Different Partial Quotients

10 min

Narrative

The purpose of this Warm-up is to elicit the idea that multiplication and division both can be used to represent partial quotients. Students compare two representations, both familiar from grade 4, of partial quotients. This will be helpful when students record partial quotients with division expressions later in the lesson.

Launch

  • Groups of 2
  • Display the image.
  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Task

What do you notice? What do you wonder?

Clare’s strategy

divide. three hundred sixty four divided by 13. With two image underneath.
divide. three hundred sixty four divided by 13. With two image underneath. Left Image, 4 rows. First row: 13 times 10 equals one hundred thirty. Second row: 13 times 20 equals two hundred sixty. Third row: 13 times 5 equals sixty 5. Fourth row: 13 times 3 equals 39. Right image. 7 rows. First row: three hundred sixty four. Second row: minus two hundred sixty. Horizontal line. Third row: one hundred four. Fourth row: minus sixty 5. Horizontal line. Fifth row: 39. Sixth row: minus 39. Horizontal line. Seventh row 0.

Jada’s strategy

\begin{align} 130\div 13&= 10\\ 130\div 13 &= 10\\ 65 \div 13 &= \phantom{0} 5\\ 39\div 13 &= \phantom{0} 3\\ \overline {\hspace{5mm}364 \div 13} &\overline{\hspace{1mm}=  28 \phantom{000}}\end{align}

Sample Response

Students may notice:

  • They used some of the same numbers, but Clare multiplied and Jada divided. 
  • Clare used subtraction to figure out how much she had left after she found part of the quotient.
  • Both have numbers that sum to 28.

Students may wonder:

  • What is the answer?
  • Did they do the same thing?
  • Why did Clare write 13×10=13013\times 10 = 130, but she did not circle the 10?
Activity Synthesis (Teacher Notes)
  • “How are these strategies the same as, and how are they different from, the way you found quotients in the previous lesson?” (They use multiplication and division. They break the problem into smaller parts, with numbers that are easier to calculate.)
  • “Multiplication can help us think about partial quotients.”
Standards
Addressing
  • 5.OA.2·Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <em>For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.</em>
  • 5.OA.A.2·Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <span>For example, express the calculation “add <span class="math">\(8\)</span> and <span class="math">\(7\)</span>, then multiply by <span class="math">\(2\)</span>” as <span class="math">\(2 \times (8 + 7)\)</span>. Recognize that <span class="math">\(3 \times (18932 + 921)\)</span> is three times as large as <span class="math">\(18932 + 921\)</span>, without having to calculate the indicated sum or product.</span>
Building Toward
  • 5.NBT.6·Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
  • 5.NBT.B.6·Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

20 min

15 min