Expanded Notation for Dividing Hundreds

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SWBAT divide three digit dividends by one digit divisors using expanded notation to build quotients.

Big Idea

Students practice a division strategy with a partner.

Warm Up

5 minutes
Being able to multiply and divide fluently within 100 is essential during the early years of schooling, and the basic facts of addition subtraction and multiplication/division are a critical baseline, not only then but also during later work with fractions, decimals, ratio, proportion, and more. 

Researchers have gained insights from brain research about demands on the working brain. As students begin to learn math facts, their brains are focused on those basic computations, but as students become automatic with basic facts, their brains are then able to focus on other aspects of the task like the challenges of place value, decimals, or fractions. Being automatic with basic facts frees the brain to focus on other math processes. Committing basic math facts to memory speeds up math tasks. As math tasks increase in complexity, they often require multiple steps to find the solution.

Because basic facts are so important for my students to master, and more than half of my classdoes not have them mastered, I chose an oral division facts practice for this warm up.  I change the PowerPoint slides every 3 seconds. I encourage my students to yell their answers "loud and proud."  Click here to access the facts practice PowerPoint. 

Concept development

45 minutes

I begin this lesson with the following video about expanded notation and long division. I chose to use a video as an introduction simply for a change of pace. The last few lessons have entailed me doing a lot of modeling and explicit instruction.  Students can become disengaged with lectures, sitting, and listening.  I try to keep this at the forefront of my brain to keep students interested and tuned into the lesson. This lesson also incorporates modeling and explicit teaching, but by using this video as an introduction, I can engage students immediately simply by varying my format of modeling and explicit instruction.



While the Common Core State Standards ask student to think critically and reason about topics and concepts, students still must have a balance of explicit or direct instruct, application of skills learned and building conceptual understanding. Explicit teaching involves directing student attention toward specific learning in a highly structured environment. It is teaching that is focused on producing specific learning outcomes.

I also believe that another important characteristic of explicit teaching involves modeling skills and behaviors and modeling thinking. This involves me thinking out loud when working through problems and demonstrating processes for students. The attention of students is important and listening and observation are key to success.

Students have spent the previous 4 lessons with practicing division strategies.  After this lesson, students will spend one more lesson practicing a specific strategy, and then students will engage in a task in which they must apply their division and multiplication skills.  As defined in the common core standards, rigor is the balance of procedural practice, application of skills, and conceptual understanding.  Students have spent much time lately practicing procedures, however, they will have opportunities in future lessons to apply these skills.

For the remainder of the lesson, students sit on the floor or find an area in the classroom to work with their learning partner to solve division problems. Students take turns listing and writing a three digit dividend divided by a one digit divisor division problem. Allowing students this choice in choosing their problems is a deliberate engagement decision.  Students are naturally more engaged when they get to choose the problems they work on. Also, students naturally pick numbers they are more comfortable with.  By observing students, I can gauge which students are comfortable and proficient with dividing numbers by 7, 8, and 9, and also which students are always choosing divisors like 5 and/or lower numbers.

 This lesson incorporates Math Practice Standard 7 as students look for patterns of structure in the expanded notation method.  Students  begin to recognize the significance in concepts and models and they make connections between the area model and expanded notation model, and can then in future lessons apply strategies for solving related problems.

 Listen to this video to observe a student attempting to make connections between the area model and multiplication and division.

See this same students work sample (taken just before this video) and see how he is using both the area model and expanded notation method to make use of structure and connections between the concept of division and the two strategies.  Click here to see the work and hear my thoughts about this student's work sample.

Student debrief - Wrap up

7 minutes

For this lesson's wrap up, I have all students solve a division problem on their personal whiteboard. This serves as a formative assessment for me and I can make instructional decisions immediately based on answers and strategies I observe.  When students finish solving their division problem, I ask them to quietly show their whiteboard to me by holding it above their heads. I ask them to keep their boards up until I give them a signal to put them down. By allowing students to hold their boards up, I am scanning and watching students in the class who do not have boards up and watching to see which students are glancing up at other students board for help.  This is a great clue to me to be aware of so I can make sure I conference with these students, small group these students, provide extra practice or scaffolds, and re-teach these students as necessary.

You can see in this video there are several students who finish way ahead of the other students. This is also a sign to me that these students may need an extension activity and could also serve as peer teachers when I design group and partner tasks.