# Lesson 15

Estimemos y restemos

## Warm-up: Conversación numérica: Uno y cinco décimas (10 minutes)

### Narrative

The purpose of this number talk is for students to develop mental strategies for subtracting decimals where decomposition of one or more units is needed when subtracting by place value. These strategies will be useful as students subtract more complex decimals and need to pay close attention to place value.

### Launch

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$1.50 - 0.51$$
• $$1.50 - 0.52$$
• $$1.50 - 0.60$$
• $$1.50 - 0.62$$

### Activity Synthesis

• “¿Cómo usaron el valor de la tercera expresión para encontrar el valor de la última expresión?” // “How did you use the value of the third expression to find the value of the last expression?” (I know 0.62 is two hundredths more than 0.60 so I subtracted 2 hundredths from 0.90 and that left 0.88.)

## Activity 1: Retomemos el algoritmo (15 minutes)

### Narrative

The purpose of this activity is for students to understand that the standard algorithm for subtraction can be used with decimals. Students first find the value of a difference of decimals using a strategy that makes sense to them and then see calculations organized using the standard algorithm. When students calculate using a strategy of their choice, they may or may not show decomposition of a whole into 10 tenths. The standard algorithm explicitly shows this decomposition.

When students share their explanation of Han's calculations with a partner and revise their work after receiving feedback they critique the reasoning of others and improve their arguments (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Conversing.

### Launch

• Groups of 2
• “Trabajen individualmente en los dos primeros problemas” //  “Work on the first two problems on your own.”

### Activity

• 5–6 minutes: independent work time
MLR1 Stronger and Clearer Each Time
• “Compartan con su compañero su descripción del trabajo de Han. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your description of Han’s work with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: structured partner discussion
• Repeat with 2–3 different partners.
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

### Student Facing

1. Encuentra el valor de $$8.53 - 2.7$$. Muestra cómo razonaste.
2. Han usó el algoritmo estándar para restar números decimales. Este es el trabajo de Han. Describe qué hizo en cada paso.

3. Usa el algoritmo estándar para encontrar el valor de $$6.62 - 3.71$$.

### Activity Synthesis

• “Resuelvan individualmente el último problema” // “Solve the last problem on your own.”
• 2-3 minutes: independent work time
• “¿En qué se parecen el algoritmo estándar de resta y el algoritmo estándar de suma?” // “How is the standard algorithm for subtraction the same as the standard algorithm for addition?” (I work from right to left. I line up all of the place values and make sure the digits I am subtracting have the same place value.)
• “¿En qué son diferentes el algoritmo estándar de resta y el algoritmo estándar de suma?” // “How is the standard algorithm for subtraction different than the standard algorithm for addition?” (Sometimes I need to decompose a unit. I need to decompose a whole into 10 tenths. With addition, I sometimes need to compose a new unit.)

## Activity 2: Estimemos y restemos (20 minutes)

### Narrative

The purpose of this activity is for students to estimate and then find differences. Students just learned the standard algorithm for subtraction can be used with decimals in the previous activity and may use it here. All students should be encouraged to find the differences in a way that makes sense to them. Any subtraction strategy they use will require close attention to place value and this is the focus of the synthesis.

Action and Expression: Develop Expression and Communication. Provide alternative options for expression. Students can share their thinking using a place value chart, diagram, or a picture to show how they decomposed in order to subtract.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

• Groups of 2

### Activity

• 8 minutes: independent work time
• 2 minutes: partner work time
• Monitor for students who:
• use place value understanding to subtract by place
• use the standard algorithm
• use a compensation strategy to subtract 9.8

### Student Facing

1. ¿Qué número entero está más cerca de la diferencia $$15.27 - 4.3$$? Explica o muestra cómo razonaste.
2. ¿Cuál es el valor de $$15.27 - 4.3$$? Explica o muestra cómo razonaste.
3. ¿Qué número entero está más cerca de la diferencia $$16.14 - 9.8$$? Explica o muestra cómo razonaste.
4. ¿Cuál es el valor de $$16.14-9.8$$? Explica o muestra cómo razonaste.

