Lesson 3

This warm-up prompts students to reason about regrouping and about when the zeros in a decimal affect the number that it represents. The mathematical work of interest is how students combine two decimals (e.g., in analyzing \(1.009 + 0.391\), do they see that \(0.009 + 0.001 = 0.010\)?) and how they write the sum (e.g., do they know \(1.4=1.40=1.400\)?).

Arrange students in groups of 2. Give students 1 minute of quiet time to mentally add the decimals in the first problem and then another minute to discuss their answer and strategy with a partner. Briefly discuss their strategies as a class, and then ask students to complete the true-or-false questions.

Some students may say, “You can take the zeros away after the decimal point and the number stays the same.” Although students could mean, for instance, that 12.9 is equal to 12.90, they might also mistakenly think 12.09 is equal 12.9. Ask these students to be more precise in their statement. Ask if the zero can be in any place after the decimal point or only in certain places.

Ask students to indicate whether they think each equation 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 greater than, less than, or equal to the other. If students do not notice that the two numbers in the true-or-false questions have the same digits except for the missing zeros, point that out after each question.

If not mentioned by students in their explanations, ask:

• “Can zeros be written at the end of a decimal without changing the number that it represents?”
• “Can zeros be eliminated from the end of a decimal without changing the value?”
• “Can zeros be written or erased in the middle of a decimal without changing the value?”

Here students continue to use diagrams to represent sums of decimals, but they also transition to writing addition calculations vertically. They think about the alignment of the digits in vertical calculations to help ensure that correct values are combined. This activity is optional, so students have the option to spend more time subtracting decimals in the next activity.

As in the previous activity, consider having students use physical base-ten blocks (if available), a paper version of the base-ten figures (from the blackline master), or this digital applet ggbm.at/FXEZD466, as alternatives to drawing diagrams.

The goal of this discussion is to help students understand that vertical calculation is an efficient way to find the sums of decimals. Discuss:

• “When finding \(0.008 + 0.07\), why do we not combine the 8 thousandths and 7 hundredths to make 15?” (Hundredths and thousandths are different units. If each hundredth is unbundled into 10 thousandths, we can add 70 thousandths and 8 thousandths to get 78 thousandths).
• “How do we use representations of base-ten numbers to add effectively and efficiently?” (Make sure to put together tenths with tenths, hundredths with hundredths, etc. Also, make sure that if a large square represents \(\frac{1}{10}\) for one summand, it also represents \(\frac{1}{10}\) for the other.)
• “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 like units. 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.)

Additionally, consider using color coding to help students visualize the place-value structure, as shown here.

In this activity, students encounter two variations of decimal subtraction in which regrouping is involved. They subtract a number with more decimal places from one with fewer decimal places (e.g., \(0.1 - 0.035\)), and subtract two digits that represent the same place value but where the value in the second number is greater than that in the starting number (e.g., in \(1.12 - 0.47\), both the tenth and hundredth values in the second number is 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 of “unbundling” or of decomposing a unit with 10 of another unit that is \(\frac{1}{10}\) its size to make it easier to subtract. In some cases, students would need to decompose twice before subtracting (e.g., a tenth into 10 hundredths, and then 1 hundredth into 10 thousandths).

Use the whole-class discussion to highlight the correspondences between the two methods and to illustrate how writing zeros at the end of a decimal helps us perform subtraction.

As in previous activities, consider having students use physical base-ten blocks (if available), a paper version of the base-ten figures (from the blackline master), or this digital applet https://ggbm.at/FXEZD466, as alternatives to drawing diagrams.

If students struggle to subtract two numbers that do not have the same number of decimal digits (such as in questions 3a and 3d), consider representing the subtraction with base-ten diagrams. For example, to illustrate \(0.3 - 0.05\), start by drawing 3 large rectangles to represent 3 tenths. Replace 1 rectangle with 10 squares, each representing 1 hundredth. Cross out 5 squares to show subtraction of 5 hundredths. Alternatively, replace 3 rectangles (3 tenths) with 30 squares (30 hundredths), cross out 5 squares to show subtraction of 5 hundredths.

The purpose of this discussion is for students to make connections between two different and correct ways to subtract decimals when unbundling is required. Ask:

• “What is the difference between Diego’s method and Elena’s method?” (Diego only breaks up 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.)

Consider connecting the diagrams for Diego’s and Elena’s work with numerical equations as shown here.

We can also show both calculations by arranging the numbers vertically. On the left, the 3 and 10 in red show Diego’s unbundling 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.

To deepen students’ understanding, consider asking:

• “In the problem \(2.1 - 0.4\), how does unbundling in the diagram show up in the vertical calculation with numbers?” (In the diagram, we unbundle 1 whole to make 10 tenths. With the calculation, we rewrite 1 whole as 10 tenths (over the tenth place) and combine it to the given 1 tenth before subtracting 4 tenths.)
• “When might it be really cumbersome to subtract using base-ten diagrams? Can you give examples?” (When the numbers involve many decimal places, such as \(113.004 - 6.056802\), or a problem with large digits, such as \(7.758 - 0.869\).)

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.