# Lesson 5

Otro algoritmo de suma

## Warm-up: Observa y pregúntate: Otra tabla interesante (10 minutes)

### Narrative

The purpose of this warm-up is to elicit observations about patterns in sums of two- and three-digit addends in an addition table. The table is partially filled out to highlight some properties of operations. For example, the sums in the table can illustrate the commutative property ($$99 + 98$$ and $$98 + 99$$ both give 197). The numbers also prompt students to notice patterns in sums of odd and even numbers. For example, the sum of an odd number and an even one is always odd.

While students may notice and wonder many things about the addition table, focus the discussion on the patterns in the table and possible explanations for them. When students make sense of patterns in sums and explain them in terms of the features of the addends and how they are added, they notice and use regularity in repeated reasoning (MP7).

### Launch

• Groups of 2
• Display the table.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

+ 98 99 100 101 102
98 197 199
99 197 199 201
100 ? ?
101 199 201 203
102 201 203

### Activity Synthesis

• “¿Cómo creen que funciona la tabla?” // “How do you think the table works?” (The table shows addition. Each number in the row at the top is added to each number in the first column on the left.)
• “¿Qué números deberían ir en las celdas que tienen un signo de interrogación?” // “What numbers would go in the cells with a question mark?” (199 and 201.) “¿Cómo lo saben?” // “How do you know?”
• For each of the following questions, give students a minute of quiet think time. Illustrate their responses with equations, if possible.
• “¿Por qué estas sumas aumentan de 2 en 2 de izquierda a derecha y de arriba a abajo?” // “Why do the sums go up by 2 from left to right and top to bottom?” (Because of the skipping, one number being added goes up by 2, so the sum also goes up by 2.)
• “¿Por qué las sumas desde la esquina superior izquierda hasta la esquina inferior derecha aumentan de 2 en 2?” // “Why do the sums from upper left corner to lower right corner increase by 2?” (Each of the two numbers being added go up by 1, so the sum goes up by 2.)
• “¿Por qué las sumas que forman una línea desde la esquina inferior izquierda a la esquina superior derecha son el mismo número?” // “Why are the sums that make a line from the lower left corner to the upper right corner the same number?” (Each time, the first number being added goes up by 1 and the second number goes down by 1, so the sum stays the same.)
• Consider asking: “¿Las sumas o los sumandos son pares o impares? ¿Qué observan?” // “What do you notice about whether the addends or the sums are even or odd?” (In each pair of addends, one number is even and the other number is odd. All the sums are odd.)

## Activity 1: Un nuevo algoritmo de suma (15 minutes)

### Narrative

The purpose of this activity is for students to learn an algorithm in which a single digit is recorded as each place value position is added. Students use an algorithm from a prior lesson to make sense of the new algorithm. They learn that single digits can be used to represent the sum in each place value position because of what we know about place value.

Action and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches.
Supports accessibility for: Conceptual Processing

### Launch

• Groups of 2
• Display the algorithms.
• “Han y Elena usaron dos algoritmos diferentes para resolver este problema. Los pasos que siguieron están marcados en orden” //“Han and Elena used two different algorithms to solve this problem. The steps they took are labeled in order.”
• Point to the different steps.
• “Vimos el algoritmo de Han en una lección anterior. Piensen cómo funciona el algoritmo de Elena” // “We saw Han's algorithm in an earlier lesson. Think about how Elena’s algorithm works.”
• 2 minutes: quiet think time

### Activity

• “Ahora discutan con su pareja en qué se diferencia el algoritmo de Elena del algoritmo de Han. También discutan por qué ambos algoritmos funcionan” // “Now discuss with your partner how Elena’s algorithm is different from Han’s algorithm. Also discuss why both algorithms work.”
• 5-7 minutes: partner discussion

### Student Facing

Estos son dos algoritmos para sumar $$367 + 231$$.

El algoritmo de Han

El algoritmo de Elena

Discute con tu compañero:

1. ¿En qué son diferentes los algoritmos de Elena y de Han?
2. ¿Por qué ambos algoritmos funcionan?

### Advancing Student Thinking

If students do not explain how the algorithms are different, consider asking:

• “¿Cuáles son los pasos de cada algoritmo?” // “What are the steps for each algorithm?”
• “¿Por qué el algoritmo de Han tiene un paso extra?” // “Why does Han's algorithm have an extra step?”

### Activity Synthesis

• Invite students to share their analyses of Elena and Han's algorithms.
• Display the algorithms and annotate them to illustrate students' explanations.
• “¿En dónde ven 8, 90 y 500 en el algoritmo de Elena?” // “Where do you see the 8, 90, and 500 in Elena’s algorithm?”
• “¿Por qué el algoritmo de Han tiene un paso 4?” // “Why does Han’s algorithm have a step 4?”
• Select other students to share their thinking on why both algorithms work.

