The purpose of this activity is for students to develop an understanding of the vertical method of recording partial quotients and use it to divide. In the Launch, students look at an algorithm that uses partial quotients and annotate it to show the multiplication that takes place. Then students find the value of several division expressions.

• Display responses that show variation in how students decomposed 392.
• Highlight that there are countless ways of using partial quotients to divide a number, but it may be more efficient to divide larger groups than smaller groups. (For example, removing 50 groups of 7 or 350 once, instead of removing 70 five times. Removing smaller groups is just as valid.)
• Invite students to share their answers and reasoning for the last problem. To help students see the relationship between corresponding partial products and partial quotients, consider illustrating them visually. Examples:

  

  

  

The purpose of this activity is for students to apply their understanding of partial quotients and the vertical recording method to divide four-digit numbers. They also identify some errors that are common when finding quotients this way. When students determine where the errors are and correct them, they critique the reasoning of others and construct viable arguments (MP3).

• Select students to share their responses to the last question.
• “What are some ways to check our answers and avoid the mistakes Andre and Elena made? For example, how can we tell if dividing 2,315 by 5 gives a result in the 100s or in the 400s?” (We can estimate by multiplying the result by 5: is a little over 500, and is 2,000.)
• To reinforce the importance of keeping track of the partial quotients during division, consider annotating a corrected version of Elena’s algorithm. Record the product that corresponds to each number being subtracted from the dividend (1,500, 500, 300, and 15).

• Compare and contrast the four calculations. “How are the four strategies alike? How are they different?” (The first three are similar in that they involve partial quotients. The last one involves estimation.)
• “Which strategy or strategies do you find easy to follow? Hard to follow?”
• “Which strategy seems the most efficient? The least efficient?” (Calculations B and C seem lengthy and could be shortened by using larger multiples of 3. The last strategy seems most efficient.)