# Lesson 1

Comparemos, contemos hacia adelante y contemos hacia atrás

## Warm-up: Conversación numérica: Contemos hacia atrás (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for counting back as a strategy for finding the value of differences. These understandings help students develop fluency and will be helpful later in this lesson when students subtract within 1,000. As students share their thinking, represent it on an open number line to help them make connections.

### 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

• Record answers and strategies on an open number line.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$586 - 6$$
• $$586 - 8$$
• $$434 - 5$$
• $$352 - 4$$

### Activity Synthesis

• “Aunque había números de tres dígitos, me di cuenta de que algunos de ustedes usaron las mismas estrategias que usaron antes. ¿Por qué funciona eso?” // “Even though there were three-digit numbers, I noticed that some of you used the same strategies you’ve used before. Why does that work?” (If you are subtracting a small amount, you can count back by ones to find the answer.)
• “Hoy vamos a examinar otras formas en las que podemos sumar y restar números de tres dígitos usando estrategias que hemos usado antes para sumar y restar números de dos dígitos” // “Today we are going to look at other ways we can use strategies we've used for adding and subtracting two-digit numbers to add and subtract with three-digit numbers.”

## Activity 1: Observemos la diferencia (20 minutes)

### Narrative

The purpose of this activity is for students to compare three-digit numbers and find the value of their difference. Students used the number line to compare three-digit numbers in an earlier unit. They build on this understanding as they find the difference between 2 three-digit numbers that are within 10 of one another. Students notice that when the numbers in a subtraction expression are close together, they can count on or count back to find the value of the difference (MP7, MP8).

Engagement: Provide Access by Recruiting Interest. Optimize meaning and value. Invite students to represent the jumps/movement on the number line using an animal (frog, rabbit, grasshopper, etc.). Focus on the minimal jumping the animal does when the numbers are close on the number line.
Supports accessibility for: Conceptual Processing, Attention

• Groups of 2

### Activity

• “A Tyler y a Elena les pidieron que encontraran el valor de $$81 - 79$$. Este es su trabajo” // “Tyler and Elena were asked to find $$81 - 79$$. This is their work.”
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (The numbers are really close on the number line. Tyler drew out all the tens and ones for 81 and subtracted almost all the tens and ones. Why did Elena put a point on 79 and count up instead of subtracting 79?)
• “En una lección anterior aprendieron que pueden restar quitando una cantidad. También aprendieron que pueden contar hacia adelante de un número a otro o contar hacia atrás de un número a otro para encontrar la diferencia entre 2 números” // “You have learned in an earlier lesson that you can subtract by taking an amount away. You also learned you can count on or count back from one number to the other to find the difference between the 2 numbers.”
• “Esto también sirve con números de tres dígitos” // “This works with three-digit numbers too.”
• “Resuelvan los siguientes problemas solos y después comparen con un compañero cómo pensaron” // “Do the next few problems on your own, and then compare your thinking with a partner.”
• 8 minutes: independent work time
• 4 minutes: partner discussion
• Monitor for students who count on or count back.

### Student Facing

A Tyler y a Elena les pidieron que encontraran el valor de $$81 - 79$$.
Este es su trabajo.

Tyler

\begin{align} 81-70 &=11 \\ 11-9&= 2 \\ \end{align}

Elena

$$81 - 79 = 2$$

¿Qué observas? ¿Qué te preguntas?

1. Ubica y marca 203 y 198 en la recta numérica.

Usa un >, un < o un = para compararlos.

$$\underline{\hspace{1.5cm}} \quad\boxed{\phantom{\huge00}} \quad \underline{\hspace{1.5cm}}$$

Encuentra el valor de $$203 - 198$$. Muestra cómo pensaste.

2. Ubica y marca 673 y 680 en la recta numérica.

Usa un >, un < o un = para compararlos.

$$\underline{\hspace{1.5cm}} \quad\boxed{\phantom{\huge00}} \quad \underline{\hspace{1.5cm}}$$

Encuentra el valor de $$680 - 673$$. Muestra cómo pensaste.

3. Marca y ubica 501 y 499 en la recta numérica.

Encuentra el valor de $$501 - 499$$. Muestra cómo pensaste.

