Lesson 10

The purpose of this warm-up is for students to connect a base-ten diagram to addition on the number line. This will support their work later in the lesson when they connect place value methods to representations of addition and subtraction on the number line.

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

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

• “¿Qué ecuación de suma puede estar representada aquí?” // “What addition equation could be represented here?” (\(28 + 10 + 5 = 43\) or \(28 + 15=43\))
• “Vamos a seguir pensando en qué se parecen y en qué se diferencian los diagramas en base diez y las rectas numéricas” // “We are going to keep thinking about what is the same and what is different between base-ten diagrams and number lines.”

The purpose of this activity is for students to connect a subtraction method based on place value to representations on the number line. Students compare representations of a subtraction method using base-ten diagrams, equations, and number lines (MP2). They notice that, just like with base-ten blocks, they can think about subtracting or counting by tens first or by ones first on the number line.

• Groups of 2
• Give students access to base-ten blocks.
• Display image of Clare’s base-ten diagram.

• “Clare restó y representó en un diagrama en base diez cómo pensó” // “Clare subtracted and represented her thinking with a base-ten diagram.”
• “¿Qué nos dice este diagrama?” // “What does this diagram tell us?” (She started with 46. She took away 35. She has 1 ten and 1 one left.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses
• “Van a escribir una ecuación para representar el trabajo de Clare. Luego, representarán en la recta numérica el método de Clare” // “You are going to write an equation to represent Clare’s work. Then you will represent Clare's method on a number line.”
• “Con su compañero, decidan cómo representar de la mejor manera lo que ustedes piensan que Clare hizo. Pueden discutir sobre dónde empezar, cuántos saltos deben dibujar, qué tan largo debe ser cada salto y a dónde llegar” // “Work with your partner to decide how to best represent what you think Clare did. You may discuss where to start, how many jumps you should draw, how long each jump should be, and where to land.”
• 5 minutes: partner work time
• “Ahora, intenten uno solos” // “Now try one on your own.”
• 8 minutes: independent work time
• Monitor for students who:
  • start at 58 and jump 20 and then 4
  • start at 58 and jump 4 and then 20
  • start at 58 and jump 10, 10, and then 4
  • start at 58 and show a jump for each ten and each one (6 total jumps)

If students represent their thinking using a base-ten diagram, but the number line doesn't match their thinking, consider asking::

• “¿Puedes decirme más acerca de tu recta numérica? ¿Cómo supiste dónde empezar?” // “Can you tell me more about your number line? How did you decide where to start?”
• “¿Cómo se relaciona la recta numérica con tu diagrama en base diez?” // “How does the number line connect to your base-ten diagram?”

• Invite previously identified students to share.
• Display or record their methods.
• Consider asking each student:
  • “¿Cómo decidieron dónde empezar, cuántos saltos dar y la longitud de cada salto?” // “How did you decide where to start, how many jumps to make, and the length of each jump?”
• “¿En qué se parecen estos métodos? ¿En qué son diferentes?” // “How are these methods the same? How are they different?” (They started at 54. Some show subtracting tens first, some show ones first. Some show subtracting the value of all the tens or all the ones. Some show subtracting each ten or each one.)
• “¿De qué manera la recta numérica les ayuda a ver en qué se parecen estos métodos?” // “How does the number line help you see how these methods are the same?” (It helps you see that it doesn't matter if you subtract tens first or ones first. They both show subtracting 24.) 

The purpose of this activity is for students to represent addition and subtraction within 100 on a number line. Students make connections to strategies based on counting on or back by place. The numbers in each subtraction equation are designed to elicit methods that do not require students to explicitly decompose a ten. For example, when finding the value of \(50 - 32\), students may first add on to make a ten (\(32 + 8 = 40\)), then add on more tens to reach the total (\(40 + 10 = 50\)). Others may see they can count back 2, then subtract the tens. In the synthesis, students share their thinking and discuss how the number line helps see how they can use what they know about the structure of counting sequence and what they know about tens and ones to add and subtract (MP7).

• Groups of 2
• Give students access to base-ten blocks.
• Display images of Diego’s number line.
• “Diego encontró el valor de \(33 + 45\). Él usó una recta numérica para representar cómo pensó” // “Diego found the value of \(33 + 45\). He used a number line to represent his thinking.”
• “¿En qué parte de su recta numérica ven 33 y en qué parte ven 45?” // “Where do you see 33 and 45 on his number line?” (On the number line there is a point on 33 and 4 jumps of 10 and 1 jump of 5.)
• 30 seconds: quiet think time
• Share responses

• “Van a encontrar el valor de expresiones y a usar rectas numéricas para representar cómo pensaron” // “You will be finding the value of expressions and representing your thinking on the number line.”
• “Si les ayuda, dibujen diagramas en base diez” // “Draw base-ten diagrams if it helps.”
• 8–10 minutes: independent work time

If students use base-ten diagrams or blocks to show decomposing a ten to subtract, validate their reasoning and consider asking:

• “Después de descomponer una decena, ¿restaste decenas o unidades primero?” // “After you decomposed a ten, did you subtract tens or ones first?”
• “Ubica 50 (o 40) en la recta numérica. Si usaras la recta numérica para mostrar cómo contar hacia atrás, ¿contarías hacia atrás, primero de decena en decena o de unidad en unidad? ¿Por qué?” // “Locate 50 (or 40) on the number line. If you used the number line to show counting back, would you count back by tens first or ones first? Why?”
• “¿Qué debes hacer ahora?” // “What should you do next?”
• “¿En qué se parece este método a lo que hiciste con el bloque (o el diagrama en base diez)? ¿En qué es diferente?” // “How is this method like what you did with the block (or base-ten diagram)? How is it different?”

• Invite 2–3 previously selected students to share.
• Consider asking each student:
  • “¿Cómo decidieron dónde empezar?” // “How did you decide where to start?”
  • “¿Cómo decidieron cuánto sumar/restar primero?” // “How did you decide how much to add/subtract first?”
  • “¿De qué manera su recta numérica muestra el valor de la diferencia?” // “How does your number line show the value of the difference?”
• “¿Qué otras preguntas tienen sobre la recta numérica de _____?” // “What other questions do you have about _____'s number line?”
• “¿De qué manera la recta numérica les ayudó a entender el método de _____?” // “How does the number line help you make sense of _____'s method?”