This Warm-up draws students’ attention to the meaning of zeros in a decimal, in particular, whether and when a zero affects the value of the decimal.
Students first think about zeros in the context of addition. When adding the thousandths in , students may write 0.010 or 0.01 for the sum of 0.009 and 0.001. When recording the sum of 1.009 and 0.391, they may write 1.400, 1.40, or 1.4, depending on whether they think of 10 thousandths as 1 hundredth and 10 hundredths as 1 tenth. Then students reason about whether two decimals with different numbers of digits—one with more zeros than the other—could have the same value.
As they reason about the place values of the digits and the meaning of zeros in decimals, students practice looking for and making use of structure (MP7).
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Student Lesson in Spanish
Launch
Arrange students in groups of 2. Give students 1 minute of quiet time to mentally add the decimals in the first problem and another minute to discuss their answer and strategy with a partner. Then ask students to pause and write down the sum. Poll the class on the decimal that they wrote: 1.4, 1.40, or 1.400. Then ask students to complete the rest of the Warm-up.
Activity
None
Find the value mentally:
Decide if each statement is true or false. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share whether they think each statement in the last problem is true or false, and ask for an explanation for each. Students may simply say that we can or cannot just remove the zeros. Encourage them to use what they know about place values or comparison strategies to explain why one number is or is not equal to the other.
If not mentioned in students’ explanations, highlight, for instance, that 34.560 (thirty-four and five hundred sixty thousandths) is equal to 34.56 (thirty-four and 56 hundredths) because in both numbers, there are 3 tens, 4 ones, 5 tenths, 6 hundredths, and no thousandths, ten-thousandths, or hundred-thousandths.
If time permits, use Critique, Correct, Clarify to give students an opportunity to improve a sample written claim about the zeros in a decimal by correcting errors, clarifying meaning, and adding details.
Display this first draft: “We can take the zeros away after the decimal point and the value of the number would stay the same.”
Ask, “What parts of this response are unclear, incorrect, or incomplete?”
Give students 2–4 minutes to work with a partner to revise the first draft.
Tell students that they can use examples in their revision if it would help to improve the draft.
Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, and then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
If not illustrated in students’ revised statements, consider presenting an example that can counter the claim in the first draft. For instance, 12.90 is equal to 12.9, but 12.09 is not equal to 12.9.
3.2
Activity
Optional
Standards Alignment
Building On
Addressing
6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
In this activity, students continue to use base-ten diagrams and vertical calculations to represent sums of decimals and to connect the two representations of addition. They think about the alignment of the digits in vertical calculations to help ensure that correct values are combined.
In the digital version of the activity, students use an applet to create base-ten diagrams that represent sums of decimals. The applet allows students to generate blocks to represent the numbers in the activity, but students must first define the value of each block. Students can also compose 10 of a smaller base-ten units into a larger unit. The digital version may reduce barriers for students who need support with fine-motor skills.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Activity Synthesis
The goal of this discussion is to highlight the importance of attending to place values when adding decimals using vertical calculations or the standard algorithm for addition. Discuss questions such as:
“When finding , why do we not combine the 8 thousandths and 7 hundredths to make 15?” (Hundredths and thousandths are different units.)
“When adding numbers without using base-ten diagrams or other representations, what can we do to help add them correctly?” (Pay close attention to place value so we combine only units of the same place value. When using vertical calculations, it is helpful to line up the digits of the numbers so that numerals that represent the same place value are placed directly on top of one another.)
Consider using color coding to help students visualize the place-value structure, as shown here.
3.3
Activity
Standards Alignment
Building On
Addressing
6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
In this activity, students encounter two variations of decimal subtraction in which regrouping is involved:
Subtracting a number with more decimal places from one with fewer decimal places (such as )
Subtracting a larger value from a smaller value in the same place (for example, in , both the tenths and hundredths in the second number are larger than those in the first).
Students represent these situations with base-ten diagrams and study how to perform them using vertical calculations. The big idea here is that decomposing a unit with 10 of another unit that is its size makes it easier to subtract. In some cases, students would need to decompose twice before subtracting (for instance, decomposing 1 tenth into 10 hundredths, and then 1 hundredth into 10 thousandths).
In the digital version of the activity, students have the option to use an applet to reason about subtraction of decimals. The applet allows students to generate blocks to represent the numbers in the activity, but students must first define the value of each block. Students can also decompose a base-ten unit into 10 smaller units.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Activity Synthesis
The purpose of this discussion is for students to make connections between two different ways to subtract decimals when decomposing one or more base-ten units is necessary.
Display Diego’s and Elena’s diagrams and the numerical calculations that correspond to them, as shown here.
