Multi-Digit Division

1. Understand that a remainder is the number left over when one number is divided by another. 
2. Solve division word problems within 100 that involve a remainder, using an array, an area model, or a tape diagram to represent the problem (MP.4).
3. Check the solution to division problems using inverse operations, multiplying the quotient with the dividend and adding the remainder. 
4. Understand that the units of the quotient and remainder are different, since the quotient represents either the number of groups or the size of the groups, but the remainder represents what remains of the whole.

• Students have likely come across division situations in real life that do not result in a simple whole number quotient (especially considering that problems with 6 as a divisor, for example, will result in a whole number only one time out of six). Because of this, students may choose to interpret the remainder in the problems they encounter in this lesson. You don’t necessarily want to discourage students from doing so, but note that this will be the focus of Lesson 2.
• “To compute and interpret remainders in word problems (4.OA.A.3), students must reason abstractly and quantitatively (MP.2), make sense of problems (MP.1), and look for and…search for the structure (MP.7) in problems with similar interpretations of remainders” (PARCC Model Content Frameworks, Mathematics, Grades 3–11).
• “The result of division within the system of whole numbers is frequently written as:

$$ 84 \div10 = 8\space \mathrm{R} \space 4$$ and $$44 \div 5 = 8 \space \mathrm{R}\space 4$$

Because the two expressions on the right are the same, students should conclude that $$84\div10$$ is equal to $$44\div5$$, but this is not the case. (Because the equal sign is not used appropriately, this usage is a non-example of Standard for Mathematical Practice 6.) Moreover, the notation $$8\space \mathrm{R} \space 4$$ does not indicate a number. Rather than writing the result of division in terms of a whole-number quotient and remainder, the relationship of whole-number quotient and remainder can be written as:

$$ 84 = 8 \times 10 + 4$$ and $$44 = 8\times 5 + 4$$”

(Progressions for the Common Core State Standards in Mathematics, Number and Operations in Base Ten, K-5, p. 16). Thus, you should avoid using the “R” notation immediately following the equal sign.

• Problem Set

• Problem Set Answer Key