Using Long Division
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to make sense of a long division calculation, by familiarizing themselves with the structure and the mathematics that might be involved (MP1), before they later learn to use the algorithm to calculate quotients.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Launch
Arrange students in groups of 2. Display the images of Lin’s calculations for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing that they notice and one thing that they wonder about. Give students another minute to discuss their observations and questions.
Student Task
Here are Lin’s calculations for finding 657÷3.
What do you notice? What do you wonder?
Sample Response
Students may notice:
- Lin arranged her calculations vertically.
- There are no partial quotients being added, but the quotient is still 219, just as in other calculations.
- The quotient is shown one digit at a time, starting with the hundreds, then the tens, then the ones.
- Lin subtracted 6, then 3, and then 27.
- There are arrows pointing down in the tens place and ones place.
Students may wonder:
- What do the arrows mean?
- Why did Lin divide one number at a time?
- Why did Lin subtract one-digit numbers at first, 6 and 3, but then subtract a two-digit number, 27, at the end?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the calculations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the idea of dividing by place value does not come up during the conversation, ask students to discuss this idea.
Standards
- 4.NBT.6·Find whole-number quotients and remainders with up to four-digit dividends and one-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.
- 4.NBT.B.6·Find whole-number quotients and remainders with up to four-digit dividends and one-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.
- 6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.
- 6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.
25 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Using Long Division
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to make sense of a long division calculation, by familiarizing themselves with the structure and the mathematics that might be involved (MP1), before they later learn to use the algorithm to calculate quotients.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Launch
Arrange students in groups of 2. Display the images of Lin’s calculations for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing that they notice and one thing that they wonder about. Give students another minute to discuss their observations and questions.
Student Task
Here are Lin’s calculations for finding 657÷3.
What do you notice? What do you wonder?
Sample Response
Students may notice:
- Lin arranged her calculations vertically.
- There are no partial quotients being added, but the quotient is still 219, just as in other calculations.
- The quotient is shown one digit at a time, starting with the hundreds, then the tens, then the ones.
- Lin subtracted 6, then 3, and then 27.
- There are arrows pointing down in the tens place and ones place.
Students may wonder:
- What do the arrows mean?
- Why did Lin divide one number at a time?
- Why did Lin subtract one-digit numbers at first, 6 and 3, but then subtract a two-digit number, 27, at the end?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the calculations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the idea of dividing by place value does not come up during the conversation, ask students to discuss this idea.
Standards
- 4.NBT.6·Find whole-number quotients and remainders with up to four-digit dividends and one-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.
- 4.NBT.B.6·Find whole-number quotients and remainders with up to four-digit dividends and one-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.
- 6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.
- 6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.