Ask students to discuss how each number in the calculations is represented in the diagrams. Highlight the ways in which the decomposition of a larger base-ten unit into 10 smaller ones can be represented numerically.
We can also show both calculations by arranging the numbers vertically. On the left, the 3 and 10 in red show Diego’s decomposition of the 4 hundredths. The calculation on the right illustrates Elena’s representation: the 0 written in blue helps us see 4 tenths as 40 hundredths, from which we can subtract 3 hundredths to get 37 hundredths. Use this example to reinforce that writing an additional zero at the end of a non-zero decimal does not change its value.
If time permits, consider asking:
“For , how do we show 1 one being decomposed into 10 tenths using a diagram?” (Decompose a large square into 10 smaller rectangles.)
“How do we show the same thing in a vertical calculation?” (Write 1 one as 10 tenths over the tenth place, combine it with the given 1 tenth, and then subtract 4 tenths.)
“When might it be really cumbersome to subtract using base-ten diagrams? Can you give an example?” (When the numbers involve many decimal places, such as , or a problem with large digits, such as .)
Emphasize that the algorithm (vertical calculation) for subtraction of decimals works like the algorithm for subtraction of whole numbers. The only difference is that the values involved in the subtraction problems can now include tenths, hundredths, thousandths, and so on. The key in both cases is to pay close attention to the place values of the digits in the two numbers.
Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on their own progress. For example, ask students how comfortable they are composing and decomposing base-ten units with and without manipulatives. Supports accessibility for: Organization, Conceptual Processing
Lesson Synthesis
Two key takeaways from this lesson are:
When subtracting decimals, it is important to subtract the values that are in the same decimal place, as is the case when subtracting whole numbers.
We can decompose a base-ten unit into 10 of a unit that is its size to help us subtract. For instance, 1 hundredth (0.01) can be decomposed into 10 thousandths (0.010).
To emphasize these ideas, discuss questions such as:
“To find , how do we remove 7 tenths from 5 tenths?” (We can decompose 1 one into 10 tenths, and subtract 7 tenths from a total of 15 tenths. If using base-ten blocks, we can exchange a 1 with 10 tenths, which also allows us to subtract 7 tenths.)
“When subtracting decimals, why is it helpful to line up the decimal points?” (It allows us to line up the same base-ten units and to subtract correctly.)
“We saw that zeros can be written to or removed from the end of a decimal without changing the value of the number. For instance, 3.2, 3.20, and 3.200 are equal. Why is that?” (The 2 tenths can be decomposed into 20 hundredths and then into 200 thousandths, so 3.2, 3.20, and 3.200 all represent the same amount.)
Student Lesson Summary
Base-ten diagrams can help us understand subtraction. Suppose we are finding . Here is a diagram showing 0.23, or 2 tenths and 3 hundredths.
Subtracting 7 hundredths means removing 7 small squares, but we do ;not have enough to remove. Because 1 tenth is equal to 10 hundredths, we can decompose one of the tenths (1 rectangle) into 10 hundredths (10 small squares).
Base ten diagram. 0 point 23. Two rectangles in the tenths column. 3 small squares in the hundredths column. A dotted rectangle is drawn around one of the rectangles with an arrow to 10 small squares. The arrow is labeled decompose.
We now have 1 tenth and 13 hundredths, from which we can remove 7 hundredths.
Base ten diagram. 0 point 23. One rectangle in the tenths column. 13 small squares in the hundredths column. 7 small squares have an X through them. The words subtract 0 point 0 7 is below the small squares.
We have 1 tenth and 6 hundredths remaining, so .
Here is a vertical calculation of .
Vertical subtraction. First line. 0 point 23. The 2 is crossed out and has a 1 above it. The 3 is crossed out and has 13 above it. Second line. Minus 0 point 0 0 7. Horizontal line. Third line. 0 point 17.
Notice how this representation also shows that a tenth is decomposed into 10 hundredths in order to subtract 7 hundredths.
This works for any decimal place. Suppose we are finding . Here is a diagram showing 0.023.
We want to remove 7 thousandths (7 small rectangles). We can decompose one of the hundredths into 10 thousandths.
Base 10 diagram. 0 point 0 2 3. Two small squares in the hundredths column. Three small rectangles in the thousandths column. A square is drawn around 1 small square. An arrow is drawn to 10 small rectangles. The arrow is labeled decompose.
Now we can remove 7 thousandths.
Base 10 diagram. 0 point 0 2 3. One small square in the hundredths column. 13 small rectangles in the thousandths column. 7 small rectangles have an X through them. Below the small rectangles are the words subtract 0 point 0 0 7.