## Activity 2: Compongamos nuevas unidades (20 minutes)

### Narrative

The purpose of this activity is for students to consider how the composition of new tens and hundreds are recorded in the algorithm they saw in the previous activity. Students interpret the work and thinking of others and discuss the similarities and differences in two different strategies for finding a sum (MP3). Students see that in Elena’s algorithm, when the sum of the digits in a place has more than one digit, a newly composed ten or hundred is recorded as “10” or “100” above the addends, while the remaining value is recorded as a single digit below the addends. The synthesis focuses on clarifying how to record newly composed units when adding two numbers.

MLR8 Discussion Supports. Synthesis: Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.

### Launch

• Groups of 2
• Display Elena’s algorithm.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: Elena is adding from right to left. There is a 100 at the top. She adds ones to ones, tens to tens, and hundreds to hundreds. Students may wonder: Why is there a 100 at the top? Why does she only record one number in each place?)
• 1 minute: quiet think time
• Share and record responses.
• “El primer problema menciona ‘14 decenas’. ¿De dónde podrían venir 14 decenas?” // “The first problem mentions '14 tens.' Where might the 14 tens come from?” (From adding 6 tens and 8 tens, or 60 and 80.)

### Activity

• “Trabajen con su pareja para responder la primera pregunta. Hagan una pausa antes de seguir con el segundo grupo de problemas” // “Work with your partner to answer the first question. Then, pause before moving on to the second set of problems.”
• 2 minutes: partner discussion
• Invite students to share how 14 tens are recorded differently in the two algorithms.
• If no students mention that the 100 at the top represents 10 of the 14 tens, or 100 of the 140, bring this to their attention.
• “Ahora prueben el algoritmo de Elena para encontrar el valor de cada suma” // “Now try using Elena's algorithm to find the value of each sum.”
• 7-10 minutes: partner work time
• Monitor for how students record multiple compositions of tens or hundreds when finding the value of $$354 + 198$$.

### Student Facing

Estos son dos algoritmos para sumar $$365 + 182$$.

El algoritmo de Han

El algoritmo de Elena

1. ¿Cómo muestran los algoritmos las 14 decenas de forma diferente?
2. Prueba el algoritmo de Elena para encontrar el valor de cada suma.

1. $$174 + 352$$
2. $$273 + 619$$
3. $$354 + 198$$
4. $$525 + 376$$

### Advancing Student Thinking

If students do not record the newly composed tens or hundreds in the second and third problems, consider asking:

• “¿Qué unidades nuevas tuviste que componer en este problema?” // “What new units did you have to compose in this problem?”
• “¿En dónde tendría más sentido para ti anotar el nuevo 10 que se compuso? ¿En dónde tendría más sentido anotar el nuevo 100 que se compuso?” // “Where would it make the most sense to you to record the newly composed 10? Where would it make the most sense to you to record the newly composed 100?”

### Activity Synthesis

• Select students to display their work for finding the value of $$354 + 198$$.
• “¿Cómo decidieron en dónde anotar la nueva decena y la nueva centena?” // “How did you decide where to record the new ten and hundred?” (It made the most sense to me to stack the hundred on the ten since the ten was already there.)
• “En este algoritmo, por lo general apilamos las nuevas decenas y centenas que componemos en el orden en el que aparecen al sumar de derecha a izquierda. Es decir, desde la posición de unidades a la de las decenas y después a la de las centenas” // “In this algorithm, we typically stack the newly composed tens or hundreds in the order they happen as we add from right to left, or from the ones place to the tens place to the hundreds place.”

## Lesson Synthesis

### Lesson Synthesis

Display an expression from the last activity, such as, $$174 + 352$$.

“En esta lección, hemos sumado de derecha a izquierda, empezando en la posición de las unidades. Examinemos esta expresión otra vez. Pensemos qué pasa si sumamos de izquierda a derecha, empezando en la posición de las centenas” // “In this lesson, we have been adding from right to left, starting with the ones place. Let’s look at this expression again. Let’s consider what would happen if we started adding from the left with hundreds place.”

Work with the class to find the sum, setting it up like Elena's algorithm, but start by adding the hundreds.

“¿Qué pasaría después? ¿Podemos sumar de izquierda a derecha” // “What would happen next? Can we add from left to right?” (If we worked from left to right, we would have to add the hundreds, then add them again if the tens add up to make a new hundred. It's the same with the tens. If we added them before the ones, we would have to add the tens again if the sum of the ones made a new ten.)