Lesson 7

The purpose of this warm-up is to elicit the idea that addition and subtraction can be represented on the number line. Students have learned that numbers farther to the right are larger and numbers to the left are smaller. In this warm-up, students see two number lines with arrows that connect the same numbers. However, one arrow starts at the lesser number and points at the greater number and the other starts with the greater number and points at the lesser number. Noticing the difference in these “jumps” will be useful when students match equations to representations on number lines in a later activity. While students may notice and wonder many things about these images, it is important to discuss how the arrows represent increases and decreases in value 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.

• “¿Cómo describirían lo que ocurre en la recta numérica?” // “How would you describe what’s happening on the number line?” (Start at 8, add/jump 4, and land on 12. Start at 12, subtract/jump back 4, and land on 8.)
• If needed, “¿Cuántos espacios nos movimos en la recta numérica?” // “How many spaces did we move on the number line?”
• “Algunas veces marcamos el salto con un número para mostrar cuánto saltamos” // “Sometimes, we label the jump with a number to show how far we jumped.”
• Record 4 above the jump.
• “Hoy vamos a pensar sobre cómo podemos mostrar la suma y la resta en la recta numérica” // “Today, we are going to think about how we can show addition and subtraction on the number line.”

In previous lessons, students interpreted and represented numbers on the number line. They understand that numbers are represented as lengths from 0, consecutive numbers on the number line must be spaced equally, numbers can be represented with tick marks, and specific numbers can be identified on the number line using a point.

The purpose of this activity is for students to make sense of representations that show addition and subtraction on the number lines. They reason that an arrow pointing to the right represents addition because numbers to the right represent greater numbers, while an arrow pointing to the left represents subtraction because numbers to the left represent lesser numbers. Students connect the starting location, ending location, and direction of an arrow to equations (MP2). They interpret the length between the numbers (or distance traveled by the jump) as the number that was added or subtracted.

• Groups of 2
• Display image from warm-up and the equations \(4 + 8 = 12\), \(8 + 4 = 12\), \(12 - 4 = 8\), and \(12 - 8 = 4\).
• “¿Cuáles ecuaciones están representadas por estas rectas numéricas? ¿Cómo lo saben?” // “Which equations are represented by these number lines? How do you know?” (\(8 + 4 = 12\) because there is a point on the 8, a jump of 4, and the arrow is pointing to 12. \(12 - 4 = 8\) because there is a point on 12, a jump back of 4, and the arrow is pointing to 8.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.

• “Ahora van a ver más rectas numéricas que representan ecuaciones de suma y ecuaciones de resta” // “Now you are going to look at some more number lines that represent addition and subtraction equations.”
• “Marquen la ecuación que cada recta numérica representa” // “Circle the equation that each number line represents.”
• “En el último problema, expliquen por qué la otra ecuación no le corresponde a la recta numérica” // “For the last problem, explain why the other equation doesn’t match the number line.”
• 6 minutes: independent work time
• “Discutan sus elecciones y su explicación con su compañero” // “Discuss your choices and your explanation with your partner.”
• 2 minutes: partner work time
• Monitor for a student who clearly explains that the jump shows 10 for \(4 + 10 = 14\).

If a student circles the equation that is not represented by the number line, consider asking:

• “¿Puedes explicar lo que ves que ocurre en esta recta numérica?” // “Can you explain what you see happening on this number line?”
• “¿Cómo sabes si la flecha muestra suma o resta?” // “How do you know if the arrow shows addition or subtraction?”

• Display the number line for \(4 + 10 = 14\).
• “¿Hemos aprendido que \(4 + 10\) y \(10 + 4\) tienen el mismo valor. ¿Por qué \(10 + 4\) no le corresponde a esta recta numérica?” // “We have learned that \(4 + 10\) and \(10 + 4\) have the same value. Why doesn’t \(10 + 4\) match this number line?” (The arrow shows addition, but the first point is on 4 and the arrow shows moving 10 units to the right.)

The purpose of this activity is for students to match addition and subtraction equations to number line representations. Some equations use the same numbers, requiring students to look for the direction of the arrow to see if the number line is representing addition or subtraction (MP7). There is one equation that does not have a matching number line representation. Students represent this equation on the number line. This problem can be used as a formative assessment of student understanding of the connection between equations and their representations on the number line. Teachers can use this information to plan for any additional support that may be needed in the following lesson where students represent different equations on number lines. In the synthesis, students consider how addition and subtraction can look alike on the number line, and how they are different.

• Groups of 2
• Give students glue and scissors.

• “Ahora, con su compañero, van a emparejar ecuaciones con representaciones en la recta numérica” // “Now you will work with your partner to match equations to representations on the number line.”
• “Recorten las ecuaciones y peguen cada una al lado de la recta numérica que ella representa” // “Cut out the equations and glue them next to the number line that represents it.”
• “Antes de pegar sus respuestas, asegúrense de comparar con su compañero” // “Before gluing your answers, be sure to compare with your partner.”
• “Hay una ecuación que no tiene una recta numérica que le corresponda. Peguen esa ecuación en el espacio extra y usen la recta numérica que no tiene saltos para representarla” // “There is one equation that doesn’t have a number line to match. Glue it in the extra box and represent that equation on the blank number line.”
• 12 minutes: partner work time

If students match an equation to a number line that does not represent the equation, invite them to read the equation. Consider asking:

• “¿Cómo puedes mostrar esta ecuación en la recta numérica?” // “How could you act out this equation on the number line?”
• “¿Dónde debes empezar? ¿En qué dirección debes ir?” // “Where should you start? What direction should you go?”
• “¿Cuánto debes recorrer? ¿Dónde debes detenerte?” // “How far should you go? Where should you stop?”

• Display the image of the number lines for \(20 - 3 = 17\) and \(3 + 17 = 20\).
• “¿En qué se parecen estas rectas numéricas? ¿En qué son diferentes?” // “What is the same about these number lines? What is different?” (They both have 3, 17, and 20. One has a long jump of 17, but the other has a small jump of 3. The difference between 17 and 20 is 3 and 20 is 17 away from 3.)