# Lesson 3

Contemos hacia adelante o hacia atrás para restar

## Warm-up: Conversación numérica: Decenas y centenas (10 minutes)

### Narrative

This Number Talk encourages students to think about adding multiples of 10 or 100. The expressions in this activity help students recognize that adding multiples of 10 to a number only changes the number in the tens place and adding multiples of 100 to a number only changes the number in the hundreds place when there is no need to compose a new ten or hundred. As students share their thinking, consider recording on an open number line.

### 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 students’ thinking on an open number line and with equations.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$120 + 20$$
• $$120 + 200$$
• $$124 + 30$$
• $$124 + 300$$

### Activity Synthesis

• “¿Qué patrones observaron en estas expresiones?” // “What patterns did you notice with these expressions?” (We added numbers that changed the tens place and numbers that changed the hundreds place. Only the number in the tens place or hundreds place changed depending on the number.)

## Activity 1: Saltemos hacia atrás, atrás, atrás (20 minutes)

### Narrative

The purpose of this activity is for students to consider counting back by place as a method for subtracting. Students analyze different methods for finding the value of a difference. After making connections between subtracting by place and counting back by decomposing the subtrahend, students try using an open number line to show their thinking. There is not an expectation for precision with the length of the jumps, as this representation is used as a tool to get students thinking about the idea of moving along the number line. The numbers are selected so that no decomposition is needed when subtracting by place value, allowing students to focus on the structure of subtracting tens from tens and ones from ones using base-ten blocks, base-ten diagrams, or the number line (MP7).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display the images for Jada's work and Andre's work.
• “Tanto Jada como Andre encontraron el valor de $$375 - 24$$, pero pensaron sobre esto de forma un poco diferente” // “Jada and Andre both found the value of  $$375 - 24$$, but thought about it a little differently.”
• “Tómense un minuto para examinar su trabajo. ¿Qué observan? ¿Qué se preguntan?” // “Take a minute to look at their work. What do you notice? What do you wonder?”
• 12 minutes: quiet think time
• 2 minutes: partner discussion
• Share responses.
• As needed:
• “¿Qué observaron acerca de las representaciones?” // “What did you notice about the representations?” (Jada used a base-ten diagram and Andre used a number line and equations. They showed subtraction with jumps and by crossing out.)
• “¿En qué parte de la representación de Jada pueden ver que ella resta 4 unidades?” // “Where in her representation can you see Jada subtracting 4 ones?” (She subtracts 4 by crossing out 4 ones.)
• “¿En qué parte de la representación de Andre pueden ver que él resta 4 unidades?” // “Where in his representation can you see Andre subtracting 4 ones?” (He jumped 20 and then 4. So his second jump shows subtracting 4.)
• “Hemos usado bloques en base diez y diagramas en base diez para encontrar diferencias” // “We have been using base-ten blocks and base-ten diagrams to find differences.”
• “Cuando usamos estas representaciones, restamos de acuerdo al valor posicional, quitando unidades a las unidades y decenas a las decenas” // “When we use these representations, we subtract by place, taking ones from ones and tens from tens.”
• “Otra forma en la que podemos encontrar la diferencia es contando hacia atrás. Podemos usar lo que sabemos sobre el valor posicional para contar hacia atrás en algunas partes para hacerlo más fácil” // “Another way we can find the difference is by counting back. We can use what we know about place value to count back in parts to make it easier.”

### Activity

• “Ahora intenten encontrar el valor de $$189 - 73$$ de la forma en la que Andre restó. Después, encuentren el valor de $$647 - 46$$ de la forma que quieran” // “Now, try using Andre’s way to find the value of $$189 - 73$$. Then, find the value of $$647 - 46$$  your own way.”
• 8 minutes: independent work time
• “Comparen su trabajo con un compañero” // “Compare your work with a partner.”
• 2 minutes: partner discussion
• Monitor for students who:
• represent their thinking using base-ten diagrams
• represent their thinking using a number line
• represent their thinking using equations

### Student Facing

Jada y Andre encontraron el valor de $$375 - 24$$.

Este es su trabajo.

$$375 - 24 = 351$$

El trabajo de Andre

$$375 - 20 = 355\\ 355 - 4 = 351\\ 375 - 24 = 351$$

¿Qué observas? ¿Qué te preguntas?

1. Intenta encontrar el valor de $$189 - 73$$ de la forma en la que Andre restó.

Muestra cómo pensaste. Si te ayuda, usa una recta numérica.

2. Encuentra el valor de $$647 - 46$$ a tu manera.

Muestra cómo pensaste. Si te ayuda, usa una recta numérica.

