ultimate goal when teaching these strategies.
Here’s a quick overview of the strategies I teach my 4th graders:
- Place Value Strategies: Using multiples of 10 (This should have already been introduced with multiplication, so this is the first place I start with students)
- 6400 / 8 is actually 64 / 8 x 100
- Expanded Form or the Distributive Property (246 / 2) = (200 +400+6) / 2
- Area Model
- Partial Quotients
- Standard Algorithm — Don’t throw tomatoes, but I always introduce the standard algorithm, even in 4th grade, because I find that once they have these strategies in place, it tends to come fairly quickly for some students and becomes another strategy they can use. Our end goal is total and complete conceptual understanding.
I have created these free long division posters to help you and your students master these long division strategies!
Beyond Long Division Strategies
You’re not done once your students have developed an understanding of standard algorithm or another strategy! There are so many other pieces and parts of division that are critical to teach. Let’s dive into two areas that are critical for a deep understanding of division and are often forgotten. Incorporating error analysis and interpreting remainders into your division instruction is critical. Whether you have taught just strategies or the standard algorithm, I implore you to incorporate these two components into your division unit.
Long Division Error Analysis
I am a huge fan of error analysis to increase critical thinking and understanding. You can read all about error analysis here. Marzano specifically calls out error analysis as crucial to building critical thinking and conceptual understanding in students. Being able to judge, defend, interpret, and correct fallacies requires far more thought and understanding than just solving a problem.
Using Error Analysis with Long Division
- Once students show proficiency in the standard algorithm (or strategies), take it a step further and have them dive into error analysis where they can show a “reverse” understanding as they evaluate mistakes made and fix them. Being able to identify an error in someone else’s work requires higher order thinking not found in most other projects or activities and certainly not found in basic math fact completion.
- First, teach students the difference between a computational error and a conceptual error.
- Computational is when they make a mistake in basic math facts. This might look as simple as 64/8 does not equal 7. Oops!
- A Conceptual or Procedural Error is when they make a mistake in the procedure or concept. This usually happens when students don’t have a good base of why strategies or algorithms work and have just memorized steps.
- I can’t tell you how many times students show as not proficient on a topic when the mistakes they are making are COMPUTATIONAL and not conceptual or procedural. They don’t need more review on how to use a strategy… they need to slow down and pay closer attention to their math facts.
- Once we’ve introduced the types of errors (conceptual and computational) students should be looking out for, we move on to actually analyzing these errors in someone else’s work and fixing the mistake.
- I have created error analysis division tasks for you to use with you students so they can identify the errors, types of errors, rework the problem, and create their own version of the problem and solve it. I have seen great success with incorporating these tasks into ALL of my math units. I even have kids beg to take their error analysis tasks out to recess to finish!
- First, teach students the difference between a computational error and a conceptual error.
- The final step in using error analysis is actually having students correct their OWN mistakes. Once I have instructed on types of errors, I will start by simply telling them, “Oops! You’ve made a computational error here!” That way they aren’t furiously looking through the procedure for a mistake, instead, they are looking to see where they computed incorrectly. Conversely, I’ll tell them if they’ve made a procedural mistake, and that can guide them in figuring out what they need to look for.
- Looking at the different types of errors students are making is essential to guiding my instruction as well, so even though it takes a bit longer to grade things like this, it is immensely helpful to me as I make adjustments to my instruction.
Division Word Problems
One more quick tip to to encourage even more critical understanding of division… Have your students work to generate their own problems! Once they can solve and check their work, it’s time for them to generate their own division scenarios/word problems!
- This is less about the algorithm and more about having a critical understanding of the purpose of division and being able to generate real-world scenarios that apply to division.
- It is really challenging for students if we tell them to simply go create a division problem, so give them some guidance. Give them numbers and topics to guide their generation of problems so that they don’t get stuck on two rather unimportant parts of this process. For example, I created these Write and Solve Long Division Task Cards. It gives them numbers and a scenario and students must generate and solve the division problem.
- Remainders are a serious pain point for students, particularly in 4th grade. Helping them to interpret their remainders and evaluate them for reasonableness is key to their success with division. I have several remainder reminders that I ALWAYS push for my students. This poster is also included in the free download above!
- Hot tip remainder reminders for students:
- Don’t be afraid of Domain-Specific Vocabulary! Point out that the word “remain” is in the word remainder. This is the value that remains and can’t be evenly divided into your divisor.
- Your remainder can’t have a value greater than your divisor. I am stickler for math vocabulary, so it’s really important to me that students can use that specific language (divisor, quotient, dividend).
- The remainder CAN have a value greater than your quotient. This happens, and it happens often, so be sure to give a lot of examples of this.
- When a number divides evenly into another number, the remainder is technically ZERO.
- An even number divided by 2 should never have a remainder.
- Your remainder always serves a purpose. It’s not always just the “leftovers.” We’ll talk more about that shortly.
- Visualizing Remainders:
- Use arrays to visualize remainders as leftovers.
- Use manipulatives for visualization.
- Use your easiest manipulative: human bodies! Do a ton of scenarios using the students in your class, including problems with remainders… this is a great time to do a transportation problem. For Example: Ms. Smith’s class was going on a field trip. There were 24 kids in the class, and each van holds 7 students. How many vans will they need?
- Use number lines to help represent remainders
- Interpreting Remainders: When I started adding a lesson or two on interpreting remainders, my students’ understanding of not only remainders, but of understanding word problems, increased dramatically. The act of interpreting remainders should be taught after students have a good understanding of performing division using strategies or the algorithm. We focus on three different ways remainders can be used:
- Use the remainder — This happens when the remainder is actually the answer.
- Round Up – When you use the remainder to round up your answer. The quotient is not your answer, but it’s one greater because you have a remainder. (For example: when you are splitting children among cars to go to a summer camp. You can’t simply leave two students out, so you need to add another car to transport everyone)
- Ignore – Sometimes we ignore the remainder. This usually happens when we are finding truly equal groups.
- Share the Remainder as a Fraction or Decimals – I MAY give my students a little blurb about this option, particularly students ready for this information, but in general, I do not teach this to 4th graders as it begins to muddy their understanding.
Giving students AMPLE opportunities to interpret the function of a remainder (without even requiring students to solve the problem) is critical to their understanding.
I hope these tips have helped you as you get started with your division unit!
I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.