Division Principles and Methods

Overview

This lecture covers the core principles of division, its interpretations, word problem types, division with remainders, standard and scaffold algorithms, and division involving fractions and decimals.

Division Basics & Symbols

• Division can be shown as a ÷ b, a/b, or the long division "house" symbol.
• a (dividend) ÷ b (divisor) = quotient (result).
• Division has multiple standard symbols, unlike addition or subtraction.

Interpretations of Division

• "How many groups?" asks how many groups of size b can be made from a.
• "How many units in one group?" asks how many units per group when a is shared among b groups.
• Example: 7 cups for 3 pints of potion—find cups per pint ("how many units in one group").
• Example: $7 for $3/pound—find pounds you can buy ("how many groups").

Division Word Problems

• Division problems can be set as arrays, area questions, or real-world stories (e.g., penguins, candies).
• Multiple contexts can model the same division operation (e.g., 54 ÷ 6).

Zero and Division

• Dividing by zero is undefined (e.g., 8 ÷ 0).
• Zero divided by any number (except zero) is zero (0 ÷ 3 = 0).

Division with Remainders

• Remainder: amount left after groups are made evenly.
• "How many groups?"—remainder is what's left that can't form a full group.
• "How many units in one group?"—remainder is leftover after distribution.

Standard Algorithm vs. Scaffold Method

• Standard algorithm: uses place value alignment for multi-digit division.
• Scaffold method: repeatedly subtracts large multiples of the divisor, similar to partial sums in multiplication.
• Scaffold method can be flexible but less efficient; standard algorithm is streamlined.

Division with Decimals & Fractions

• For decimals: align the decimal points and keep place values straight.
• To express a fraction as a decimal, divide numerator by denominator.
• "How many groups?" problems with fractions use tables or diagrams.
• Divide fractions by inverting the second fraction and multiplying ("keep, change, flip").

Key Terms & Definitions

• Dividend — the number being divided.
• Divisor — the number you divide by.
• Quotient — the result of division.
• Remainder — what's left after division.
• Scaffold Method — division by repeated subtraction of large multiples.
• Standard Algorithm — traditional stepwise division method.
• Undefined — result of dividing by zero.

Action Items / Next Steps

• Watch the video linked in Activity 6B on types of division problems.
• Complete Activity 6B and double-check your answers.
• Review pages 248–259 for detailed division situations.
• Complete reflections for Activities 6F and 6M on Canvas.
• Read pages 279–281 for explanations on dividing fractions ("invert and multiply").