### Student Response

If students use base-ten diagrams without making the connection to counting on the number line, show jumps of 10 instead of a multiple of 10. For example, show 70 as 7 jumps of 10. Consider asking:

• “¿En qué parte de la representación en esta recta numérica ves las 7 decenas?” // “Where do you see the 7 tens on this number line representation?”
• “¿Cómo se relaciona esto con tu representación en base diez?” // “How does this connect to your base-ten representation?”

### Activity Synthesis

• Invite previously identified students to share for each expression.
• Consider sharing work from students who use base-ten diagrams and students who use number lines or equations to make connections and show reasoning about place value.
• “¿En qué se parecen y en qué son diferentes la representación de ____ y la representación de _______?” // “What is the same and what is different between _____’s representation and _____’s representation?”
• Highlight connections based on place value across representations.

## Activity 2: ¿Quién regó pintura? (15 minutes)

### Narrative

The purpose of this activity is for students to analyze equations to determine the unknown value. Students use their understanding of place value and counting within 1,000 to find the value that makes each equation true. Students are given equations with an unknown addend and a sum that is a multiple of 100 or equations with a number that is subtracted from a multiple of 100. They use any method that works for them to find the numbers that make each equation true. Monitor for ways students use what they know about sums of 10 and 100 and counting on or back by place. They may also use an open number line or base-ten diagrams as needed to make sense of each equation or show their thinking. If they do use base-ten diagrams, look for ways they consider composing or decomposing units. All students will make sense of this method in upcoming lessons.

MLR8 Discussion Supports. Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain how they found the values hidden by the spilled paint. Invite groups to rehearse what they will say when they share with the whole class.
Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Begin with a slightly more accessible problem. For example, ??? + 400 = 1,000. Discuss the strategy used to determine the missing number.
Supports accessibility for: Attention, Organization, Language

### Launch

• Groups of 2
• Display the image for Diego’s equation.
• “¡Oh, no! Diego regó pintura en su hoja y ahora no puede ver todos los números” // “Oh no! Diego spilled paint on his paper and now he can’t see all the numbers.”
• “¿Qué número creen que se manchó? ¿Cómo lo saben?” // “What number do you think got smudged? How do you know?” (460 because $$540 + 60 = 600$$ and $$600 + 400 = 1,\!000$$)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.

### Activity

• “Ahora van a trabajar en otras ecuaciones en las que Diego regó algo” // “Now you are going to work on a few more equations where Diego made a mess.”
• “¿Cómo pueden ayudarles las sumas de 10 a pensar en estas ecuaciones?” // “How can sums of 10 help you think about these equations?” ($$3 + 7 = 10$$ and $$30 + 70 = 100$$)
• “Primero van a trabajar individualmente y después tendrán tiempo para compartir con un compañero” // “You’ll work on your own first and then have time to share with a partner.”
• 4 minutes: independent work time
• “Yo sé que posiblemente no terminaron todos los problemas. Eso está bien” // “I know you may not have finished all of the problems. That is okay.”
• “Comparen con un compañero y resuelvan juntos el resto si quieren” // “Compare with a partner and solve the rest together if you’d like.”
• 6 minutes: partner work
• Monitor for students who find $$900 - 370 = 530$$ by:
• counting on by hundreds then tens
• counting on by tens then hundreds
• counting back by hundreds then tens
• using a number line to keep track of their method

### Student Facing

¡Oh, no! Diego regó pintura en su hoja y ahora no puede ver todos los números. Encuentra el número que la pintura está cubriendo.

En cada caso, encuentra el número que hace que la ecuación sea verdadera.

1. __________

2. __________

3. __________

4. __________

5. __________

### Activity Synthesis

• Invite previously identified students to share how they found the value of $$900 - 370$$.
• Display $$370 + {?} = 900$$.
• “¿Cómo nos puede ayudar esta expresión a pensar en este problema?” // “How could this expression help us think about this problem?” (You can start at 370 and count up to 900 to find the difference.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy usaron la relación que hay entre la suma y la resta para encontrar el valor de diferencias y de números que hacen que las ecuaciones sean verdaderas” // “Today you used the relationship between addition and subtraction to find the value of differences and numbers that make equations true.”

“Piensen en todas las maneras que vieron que se usaron para encontrar el valor de las diferencias y el de los sumandos desconocidos. Cuéntenle a su pareja sobre una manera que se sientan seguros de usar y díganle por qué se sienten seguros” // “Think about all of the ways you saw used to find the value of differences and unknown addends. Tell your partner about one that you feel really confident about using and why.”

Share responses.