Unit 5, Lesson 9
Using Base-Ten Diagrams to Divide (Optional)
Grade 6
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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:
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.
Elena used base-ten diagrams to find
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
Discuss with a partner:
Elena’s diagram for 372 has 7 tens. The one for
Where did the extra ones (small squares) come from?
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
Highlight Elena’s process of separating base-ten units into equal groups. Discuss questions such as:
Tell students that they will use base-ten representations to explore division of other numbers.
Optional
In this activity, students use base-ten diagrams to divide two whole numbers that result in terminating decimal quotients. In the given problems, the ones cannot be placed into equal groups of whole numbers without a remainder and must be decomposed into tenths. Students see that this process is conceptually no different than decomposing hundreds into tens or tens into ones so that all the pieces can form equal-size groups.
Arrange students in groups of 2. Display Mai’s diagrams and read aloud the problem stems. Give students 2 minutes of quiet time to analyze Mai’s work and think about the first set of questions, followed by 2–3 minutes to discuss their observations with their partner. Pause for a whole-class discussion, making sure that all students understand how Mai dealt with the remainder.
Give students 5–7 minutes to complete the final two questions. Provide access to base-ten blocks or paper cutouts of base-ten representations (from the blackline master). Follow with a whole-class discussion.
Mai used base-ten diagrams to calculate
She then made 5 groups, each with 1 ten. There was 1 ten left. She decomposed it into 10 ones and distributed the ones across the 5 groups.
Here is Mai’s diagram for
Discuss these questions with a partner:
Mai should have a total of 12 ones, but her diagram shows only 10. Why?
She did not originally have tenths, but in her diagram each group has 4 tenths. Why?
What value has Mai found for
Four students share a $271 prize from a science competition. How much does each student get if the prize is shared equally? Show your reasoning.
The goal of the discussion is to highlight that the same reasoning process is involved when we divide two numbers, regardless of whether the quotient is or is not a whole number.
Consider displaying Elena’s diagram for
Ask questions such as:
The big new idea here is that sometimes division of whole numbers does not end when we get to the ones place. When the whole-number amount can no longer be distributed into equal groups, we can first decompose them into tenths, hundredths, or other smaller base-ten units and then distribute them.
Advances: Representing, Conversing
In this activity, students analyze several diagrams that represent a decimal being divided by a whole number. They interpret and explain the presence or arrangement of base-ten units in several stages of reasoning, including in a final diagram, which shows the quotient. As students decompose and distribute base-ten pieces into equal-size groups and think about the meaning of each piece, students practice reasoning abstractly and quantitatively (MP2).
Keep students in groups of 2. Tell students that in this activity they will try to follow Elena’s reasoning as she used base-ten diagrams to represent a division of a decimal, 53.8, by a whole number, 4. Give students 2–3 minutes of quiet time to read the problem stem and to analyze Elena’s work, and another 2–3 minutes to discuss the questions with their partner.
To find
She placed 1 ten into each group, decomposed the remaining 1 ten into 10 ones, and went on distributing the units.
This diagram shows Elena’s initial placement of the units and the decomposition of 1 ten.
4 groups of base-ten diagrams. Group 1, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 2, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 3, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 4, 1 rectangle labeled, tens, 2 small rectangles labeled, tenths. Below the groups, one rectangle decomposed into 10 squares.
Here’s Elena’s finished diagram, showing the quotient of
Discuss with a partner:
Some students may stop dividing when they reach a remainder rather than decomposing the remainder into smaller units. Remind them that they can continue to divide the remainder by decomposing and to refer to Elena’s worked-out example or those from earlier lessons, if needed.
Invite a student or a group to share their explanation of Elena’s diagrams. If possible, display and mark up the second given diagram to reflect students’ explanation and to arrive at the same result as shown in the final diagram.
Emphasize that anytime Elena had base-ten units that can’t be distributed equally into 4 groups, she decomposed each into 10 of the next smaller unit and distributed those. She kept going until there were no pieces remaining.
If time permits, use Critique, Correct, Clarify to give students an opportunity to improve a sample written prediction about the value of
Display this first draft:
“The value ofOne way to find the quotient of two numbers, such as
We can think of the division by 3 as splitting up 345 into 3 equal groups.
Each group has 1 hundred, 1 ten, and 5 ones, so
Base-ten diagrams can also help us think about division when the result is not a whole number. Let’s look at
4 groups of base-ten diagrams. Each group contains 2 rectangles labeled, tens, and 1 square labeled, ones. 2 squares are decomposed into 20 small rectangles labeled, tenths.
We can see that there are 4 groups of 21 in 86 with 2 ones left over. To find the quotient, we need to distribute the 2 ones into the 4 groups. To do this, we first need to decompose the 2 ones into 20 tenths and then put 5 tenths in each group.
Once the 20 tenths are distributed, each group will have 2 tens, 1 one, and 5 tenths, so
For some division problems, such as