4. Encuentra el valor de $$400 - 396$$. Muestra cómo pensaste.

### Student Response

If students find the difference by subtracting the second number from the first using base-ten blocks or base-ten diagrams, consider asking:
• “¿Qué observaste cuando ubicaste los números en la recta numérica? ¿Cómo puede ayudarte eso a pensar sobre encontrar la diferencia sin usar una recta numérica?” // “What did you notice when you located the numbers on the number line? How could that help you think about finding the difference without a number line?”
• “¿Cómo podrías usar una de las estrategias que compartimos en el calentamiento para encontrar la diferencia?” // “How could you use one of the strategies we shared in the warm-up to find the difference?”
• “¿Cómo puedes pensar en esta diferencia como una ecuación que tiene un sumando desconocido?” // “How could you think about this difference as an unknown addend equation?”

### Activity Synthesis

• Invite previously identified students to share how they found the value of $$400 - 396$$.
• As time permits, invite students to share their comparisons and how they found the value of each difference.
• “¿Qué observaron sobre los números cuándo los ubicaron en la recta numérica?” // “What did you notice about the numbers when you located them on the number line?” (They were close together. It was easy to see a way to just count from one number to the other.)

## Activity 2: ¿Cuál es la gran diferencia? (15 minutes)

### Narrative

The purpose of this activity is for students to use what they know about counting within 1,000 to make sense of number lines that show counting on or counting back by 10 or 100. This work helps build fluency with counting within 1,000 and connects to an upcoming lesson where students add and subtract multiples of 10 or 100 using equations.

MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, “¿Qué patrón observan?” // “What pattern do you notice?” This gives both students an opportunity to produce language.

### Launch

• Groups of 2
• Display the number line with jumps of 100.
• “En esta recta numérica hay 5 marcas, pero solo 3 tienen el número escrito debajo. ¿Cuáles son los números desconocidos y cómo lo saben?” // “On this number line there are 5 tick marks, but only 3 are labeled. What are the missing numbers and how do you know?” (534 and 634 because the jumps are the same length and they must be jumps of 100. The hundreds place is increasing by 1, but the other places are not changing.)
• 30 seconds: quiet think time
• 30 seconds: partner discussion
• Share and record responses.
• “Ustedes reconocieron que la longitud entre cada marca debe representar 100. Las marcas y flechas muestran que se está contando hacia adelante de 100 en 100 en la recta numérica” // “You recognized that the length between each tick mark must represent 100. The tick marks and arrows show counting on by 100 on the number line.”

### Activity

• “En cada uno de los problemas, usen lo que saben sobre contar, representar números en la recta numérica y el valor posicional para encontrar los números desconocidos” // “For each of the problems, use what you know about counting, representing numbers on the number line, and place value to find the missing numbers.”
• 10 minutes: partner work time
• Monitor for a student to share their reasoning for _____, _____, 332, 342, 352

### Student Facing

1. Completa los números desconocidos.

¿Esta recta numérica muestra que se está contando hacia adelante de 10 en 10 o de 100 en 100?

2. Completa los números desconocidos.

¿Esta recta numérica muestra que se está contando hacia adelante de 10 en 10 o de 100 en 100?

3. Completa los números desconocidos.

¿Esta recta numérica muestra que se está contando hacia adelante de 10 en 10 o de 100 en 100?

4. Completa los números desconocidos para mostrar que se está contando de 10 en 10.

739, 749, __________, 769, __________

5. Explica cómo puedes saber que tus números muestran que se está contando hacia adelante de 10 en 10 y no de 100 en 100.

### Student Response

If students find the missing numbers by counting by 1 instead of using patterns increasing by 10 or 100, consider asking:
• “¿Qué dígito está cambiando? ¿Está aumentando o disminuyendo?” // “What digit is changing? Is it going up or down?”
• “¿Cómo puede ayudarte eso a descifrar el conteo que se está haciendo?” // “How can that help you figure out the count?”

### Activity Synthesis

• Invite a student to share their reasoning for _____, _____, 332, 342, 352.
• “¿Cómo descifraron cuál era el primer número?” // “How did you figure out the starting number?” (I saw that it was going up by 10, so I knew I had to count back by 10 to find the starting numbers.)
• Record responses with an open number line and equations.

## Lesson Synthesis

### Lesson Synthesis

“Hoy compararon números de tres dígitos y encontraron la diferencia entre ellos” // “Today you compared three-digit numbers and found the difference between them.”

“También le dieron sentido a contar hacia adelante y contar hacia atrás de 10 en 10 o de 100 en 100 en la recta numérica” // “You also made sense of counting on and counting back by 10 or 100 on the number line.”

“¿Cómo les puede ayudar esto a pensar en ​​​​​​$$234 + 200$$?” // “How can this help you think about $$234 + 200$$?” (It is like 2 jumps of 100 on the number line, so it would be 434.)