# Lesson 14

Make Sense of Decimal Subtraction

## Warm-up: True or False: Decimal Differences (10 minutes)

### Narrative

The purpose of this True or False is for students to apply their understanding of place value to subtraction equations with decimals. The names of the decimals will likely suggest a strategy using whole number subtraction. For example \(0.61 - 0.02 = 0.59\) can be read as “61 hundredths minus 2 hundredths equals 59 hundredths,” which students will recognize is true. Alternatively they can think about place value. For example, 0.5 is 5 tenths or 50 hundredths and 50 hundredths minus 1 hundredth is 49 hundredths.

### Launch

- Display one statement.
- “Give me a signal when you know whether the statement is true and can explain how you know.”
- 1 minute: quiet think time

### Activity

- Share and record answers and strategy.
- Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

- \(0.5 - 0.01 = 0.4\)
- \(0.61 - 0.02 = 0.59\)
- \(1 - 0.07 = 0.93\)

### Student Response

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### Activity Synthesis

- Display first equation.
- “How did you use what you know about place value to decide if the equation is true?” (I know 5 tenths is 50 hundredths and if I subtract 1 hundredth that’s 49 hundredths so it’s false.)

## Activity 1: The Difference (20 minutes)

### Narrative

The purpose of this activity is for students to subtract decimals in a way that makes sense to them. Students should be encouraged to use whatever strategies make sense to them, including using place value understanding and the relationship between addition and subtraction. Strategies students may use include

- using hundredths grids (MP5)
- using place value and writing equations

This activity uses *MLR7 Compare and Connect*. Advances: Representing, Conversing.

*Representation: Develop Language and Symbols.*Synthesis: Invite students to explain their thinking orally, using a visual from the gallery walk that was similar to their approach to find the value of the subtraction expression.

*Supports accessibility for: Conceptual Processing, Language, Attention*

### Required Materials

### Launch

- Groups of 2
- Give each group of students a piece of chart paper, colored pencils, crayons or markers, and access to grids.
- 5 minutes: independent work time

### Activity

**MLR7 Compare and Connect**

- “Create a visual display that shows your thinking. You may want to include details such as notes, diagrams, and drawings to help others understand your thinking.”
- 2–5 minutes: partner work time
- 5–7 minutes: gallery walk
- Monitor for students who:
- use strategies like the ones listed in student solutions

### Student Facing

- Find the value of \(2.26 - 1.32\). Explain or show your reasoning.
- What questions do you have about subtracting decimals?

### Student Response

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### Advancing Student Thinking

If students do not have a strategy to get started, ask, “Is the difference greater than or less than 1? How do you know?”

### Activity Synthesis

- Refer to visual displays that students created.
- “What is the same and what is different about the strategies used to find the value of the difference?” (They all pay close attention to place value and the meaning of each digit in the numbers. Some of the displays use subtraction while some of them use addition. They add and subtract different pieces sometimes in different order. Some of the displays have diagrams and some don’t.)
- Display image from students solution.
- “How does the diagram show the difference \(2.26 - 1.32\)?” (It shows 2.26 and then it takes away 1.32. What’s left is the difference.)
- “How is the diagram different from using equations?” (The 1 and 32 hundredths can be crossed out anywhere that’s convenient and then you just count what’s left. You don't have to think as strategically about what to take away or in what order.)

## Activity 2: Target Numbers: Subtract Tenths or Hundredths (15 minutes)

### Narrative

The purpose of this activity is for students to practice subtracting decimals to the hundredth. Students play a new stage of the center Target Numbers. Students played a previous stage that involved adding tenths or hundredths. The game begins when students roll a number cube and decide whether they want the number to represent tenths or hundredths. Then, they subtract the number from 2. They roll the number cube six times and try to make a final difference that is as close to 1 as possible, without being less than 1. Students make strategic choices about which value to assign the number they rolled and adapt their strategy throughout the game. For example, here is a sample record of a game.

number rolled |
0.1 | 0.01 | equation to represent the difference |
---|---|---|---|

4 | 0.4 | \(2-0.4=1.6\) | |

1 | 0.1 | \(1.6-0.1=1.5\) | |

3 | 0.3 | \(1.5-0.3=1.2\) | |

5 | 0.05 | \(1.2-0.05=1.15\) | |

2 | 0.02 | \(1.15-0.02=1.13\) | |

5 | 0.05 | \(1.13-0.05=1.08\) |

Some students may notice that the strategy of this game is identical to the addition game they played in an earlier lesson, the only difference being that here they start at 2 and subtract whereas with the addition game they started at 0 and add (MP8).

### Required Materials

### Launch

- Groups of 2
- Give each group 1 number cube.
- “We’re going to play a new stage of the game called Target Numbers. Let’s read through the directions and play one round together.”
- Read through the directions with the class and play a round with the class:
- Display each roll of the number cube.
- Think through your choices aloud.
- Record your move and score for all to see.

- “Now, play the game with your partner.”

### Activity

- 8–10 minutes: partner work time
- Monitor for students who:
- go under 1 before their sixth roll and describe how they could have changed the placement of a number from the tenths place to the hundredths place to possibly impact the outcome of the game
- have a final difference that is not close to 1 and describe how they could have strategically changed the placement of a number from the hundredths place to tenths place to possibly impact the outcome of the game

### Student Facing

Directions:

- Play one round of Target Numbers.
- Partner A:
- Start at 2. Roll the number cube. Choose whether to subtract that number of tenths or hundredths from your starting number.
- Write an equation to represent the difference.

- Take turns until you’ve played 6 rounds.
- Each round, the difference from the previous equations becomes the starting number in the new equation.
- The partner to get a difference closest to 1 without going under wins.

- Partner A:
- Describe a move that you could have made differently to change the outcome of the game.

### Student Response

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### Advancing Student Thinking

If students are not being strategic about their placement of the digits, ask, “Why did you choose that value for the number you rolled?”

### Activity Synthesis

- Ask previously selected students to share their reflections on the game.
- Recall the addition game students played three lessons earlier, choosing tenths or hundredths with a target number of 1.
- “How is the subtraction game the same as the addition game?” (I have 6 rolls of the dice and I get to choose tenths or hundredths each time. The strategy is the same because in both cases I want the 6 decimals to add up to close to 1 without going over.)
- “How is the subtraction game different than the addition game?” (Instead of adding the decimals starting with 0, I am subtracting them starting with 2. Sometimes I have to break up a 1 or a tenth. In the addition game I sometimes made a 1 or a tenth.)
- “Which game did you prefer?” (I liked the addition game because it was easier to add the decimals than to subtract.)

## Lesson Synthesis

### Lesson Synthesis

Create a large chart titled “Decimal Subtraction”.

“How is subtracting decimals the same as subtracting whole numbers?” (You have to pay attention to the place values and make sure you are subtracting digits with the same place value. Sometimes you have to decompose a unit.)

“How is it different?” (There are more places to pay attention to and I have to remember how many tenths are in a whole, how many hundredths are in a tenth.)

"How is subtracting decimals the same as adding decimals? How is it different?” (I have to think about place value in the same way. The only difference is that I’m subtracting now instead of adding. So I have to decompose sometimes while with addition I compose sometimes.)

Record responses on poster.

## Cool-down: Subtract (5 minutes)

### Cool-Down

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