Adding and Subtracting Decimals with Many Non-Zero Digits
10 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to review the alignment of the digits when using a standard algorithm to subtract two numbers in base-ten. The Notice and Wonder routine in the launch gives students a chance to think about whether or how different placements of the 5 (the first number) affects the subtraction. It also gives the teacher insight about how students interpret the 5 and its value. For instance:
- Would students place the decimal point directly after the 5?
- When placing 5 above 7 (in the tenths place), do they see it as 5 ones or as 5 tenths?
- Do they wonder if the 5 is missing a decimal point in the second and third case?
In the Student Task Statement, students see the same subtraction in the context of a situation, allowing them to see more clearly the equivalence of 5 and 5.00.
Launch
Tell students to close their books or devices (or to keep them closed). Display the three ways of writing a subtraction calculation for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder about. Record and display responses without editing or commentary. If possible, record the relevant reasoning on or near the calculation setup referred to.
If the question of whether the placement of the 5 affects the result of subtraction does not come up during the conversation, ask students to discuss it.
Tell students to open their books or devices. Give students 2 minutes of quiet work time, and follow that with a whole-class discussion.
Student Task
-
Clare bought a photo for 17 cents and paid with a $5 bill. Which of these three ways of writing the numbers could Clare use to find the change she should receive? Be prepared to explain your reasoning.
-
Find the amount of change that Clare should receive. Explain or show your reasoning.
Sample Response
- The first setup (in which 5 and 0 line up vertically) is most conducive to correct subtraction. Sample reasoning: The 5 means $5.00, and it helps line up the dollars (the ones) and the cents (the tenths and hundredths) when subtracting.
- $4.83. Students may or may not use vertical calculation. Sample reasoning:
- Adding 0.17 to 4.83 gives 5.
- Subtracting $0.10 (10 cents) from $5 leaves $4.90. Subtracting another $0.07 (7 cents) leaves $4.83.
Activity Synthesis (Teacher Notes)
Invite students to share which way they think Clare could write the calculation to find the amount of change. Clarify that in this particular case, the first setup is the most conducive to correct computation, but there is not one correct answer. Students could find the answer with any one of the setups as long as they understand that the 5 represents $5.00.
If any students choose the second or third setup because they can mentally subtract the values without lining them up by place values, invite them to share their reasoning. Ask if they would use the same strategy for dealing with longer decimals (such as 5.23−0.4879 ) and, if not, ask them what approach might be more conducive to correct calculation in those cases.
Encourage more students to be involved in the conversation, by asking questions such as:
- “Do you agree or disagree? Why?”
- “Can anyone explain ’s reasoning in their own words?”
- “What is important for us to think about when subtracting this way?”
- “How could we have solved this problem mentally?”
Standards
- 6.NS.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
- 6.NS.B.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Adding and Subtracting Decimals with Many Non-Zero Digits
10 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to review the alignment of the digits when using a standard algorithm to subtract two numbers in base-ten. The Notice and Wonder routine in the launch gives students a chance to think about whether or how different placements of the 5 (the first number) affects the subtraction. It also gives the teacher insight about how students interpret the 5 and its value. For instance:
- Would students place the decimal point directly after the 5?
- When placing 5 above 7 (in the tenths place), do they see it as 5 ones or as 5 tenths?
- Do they wonder if the 5 is missing a decimal point in the second and third case?
In the Student Task Statement, students see the same subtraction in the context of a situation, allowing them to see more clearly the equivalence of 5 and 5.00.
Launch
Tell students to close their books or devices (or to keep them closed). Display the three ways of writing a subtraction calculation for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder about. Record and display responses without editing or commentary. If possible, record the relevant reasoning on or near the calculation setup referred to.
If the question of whether the placement of the 5 affects the result of subtraction does not come up during the conversation, ask students to discuss it.
Tell students to open their books or devices. Give students 2 minutes of quiet work time, and follow that with a whole-class discussion.
Student Task
-
Clare bought a photo for 17 cents and paid with a $5 bill. Which of these three ways of writing the numbers could Clare use to find the change she should receive? Be prepared to explain your reasoning.
-
Find the amount of change that Clare should receive. Explain or show your reasoning.
Sample Response
- The first setup (in which 5 and 0 line up vertically) is most conducive to correct subtraction. Sample reasoning: The 5 means $5.00, and it helps line up the dollars (the ones) and the cents (the tenths and hundredths) when subtracting.
- $4.83. Students may or may not use vertical calculation. Sample reasoning:
- Adding 0.17 to 4.83 gives 5.
- Subtracting $0.10 (10 cents) from $5 leaves $4.90. Subtracting another $0.07 (7 cents) leaves $4.83.
Activity Synthesis (Teacher Notes)
Invite students to share which way they think Clare could write the calculation to find the amount of change. Clarify that in this particular case, the first setup is the most conducive to correct computation, but there is not one correct answer. Students could find the answer with any one of the setups as long as they understand that the 5 represents $5.00.
If any students choose the second or third setup because they can mentally subtract the values without lining them up by place values, invite them to share their reasoning. Ask if they would use the same strategy for dealing with longer decimals (such as 5.23−0.4879 ) and, if not, ask them what approach might be more conducive to correct calculation in those cases.
Encourage more students to be involved in the conversation, by asking questions such as:
- “Do you agree or disagree? Why?”
- “Can anyone explain ’s reasoning in their own words?”
- “What is important for us to think about when subtracting this way?”
- “How could we have solved this problem mentally?”
Standards
- 6.NS.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
- 6.NS.B.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.