Lesson 4
Conozcamos los algoritmos de suma
Warm-up: Cuál es diferente: 247 (10 minutes)
Narrative
This warm-up prompts students to compare three expressions and one three-digit number. During the synthesis, ask students to explain the meaning of any terminology they use, such as the value of each expression and ways that place value was used to write the number 247 in different ways.
Launch
- Groups of 2
- Display the expressions and number.
- “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
- 1 minute: quiet think time
Activity
- “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
- 2-3 minutes: partner discussion
- Share and record responses.
Student Facing
A. \(200 + 30 + 17\)
B. 247
C. \(200 + 47 + 10\)
D. \(100 + 140 + 7\)
Student Response
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Activity Synthesis
- “¿Cómo supieron que A y D eran iguales a 247?” // “How did you know that A and D were equal to 247?” (In A there were 2 hundreds, 4 tens, and 7 ones, but some of the tens were with the ones. In D there were 2 hundreds, 4 tens, and 7 ones, but some of the hundreds were with the tens.)
- Consider stating: “Encontremos por lo menos una razón por la que cada una es diferente” // “Let’s find at least one reason why each one doesn’t belong.”
Activity 1: ¿Qué es un algoritmo? (20 minutes)
Narrative
In this activity, students use their knowledge of base-ten representations and place value to make sense of two addition algorithms. One algorithm shows the addends in expanded form. Both algorithms show the sums of ones, tens, and hundreds separately, but display these partial sums differently. Students notice that both algorithms show hundreds added to hundreds, tens to tens, and ones to ones, regardless of order. In the synthesis, introduce the term “algoritmo” // “algorithm.”
Advances: Representing, Conversing
Required Materials
Materials to Gather
Launch
- Groups of 2
- Give students access to base-ten blocks
- En una lección anterior, exploramos muchas formas de encontrar el valor de una suma. Por un minuto, observen cómo estos 3 estudiantes sumaron \(362+354\)” // “In an earlier lesson, we saw many ways to find the value of a sum. Take a minute to look at how these 3 students added \(362+354\).”
- 1 minute: quiet think time
Activity
- “Con su pareja, expliquen cómo funciona cada dibujo o método” // “Work with your partner to explain how each method works.”
- 7-10 minutes: partner work time.
Student Facing
Tres estudiantes encontraron el valor de \(362 + 354\). Este es su trabajo. Explica cómo funciona cada dibujo o método.
- El dibujo de Tyler
- El método de Lin
- El método de Han
Student Response
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Advancing Student Thinking
If students don't explain one of the written algorithms, consider asking:
- “¿Qué hizo Lin (o Han) para sumar los números?” // “What did Lin (or Han) do to add the numbers?”
- “¿Cómo se relaciona el trabajo de ellos con el dibujo de Tyler?” // “How is their work related to Tyler's drawing?”
Activity Synthesis
- For each method, ask a student share their explanation. As students share, record the sequence of steps they describe in their explanation.
- Consider asking:
- “¿Alguien puede expresar el razonamiento de _______ de otra forma?” // “Who can restate _______ 's reasoning in a different way?”
- “¿Alguien tuvo una idea similar, pero la explicaría de otra forma?” // “Did anyone have a similar idea but would explain it differently?”
- “¿Alguien explicó el método de otra forma?” // “Did anyone explain the method in a different way?”
- “¿Alguien quiere agregar algo a la explicación de ____?” // “Does anyone want to add on to____’s explanation?”
- As students add on, edit the steps so the class is in agreement about how each method works.
- “Lin y Han usaron algoritmos para resolver este problema. Un algoritmo es una serie de pasos que, si se siguen correctamente, siempre funciona” // “Lin and Han used algorithms to solve this problem. An algorithm is a set of steps that works every time as long as the steps are carried out correctly.”
- “¿En qué se parecen los algoritmos de Lin y Han?” // “How are Lin and Han’s algorithms the same?” (They both add ones to ones, tens to tens, and hundreds to hundreds.)
- “¿En qué son diferentes los algoritmos?” // “How are the algorithms different?” (Lin writes the number in expanded form, but Han didn’t. Lin hasn't added the sums of hundreds, tens, and ones, but Han has.)
