Using Base-Ten Diagrams to Divide
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to divide two whole numbers by reasoning about place value and using base-ten diagrams. The work here builds on students’ prior experience with base-ten representations and on their understanding that division can be interpreted in terms of creating equal-size groups.
The divisor and dividend are chosen so that the hundreds in the dividend can be partitioned into equal groups of whole numbers without a remainder but the tens cannot. The quotient, however, is a whole number. The key ideas that would enable students to ultimately divide a decimal by a decimal are present in this example:
- A number can be decomposed to make the division convenient. For example, 372 can be viewed as 300 + 60 + 12.
- Place value, expressed in the form of base-ten diagrams, plays a very important role in division.
Launch
Arrange students in groups of 2. Display the diagrams showing Elena’s method, and read aloud the accompanying paragraphs.
Give students 1 minute of quiet think time and another minute to discuss with a partner. Follow with a whole-class discussion.
Student Task
Elena used base-ten diagrams to find 372÷3.
She started by representing 372.
She made 3 groups, each with 1 hundred. Then, she put the tens and ones in each of the 3 groups. Here is her diagram for 372÷3.
Discuss with a partner:
-
Elena’s diagram for 372 has 7 tens. The one for 372÷3 has only 6 tens. Why?
-
Where did the extra ones (small squares) come from?
Sample Response
Sample reasoning: Elena first put the 3 hundreds into 3 groups, placing 1 hundred in each group. Then she put 6 of the 7 tens into 3 groups, giving 2 tens to each group. She traded the remaining ten for 10 ones. Combining these 10 ones with the original 2 ones, she then has 12 ones. Elena put the 12 ones into 3 groups, putting 4 ones in each group. Each group then has 124, so 372÷3=124.
Activity Synthesis (Teacher Notes)
Highlight Elena’s process of separating base-ten units into equal groups. Discuss questions such as:
- “Which base-ten unit(s) did Elena decompose or break up?” (She decomposed a tens unit.)
- “What did that accomplish?” (She had only 1 ten left and there were 3 equal groups. Breaking up the ten into smaller units made it possible to place parts of the ten in the 3 groups.)
- “How might one find 378÷3 using Elena’s method?” (By thinking of 378 as 3 hundreds, 6 tens, and 18 ones and placing them into 3 equal groups.)
Tell students that they will use base-ten representations to explore division of other numbers.
Anticipated Misconceptions
If students have difficulty making sense of Elena’s method, consider providing students with actual base-ten blocks or paper cutouts and asking them to use them to represent 372÷3.
Standards
- 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.
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Using Base-Ten Diagrams to Divide
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to divide two whole numbers by reasoning about place value and using base-ten diagrams. The work here builds on students’ prior experience with base-ten representations and on their understanding that division can be interpreted in terms of creating equal-size groups.
The divisor and dividend are chosen so that the hundreds in the dividend can be partitioned into equal groups of whole numbers without a remainder but the tens cannot. The quotient, however, is a whole number. The key ideas that would enable students to ultimately divide a decimal by a decimal are present in this example:
- A number can be decomposed to make the division convenient. For example, 372 can be viewed as 300 + 60 + 12.
- Place value, expressed in the form of base-ten diagrams, plays a very important role in division.
Launch
Arrange students in groups of 2. Display the diagrams showing Elena’s method, and read aloud the accompanying paragraphs.
Give students 1 minute of quiet think time and another minute to discuss with a partner. Follow with a whole-class discussion.
Student Task
Elena used base-ten diagrams to find 372÷3.
She started by representing 372.
She made 3 groups, each with 1 hundred. Then, she put the tens and ones in each of the 3 groups. Here is her diagram for 372÷3.
Discuss with a partner:
-
Elena’s diagram for 372 has 7 tens. The one for 372÷3 has only 6 tens. Why?
-
Where did the extra ones (small squares) come from?
Sample Response
Sample reasoning: Elena first put the 3 hundreds into 3 groups, placing 1 hundred in each group. Then she put 6 of the 7 tens into 3 groups, giving 2 tens to each group. She traded the remaining ten for 10 ones. Combining these 10 ones with the original 2 ones, she then has 12 ones. Elena put the 12 ones into 3 groups, putting 4 ones in each group. Each group then has 124, so 372÷3=124.
Activity Synthesis (Teacher Notes)
Highlight Elena’s process of separating base-ten units into equal groups. Discuss questions such as:
- “Which base-ten unit(s) did Elena decompose or break up?” (She decomposed a tens unit.)
- “What did that accomplish?” (She had only 1 ten left and there were 3 equal groups. Breaking up the ten into smaller units made it possible to place parts of the ten in the 3 groups.)
- “How might one find 378÷3 using Elena’s method?” (By thinking of 378 as 3 hundreds, 6 tens, and 18 ones and placing them into 3 equal groups.)
Tell students that they will use base-ten representations to explore division of other numbers.
Anticipated Misconceptions
If students have difficulty making sense of Elena’s method, consider providing students with actual base-ten blocks or paper cutouts and asking them to use them to represent 372÷3.
Standards
- 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.