We have 1 hundredth and 6 thousandths remaining, so .
Here is a vertical calculation of .
Vertical subtraction. First line. 0 point 0 2 3. The 2 is crossed out and has a 1 above it. The 3 is crossed out and has 13 above it. Second line. Minus 0 point 0 0 7. Horizontal line. Third line. 0 point 0 1 6.
Tell students to close their books or devices (or to keep them closed). Display the image of Andre’s and Jada’s diagrams 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 diagrams.
If the idea of the diagram representing base-ten units does not come up during the conversation, ask students to discuss this idea. Then, tell students to open their books or devices.
Arrange students in groups of 2. Give students 8–10 minutes of quiet work time, but encourage them to briefly discuss their responses with their partner after completing the second question and before continuing with the rest. Follow with a whole-class discussion.
Use Collect and Display to direct attention to words collected and displayed from an earlier activity. Collect the language that students use to describe 10 base-ten units being composed into 1 larger unit as they add decimals. Display words and phrases, such as “bundle,” “group,” “put together,” and “compose.”
Activity
None
Andre and Jada drew base-ten diagrams to represent .
Andre drew 11 small rectangles.
Jada drew only two figures: a square and a small rectangle.
If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
Draw or describe a diagram that could represent the sum .
Here are two calculations of . Which is correct? Explain why one is correct and the other is incorrect.
The calculation on the left adds zero point 2 and zero point zero 5 by aligning the ones units, tenths unit, and hundredths unit. The sum is zero point 2 5.
The calculation on the right adds zero point 2 and zero point zero five by aligning the hundredths unit under the tenths unit. The sum is zero point zero 7.
Compute each sum. If you get stuck, consider drawing base-ten diagrams to help you.
Student Response
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Building on Student Thinking
Students might not recall how to name decimals in thousandths such as 0.209. Consider displaying a place-value chart for reference, or asking students to name a sequence of decimals starting from 1 thousandth: 0.001, 0.009, 0.010, 0.047, 0.099, 0.100, 0.200, and 0.209.
Launch
Keep students in groups of 2. Give partners 4–5 minutes to complete the first two questions about Diego’s, Noah’s, and Elena’s diagrams for . Then pause for a brief discussion. Invite 1–2 groups to share their responses. Ask questions such as:
“What is the difference between Diego’s method and Elena’s method?” (Diego breaks up only 1 tenth into 10 hundredths, whereas Elena breaks up all 4 tenths into hundredths.)
“What are some advantages to Diego’s method?” (Diego’s method is quicker to draw. It shows the 3 tenths and 7 hundredths. Elena would need to count how many hundredths she has.)
“What are some advantages to Elena’s method?” (Elena’s diagram shows a difference of 37 hundredths, which matches how we say 0.37 in words.)
Next, give students 4–5 minutes of quiet time to complete the last question. Provide access to graph paper in case students wish to use it for aligning the digits when using vertical calculations.
Use Collect and Display to direct attention to words collected and displayed from an earlier activity. Collect the language that students use to describe 1 base-ten unit being decomposed into 10 smaller units as they subtract decimals. Display words and phrases, such as “unbundle,” “take apart,” “separate,” “regroup,” and “decompose.”
Activity
None
Diego and Noah drew different diagrams to represent . Each rectangle represents 0.1. Each square represents 0.01.
Diego started by drawing 4 rectangles to represent 0.4. He then replaced 1 rectangle with 10 squares and crossed out 3 squares to represent subtraction of 0.03, leaving 3 rectangles and 7 squares in his diagram.
A base-ten diagram labeled “Diego’s Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 10 squares in that column. The last rectangle is circled with a dashed line and an arrow pointing from the rectangle to the column of squares is labeled “decompose.” The last three squares are crossed out.
Noah started by drawing 4 rectangles to represent 0.4. He then crossed out 3 rectangles to represent the subtraction, leaving 1 rectangle in his diagram.
Do you agree that either diagram correctly represents ? Discuss your reasoning with a partner.
Elena also drew a diagram to represent . She started by drawing 4 rectangles. She then replaced all 4 rectangles with 40 squares and crossed out 3 squares to represent subtraction of 0.03, leaving 37 squares in her diagram. Is her diagram correct? Discuss your reasoning with a partner.
A base-ten diagram labeled “Elena's Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 40 squares in that column. All four rectangles are circled with a dashed line and an arrow pointing from the rectangles to the column of squares is labeled “decompose.” The last three squares are crossed out.
Find each difference. Be prepared to explain your reasoning. If you get stuck, you can use base-ten blocks or diagrams to represent each expression and find its value.