### Activity Synthesis

• Invite students to share their strategies for finding the difference $$15.27 - 4.3$$.
• “¿En qué se parecen las estrategias? ¿En qué son diferentes?” // “How are the strategies the same? How are they different?” (They both broke up 4.3 into pieces and then subtracted each piece. Some students used the standard algorithm and some subtracted by place value. The standard algorithm works in one way. There are different ways to subtract 4.3 working by place value.)
• Invite students to share their estimate for $$16.14 - 9.8$$.
• “¿Cómo supieron que el valor estaría cerca de 6?” // “How did you know that the value would be close to 6?” (16.14 is just a little more than 16 and 9.8 is almost 10 so I used whole number subtraction to estimate, $$16 - 10 = 6$$.)
• “¿La estimación les ayudó a hacer el cálculo?” // “Did the estimate help you with the calculation?” (Yes, I used the standard algorithm and got an answer that was just a little more than 6 and that agreed with my estimate. Yes, the estimate made me think of subtracting 10 which I could do in my head and then adding back 0.2 which I could also do in my head.)

## Activity 3: Comparemos cálculos para evaluar una diferencia [OPTIONAL] (10 minutes)

### Narrative

The purpose of this activity is for students to analyze a common error when using the standard algorithm to subtract decimals. This activity is recommended if students need additional practice with the standard algorithm when the two numbers do not have the same number of decimal places. The standard addition algorithm requires students to subtract digits with the same place value. In the given example the two numbers are “right aligned” as when subtracting whole numbers and this leads to the error of subtracting hundredths from tenths, tenths from ones, and so on as if these digits had the same place value.

When students identify and correct Elena's error they construct viable arguments and critique the reasoning of others (MP3).

This activity uses MLR3 Clarify, Critique, and Correct. Advances: Reading, Writing, Representing.

• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
MLR3 Clarify, Critique, Correct
• Display Elena’s solution and explanation.
• “¿Qué creen que quiere decir Elena? ¿Hay algo que no sea claro?” // “What do you think Elena means? Is anything unclear?” (She says the answer will be less than 622.3, but she doesn’t say how much less. The answer will be a lot less than 615.21.)
• “Escriban con su compañero una solución y una explicación ajustadas” // “With your partner, work together to write a revised solution and explanation.”
• (Optional) Display and review the following criteria:
• Explanation of mistakes
• Specific words and phrases
• Decimal point
• Place value
• Labeled diagram
• Correct solution
• 3–5 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class.
• “¿En qué se parecen y en qué son diferentes las soluciones y explicaciones ajustadas de estos grupos?” // “What is the same and different about the revised solutions and explanations?”

### Student Facing

1. Encuentra el valor de $$622.35 - 71.4$$. Explica o muestra cómo razonaste.
2. Elena y Andre encontraron el valor de $$622.35 - 71.4$$. ¿Con quién estás de acuerdo? Explica o muestra cómo razonaste.

### Student Response

If a student does not have a strategy to solve problem 1, ask them to explain how they would find the value of $$622 - 51$$.

### Activity Synthesis

• Display Andre’s solution and explanation.
• “¿Por qué creen que Andre escribió 71.40 en vez de 72.4?” // “Why do you think Andre wrote 71.40 instead of 72.4?” (It helps him keep the place values and decimals in line so he doesn’t mix up the places when he is subtracting. 71.40 is equivalent to 71.4.)
• Display Elena’s solution.
• “¿Cómo pueden darse cuenta, sin encontrar la diferencia, de que la respuesta de Elena no es correcta?” // “How can you see without finding the difference that Elena’s answer is not correct?” (Taking 71 away from 622 is a lot less than 615. $$620-70=550$$ so the answer should be around 550.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy restamos números decimales y usamos el algoritmo estándar” //  “Today we subtracted decimals and we used the standard algorithm.”

“¿En qué se parece el algoritmo estándar cuando se restan números enteros al algoritmo estándar cuando se restan números decimales? ¿En qué se diferencia?” // “How is the standard algorithm for subtracting the same and different when you subtract whole numbers or decimals?” (When I can’t subtract in one place, I need to move to a larger place and break up a larger unit. I have to keep track of the decimal points and make sure that I am subtracting the same place values.)

“¿Hay otras estrategias para restar números enteros que también funcionan con números decimales?” // “Are there other subtraction strategies that you use for whole numbers that also work with decimals?” (Yes. I can use addition. I can add on to the smaller number. I can also subtract by place value or use compensation.)