Prior to grade 6, students reasoned about division of whole numbers and decimals to the hundredths in different ways. In this first lesson on division, they revisit two methods for finding quotients of whole numbers without remainder: using base-ten diagrams and using partial quotients. Reviewing these strategies reinforces students’ understanding of the underlying principles of base-ten division—which are based on the structure of place value, the properties of operations, and the relationship between multiplication and division—and paves the way for understanding the long division algorithm. Here, partial quotients are presented as vertical calculations, which also foreshadows long division.
In a previous unit, students revisited the two meanings of division—as finding the number of equal-size groups and finding the size of each group. Division is likewise interpreted in both ways here (MP2). When using base-ten diagrams or dividing by a small whole-number divisor, it is often natural to think about finding the size of each group. When using partial quotients, it may be more intuitive to think of division as finding the number of groups (e.g., \(432 \div 16\) can be viewed as “how many 16s are in 432?”).
Lesson overview
- 9.1 Warm-up: Using Base-Ten Diagrams to Calculate Quotients (5 minutes)
- 9.2 Activity: Using the Partial Quotients Method to Calculate Quotients (15 minutes)
- 9.3 Activity: What’s the Quotient? (15 minutes)
- Lesson Synthesis
- 9.4 Cool-down: Dividing by 11 (5 minutes)
Learning goals:
- Comprehend that the phrase “partial quotients” (in spoken and written language) refers to decomposing a division problem.
- Divide whole numbers that result in a whole-number quotient, and explain the reasoning (using words and other representations).
- Interpret different methods for computing the quotient of whole numbers, i.e., base-ten diagrams and partial quotients, and evaluate (orally) their usefulness.
Learning goals (student facing):
- Let’s divide whole numbers.
Learning targets (student facing):
- I can use the partial quotients method to find a quotient of two whole numbers when the quotient is a whole number.
Required materials:
- Graph paper
Required preparation:
- Students draw base-ten diagrams in this lesson.
- If drawing them is a challenge, consider giving students access to:
- Commercially produced base-ten blocks, if available.
- Paper copies of squares and rectangles (to represent base-ten units), cut up from copies of the blackline master of the second lesson in the unit.
- Digital applet of base-ten representations, such as: https://www.geogebra.org/m/FXEZD466
- Some students might find it helpful to use graph paper to help them align the digits as they divide using the partial quotients method.
- Consider having graph paper accessible throughout the lesson.
Glossary:
- Access the complete Grade 6 glossary.
Standards
- This lesson builds on the standard: CCSS.5.NBT.B.6MS.5.NBT.6MO.5.NBT.A.8
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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