Lesson 2

The purpose of this warm-up is for students to review place value when working with decimals. There are many ways students might find the numbers represented by the large rectangle and large square. However, the focus as students work should be on understanding that each place represents a unit that is 10 times larger than the unit immediately to its right. This understanding can be represented by diagrams or by multiplication expressions (MP8).

Arrange students in groups of 2. Tell students to look for patterns as they work. Give students 1–2 minute of quiet think time, followed by a brief partner discussion. Tell the partners to share their responses, come to an agreement on each answer, and discuss any patterns they noticed. Select students with correct responses to share during the whole-class discussion.

Some students may continually use skip counting (by 10, by 0.1, etc.) to find the value of the rectangle and the square, rather than making connections to place value. To help these students see a pattern connected to their skip counting, ask them to also write the multiplication expression that relates the value of each of the smaller units to that of the larger unit they compose.

Ask selected students to share their values for the diagrams, how they found them, and what patterns they noticed. Record and display these values for all to see. Ask other students for the multiplication equation that represents each response to highlight that each decimal place value is 10 times the value of the unit to its right. For example, the recording for the first question should look like this:

\(10 \boldcdot 1=10\)

\(10 \boldcdot 0.1=1\)

\(10 \boldcdot 0.01=0.1\)

\(10 \boldcdot 0.001=0.01\)

If time permits, discuss some of these questions:

• “When you change what each small square represents, how did the change affect the value of the large rectangle?”
• “What might be some other numbers that the small square or long rectangle could represent?”
• “Why could these representations be called ‘base-ten diagrams’?”

In this activity, students reinforce their understanding of the meaning of place value, i.e., that the value in ones place is 10 times the value of the place to its right (and \(\frac {1}{10}\) the value of the place to its left). They draw base-ten diagrams to represent addition of decimals; they connect the process of regrouping numbers in the addition algorithm to the “bundling” of pieces in the diagram. For example, they see that 10 medium squares (0.01) can be composed or “bundled” to make 1 medium rectangle (0.1). Likewise, when the numbers in the ones place add up to be at least 10, we can group them and add 1 to the tens place.

As alternatives to drawing diagrams, 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. 

Students will need a solid understanding of place value to be successful with the activity. If they are unclear about the terms tenths, hundredths, thousandths, and ten thousandths, review place value before proceding with the activity. Consider having students practice by counting by a certain base-ten unit (6 and 8 tenths, 6 and 9 tenths, 7, 7 and 1 tenth. . .), completing place value charts, reading decimals aloud, or by matching a decimal that is read aloud to its written numerical representation.

Select a few students to share their responses to the questions, or display the correct answers for all to see. Discuss the following questions:

• “Is it always helpful or important to use as few base-ten figures as possible when representing addition of two numbers?” (Yes: fewer pieces mean less counting and smaller likelihood of making a counting mistake. No: once the pieces are bundled, it is harder to see the two numbers that make up the sum.)
• “When can you bundle a base-ten figure?” (When there are at least 10 of a unit.)
• “What would a figure that represents 10 look like?” (An extra-large rectangle, composed of 10 big squares.) “What about 100?” (A giant square made from 10 of the extra-large rectangles representing 10.)
• “Why might using base-ten diagrams for addition be cumbersome with larger multi-digit numbers?” (We would have to draw a lot of squares and rectangles. When there are more units, the diagrams become more complex.)

In this activity, students use symbols and diagrams to find a sum that requires regrouping of base-ten units. Use this activity to give students more explicit instruction on how to bundle smaller units into a larger one and additional practice on using addition algorithm to add decimals.

Again, consider having physical base-ten blocks, a paper version of the base-ten figures, or this digital applet ggbm.at/n9yaWPQj available as alternatives to diagram drawing, or to more concretely illustrate the idea of bundling and unbundling.

If students have difficulty drawing the diagrams to represent bundling, it might be helpful for them to work with actual base-ten blocks, paper printouts of base-ten blocks, or digital base-ten blocks so that they can physically trade 10 hundredths for 1 tenth or 10 tenths for 1 whole.

Focus the whole-class debriefing on the idea of choosing appropriate tools to solve a problem, which is an important part of doing mathematics (MP5). Highlight how drawings can effectively help us understand what is happening when we add base-ten numbers before moving on to a more generalized method. Discuss:

• “In which place(s) did bundling happen when adding 0.38 and 0.69?” (In the hundredth and tenth places.) “Why?” (There is a total of 17 hundredths, and 10 hundredths can be bundled to make 1 tenth. This 1 tenth is added to the 3 tenths and 6 tenths, which makes 10 tenths. Ten tenths can be bundled into 1 one.) 
• “How can the bundling process be represented in vertical calculations?” (We can show that the 8 hundredths and 9 hundredths make 1 tenth and 7 hundredths by recording 7 hundredths and writing a 1 above the 3 tenths in 0.38.)
• “Which method of calculating is more efficient?” (It depends on the complexity and size of the numbers. The drawings become hard when there are lots of digits or when the digits are large. The algorithm works well in all cases, but it is more abstract and requires that all bundling be recorded in the right places.)

In this activity, students use base-ten diagrams and vertical calculations to perform subtraction. As with addition of decimals, students need to pay close attention to place value when calculating differences. They identify the need to pair the digits of like base-ten units when subtracting decimals and why it is helpful to line up the decimal points. 

There is no decomposition or “unbundling” of a base-ten unit into smaller units in this activity, that will come up in the next activity.

As in previous activities, consider having students use physical base-ten blocks (if available), a paper version of the base-ten diagrams (from the blackline master), or this digital applet https://ggbm.at/n9yaWPQj, as alternatives to drawing diagrams.

Arrange students in groups of 2. Review that the term “difference” means the result of a subtraction. Tell students that in this lesson they will use base-ten diagrams to determine differences and, in the diagrams, use X’s to indicate what is being taken away.

Give students 10 minutes of quiet work time. Ask them to pause after the third question, share their diagrams and calculations with a partner, and resolve any differences before finishing the activity.

The goal of the whole-class discussion is to make sure students understand that when we perform subtraction without diagrams, it is essential to pay close attention to place value in the numbers. Select a few students to share their responses and reasonings for the last two questions. Highlight how the different sizes of the base-ten units in the diagram informs how we subtract one decimal from another. Then, discuss:

• How are addition and subtraction of decimal numbers similar? (It is important to attend to place value and to add or subtract numbers that represent the same base-ten units.)
• Did anyone find different results when using diagrams versus when calculating vertically? If so, where did the error happen and what might have caused it?
• Why is it helpful to line up the decimal points when calculating differences of decimals? (Aligning the points helps us align digits with the same place value.)  
• Which is more efficient, using base-ten blocks or calculating the difference? (For some numbers, such as \(1.26 - 0.14\), both methods are efficient. If the numbers contain more decimal places or larger digits, the diagrams would take a lot of time to draw.)