- Consider asking:
- “¿Podemos saber en qué posición empezó Lin? ¿Por qué sí o por qué no?” // “Can we tell which place Lin started with? Why or why not?” (We can’t really tell with Lin’s method because of how the numbers are next to each other. She might have started with the ones or the hundreds. No matter which place she starts with she would get the same sum.)
Activity 2: Probemos un algoritmo (15 minutes)
Narrative
The purpose of this activity is for students to try the algorithms they saw earlier in the lesson. The important thing is that they combine hundreds and hundreds, tens and tens, and ones and ones, which should be a familiar idea from grade 2. The synthesis provides an opportunity to show a different way of recording newly composed tens and hundreds when compositions are required, which will be discussed in more detail in subsequent lessons. Provide access to base-ten blocks for students to use to support their reasoning about the algorithms, in case requested.
Students analyze and improve a given explanation of how to find a sum, filling in details and using more precise language to explain the calculation more fully (MP3, MP6).
This activity uses MLR3 Clarify, Critique, Correct. Advances: reading, writing, representing
Supports accessibility for: Organization, Social-Emotional Functioning
Required Materials
Materials to Gather
Launch
- Groups of 2
- Give students access to base-ten blocks.
- “Ahora vamos a tener la oportunidad de probar los algoritmos que Lin y Han usaron en la actividad anterior” // Now you are going to have a chance to try the algorithms that Lin and Han used in the last activity. Take a minute to think about which algorithm you want to use for each problem.”
- 1 minute: quiet think time
Activity
- Con su pareja, prueben un algoritmo para encontrar el valor de cada suma” // “Work with your partner to try an algorithm to find the value of each sum.”
- 5 minutes: partner work time
- Monitor for students who use Lin’s algorithm and Han's algorithm on the last problem.
Student Facing
Prueba usar un algoritmo para encontrar el valor de cada suma. Muestra cómo pensaste. Organiza tu trabajo para que los demás puedan entenderlo.
- \(475 + 231\)
- \(136 + 389\)
- \(670 + 257\)
Student Response
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Activity Synthesis
- Select previously identified students to share their work on the last expression.
MLR3 Clarify, Critique, Correct
- Display the following partially correct answer and explanation: I added the ones, the tens, and then hundreds.
- Read the explanation aloud.
- “¿Qué no está claro?” // “What is unclear?”
- 1 minute: quiet think time
- 2 minute: partner discussion
- “Con su pareja, escriban una explicación ajustada” // “With your partner, work together to write a revised explanation.”
- Display and review the following criteria:
- Explanation for each step
- Words such as: first, next, then
- 3–5 minute: partner work time
- Select 1–2 groups to share their revised explanation with the class. Record responses as students share. (Sample explanation: First, I stacked the numbers vertically. Then, I added the ones to get 7 and recorded a 7 below the ones place. Next, I added 70 and 50 and got 120. I recorded 20 below the 7 and 100 below the 20. I added 600 and 200 to get 800 and recorded that below the 100. Then I added and wrote 927 as the answer.)
- “¿En qué se parecen y en qué se diferencian estas explicaciones?” // “What is the same and different about the explanations?” (Both explanations say that they added the ones, tens, and hundreds, but the revised explanation gives more detail about how to record each step.)
- “Comparen esta forma de registrar el trabajo con el método de Han de la primera actividad. ¿En qué se parecen? ¿En qué son diferentes?” // “How is this way of recording this work the same or different from Han’s method in the first activity?” (Han would record 120 in the second row where we record the tens. It was 12 tens, but that’s the same as 1 hundred and 2 tens. We could record 100 on one line and 20 on the next line.)
Lesson Synthesis
Lesson Synthesis
Display Lin and Han’s algorithms.
“Hoy aprendimos sobre dos algoritmos diferentes, es decir, dos series diferentes de pasos para encontrar el valor de una suma. ¿En qué se parecen los dos algoritmos? ¿En qué son diferentes?” // “Today we learned about two different algorithms or two different sets of steps for finding the value of a sum. How are the 2 algorithms alike? How are they different?” (Alike: They give the same result at the end. They both involve using place value and stacking the numbers being added. It doesn’t matter which place value unit we add first. Different: In one algorithm the numbers being added are written in expanded form.)
Cool-down: Escoge un algoritmo (5 minutes)
Cool-Down
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