Add and Subtract Decimals with Paper and Pencil

Est. Class Sessions: 2–3

Developing the Lesson

Part 1: Adding and Subtracting Decimals with Paper and Pencil

Create Strategies Menus. Understanding place value is crucial to proficiency with adding and subtracting decimals. Students must understand that they can do these operations correctly only when like places are aligned. Display the following addition problem vertically, carefully aligning the decimal points:

0.5

+ 0.25

Ask:

If I place a 0 behind the 5 so the problem reads 0.50 0.25, does the value of 0.5 change? (no)
How did I write the problem? (Possible response: You carefully aligned the like place values. The hundredths are above the hundredths and the tenths are above the tenths. There are no ones. The decimal points are lined up, too. That way you can add the like values.)
Estimate the sum with a partner. (Possible response: between 0.7 and 0.8)
Work with your partner to find the exact sum. Is it reasonable? (0.75)

Encourage students to use base-ten pieces, Hundredths Circle Wheels, number lines, and hundredths grids. Have copies of the Decimal Number Lines, Decimal Grids, and Hundredths Circles Masters readily available for students to use. Students might also think about the decimals in terms of fractions ( 1/2 liter + 1/4 liter is 3/4 liter) or money (50¢ + 25¢ = 75¢ or 0.75).

After students have had time to solve the problem, select several volunteers and use displays of the Decimal Number Lines, Decimal Grids, and Hundredths Circles Masters to present a variety of solution strategies.

Direct students to the Adding Decimals section of the My Decimal Strategies Menus pages in the Student Activity Book. Tell students that they will create their own strategy menu. They will work with a partner to add examples to the addition menu for solving 0.5 + 0.25. Students will complete the Estimating, Number Line or Counting On, and Using Models sections of the menu, saving the Using Paper and Pencil section for later. Upon completion, have several student volunteers display and discuss the examples they added to their menus.

Next display the following subtraction problem:

0.35

− 0.18

Ask:

How did I write the problem? (Possible response: The same way you wrote the addition problem. You carefully aligned the like place values. The hundredths are above the hundredths and the tenths are above the tenths. You can subtract like values. There are no ones. The decimal points are lined up, too.)
Estimate the difference with a partner. (Possible response: I know 18 + 18 = 36, so 0.35 − 0.18 is close to 0.18.)
Work with your partner to find the exact difference. Is it reasonable? (0.17)

After students have had time to solve the problem, select several volunteers to present a variety of solution strategies including those that use models such as base-ten pieces, hundredths circles, Hundredths Circle Wheels, number lines, and hundredths grids.

Direct students to the Subtracting Decimals section of the My Decimal Strategies Menus pages in the Student Activity Book. Students will work with a partner to add examples to the subtraction menu for solving 0.35–0.18, again completing all but the Using Paper and Pencil section of the menu. Upon completion, have several student volunteers display and discuss the examples they added to their menus.

Ask:

How is using a grid to add or subtract decimals similar to using base-ten pieces? (Possible responses: It is very similar to using base-ten pieces. You are either putting similar pieces together or taking pieces away. You have to add the bits to the bits and the skinnies to the skinnies. On a grid, you add squares when you are adding hundredths or rows when you are adding tenths. Using a grid is just like base-ten shorthand but there is less drawing.)
How is using a number line to add or subtract decimals similar to using it to add or subtract whole numbers? (Possible response: You do the same thing. The size of your hops is just different. For example, instead of hopping hops of one or ten, you might take hops of one-tenth or one-hundredth.)

Introduce Paper-and-Pencil Methods. Ask students to temporarily put their strategies menus aside and direct them to the Add and Subtract Decimals with Paper and Pencil pages in the Student Guide. Read the vignette describing Mr. Moreno's class at the grocery store to introduce paper-and-pencil addition and subtraction methods aloud. The vignette reinforces the idea of aligning like places and regrouping across the decimal. It also reinforces estimation as a means to verify the reasonableness of a solution.

After reading about how Jessie and Roberto solve 0.5 + 0.25 in the Adding Decimals section, ask:

Study the way that Jessie writes the addition problem. How is the addition of decimals with paper and pencil like adding whole numbers with paper and pencil? (Possible response: It is the same. You put the numbers that are in the same places together. You just have to remember the decimal point.)
Why do you think the mouse on the page says, “You're all out of line,” to Roberto? (Possible response: You have to line up the numbers according to their place value when adding decimals so that you can add the hundredths to the hundredths and the tenths to the tenths and Roberto didn't do that.)
What might Roberto have done incorrectly when adding 0.5 and 0.25 liters to get 0.30 liter? (Roberto added the 5 in the tenths place to the 5 in the hundredths place.)

Continue reading the vignette in the Subtracting Decimals section. Shannon and Roberto solve 0.35 − 0.18. Discuss how the students regroup across the decimal.

Ask:

How did Shannon write the subtraction problem? (She carefully lined up the digits by their correct values.)
Shannon used the compact method. How did she subtract 8 hundredths from 5 hundredths? (She traded one tenth for ten hundredths so she could take 8 hundredths from 15 hundredths.)
Roberto used expanded form to subtract. How did he show trading? (Roberto made 0.3 + 0.05 into 0.2 + 0.15. He traded one tenth for ten hundredths so he could subtract 0.08 from 0.15.)
How is trading with whole number subtraction and trading with decimal subtraction similar? (Possible response: It is the same. You trade 1 of the larger units for 10 of the smaller units, like 1 hundred for 10 tens, or 1 tenth for 10 hundredths.)

Remind students that the ten-to-one relationship extends to include numbers that are less than one. Places to the right of the decimal point have the same ten-to-one relationship: tenths, hundredths, thousandths, and so on.

Read the Adding and Subtracting with Larger Numbers section and discuss how the addition and subtraction of decimals is the same as addition and subtraction of whole numbers, no matter how large the whole number is. The key is to align like place values.

Complete Strategies Menus. Students can now complete the Using Paper and Pencil sections of both the addition and subtraction menus on the My Decimal Strategies Menus pages in the Student Activity Book. See Figure 1 for completed sample strategies menus.

Assign Questions 1–3 on the Add and Subtract Decimals with Paper and Pencil pages in the Student Guide to student pairs. Upon completion, discuss the necessity of aligning like places and decimal points when adding or subtracting decimals (Question 1). Have students demonstrate ways to solve the problem using paper and pencil. Do the same for the subtraction problem in Question 2. See Figures 2–3 for example paper and pencil solutions. Stress that students should use the addition and subtraction paper-and-pencil strategy they feel most confident using, just as they did when adding and subtracting whole numbers.

Have students share the addition number sentences they created in Question 3. Then discuss how the placement of the decimal point affects the value of the number. Remind students that the numbers to the right of the decimal point are smaller than one. Finally, discuss that placing a decimal behind a whole number doesn't change the value of the number (e.g., 250 = 250. = 250.0), but moving a decimal point within a number will change the value of the number (e.g., .250 and 2.50).