# Lesson 21

Zeros in the Standard Algorithm

## Warm-up: Which One Doesn’t Belong: Numbers with 0, 2, and 5 (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare features of multi-digit numbers. In making comparisons, students have a reason to use language precisely (MP6), especially place value names. The activity also enables the teacher to hear how students talk about the meanings of non-zero digits in different places of a multi-digit number.

Students observations will support their reasoning in the next activity when they subtract a number with non-zero digits from the four numbers listed.

### Launch

• Groups of 2
• Display numbers.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn’t belong?

1. 2,050
2. 2,055
3. 205.2
4. 20,005

### Activity Synthesis

• “What if we subtracted 44 from each whole number?” Record “– 44” under each whole number.
• “Which number would it be easiest to subtract 44?” $$(2,\!055 - 44$$ because you don’t have to decompose units.)
• “How could we subtract 44 from the other whole numbers?” (We would have to decompose other place values.)

## Activity 1: What If There is Nothing to Decompose? (20 minutes)

### Narrative

The purpose of this activity is to examine subtraction cases in which non-zero digits are subtracted from zero digits. In some cases, students could simply look at the digit to the left of a 0 and decompose 1 unit of that number. But in other cases, the digit to the left is another 0 (or more than one 0), which means looking further to the left until reaching a non-zero digit. Students learn to decompose that unit first, and then move to the right, decomposing units of smaller place values until reaching the original digits being subtracted. The problems are sequenced from fewer zeros to more zeros to allow students to see how to successively decompose units.

Recording all of the decompositions can be challenging. For the last problem, two sample responses are given to show two different ways of recording the decompositions. The important point to understand is that because there are no tens, hundreds, or thousands to decompose, a ten-thousand must be decomposed to make 10 thousands. Then, one of the thousands is decomposed to make 10 hundreds, and so on, until reaching the ones place. Those successive decompositions can be lined up horizontally, but this can make it hard to see what happened first. A second way shows more clearly the order in which the decompositions happen, but it may be challenging to see which place the successive decomposed units are in.

To add movement to this activity, the second problem could be done as a gallery walk where each group completes one problem and then walk around the room to look for similarities and differences in others’ posters.

MLR8 Discussion Supports. Display sentence frames to support partner discussion: “I noticed _____ so I . . .”, and “I wonder if . . . .”
Representation: Develop Language and Symbols. Maintain a visible display to record language that students can use to explain their work. Include important vocabulary, such as the name of each place value category, decompose, and exchange. Also include sentence frames, such as: “I looked at the _____ place and saw that I did not have enough ____ to subtract.” and “I exchanged one ______ for 10 _____.”
Supports accessibility for: Conceptual Processing, Language, Organization

### Launch

• Groups of 2–4
• 5 minute: independent think time
• 3 minute: partner share

### Activity

• “Take some time to think about the first problem and then discuss it with your partner.”
• 2 minutes: quiet think time
• 3 minutes: partner discussion
• Pause for clarifying questions, as needed.
• “Now, work with your partner to complete the next 2 problems.”
• 5 minutes: partner work time

### Student Facing

Here are some numbers you saw earlier. Each number has at least one 0. From each number, 1,436 is being subtracted.

1. Make sense of the problems and explain to a partner.

2. Use the approach in the first problem to find these two differences:

3. Find the value of each difference. Be prepared to explain your reasoning. If you get stuck, try subtracting using the expanded form.

### Student Response

Students may lose track of the units when multiple rounds of decompositions are required. Offer base-ten blocks and prompt students to show multiple decompositions when finding the value of $$100 - 1$$ using a large square base-ten block that represents 100. Consider asking:

• “How can you show subtraction of 1?” (Exchange it for 100 ones.)
• “What if we could only exchange 1 unit for 10 units at a time?” (Exchange the hundred for 10 tens, then exchange 1 ten for 10 ones, so that we’d have 9 tens and 10 ones. Then, we can remove 1.)
• “How would you show the subtraction of 1 from 100 using the standard algorithm?”

### Activity Synthesis

• Ask students to share responses and to demonstrate consecutive decomposing when multiple zeros are involved.
• If needed, use expanded form to represent the decompositions students are explaining.

## Activity 2: What is Your Age? (15 minutes)

### Narrative

In this activity, students solve contextual problems that involve subtracting numbers with non-zero digits from numbers with one or more zero digits. Students may choose other ways to find the difference (for example, adding up and using a number line to keep track), but are asked to use the standard algorithm at least once.

In the launch, students subtract their age from the current year. This provides an opportunity for students to notice the relationship between this difference and their current age (MP7).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “Write the year that you were born down and subtract that year from the current year.”
• “Share with a neighbor what you found out.” (Students will notice that the number that results is their current age or the age they will be on their upcoming birthday.)
• “Jada used this method to find the age of some of her relatives.”

### Activity

• 10 minutes: partner work time

### Student Facing

Jada recorded the birth year of some of her maternal grandparents for a family history project.
family member birth year
grandmother 1952
grandfather 1948
great-grandmother      1930
great-grandfather 1926
As of this year, what is the age of each family member? Show your reasoning. Use the standard algorithm at least once.

### Activity Synthesis

• Ask students to share responses.
• “Which years did you find easiest to subtract? Which were more difficult? Why?” (Subtracting a year without decomposing any units was easier than subtracting a year that required decomposing.)
• Prompt students to check and compare ages found using the standard algorithm and those found using other ways of reasoning.

## Lesson Synthesis

### Lesson Synthesis

Display these expressions. “Here are three expressions.”

“How are the expressions alike?” (They all involve 2,222 that is subtracted from a six-digit number with three 5s and three 0s. Finding each difference requires multiple regroupings.)

“How are they different?” (The fives and zeros are in different positions in each number. In the first expression, only one unit needs to be decomposed before 2 ones could be subtracted from it. In the second expression, two units need to be decomposed. In the third expression, three units need to be decomposed before 2 ones could be subtracted.)

“A friend is unsure how to solve the second expression, $$505,\!500 - 2,\!222$$. Explain to a partner how you would use the standard algorithm to find the value of the difference.”

2 minutes: partner discussion

2 minutes: whole-group discussion