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The Division Process: Division with Remainders: Abstract Level

More Teaching Plans on this topic: Concrete, Representational


Phase 1

Initial Acquisition of Skill

Phase 2

Practice Strategies

Phase 3


Phase 4


Download printable version of this teaching plan, with additional detailed descriptions


Math Skill/Concept: Division process and division with remainders using numbers and symbols.

Prerequisite Skills:

  • Concrete understanding of the division process.
  • Ability to solve division equations using concrete materials.
  • Ability to draw solutions for division problems.
  • Ability to use the FASTDRAW Strategy to find the important information in division story problems, to set up a division equation, and to a draw solution.
  • Ability to skip count.
  • Mastery of multiplication facts/ability to use a times table key or calculator to figure multiplication facts.

Learning Objectives:

1) Solve division story problems/equatons without the use of concrete objects or drawings.

Important Ideas for Implementation:

1) This teaching plan provides descriptions of instructional methods and activities that teach students to use the steps of the FASTDRAW Strategy and the DRAW Strategy to solve both division story problems and division equations (without drawing). The plan also provides suggestions for how to help students increase their fluency for basic division facts.

2) The three methods described to teach students how to “answer” division equations at the abstract level can all be helpful strategies for students. Whether you teach one or more than one method is dependent on the needs of your students. The order in which they are presented represents a natural progression from drawing solutions for division equations. The first strategy described, “visualizing pictures,” can be an effective method for students with strong visual memory skills but may prove troublesome for those students who have trouble with visual memory or visual imagery. The second method, “repeated addition,” can be very effective, particularly for students who understand the repeated addition process. Linking division to the multiplication process of repeated addition provides students a solid foundation to practice division at the abstract level. The third method is also an important link to the multiplication process but requires a more sophisticated understanding of what the multiplicands/factors represent and what the product represent. Explicitly teaching what each of these represent and associating this to the division process can be very empowering to students. However, students who have learning problems or who have limited prior knowledge of multiplication need very explicit instruction regarding these relationships.

3) Students who have learning problems will need ample opportunities to practice solving division problems at the abstract level before they become fluent. For students who understand the concept of division but have difficulty remembering the steps in dividing, continued access to the steps of the DRAW Strategy will allow them to independently solve division equations without continual teacher prompting regarding which step to do next. With multiple opportunities to respond, students’ memory of the necessary steps will improve.

4) Allow students to draw solutions as a “back-up” when needed. Providing students this alternative will allow them to work independently. Again, multiple independent practice opportunities are the best chance for students to move to abstract level success!

5) Fading student use of both the DRAW Strategy and drawings is an important final step, however the fading process should occur only for those division facts they demonstrate ability to solve “abstractly.” It is perfectly reasonable and sound instruction to encourage students to use cues for those division problems/facts they have not yet acquired at the abstract level, but to require students not to use cues for those division problems/facts they demonstrate proficiency or mastery of.

6) Providing daily one-minute timings and charting student progress is an excellent method to increase students’ fluency and mastery of division facts (See Instructional Phase 4 – Continuously Monitor and Chart Student Performance in this abstract level instructional plan).

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